C^nrnell Hnicctattg Sithtatg 3tljata, New $ork .Jj3.hn..ALenr..\x..I£a.n.rLe.ir... MATHEMATICS Cornell University Library QA 37.M58 Higher mathematics.A text-book for class 3 1924 001 518 483 The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001518483 HIGHER MATHEMATICS. A TEXT-BOOK FOR CLASSICAL AND ENGINEERING COLLEGES. EDITED BY MANSFIELD MERRIMAN, Professor of Civil Engineering in Lehigh University, AND ROBERT S. WOODWARD, Professor of Mechanics in Columbia University. FIRST EDITION, FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. London : CHAPMAN & HALL, Limited. 1896. Copyright, 1896, BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD. ROBERT DRUMMOND, ELECTROTYPEE AND PRINTER, NEW YORK. PREFACE. In the early part of this century it was possible for an in- dustrious student to acquire a comprehensive if not minute knowledge of the entire realm of mathematical science. The more eminent minds of that time, like Lagrange, Laplace, and Gauss, were about equally familiar with all branches of pure and applied mathematics. Since that epoch the tendency has been constantly towards specialization ; and additions to pure theory along with extensions of applications have been made with increasing rapidity, until now the mere quantity of in- formation available presents a formidable obstacle to the simul- taneous attainment of the breadth and depth of knowledge which characterized the mathematician of a generation ago. It would appear, however, that this obstacle is due to the bewildering mass of details rather than to any considerable increase in the number of fundamental principles. Hence the student who seeks to gain a comprehensive view of the mathe- matics of the present day needs most of all that sort of guid- ance which fixes his attention on essentials and prevents him from wasting valuable time and energy in the pursuit of non- essentials. During the past twenty years a marked change of opinion has occurred as to the aims and methods of mathematical instruction. The old ideas that mathematical studies should be pursued to discipline the mind, and that such studies were ended when an elementary course in the calculus had been covered, have for the most part disappeared. In our best classical and engineering colleges the elementary course in IV PREFACE. calculus is now given in the sophomore year, while lectures and seminary work in pure mathematics are continued during the junior and senior years. It is with the hope of meeting the existing demand for a suitable text to be used in such upper- class work that the editors enlisted the cooperation of the authors in the task of bringing together the chapters of this book. It was the intention of the editors to include a chapter on elliptic integrals and functions ; much to their regret, how- ever, it was found impracticable to obtain the manuscript in time for publication. Notwithstanding this omission the volume contains about one-fifth more matter than was originally con- templated. Each chapter, so far as it goes, is complete in itself, and is intended primarily to give a clear idea of the leading prin- ciples of the subject treated. While the authors have been guided by general instructions issued by the editors, each has been free to follow his own plan of treatment. It will be found that certain chapters adopt the formal method usual in text- books, while others employ what may be called the historical and intuitive method. A glance at the table of contents will show that the chapters of the work present a considerable variety of subjects, thus affording teachers and students an opportunity to select such topics as maybe suited to their time and tastes. Numerous problems are given for solution, numer- ical examples of the application of theory to physical science are freely introduced, and the foot-notes set forth much sug- gestive matter of a historical and critical nature. The Editors. June 30, 1896. CONTENTS. Chapter I. THE SOLUTION OF EQUATIONS. By Mansfield Mereiman, Professor of Civil Engineering in Lehigh University. Article i. Introduction Page r 2. Graphic Solutions o 3. The Regula Falsi e 4. Newton's Approximation Rule 6 5. Separation of the Roots g 6. Numerical Algebraic Equations I0 7. Transcendental Equations ™ 8. Algebraic Solutions s c 9. The Cubic Equation T y 10. The Quartic Equation X n 11. Quintic Equations 21 12. Trigonometric Solutions 24 13. Real Roots by Series 27 14. Computation of All Roots 2 S 15. Conclusion j r Chapter II. DETERMINANTS. By Laenas Gifford Weld, Professor of Mathematics in the State University of Iowa. Article 1. Introduction 33 2. Permutations ,4 3. Interchange of Two Elements 35 4. Positive and Negative Permutations 36 5. The Determinant Array 36 6. Determinant as Function of ri l Elements 37 7. Examples of Determinants 38 8. Notations 30 9. Second and Third Orders , 10. Interchange of Rows and Columns ^2 11. Interchange of Two Parallel Lines j~ 12. Two Identical Parallel Lines ^3 v VI CONTENTS. Article 13. Multiplying by a Factor Page 44 14. A Line of Polynomial Elements. .. ... 44 15. Composition of Parallel Lines 45 16. Binomial Factors 4° 17. Co-factors; Minors 47 18. Development in Terms of Co-factors 49 19. The Zero Formulas 5 1 20. Cauchy's Method of Development 5 2 21. Differentiation of Determinants 54 22. Raising the Order 55 23. Solution of Linear Equations 5° 24. Consistence of Linear Systems 58 25. The Matrix 60 26. Homogeneous Linear Systems 60 27. Co-factors in a Zero Determinant 62 28. Sylvester's Method of Elimination 63 29. The Multiplication Theorem 65 30. Product of Two Rectangular Arrays 68 31. Reciprocal Determinants 69 Chapter III. PROJECTIVE GEOMETRY. By George Bruce Halsted, Professor of Mathematics in the University of Texas. Article 1. The Elements and Primal Forms 70 2. Projecting and Cutting -72 3. Elements at Infinity 72 4. Correlation and Duality 74 5. Polystims and Polygrams 74 6. Harmonic Elements 77 7. Projectivity 80 8. Curves of the Second Degree 82 9. Pole and Polar 87 10. Involution 88 11. Projective Conic Ranges qj 12. Center and Diameter _. 94 13. Plane and Point Duality „g 14. Ruled Quadric Surfaces q8 15. Cross Ratio • • 104 Chapter IV. HYPERBOLIC FUNCTIONS. By James McMahon, Associate Professor of Mathematics in Cornell University. Article 1. Correspondence of Points on Conies Ior 2. Areas of Corresponding Triangles I0Q 3. Areas of Corresponding Sectors IOQ 4. Characteristic Ratios of Sectorial Measures j 10 CONTENTS. Vll Article 5. Ratios expressed as Triangle-measures Page no 6. Functional Relations for Ellipse in 7. Functional Relations for Hyperbola in 8. Relations between Hyperbolic Functions 112 9. Variations of the Hyperbolic Functions 114 10. Anti-hyperbolic Functions 116 n. Functions of Sums and Differences 116 12. Conversion Formulas 11S 13. Limiting Ratios 119 14. Derivatives of Hyperbolic Functions 120 15. Derivatives of Anti-hyperbolic Functions 122 16. Expansion of Hyperbolic Functions 123 17. Exponential expressions 124 18. Expansion of Anti-functions 125 19. Logarithmic Expression of Anti-functions 127 20. The Gudermanian Function 128 21. Circular Functions of Gudermanian 128 22. Gudermanian Angle 129 23. Derivatives of Gudermanian and Inverse 130 24. Series for Gudermanian and its Inverse 131 25. Graphs of Hyperbolic Functions 132 26. Elementary Integrals 135 27. Functions of Complex Numbers 138 2S. Addition Theorems for Complexes 140 29. Functions of Pure Imaginaries 141 30. Functions of x -f- iy in the Form X-\-iY 143 31. The Catenary 145 32. The Catenary of Uniform Strength 147 33. The Elastic Catenary 148 34. The Tractory 149 35. The Loxodrome 150- 36. Combined Flexure and Tension 151 37. Alternating Currents 153 38. Miscellaneous Applications 158 39. Explanation of Tables 160 Chapter V. HARMONIC FUNCTIONS. By William E. Byerly, Professor of Mathematics in Harvard University. Article 1. History and Description 169 2. Homogeneous Linear Differential Equations 172 3. Problem in Trigonometric Series 174 4. Problem in Zonal Harmonics 177 5. Problem in Bessel's Functions 183 6. The Sine Series 188 7. The Cosine Series 192 8. Fourier's Series 194 Vlll CONTENTS. Article -9. Extension of Fourier's Series Page 196 10. Dirichlet's Conditions 198 11. Applications of Trigonometric Series '. 200 12. Properties of Zonal Harmonics 202 13. Problems in Zonal Harmonics 205 14. Additional Forms 207 15. Development in terms of Zonal Harmonics 208 16. Formulas for Development 209 17. Formulas in Zonal Harmonics 212 18. Spherical Harmonics. 213 19. Bessel's Functions. Properties 213 20. Applications of Bessel's Functions 215 21. Development in Terms of Bessel's Functions 217 22. Problems in Bessel's Functions 220 23. Bessel's Functions of Higher Order 221 24. Lame's Functions 221 25. Tables 222 Chapter VI. FUNCTIONS OF A COMPLEX VARIABLE. By Thomas S. Fiske, Adjunct Professor of Mathematics in Columbia University. Article 1. Definition of Function 226 2. Representation of Complex Variable 227 3. Absolute Convergence 228 4. Elementary Functions 229 5. Continuity of Functions 230 6. Graphical Representation of Functions 232 7. Derivatives 233 8. Conformal Representation 236 9. Examples of Conformal Representation 238 10. Conformal Representation of a Sphere 244 11. Conjugate Functions 245 12. Application to Fluid Motion 246 13. Critical Points 250 14. Point at Infinity 256 15. Integral of a Function 257 16. Reduction of Complex Integrals to Real 261 17. Cauchy's Theorem 262 18. Application of Cauchy's Theorem 264 19. Theorems on Curvilinear Integrals 267 20. Taylor's Series 269 21. Laurent's Series 271 22. Fourier's Series 273 23. Uniform Convergence 274 24. One-valued Functions with Critical Points 278 25. Residues 282 26. Integral of a One-valued Function 284 CONTENTS. IX Article 27. Weierstrass's Theorem Page 287 28. Mittag-Leffler's Theorem 292 29. Critical Lines and Regions 298 30. Functions having n Values 300 Chapter VII. DIFFERENTIAL EQUATIONS. By W. Woolsey Johnson, Professor of Mathematics in United States Naval Academy. Article I. Equations of First Order and Degree 303 2. Geometrical Representation 305 3. Primitive of a Differential Equation 307 4. Exact Differential Equations 308 5. Homogeneous Equation 311 6. The Linear Equation 312 7. First Order and Second Degree 314 8. Singular Solutions 317 9. Singular Solution from the Complete Integral 320 10. Solution by Differentiation 322 11. Geometric Applications; Trajectories 325 12. Simultaneous Differential Equations 327 13. Equations of the Second Order 330 14. The First Two Integrals 333 15. Linear Equations 336 16. Linear Equations with Constant Coefficients 338 17. Homogeneous Linear Equations 342 18. Solutions in Infinite Series 344 19. Systems of Differential Equations 349 20. First Order and Degree with Three Variables 352 21. Partial Differential Equations of First Order and Degree... 355 22. Complete and General Integrals 359 23. Complete Integral for Special Forms 362 24. Partial Equations of Second Order 365 25. Linear Partial Differential Equations 368 Chapter VIII. GRASSMANN'S SPACE ANALYSIS. By Edward W. Hyde, Professor of Mathematics in the University of Cincinnati. Article I. Explanations and Definitions 374 2. Sum and Difference of Two Points 375 3. Sum of Two Weighted Points 378 4. Sum of any Number of Points 381 5. Reference Systems 3 g6 6. Naturs of Geometric Multiplication 3go 7. Planimetric Products 3 q 2 8. The Complement , Q0 X CONTENTS. Article 9. Equations of Condition and Formulas Page 405 10. Stereometric Products 4 10 11. The Complement in Solid Space 4 10 12. Addition of Sects in Solid Space 4 J 9 Chapter IX. VECTOR ANALYSIS and QUATERNIONS. By Alexander Macfaelane, Lecturer in Electrical Engineering in Lehigh University. Article 1. Introduction 4 2 5 2. Addition of Coplanar Vectors 4 20 3. Products of Coplanar Vectors 43 2 4. Coaxial Quaternions 439 5. Addition of Vectors in Space 443 6. Product of Two Vectors 444 7. Product of Three Vectors 449 8. Composition of Located Quantities 453 9. Spherical Trigonometry 457 10. Composition of Rotations 463 Chapter X. PROBABILITY and THEORY OF ERRORS. By Robert S. Woodward, Professor of Mechanics in Columbia University. Article 1 . Introduction 467 2. Permutations 471 3. Combinations 473 4. Direct Probabilities 476 5. Probability of Concurrent Events 479 6. Bernoulli's Theorem 482 7. Inverse Probabilities 484 8. Probabilities of Future Events 487 9. Theory of Errors 490 10. Laws of Error 491 11. Typical Errors of a System 493 12. Laws of Resultant Error 494 13. Errors of Interpolated Values 497 14. Statistical Test of Theory 504 Chapter XI. HISTORY OF MODERN MATHEMATICS. By David Eugene Smith, Professor of Mathematics in Michigan State Normal School. Article I. Introduction 508 2. Theory of Numbers 511 3. Irrational and Transcendent Numbers 513 4. Complex Numbers 515 5. Quaternions and Ausdehnungslehre 517 CONTENTS. XI Article 6. Theory of Equations Page 519 7. Substitutions and Groups 524 8. Determinants 526 9. Quantics 528 10. Calculus 531 11. Differential Equations 535 12. "«afinite Series 539 13. Theory of Functions 543 14. Probabilities and Least Squares 550 15. Analytic Geometry 552 16. Modern Geometry 558 17. Trigonometry and Elementary Geometry 563 18. Non-Euclidean Geometry 565 19. Bibliography ° 568 Index 571 HIGHER MATHEMATICS Chapter I. THE SOLUTION OF EQUATIONS. By Mansfield Merriman, Professor of Civil Engineering in Lehigh University. Art. 1. Introduction. In this Chapter will be presented a brief outline of methods, not commonly found in text-books, for the solution of an equation containing one unknown quantity. Graphic, numeric, and algebraic solutions will be given by which the real roots of both algebraic and transcendental equations may be ob- tained, together with historical information and theoretic discussions. An algebraic equation is one that involves only the opera- tions of arithmetic. It is to be first freed from radicals so as to make the exponents of the unknown quantity all integers ; the degree of the equation is then indicated by the highest ex- ponent of the unknown quantity. The algebraic solution of an algebraic equation is the expression of its roots in terms of the literal coefficients ; this is possible, in general, only for linear, quadratic, cubic, and quartic equations, that is, for equations of the first, second, third, and fourth degrees. A numerical equation is an algebraic equation having all its coefficients real numbers, either positive or negative. For the four degrees 2 THE SOLUTION OF EQUATIONS. [CHAP. I. above mentioned the roots of numerical equations may be computed from the formulas for the algebraic solutions, unless they fall under the so-called irreducible case wherein real quantities are expressed in imaginary forms. An algebraic equation of the n th degree may be written with all its terms transposed to the first member, thus: x" + a x x n - 1 + a,x"- 2 + . . . + a n .,x + a„ = O, and, for brevity, the first member will be called fix) and the equation be referred to as/(^) = 0. The roots of this equa- tion are the values of x which satisfy it, that is, those values of x that reduce f(x) to o. When all the coefficients a lt a t ,. ..a n are real, as will always be supposed to be the case, Sturm's theorem gives the number of real roots, provided they are un- equal, as also the number of real roots lying between two assumed values of x, while Horner's method furnishes a con- venient process for obtaining the values of the roots to any required degree of precision. A transcendental equation is one involving the operations of trigonometry or of logarithms, as, for example, x -\- cos.r = o, or A 2 * -[~ xb* = °- No general method for the literal solution of these equations exists ; but when all known quantities are expressed as real numbers, the real roots may be located and computed by tentative methods. Here also the equation may be designated as f{x) = o, and the discussions in Arts. 2-5 will apply equally well to both algebraic and transcendental forms. The methods to be given are thus, in a sense, more valuable than Sturm's theorem and Horner's process, although for algebraic equations they may be somewhat longer. It should be remembered, however, that algebraic equations higher than the fourth degree do not often occur in physical problems, and that the value of a method of solution is to be measured not merely by the rapidity of computation, but also by the ease with which it can be kept in mind and applied. Prob. 1. Reduce the equation [a + x)i + [a — x)l = 2b to an equation having the exponents of the unknown quantity all integers. Art. 2.] GRAPHIC SOLUTIONS. Art. 2. Graphic Solutions. Approximate values of the real roots of two simultaneous algebraic equations may be found by the methods of plane analytic geometry when the coefficients are numerically expressed. For example, let the given equations be x 1 -\- y = a\ x' — bx = y — cy, the first representing a circle and the second a hyperbola. Drawing two rectangular axes OX and OY, the circle is de- scribed from O with the radius a. The coordinates of the center of the hyperbola are found to be OA = \b and AC = \c, while its diameter BD =\/b* — c', from which the two branches may be described. The intersections of the circle with the hyperbola give the real values of x and y. If a = i, b = 4, and c = 3, there are but two real values for x and two real values for y, since the circle intersects but one branch of the hyperbola ; here Om is the positive and Op the negative value of x, while mn is the positive and pq the negative value of y. When the radius a is so large that the circle intersects both branches of the hyperbola there are four real values of both x and y. By a similar method approximate values of the real roots of an algebraic equation containing but one unknown quantity may be graphically found. For instance, let the cubic equation x a + ax — b — o be required to be solved.* This may be written as the two simultaneous equations y = x s , y = — ax -f- b, *See Proceedings of the Engineers' Club of Philadelphia, 1884, Vcl. IV, PP. 47-49 4 THE SOLUTION OF EQUATIONS. [CHAP. I. and the giaph of each being plotted, the abscissas of their points of intersection give the real roots of the cubic. The curve y = x° should be_ plotted upon I cross-section paper by the help of a \ / table of cubes ; then OB is laid off "" /*"\ equal to b, and OC equal to a/b, tak- ^ _XjQ ing care to observe the signs of a and b. The line joining B and C cuts the curve at p, and hence qp is the real root of x' -J- ax — b = O. If the cubic equation have three real roots the straight line BC will intersect the curve in three points. Some algebraic equations of higher degrees may be graphic- ally solved in a similar manner. For the quartic equation z' -\- Az 1 -\- Bz — C=O t it is best to put z=A' k x, and thus reduce it to the form x' -\- x' -\- bx — c — O; then the two equations to be plotted are y = x* + x\ y=—bx-\-c, the first of which may be drawn once for all upon cross-section paper, while the straight line represented by the second may be drawn for each particular case, as described above.* This method is also applicable to many transcendental equa- tions ; thus for the equation Ax — Bs'm x — o it is best to write ax — sinx=o; then y = sin* is readily plotted by help of a table of sines, while y = ax is a straight line passing through the origin. In the same way a* — x' = o gives the curve represented by y = a* and the parabola represented by y = x\ the intersections of which determine the real roots of the given equation. Prob. 2. Devise a graphic solution for finding approximate values of the real roots of the equation x"-\- ax*-\- dx'+ ex + d = o. Prob. 3. Determine graphically the number and the approximate values of the real roots of the equation arc x - 8 sin x = o (Ans.— Six real roots, x = ± 159°, ± 430°, and ± 456 .) *For an extension of this method to the determination of imaginary roots, see Phillips and Beebe's Graphic Algebra, New York, 1S82. Art. 3.] the regula falsi. 5 Art. 3. The Regula Falsi. One of the oldest methods for computing the real root of an equation is the rule known as "regula falsi," often called the method of double position.* It depends upon the princi- ple that if two numbers x^ and x^ be substituted in the expres- sion/^), and if one of these renders /(V) positive and the other renders it negative, then at least one real root of the equation f{x) = o lies between x l and x,. Let the figure represent a part of the real graph of the equation y =/(x). The point X, where the curve crosses the axis of abscissas, gives a real root OX of the equation /(;r) = o. Let OA and OB be inferior and superior limits of the root OX which are determined either by trial or by the method of Art. 5. Let Aa and Bb be the values of f(x) corresponding to these limits. q j^ Join ab, then the intersection C of the straight line ab with the axis OB gives an approximate value OC for the root. Now compute Cc and join ac, then the intersection D gives a value OD which is closer still to the root OX. Let x x and x^ be the assumed values OA and OB, and let f( x \) and_/(^ 2 ) be the corresponding values of f{x) represented by Aa and Bb, these values being with contrary signs. Then from the similar triangle AaC and BbC the abscissa OC is xjixj - ■*•,/(•<> {x-x,)f{x,) _. (*, — -*,)/Q 3 ) x '~ /(*,) - /OO 1+ A*>) -A*.) * + /(*,) - /K) " By a second application of the rule to x^ and x s , another value x t is computed, and by continuing the process the value of x can be obtained to any required degree of precision. As an example let fix) — x * + 5^ + 7 = °- Here lt ma -Y be found by trial that a real root lies between —2 and — 1.8. *This originated in India, and its first publication in Europe was by Abra- ham ben Esra, in 1130. See Matthiesen, Grundziige der antiken und moder- nen Algebra der litteralen Gleichungen, Leipzig, 1878. 6 THE SOLUTION OF EQUATIONS. [CHAP. I- For.*-, = - 2 ,f(x) = -5, and for x,= - 1.8, /«) = + 4-3°4; then by the regula falsi there is found x s = — 1.90 nearly. Again, for x % = — 1.90, f(x a ) = + 0.290, and these combined with x l and /(#,) give * 4 = — 1.906, which is correct to the third decimal. As a second example let f[x) = arc.*- — sin x — o. 5 = 0. Here a graphic solution shows that there is but one real root, and that the value of it lies between 85 and 86° For jt,= 85°, flx t ) = - 0.01266, and for x, = 86°, f(x,) = + 0.00342 ; then by the rule x 3 = 85" 44', which gives f(x 3 ) = — 0.00090. Again, combining the values for x^ and x s there is found x t = 85° A7 • which gives f(x t ) = — 0.00009. Lastly, combining the values for x^ and x t there is found x 6 = 85- 47'.^, ft'hich is as close an approximation as can be made with five-place tables. In the application of this method it is to be observed that the signs of the values of x and f(x) are to be carefully re- garded, and also that the values of f(x) to be combined in one operation should have opposite signs. For the quickest approximation the values of /(.*•) to be selected should be those having the smallest numerical values. Prob. 4. Compute by the regula falsi the real roots of .X s — 0.25 = 0. Also those of x* -f- sin 2X = o. Art. 4. Newton's Approximation Rule. Another useful method for approximating to the value of the real root of an equation is that devised by Newton in 1666.* If y =/(x) be the equation of a curve, OX in the figure represents a real root of the equation f(x) = o. Let OA be an approximate value of OX, and Aa the corresponding value /b o(f(x). At a let aB be drawn tangent to the curve; then OB is another approximate value of OX. * See Analysis per equationes numero terminorum infinitas, p. 269, Vol. I of Horsely's edition of Newton's works (London, 1779), where tne method is given in a somewhat different form. Art. 4.] newton's approximation rule. 7 Let Bb be the value of f{x) corresponding to OB, and at b let the tangent bC be drawn ; then OC is a closer approxima- tion to OX, and thus the process may be continued. Let/'(>) be the first derivative o\ f(x)\ or,f'{x) = df[x)/dx. For x = x 1 = OA in the figure, the value of f(x^) is the ordi- nate Aa, and the value of /'(#,) is the tangent of the angle aBA ; this tangent is also Aa/AB. Hence AB = f{x J )/f'(x 1 ), and accordingly OB and CC are found by which is Newton's approximation rule. By a third application to x^ the closer value .r 4 is found, and the process may be con- tinued to any degree of precision required. For example, let f(x) = x" -)- ^x" 1 -(-7 = 0. The first deriv- ative is/~'(;r) = $x' -(- 10^. Here it may be found by trial that — 2 is an approximate value of the real root. For x s = — 2 f(x^) = — 5, and f'{x^) = 60, whence by the rule x^ — — 1.92. Now for ;r 2 = — 1.92 are found /(^ 2 ) = — 0.6599 and f(z t ) = 29.052, whence by the rule x 3 = — 1.906, which is correct to the third decimal. As a second example let fix) = x 1 -\~ 4sin x = o. Here the first derivative is f'{x) = 2x -\- 4 cos x. An approximate value of x found either by trial or by a graphic solution is ;r= — 1.94, corresponding to about — in°09'. For ^, = — 1.94, /(x^ = 0.03304 and f\x,) = — 5.323, whence by the rule .*•,= — I.934. By a second application x s = — 1.9328, which corresponds to an angle of — 1 10° 54^'. In the application of Newton's rule it is best that the assumed value of x l should be such as to render f{x^) as small as possible, and also /"'(.*■,) as large as possible. The method will fail if the curve has a maximum or minimum between a and b. It is seen that Newton's rule, like the regula falsi, applies equally well to both transcendental and algebraic equa- tions, and moreover that the rule itself is readily kept in mind by help of the diagram. 8 THE SOLUTION OF EQUATIONS. [CHAP. I. Prob. s- Compute by Newton's rule the real roots of the alge- braic equation x" — ■jx + 6 = o. Also the real roots of the trans- cendental equation sin x -f- arc x — 2 = o. Art. 5. Separation of the Roots. The roots of an equation are of two kinds, real roots and imaginary roots. Equal real roots may be regarded as a spe- cial class, which lie at the limit between the real and the imagi- nary. If an equation has^> equal roots of one value and q equal roots of another value, then its first derivative equation has p— 1 roots of the first value and ^ — 1 roots of the second value, and thus all the equal roots are contained in a factor common to both primitive and derivative. Equal roots may hence always be readily detected and removed from the given equation. For instance, let x i — 9^' -j- \x -\- 12 == o, of which the derivative equation is 4x° — i8x -\- 4 = o ; as x — 2 is a factor of these two equations, two of the roots of the primitive equation are -f- 2. The problem of determining the number of the real and imaginary roots of an algebraic equation is completely solved by Sturm's theorem. If, then, two values be assigned to x the number of real roots between those limits is found by the same theorem, and thus by a sufficient number of assumptions limits may be found for each real root. As Sturm's theorem is known to all who read these pages, no applications of it will be here given, but instead an older method due to Hudde will be presented which has the merit of giving a comprehensive view of the subject, and which moreover applies to transcendental as well as to algebraic equations.* If any equation y = j\x) be plotted with values of x as abscissas and values of y as ordinates, a real graph is obtained whose intersections with the axis CUT give the real roots of the * Devised by Hudde in 1659 and published by Rolle in 1690. See OSuvres de Lagrange, Vol. VIII,. p. 190. Art. 5.] SEPARATION OF THE ROOTS. equal ion fix) = o. Thus in the figure the three points marked Xg'vre three values OX for three real roots. The curve which represents y = f(x) has points of maxima and minima marked A, 3-nd inflection points marked B. Now let the first deriva- tive equation dy/dx— f'{x) be formed and be plotted in the same manner on the axis O'X'. The condition f'{x) = o gives the abscissas of the points A, and thus the real roots O'X' give limits separating the real roots of fix) = o. To ascertain if a real root OX lies between two values of O'X' these two values are to be substituted in/(;tr): if the signs of/(;tr) are unlike in the two cases, a real root of fix) = o lies between the two limits ; if the signs are the same, a real root does not lie between those limits. In like manner if the second derivative equation, that is, d*y/dx l = f"ix), be plotted on 0"X", the intersections give limits which separate the real roots of f'(x)=o. It is also seen that the roots of the second derivative equation are the abscissas of the points of inflection of the curve y = f{x). To illustrate this method let the given equation be the quintic fix) = x & — c,x 3 -f- 6x -f- 2 — o. The first derivative equation is fix) = 5 x" — 1 5 x" 1 + 6 = o, the ro'ots of which are approximately — 1.59, —0.69, +0.69, + 1.59. Now let each of these values be substituted for x in the given quintic, as also the values — 00 , o, and + °° > and l et the corresponding values of fix) be determined as follows : 10 THE SOLUTION OF EQUATIONS. [Chap. CO CO , x = — °o, —1.59, —0.69, o, +0.69, +1.59, + /(*)=- 00, +2.4, -0.6, +2, +4.7, +1.6, + Since f(x) changes sign between x =— 00 and x, — — 1.59, one real root lies between these limits ; since f(x) changes sign between^, = — 1.59 and x^ = — 0.69, one real root lies between these limits ; since fix) changes sign between x, = — 0.69 and x, = 0, one real root lies between these limits; since f(x) does not change sign between x 3 = and x t = 00 , a pair of imagi- nary roots is indicated, the sum of which lies between + 0.69 and 00 . As a second example let f(x) = r c — ?* — 4 = 0. The first derivative equation is f'(x) = e" — 2e* x = O, which has two roots e" = i and e" = o, the latter corresponding to x = — 00 . For x — — 00 , f(x) is negative; for e* — i, f{x) is negative ; for x — + 00 , f[x) is negative. The equation ^—^ — 4 = has, therefore, no real roots. When the first derivative equation is not easily solved, the second, third, and following derivatives may be taken until an equation is found whose roots may be obtained. Then, by working backward, limits may be found in succession for the roots of the derivative equations until finally those of the primative are ascertained. In many cases, it is true, this proc- ess may prove lengthy and difficult, and in some it may fail entirely; nevertheless the method is one of great theoretical and practical value. Prob. 6. Show that e* + e' Sx —4 = has two real roots, one positive and one negative. Prob. 7. Show that x° + * + 1 = o has no real roots; also that x° — x — 1 = o has two real roots, one positive and one negative. Art. 6. Numerical Algebraic Equations. An algebraic equation of the n ih degree may be written with all its terms transposed to the first member, thus : j- + fli *--i + a^*-* + . . . + a n .,x + a n = O ; Art. 6.] NUMERICAL ALGEBRAIC EQUATIONS. 11 and if all the coefficients and the absolute term are real num- bers, this is commonly called a numerical equation. The first member may for brevity be denoted by/(^) and the equation itself by/(V) = o. The following principles of the theory of algebraic equations with real coefficients, deduced in text-books on algebra, are here recapitulated for convenience of reference : (i) If x t is a root of the equation, /(x) is divisible by x — x t ; and conversely, if f{x) is divisible by x — x s , then x 1 is a root of the equation. (2) An equation of the n th degree has n roots and no more. (3) If x t , x 2 , . . . x„ are the roots of the equation, then the prod- uct (x — x^){x — x,) . . . (x — x„) is equal to/(x). (4) The sum of the roots is equal to — a,; the sum of the prod- ucts of the roots, taken two in a set, is equal to + ,) = 0.486 and /'fa) =0.206, whence by Newton's rule (Art. 4) ^=13 nearly. Next for ^ = 13. /«) = - 0.0298 and /'(*,) = 0.322, whence z, = 13.1. Lastly for z, = 13.1 f(z 3 ) =0.0012 and /'(a,) = 0.3142, whence z* = T 3-096, which is a sufficiently close approximation. The horizontal tension in the given catenary is hence 1 30.96 pounds* * Since e — e~ =2sinh8. this equation may be written iiS-iosinh6, where 6 = ios"*, and the solution rmy be expedited by the help of tables of hyperbolic functions. See Chapter IV Art. 8.] algebraic solutions. 15 Prob. 9. Show that the equation 3 sin x — 2x — 5 = o has but one real root, and compute its value. Prob. 10. Find the number of real roots of the equation 2X -(- log x — 10 000 = o, and show that the value of one of them is x — 4995-74- Art. 8. Algebraic Solutions. Algebraic solutions of complete algebraic equations are only possible when the degree n is less than 5. It frequently happens, moreover, that the algebraic solution cannot be used to determine numerical values of the roots as the formulas expressing them are in irreducible imaginary form. Neverthe- less the algebraic solutions of quadratic, cubic, and quartic equations are of great practical value, and the theory of the subject is of the highest importance, having given rise in fact to a large part of modern algebra. The solution of the quadratic has been known from very early times, and solutions of the cubic and quartic equations were effected in the sixteenth century. A complete investiga- tion of the fundamental principles of these solutions was, how- ever, first given by Lagrange in 1770.* This discussion showed, if the general equation of the « th degree, f(x) =0, be deprived of its second term, thus giving the equation f(y) = o, that the expression for the root y is given by y= cos, + go's, -f . . . + oo"-'s„_. 1 , in which n is the degree of the given equation, go is, in suc- cession, each of the n th roots of unity, 1, e, e 2 , . . . e"'\ and jr,, s,, . . . s n _, are the so-called elements which in soluble cases are determined by an equation of the n — I th degree. For instance, if n = 3 the equation is of the third degree or a cubic, the three values of go are co, = 1, go= — £ + £-/— 3 = e- <» = — i — iV— 3 = £2 > *Memoirs of Berlin Academy, 1769 and 1770; reprinted in CEuvres de Lagrange (Paris, 1868), Vol. II, pp. 539-562. See also Traite de la resolution des Equations numeriques, Paris, 1798 and 1808. 16 THE SOLUTION OF EQUATIONS. [CHAP. I. and the three roots are expressed by .Pi = ■*. + *. > ^=es l -{-e\, y t = e\+es a , in which s' and s, s are found to be the roots of a quadratic equation (Art. 9). The n values of go are the n roots of the binomial equation gj" — 1 = o. If n be odd, one of these is real and the others are imaginary ; if n be even, two are real and n — 2 are imagi- nary* Thus the roots of w 1 — 1 = O are -(- l and — 1 '< those of go 3 — 1 = o are given above ; those of go* — 1 = O are + 1, + i, — 1, and — i where i is-v/— 1. For the equation oo b — 1 = o the real root is -f- 1, and the imaginary roots are denoted by e, e\ e 3 , e 4 ; to find^these let go" — 1 =0 be divided by (»— 1, giving 00* -\- go 3 -\- go' + 00 -\- I = O, which being a reciprocal equation can be reduced to a quad- ratic, and the solution of this furnishes the four values, e =— i(i- Vs + ^-10-2^5), b » = _j(i + V5 + 4/_ 10 + 2^5)^ e' = -i(l-V5_ iZ-io-aVs), e s = -i(l+ V5_ V-io + 2V5), where it will be seen that e.e* = 1 and e 2 .e 3 = 1, as should be the case, since e 6 = 1. In order to solve a quadratic equation by this general method let it be of the form x* -j- 2ax -f- b = o, and let x be replaced by y — a, thus reducing it to / - (a' -b) = o. Now the two roots of this are y x = -\- s, and y a = — s lt whence the product of (y — s x ) and (y -\- s t ) is f - j» = o. Thus the value of s' is given by an equation of the first degree, * The values of a> are, in short, those of the n " vectors " drawn from the center which divide a circle of radius unity into n equal parts, the first vector co 1 = J being measured on the axis of real quantities. See Chapter X. Art. 9.] the cubic equation. 17 s' — d 2 — b; and since x = — a-\-y, the roots of the given equation are x 1 = — a -j- yd' — b, x^— — a — yd' — b, which is the algebraic solution of the quadratic. The equation of-the n — I th degree upon which the solution of the equation of the n th degree depends is called a resolvent. If such a resolvent exists, the given equation is algebraically solvable ; but, as before remarked, this is only the case for quadratic, cubic, and quartic equations. Prob. ii. Show that the six 6 th roots of unity are -j- i, +i(i+ ^~3), -t(r~ ^~3), -i, -*(i+ ^~3), -i(i~ V~2,)- Art. 9. The Cubic Equation. All methods for the solution of the cubic equation lead to the result commonly known as Cardan's formula.* Let the cubic be x % -\- lax* -\- 2,bx -(- 2c = o, (i) and let the second term be removed by substituting y — a for x, giving the form, y + 3By+2C=o, (i') in which the values of B and C are B = - d + b, C '- a' — %ab + c. (2) Now by the Lagrangian method of Art. 8 the values of y are y, = *i + *. . y% = «"i + e 2 s, , y % = e's, + es, , in which e and e 2 are- the imaginary cube roots of unity. Forming the products of the roots, and remembering that e s — 1 and e 3 -\- y, and this is the well-known formula of Cardan. The algebraic solution of the cubic equation (i) hence con- sists in finding B and C by (2) in terms of the given coefficients, .and then by (3) the elements ^ and s, are determined. Finally, x, — — a + (j, + s,), * t = — *-ifo + J i) + *•/""— 3( J . -0> (4) x, = - a - iO. -4- j,) — h V - 3(^1 — •*,). which are the algebraic expressions of the three roots. When B 3 -j- C 2 is negative the numerical solution of the cubic is not possible by these formulas, as then both s 1 and s, are in irreducible imaginary form. This, as is well known, is the case of three real roots, 5, -f- s, being a real, while s l — j 2 is a pure imaginary.* When B 3 -\- C* is o the elements J, and s, are equal, and there are two equal roots, x t = x 3 = — « -(- C*, while the other root is x^ = — a — 2(7*. When B" -f- C" is positive the equation has one real and two imaginary roots, and formulas (2), (3), and (4) furnish the numerical values of the roots of (1). For example, take the cubic x' — 4.5*' 4- 12X —5 = 0, whence by comparison with (1) are found a = — 1.5, b = 4- 4, c = —2.5. Then from (2) are computed B = 1.75, £"=4-3.125. These values inserted in (3) give s l = -|- 0.9142, s 2 = — 1. 9142 ; thus j, 4- s, = — 1.0 and s,— s t = -\- 2.8284. Finally, from (4) x, = 1.5 - 1.0 = +0.5, *, = i-5 +o.S + I-4H2 V^~3 = 2 -f 2.4495?, ** = i-5 + 0.5 - 1.4142 \ /zr i = 2 — 2.44952, which are the three roots of the given cubic. * The numerical solution of this case is possible whenever the angle whdse cosine is — C/ V '— B % ran be eeomeirically trisected. ART. 10.] THE QUARTIC EQUATION. 19 Prob. 12. Compute the roots of x' — 2x — 5 = o. Also the roots of x 3 -\- o.6x 2 — 5.76.x -|- 4.32 = o. Prob. 13. A cone has its altitude 6 inches and the diameter of its base 5 inches. It is placed with vertex downwards and one fifth of its volume is filled with water. If a sphere 4 inches in diameter be then put into the cone, what part of its radius is immersed in the water ? (Ans. 0.5459 inches). Art. 10. The Quartic Equation. The quartic equation was first solved in 1545 by Ferrari, who separated it into the difference of two squares. Lagrange in 1637 resolved it into the product of two quadratic factors. Tschirnhausen in 1683 removed the second and fourth terms. Euler in 1732 and Lagrange in 1767 effected solutions by assuming the form of the roots. All these methods lead to cubic resolvents, the roots of which are first to be found in order to determine those of the quartic. The methods of Euler and Lagrange, which are closely similar, first reduce the quartic to one lacking the second term, / + 6Bf -\-4Cy + D = o\ and the general form of the roots being taken as y x = + V7, + V7, + Vs„ J>,= -V7. + Vs.- Vi„ y t = 4- VI, - V7, - v7 M y t = - VI, - V7 2 + V7 % , the values s lt s„ s s , are shown to be the roots of the resolvent, 1 s 3 + 3Bs' + i(gB* - D)s - \0 = o. Thus the roots of the quartic are algebraically expressed in terms of the coefficients of the quartic, since the resolvent is solvable by the process of Art. 9. Whatever method of solution be followed, the following final formulas, deduced by the author in 1892, will result* Let the complete quartic equation be written in the form x l +4ax' + 66x* + $cx + d = o. (1) * See American Journal Mathematics, 1892, Vol. XIV, pp. 237-245. 20 THE SOLUTION OF EQUATIONS. [CHAP. I. First, let g, h, and k be determined from g = a*-b, h = b*-\-c'-2abc + dg, k = iac-P-\d. (2) Secondly, let / be obtained by / = \(]i + VW+P? + Xh - VFTT') 1 (3) Thirdly, let u, v, and w be found from U=g+l, V=2g-l, W = \l? + lk - \2gl. (4) Then the four roots of the quartic equation are x^-a + VZ+^v+V w x t = — a-\-Vu — ^v-\- Vw, x. = — a — Vu 4- V v — Vzv, (5) x t = — a — Vu — 'v — Vw, in which the signs are to be used as written provided that 2 a 2 — 3' + q 2 + 2m')x* + tyqmx — (p* + tf — m^m 1 = o, and thus the problem is numerically solvable by the above formulas if two roots are real and two imaginary. As a special case let p = 4 feet, q — 3 feet, and m = 1 foot ; then x* — 2jx'' -\- 48* — 24 = o. ART. 11.] QUINTIC EQUATIONS. 21 By comparison with (i) are found a = o, b = — 4%, c = + 12, and jx -\- 6 = o. (Ans. — 1.388, — 1. 000, 1. 194 ± 1. 701/.) Art. 11. Quintic Equations. The complete equation of the fifth degree is not algebraic- ally solvable, nor is it reducible to a solvable form. Let the equation be x h + $ax' + S&x* + $cx* + $dx + 2e — o, and by substituting y — a for x let it be reduced to / + $Bf + SC/ + $Dy + 2E = o. The five roots of this are, according to Art. 8, y, = es t + e\ + e\ + e\y„ y, = e% + e's, + es, + e 3 s lt y t = e's, + es t -\- e 4 s 3 + e% y*. = e% + e\ + e*s 3 + es t , in which e, e', e 3 e 4 are the imaginary fifth roots of unity. Now if the several products of these roots be taken there will be * This example is known by civil engineers as the problem of finding the length of a strut in a panel of the Howe truss. 22 THE SOLUTION OF EQUATIONS. [CHAP. I. found, by (4) of Art. 6, four equations connecting the four ele- ments j„ s 2 , s 3 , and s t , namely, — B — s,s t + s,s„ — C = s*s 3 + s^s, + s,\ + s,\, - D = s,\ + s t \ + sfs, + j,*j, - J.V - J,V, + WA. - 2 £ = J 1 , + J, , + *,' + */ + 5(*,V*.-W.V*,+ ^, + V^,) - 50.V, + Vv. + Vv* + *. V.} ; but the solution of these leads to an equation of the 120th degree for s, or of the 24th degree for s\ However, by taking s,s t — v, or j, 6 + s," + j 3 6 + *," as the unknown quantity, a resolvent of the 6th degree is obtained, and all efforts to find a resolvent of the fourth degree have proved unavailing. Another line of attack upon the quintic is in attempting to remove all the terms intermediate between the first and the last. By substituting y* + py -\-g for x, the values of p and q maybe determined so as to remove the second and third terms by a quadratic equation, or the second and third by a cubic equation, or the second and fourth by a quartic equation, as was first shown by Tschirnhausen in 1683. By substituting y ~r~/J / * ~t~ iy "4" r f° r x i three terms may be removed, as was shown by Bring in 1786. By substituting y*-\-py'-\- qy* -\-ry-\-t for x it was thought by Jerrard in 1833 that four terms might be removed, but Hamilton showed later that this leads to equations of a degree higher than the fourth. In 1826 Abel gave a demonstration that the algebraic solu- tion of the general quintic is impossible, and later Galois published a more extended investigation leading to the same conclusion.* Although these discussions are complex, and not devoid of points of doubt,-|- they have been generally accepted as conclusive. Moreover, the fact that the quintic is still un- solved in spite of the enormous amount of work done upon it during the past two centuries, is strong evidence that the prob- lem is an impossible one. *See Jordan's Trait§ des substitutions et des Equations algGbriques, 1870. fSee Kronecker, Verhandlungen der Berliner Akademie, 1853, p. 3S; also- Cockle, Philosophical Magazine, 1854, Vol. VII, p. 134. Art. 11.] QUINTIC EQUATIONS. 23 There are, however, numerous special forms of the quintic whose algebraic solution is possible. The oldest of these is the quintic of De Moivre, f + sB/ + $By + 2E = o, which is solved at once by making s, = s 3 = o in the element equations ; then — B — s,s t and — 2E = s* + s t \ from which j, and s t are found, and y x = j, -f- s t , or y, =(- E + VB' + E'Y + (- E - V '&+£')*, while the other roots are jj/ 2 = es, -(- e's t , y z = e^, + e 3 s i , y,= e's, + e\ , and j s = e , s 1 + 6^, . If ^' 4- £ a be negative, this quintic has five real roots; if positive, there are one real and four imaginary roots. When any relation, other than those expressed by the four element equations, exists between s t ,s t , s t , s t , the quintic is solvable algebraically. As an infinite number of such relations may be stated, it follows that there are an infinite number of solvable quintics. In each case of this kind, however, the co- efficients of the quintic are also related to each other by a certain equation of condition. The complete solution of the quintic in terms of one of the roots of its resolvent sextic was made by McClintock in 1884.* By this method j, 6 , s,*, s s \ and s t * are expressed as the roots of a quartic in terms of a quantity t which is the root of a sextic whose coefficients are rational functions of those of the given quintic. Although this has great theoretic interest, it is, of course, of little practical value for the determination of numer- ical values of the roots. By means of elliptic functions the complete quintic can, however, be solved, as was first shown by Hermite in 1858. For this purpose the quintic is reduced by Jerrard's transfor- mation to the form x" -\- $dx-\-2e = o, and to this form can also be reduced the elliptic modular equation of the sixth degree. Other solutions by elliptic functions were made by * American Journal of Mathematics, 1886, Vol. VIII, pp. 49-83. 2-1 THE SOLUTION OF EQUATIONS. [CHAP. I. Kronecker in 1861 and by Klein in 1884* These methods, though feasible by the help of tables, have not yet been sys- tematized so as to be of practical advantage in the numerical computation of roots. Prob. 15. If the relation s,s t = s,s, exists, between the elements show that V + s t ' + i- 3 6 + s, 6 = - zE. Prob. 16. Compute the roots of y" + 10/ + 2oy + 6 = o, and also those of y" — ioy 3 + 207 + 6 = 0. Art. 12. Trigonometric Solutions. When a cubic equation has three real roots the most con- venient practical method of solution is by the use of a table of sines and cosines. If the cubic be stated in the form (1) of Art. 9, let the second term be removed, giving y SJ rZBy+2C=o. Now suppose y = 2r sin 8, then this equation becomes B C 8 sin 3 84-6-, sin 0+2-5 = 0, r r and by comparison with the known trigonometric formula 8 sin 3 8 — 6 sin 0-f- 2 sin 36 = o, there are found for r and sin 38 the values r = V- B, sin 38 = C/ V— B\ in which B is always negative for the case of three real roots (Art. 9). Nowsin 38 being computed, 38 is found from a table of sines, and then 6 is known. Thus, jj/, = 2r sin 8, y, = 2r sin (120° -\- 6), y, = 2r sin (240 + 8), are the real roots of the cubic in y.f * For an outline of these transcendental methods, see Hagen's Synopsis der hoheren Mathematik, Vol. I, pp. 339-344. f When B* is negative and numerically less than C 2 , as also when B s is positive, this solution fails, as then one root is real and two are imaginary. • In this case, however, a similar method of solution by means of hyperbolic sines is possible. See Grunert's Archiv fur Mathematik und Physik, Vol. xxxviii, pp. 48-76. ART. 12.] TRIGONOMETRIC SOLUTIONS. 25 For example, the depth of flotation of a sphere whose diam- eter is 2 feet and specific gravity 0.65, is given by the cubic equation x 3 — $x' + 2.6 — (Art. 6). Placing x = y + 1 this reduces to/— 37 + 0.6 = o, for which B= — i and C =+0.3. Thus r= 1 and sin $0 = + 0.3. Next from a table of sines, 36/ = 17" 27', and accordingly = 5 49'. Then y t — 2 sin 5" 49' = +0.2027, y t = 2 sin 125 49' = + 1.6218, y, = 2 sin 245 49' = — 1.8245. Adding I to each of these, the values of x are x, = + 1.203 feet, #„ = + 2.622 feet, x, = —0.825 feet ; and evidently, from the physical aspect of the question, the first of these is the required depth. It may be noted that the number 0.3 is also the sine of 162° n', but by using this the three roots have the same values in a different order. When the quartic equation has four real roots its cubic re- solvent has also three real roots. In this case the formulas of Art. 10 will furnish the solution if the three values of / be ob- tained from (3) by the help of a table of sines. The quartic being given, g, h, and k are found as before, and the value of k will always be negative for four real roots. Then r = V— k, sin 36* = — -h/r*, and 3# is taken from a table ; thus is known, and the three values of /are /, = r sin 0, h = r sin ( 1 20° + 0), l,~r sin (240 + 0). Next the three values of u, of v, and of w are computed, and those selected which give u, w, and v — Vw all positive quanti- ties. Then (5) gives the required roots of the quartic. As an example, take the case of the inscribed rectangle in Art. 10, and let/ = 4 feet, q = 3 feet, m = V13 feet; then the quartic equation is X* — 5 !■*•' + 48 Vl3 X — 156 = 0. 26 THE SOLUTION OF EQUATIONS. [Chap. I. Here a = o, b = — 8J, c = + 12 V13", and a i + (2t>'-£)a"-($6 ! '-5fc + d)a< + (i4b i -2iFi:+6?>d+3t: i — e)a> -(42^-84^+28^+28^- ibe- yed + /)a'+. . ., is an expression of one of the roots of the equation. In order that this series may converge rapidly it is necessary that a should be a small fraction.* To apply this to a cubic equation the coefficients d, e,f, etc., are made equal to o, For example, let x 3 — $x + 0.6 = o ; this reduced to the given form is 0.2 = x — \x*, hence a — 0.2, b = O, c — — \, and then x = 0.2 + i . 0.2 3 + i . 0.2 6 + etc. = + 0.20277, which is the value of one of the roots correct to the fourth decimal place. This equation has three real roots, but the series gives only one of them ; the others can, however, be found if their approximate values are known. Thus, one root is about +1.6, and by placing x=y-{-i.6 there results an equation my whose root by the series is found to be + 0.0218, and hence + 1.6218 is another root of x* — 3* + 0.6 = o. *This method is given by J. B. Molt in The Analyst, 1882. Vol. IX, p. 104. '.28 THE SOLUTION OF EQUATIONS. [CHAP. I. Cardan's expression for the root of a cubic equation can be ■expressed as a series by developing each of the cube roots by the binomial formula and adding the results. Let the equa- tion be y -j- T,By -f- 2C = o, whose root is, by Art. 9, y = (- C + V£> + C 1 )* + (- C- V£° + C>)\ then this development gives the series, . _,V 2 2.5.8, 2. 5.8. 11 . 14 , \ y = 2(— C)Hi r — r 1 r~tr r — ■ ■ • < K ' \ 2 2.3.4 2.3.4.5.6 /' in which r represents the quantity (J? -f- C*)/t,C*. If r = O the equation has two equal roots and the third root is 2(— C fi. If r is numerically greater than unity the series is divergent, and the solution fails. If r is numerically less than unity and sufficiently small to make a quick convergence, the series will serve for the computation of one real root. For example, take the equation x a — 6x -\- 6 = o, where j5= — 2 and C = 3 ; hence r = 1/8 1, and one root is y — — 2.8845(1 — 0.01235 — 0.00051 — 0.00032—) = — 2.846, which is correct to the third decimal. In comparatively few cases, however, is this series of value for the solution of cubics. Many other series for the expression of the roots of equa- tions, particularly for trinomial equations, have been devised. One of the oldest is that given by Lambert in 1758, whereby the root of x n -\- ax — b = o is developed in terms of the .ascending powers of b/a. Other solutions were published by Euler and Lagrange. These series usually give but one root, and this only when the values of the coefficients are such as to render convergence rapid. Prob. 19. Consult Euler's Anleitung zur Algebra (St. Petersburg, 1771), pp. 143-150, and apply his method of series to the solution of a quartic equation. Art. 14. Computation of all Roots. A comprehensive and valuable method for the solution of equations by series was developed by McClintock, in 1894, by Art. 14.] computation of all roots 29 means of his Calculus of Enlargement.* By this method all the roots, whether real or imaginary, may be computed from a single series, The following is a statement of the method as applied to trinomial equations : Let x" = nAx n ~ k + B" be the given trinomial equation. Substitute x = By and thus reduce the equation to the form y" = nay n ' k -\-i where a = A/B h . Then if B" is positive, the roots are given by the series y = 00 -L-03 1 -* a + atf~*\\ — 2k + ri)c?/2 ! +a> , -s*(i -3*+»)(i -3^+2«K/3 ! +<»-^(i-4^+«)(i-4/&+2«)(i-4/§+3«K/4! +. . ., in which w represents in succession each of the roots of unity. If, however, B" is negative, the given equation reduces to y" = nay"~ k —I, and the same series gives the roots if go be taken in succession as each of the roots of — I. In order that this series may be convergent the value of a" must be numerically less than k~ k (n — k)*~ n ; thus for the quar- tic y i = 4ax -\- I, where n = 4 and k = 3, the value of a must be less than 27-*. To apply this method to the cubic equation x 3 =^Ax±B^, place n — 3 and k = 2, and put ^ = .##. It then becomes y' = $ay ± I where « = A/B 2 , and the series is y = go -f- g/# — -Jftja 3 -f- ita'a* -(-..., in which the values to be taken for go are the cube roots of 1 or — 1, as the case may be. For example, let x 3 — 2x — 5 =0. Placing y=$ix, this reduces to y* =0.684 y-\-i. Here #=0.228, and as this is less than 4-1 the series is convergent. Making go = 1, the first root is y = 1 +0.2280 — O.OO39 + O.OOO9 = I.22SO. *See Bulletin of American Mathematical Society, 1894, Vol. I, p. 3; also> American Journal of Mathematics, 1895, Vol. XVII, pp. 89-110. 30 THE SOLUTION OF EQUATIONS. [CkAP. I. Next making co = — \ + £ V— 3, a? is — \ —\ V— 3, and the corresponding root is found to be y = - 0.6125 + 0.3836 4/^3. Again, making 00 = — i — i V— 3 the third root is found to be the conjugate imaginary of the second. Lastly, multiplying each value of y by 5*, ^ = 2.095, X = — I.O47 ± I-I36 V— I, which are very nearly the roots of x* — 2x — 5=0. In a similar manner the cubic x" -f- 2x -\- 5 =0 reduces to y = — 0.6847 —1, for which the series is convergent. Here the three values of oa are, in succession, — 1, \ -\-\ V — 3, — £ -j- £ V— 3, and the three roots are y =■ — 0.777 and y = 0.388 ± 1.137*- When all the roots are real, the method as above stated fails because the series is divergent. The given equation can, however, be transformed so as to obtain n — k roots by one application of the general series and k roots by another. As an example, let x* — 243* -\- 330 = o. For the first applica- tion this is to be written in the form x=z — + ?32 243^243' for which n = 1 and k — — 2. To make the last term unity 330 place x = — — y, and the equation becomes whence a = 33073.243'. These values of n, k, and a are now inserted in the above general value of y, and 00 made unity; thus y = 0.9983, whence x, =1.368 is one of the roots. For the second application the equation is to be written Art. 15.] conclusion. 31 for which n — 2 and k = 3. Placing x = 243^, this becomes ' — &■ + '■ whence a = — 110/2435, and the series is convergent. These values of n, k, and a are now inserted in the formula for y, and 00 is made + 1 and — 1 in succession, thus giving two values for y, from which x, = 14.86 and x, = — 16.22 are the other roots of the given cubic. McClintock has also given a similar and more general method applicable to other algebraic equations than trinomials. The equation is reduced to the form y" = no. . y ± 1, where na . cpy denotes all the terms except the first and the last. Then the values of y are expressed by the series y-=.oo -4- GD x - n 4>oo . a-\-G) 1 -"-7-Go 1 ~''(cj)Goy . — -J- d +( CBl ~"^) &,I ~ K ^ a5 ) S - 3 l+- in which the values of go are to be taken as before. The method is one of great importance in the theory of equations, as it enables not only the number of real and imaginary roots to be determined, but also gives their values when the conver- gence of the series is secured. Prob. 20. Compute by the above method all the roots of the quartic x* + x -f- 10 = o. Art. 15. Conclusion. While this Chapter forms a supplement to the theory of equations as commonly given in college text-books, yet the brief space allotted to it has prevented the discussion and de- velopment of many interesting branches. Chief among these is the topic of complex or imaginary roots, particularly of their graphical representation and their numerical computation. Although such roots rarely, if ever, are required in the solution of problems in physical science, their determination is a matter of much theoretic interest. It may be mentioned, however, 32 THE SOLUTION OF EQUATIONS. [CHAP. I. that both the regula falsi and Newton's approximation rule may, by a slight modification, be adapted to the computation of these imaginary roots, approximate values of them being first obtained by trial. A method of solution of numerical algebraic equations, which may be called a logarithmic process, was published by Graffe in 1837, and exemplified by Encke in 1841.* It consists in deriving from the given equation another equation whose roots are high powers of those of the given one, the coefficients of the latter then easily furnishing the real roots and the moduluses of the imaginary roots. The method, although little known, is without doubt one of high practical values, as logarithmic tables are used throughout; moreover, Encke states that the time required to completely solve an equation of the seventh degree with six imaginary roots, as accurately as can be done with seven-place tables, is less than three hours. The algebraic solutions of the quadratic, cubic, and quartic equations are valid not only for real coefficients, but also for imaginary ones. In the latter case the imaginary roots do not necessarily occur in pairs. The method of McClintock has the great merit that it is applicable also to equations with imagi- nary coefficients ; it constitutes indeed the only general method by which the roots in such cases can be computed. Prob. 21. Compute by McClintock's series the roots of the equa- tion x* — ix — 1 = o. Prob. 22. Solve the equation cos x coshx + i = o, and also the equation x — e x = o. (For answers see Crelle's Journal fur Mathe- matik, 1841, Vol. XXII, pp. 1-62.) * See Crelle's Journal fiir Mathematik, 1841, pp. 193-248. Art. 1.] INTRODUCTION. 33 Chapter II. DETERMINANTS. By Laenas Gifford Weld, Professor of Mathematics in the State University of Iowa. Art. 1. Introduction. As early as 1693 Leibnitz arrived at some vague notions regarding the functions which we now know as determinants. His researches in this subject, the first account of which is contained in his correspondence with De L'Hospital, resulted simply in the statement of some rather clumsy rules for elimi- nating the unknowns from systems of linear equations, and exerted no influence whatever upon subsequent investigations in the same direction. It was over half a century later, in 1750, that Gabriel Cramer first formulated an intelligible and general definition of the functions, based upon the recognition of the two classes of permutations, as presently to be set forth. Though Cramer failed to recognize, even to the same extent as Leibnitz, the importance of the functions thus defined, the development of the subject from this time on has been almost continuous and often rapid. The name " determinant" is due to Gauss, who, with Vandermonde, Lagrange, Cauchy, Jacobi, and others, ranks among the great pioneers in this development. Within recent years the theory of determinants has come into very general use, and has, in the hands pi such mathema- ticians as Cayley and Sylvester, led to results of the greatest interest and importance, both through the study of special forms of the functions themselves and through their applica- tions.* * A list of writings on Determinants is given by Muir in Quarterly Journal of Mathematics, 1881, Vol. XVIII, pp. 110-149. 34 DETERMINANTS. [CHAP. II. Art. 2. Permutations. The various orders in which the elements of a group may be arranged in a row are called their permutations. Any two elements, as a and b, may be arranged in two orders : ab and ba. A third, as c, may be introduced into each of these two permutations in three ways : before either element, or after both ; thus giving 3X2 = 6 permutations of the three elements. In like manner an additional element may be intro- duced into each of the permutations of i elements in (z'-f- 1) ways: before any one of them, or after all. Hence, in general, if P t denote the number of permutations of i ele- ments, P i+1 = (Y-f- i)P { . Now, /> 3 = 3 X 2 X 1=3!; hence P t = 4 X 3 ! = 4 ! ; and, n being any integer, P n = n(n — i){n — 2) . . . 1 = n ! . That is, the number of permutations of 11 elements is n !. For all integral values of n greater than unity, n ! is an even number. If the elements of any group be represented by the differ- ent letters, a, b, c, . . ., the alphabetical order will be considered as the natural order of the elements. If represented by the same letter with different indices, thus : a lt a 2 , a 3 , . . . ; or thus : a', a", a'", . . ., the natural order of the elements is that in which the indices form a continually increasing series. Any two elements, whether adjacent or not, standing in their natural order in a permutation constitute a permanence ; standing in an order which is the reverse of the natural, an inversion. Thus, in the permutation daecb, the permanences are de, ae, ab, ac ; the inversions, da, dc, db, ec, eb, cb. The permutations of the elements of a group are divided into two classes, viz.: even or positive permutations, in which the number of inversions is even ; and odd or negative permu- tations, in which the number of inversions is odd. ART. 3.] INTERCHANGE OF TWO ELEMENTS. 35 When the elements are arranged in the natural order the number of inversions is zero — an even number. Thus, the even or positive permutations of the elements a lt a v a, are while the odd or negative permutations are a 3 permuta- tion of the old set has been converted into an I even f one in the new. Hence, in either set, there are as many even permu- tations as odd ; that is, one half are even and one half odd. Prob. i. Classify the following permutations: (i)dcdea; (2) 111 v 1 11 iv; (3) knimlj; (4) a" a? a' a™ a'"; (5) fieyZad- (6)52413; (7) x^x^x.x^^x,; (8) F. Tu. M. Th. W.; (9) jx k v iX. Prob. 2. Derive the formula for the number of permutations of n elements taken m at a time. (Ans. n\/{n — m)\.) Prob. 3. How many combinations of m elements arranged in the natural order may be selected from a group of n elements? (Ans. n\/m\(n — *»)!.) Prob. 4. Show that o! = 1. Art. 5. The Determinant Array. Assume n % elements arranged in n vertical ranks or columns, and n horizontal ranks or rows, thus : a; a," . . . « « a' a," . . . a ™ a' a" n <"' Art. 6.] determinant as function of « a elements. 37 Ins this array all the elements in the same column have the same superscript, and those in the same row the same subscript. The columns being arranged in order from left to right, and the rows likewise in order from the top row downward, the position of any element of the array is shown at once by its indices. Thus, «/" is in the third column and the fifth row of the above array. The diagonal passing through the elements a/, a", . . . a„ ln) is called the principal diagonal of the array ; that passing through a„', «„_,", . . . «/"', the secondary diagonal. The posi- tion occupied by the element a x ' is designated as the leading position. Art. 6. Determinant as Function of tf Elements. The array just considered, inclosed between two vertical bars, thus : a x a t . . . a a' a" . ^ ■ («) a n a„ is used in analysis to represent a certain function of its rc 2 ele- ments called their determinant.* This function may be defined as follows : Write down the product of the elements on the principal diagonal, taking them in the natural order ; thus : in («> a: a, a, This product is called the principal term of the determinant. Now permute the subscripts in this principal term in every possible way, leaving the superscripts undisturbed. To such of the n ! resulting terms as involve the even permutations of the subscripts give the positive sign ; to those involving the odd *This notation was first employed by Cauchy in 1815. See Dostor's Theorie des determinants, Paris, 1877. 38 DETERMINANTS. [Chap. II. permutations, the negative sign. The algebraic sum of all the terms thus formed is the determinant represented by the given array. Art. 7. Examples of Determinants. Applying the process above explained to the array of four elements gives «/«," = *,'*," -a: a,". (1) As an example of a determinant of nine elements, with its ex- pansion, may be written — 1 « / ~ a ~ hi 1 « 1 ~ " ~ "1 1 « ' « "„ "/ z= -\- a, a^ a, -\- a, a 3 a, -\- a, a^ a, a„ a„ a„ s, I r, " s, I" 'l\ It is evident, from the mode of its formation, that each term of the expansion of a determinant contains one, and only one, element from each column and each row of the array. It follows that every complete determinant is a homoge- neous function of its elements. The degree of this function, with respect to its elements, is called the order of the deter- minant. Thus, (1) and (2) are of the second and third order respectively. The definition of a determinant given in the preceding article is once more illustrated by the following example of a determinant of the fourth order with its complete development : a, b t c l -fA — <*AcA + a,*M + aA^A — apsA — "Ac A + a Ac A, + aA c A - <*A C A> — « 3 VX + "AcA + a Ac A — a Ac A - a A c A + a Ac A, + a Ac A ~ a Ac A — aAcAi + aAcA + aAcA — aA c Ai — a A c A, + <*AcA> \ (3) Art. 8.] NOTATIONS. 39 It will be noticed that, in this case, the columns are ranked alphabetically instead of by the numerical values of a series of indices. Art. 8. Notations. Besides the notations already employed, the following is very extensively used : «... This is called the double-subscript notation ; the first subscript indicating the rank of the row, the second that of the column. Thus the element « 23 is in the second row and the third column. The letters are sometimes omitted, the elements being thus represented by the double subscripts alone.* Instead of writing out the array in full, it is customary, when the elements are merely symbolic, to write only the prin- cipal term and enclose it between vertical bars. This is called the umbral notation. Thus, the determinant of the «th order is written I a' a' 1 . . . a„ c«) or, using double subscripts, i ®nn \ These last two forms are sometimes still further abridged to («) and respectively. Prob. 5. Write out the developments of the following determi- nants: (1) * Leibnitz indicated the elements of a determinant in this same manner, though he made no use of the array. «. b x ; (*) /'/' ; (3) p> q > ; (4) a b a, b t q'v" P" 1" a/3 40 (5) I «//= (6) p p p DETERMINANTS ; (7) ?VV" P ' q' t>" q" r" f>'"q'"r'" [Chap. II. (8) a b c a ji y x y z (9) I 11, 22 ] ; (10) I a U2 I ; (11) I l m,n, | ; (12) | a u a„a„a tt \ . Prob. 6. How many terms are there in the development of the determinant | a™ | ? In the above determinant tell the signs of the terms : (1) «,W'W«, Tl ; (*) *>,' WXX Ti ; ( 3 ) a 6 w'Vxx vi - Prob. 7. Show that in the expansion of any determinant, all of whose elements are positive, one half the terms are positive and one half negative. Prob. 8. In determinants of what orders is the term containing the elements on the secondary diagonal (called the secondary term) positive ? Prob. 9. What is the order of the determinant whose secondary term contains 10 inversions ? 36 inversions ? Prob. 10. In the expansion of a determinant of the «th order, how many terms contain the leading element ? Art. 9. Second and Third Orders. Simple rules will now be given for writing out the expan- sions of determinants of the second and third orders directly from the arrays by which they are represented. To expand a determinant of the second order, write the product of the elements on the principal diagonal minus the product of those on the secondary diagonal, thus : = ad — be. Likewise, a c ~9 — 2 5| = -3 + 10 il The following method is applicable to determinants of the third order:* * This method was first given by Sarrus, and is often called the rule of Sarrus; sec Finck's Elements d'Algebre, 1846, p. 95. Art. 9] SECOND AND THIRD ORDERS. 41 Beneath the square array let the first two rows be repeated in order, as shown in the figure. Now write down six terms, each the product of the three ele- ments lying along one of the six oblique lines parallel to the diagonals of the original square. Give to those terms whose ele- ments lie on lines parallel to - the principal diagonal the posi- tive sign ; to the others, the - negative sign. The result is the required expansion. Ap- plying the method to the determinant just written gives I «,V. I = «iVs + a A c i + «sV 2 — «,*/. — fl iV. — a A c *- After a little practice the repetition of the first two rows will be dispensed with. The above methods are especially useful in expanding determinants whose elements are not marked with indices, or in evaluating those having numerical elements. No such sim- ple methods can be given for developing determinants of higher orders, but it will be shown later that these can always be resolved into determinants of the third or second order. Prob. ii. Develop the following determinants: (i) (4) (7) a h g h b f g f c x, y, i x, y, i i cos a cos oe i (5) (8) Prob. 12. Evaluate the following: (i) i 2 3 3 i 2 2 3 I (*) — 2 — 2 O — 2 12 2 o — n—m > (3) A c b > 11 o — / c £ a m I o b a C i P Q i (6) cos a sin /? o cos a sin /3 sin a cos yS o s i in a cos /? V - i ) (9) a b c cab b c a • 4^-2 no i wing: ; (3) - i _ v" - 4/"=! o V - - i - i - V - i I V - - i V^l - 1 (An 5. 18; [6 2.) 42 DETERMINANTS. [Chap. II. Art. 10. Interchange of Rows and Columns. Any term in the development of the determinant | a™\ may- be written a h «i a y ... a? ', in which hij. . ./is some permutation of the subscripts 1,2, 3,. . .n. Designate by u the number of inversions in hij . . . I. Also, let v be the number of interchanges of two elements necessary to bring the given term into the form ± a « a ,«*> a^ . . . a„«>, in which the subscripts are arranged in the natural order, while pqr ... t is a certain permutation of the superscripts ', ", '", . . . (B) . This permutation is even or odd according as v is even or odd. But u and v are obviously of the same class ; that is, both are even or both odd. Hence the permutations hij . . . I and pqr . . . t are of the same class ; and the term will have the same sign, whether the sign be determined by the class of the permutation of the subscripts when the superscripts stand in the natural order, or by the class of the permutation of the superscripts when the order of the subscripts is natural. It follows that the same development of the determinant array will be obtained if, instead of proceeding as indicated in Art. 6, the superscripts of the principal term be permuted, the subscripts being left in the natural order, and the sign of each of the resulting terms written in accordance with the class of the permutations of its superscripts. Passing from one of these methods of development to the other amounts to the same thing as changing each column of the array into a row of the same rank, and vice versa. Hence, a determinant is not altered by changing the columns into cor- responding rows and the rows into' corresponding columns. Thus : a' (») , («) a, a t a.' a„ a K a n (") <») n («) («) ART. 11.] TWO IDENTICAL PARALLEL LINES. 43 Whatever theorem, therefore, is demonstrated with reference to the rows of a determinant is also true with reference to the columns. The rows and columns of a determinant array are alike called lines. Art. 11. Interchange of Two Parallel Lines. If any two parallel lines of a determinant be interchanged, the determinant will be changed only in sign. For, interchanging any two parallel lines of a determinant array amounts to the same thing as interchanging, in every term of the expansion, the indices which correspond to these lines. Since this changes the class of each permutation of the indices in question from odd to even or from even to odd, it changes the sign of each term of the expansion, and therefore that of the whole determinant. It follows from the above that if any line of a determinant be passed over m parallel lines to a new position in the array the new determinant will be equal to the original one multi- plied by (— i) M . The element a^ may be brought to the leading position by passing the kxh. row over the (k — i) preceding rows, and the ^th column over the (s — t) preceding columns. This being done the determinant is multiplied by (- I) 4 " 1 . (- i)-« = (- i)** which changes its sign or not according as (/£ + *) is odd or even. The position occupied by a k is) is called a positive position when {k -+- s) is even ; a negative position when (k -j-s) is odd. Art. 12. Two Identical Parallel Lines. A determinant in which any two parallel lines are identical is equal to zero. For the interchange of these two parallel lines, while it 44 DETERMINANTS. [Chap. II. changes the sign of the determinant, will in no way alter its value. The value then, if finite, can only be zero. Art. 13. Multiplying by a Factor. Multiplying each element of a line of a determinant by a given factor multiplies the determinant by that factor. Since each term of the development contains one and only one element from the line in question (Art. 7), then multiply- ing each element of this line by the given factor multiplies each term of the development, and therefore the whole deter- minant, by the same factor. It follows that, if the elements of any line of a determinant contain a common factor, this factor may be canceled and written outside the array as a factor of the whole determinant ; thus : a n . . m «„• . . . a in = m \ a„ a„ a„ . . m a ti . . . a tn «„, . . m a ni . . . a nn A determinant in which the elements of any line have a ■common ratio to the corresponding elements of any parallel line is equal to zero. For this common ratio may be written outside the array, which will then have two identical lines. Its value is therefore zero (Art. 12). A determinant having a line of zeros is equal to zero. Art. 14. A Line of Polynomial Elements. A determinant having a line of elements each of which is the sum of two or more quantities can be expressed as the sum of two or more determinants. Let a t {b-b/ + b/' ±...) *,... =A (1) *, (*,- K + K ± • • • ) c *. (h- 1>; + b:' ± . . . ) c. be such a determinant. Then, if Art. 15.] COMPOSITION OF PARALLEL LINES. 45 any term of the expansion of the determinant A is ± a h BiCj . . . = ± a h b t cj . . . =F a h b/ Cj . . . ± a h b," cj . , . ± . . . ( 2 ) The terms in the expansion of A are obtained by permuting the subscripts h, i,j, ... of a h B { Cj . . . . But permuting at the same time the subscripts of the terms in the second mem- ber of (2), and giving to each term thus obtained its proper sign, there results A = I ai Bf, . . . J = I a, V, ••• I - I aftc % . . . | + | aft'c, . . . | ± . . . ,. which proves the theorem. Art. 15. Composition of Parallel Lines. If each element of a line of a determinant be multiplied by a given factor and the product added to the corresponding ele- ment of any parallel line, the value of the determinant will not be changed ; thus: . «, *"n\ ^ni t *«3 ' This wiil appear upon resolving the second member into two determinants (Art. 14), one of which will be the given de- terminant, while the other, upon removal of the given factor, will vanish because of having two identical lines. In like manner any number of parallel lines may be com- bined without changing the value of the determinant, care being taken not to modify in any way the elements to which are added multiples of corresponding elements from other parallel lines. For example, | a u „ | is equivalent to «„ (la tt + a u - ma lt + . . .) a lt . . . a in fo.+AO (« M -R«..) . . . (a,„+Aa, n ) — w(a„ + A« 18 ) + . . .) «l (^«. + a m — ma n 3 + • • •) ««s •••««» 46 DETERMINANTS. Art. 16. Binomial Factors. [Chap. II. A determinant which is a rational integral function of a and of b, such that if b is substituted for a the determinant vanishes, contains {a — b) as a factor. For example, A = a' — /> s a — q a-\-r b* —f b — q b + r p q r is divisible by (a — b). To prove this, let the expansion of any such determinant be written in the form A = m a -J- m x a -J- **,«' -)-..., the coefficients m a , m lt ;«,, . . . being independent of «,£ + ;/z s £ 2 -j- . . . Subtracting this from the preceding gives A = mla - b) + m,{a' - b*) + . . . This being divisible by (a — b), the theorem is proven. Prob. 13. Prove the following without expansion : (1) — X X my -y W27Z2 «£ (3) (4) = o; b -\- c a a b c -\- a b c c a -\-b 6* + S (2) = 2 C b o — c c o —a = o; a -a a b b b c , of the expansion of (1) is the same as the sign of the corre- sponding term, a/'a/" . . .«/">, of the expansion of (2). Hence, ,<») (3) «, o a' a' . . .0 («) u n d n («) a„" a n '"...aW The determinant (2) is called the co-factor or complement of the element a/ in the determinant (n) |. It is obtained from this determinant by deleting the first column and the first row. The co-factor of any element a k (s) maybe found in the same manner upon transposing this element to the leading position. But by this transposition the sign of the determinant will be changed or not according as a t ® occupies a negative or a posi- tive position (Art. n). Hence, to find the co-factor of any element a™ of the determinant | «,<*> |, delete the row and the column to which the element belongs, giving the resulting determinant the i P 0Slti . ve \ s i gn w h en (k + s) is j even ( negative j & v ~ ; | odd Art. 18.] DEVELOPMENT IN TERMS OF CO-FACTORS. 49 The co-factor thus obtained is represented by the symbol A <*) • the sign-factor of which, (- if+% is intrinsic, i.e., included in the symbol itself, which is accordingly written as positive. The co-factors of the various elements of \a il a n a it \ are as follows : A u = a n « 3 s «3, «33 A u = — «l» *.3 a z, ^33 ^31 «„ «13 «» a K A,. — ^..= -4.. s «>■ «,3 <*3. «33 «U «>. «3, «33 «» «.3 «»i «33 ^„ ^„ = A,, — n *.. «3 The result obtained by deleting the kih. row and the sth column of A = | «/"' | is called the minor of the determinant with respect to the element «£>, and is written A%. This minor is the same as the co-factor of the same element without its sign-factor ; thus : A h <* = (- if* 4% Similarly A[£;l is the result obtained by deleting the ^th and £th rows and the /th and jth columns of A, and is called a second minor of the given determinant. Minors of still lower orders are obtained in a similar manner, and expressed by a similar notation. The /£th minors are determinants of the order (« — k). Art. 18. Development in Terms of Co-factors. The (n — i) ! terms of | «, (M) | which contain a^ are repre- sented in the aggregate by a k {s) A k (s) (Eq. 3, Art. 17). In like manner the groups of terms containing the successive elements a k ', a k ", . . . a k (n) are respectively „ I A I „ II A " n (") A <") Each one of these n groups includes (« — 1) ! terms of the determinant | a™ \ , no one of which is found in any other 50 DETERMINANTS. [Chap. II. group. In all of them, then, there are «X(« — i) ! or n\ dif- ferent terms of the determinant, which is the whole number. Hence, | a « | = a,'A k ' + a k "A t " + . . . + a^Af\ (i) Similarly (Art. 10), , («) = a^A^ -\-a^A^ + . . . + «„ a 1 b t d 1 -< a* K c, b, c, d 3 a 2 c % d, a % b s d, « s b* c, b, c, d t a < C A d < a t b t d t a < b, c t Upon a second application of the same formula this becomes = «, K c* d, — a i c* b,d, + a x d, b s c. c d, a, b 3 b t d t a t d t a t b t — a,dA b, c % + b % d. a, c, - c* d x a,b, b t c K a t c, a, b t The complete development may be written out directly from the above. It is given in Eq. 3, Art. 7. Art. 19.] THE ZERO FORMULAS. 51 Prob. 19. Develop the following determinants: (1) I X 1 y X I y 1 I y 1 X y 1 X I (2) a X y a X y y X a y x a (3) o q r s p o r s p q o s p q r o (Axis. (x-yY((x+yY- 4 ); (*'-/)'; tfqrs.) Prob. 20. Find the values of the following determinants: U) (4) 1 2 3 4 2 34i 3 4i2 4123 1 1 1 1 O I I I I O I I I I o (2) 0102 1020 0201 2010 (3) 3 - 1 5 o 9 -3 8 3 (S) 3 3 3 3 ; (6) o o o 3 3222 1002 2211 OIOI IIIO OOIO (Ans. 160; 9; o; 3; 3; 3.) Prob. 21. Obtain the determinants in Exs. 5 and 6 of the pre ceding problem from that in Ex. 4. Prob. 22. Evaluate Oil 1 o 1 1 1 o of the «th order. (Ans. (n - i)(- 1)"-'.) Prob. 23. Show that abed b a —d c c d a —b d —c b a = {a'+b'+c'+d 2 )' Art. 19. The Zero Formulas. If in the determinant | «,<"> | the ^th and /£th rows be sup- posed identical, the elements a k ', a", . . . a k {n) in the formula {1) of the last article may be replaced by a,/, a,/', . . . a h w re- spectively. But in this case the value of the determinant is zero (Art. 12). Hence, in reference to the determinant I a t in) I , h and k being different subscripts, a h 'A k ' + a h "A t " + . . . + of>Af> = O. 52 DETERMINANTS. [CHAP. II. Similarly, p and s being different superscripts, a/^w + a^A^ + . . . + ai»Al» = o. Art. 20. Cauchy's Method of Development. It is frequently desirable to expand a determinant with reference to the elements of a given row and column. Let the determinant be A == | a^ n) \ , and the given row and column the /ith and pth respectively. Then is A k (/,) the co-factor of a h {p \ the element at the intersection of the two given lines. The co-factor of any element a k (s) of A h ip) will be designated by B k {s) , this being a determinant of the order {n — 2). The required expansion may now be obtained by means of the following formula, due to Cauchy : in which k = 1, 2, . . . h — 1, A -(- 1, . .. k, and $ = 1, 2, .. ./ — i, / + 1, . . . n, successively. To prove this, consider that B t {s) is the aggregate of all terms of the expansion of A which contain the product a h {p) a k [s) . These terms are included in a k m A h w . Now, every term in the expansion which does not contain a h ip) must contain some other element a h ls) from the Ath row and also some other element a k w from the pth. column, and thus contains the prod- uct a,^a k (p K But this product differs from « A ( % (s) only in the order of the superscripts ; and is, therefore, in the expansion of A, multiplied by an aggregate of terms differing in sign only from that multiplying a,^a^\ Hence, — B k ® is the coefficient of a^> a k (p) in the required expansion. In the formula a h lp) A k ip) gives (n — 1) ! terms of A. There are also (« - 1)' such aggregates as — a^a^B^, each con- taining (n— 2) ! terms. The formula therefore gives (« — 1) ! + (« — i) J (n - 2) ! = n ! terms, which is the complete expansion. When the expansion is required with reference to the ele- Art. 20.] CAUCHY S METHOD OF DEVELOPMENT. 53 merits of the first column and the first row the formula, written explicitly, becomes |« W| = a/ A/- aJa/'B," - a^a^'B,'" - ... - «,'«w/|W — a n a i -»« — «» «i ^>k — a u 'a™B*\ (2) in which B£ s) has intrinsically the sign (— i) k+s . Cauchy's formula is particularly useful in expanding deter- minants which have been bordered ; such as -Q = o «, u„ u. W 3 «.l «33 «93 (3) Applying formula (2) to this determinant gives -Q = + »,«, <^ S j ^23 + «,«„ «,. «33 — U,U, a u a„ #33 «33 *»■ «33 tz si # 3 , «1. « 1S - «,' «.. «13 + u,u, «,. «.. j #33 #33 «>1 ^33 «S, «S3 1 «1. «>s + «,«. «.I «l. -K «11 «1. #, 2 #33 «„ «33 ^31 ^33 Letting a hs = a sk , and writing ^ n , ^„, ... for the co-factors of the elements of | a 11 a„a„ | , the above becomes <2 = A n U* + ^..W,* + ^..w.' + 2A„U,U S + 2^t 11 «,«, + 2A„U 1 U,, Prob. 24. Develop the following determinants by Cauchy's formula: (0- a h g u hb fv gfcw U V wo (2) o yz zx xy yz o 1 1 zx 1 o 1 xy 1 1 o (3) 1 1 1 1 o jy z.r 1 xy o_yz 1 zx yz o 54 (4) DETERMINANTS. [CHAI 1L — I — X i - y X o I — z I -I y i —i ; (5) I I I X x y z o i i i y I I I z ; (6) o a b — a sin A sin.5 — b — cos A cosi? Art. 21. Differentiation of Determinants. By the formula (i) of Art. 18 A = | y 1>n | = Y kl y tl + Y k ,y tt + . . . F*„.n„. (i) Considering the elements of the determinant as independent variables and differentiating with respect to y ki gives 6 kl A = Y ks dy h , or Substituting in (i), SA , dJ Similarly ^Jy-+^dy- v + Y -^ Yk *- dy k ; ■ +y. +y« SA dy, kn SA dy H i (2) (3) (4) Again differentiating (i), this time with respect to all the ele- ments of the kih row, there results S k A = Y kl dy kl + Y k ,dy H + • • • + Y kH dy kK . (5) In the total differential of A there are obviously n such ex- pressions as (5), each of which may be obtained from A by replacing the elements of some one of the rows by their differ- entials ; thus : dA = dy„- y*> ■ ■ • dy in • ■ y** + 7,. • dy n . • ■y.n • ■ dy, n + • • + y^ y,i- ■ y,« ■ y w y*> ■ ■ -y»n y nx • ■ ■ y«n d y« x - • • dy„„ .(6) If all the elements are functions of one independent variable x, then, representing -^ by y ks ', dA dx y* • y>« y» ■ y« + y„ ■ ■ -y\ yj-- • y v y«> y« + •■• + y» yJ y>« y m • y»» • (7) Art. 22.] RAISING THE ORDER. 55 Prob. 25. Show that Cauchy's formula may be written SA J = | «,<">| =« A «- njr -2 ai oi Si 6' J ip) da h w d a ^ Art. 22. Raising the Order. Since, in the expansion of the determinant (1) of Art. 17 the elements a t ', . . . a„' do not appear, these may be replaced by any quantities whatever, as Q, . . . T, without changing the value of the determinant ; thus: a/ o o . . . o Q a," <" . . . «,<"> #„ («> Similarly, «/ O O «,' «," O ^3 "3 o o -*« w « tc » . (II) r««"a. -<») . (») w « . . . tt M 0/0 o . . . o 0«," o ... o R L a,'". . . a*> T Na H '"...a™ in which Q, R, . . . T and L, . . . iVare any quantities whatever. Finally, a,' o . . . o G *,"■•■ o o s Af . . .^ir" o T N ... C *<"> a l a' a . . . o . . . o o o ■ a n ±T" o (»-l)/Y (")- *•» — I M M that is, if all the elements on one side of the principal diagonal are zeros the determinant is equal to its principal term, and the elements on the other side of this diagonal may be replaced by any quantities whatever. By what precedes, («) a ' a (B) u n . . . u n 1 I O . . . O Q a/ ... a™ Ta'.. a <*> 5G DETERMINANTS. [Chap. II. Hence, a determinant of the nth order may be expressed as a determinant of the order (n -+- i) by bordering it above by a row (to the left by a column) of zeros, to the left by a column (above by a row) of elements chosen arbitrarily, and writing i at the intersection of the lines thus added. By con- tinuing this process any determinant may be expressed as a determinant of any higher order. Prob. 26. If all the elements on one side of the secondary diag- onal are zeros, what is the value of the determinant ? Prob. 27. Develop the determinant Prob. 28. A determinant in which a k a h g u o h b f v o g f c w o u v w o t o o o t s w ,.(*> said to be skew-symmetric. Prove that every skew-symmetric deter- minant of odd order is equal to zero. Art. 23. Solution of Linear Equations. Of the many analytical processes giving rise to determinants the simplest and most common is the solution of systems of simultaneous linear equations. Thus, solving the equations a/x' + < V = Ki , I by the methods of ordinary algebra gives : a x *-, - a//f, x X In the notation of determinants these are written : / , x" = < «•, / < < < K, It will be noted that the two fractions expressing the values of x' and x" have a common denominator, this being the de- terminant whose elements are the coefficients of the unknowns arranged in the same order as in the given equations. The .Art. 23.] SOLUTION OF LINEAR EQUATIONS. 57 numerator of the fraction giving the value of x' is formed from this denominator by replacing each coefficient of x' by the corresponding absolute term. Similarly for x". The difficulty of solving systems of linear equations by the ordinary processes of elimination increases rapidly as the num- ber of equations is increased. The law of formation of the roots explained above is. however, capable of generalization, being equally applicable to all complete linear systems, as will .now be shown. Let such a system be written «, V -f «/ V + . a;x' + al'xf' + . -f a^x™ — K a„'x' + a„"x" + . . . +'«, MjpM — K (I) Now form the determinant of the coefficients of these •equations ; thus : D = a'a' . a. ■ («) i (») .a, (a) and let Atf 1 be the co-factor of « A W in this determinant. The function w is equal to D when/ = ^ (Art. 18) ; to zero when p and s are different superscripts (Art. 19). Then, multiplying the given equations by A®, Af\ . . . A® respectively, the sum of the resulting equations is a linear equation in which the coefficient of x {s) is equal to D, while those of all the other unknowns vanish. The sum is, therefore, DxP = M« + k^ + . . . + K n AJ*\ (2) But the second member of this equation is what D becomes upon replacing the coefficients a, is) , a< s \ . . . a} s) of the unknown jr (t - by the absolute terms k, , /c, , . . . k„ in order. Hence, 58 DETERMINANTS. [Chap. U, x» = . a a, . . . a. (s -I) KM (i -i) /f,«. M-i) (H-i) . a (») («) . («-i) K n a (s+i) . . a <«) . («) . a («> a„ «„ . («) (3). This result ma}' be stated as follows : (a) The common denominator of the fractions expressing- the values of the unknowns in a system of n linear equations involving n unknown quantities is the determinant of the coefficients, these being written in the same order as in the- given equations. (&) The numerator of the fraction giving the value of any one of the unknowns is a determinant, which may be formed from the determinant of the coefficients by sub- stituting for the column made up of the coefficients of the unknown in question a column whose elements are the absolute terms of the equations taken in the same order as the coeffi- cients which they displace. Prob. 29. Solve the following systems of equations : (1) 3x + 5y=2i, 6x + 2y=i$; 37 _ 1 \ x (2) 7 2X , 1^ (3) 3* + J 7 + 2Z = 50, x + 2y - 32 = 15, 2.X+2J'- 32 = 25; (4) - + -=/, -+- = ?, ~+ ~ = r; y z z x x y (5) T+f+-:+^ 3579 5 7 9 11 2I44r w . x y , z ui x , y , z 1 = 1744, b—--\ = 1472. 7 9 11 13 9 11 13 IS Prob. 30. Show that the three right lines y = x -f- 1, y = — 2.x -\- 16, y intersect in a common point. 3* Art. 24. Consistence of Linear Systems. When the number of given equations is greater than the number of unknowns their consistency with one another must Art. 24.] CONSISTENCE OF LINEAR SYSTEMS. 59 obviously depend upon some relation among the coefficients. This relation will now be investigated for the case of (n + i) linear equations involving n unknowns. Let the equations be + ... + «,<">*« =/c 1) "I «, x' ■ + «, (»M«) V a, l+l 'x + . . . + a&zf* - K n+l . If the above equations are consistent the values of the unknowns obtained by solving any n of them must satisfy the remaining equation. Solving the first n equations by the method of the preceding article, substituting the values of x' , x", . . . x w thus obtained in the last equation, and clearing of fractions, the result reduces to (Art. 18) (>o a„ («) M. = o, which is the condition to be fulfilled by the coefficients in order that the given equations may be consistent. Hence the condition of consistency for a set of linear equa- tions involving a number of unknowns one less than the number of equations is that the determinant of the coefficients and absolute terms, written in the same order as in the given equa- tions, shall be zero. This determinant is called the resultant* or eliminant of the equations. Thus the equations x -\-y — z — o, x —y~\-s= 2, — x-\-y + £ = 4, x-\-y-\-z = 6 are consistent, for the reason that I I — I o I — I I 2 I I I 4 I I I 6 * This term was introduced by Laplace in 1772. The term eliminant is due to Sylvester. 60 DETERMINANTS. [Chap. II. Art. 25. The Matrix. Assume r linear equations involving n unknowns, r being greater than n. as follows : a„'x' + . . • + a™* («M») ^«i tf r V + • ■ • + « r W ^ = K r . j The consistency of these equations requires that every deter- minant of the order {n + i), formed by selecting (n + i) rows from the array whose elements are the coefficients and abso- lute terms written in order, shall be zero. If the elements of the array fulfill this condition the fact is expressed thus : a,' ...«„' ...a r ' =o; (n) . .a ■ K <«) . a («> K r the change of rows into columns being purely arbitrary. The above expression is called a rectangular array, or a matrix. Art. 26. Homogeneous Linear Systems. Let the equations of the given system be both linear and homogeneous ; thus : «/#'+. . . + a^x™ = o, «„'*'+...+ *„<•>*<"> = o. (I) Representing the determinant of the coefficients by E, the general solution, as given by the formula (3) of Art. 23, is *w = o/£. That is, all the unknowns are equal to zero, and the equa- tions have no other solution than this unless E = o. Art. 26.] HOMOGENEOUS LINEAR SYSTEMS. 61 But in this case the value of each unknown is obtainable only in the indeterminate form o/o. The ratios of the un- knowns may be readily obtained, however. For, dividing each equation through by any one of these, as x (s \ the system (i) becomes „'* 4- +«; <•-!)" ,(*-.) . ( S +ii: Us+i) x v 4-/7 <">_ — . . M *' W h (2) •+«.' (i -i) ^r l (s-i) 7 (s+l)_ -w + ...+«; M ,.{») Now the condition £ = o establishes the consistency of the n equations (2) involving the (n — 1) unknown ratios (Art. 24), (S-I) r U)' jr' Hence, if .£ = O the given equations (1) are consistent ; that is, the values of the above (« — 1) ratios obtained by solving any ( n — j) of them will satisfy the remaining equation. Any n quantities having among themselves the ratios thus determined will satisfy the given equations. Thus, if xj, x„", . . . x M are n such quantities, so also are Xx„', A x a " , . . . Xx (u) , A. being any factor whatever. The determinant E of the coefficients of the given homo- geneous linear equations is called the resultant or eliminant of the system. When the number of equations is greater than the number of unknowns the conditions of consistency are expressible in the form of a rectangular array, as in Art. 25. As an example, consider the five equations 2X — 3j 4- z — o, 4* — y — z = o, — ix + 3j 4- z = o, x+y — z = o, 5^-57 + ^ = 0. Dividing each of the first two equations by z and solving x y for the two unknowns - and - gives z z x z -1-3 1 — 1 2- 3 4- 1 y z 2 — I 4 1 2 — 3 4— 1 62 DETERMINANTS. [Chap. II. or x:y:z::2 = 3 : 5; and any three quantities having these ratios will satisfy the two equations, as 10, 15, and 25. That the third equation is consistent with the first two is shown by the vanishing of the determinant 2-3 1 4- 1 - 1 -7 3 1 If all the equations are consistent the determinant of the .coefficients of any three of them must vanish ; that is, 2 4- 7 1 5 -3-1 3 1 - - 5 1 — 1 1 — 1 1 Art. 27. Co-factors in a Zero Determinant. If, in the preceding article, E = o, it follows from Arts. 18 ; and 19 that a/A,' + ai "A k " + . . . + *«A<"> = o, a t 'A„' + a k "A t " + . . . + a^Af* = o = E, a n 'A k ' + a n "A k " + . . . + a™A£* = o. These equations obviously give for the ratios Ajf_ A_t^ AJ*+v AJ* AP' ' "~A k * ' Af ' ' ' ' A? -values which are identical with those obtained for the ratios M' j-(s+l) from the equations (1) of Art. 26. It follows that x' , x", . . . x lK) are proportional to A k ', A k ", . . . A k ™, whatever the value of k. Thus, giving to k the successive values 1, 2, . . . n, there result x' : x" : . . . : *<"> : : ^/ : ^/' : . . . :A™ ::A,': A," : . . . : 4 ,<«> '.'. A„ : A„ :A (B) Art. 28.] Sylvester's method of elimination. 63 Hence, when a determinant is equal to zero, the co-factors of the elements of any line are proportional to the co-factors of the corresponding elements of any parallel line. Art. 28. Sylvester's Method of Elimination.* Let it be required to eliminate the unknown from the two -equations ay -\- ay -f- a x x -\- #„ = o, by + b,x + b = o. This will be done by what is called the dialytic method, the invention of which is due to Sylvester. Multiplying the first of the given equations by x, and the second by x and x'' suc- cessively, the result is a system of five equations, viz.: ay -\- ay -f- a x x -f- a a — o, ' ay -j- &y + ay + a t x = o, by -L- b t x + b a = o, by + by -f b x - o, by + by + by =o. . The eliminant of these five equations, involving the four unknowns x, x', x 3 , and x' is (Art. 24) E = o a 3 a, a t a a ■- < >. a z a * a i a „ o o o b, b x b o b t b x b, o b, b t b„ o o 'If the given equations be not consistent this determinant will not vanish. The above method is a general one. Thus, let the two given equations be a m x m + . . . + a x x + a„ = o, b n x n + + b,x + b = o. Multiplying the first equation (« — 1) times in succession by x, and the second (m — 1) times, (m -f- n) equations are * Philosophical Magazine, 1840, and Crelle's Journal, Vol. XXI. 64 DETERMINANTS. [Chap. II. obtained which involve as unknowns the first (m -\- n — i) powers ot x. The eliminant of these equations is a determinant of the order {m -f- n), which is of the wth degree in terms of the coefficients of the equation of the mth degree, and vice versa. The law of formation of the eliminant is obvious. The same method may be used in eliminating one or both the variables from a pair of homogeneous equations. As an example, let it be required to eliminate the variables from the equations 2x 3 — $xy — gy* = o and 3^ 2 — "jxy — 6y' = o. Dividing the first by y 3 , and multiplying by — ; the second y x byy\ and multiplying by— twice in succession, there result, X X X X in all, five equations involving — , — , -r, and -r-. y y y y these four ratios gives Eliminating E~ 2 — 5 o 5 0-9 o 3-7 3-7-6 3-7 1; the vanishing of which shows that the two given equations are consistent. Prob. 31. Test the consistency of each of the following systems of equations: (1) x+y+ 2Z=g, x+y — z—o, 2X — y + z =3, x— $y + 2 z- (2)x—y — 2z=o, x — 2y+z = o, 2X — $y — z = o; (3) sx'y - x? = o, 8x'y + 8x/ - 5/ = o. Prob. 32. Find the ratios of the unknowns in the equations 2X+y — 2z = o, 4W—y—4z = o, 2W + x — 5V + z = o. Prob. 33. In the equations a k 'x' + . . . +«*<•>*« + a k <" +"*(»+" = o, [* = 1, 2, . . . «] prove that x' : . . . : x [ (n) . x (n+i) .. M' : . . . : M {H) : M<»+» , where Art. 29.] THE MULTIPLICATION THEOREM. 65 (— i) 1 1 M<- i> is the determinant obtained by deleting the z'th column from the rectangular array M = «,™ 0,' a,'... a,™ «,<"+»> Prob. 34. From lx + v y + P* _ Vx + m y + Xz _ /**+^y+«* ^ ? r deduce * y r V /*/ I p.p I v p m\ q v A q v m q \ 11 r )a n r fxX r Prob. 35. Show that the three straight lines a'x + b'y + / = o, a"x + b"y + c" = o, and a'"x + b'"y + c'" = o, are concurrent when I a'b"c'" \ = o. Prob. 36. Prove that the medians of a triangle are concurrent. Prob. 37. Show that the points (x ,yj, (x,.y,), and (x t ,y,) are collinear when x y„ 1 *,>, 1 x t y, 1 Prob. 38. Write the conditions that all the points {x u y^) 7 (x,_,y^), . . . (x n ,y„) shall be collinear in the form of a matrix. Prob. 39. Obtain the equation of a right line through (x 1 ,y l ) and (* 2 ,Jj) in the form of a determinant. Prob. 40. Show that the equation x y z 1 = o x, y, z i 1 x , y* z i J x, y, z, 1 represents a plane through (x t , y t , z,), (x, , y 2 , z t ), and (x, , y s , z,). Art. 29. The Multiplication Theorem. Let the two homogeneous linear equations «U*1 + V S = > ] <*«*. + «= .*« = O, J be subjected to linear transformation by substituting ■*, = b n u, + £„«„ } x, = b n ii s -f £„?*,. I (1) (2) 66 DETERMINANTS. [Chap. II. (3) (4) The result of such transformation is (tf.Ai + «,A>. + 0. A. + «,A.K = °. (fl,/„ + tf 2 A>, + («,A. + «,A>, = o. The vanishing of the determinant *.A, + «.A* «,A. + fl .A «*Ai + «,A» a »Ai + a„b. is the condition that the equations (3) may be consistent ; that is, the condition that they may have solutions other than u x — o = u, (Art. 26). Now the equations (3) may be consist- ent because of the consistency of the equations (1), in which case the determinant (5) vanishes. Or, this condition failing, and the equations (1) thus having no solution other than x y — o = x^, the equations (3) will still be consistent if the equations (2) are so ; that is, if the determinant K K < 6 > vanishes. The vanishing of either of the determinants (5) or (6), therefore, causes the determinant (4) to vanish. It follows that (5) and (6) are factors of (4) ; and since their product and the determinant (4) are of the same degree with respect to the coefficients a n , . . . , 6 U , . . . , they are the only factors. Hence, «,A. + «■ A «.A. + a J™ «.A. + «>A, <*,Ai + «,A, (7) The above method is equally applicable to the formation of the product of any two determinants of the same order. Hence results the following general formula: I an a-ii . ■ . u„ n I " I in bw . . . 6„„ I = #11^11+ . . . -\-ci\ n b\ n awb?.\ + . . . +U\abi n ■ Cttibn + . . . -\-dinb\n &l\0?l + • ■ • +#2n*2n . aubn\+ . . . +ambnn u n \bii 4- • ■ • +&nnbln (l n ^-i\ + . - . + &tinbm .... flnlPnl + . . . fflnn^n (8) Art. 29.] THE MULTIPLICATION THEOREM. 67 The process indicated by this formula may be described as follows : * To form the determinant \p ln \ and | b ua \ . Then place the first row of \a un \ upon each row of \b h „\ in turn and let each two elements as they touch become products. This is the first row of \p 1>n \. Perform the same operation upon \b h „\ with the second row of \a un \ to obtain the second row of \p u „\; and again with the third row of |fl,, K |to obtain the third row of|/ 1>M |; etc. Any element of this product is Pi, = <*hK + a H d st + . . . + ajb m . (9) When the two determinants to be multiplied together are of different orders the one of lower order should be expressed as a determinant of the same order as the other (Art. 22), after which the above rule is applicable. The product of two determinants may be formed by columns, instead of by rows as above. In this case the result is obtained in a different form. Thus the product of the de- terminants (5) and (6) by columns is «,A + *sA a Ju + «J>» Prob. 41. Form the following products : (2) (3) (1) a h g a g h b f g c g f c <*n «u a n _ A u A,, a i\ a ii a ?i A A «n a * 1 a 33 A 31 ^32 b f a g\- a h f c g c\ h b (4) a x b l c x Oil <*, b* ^ 1 1 a, b, c, 1 1 Prob. 42. Generalize the last example (see Prob. 22, Art. 18). Prob. 43. By forming the product a + bV- e + dV~- c + dV- 1 a — b V^l j+kV- ■ 1+mV- 1 j-k V^l * Carr's Synopsis of Pure Mathematics, London, 1886, Article 570. 68 DETERMINANTS. [Chap. II. show that the product of two numbers, each the sum of four squares, is itself the sum of four squares. Art. 30. Product of Two Arrays. The process explained in the preceding article may be ap- plied to form what is conventionally termed the product of two rectangular arrays. It will appear, however, that multi- plying two such arrays together by columns leads to a result radically different from that obtained when the product is formed by rows. Let the two rectangular arrays be and KAA* The product of these by columns is A = «,A. + *.A. «,A. + «=A, «.Ai + a»K «iA» + ",A, «.A» + a ,A, «i3*» + a, A «.As -f a tl b„ a„b ia + a„b„ a ls b ls + a J, The determinant A is plainly equal to zero, being the prod- uct of two determinants formed by adding a row of zeros to one of the given rectangular arrays and a row of elements chosen arbitrarily to the other. In general, the product by columns of two rectangular arrays having m rows and n columns, m being less than n, is a determinant of the n tb order whose value is zero. Multiplying together the above rectangular arrays by rows, the result is A' ; «.Ai + a >A, + ".A. «.A. + a lt b„ + a ia b„ 0.A. + a m b n + ajb„ a n a n KK + Vi. • KK 4- 1 «.A, . M„ a ii a w b n b n a ix a t , b *A* «»!«« ^A, In the same manner it may be shown that the product by rows of two rectangular arrays having in rows and n columns, m being less than n, is a determinant of the m tt ' order, which may be expressed as the sum of the n \/m ! (n — m) ! determinants Art. 31.] RECIPROCAL DETERMINANTS. 69 formed from one of the arrays by deleting (n — m) columns, each multiplied by the determinant formed by deleting the same columns from the other array. Art. 31. Reciprocal Determinants. The determinant formed by replacing each element of a given determinant by its co-factor is called the reciprocal of the given determinant.* Thus, the reciprocal of 6 = is A^A,, . . . A Y A ix A M . . . A, The product of these two determinants is 6.A= on^n+ . . . -\-amAi n anAii-\- . . .-\-ai n Ai "ai^n+ . . . -\-a^ n Ai n a^\Ai\-\- . . .-\-a in Ai . &iiA n i-^-. . .-\-Oi n Ann &2lAni-{-' . .-\~QlnA nn a n iA u+ . . . .-\-a nn Am a n iAn-{-. . .-\-a nn Ai n - • • • 0ni^rn+. • --YannA Each element on the principal diagonal of this product is equal to 8 (Art. 18), while all the other elements vanish (Art. 19). Hence, d. A = do., o d .. -,(») = d", or 0 - i)/8]|y - n - 2 - 4(« - 2)] = «(« — \)(n — 2)(n — 3)/8. 47'. A set of n fans may be selected in several ways so that two and only two contain each one of the n sides. Such a set of fans is called a ' complete set ' of fans. 47,. A set of n connectors may be selected in several ways so that two and only two contain each one of the n dots. Such a set of connectors is called a ' complete set ' of connectors. 48,. There are (« — 1) 1/2 48'. There are (« — 1) \h complete sets of fans. complete sets of connectors. Proof of 48,. In a polystim of n dots there are through any single dot (n — 1) connectors, and hence (n — i)(n — 2)/z pairs of connectors. Consider one such pair, as BC and BE. 76 PROJECTIVE GEOMETRY. [CHAP. III. The number of different sets (each of n — 2 connectors) from C to E through A, D, F, G, etc. [there being (« — 3) such dots], is (n — 3) !, i.e. the number of permutations of {11 — 3) things. Hence the number of complete sets of con- nectors having the pair BC and BE is {n — 3) ! Therefore the whole number of complete sets of connectors is (« - i)(» - 2)[(» - 3) !]/2 = (» - 1) !/2- 49,. In any complete set of 49'- I n any complete set of connectors, when « is even, the fans, when « is even, the first and first and the («/2+i)th are the (»/2 + i)th are called 'op- called ' opposite '. posite.' 50,. A ' tetrastim ' is a system 50'. A ' tetragram ' is a system of four dots with their six con- of four straights with their six nectors. Each pair of opposite fans. Each pair of opposite fans connectors intersect in a codot. concur in a diagonal. These These three codots determine three diagonals determine the the ' codot-tristim ' of the tetra- ' diagonal-trigram ' of the tetra- stim. gram. 51. Two correlated polystims whose paired dots and co- dots have their joins copunctal are called ' copolar.' 52. Two correlated polystims whose paired connectors in- tersect and have their intersections costraight are called ' coaxal.' 53. If two non-coplanar tristims be copolar, they are coaxal. For since AA' crosses BB', therefore AB and A'B' intersect on the meet of the planes of the tristims. 54. If two non-coplanar tristims be coaxal, they are copolar. For since AB intersects A'B', these four points are coplanar. The three planes ABA'B', ACA'C , BCB'C are copunctal. Hence so are their meets AA', BB', CC. 55. By taking the angle between the planes evanescent, is seen that coplanar coaxal tristims are copolar ; and then by reductio ad absurdum that coplanar copolar tristims are coaxal. 56. If two coplanar polystims are copolar and coaxal they are said to be ' complete plane perspectives.' Their pole and Art. 6.] harmonic elements. 77* axis are called the ' center of perspective' and the ' axis of perspective.' 57. If two coplanar tristims are copolar or coaxal, they are complete plane perspectives. 58. If two coplanar polystims are images of the same poly- stim from different projection vertices V l , V^, they are com- plete plane perspectives. For the joins of pairs of correlated points are all copunctal (on the pass of the straight V, V, with the picture plane), and the intersections of paired con- nectors are all costraight (on the meet of the picture plane and the plane of the original). Prob. 5. In a hexastim there are 15 connectors and 45 codots. In a hexagram there are 15 fans and 45 diagonals. Prob. 6. If the vertices of three coplanar angles are costraight, their sides make three tetragrams whose other diagonals are copunc- tal by threes four times. [Prove and give dual.] Prob. 7. The corresponding sides of any two funiculars of a given system of forces cross on a straight parallel to the join of the poles of the two funiculars. Art. 6. Harmonic Elements. 59. Fundamental Theorem. — If two correlated tetrastims- lie on different planes whose meet is on no one of the eight dots, and if five connectors of the one intersect their mates,, then the tetrastims are coaxal. For the two pairs of tristims fixed by the five pairs of intersecting connectors being coaxal are copolar. Hence the sixth pair of connectors are coplanar. 60. If the tetrastims be coplanar, and if five intersections of pairs of correlated connectors are costraight, this the coplanar case can be made to depend upon the other by substituting for one of the tetrastims its image on a second plane meeting the first on the bearer of the five intersections. 61. If the axis in is a figurative straight, the theorem reads : If of two correlated tetrastims five pairs of mated connectors are parallel, so are the remaining pair. 62. Four costraight points are called ' harmonic points,' or 78 PROJECTIVE GEOMETRY. [Chap. III. a ' harmonic range,' if the first and third are codots of a tetra- stim while the other two are on the connectors through the third codot. 63. By three costraight points and their order the fourth harmonic point is uniquely determined. For if the three points in order are A, B, C, draw any two straights through A, and a third through B to cross these at K and M respectively. Join CK, crossing AM at N. Join CM, crossing AK at L. Then the join LN crosses the straight ABC, always at the same point D, the fourth harmonic to ABC separated from B. 64. In projecting from a point not coplanar with it a tetrastim defining a harmonic range, the four harmonic points are projected by four coplanar straights, called ' harmonic straights ' or a ' harmonic flat-pencil.' 65. The four planes projecting harmonic points from an axis not coplanar with them are called ' harmonic planes,' or a ' harmonic axial-pencil.' 66. Projecting or cutting a harmonic primal figure gives always again a harmonic primal figure. 67. By three elements of a primal figure, given which is the second, the fourth harmonic is completely determined. 68. Defining harmonic points by the tetrastim distinguishes Art - •■] HARMONIC ELEMENTS. 79 two points made codots from the other two. Yet it may be shown that the two pairs of points play identically the same role. First, from the definition of four harmonic points each sep- arated two may be interchanged without the points ceasing to be harmonic [or, if ABCD is a harmonic range, so is also ADCB, CBAD, and CDAB]. For the first and third remain codots. Second, to prove that in a harmonic range the two pairs of separated points may be interchanged without the four points ceasing to be harmonic [or, if ABCD is a harmonic range (and therefore ADCB, CBAD, and CDAB), then also is BADC, DABC, BCDA, and DCBA] : Through the third codot O draw the joins AO and CO. These determine on the connectors NK, KL, LM, and MN four new points, 5, T, U, V, respec- tively. The tetrastim KTOS has for two codots A and C, and has a connector though B; hence its remaining connector TS must pass though D. In like manner, the connector UV of the tetrastim MVO U must pass through D, and a connector of each of the tetrastimsZt/CTand VNSO through B. There- fore B and D are codots of a tetrastim STUV with the remain- ing connectors, one through A, one through C. 69. The separated points A and C are called ' conjugate points,' as also are B and D. Either two are said to be ' har- monic conjugates ' with respect to the other two. Prob. 8. To determine the join of a given point M with the in- accessible cross X of two given straights n and ri . 80 PROJECTIVE GEOMETRY. [Chap. Ill, Through Mdraw any two straights crossing n at B and B' , and n r at D and D' . Join DB and D' B' , crossing on A. Through A draw any third straight crossing. « at j?" and «' at Z>". Join ^'Z>" and D'B", crossing at L. Then LM is the join required. Proof. The tetrastim XBMD makes AB'C'D' a harmonic range, as XB'LD' does AB"C"D". But projecting AB"C"D" from X, and cutting the eject by AB'D' gives a harmonic range. Therefore C", C, and X are costraight.* • Prob. 9. Through a given point to draw with the straight-edge a straight parallel to two given parallels. Prob. ro. To determine the cross of a given straight m with the inconstructible join x of two given points JV and N' . Join any two points on m with N and N' , giving b and b' on N, d and d'onJV'. Join the crosses db and d'b' by a. On a take any third point join- ing with N in b" and with N' in a?". Join the crosses b'd" and d'b" by /. Then /#z is the cross re- quired. [From Prob. 8, by duality.] Prob. 11. Cut four coplanar non-copunctal straights in a har- monic range. Prob. 12. On a given straight determine a point from which the ejects of three given points form with the given straight a harmonic pencil. Art. 7. Projectivity. 70. Two primal figures of three elements are always pro- jective. — If one be a pencil, take its cut by a transversal. If the bearers of ABC and A'B'C be not coplanar, join AA', BB', CC, and cut these joins by a transversal, m. Then ABC and A'B'C are two cuts of the axial mAA', mBB', mCC '. * Numerous problems in Surveying may be solved by the application of the preceding principles, but such application has not been found advantageous in practice. See Gillespie's Treatise on Land Surveying, New York, 1872. Art. 7.] projectivity. 81 If the bearers are coplanar, take on the join A A' any two projection vertices M and M' . Join MB and M'B', crossing at B"; join MC and M'C, crossing at C" . Join B"C" crossing A A' at A". Then ABC and A'B'C are images of A" B" C" . 71. If any four harmonic elements are taken in one of two projective figures, the four elements correlated to these are also harmonic. For both ejects and cuts of harmonic figures are themselves harmonic. 72. Two primal figures are projective if they are so corre- lated that to every four harmonic elements of the one are correlated always four harmonic elements of the other. For the same projectings and cuttings which derive A'B'C from. ABC vt\\\ give Z>, from D. Therefore A'B'C'D, is harmonic. But by hypothesis A'B'C'D' is harmonic. Therefore D l is D'. 73. If two primal figures are projective, then to every con- secutive order of elements of the one on a bearer corresponds a consecutive order of the correlated elements of the other on a bearer. 74. Two projective primal figures having three elements self-correlated are identical. For two self-correlated elements cannot bound an interval containing no such element, since they must harmonically separate one without it from one within. 75. Two ranges are called ' perspective ' if cuts of the same flat pencil. Two flat pencils are perspective if cuts of the same axial pencil, or ejects of the same range. Two axials are perspective if ejects of the same flat pencil. A range and a flat pencil, a range and an axial pencil, or a flat pencil and an axial are perspective if the first is a cut of the second. 76,. If two projective ranges 76'. If two coplanar projective not costraight have a self-corre- flat pencils not copunctal have lated point A, they are perspec- a self-correlated straight a, they tive. are perspective. 82 PROJECTIVE GEOMETRY. [Chap. III. Let the join of any pair of correlated points BB' cross the join of any other pair CC at V. Projecting the two given ranges from V, their ejects are identical, since they are projec- tive and have the three straights VA, VBB', VCC self-corre- Jated. Let the cross of any pair of correlated straights bb' join the cross of any other pair cc' by m. Cutting the two given flat pen- cils by m, their cuts are identical, since they are projective and have the three points ma, mbb' , mcc' self-correlated. Art. 8. Curves of the Second Degree. 77,. If two coplanar non- •copunctal flat pencils are pro- jective but not perspective, the crosses of correlated straights form a ' range of the second de- gree,' or 'conic range.' 77'. If two coplanar non- costraight ranges are projective but not perspective, the joins of correlated points form a ' pencil of the second class,' or 'conic pencil.' 78,. If two copunctual non- 78'. If two copunctal non- costraight axial pencils are pro- coplanar flat pencils are projec- jective but not perspective, the tive but not perspective, the meets of correlated planes form planes of correlated straights a ' conic surface of the second form a ' pencil of planes of the order,' or ' cone.' second class,' or ' cone of planes.' 79. All results obtained for the conic range or the conic pencil are interpretable for the cone or cone of planes, since the eject of a conic is a cone and the cut of a cone is a conic. 8o,. On the cross A of any pair 80'. On the join a of any pair -of correlated straights a and a 1 of correlated points A and A of Art. 8.] CURVES OF THE SECOND DEGREE. 83 of the projective flat pencils V and Vj draw two straights u and »j. The cuts ABC and A.B^C, being projective and having a pair of correlated points A, A l coincident, are perspective, both being cuts of the pencil on V t , the cross of the joins BB X and CC V Any straight d of V, crossing u at D, is then correlated to the join of V x with the cross D x of «, and the join DV V Any d crosses its d x so deter- mined, at P, a point of the conic range k. 8 1,. The pencil-points V, Vi of the generating pencils pertain to the conic, since their join VF 1 is crossed by the element correlated to it in either pencil at its pencil-point. the projective ranges u and », take two points V and V v The ejects abc and «//, being projective and hav- ing a pair of correlated straights a, #, coincident, are perspective, both be- ing ejects of the range on u v the join of the crosses bb x and cc x . Any point D of u, joined with V by d, is then correlated to the cross of u x with the join d^ of V x and the cross du t . Any -D joined to its D, so de- termined, gives p a straight of the conic pencil K. 8i\ The bearers u, «, of the generating ranges pertain to the conic, since their cross ««, is joined to the element correlated to it in either range by its bearer. S4 PROJECTIVE GEOMETRY. [Chap. Ill 82,. The straight on V corre- lated to V x V is called the ' tan- gent' at V. Every other straight on V is its join with a second point of the conic. 83;. On any straight, as u, on any point A of the conic, its second element is its cross M with the join V^V V 84^ From the five given points Wj AMZ, of k construct a sixth, P. The cross D of n with the join VP, and the cross Z>, of u x with the join V X P are costraight with V v Therefore* the three opposite pairs in every complete set of connectors of a hexastim whose dots are in a conic inter- sect in three costraight codots whose bearer is called a ' Pascal straight.' This hexastim has sixty Pascal straights, since it has sixty com- plete sets of connectors. 85!. The ejects of the points of a conic from any two are pro- jective. 86j. By five of its points a conic is completely determined. 87^ Instead of five points may be given the two pencil- points and three pairs of corre- lated straights. If one given straight is the join of the pencil- points, then four points and a tangent at one of them are given. Thus by four of its points and the tangent at one of them a * Pascal, 1640. 82'. The point on u correlated to u x u is called the ' contact ' on u. Every other point on u is its cross with a second straight of the conic. 83'. On any point, as V, on any straight a of the conic, its second element is its join q with the cross »,«,. 84'. From the five given straights u, u u a, q, r u of K con- struct a sixth -D-D , or p. The join d of V with the cross up, and the join d x of V, with the cross u x p are copunctal with « 3 . Therefore f the three opposite pairs in every complete set of fans of a hexagram whose sides are in a conic concur in three copunctal diagonals whose bearer is called a ' Brianchon point.' This hexagram has sixty Brian- chon points, since it has sixty complete sets of fans. 85'. The cuts of the straights of a conic by any two are pro- jective. 86'. By five of its straights a conic is completely determined. 87'. Instead of five straights may be given the two bearers and three pairs of correlated points. If one given point is the cross of the bearers, then four straights and a contact point on one of them are given. Thus by four of its straights and a contact-point on one of t Brianchon, 1806. Art. 8.] CURVES OF THE SECOND DEGREE. 85 them a conic is completely de- termined. 88'. By three of its straights and the contact-points on two of them the conic is completely determined. 89'. Interpreting a pentagram as a hexagram with two sides coinciding gives: In every com- plete set of fans of a pentagram whose sides are in a conic, two pairs of non-consecutive fans determine by their two concurs a point on which is the join of the fifth fan-point with the con- tact-point on the opposite side. conic is completely determined. 88j. By three of its points and the tangents at two of them the conic is completely deter- mined. 89,. Interpreting a pentastim as a hexastim with two dots coinciding gives: In every com- plete set of connectors of a pen- tastim whose dots are in a conic, two pairs of non-consecutive connectors determine by their two intersections a straight on which is the cross of the fifth connector with the tangent at the opposite dot. Thence follows the solution of the problems : 90,. Given five points of a conic, to construct tangents at the points, using the ruler only. 91,.* The hexastim with a pair of opposite connectors re- placed by tangents gives: The intersections of the two opposite pairs in every complete set of connectors of a tetrastim with dots in a conic are both costraight with the crosses of the two pairs of tangents at opposite dots. Or: A tetrastim with dots in a conic has each pair of codots costraight with a pair of fan- points of the tetragram of tan- 90'. Given five straights of a conic, to find contact-points on the straights, using the ruler only. 91'. The hexagram with a pair of opposite fans replaced by con- tact-points gives: The concurs of the two opposite pairs in every complete set of fans of a tetra- gram with sides in a conic are both copunctal with the joins of the two pairs of contact-points on opposite sides. Or: A tetragram with sides in a conic has each pair of diagonals copunctal with a pair of con- nectors of the tetrastim of con- gents at the dots. tacts on the sides. The figure for 91, and that for 91' are identical, and called Maclaurin's Configuration. (See page 86.) 92,. The tangents of a conic 92'. The contact-points of a range are a conic pencil. conic pencil are a conic range. * Due to Maclaurin, 1748. 86 PROJECTIVE GEOMETRY. [Chap. III. 93. The points of a conic range may now be conceived as all on a curve, a ' conic curve,' their bearer. The straights of the corresponding conic pencil, tangents of this conic range, may now also be conceived as all on this same conic curve on which are their contact-points. Conse- quently the conic curve is dual to itself, and so the principle of dual- ity on a plane receives an impor- tant extension. 94. It follows immediately from their generation that all conies are closed curves, though they may be compendent through one or two points at infinity. With two points at infinity the curve is called ' hyperbola ;' with one, ' parabola ; ' with none, ' ellipse,' * 95. If a point has on it tan- gents to the curve, it is called 'without' the curve; if none, ' within ' the curve. The contact- point on a tangent is ' on ' the curve ; all other points on a tan- * The generation shows that a straight cuts the curves in two points and that from any point two tangents to the curves may be drawn. Hence the curves are of the second order and of the second class, that is they are identical with the conies of analytic geometry. Analytically the equations P-\- XQ = o, P -\-\Q' = 0, where P, Q, P', Q are linear functions of point coordinates, represent two projective pencils, the correlated rays corresponding to the same value of X. Hence the locus of the intersection of correlated rays is repre- sented by PQ' — P'Q = o, a second-degree point equation. Projective ranges are represented by R + XS = o, R' 4- ^S' = o, where R, S, R', S' are linear functions of line coordinates. The envelope of the joins of correlated points is represented by R S' — R'S= o, a second-degree line equation. The projective generation of conies is developed synthetically in Steiner's Theorie der Kegelschnitte, 1866, and in Chasles' Geometrie superieure, 1852. For the analytic treatment see Clebsch, Geometrie, vol. I, 1876. Art. 9.] pole and polar. 87 gent are without the curve. Every straight in its plane con- tains innumerable points without the curve, since the straight crosses every tangent. Prob. 13. Given four points on a conic and the tangent at one of them, draw the tangent at another. Prob. 14. If the n sides of a polygram rotate respectively about n fixed points not costraight, while (« — 1) of a complete set of fan- points glide respectively on (» — 1) fixed straights, then every remain- ing fan-point describes a conic* Prob. 15. In any tristim with dots on a conic the three crosses of the connectors with the tangents at the opposite dots are costraight. f Prob. 16. If two given angles rotate about their fixed vertices so that one cross of their sides is on a straight, either of the other three crosses describes a conic. % , Prob. 17. Construct a hyperbola from three given points, and straights on its figurative points. Art. 9. Pole and Polar. 96. Taking every tangent to a conic as the dual to its own contact-point fixes as dual to any given point in the plane one particular straight, its ' polar,' of which the point is the ' pole.' 97. With reference to any given conic, to construct the polar of any given point in its plane. Put on the given point Z two secants crossing the curve, one at A and D, the other at B and C. The join of the other codots X and Y of ABCD is the polar of Z. Varying either secant, as ZBC, does not change this polar, since on it must always be the cross 5 of the tangents at A and D, and also the point which D and A harmonically separate from Z (given by each of the variable tetrastims BXCY). 98. The join of any two codots of a tetrastim with dots on a conic is the polar of the third codot with respect to that * Due to Braikenridge, 1735. f From Pascal ; dual from Brianchon. \ Given by Newton in Principia, Book I, lemma xxi, under the name of "the organic description " of a conic. 88 PROJECTIVE GEOMETRY. [Chap. III. conic, and either codot is the pole of the join of the other two. Any point is harmonically separated from its polar by the conic. 99. To draw with ruler only the tangents to a conic from a point without, join it to the crosses of its polar with the conic. ioo,. Two points are called 100'. Two straights are called ' conjugate ' with reference to a ' conjugate ' with reference to a conic if one (and so each) is on conic if one (and so each) is on the polar of the other. the pole of the other. ioij. All points on a tangent 101'. All straights on a con- are conjugate to its contact- tact-point are conjugate to its point. tangent. 102,. The points of a range 102'. The straights of a flat are projective to their conjugates pencil are projective to their on its bearer. conjugates on its bearer. 103,. With reference to a given 103'. With reference to a given conic, the 'kerncurve,' the conic, the ' kerncurve,' the poles polars of all points on a second of all tangents on a second conic conic make a conic pencil, whose make a conic range, whose bearer bearer is the ' polarcurve ' of is the ' polarcurve ' of the second the second conic. conic. Prob. 18. Either diagonal of a circumscribed tetragram is the polar of the cross of the others. Prob. 19. A pair of tangents from any point on a polar harmoni- cally separate it from its pole. Prob. 20. A pair of tangents are harmonic conjugates with respect to any pair of straights on their cross which are conjugate with respect to the conic. Art. 10. Involution. 104. If in a primal figure of four elements (a 'throw ') first any two be interchanged, then the other two, the result is pro- jective to the original. [That is, ABCD a BADC a CDAB a DCS A.] Let ABCD be a throw on m. Project itfram V. Cut this eject by a straight {m') on A. The cut is AB'C'D'. Now project ABCD from C. The cut of this latter eject by V B is Art. 10.] involution. 89 B'B VH. Project B'B VH from D and cut the eject by m'. The cut is B'AD'C, which is perspective to BADC. 105. Two projective primal figures of the same kind of ele- ments and both on the same bearer are called ' conjective.' When in two conjective primal figures one particular element has the same mate to whichever figure it be regarded as be- longing, then every element has this property. If AA'BB' is projective to A'AB'X, then by § 104, AA'BB' is projective to AA'XB', and having three elements self-corre- lated, they are identical. 106. Two conjective figures such that the elements are mutually paired (' coupled ') form an ' Involution.' For exam- ple, the points of a range, and, on the same bearer, their con- jugates with respect to a conic, form an involution. Every eject and every cut of an involution is an involution. 107. When two ranges are projective, the point at infinity of either one is correlated to a point of the other called its ' vanishing point.' 108. When two conjective ranges form an involution the two vanishing points coincide in a point called the ' center ' of the involution. 109. If two figures forming an involution have self-corre- lated elements, these are called the ' double ' elements of the involution. An involution has at most two double elements ; for were three self-correlated, all would be self-correlated. no. If a primal figure of four elements is projective with a second made by interchanging two of these elements, they harmonically separate the other two. For project the range ABCD from Fand cut the eject by a 90 PROJECTIVE GEOMETRY. [Chap. III. straight on A. The cut AB'C'D' is projective to ABCD, which by hypothesis is projec- tive to ADCB. Therefore ADCB is perspective to AB'C'D'. So VC'C is on the cross X of the joins DB' and BD'. So B and D are codots of the tetrastim VD'XB', while A and C are on the connectors through C, the third codot. in. If an involution has two double elements these sepa- rate harmonically any two coupled elements. Let A and C be the double elements. Then ABCB' is projective to AB'CB ; therefore by § no ABCB' is harmonic. 1 1 2. An involution is completely determined by two couples. For the projective correspondence AA'B . . . a A'AB' ... is completely determined by the three given pairs of correlated elements, and since among them is one couple, so are all corre- lated elements couples. 113. When there are double elements, then the elements of no couple are separated by those of another couple. In- versely, when the elements of one couple separate those of another, then the elements of every couple are separated by those of every other, and there are no double elements. 114'. The three pairs of op- posite fan-points of a tetragrara are projected from any projec- tion-vertex by three couples of an involution of straights. 114,. The three pairs of op- posite connectors of a tetrastim are cut by any transversal in three couples of a point involu- tion.* f Due to Desargues, 1639. ART. It] PROJECTIVE CONIC RANGES. 91 Let QRST be a tetrastim of which the pairs of opposite connectors ^7* and QS, ST and QR, QT and RS are cut by any transversal respectively in A and A', B and B', C and C. From the projection-vertex Q, the ranges A TPR and ACA'B' are perspective. But A TPR and ABA'C are perspective from 5. Therefore ACA'B' is projective to ABA'C, and therefore K.oA'C'AB<$\o£). Since thus^ and A' are coupled, so (§105) are B and j5', and C and C\ 115. To construct the sixth point C of an involution of which five points are given, draw through C any straight, on which take any two points Q and T. Join A T, B'Q crossing at R. Join BT, A'Q crossing at S. The join RS cuts the bearer of the involution in C. Prob. 21. Find the center O of a point involution of which two couples AA' BB' are given. Prob. 22. If two points M and iV* on m are harmonically sepa- rated by two pairs of opposite connectors of a tetrastim, then so are they by the third pair. Prob. 23. To construct a conic which shall be on three given points, and with regard to which the couples of points of an involu- tion on a given straight shall be conjugate points. Art. 11. Projective Conic Ranges. 116. Four points on a conic are called harmonic if they are projected from any (and so every) fifth point on the conic by four harmonic straights. 117. A conic and a primal figure or two conies are called projective when so correlated that every four harmonic ele- ments of the one correspond to four harmonic elements of the other. 118. If a conic range and a flat pencil are projective, and every element of the one is on the correlated element of the other, they are called perspective. A conic is projected from every point on it by a flat pencil perspective to it. Inversely the pencil-point of every flat pencil perspective to a conic is on the conic. 92 PROJECTIVE GEOMETRY. [CHAP. III. 119. Two conies are projective if flat pencils respectively perspective to them are projective. Therefore any three elements in one can be correlated to any three elements in the other, but this completely pairs all the elements. 120. Two different conic ranges on the same bearer have at most two self-correlated elements. 121. Two different coplanar conic ranges with a point V in common are projective if every two points costraight with V are correlated. For both are then perspective to the flat pencil on V. Every common point other than V is self-corre- lated ; but V only when they have there a common tangent. They can have at most three self-correlated points. 122. If a flat pencil V and conic range k are coplanar and projective but not perspective, then at most three straights of the pencil are on their correlated points of the conic; but at least one. For any flat pencil M perspective to k is projective to V, and with it determines in general a second conic range which must have in common with k every point which lies on its correlated straight of V. So if more than three straights of V were on their correlated points of k, the conies would be iden- tical and V perspective to k. Again, since every conic is compendent, and so divides its plane into two severed pieces, therefore the two different conies if they cross at their common point M must cross again, say at P. In this case the straights VP and MP are correlated, and so VP is on the point P correlated to it on k. In case they do not cross at their common point M, the straight VM corresponds to the common tangent at M, and so to the point M correlated to it on k. 123. Two projective conic ranges on the same curve form an involution if a pair of points are doubly correlated. Besides the couple AA V let B and B l be any other two correlated points, so that AA X B corresponds to A t AB r The cross of A A, and BB l call U, and its polar u. Project AA^B from B t . ART- 11.] PROJECTIVE CONIC RANGES. 93 Project A X AB X from B. The ejects B X (AA X B) and B(A X AB X ) are projective, and having the straight B X B (or BB X ) self-corre- lated, so are perspective. The crosses of their correlated ele- ments are therefore costraight. But the cross of B X A with its correlated straight BA l is known to be on u, the polar of U, the \\ \ \ / / / / ! Jh sy I n. 1/7% JSTj rp^ jS ^A / '■ ]££ " 'C, cross of AA 1 with BB X . Likewise the cross of B^A X with BA is on u. Therefore the point C x correlated to C is the cross of CU with the curve. So C and C x are coupled. 124. If two conic ranges form an involution, the joins of "coupled paints are all copunctal on the ' involutioncenter.' 125. Calling projective the conic pencils dual to projective conic ranges, if these ranges form an involution, so do the pencils, and the crosses of coupled tangents are all costraight on the ' involutionaxis.' So two conic pencils forming an involution are cut by each of their straights in two ranges forming an involution. Two conic ranges forming an involution are projected from each of their points in two flat pencils, forming an involution. 126. If the involutioncenter lies without the conic bearer of an involution, it has two double elements where it is cut by the involutionaxis. 127. To construct the self-correlated points of two pro- jective conic ranges on the same conic. — Let A, B, C be any three points of k, and A,, B x , C x their correlated points of k x . The projective flat pencils A{A X B X C X ) and A X (ABC) have AA„ self- corresponding, hence they are perspective to a range on the join u of the cross of AB X and A X B with the cross of AC X 94 PROJECTIVE GEOMETRY. [CHAP. III. and A X C. The crosses of the conic and this join n are the self-correlated points of k and k x . 128. If the dots of a tetrastim are on a conic, the six points where a straight not on a dot cuts the conic and two pairs of opposite connectors form an involution. For the two flat pencils in which the two crosses of m with the conic, P,P lt and two opposite dots R, T, are pro- jected from the other two dots Q, S, are projective, and con- sequently so are the cuts of these flat pencils by m; that is, PBP l A7\PA,P,B l . But PA l P l B l hP l B l PA 1 . Therefore PBP X A a PfrPAr 129,. Conies on which are the 129'. Copunctal tangents to dots of a tetrastim are cut by a conies on which are the sides of transversal in points of an involu- a tetragram form an involution, tion. At its double points the The double straights touch two transversal is tangent to two of of those conies at the pencil- those conies. point. Prob. 24. The pairs of points in which a conic is cut by the straights of a pencil whose pencil-point is not on the conic form an involution. Art. 12. Center and Diameter. 130. The harmonic conjugate of a point at infinity with respect to the end points of a finite sect is the ' center ' of that sect. 131. The pole of a straight at infinity with respect to a certain conic is the ' center ' of the conic. 132. The polar of any figurative point is on the centre of the conic, and is called a ' diameter.' 133. If a straight crosses a conic the sect between the crosses is called a ' chord.' The center of a conic is the center of all chords on it. 134. The centers of chords on straights conjugate to a diameter are all on the diameter. 135. Two diameters are conjugate when each is the polar of the figurative point on the other. Art. 12.] center and diameter. 95 136. The tangents at the crosses of a straight with a conic cross on the diameter which is a conjugate to that straight. 137. The joins of any point on the conic to the crosses of a diameter with the conic are parallel to two conjugate diameters. 138. Of two conjugate diameters, each is on the centers of the chords parallel to the other ; and if one crosses the conic, the tangents at its crosses are parallel to the other diameter. 139. The center of an ellipse is within it, for its polar does not meet the curve, and so there are no tangents from it to the curve. The centre of a parabola is the contact point of the figurative straight. The centre of a hyperbola lies without the curve, since the figurative straight crosses the curve. The tan- gents from the center to the hyperbola are called * asymptotes.' •Their contact-points are the two points at infinity on the curve. 140. If a diameter which cuts the curve be given, the tan- gents at its crosses can be constructed with ruler only, and so however many chords on straights conjugate to the diameter. 141. Every flat pencil is an involution of conjugates with respect to a given conic. Hence the pairs of conjugate diam- eters of a conic form an involution. If the conic is a hyperbola, the asymptotes are the double straights of the involution. Hence any two conjugate diam- eters of a hyperbola are harmonically separated by the asymp- totes ; and since the hyperbola lies wholly in one of the two explemental angles made by the asymptotes, one diameter cuts the curve, the other does not. 142. Any one pair of conjugate diameters of an ellipse is always separated by any other pair. Any one pair of conjugate diameters of a hyperbola is never separated by any other pair. 143. If a tangent to a hyperbola cuts the asymptotes at A and C, then the contact-point B is the center of the sect AC, since the tangent cuts the harmonic pencil made by the diame- ter through B, the conjugate diameter and the asymptotes, in the harmonic range ABCD where D is at infinity. Just so the 96 PROJECTIVE GEOMETRY. [Chap. III. center of any chord is the center of the costraight sect bounded by the asymptotes. 144. If a point is the center of two chords it is the center of the conic, for its polar is the figurative straight. 145. As many points as desired of a conic may be con- structed by the ruler alone. With the aid of one fixed conic all problems solvable by ruler and compasses can be solved by ruler alone, that is, by pure projective geometry. For example : Of two projective primal figures (say ranges) on the same bearer, given three pairs of correlated elements to find the self-corresponding ele- ments, if there be any. Project the two ranges from any point V of the given conic. These ejects are cut by the conic in projective conic ranges. Of these determine the self-correlated ' points by § 127. Project these from V. The ejects cut the bearer of the original ranges in the required self-correlated points. Prob. 25. Find the crosses of a straight with a conic given only by five points. Prob. 26. Given a conic and its center, find a point B such that for two given points A, C, the center of the sect AB shall be C. Prob. 27. The join of the other extremities of two coinitial sects is parallel to the join of their centers. Prob. 28. In an ellipse let A and B be crosses of conjugate diam- eters CA, CB with the curve. Through A' the cross of the diameter conjugate to CA with the curve draw a parallel to the join AB. Let it cut the curve again at B' . Then CB' is the diameter conjugate to CB, Art. 13. Plane and Point Duality. 146,. On a plane are 00 'points, 146'. On a point are o° 2 planes, a ' point-field.' a ' plane-sheaf.' 147,. The oo 1 planes of a sin- 147'. The 00 ' points of a sin- gle axial pencil have on them all gle range have on them all the the points of point-space; so planes of plane-space; so there there are just oo 3 points. are just 00 3 planes. Point-space is tridimensional. Plane-space is tridimensional. Art. 13] PLANE AND POINT DUALITY. 97 148. With the straight as element, space is of four dimen- sions. On a plane are oo 2 straights, a 'straight-field.' On a straight are 00 ' planes, and so 00 3 straights. On each of the oo 2 points on a plane are the 00 2 straights of a straight-sheaf; so there are just 00 * straights. 149,. Two planes determine a straight, their meet. 150,. Two planes determine an axial-pencil on their meet. 151,. Two bounding planes determine an axial angle. 152,. A plane and a straight not on it determine a point, their pass. 153,. An axial pencil and a plane not on its bearer deter- mine a flat pencil. 154,. Three planes determine a point, their apex. i55j. Three planes determine a plane-sheaf. 156,. Two coplanar straights are copunctal. On a point are c° 2 straights, a ' straight-sheaf.' On a straight are oo l points, and so oo 3 straights. On each of the oo 2 planes on a point are the 00 2 straights of a straight-field; so there are just oo 4 straights. 149'. Two points determine a straight, their join. 150'. Two points determine a range on their join. 151'. Two bounding points determine a sect. 152'. A point and a straight not on it determine a plane. A range and a point not bearer determine a flat 153 on its pencil. 154'. Three points determine a plane, their junction. 155'. Three points determine a point-field. 156'. Two copunctal straights are coplanar. 157. Any figure, or the proof of any theorem of configu- ration and determination, gives a dual figure or proves a dual theorem by simply interchanging point with plane. Thus all the pure projective geometry on a plane may be read as geom- etry on a point. Prob. 29. If of straights copunctal in pairs not all are copunctal, then all are coplanar. Prob. 30. On a given point put a straight to cut two given straights. Prob. 31. If two triplets of planes ocfty, a'ft'y' are such that the meets fty and ft'y' , yen and y'ot', aft and a' ft' lie on three planes a'\ ft", y" which are costraight, then the meets aa', ft ft', yy' are coplanar. 98 PROJECTIVE GEOMETRY. [Chap. III. Prob. 32. Describe the figures in space dual to the polystim and the polygram. Art. 14. Ruled Quadric Surfaces. 158. The joins of the correlated points of two projective ranges whose bearers are not coplanar form a ' ruled system ' of straights no two coplanar. For were two coplanar, then two points on the bearer m and two on the bearer m x would all four be on this plane, and so m and ;«, coplanar, contrary to hypothesis. 159. Let the straights n, «,, » 3 be any three of the elements of a ruled system, and N^ any point on « 3 . Put a plane on N, and the straight n. x , and let its pass with n be called N. The straight NN t cuts n, n lt «, all three. Projecting the generating ranges of the ruled system (on the bearers m and ;«,) from the straight NN t (or m^} as axis produces two projective axial pencils, which having three planes mjt, vtji^ , m,n, self-corre- sponding, are identical. Therefore every pair of correlated points of the ranges on m and «z, is coplanar with ;« 2 ; that is, ;« 2 cuts every element of the ruled system. By varying the point JV, 00 ' straights are obtained, all cutting all the oo 1 straights of the original ruled system and making on every two projective ranges. Of the straights so obtained no two cross, for that would make two of the first ruled system coplanar. Either of these two systems may be considered as generating a 'ruled surface,' which is the bearer of both. Each of the two systems is completely determined by any three straights of the other, and therefore so is the ruled surface also. From the construction follows that the straights of either ruled system cut all the straights of the other in projective ranges. So any two straights of either system may be considered as bearers of projective ranges generating the other system, or indeed the ruled surface. 160. On each point of this ruled surface are two and only two straights lying wholly in the surface (one in each ruled RULED QUADRIC SURFACES. 99 Art. 14.] system). So a plane on one straight of the ruled surface is also on another straight of this surface. 161. If in the two generating projective ranges the point at infinity of one is correlated to the point at infinity of the other, the ruled surface is called a ' hyper- bolic-paraboloid.' The join of these figurative points is on the figurative plane. Therefore the plane at infinity cuts the surface in a straight and so has a second straight in common with the ruled surface. That a hyperbolic-paraboloid has two straights in common with the plane at infinity may also be proved as follows: Call the bearers of the generating ranges m and m x , and let n, ;z, be any two elements, and /"the element at infinity. By § 159 the ruled surface may be considered as generated by the straights on the three elements n, n, , f. But all these straights must be parallel to the same plane, namely, to any plane on f. On f and each one of these straights put a plane ; these planes make a parallel-axial-pencil, and cut any two of the original elements in projective ranges with the figurative points corre- lated. Therefore the figurative straight joining the figurative points of n and », is wholly on the ruled surface. 162. From § 161 follows that all straights pertaining to the same ruled system on a hyperbolic-paraboloid are parallel to the same plane. Such planes are called ' asymptote-planes.' A hyperbolic-paraboloid is completely determined by two non- coplanar straights and an asymptote-plane cutting them. To get an element cut the two given straights by any plane par- allel to the asymptote-plane, and join the meets. 163. Three non-crossing straights, all parallel to the same plane, completely determine a hyperbolic-paraboloid. Let tn, m lt m, be the given straights. The passes of planes on m, 100 PROJECTIVE GEOMETRY. [Chap. III. with m and m 1 are projective ranges whose joins are a ruled system. But from the hypothesis one of these planes is parallel to both m and m. . Therefore their points at infinity are corre- lated and the ruled surface is a hyperbolic-paraboloid. 164. If two non-coplanar projective ranges be each axially projected from the bearer of the other, two projective axial pencils are formed, with those planes correlated on which are the correlated points of the ranges. If A, A x be correlated points, then the straight AA l is the meet of correlated planes. Thus two projective axial pencils with axes not coplanar gen- erate a ruled system. If the whole figure be cut by a plane, this will cut these axial pencils in two projective flat pencils, and the conic generated by these will be the cut of the ruled surface. So every plane cuts it in a conic or a pair of straights. Hence no straight not wholly on the surface can cut it in' more than two points. The surface is therefore of the second degree (quadric). If the plane at infinity cuts the ruled surface in a pair of straights, it is a hyperbolic-paraboloid. If not, it is called a ' hyperboloid of one nappe,' a fig- ure of which is here shown. 164I. Copunctal straights par- allel to the generating elements of a hyperboloid of one nappe are on a cone. Copunctal straights par- allel to the generating elements of a hyperbolic-paraboloid are on a system of two planes. For the figurative plane cuts the hyperboloid of one nappe in a conic curve, but cuts the hyper- bolic-paraboloid in two straights; and each of the copunctal straights goes to a point of the figurative cut. 165. Each straight in one ruled system of a hyperboloid of ART. 14.] RULED QUADRIC SURFACES. 101 c • one nappe is parallel to one, but only to one, straight in the other ruled system. Of the straights on a hyperbolic-parabo- loid no two are parallel. Let n and n l , any two elements of one ruled system, be the bearers of the generating ranges R and R v . If Fis the vanishing point of R, then the straight on V parallel to «, is an element of the other ruled system. But for the hyperbolic-paraboloid V\% itself a figurative point. 166. Any straight of one ruled system on a ruled surface is called a ' guide-straight ' of the other ruled system. 167,. A ruled system is cut by 167'. A ruled system is pro- any two of its guide-straights in jected from any two of its guide- projective ranges. straights in projective axial pen- cils. For if m, m,, m, be any three guide-straights of the ruled system, the planes on m 2 cut m and k, in projective ranges the joins of whose correlated points are the elements of the ruled system. Again, if the points on «, be projected axially from m and «„ the meets of the planes so correlated are the ele- ments of the ruled system. 168. Four straights of a ruled system are called harmonic straights if they are cut in four harmonic points by one (and so by every) guide-straight. By three straights, no two coplanar, a fourth harmonic is determined lying in a ruled system with the given three and on a fourth harmonic point to any three costraight points of the given three. 169. A plane cutting the ruled surface in a straight m of one ruled system and consequently also in a straight n of the other ruled system has in common with the surface no point not on one of these straights. For any straight from such a point cutting both these straights would lie wholly on the ruled sur- face ; and so therefore would their whole plane, which is im- possible. Any third straight coplanar with m and n on their cross has no second point in common with the surface and so is a tangent, and the plane of m and n is called tangent at their cross, the point mn. 102 PROJECTIVE GEOMETRY. [Chap. III. The number of planes tangent to the ruled surface and on a given straight equals the number of points the straight has in common with the ruled surface, that is two ; so the ruled surface is of the second class. 170. Project the two generating ranges of a ruled system from any projection-vertex J^not on it. The eject consists of two copunctal projective fiat pencils. The plane of any two correlated straights is on an element of the ruled system. All such planes form a cone of planes. The points of contact of these planes with the ruled surface are a conic range. The planes tangent to a ruled surface at the points on its cut with a plane form a cone of planes. 171. The cut of a hyperbolic-paraboloid by a plane not on an element has on it the meets of the plane with the two figu- rative elements, and so is a hyperbola except when their cross is on the plane, in which case it is a parabola. The figurative plane is a tangent plane. 172. The planes tangent at the figurative points of a hyper- boloid of one nappe are all proper planes, copunctal and form- ing a cone of planes tangent to the ' asymptote-cone ' of the hyperboloid. Each element to the asymptote-cone is parallel to one element of each ruled system. Any plane not on an element of the hyperboloid of one nappe cuts it in a hyperbola, parabola, or ellipse, according as it is parallel to two elements, one, or no element of the asymp- tote-cone, that is, according as it has in common with the figu- rative conic on the hyperboloid two points, one, or no point. 173. If an axial pencil and a ruled system are projective, they generate in general a ' twisted cubic curve,' which any plane cuts in one point at least and three at most. For a plane cuts the ruled system in a conic range perspective to it, of which in general three points at most lie on the correspond- ing planes of the pencil. 174. The ruled quadric surface is the only surface doubly Art. 14.] RULED QUADRIC surfaces. 103 ruled. The figure of two so united ruled systems is one of the most noteworthy discovered by the modern geometry.* 175. To find the straights crossing four given straights. — Let «,, u. t , u„ u t be the given straights. Projecting the range R t on «, from the axes u, and u, gives two axial pencils, each perspective to R„ and consequently projective. The meets of their correlated planes are all the oo 1 straights on «,, u^, u 3 , and form a ruled system of which «,, u„ u % are guide-straights. The two projective axial-pencils cut the fourth straight u k in two ' conjective ' ranges. [Two projective primal figures of the same kind and on the same bearer are called conjective.J If now a straight m of the ruled system crosses u t , then the two correlated planes of which this straight tn is the meet must cut u t in the same point, which consequently is a self-correspond- ing point of the two conjective ranges. Since there are two such (the points common to u t and the ruled surface), so there are two straights (real or conjugate imaginary) crossing four given straights. Their construction is shown to depend on that for the two self-correlated points of two conjective ranges. This important problem in the four-dimensional space of straights, ' what is common to four straights ? ' is the analogue of the problem in the space of points, ' what is common to three points?' and ,its dual in the space of planes, ' what is common to three planes? ' It shows not only their fundamental diversity, but also, as compared to points-geometry and planes-geometry, the inher- ently quadratic character of straights-geometry. Prob. 33. Find the straights cutting two given straights and parallel to a third. Prob. 34. Three diagonals of a skew hexagram whose six sides are on a ruled surface are copunctal. Prob. 35. If a flat pencil and a range not on parallel planes are projective, then straights on the points of the range parallel to the correlated straights of the pencil form one ruled system of a hyper- bolic-paraboloid. *See Monge, Journal de l'gcole polytechnique, Vol. I. 104 PROJECTIVE GEOMETRY. [CHAP. III. Prob. 36. What is the locus of a point harmonically separated from a given point by a ruled surface ? Art. 15. Cross-Ratio. 176. Lindemann has shown how every one number, whether integer, fraction, or irrational, -(- or — , may be correlated to one point of a straight, without making any use of measure- ment, without any comparison of sects by application of a unit sect.* He gets an analytic definition of the ' cross-ratio ' of four copunctal straights. Then this expression is applied to four costraight points. Then is deduced that the number pre- viously attached to a point on a straight is the same as the cross-ratio of that point with three fixed points of the straight. Thus analytic geometry and metric geometry may be founded without using ratio in its old sense, involving measurement. Thus also the non-Euclidean geometries, that of Bolyai-Loba- cheVski in which the straight has two points at infinity, and that of Riemann in which the straight has no point at infinity, may be treated together with the limiting case of each between them, the Euclidean geometry, wherein the straight has one but only one point at infinity. Relinquishing for brevity this pure projective standpoint and reverting to the old metric usages where an angle is an in- clination, a sect is a piece of a straight, and any ratio is a number; distinguishing the sect AC from CA as of opposite 'sense,' so that AC = — CA, the ratio [AC/BC]/\AD/ BD\ is called the cross-ratio of the range ABCD and is written [ABCB] where A and B, called conjugate points of the cross-ratio, may be looked upon as the extremities of a sect divided internally or externally by C and again by D.\ * Von Staudt in Beitrgge zur Geometrie der Lage, 1856-60, determines the projective definition of number, and thus makes the metric geometry a conse- quence of projective geometry. f The fundamental property of cross-ratio is stated in the Mathematical Col- lections of Pappus, about 400 B.C. The cross-ratio is the basis of Poncelet's Traite des proprietes projectives, 1822, which distinguishes sharply the projec- tive and metric properties of curves. .Art. 15.] CROSS-RATIO. 105 177. If on ABCD respectively be the straights abed co- punctal on V, then A C/BC= A A VC/AB VC or A C/BC = \A V. VC sin (ac)/\B V. VC sin (be). AD/BD = A A VD/ABVD = \A V. VD sin {ad)/%B V. VD sin {fid). Therefore [ABCD] = [sin (ae)/sin (be)]/[sin ad/sin (bd)] . Thus as the cross-ratio of any fiat pencil V[abed] or axial pencil u(afiyd) may be taken the cross-ratio of the cut ABCD on any transversal. 178. Two projective primal figures are 'equicross;' and inversely two equicross primal figures are projective. 179. As D approaches the point at infinity, AD/BD ap- proaches 1. The cross-ratio [ABCD] when D is figurative equals AC/BC. 180. Given three costraight points ABC, to find D so that [ABCD] may equal a given number n (-)- or — ). On any straight on C take A' and B' such that CA'/CB' = n; A' and B' lying on the same side of C if n be positive, but on opposite sides if n be negative. Join A A', BB', crossing in V. The parallel to A'B' on Twill cut AB in the required D. For if D' be the point at infinity on A'B', and ABCD be projected from V, then A'B' CD' is a cut of the eject ; so [ABCD] = [A'B' CD'] =A'C/B'C = n. 181. If [ABCD] = [ABCD,], then D x coincides with D. 182. If two figures be complete plane perspectives, four costraight points (or copunctal straights) in one are equicross with the correlated four in the other. Let O be the center of perspective. Let M and M' be any pair of correlated points of the two figures, iVand N' another pair of correlated points lying on the straight OMM' whose cross with the axis of per- spective is X. Then [OXMN] = [OXM'N 1 ]. That is, [OMfXM]/[ON/XN] = [OM'/XM']/[ON'/XN']. Therefore [OM/XM]/[OM'/XM'] = [ON/XN]/[ON' /XN']. That is, [OXMM' ] = [OXNN 1 ] ; or the cross-ratio [OXMM 1 ] 106 PROJECTIVE GEOMETRY. [Chap. III. is constant for all pairs of correlated points M and M' taken on a straight OX on the center of perspective. Next let L and L' be another pair of correlated points and Y the cross of OLL with the axis of perspective. Since LM and L'M' cross on some point Z of the axis XY, therefore if OXMM' be projected from Z, the cut of the eject by OY is YLL'. So \6XMM' ] = [<9 YLL'] ; or the cross-ratio [OXMM'] is constant for all pairs of correlated points. It is called the ' parameter ' of the correlation. When the parameter equals — I, the range OXMM' is harmonic, and two correlated elements correspond doubly, are coupled, and the correlation is 'involutorial.' 183. When the correlation is involutorial and the center of perspective is the figurative point on a perpendicular to the axis of perspective, this is called the 'axis of symmetry,' and the complete plane perspectives are said to be ' symmetrical.' 184. When the correlation is involutorial and the axis of perspective is figurative, then the center of perspective is called the 'symcenter,' and the complete plane perspectives are said to be ' symcentral.' Prob. 37. In a plane are given a parallelogram and any sect. With the ruler alone find the center of the sect and draw a parallel to it. Prob. 38. The locus of a point such that its joins to four given points have a given cross-ratio is a conic on which are the points. Prob. 39. If the sides of a trigram are tangent to a conic, the joins of two of its fan-points to any point on the polar of the third are conjugate with respect to the conic. Prob. 40. If from any point of the sect between the contact- points of a pair of tangents to a parabola straights be drawn parallel to these tangents, the join of their proper crosses with the tangents will be a tangent. ART. 1.] CORRESPONDENCE OF POINTS ON CONICS. 107 Chapter IV. HYPERBOLIC FUNCTIONS. By James McMahon, Assistant Professor of Mathematics in Cornell University. Art. 1. Correspondence of Points on Conics. To prepare the way for a general treatment of the hyper- bolic functions a preliminary discussion is given on the relations between hyperbolic sectors. The method adopted is such as to apply at the same time to sectors of the ellipse, including the circle; and the analogy of the hyperbolic and circular functions will be obvious at every step, since the same set of equations can be read in connection with either the hyperbola or the ellipse.* It is convenient to begin with the theory of correspondence of points on two central conics of like species, i.e. either both ellipses or both hyperbolas. To obtain a definition of corresponding points, let O l A l , 0,B i be conjugate radii of a central conic, and 0,A, , 0,B 2 conjugate radii of any other central conic of the same species ; let P x , P 2 be two points on the curves; and let their coordi- nates referred to the respective pairs of conjugate directions be {x l , y x ), (x t , y,); then, by analytic geometry, * The hyperbolic functions are not so named on account of any analogy with what are termed Elliptic Functions. " The elliptic integrals, and thence the elliptic functions, derive their name from the early attempts of mathemati- cians at the rectification of the ellipse. ... To a certain extent this is a disadvantage; . . because we employ the name hyperbolic function to de- note cosh «, sinh «, etc., by analogy with which the elliptic functions would be merely the circular functions cos ,, O.A, =a lt and the conjugate radius £>,.£>, = b x ; then the ratios xja x , yjb x are called the charac- teristic ratios of the given sectorial measure S,/K r These ratios are constant both in magnitude and sign for all sectors of the same measure and species wherever these may be situ- ated (Art. 3). Hence there exists a functional relation be- tween the sectorial measure and each of its characteristic ratios. Art. 5. Ratios Expressed as Triangle-measures. The triangle of a sector and its complementary triangle are measured by the two characteristic ratios. For, let the triangle A,O l P l and its complementary triangle P.O.B, be denoted by T„ 7V; then T\ _ W. sin&? . _h (5) K i«A sin £»j b 1 t; #,*, sin gd 1 x K, ¥ rt A sin as, a ART. 7.] FUNCTIONAL RELATIONS FOR ELLIPSE. Ill Art. 6. Functional Relations for 'Ellipse. The functional relations that exist between the sectorial measure and each of its characteristic ratios are the same for all elliptic, in- cluding circular, sec- tors (Art. 4). Let/ 3 ,, P, be corresponding points on an ellipse and a circle, referred 6 to the conjugate di- rections O t A lt O^B^, and 0, A„O^B a , the latter pair being at right angles ; let the angle A,0,P^ — in radian measure; then s * — \ a * \ u * -* = cos ' a, JC, e. (6) *r sm it; [«• = K hence, in the ellipse, by Art. 3, — = cos K b. sin k: (7) Prob. 4. Given x t = lar, find the measure of the elliptic sector A x OiPi. Also find its area when ir **% and these express that the ratio of the two lines on the left is a certain definite function of the ratio of the two areas on the right. These functions are called by analogy the hyperbolic 112 HYPERBOLIC FUNCTIONS. [CHAP. IV. cosine ana the nyperbolic sine. Thus, writing u for SJK V the two equations x y — = cosh u, v — sinh u (8> «. *. serve to define the hyperbolic cosine and sine of a given secto- rial measure u ; and the hyperbolic tangent, cotangent, secant, and cosecant are then defined as follows : sinh u , cosh« tanh u = = — , coth u = -^—. — , cosh w sinh u sech u = — : — , csch « cosh «' sinh u (9) The names of these functions may be read " h-cosine,"' " h-sine," "h-tangent," etc. Art. 8. Relations between Hyperbolic Functions. Among the six functions there are five independent rela- tions, so that when the numerical value of one of the functions is given, the values of the other five can be found. Four of these relations consist of the four defining equations (9). The fifth is derived from the equation of the hyperbola a, b, giving cosh J « — sinh"« = 1. (10) By a combination of some of these equations other subsidi- ary relations may be obtained; thus, dividing (10) successively by cosh 2 u, sinh 2 u, and applying (9), give 1 — tanh" u — sech a u, ) coth 3 u — 1 = csch 2 u. ) Equations (9), (10), (11) will readily serve to express the value of any function in terms of any other. For example^ when tanh u is given, coth u = - — — , sech u = *J 1 — tanh 2 «, tanh u ' Art. 8.] relations between hyperbolic functions. 113 , I . , tanh u cosh u = — , sinh u csch u -y/ I — tanh" u -v/ I — tanh 2 _ y I — tanh 2 u tanh u The ambiguity in the sign of the square root may usually be removed by the following considerations : The functions cosh u, sech u are always positive, because the primary char- acteristic ratio x l /a 1 is positive, since the initial line O x A^ and the abscissa O l M 1 are similarly directed from 1 , on which- ever branch of the hyperbola P l may be situated; but the func- tions sinh u, tanh u, coth u, csch u, involve the other charac- teristic ratio yjb^ , which is positive or negative according as y l and b 1 have the same or opposite signs, i.e., as the measure u is positive or negative ; hence these four functions are either all positive or all negative. Thus when any one of the func- tions sinh u, tanh u, csch w, coth u, is given in magnitude and sign, there is no ambiguity in the value of any of the six hyperbolic functions ; but when either cosh u or sech u is given, there is ambiguity as to whether the other four functions shall be all positive or all negative. The hyperbolic tangent may be expressed as the ratio of two lines. For draw the tangent line AC=t\ then *. u y . x a y tanh u =■ - :-=—.— b a b x a t / b ' a b (12) o AM The hyperbolic tangent is the measure of the triangle OAC. For OAC at t . . . — — — = —-=— = tanh u. (\X) OAB ab b v *' Thus the sector AOP, and the triangles AOP, FOB, AOC, are proportional to u, sinh u, coshu, tanh u (eqs. 5, 13) ; hence sinh« > «> tanha. (i4> 114 HYPERBOLIC FUNCTIONS. [Chap. IV. Prob. 7. Express all the hyperbolic functions in terms of sinh u. Given cosh u — 2, find the values of the other functions. Prob. 8. Prove from eqs. 10, 11, that coshw> sinh u, cosh«>i, tanh u < 1, sech u < 1. Prob. 9. In the figure of Art. 1, let OA — 2, OB—\, AOB = 6o°, and area of sector AOP = 3; find the sectorial measure, and the two characteristic ratios, in the elliptic sector, and also in the hyper- bolic sector; and find the area of the triangle A OP- (Use tables of cos, sin, cosh, sinh.) Prob. 10. Show that coth u, sech u, csch u may each be ex- pressed as the ratio of two lines, as follows: Let the tangent at P make on the conjugate axes OA, OB, intercepts OS = m, OT = ?i\ let the tangent at B, to the conjugate hyperbola, meet OP in R, making BR = /; then coth u = l/a, sech u = m/a, csch u = njb. Prob. 11. The measure of segment AMP is sinh u cosh u — u. Modify this for the ellipse. Modify also eqs. 10-14, an d probs. 8, 10. Art. 9. Variations of the Hyperbolic Functions. Since the values of the hyperbolic functions depend only on the sectorial measure, it is convenient, in tracing their vari- ations, to consider only sectors of one half of a rectangular hyperbola, whose conjugate radii are equal, and to take the principal axis OA as the common initial line of all the sectors. The sectorial measure u assumes every value from — 00, through o, to -f- 00 , as the terminal point P comes in from infinity on the lower branch, and passes to infinity on the upper branch ; that is, as the terminal line OP swings from the lower asymptotic posi- tion y — — x, to the upper one, y = x. It is here assumed, but is proved in Art. 17, that the sector AOP becomes infinite as P passes to infinity. Since the functions cosh u, sinh u, tanh u, for any position ART. 9.] VARIATIONS OF THE HYPERBOLIC FUNCTIONS. 115 of 0P l are equal to the ratios of x, y, t, to the principal radius a, it is evident from the figure that cosh 0=1, sinh = 0, tanh 0=0, (15) and that as u increases towards positive infinity, cosh u, sinh u are positive and become infinite, but tanh& approaches unity as a limit ; thus cosh 00 = 00 , sinh 00 = 00 , tanh 00 = 1. (16) Again, as u changes from zero towards the negative side, cosh u is positive and increases from unity to infinity, but sinh u is negative and increases numerically from zero to a negative infinite, and tanh u is also negative and increases numerically from zero to negative unity ; hence cosh (— 00) = 00 , sinh (— 00) =— 00 , tanh (— 00 ) =— 1. (17) For intermediate values of u the numerical values of these functions can be found from the formulas of Arts. 16, 17, and are tabulated at the end of this chapter. A general idea of their manner of variation can be obtained from the curves in Art. 25, in which the sectorial measure u is represented by the abscissa, and the values of the functions cosh u, sinh u, etc., are represented by the ordinate. The relations between the functions of — u and of u are evident from the definitions, as indicated above, and in Art. 8. Thus cosh (—«) = + cosh u, sinh (—#)= — sinh u, \ sech (—«)=-{- sech u, csch (—«) = — csch u, > (18) tanh (—«)=— tanh u, coth (— u) = — coth u. ) Prob. 12. Trace the changes in sech u, coth u, csch u, as u passes from — 00 to + 00 . Show that sinh u, cosh u are infinites of the same order when u is infinite. (It will appear in Art. 17 that sinh u, cosh u are infinites of an order infinitely higher than the order of u.) Prob. 13. Applying eq. (12) to figure, page 114, prove tanh u, = tan A OP- 116 HYPERBOLIC FUNCTIONS. [CHAP. IV. Art. 10. Anti-hyperbolic Functions. x y t The equations - = cosh u, j = sinh u, -r = tanh u, etc., may also be expressed by the inverse notation « = cosh -1 — , y t u= sinh _1 -r. « = tanh _1 -T, etc., which may be read: "« is the sectorial measure whose hyperbolic cosine is the ratio x to a," etc. ; or " u is the anti-h-cosine of x/a," etc. Since there are two values of u, with opposite signs, that correspond to a given value of cosh u, it follows that if u be determined from the equation cosh u = m, where m is a given number greater than unity, u is a two-valued function of m. The symbol cosh~' m will be used to denote the positive value of 11 that satisfies the equation cosh u = m. Similarly the symbol sech _1 ;« will stand for the positive value of u that satisfies the equation sech u = m. The signs of the other functions smhr^m, tanh" 1 ;^, coth -1 m, csch -1 m, are the same as the sign of nt. Hence all of the anti-hyperbolic functions of real numbers are one-valued. Prob. 14. Prove the following relations : cosh _1 OT = sinh" 1 Vm* — 1, sinh" 1 #2 = ± cosh" 1 Vm' + 1, the upper or lower sign being used according as m is positive or negative. Modify these relations for sin"" 1 , cos -1 . Prob. 15. In figure, Art. i,let OA — 2, OB = i,AOB — 6o°; find the area of the hyperbolic sector AOP, and of the segment AMP, if the abscissa of P is 3. (Find cosh" 1 from the tables for cosh.) Art. 11. Functions of Sums and Differences. (a) To prove the difference-formulas sinh (u — v) = sinh u cosh v — cosh u sinh v, ) cosh (u — v) = cosh u cosh v — sinh u sinh v.) Let OA be any radius of a hyperbola, and let the sectors AOP, AOQ have the measures u, v; then u — v is the measure of the sector QOP. Let OB, OQ be the radii conjugate to OA, OQ; and let the coordinates of P, Q, Q' be (x 1 , _y,), (x, y), (x', y') with reference to the axes OA, OB; then Art. 11.] FUNCTIONS OF SUMS AND DIFFERENCES. 117 sinh («_*) = sinh ^SLQOP = tria "g le Q 0P [Art s . _ |(.yy,— jfj/) sin go _y,x_ yx t \a x b x sin a> 3, a, £, a t = sinh « cosh z/ — cosh ?< sinh v : cosh (« but . , sector £><9/> triangle POO' r A z>) = cosh — ^ = a— S [Art. 5. _ l(^y— jV.-yQ sin 00 _ y x, _J^£^. (20) 2^,3, sin a? 7' X a t b' since (2, <2' are extremities of conjugate radii ; hence cosh (u — v) = cosh u cosh v — sinh « sinh v. In the figures & is positive and v is positive or negative. Other figures may be drawn with u negative, and the language in the text will apply to all. In the case of elliptic sectors, similar figures may be drawn, and the same language will apply, except that the second equation of (20) will be x' /a y = —y/b;, therefore sin {u — v) = sin u cos v — cos u sin v, cos {u — v) = cos u cos v -f- sin u sin v. (b) To prove the sum-formulas sinh (u -(- v) = sinh ?e cosh z> -{- cosh u sinh #, ) \ (21) cosh (« -)- v) = cosh ?* cosh ^ -f- sinh & sinh v . ) These equations follow from (19) by changing v into — z<, 118 HYPERBOLIC FUNCTIONS. [CHAP. IV. and then for sinh(— v), cosh(— v), writing — sinh v, cosh v (Art. 9, eqs. (18)). , . _, A , , . tanh u ± tanh v , , (c) 1 o prove that tanh (u ±v) = — . (22} w ^ v ; i±tanh?^tanh^ v ' Writing tanh (u + v) = -4 -,, expanding and dividing s v • ^ ; cosh (u±v) v s s numerator and denominator by cosh u cosh v, eq. (22) is ob- tained. Prob. 16. Given cosh u = 2, cosh z> = 3, find cosh (u -)- z»). Prob. 17. Prove the following identities: 1. sinh 211 = 2 sinh u cosh u. 2. cosh 2« = cosh 2 # -f- sinh 2 & =1+2 sinh 3 u = 2 cosh 2 u — 1. 3. 1 + cosh u = 2 cosh 5 ^«, cosh u — 1 = 2 sinh 2 ■£#. , , sinh & cosh « — 1 /cosh u — i\i 4. tanh \u = — ■ — - = — = : — . 1 -+- cosh u sinh u \cosh u -\- 1/ . , 2 tanh u t 4- tanh 2 u 5. Sinh 2« = r5 — , COsh 2« ~ r-s— . 1 — tanh & i -- tanh « 6. sinh 3« = 3 sinh « + 4 sinh 3 w, cosh 3^ = 4 cosh 8 « —3 cosh «. , . 1 + tanh £» 7. cosh « + sinh a = : — .- . 1 — tanh -$u 8. (cosh « + sinh «)(cosh v -f- sinh z>)=cosh (u -4- v) + sinh (z* + z-). 9. Generalize (8); and show also what it becomes when #=:»= . . . 10. sinh 2 .* cos 2 ^ + cosh 2 * sin 2 ^ = sinh 2 x -j- sin 2 jy. 11. cosh _1 »2 ± cosh _1 « = cosh _1 Lw« ± y (m' — i)(« 2 — i)J. 1 2. sinh -1 #z ± sinh -1 n = sinh -1 \_m y 1 -\- n 7 ± »yi -(- m'\. Prob. 18. What modifications of signs are required in (21), (22), in order to pass to circular functions ? Prob. 19. Modify the identities of Prob. 17 for the same purpose. Art. 12. Conversion Formulas. To prove that cosh Mj-f- cosh « a = 2 cosh i(Wj+ ti^) cosh ^{u 1 — «,), cosh «,— cosh u, = 2 sinh $(u, -f- «,) sinh £(«,— a,), sinh w, + sinh «, = 2 sinh £(«, -f- «,) cosh £(«,— « 3 ), J sinh u x — sinh u,=2 cosh J(«, -(- «,) sinh |(«, — u,). 1 Art. 13.] limiting ratios. 119 From the addition formulas it follows that cosh (u -\- v)-\- cosh (u — v) = 2 cosh u cosh v, cosh (u -\- v) — cosh (u — v) = 2 sinh & sinh z/, sinh (u -\-v)-\- sinh (u — v) = 2 sinh w cosh w, sinh (« -(- v) — sinh (« — v) = 2 cosh w sinh v, and then by writing u -\- v = u 1 , u — v — u^, u = ^(«, -}- «,)> z< = £(«, — « 2 ), these equations take the form required. Prob. 2o. In passing to circular functions, show that the only modification to be made in the conversion formulas is in the alge- braic sign of the right-hand member of the second formula. _. . _. ,. r cosh 2U + cosh av cosh 2« + cosh AV Prob. 2i. Simplify -^—. ; — —. , ; : sinh 2U -f- sinh av cosh 2U — cosh 4V Prob. 22. Prove sinh 2 * — sinh 2 j> = sinh (x -\-y) sinh (x — y). Prob. 23. Simplify cosh 2 * cosh'j 1 ± sinh 2 * sinh 2 j\ Prob. 24. Simplify cosh 2 * cos 2 j> -f- sinh 2 * sin 2 jc. Art. 13. Limiting Ratios. To find the limit, as u approaches zero, of sinh u tanh u u u which are then indeterminate in form. By eq. (14), sinh u > u > tanh u ; and if sinh u and tanh u be successively divided by each term of these inequalities, it follows that sinh u , 1 < < cosh u, u . _ tanh u , sech u < < v u but when u = o, cosh u = 1, sech u — 1, nence lim. sinh « _ I; lim. tanh « _ I# (24) U = O u u = o u 120 HYPERBOLIC FUNCTIONS. [Chap. IV. Art. 14. Derivatives of Hyperbolic Functions. To prove that (a) (*) (<) (d) to (/) af(sinh u) - r = cosh u, du = sinh u, Ay = sinh (u -j- ^J#) — sinh « = 2 cosh %(2u -\- Au) sinh f Au, Ay sinh £Jw -^- = cosh (u -f £zf«) — j-- . Take the limit of both sides, as Au = o, and put Ay _ dy y T-22 HYPERBOLIC FUNCTIONS. [CHAP. IV.. Art. 15. Derivatives of Anti-hyperbolic Functions. (a) (*) (?) (d) W (/) x I \x I xVi—x* (/) Similar to (e). Prob. 31. Prove ^(sin~'.s) _ 1 d(cos- 1 x)_ 1 oft Vi - x" dx V] (30 23 '245 2467 the integration-constant being zero, since sinh -1 x vanishes with x. This series is convergent, and can be used in compu- 126 HYPERBOLIC FUNCTIONS. [CHAP. IV. tation, only when x < i. Another series, convergent when x > i, is obtained by writing the above derivative in the form ^(sinh" 1 x) , , , . , if . i\~* " I _II + I3I_I3 5.1, 1 2/ T 2 4^ 2 4 6/ T 'J' .-. sinh"' * = C+log *+- -1; -- 1 -- +1 ?■ | i--. . . , (32) ' s '22^' 2 4 4*' ' 2 4 6 6* 6 VJ ' where C is the integration-constant, which will be shown in Art. 19 to be equal to log,, 2. A development of similar form is obtained for cosh - ' x\ for xL T 2« ,T 24«* r 24 6x°^'"J' hence cosh-'^= C+logx ■, 2 2.2. . (33) 1 fa 2 2* 244*' 2 46 6x" v • in which C is again equal to log, 2 [Art. 19, Prob. 46]. In order that the function cosh - '* maybe real, x must not be less than unity ; but when x exceeds unity, this series is con- vergent, hence it is always available for computation. . . <^(tanh -1 x) 1 is,.,., Again, v dx - = — —* = i+* 2 + * 4 + x +..-, and hence tanh -1 x = x -\- -x 3 -{- -x"-\ — x % + ..., (34) J J / From (32), (33), (34) are derived : sech"' x = cosh - ' — x r 1 ■*■' I-3-^* 1.3.5.^* , . S 2.2 2.4.4 2.4.6.6 ' Kib ' ART. 19 ] LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS. 127 .csch-^ = sinh-I = :I-I-L 3 + -. 3 -- 6 - i H-^---' x x 2 ix 2 4 5* 2467^ „ . . x* 1 . 3 . x" 1 . 3 . 5 . x° . ,. — C — log x -\ 2 *—h—> — • • • ; (36) S ' 2.2 2. 4.4^2.4.6.6 V ° ' coth-' x = tanh" - = - -| _|_ _L _|_ _L -f . . ., (37) x x ~ $x* ~ 5x* ' jx 1 ^ xo " Prob. 44. Show that the series for tanh -1 *, coth -1 *, sech -1 *, are always available for computation. Prob. 45. Show that one or other of the two developments of the inverse hyperbolic cosecant is available. Art. 19. Logarithmic Expression of Anti-Functions. Let x = cosh u, then Vx* — 1 = sinh u; therefore x-\- Vx' — 1 = cosh u -\- sinh u = , sech z< = cost', csch u = cot ?/. The gudermanian is sometimes useful in computation ; for instance, if sinh u be given, v can be found from a table of natural tangents, and the other circular functions of v will give the remaining hyperbolic functions of u. Other uses of this function are given in Arts. 22-26, 32-36. Prob. 49. Prove that gd u = sec - '(cosh u) — tan - '(sinh u) = cos - '(sech u) = sin" '(tanh u), Prob. 50. Prove gd -1 z> = cosh" '(sec v) = sinh" 1 (tan v) = sech"" '(cos v) — tanh" '(sin v). Prob. 51. Prove gd o = o, gd 00 = fa, gd(— 00) = —fa, gd"'o=o, gd-'(i^)=co, gd"'(-i^) = -oo. Prob 52. Show that gd u and gd"'» are odd functions of ^, v. Prob. 53. From the first identity in 4, Prob. 19, derive the rela- tion tanh \u — tan \v. Prob. 54. Prove tanh" '(tan u) = \ gd 2U, and tan" '(tanh x) = ■£ gd _1 2x Art. 22. Gudermanian Angle If a circle be used instead of the ellipse of Art. 20, the gudermanian of the hyperbolic sectorial measure will be equal to the radian measure of the angle of the corresponding circular sector (see eq. (6), and Art. 2, Prob. 2). This angle will be called the gudermanian angle ; but the gudermanian function v, as above defined, is merely a number, or ratio ; and this number is equal to the radian measure of the gudermanian angle 6, which is itself usually tabulated in degree measure ; thus 6 — i%o° v/rt (47) 130 HYPERBOLIC FUNCTIONS. [Chap. IV. Prob. 55. Show that the gudermanian angle of u may be construct- ed as follows: Take the principal radius OA of an equilateral hyperbola, as the /^ initial line, and OP as the terminal line, of the sector whose measure is u\ from M, the foot of the ordinate of P, draw MT tangent to the circle whose diameter is the transverse axis; then AOT is the angle required.* Prob. 56. Show that the angle 6 never exceeds 90 . Prob. 57. The bisector of angle AOT M bisects the sector AOP (see Prob. 13, Art. 9, and Prob. 53, Art. 21), and the line AP. (See Prob. 1, Art. 3.) Prob. 58. This bisector is parallel to TP, and the points T,P are in line with the point diametrically opposite to A. Prob. 59. The tangent at p passes through the foot of the ordinate of T, and intersects TM on the tangent at A. Prob. 60. The angle APM is half the gudermanian angle. Art. 23. Derivatives of Gudermanian and Inverse. Let v = gd u, u = gd _I v, then sec v = cosh u, sec v tan vdv = sinh u du, sec vdv = du, therefore ^(gd _1 v) = sec vdv. (48) Again, dv = cos v du — sech u du, therefore d{gd u) = sech u du. (49) Prob. 61. Differentiate: y = sinh u — gd u, y = sin v -\- gd -1 v, y = tanh u sech u -\- gd u, y = tan v sec v -\- gd -1 v. * This angle was called by Gudermann the longitude of 11, and denoted by lu. His inverse symbol was |L; thus u = ^i.(/u). (Crelle's Journal, vol. 6, 1830.) Lambert, who introduced the angle 0, named it the transcendent angle. (Hist, de l'acad, roy.- de Berlin, 1761). Hottel (Nouvelles Annales, vol. 3, 1864) called it the hyperbolic amplitude of u, and wrote it amh u, in analogy with the amplitude of an elliptic function, as shown in Prob. 62. Cayley (Elliptic Functions, 1876) made the usage uniform by attaching to the angle the name of the mathematician who had used it extensively in tabulation and in the theorv of elliptic functions of modulus unity. Art. 24.] series for gudermanian and its inverse. 131 Prob. 62. Writing the "elliptic integral of the first kind" in the form ~ d(j) ~ J^ Vi - K 3 sin 2 0' k being called the modulus, and the amplitude; that is, 4> = am u, (mod. /c), show that, in the special case when k = 1, u = gd -1 4>, am u = gd u, sin am u = tanh u, cos am u — sech u, tan am u = sinh u; and that thus the elliptic functions sin am u, etc., degenerate into the hyperbolic functions, when the modulus is unity.* Art. 24. Series for Gudermanian and its Inverse. Substitute for sech u, sec v in (49), (48) their expansions, Art. 16, and integrate, then gd u = u - \u" + ■^u" - j% f^' + . . . (50) gd-V = v + \v> + ^ 6 +^i^ 7 + • • • (5i) No constants of integration appear, since gd u vanishes with u, and gd~'v with z/. These series are seldom used in compu- tation, as gd u is best found and tabulated by means of tables of natural tangents and hyperbolic sines, from the equation gd u = tan~'(sinh u), and a table of the direct function can be used to furnish the numerical values of the inverse function ; or the latter can be obtained from the equation, gd"V = sjnh J (tan v) = cosh"'(sec v). To obtain a logarithmic expression for gd~V, let gd""!' = u, v = gd u, * The relation gd » = am «, (mod. 1), led Hoiiel to name the function gd u, the hyperbolic amplitude of «, and to write itamh u (see note, Art. 22). In this connection Cayley expressed the functions tanh u, sech «, sinh u in the form sin gd «, cos gd u, tan gd u, and wrote them sg «, eg u, tg «, to correspond with the abbreviations sn u, en u, dn u for sin am u, cos am «, tan am «. Thus tanh « = sg « = sn u, (mod. 1); etc. It is well to note that neither the elliptic nor the hyperbole functions received their names on account of the relation existing between them in a special case. (See foot-note, p. 107.) 132 HYPERBOLIC FUNCTIONS. [Chap. IV. therefore sec v = cosh u, tan v = sinh u sec v -f- tan w = cosh a -j- sinh z< = **, I -j- sin v _ I — cos (|-7r -|- z>) _ cos z> sin (£?r -|- v) u, = gd" V, = log, tan (i?r + $v). tan (iw -f £v), Prob. 63. Evaluate gd u — u gd 'g _J»=o (52) Prob. 64. Prove that gd u — sin u is an infinitesimal of the fifth order, when u = o. Prob. 65. Prove the relations \n + iv= tarry, in — \v = tan"'«"". Art. 25. Graphs of Hyperbolic Functions. Drawing two rectangular axes, and laying down a series of points whose abscissas represent, on any convenient scale, suc- cessive values of the sectorial measure, and whose ordinates represent, preferably on the same scale, the corre- sponding values of the function to be plotted, the locus traced out by this series of points will be a graphical representation of the variation of the func- tion as the sectorial meas- Art. 25.] graphs of the hyperbolic functions. 133 ure varies. The equations of the curves in the ordinary carte- sian notation are : Fig. Full Lines. Dotted Lines. A y = cosh x, y — sech x ; B y = sinh x, y — csch x ; C y = tanh x, y = coth x ; D y = gd x. Here x is written for the sectorial measure u, and y for the numerical value of cosh u, etc. It is thus to be noted that the variables x, y are numbers, or ratios, and that the equation y =■ cosh x merely expresses that the relation between the numbers x and y is taken to be the same as the relation be- tween a sectorial measure and its characteristic ratio. The numerical values of cosh n, sinh u, tanh u are given in the tables at the end of this chapter for values of u between o and 4. For greater values they may be computed from the devel- opments of Art. 16. The curves exhibit graphically the relations : sech u = — - — , csch u = — — — , coth u cosh u sinh u tanh u cosh u < 1, sech u > 1, tanh u > 1, gd u < $n, etc.; sinh (— u) = — sinh it, cosh (— u) = cosh u, tanh (— u) = — tanh u, gd (— u) = — gd u, etc.; cosh 0=1, sinh = 0, tanh = 0, csch (o) =00 , etc.; cosh (± 00 ) = 00 , sinh (± 00 ) = ±00 , tanh ( ± 00 ) = ± 1, etc. The slope of the curve jy = sinh x is given by the equation dy/dx = cosh x, showing that it is always positive, and that the curve becomes more nearly vertical as x becomes infinite. Its direction of curvature is obtained from d^y/dx* = sinh x, proving that the curve is concave downward when x is nega- tive, and upward when x is positive. The point of inflexion is at the origin, and the inflexional tangent bisects the angle between the axes. 134 HYPERBOLIC FUNCTIONS. [Chap. IV. \ The direction of curvature of the locus y = sech x is given by d'y/dx* = sech x{2 tanh 2 ;r — i), and thus the curve is con- cave downwards or upwards according as 2 tanh" x — i is negative or positive. The in- flexions occur at the points x = ± tanh- 1 . 707, = ± .881, y — .707 ; and the slopes of the inflexional tangents are ± 1/2. The curve y = csch x is asymptotic to both axes, but approaches the axis of x more rapidly than it approaches the axis of y, for when ^r = 3, jj/ is C only . 1, but it is not till y = 10 that x is so small as .1. The curves y = csch x, y = sinh x cross at the points x = ± .881, y = ± 1. -1- Prob. 66. Find the direction of curvature, the inflexional tan- gent, and the asymptotes of the curves^ = gdx,y = tanh x. Prob. 67. Show that there is no inflexion-point on the curves y = cosh x, y = coth x. Prob. 68. Show that any line y = mx -\- n meets the curve y = tanh x in either three real points or one. Hence prove that the equation tanh x = mx -\- n has either three real roots or one. From the figure give an approximate solution of the equation tanh x = x — 1. Art. 26.] elementary integrals. 135 Prob. 69. Solve the equations: cosh x = x + 2; sinh x = i x - gd x = x — \n. ' Prob. 70. Show which of the graphs represent even functions, and which of them represent odd ones. Art. 26. Elementary Integrals. The following useful indefinite integrals follow from Arts. 14, 15- 23: Hyperbolic. Circular. 1. j sinh u du = cosh «, /'sin u du = — cos a, 2. / cosh ?< ^ = sinh it, /cos u du = sin u, 3. / tanh it du = log cosh u, /"tan « du = — log cos u, 4. / coth z< du = log sinh «, /"cot « ^« = log sin u, 5. / csch it du = log tanh - , /esc z* ^ = log tan - , = — sinh-'(csch u), = — cosh -1 (csc u), 6. / sech u du = gd it, I sec u du = gd- 1 u, r dx x r c. J 4/^-7-^ = sinh_1 v" T a a a « / a ii. — dx '< x Va 2 + x 1 « /-==^==Lsech-^,/- I , x = - cot-'-, -\- x a a dx i x — - = — sec - — Vx 1 — a 2 a a- f — dx I x P — dx 12 ' J x^7T?=a CSd *~ a'J x~Vx^ 1 ,■*■ — , = - esc- - . a a a From these fundamental integrals the following may be derived: 13- /■ dx I ax-\- b = — — sinh , — =p , a positive, ac> b ; Vax' + 2bx + c Va Vac—b' I , , ax 4- b . . ,. = — =cosh , ^positive, ac J ax'-\-2bx-\-c '"" y^_# j ax -\- b VT^a~c ax -\- b tan - vW-£ ! , « negative. - I «^r 4- 3 tanh- — , ac <.b\ ax-\- b < \Zp _ ac ■ Vb*—ac — i Vb' 2 -ac Thus, / VF—ac ■ 5 dx ax 4- b , , coth- ,— — — , ac < b\ ax 4- b > ^/^ _ ^ ; 4/3 a - #<; •coth-'(^— 2) = coth- J 2 — coth _ '3 x t —4x-{-s = tanh-'(.5)—tanh- 1 (.3333) = -S494— -3466=. 2028.* / dx :— tanh -1 ^— 2 ) =tanh-'o— tanh _1 (.5) *°-4*+3 = - -5494- (By interpreting these two integrals as areas, show graph- ically that the first is positive, and the second negative.) dx 15- J (a- (a—x) Vx—b Va—b tanh-, IX ~ b *For tanh- 1 (.5) interpolate between tanh (.54) = .4930, tanh (.56) = .5080 (see tables, pp. 162, 16$; and similarly for tanh- 1 (.3333). Art. 26.] elementary integrals. 137 or — , tan . x—b 2 x—b - \ / -7 , or coth- a / r Y b—a Va — b \ a ~ b Vb—a V b—a' Va — b \ a-b ' the real form to be taken. (Put x — b = z\ and apply 9, 10.) 16. / . = . tanh~' «/ (a— *) V^— * */£— a 2 ., IT- ox . , I b—x — 2 lb—x coth- 1 */ -7 . or — tan -'a/ 7; Vb—a the real form to be taken. 17. /V - affdx = ^(;tr 2 - a'f - -a* cosh" 1 jr By means of a reduction-formula this integral is easily made to depend on 8. It may also be obtained by transforming the expression into hyperbolic functions by the assumption x — a cosh u, when the integral takes the form a i / sinh 3 udu = — / (cosh 2u — i)du = -« 2 (sinh 2u — 2u) = ^<2 2 (sinh u cosh u — 11), which gives 17 on replacing a cosh u by x, and a sinh u by ■(x* — a 2 )*. The geometrical interpretation of the result is evident, as it expresses that the area of a rectangular-hyper- bolic segment AMP is the difference between a triangle OMP and a sector OAP. x *fdx — -xid 1 - x 2 ) h 4- -a* sin" 1 -. ' 2 x ' ' 2 b . Prob. 73. Prove / sinh u tanh u du — sinh u — gd u, I u cosh 2< coth u du — cosh u + log tanh -. Prob. 74. Integrate (* 2 + 2*+5)"V.*, (x' 2 + 2% + 5)-V#, (x 2 + 2* + 5)V*. Prob. 75. In the parabola / = 4px, if s be the length of arc measured from the vertex, and

and of the curve y/a — log sec x/a. Prob. 78. Investigate a formula of reduction for^y cosh" x dx;. also integrate by parts cosh'" 1 * dx, tanh -1 * dx, (sinh' 1 x) 2 dx; and show that the ordinary methods of reduction for / cos'"xsin"xdx can be applied to / cosh'" x sinh" x dx. Art. 27. Functions of Complex Numbers. As vector quantities are of frequent occurence in Mathe- matical Physics; and as the numerical measure of a vector in terms of a standard vector is a complex number of the form x-\-iy, in which x,y are real, and i stands for-/— 1; it becomes necessary in treating of any class of functional oper- ations to consider the meaning of these operations when per- formed on such generalized numbers.* The geometrical defini- tions of cosh 11, sinh?/, given in Art. 7, being then no longer applicable, it is necessary to assign to each of the symbols *The use of vectors in electrical theory is shown in Bedell and Crehore's Alternating Currents, Chaps, xiv-xx (first published in 1892). The advantage of introducing the complex measures of such vectors into the differential equa- tions is shown by Steinmetz, Proc. Elec. Congress, 1893; while the additional convenience of expressing the solution in hyperbolic functions of these complex numbers is exemplified by Kennelly, Proc. American Institute Electrical Engineers, April 1895. (See below, Art. 37.) Art. 27,] FUNCTIONS OF COMPLEX NUMBERS. 139' cosh (x -f- iy), sinh (x -\- iy), a suitable algebraic meaning, which should be consistent with the known algebraic values of cosher, sinh^r, and include these values as a particular case when y = o. The meanings assigned should also, if possible, be such as to permit the addition-formulas of Art. 1 1 to be made general, with all the consequences that flow from them. Such definitions are furnished by the algebraic develop- ments in Art. 16, which are convergent for all values of u, real. or complex. Thus the definitions of cosh (x -\- iy), sinh [x -\- iy) are to be cosh {x + iy) = I + ±-{x + iy)' + —(x + *»< + ..., 2 ! 4 • sinh {x + iy) — (x + iy) + ^{x + iy) 3 + . . . (52) From these series the numerical values of cosh (x -\- iy), sinh (x -j- iy) could be computed to any degree of approxima- tion, when x and y are given. In general the results will come out in the complex form* cosh (x -\- iy) =. a-\- ib, sinh (x -\- iy) = c -f- id. The other functions are defined as in Art. 7, eq. (9). Prob. 79. Prove from these definitions that, whatever u may be, cosh (— u) — cosh u, sinh (— u) = — sinh u, -7- cosh u = sinh u, —sinh u = cosh u, du du -r-=cosh mu = m 1 cosh mu, - r - 5 sinh mu = >n sinh mu.\ du du ' *It is to be borne in mind that the symbols cosh, sinh, here stand for alge- braic operators which convert one number into another; or which, in the lan- guage of vector-analysis change one vector into another, by stretching and turning. t The generalized hyperbolic functions usually present themselves in Mathe- matical Physics as the solution of the differential equation d' 2

, m, u are complex numbers, the measures of vector quantities. (See Art. 37.) 140 HYPERBOLIC FUNCTIONS. [CHAP. IV. Art. 28. Addition-Theorems for Complexes. The addition-theorems for cosh (u -{- v), etc., where u, v are complex numbers, may be derived as follows. First take u, v as real numbers, then, by Art. 1 1, cosh (u -j- v) = cosh u cosh v -j- sinh u sinh v; hence I + i-,(« + vf +. . . =(i + ±f + . . .)(i + ±f+. . .) +(«+^+...)(-+ 3 v+-) This equation is true when u, v are any real numbers. It must, then, be an algebraic identity. For, compare the terms of the rth degree in the letters u, v on each side. Those on the left are — ;(«+ ») r ; and those on the right, when collected, t ! form an rth-degree function which is numerically equal to the former for more than r values of u when v is constant, and for more than r values of v when u is constant. Hence the terms of the rth degree on each side are algebraically identical func- tions of u and v* Similarly for the terms of any other degree. Thus the equation above written is an algebraic identity, and is true for all values of u, v, whether real or complex. Then writing for each side its symbol, it follows that cosh (« -L- v) = cosh u cosh v -\- sinh u sinh v; (53) and by changing v into — v, cosh (u — v) = cosh u cosh v — sinh u sinh v. (54) In a similar manner is found sinh (u ± v) = sinh it cosh v ± cosh u sinh v. (55) In particular, for a complex argument, cosh (x ± iy) = cosh ^r cosh iy ±_ sinh ^r sinh iy, ) [ (56) sinh {x ± «» = sinh x cosh z> ± cosh x sinh zj/. ) * " If two rth-degree functions of a single variable be equal for more than r values of the variable, then they are equal for all values of the variable, and are algebraically identical." Art. 29.] functions of pure imaginaries. 141 Prob. 79. Show, by a similar process of generalization,* that if sin u, cos u, exp u \ be defined by their developments in powers of u, then, whatever u may be, sin (u -f- v) = sin u cos v -\- cos u sin v, cos (« + ») = cos it cos z> — sin u sin #, exp (u -\- v) = exp » exp v. Prob. 80. Prove that the following are identities: cosh 2 u — sinh 2 » = 1, cosh ?< -+- sinh u = exp #, cosh u — sinh « = exp ( — u), cosh u = £[exp w -f- exp ( — »)], sinh « = £[exp « — exp( — u)\ Art. 29. Functions of Pure Imaginaries. In the defining identities cosh u = 1 -) — rV -I -«* + . . ., 2! 4! ' ' sinh u = u A — -u* -I — - « b + . ... 3! 5! put for u the pure imaginary ty, then cosh iy — 1 — -jpj 2 + -•/ - . . . = cosy, (57) 3 p; -t- 5 ,, sinh z> = iy -+ ■ — ( (z» 3 -f -,(«»' + = ^--^y + ^|/-- ••] =ismy, (58) and, by division, tanh iy = z tan y. (59) * This method of generalization is sometimes called the principle of the " permanence of equivalence of forms." It is not, however, strictly speaking, a " principle," but a method; for, the validity of the generalization has to be demonstrated, for any particular form, by means of the principle of the alge- braic identity of polynomials enunciated in the preceding foot-note. (See Annals of Mathematics, Vol. 6, p. 8r.) f The symbol exp « stands for "exponential function of a," which is identi- cal with e" when « is real. 142 HYPERBOLIC FUNCTIONS. [Chap. IV. These formulas serve to interchange hyperbolic and circular functions. The hyperbolic cosine of a pure imaginary is real, and the hyperbolic sine and tangent are pure imaginaries. The following table exhibits the variation of sinh u, cosh u, .tanh ;/, exp u, as u takes a succession of pure imaginary values. u sinh u cosh u tanh u exp u O O I O I \iit ■7* ■7* I •7(i+0 iijr i o oo i i \in ■7* -•7 — i •7(i - i) in o — i o — i \in -•7* -•7 i -•7(i + ¥ n — i o oo i — i \in -■7* ■7 — i •7(i-0 2in o i o i * In this table .7 is written for f 4/2, — .707 .... Prob. 81. Prove the following identities : cosy = cosh iy = 4 [exp iy -\- exp (— ty)], siny = - sinh iy — —.[exp iy — exp (— iy)], cos jc + i sin 7 = cosh iy -f- sinh zj = exp 2^, cos jc — z sin jy = cosh iy — sinh z> = exp ( — iy), cos iy — cosh jc, sin iy = z sinh y. Prob. 82. Equating the respective real and imaginary parts on -each side of the equation cos ny + i sin ny = (cos y + i sin j<)", express cos ny in powers of cos y, sin 7 ; and hence derive the cor- responding expression for cosh nv. Prob. 83. Show that, in the identities (57) and (58), y may be replaced by a general complex, and hence that sinh (x ± iy) = ± i sin (y =p ix), Art. 30.1 functions of .v -f- !> in the form J + «T. 143 cosh (x ± iy) — cos (y ^ ix), sin (x ± iy) — ± i sinh (_v =F ix), cos (ar ± ij) = cosh (y =f /.v). Prob. 84. From the product-series for sin x derive that for sinh x : «** = *{* -*?)[* -■£?){*-?„' sinh* = 4+J)( I + ^)(x + 3 ^ Art. 30. Functions of x-\-iy in the Form X-\-iY. By the addition-formulas, cosh (x -\- iy) = cosh x cosh iy -f- sinh .*- sinh zy, sinh (;tr -|- iy) = sinh ;tr cosh y/ -f- cosh ;r sinh zy, but cosh y/ = cos y, sinh (y = i sin jj/, hence cosh (x -\- iy) = cosh x cos _y -f- z sinh x sin j, , (60) sinh (x -\- iy) = sinh x cos y -\-i cosh ^ sin _y. Thus if cosh (x -\- iy) = a -f- id, sinh (x -\- iy) = c -\- id, then a = cosh .# cos ^, # = sinh ;r sin j, (61) c = sinh ;tr cos y, d = cosh ;tr sin y. From these expressions the complex tables at the end of this chapter have been computed. Writing cosh z = Z, where z = x -\- iy, Z = X-\- iY; let the complex numbers z, Z be represented on Argand diagrams, in the usual way, by the points whose coordinates are (x, y), (X, Y); and let the point z move parallel to the jj/-axis, on a given line x = m, then the point Z will describe an ellipse whose equation, obtained by eliminating y between the equa- tions X = cosh ;// cos_y, F= sinh m sin_y, is + (cosh my (sinh nif and which, as the- parameter m varies, represents a series of confocal c!''^s.es, the distance between whose foci is unity. 144 HYPERBOLIC FUNCTIONS. [CHAP. IV. Similarly, if the point z move parallel to the jr-axis, on a given, line y = n, the point Z will describe an hyperbola whose equa- tion, obtained by eliminating the variable x from the equations X= cosh x cos n, Y = sinh x sin n, is X 2 Y 2 (cos w) 2 (sin «)" and which, as the parameter n varies, represents a series of hyperbolas confocal with the former series of ellipses. These two systems of curves, when accurately drawn at close intervals on the Z plane, constitute a chart of the hyper- bolic cosine ; and the numerical value of cosh (in -f- in) can be read off at the intersection of the ellipse whose parameter is m with the hyperbola whose parameter is «.* Prob. 85. Prove that, in the case of sinh (x -f- iy), the above two systems of curves are each turned through a right angle. Compare the chart of sin (x -\- iy), and also of cos (x + iy). ■„ , „, ^ ,••, • /,-\ sinh 2X + i sin 2V Prob. 86. Prove the identity tan (x 4- iy) = = ■ J . COSH 2X + cos 2JC Prob. 87. If cosh (x -)- iy), = a -\- id, be written in the " modulus and amplitude" form as r(cos -\- i sin 0), = r exp it), then r = a 2 + ?>* — cosh 2 x — sin" y = cos 2 y — sinh 2 x, tan = b/a = tanh x tan y. Prob. 88. Find the modulus and amplitude of sinh (x -\- iy),. sin (x + iy), exp (x -f- ry). Prob. 89. The functions sinh u, cosh u have the pure imaginary period 2/77-; that is, sinh (u -\- 2in) — sinh u, cosh (a + 2in) — cosh w; also sinh (u-\-$i7z) —i cosh «, cosh (u -\- \in) = i sinh u, sinh (k-}-^) = — sinh u, cosh (u + «'t) = — cosh #. Prob. 90. The functions cosh.~'m, sinh -1 OT have multiple values at intervals of 2in, but each has a unique value (called the principal value) in which the coefficient of / lies between o and n for the former, and between — \rt and + i 71 for the latter. * Such a chart is given by Kennelly, Proc. A. I. E. E., April 1895, and is used by him to obtain the numerical values of cosh (x-\-iy), sinh (x-\-iy), which present themselves as the measures of certain vector quantities in the theory of alternating currents. (See Art. 37.) The chart is constructed for values of x and of y between o and 1.2; but it is available for all values oiy, on account of the periodicity of the functions. Art. 31.] the catenary. 145 Art. 31. The Catenary. A flexible inextensible string is suspended from two fixed points, and takes up a position of equilibrium under the action of gravity. It is required to find the equation of the curve in which it hangs. Let w be the weight of unit length, and j the length of arc AP measured from the lowest point A ; then ws is the weight of the portion AP. This is balanced by the terminal tensions, T acting in the tangent line at P, and //in the horizontal tangent. Resolving horizontally and vertically gives T'cos = H, T sin cp = ws, in which is the inclination of the tangent at P; hence ws s where c is written for H/w, the length whose weight is the constant horizontal tension ; therefore dy s ds is* dx ds s ds I s* dx c' dx y ' c* ' c \/ s l _j_ ^ x s . , x s dy y x — = sinh -1 — , sinh — = — = -=—, — = cosh — , c c c c dx c c which is the required equation of the catenary, referred to an axis of x drawn at a distance c below A. The following trigonometric method illustrates the use of the gudermanian : The " intrinsic equation," s = c tan 0, gives ds = c sec 3 d4> ; hence dx, = ds cos /"(1.80) = 1. 1804 — 1.2000 = — .0196, whence, by interpolation, z = 1.7698 and c = 8.4755. Again, y/c = logs sec x/c ; but x/c = 10/c = 1-1799 ! an d I#I 799 radians = 67° 36' 29"; hence y = 8.4755 X .41914 X 2.3026 = 8.1798, the required depth.) Prob. 96. Find the inclination of the terminal tangent. Prob. 97. Show that the curve has two vertical asymptotes. Prob. 98. Prove that the law of the tension T, and of the section w, at a distance s, measured from the lowest point along the curve, is T 00 , s ]y = ^ = cosh 7 ; and show that in the above numerical example the terminal section is 2.85 times the minimum section. Prob. 99. Prove that the radius of curvature is given by p = c cosh s/c. Also that the weight of the arc ^ is given by W = H sinh s/c, in which s is measured from the vertex. Art. 33. The Elastic Catenary. An elastic string of uniform section and density in its natu- ral state is suspended from two points. Find its equation of equilibrium. Let the element da stretch into ds ; then, by Hooke's law, ds = da(i + IT), where A is the elastic constant of the string ; hence the weight of the stretched element ds, = gpwda, — gpoods/{\ + XT). Accordingly, as before, d{T sm ) =gpwds/{\ + XT), and T cos

, — gd - >, ; hence gd" 'y„ = gd ~ l y 1 -j- (.*■, — Jr,) tan a, from which the final latitude can be found when the initial latitude and the differ- ence of longitude are given. The distance sailed is equal to (ji — ^1) csc a radii, a radius being 60 X i8o/;r nautical miles. Mercator's Chart. — In this projection the meridians are parallel straight lines, and the loxodrome becomes the straight line y' = x tan a, hence the relations between the coordinates of corresponding points on the plane and sphere are x' = x, y' = gd~'f- Thus the latitude y is magnified into gd~ 'y, which is tabulated under the name of " meridional part for latitude y " ; the values of y and of y' being given in minutes. A chart constructed accurately from the tables can be used to furnish graphical solutions of problems like the one proposed above. Prob. 103. Find the distance on a rhumb line between the points (30 N, 20° E) and (30° S, 40° E). Art. 36. Combined Flexure and Tension. A beam that is built-in at one end carries a load P at the other, and is also subjected to a horizontal tensile force Q ap- plied at the same point; to find the equation of the curve assumed by its neutral surface : Let x, y be any point of the 152 HYPERBOLIC FUNCTIONS. [CHAP. IV. elastic curve, referred to the free end as origin, then the bend- ing moment for this point is Qy — Px. Hence, with the usual notation of the theory of flexure,* EI^~ = Qy — Px, -4 = n 2 (y — tax), dx 1 ax P , Q m =Q' n =£/ which, on putting y — mx = it, ax\dd*y/dx* =d*u/dx', becomes d'u dx* = nU > whence u = A cosh nx -f- B sinh nx, that is, y = mx -j- A cosh nx -\- B sinh nx. The arbitrary constants A, B are to be determined by the terminal conditions. At the free end ;r = o, j/=o; hence A must be zero, and y = mx -f- B sinh nx, -£- = m 4- 72.5 cosh «# ; dx but at the fixed end, x = I, and dy/dx = o, hence B = — m/n cosh «/, and accordingly *# sinh «;tr y = mx j T . n cosh nl To obtain the deflection of the loaded end, find the ordinate of the fixed end by putting x = /, giving deflection = mil tanh nl\ x n ' Prob. 104. Compute the deflection of a cast-iron beam, 2X2 inches section, and 6 feet span, built-in at one end and carrying a load of 100 pounds at the other end, the beam being subjected to a horizontal tension of 8000 pounds. [In this case 7=4/3, E = 15 X io", Q = 8000, P = 100 ; hence n = 1/50, m = 1/80, deflection = ■ 8 ' Tr (72 — 50 tanh 1.44) = -5^(72 — 44.69) = .341 inches.] * Merriman, Mechanics of Materials (New York, 1895), pp. 70-77, 267-269. Art. 37.] alternating currents. 153 Prob. 105. If the load be uniformly distributed over the beam, say w per linear unit, prove that the differential equation is and that the solution is y =A cosh nx-{- B sinh nx-\- mx' -| r . n Show also how to determine the arbitrary constants. Art. 37. Alternating Currents.* In the general problem treated the cable or wire is regarded as having resistance, distributed capacity, self-induction, and leakage ; although some of these may be zero in special cases. The line will also be considered to feed into a receiver circuit of any description ; and the general solution will in- clude the particular cases in which the receiving end is either grounded or insulated. The electromotive force may, without loss of generality, be taken as a simple harmonic function of the time, because any periodic function can be expressed in a Fourier series of simple harmonics.-}- The E.M.F. and the current, which may differ in phase by any angle, will be supposed to have given values at the terminals of the receiver circuit ; and the problem then is to determine the E.M.F. and current that must be kept up at the generator terminals ; and also to express the values of these quantities at any inter- mediate point, distant x from the receiving end ; the four line-constants being supposed known, viz.: R = resistance, in ohms per mile, L = coefficient of self-induction, in henrys per mile, C '= capacity, in farads per mile, G = coefficient of leakage, in mhos per mile.:): It is shown in standard works § that if any simple harmonic * See references in foot-note Art. 27. f Chapter V, Art. 8. % Kennelly denotes these constants by r, I, c, g. Steinmetz writes s for wL, K for aoC, 6 for G, and he uses Cfor current. § Thomson and Tait, Natural Philosophy, Vol, I. p. 40; Rayleigh, Theory of Sound, Vol. I. p. 20; Bedell and Crehore, Alternating Currents, p. 214. 154 HYPERBOLIC FUNCTIONS. [Chap. IV. function a sin (oot -(- 0) be represented by a vector of length a and angle 0, then two simple harmonics of the same period 27T/C0, but having different values of the phase-angle 0, can be combined by adding their representative vectors. Now the E.M.F. and the current at any point of the circuit, distant x from the receiving end, are of the form e = e, sin (cot -\- 0), i = i, sin (oat -\- 0'), (64) in which the maximum values e lf i lt and the phase-angles 0, 6', are all functions of x. These simple harmonics will be repre- sented by the vectors eJ0, iJ0' ; whose numerical measures are the complexes e, (cos d -\-j sin If)*, z, (cos 0' -\-j sin 0'), which will be denoted by e, i. The relations between e and I may be obtained from the ordinary equations f di „ „de de „ . , di , . , dx-= Ge + c di< dx = ** + L dr < 6 5> for, since de/dt = aoe, cos (aot -j- 0) = we, sin (oat -\~ -\- ^n), then de/dt will be represented by the vector <» ~d? = m ' l > (69) the solutions of which are t e = A cosh mx -\- B sinh mx, 1 = A' cosh mx -f- 5' sinh ;«#, wherein only two of the four constants are arbitrary ; for sub- stituting in either of the equations (66), and equating coeffi- cients, give (G -\-jaoC) A = mB', (G + jooC)B = tnA', whence B' = A/m v A' =. B/m v Next let the assigned terminal values of e, 1, at the receiver, be denoted by E, I; then puttings = o gives E = A, I = A', whence B = mj, B' = E/m 1 ; and thus the general solution is e = E cosh mx -\- mj sinh mx, 1 _ 1 = I cosh ;«x -f- ~E sinh wjt. (70) * The complex constants m, m,, are written z, y by Kennelly; and the variable length x is written Z a . Steinmetz writes v for m. t See Art. 14, Probs. 28-30; and Art. 27, foot-note. 156 HYPERBOLIC FUNCTIONS. [CHAP. IV. If desired, these expressions could be thrown into the ordi- nary complex form X -\- j'Y, X' -\-jY', by putting for the let- ters their complex values, and applying the addition-theorems for the hyperbolic sine and cosine. The quantities X, Y, X', Y' would then be expressed as functions of x ; and the repre sentative vectors of e, i, would be e 1 /d,i l /&', where e' = X'-\-y, i," = X n + Y'\ tan d = Y/X, tan^ 7 =~Y , /JT. For purposes of numerical computation, however, the for- mulas (70) are the most convenient, when either a chart,* or a table, f of cosh u, sinh u, is available, for complex values of u. Prob. 106. J Given the four line-constants: R = 2 ohms per mile E = 20 millihenrys per mile, C = 1/2 microfarad per mile, c? = o; and given w, the angular velocity of E.M.F. to be 2000 radians per second; then oaL = 40 ohms, conductor reactance per mile; R -f- jooL = 2 -f- 407 ohms, conductor impedance per mile; c»C = .001 mho, dielectric susceptance per mile; C + fcoC = .001;' mho, dielectric admittance per mile; (G -\-j'ooC)' 1 = — 1 000/ ohms, dielectric impedance per mile; rri 1 = {R+jooZ)(G +J00C) -— .04 + .002;', which is the measure of .04005 /177 1 ' 8'; therefore m = measure of .2001 /88° 34' = .0050 -f- .2000/, an ab- stract coefficient per mile, of dimensions [length] "', mm^ = m/(G -f- joaC) = 200 — 57' ohms per mile. Next let the assigned terminals conditions at the receiver be: 7=o (line insulated); and E =1000 volts, whose phase may betaken as the standard (or zero) phase; then at any distance x, by (70), E e = E cosh tnx, 1 = — sinh mx, in which mx is an abstract complex. Suppose it is required to find the E.M.F. and current that must be kept up at a generator 100 miles away; then * Art. 30, foot-note. ■(■ See Table II. X The data for this example are taken from Kennelly's article. Art. 37.] alternating currents. 157' e = iooo cosh (.5 + 207), 1 = 200(40 — j)~' sinh (.5 + 20/), but, by Prob. 89, cosh (.5 + 20/) = cosh (.5 + 20/ — 6nj) = COSh (.5 + 1. 15;) = .4600 + .4750/ obtained from Table II, by interpolation between cosh (.5 + 1.17')' and cosh (.5 + 1.27); hence e — 460 + 47s/ = ^(cos +7 sin 0), where log tan = log 475 — log 460 = .0139, = 45° 55', and e l = 460 sec 6 — 625.9 volts, tl]e required E.M.F. Similarly sinh (.5 + 20/) = sinh (.5 -|- 1.157) = .2126 + 1.02807V and hence i = —■ — (100 +;')(.2i26 + 1.028/) = — - — (4046 + 2060;) 1601 1601 JJ ~ = / 1 (cos 0' +/ sin 0'), where log tan 0' = 9.70684, 0'= 26 59', ^ = 4046 sec #'/i6oi = 2.77: amperes, the phase and magnitude of required current. Next let it be required to find e at x = 8; then by subtracting $nj, and applying Prob. 89. Interpolation be- tween sinh (o + 07) and sinh (o + . 17) gives sinh (o -f- .037') = 00000 -|- .029957. Similarly sinh (.1 -f -037) = .10004 + • 3°o4/- Interpolation between the last two gives sinh (.04 + ^Zj) = .°4°°2 + .029997. Hence e = 7(40.02 +29.997')= — 29.99+40.027' =^(003 0+7 sin 0), where log tan 6 = .12530, 6 = 126° 51',?, = — 29.99 sec 12 &° S 1 ' = 5 - 01 volts. Again, let it be required to find e at x = 16; here e = 1000 cosh (.08 + 3.27) = — 1000 cosh (.08 + .067), but cosh (o + .067) = .9970 + 07, cosh (. 1 + .067) = 1.0020 + .0067; hence cosh (.08 + .067) = 1.0010 +.00487, and e— — 1001+4.87 = ^(cos #+7 sin 0), where 8 — 180° 17', e l = 1001 volts. Thus at a distance of about 16 miles the E.M.F. is the same as at the receiver, but in opposite 158 HYPERBOLIC FUNCTIONS. [CHAP. IV. phase. Since e is proportional to cosh (.005 + .2j)x, the value of x for which the phase is exactly 180 is n / '.2 = 15.7. Similarly the phase of the E.M.F. at x = 7.85 is 90°. There is agreement in phase at any two points whose distance apart is 31.4 miles. In conclusion take the more general terminal conditions in which the line feeds into a receiver circuit, and suppose the current is to be kept at 50 amperes, in a phase 40° in advance of the elec- tromotive force; then /— 5o(cos 40 +/ sin 40 ) = 38.30 + 32.14/; and substituting the constants in (70) gives i 7 — 1000 cosh (.005 + ,y')x + (7821 + 6236/') sinh (.005 + .2j)x = 460+475/— 4748+9366/'=— 4288+9841/=.?, (cos #+/sin 0), where = 113 33', e, = 10730 volts, the E.M.F. at sending end. This is 17 times what was required when the other end was insulated. Prob. 107. If the receiving end be grounded, that is if £ = o; and if a current of 10 amperes be caused to flow to ground; find the E.M.F. and current to be kept up at the generator. Also compute these quantities, and their phases, at the distances 7.85, 15.7, 31.42, 94.25 miles from the receiver. Prob. 108. If self-induction and capacity be zero, and the receiving end be insulated, show that the graph of the electromo- tive force is a catenary. Prob. 109. Neglecting leakage and capacity, prove that the solution of equations (66) is 1 = I, e = E + (R + jooL)Ix. Prob. no. If x be measured from the sending end, show how equations (65), (66) are to be modified; and prove that _ 1 _ e = E cosh mx — mj a sinh mx, 1 = I cosh mx — ~E a sinh mx, where E„ /„ refer to the sending end. Art. 38. Miscellaneous Applications. 1. The length of the arc of the logarithmic curve y = a" is j= i(cosh u-\- log tanh \y), in which M= i/log a, sinh u =y/M. 2. The length of arc of the spiral of Archimedes r = ad is j = £«(sinh 2u +- 2k), where sinh u = 6. 3. In the hyperbola x'/a 1 —y'/f — 1 the radius of curva- ture is p—{a' sinh" u-\-F cosh 2 iifi/ab; in which u is the measure of the sector AOP, i.e. cosh u = x/a, sinh u =y/b. 4. In an oblate spheroid, the superficial area of the zone ART. 38.] MISCELLANEOUS APPLICATIONS. 159 between the equator and a parallel plane at a distance y is S = 7r£ J (sinh 211 -\- 2u)/2e, wherein b is the axial radius, e the eccentricity, u = ey/p, and/ parameter of generating ellipse. 5. The length of the arc of the parabola y = 2px, measured from the vertex of the curve, is / = ^/(sinh 2u-\-2ii), in which sinh u =y/p =tan 0, where

• 1987 0.9850 01990 09817 1997 .3 0.9553 " 2955 0.9601 02960 09570 2970 .4 .9211 « .3894 .9257 .03901 .09226 .3914 .5 8776 tt 4794 8820 04802 08791 4818 .6 8253 '• 5646 8295 05656 08267 5675 .7 7648 " 6442 7687 06453 07661 6474 .8 .6967 <« .7174 .7002 .07186 .06979 .7200 .9 6216 " 7833 6247 07847 06227 7872 1.0 5403 " 8415 5430 08429 05412 8457 1.1 4536 " 8912 4559 08927 04544 8957 1.2 .3624 (f .9320 .3642 .09336 .03630 9367 1.3 2675 " 9636 2688 09652 02680 0.9684 1.4 1700 tt 9854 1708 09871 01703 0.9904 1.5 0707 " 9975 0711 09992 00709 1.0025 i* 0000 " tt 1.0000 0000 10017 00000 1.0050 y x ■= • 4 r = •5 a b c d U * c d 1.0811 .0000 .4108 .0000 1.1276 .0000 .5211 .0000 .1 1.0756 0410 4087 1079 1 . 1220 0520 5185 1126 .2 1.0595 0816 4026 2148 1.1051 1025 5107 2240 .3 1.0328 1214 3924 3195 1.0773 1540 4978 3332 .4 .9957 .1600 .3783 .4210 1.0386 .2029 .4800 .4391 .5 9487 1969 3605 5183 0.9896 2498 4573 5406 .6 8922 2319 3390 6104 0.9306 2942 4301 6367 .7 8268 2646 3142 6964 0.8624 3357 3986 7264 .8 .7532 .2947 .2862 .7755 .7856 .3738 .3631 0.8089 .9 6720 3218 2553 8468 7009 4082 3239 0.8833 1.0 5841 3456 2219 9097 6093 4385 2815 0.9489 1.1 4904 3661 1863 9635 5115 4644 2364 1.0050 1.2 .3917 .3829 .1488 1.0076 .4086 .4857 .1888 1.0510 1.3 2892 3958 1099 1.0417 3016 5021 1394 1.0865 1.4 1838 4048 0698 1.0653 1917 5135 0886 1.1163 1.5 0765 4097 0291 1.0784 0798 5198 0369 1.1248 \Tt 0000 4108 0000 1.0811 0000 5211 0000 1.1276 Art. 39.] tables. Table II. Values of cosh (x + iy) and sinh (x -f iy). 165 . = .2. X = •3 a b C d a * c d y 1.0201 .0000 .2013 .0000 1.0453 .0000 .3045 .0000 1.0150 0201 2003 1018 1.0401 0304 3030 1044 .i 0.9997 0400 1973 2027 1.0245 0605 2985 2077 .2 0.9745 0595 1923 3014 9987 0900 2909 3089 .3 .9395 .0784 .1854 .3972 .9628 .1186 .2805 .4071 .4 8952 0965 1767 4890 9174 1460 2672 5012 .5 8419 1137 1662 5760 8627 1719 2513 5903 .6 7802 1297 1540 6571 7995 1962 2329 6734 .7 .7107 .1444 .1403 .7318 .7283 .2184 .2122 .7498 .8 6341 1577 1252 7990 6498 2385 1893 8188 .9 5511 1694 1088 8584 5648 2562 1645 8796 1.0 4627 1795 0913 9091 4742 2714 1381 9316 1.1 .3696 .1877 .0730 0.9507 .3788 .2838 .1103 0.9743 1.2 2729 1940 0539 0.9829 2796 2934 0815 1.0072 1.3 1734 1984 0342 1.0052 1777 3001 0518 1.0301 1.4 0722 2008 0142 1.0175 0739 3038 0215 1.0427 1.5 0000 2013 0000 1.0201 0000 3045 0000 1.0453 in x = .6 X = • 7 a b c d a b c d' y 1.1855 .0000 .6367 .0000 1.2552 .0000 .7586 .0000 1.1795 0636 6335 1183 1.2489 0757 7548 1253 .1 1.1618 1265 6240 2355 1.2301 1542 7435 2494 .2 1.1325 1881 6082 3503 1.1991 2242 7247 3709 !3 1.0918 .2479 .5864 .4617 1.1561 .2954 .6987 .4888 .4 1.0403 3052 5587 5684 1.1015 3637 6657 6018 .5 0.9784 3595 5255 6694 1.0359 4253 6261 7087 .6 0.9067 4101 4869 7637 0.9600 4887 5802 8086 . t .8259 .4567 .4436 0.8504 .8745 .5442 .5285 0.9004 .8 7369 4987 3957 0.9286 7802 5942 4715 0.9832 .9 6405 5357 3440 0.9975 6782 6383 4099 1.0562 1.0 5377 5674 3888 1.0565 5693 6760 3441 1.1186 1.1 .4296 5934 .2307 1.1049 .4548 .7070 .2749 1.1699 1.2 3171 6135 1703 1.1422 3358 7309 2029 1.2094 1.3 2015 R274 1082 1.1 68 J 2133 7475 1289 1 . 2369 1.4 0839 6351 0450 1.18. '5 0888 7567 0537 1.2520 1.5 0000 6367 0000 1.1855 0000 7586 0000 1.2552 i« 16C HYPERBOLIC FUNCTIONS. [CHAP. IV. Table II. Values of cosh (x + iy) and sinh {x + iy). y x — .8 X = •9 a b c d a b c d .1 .2 1.3374 1.3308 1.3108 .0000 0887 1764 .8881 8837 8704 .0000 1335 2657 1.4331 1.4259 1.4045 .0000 1025 2039 1.0265 1.0214 1.0061 .0000 1431 2847 .3 1.2776 2625 8484 3952 1.3691 3034 0.9807 4235 .4 .5 .6 .7 1.2319 1.1737 1.1038 1.0229 .3458 4258 5015 5721 .8180 7794 7330 6793 .5208 6412 7552 8616 1.3200 1.2577 1.1828 1.0961 .3997 4921 5796 6613 .9455 9008 8472 7851 .5581 6871 8092 9232 .8 .9318 .6371 .6188 0.9595 .9984 .7364 .7152 1.0280 .9 1.0 1.1 8314 7226 6067 6957 7472 7915 5521 4798 4028 1.0476 1.1254 1.1919 8908 7743 6500 8041 8638 9148 6381 5546 4656 1.1226 1.2059 1.2772 1.2 1.3 1.4 1.5 .4846 3578 2273 0946 .8278 8557 8752 8859 .3218 2376 1510 0628 1.2465 1.2887 1.3180 1.3341 .5193 3834 2436 1014 0.9568 0.9891 1.0124 1.0239 .3720 2746 1745 0726 1.3357 1.3809 1.4122 1.4295 i* 0000 .8881 0000 1.3374 0000 1.0265 0000 1.4331 X = 1.2 X = i-3 y a b C d a b c d .1 .2 .3 1.8107 1.8016 1.7746 1.7298 .0000 1507 2999 4461 1.5095 1.5019 1.4794 1.4420 .0000 1808 3598 5351 1.9709 1.9611 1.9316 1.8829 .0000 1696 3374 5019 1.6984 1.6899 1.6645 1.6225 .0000 1968 3916 5824 .4 .5 .6 .7 1.6677 1.5890 1.4944 1.3849 .5878 7237 8523 9724 1.3903 1.3247 1.2458 1.1545 0.7051 0.8681 1.0224 1.1665 1.8153 1.7296 1.6267 1.5074 .6614 8142 9590 1.0941 1.5643 1.4905 1.4017 1.2990 0.7675 0.9449 1.1131 1.2697 .8 .9 1.0 1.1 1.2615 1.1255 0.9783 0.8213 1.0828 1.1824 1.2702 1.3452 1.0517 0.9383 0.8156 0.6847 1.2989 1.4183 1.5236 1.6137 1.3731 1.2251 1.0649 0.8940 1.2183 1.3304 1.4291 1.5136 1.1833 1.0557 0.9176 0.7704 1.4139 1.5439 1.6585 1.7565 1.2 1.3 1.4 1.5 .6561 4844 3078 1281 1.4069 1.4544 1.4875 1.5057 .5470 4038 2566 1068 1.6876 1.7447 1.7843 1.8061 .7142 5272 3350 1394 1.5830 1.6365 1.6737 1.6941 .6154 4543 2887 1201 1.8370 1 8991 1.9422 1.9660 \n 0000 1.5095 0000 1.8107 0000 1.6984 0000 1.9709 Art. 39.] tables. Table II. Values of cosh (x -j- iy) and sinh (x -f- iy.) 16/ X =: I.O x = i.i a b c d a b c d y 1.5431 .0000 1.1752 .0000 1.6685 .0000 1.3356 .0000 1.5354 1173 1.1693 1541 1.6602 1333 1 3290 1666 .1 1.5123 2335 1.1518 3066 1.6353 2654 1.3090 3315 .2 1.4742 3478 1.1227 4560 1.5940 3946 1.2760 4931 .3 1.4213 .4576 1.0824 .6009 1. 53U8 .5201 1.2302 6498 .4 1.3542 5634 1.0314 7398 1.4643 6403 1.1721 0.7999 .5 1.2736 6636 0.9699 8718 1.3771 7542 1.1024 0.9421 .6 1 . 1802 7571 0.8988 9941 1.2762 8604 1.0216 1.0749 .7 1.0751 0.8430 .8188 1 1069 1.1625 0.9581 .9306 1 1969 .8 0.9592 0.9206 7305 1.2087 1.0372 1.0462 S302 1.3070 .9 0.8337 0.9889 6350 1.2985 0.9015 1.1239 7217 1.4040 1.0 0.6999 1.0473 5331 1.3752 0.7568 1.1903 6058 1.4870 1.1 .5592 1.0953 .4258 1.4382 .6046 1.2449 .4840 1.5551 1.2 4128 1.1324 3144 1.4869 4463 1.2870 3573 1.0077 1.3 2623 1.1581 1998 1.5213 2836 1.3162 2270 1.6442 1.4 1092 1.1723 0831 1.5392 1180 1.3323 0945 1.6643 1.5 0000 1.1752 0000 1 5431 0000 1.3356 0000 1.6685 i* x = 1-4 X = i-5- a b c d a b c d y 2.1509 .0000 1.9043 .0000 2.3524 .0000 2.1293 .0000 2.1401 1901 1.8948 2147 2 3413 2126 2.1187 2348 .1 2.1080 3783 1.8663 4273 2.3055 4230 2.0868 4674 .2 2.0548 5628 1.8192 6356 2 2473 6293 2.0342 6951 .3 1.9811 0.7416 1.7540 0.8376 2.1667 0.8292 1.9612 0.9101 .4 1.8876 0.9130 1.6712 1.0312 2.0644 1.0208 1.8686 1.1278 .5 1.7752 1.0753 1.5713 1.2145 1.9415 1.2023 1.7574 1.3283 .6 1.6451 1.2268 1.4565 1.3856 1.7992 1.3717 1.6286 1.5155 .7 1.4985 1.3661 1.3268 1.5430 1.6389 1.5275 1.4835 1.6875 .8 1.3370 1.4917 1.1838 1.6849 1.4623 1.6679 1.3236 1.8427 .9 1.1622 1.6024 1.0289 1.8099 1.2710 1.7917 1.1505 1.9795 1.0 0.9756 1.6971 0.8638 1.9168 1.0671 1.8976 0.9659 2.0965 1.1 .7794 1.7749 .6900 2.0047 .8524 1.9846 .7716 2.1925 1.2 5754 1.8349 5094 2.0725 6293 2.0517 5696 2.2667 1.3 3656 1.8766 3237 2.1196 3998 2.0983 3619 2.3182 1.4 1522 1.S996 1347 2.1455 1664 2.1239 1506 2.3465 1.5 .0000 1.9043 0000 2.1509 .0000 2.1293 .0000 2.3524 \n 168 HYPERBOLIC FUNCTIONS. Table III. [Chap. IV. » gd « 6° u gd u e° « gd » S° 00 .0000 0.000 .60 .5669 32.483 1.50 1.1317 64.843 .02 0200 1.146 .62 5837 33.444 1.55 1.1525 66.034 .04 0400 2.291 .64 6003 34.395 1.60 1.1724 67.171 .06 0600 3.436 .66 6167 35 336 1.65 1.1913 68.257 .08 0799 0.579 .68 6329 36.265 1.70 1.2094 69.294 .10 .0998 5.720 .70 .6489 37.183 1.75 1.2267 70.284 .12 1197 6.859 .72 6648 38.091 1.80 1.2432 71.228 .14 1395 7.995 .74 6804 38.987 1.85 1.2589 72.128 .16 1593 9.128 .76 6958 39.872 1.90 1.2739 72.987 .18 1790 .1987 10.258 11.384 .78 .80 7111 .7261 40.746 41.608 1.95 1.2881 1.3017 73,805 .20 2.00 74.584 .23 2183 13.505 .82 7410 42.460 2.10 1.3271 76.037 .24 2377 13.621 .84 7557 43.299 2.20 1.3501 77.354 .26 2571 14.732 .86 7702 44.128 2.30 1.3710 78.549 .28 2764 15.837 .88 7844 44.944 2.40 1.3899 79.633 .30 .2956 16.937 .90 .7985 45.750 2.50 1.4070 80.615 .32 3147 18.030 .92 8123 46.544 2 60 1.4227 81.513 .34 3336 19.116 .94 8260 47.326 2.70 1.4366 82.310 .36 3525 20.195 .96 8394 48.097 2.80 1.4493 83.040 .38 3712 .3897 21.267 22.331 .98 8528 .8658 48.857 49.605 2.90 3 00 1.4609 1.4713 83.707 .40 1.00 84.301 .42 4083 23.386 1.05 8976 51.428 3.10 1.4808 84.841 .44 4264 24.434 1.10 9281 53.178 3.20 1.4894 85.336 .46 4446 25.473 1.15 9575 54 860 3.30 1.4971 85.775 .48 4626 26.503 1.20 9857 56.476 3.40 1.5041 ^6.177 .50 .4804 27.524 1.25 1.0127 58.026 3.50 1.5104 86.541 .52 4980 28.535 1.30 1.0387 59.511 3.60 1.5162 86.870 .54 5155 29.537 1.35 1.0635 60.933 3.70 1.5214 87 168 .56 5328 30.529 1.40 1.0873 62.295 3.80 1.5261 87.437 .58 5500 31.511 1.45 1. 1100 63.598 3.90 1.5303 87.681 Table IV. u gd u log sinh u log cosh « u gd u log sinh u log cosh M 4.0 1.5342 1.4360 1.4363 5.5 1.5626 2.08758 2.08760 4.1 1.5377 1.4795 1.4797 5.6 1.5634 2.13101 2.13103 4.2 1.5408 1.5229 1.5231 5.7 1.5641 2.17444 2.17445 4.3 1.5437 1.5664 1.5665 5.8 1.5648 2.21787 2.21788 4.4 1.5462 1.5486 1.6098 1.6532 1.6099 1.6533 5.9 1.5653 1 5658 2.26130 2.30473 2.26131 4.5 60 2.30474 4.6 1.5507 1.6967 1.6968 6.2 1.5667 2.39159 2.39160 4.7 1 5526 1.7401 1.7402 6.4 1.5675 2.47*45 2.47846 4.8 1.5543 1.7836 1.7836 6.6 1.5681 2.56531 2.56531 4.9 1.5559 1 5573 1.8270 1.8704 1.8270 1.8705 6.8 1.5686 1.5690 2.65217 2.73903 2.65217 5.0 7.0 2.73903 5.1 1.5586 1.9139 1.9139 7.5 1.5697 2.95618 3.95618 5.2 1.5598 1.9573 1.9573 8.0 1.5701 3.17333 3.17333 5.3 1.5608 2.0007 2.0007 8.5 1.5704 3.39047 3.39047 5.4 1.5618 2.0442 2.0442 9.0 1.5705 3.60762 3.60762 00 1.5708 00 CO ART. l."| HISTORY AND DESCRIPTION. 169 Chapter V. HARMONIC FUNCTIONS. By William E. Byerly, Professor of Mathematics in Harvard University. Art. 1. History and Description. What is known as the Harmonic Analysis owed its origin and development to the study of concrete problems in various branches of Mathematical Physics, which however all involved the treatment of partial differential equations of the same general form. The use of Trigonometric Series was first suggested by Daniel Bemouilli in 1753 in his researches on the musical vibrations of stretched elastic strings, although Bessel's Func- tions had been already (1732) employed by him and by Euler in dealing with the vibrations of a heavy string suspended from one end; and Zonal and Spherical Harmonics were introduced by Legendre and Laplace in 1782 in dealing with the attrac- tion of solids of revolution. The analysis was greatly advanced by Fourier in 1812-1824 in his remarkable work on the Conduction of Heat, and im- portant additions have been made by Lame (1839) and by a host of modern investigators. The differential equations treated in the problems which have just been enumerated are 170 HARMONIC FUNCTIONS. [CHAP. V. for the transverse vibrations of a musical string ; for small transverse vibrations of a uniform heavy string sus- pended from one end ; tfV ??V tfV 9? + ^ r+ 9F = °' (3 > which is Laplace's equation ; and 37 = a te + 3? + ^ (4> for the conduction of heat in a homogeneous solid. Of these Laplace's equation (3), and (4) of which (3) is a special case, are by far the most important, and we shall con- cern ourselves mainly with them in this chapter. As to their interest to engineers and physicists we quote from an article in The Electrician of Jan. 26, 1894, by Professor John Perry: " There is a well-known partial differential equation, which is: the same in problems on heat-conduction, motion of fluids, the establishment of electrostatic or electromagnetic potential, certain motions of viscous fluid, certain kinds of strain and stress, currents in a conductor, vibrations of elastic solids, vibrations of flexible strings or elastic membranes, and innumerable other phenomena. The equation has always to be solved subject to certain boundary or limiting conditions, sometimes as to space and time, sometimes as to space alone, and we know that if we obtain any solution of a particular problem, then that is the true and only solution. Further- more, if a solution, say, of a heat-conduction problem is obtained by any person, that answer is at once applicable to analogous prob- lems in all the other departments of physics. Thus, if Lord Kel- vin draws for us the lines of flow in a simple vortex, he has drawn for us the lines of magnetic force about a circular current; if Lord Rayleigh calculates for us the resistance of the mouth of an organ-pipe, he has also determined the end effect of a bar of iron which is magnetized; when Mr. Oliver Heaviside shows his match- ART. 1.] HISTORY AND DESCRIPTION. 171 less skill and familiarity with Bessel's functions in solving electro- magnetic problems, he is solving problems in heat-conductivity or the strains in prismatic shafts. How difficult it is to express exactly the distribution of strain in a twisted square shaft, for example, and yet how easy it is to understand thoroughly when one knows the perfect-fluid analogy! How easy, again, it is to imagine the electric current density everywhere in a conductor when transmitting alter- nating currents when we know Mr. Heaviside's viscous-fluid analogy, or even the heat-conduction analogy! " Much has been written about the correlation of the physical sciences; but when we. observe how a young man who has worked almost altogether at heat problems suddenly shows himself ac- quainted with the most difficult investigations in other departments of physics, we may say that the true correlation of the physical sciences lies in the equation of continuity 37 ~ a \d? + dy .r(:~+ :fg)." In the Theory of the Potential Function in the Attraction of Gravitation, and in Electrostatics and Electrodynamics,* Fin Laplace's equation (3) is the value of the Potential Func- tion, at any external point (x, y, z), due to any distribution of matter or of electricity ; in the theory of the Conduction of Heat in a homogeneous solid f V is the temperature at any point in the solid after the stationary temperatures have been established, and in the theory of the irrotational flow of an incompressible fluid % V is the Velocity Potential Function and (3) is known as the equation of continuity. If we use spherical coordinates, (3) takes the form I \r^ rV) A sinfl 90 ' sin 2 d H ]= 0; (5) * See Peirce's Newtonian Potential Function. Boston, f See Fourier's Analytic Theory of Heat. London and New York, 1878 ; or Riemann's Partielle Differentialgleichungen. Brunswick. X See Lamb's Hydrodynamics. London and New York, 1895. 172 HARMONIC FUNCTIONS. [Chap. V. and if we use cylindrical coordinates, the form d*v , idV , i d'V d*v d^^r^ + ?W + d^~ (6) In the theory of the Conduction of Heat in a homogene- ous solid,* u in equation (4) is the temperature of any point (x, y, z) of the solid at any time t, and a' is a constant deter- mined by experiment and depending on the conductivity and the thermal capacity of the solid. Art. 2. Homogeneous Linear Differential Equations. The general solution of a differential equation is the equa- tion expressing the most general relation between the primi- tive variables which is consistent with the given differential equation and which does not involve differentials or derivatives. A general solution will always contain arbitrary (i.e., undeter- mined) constants or arbitrary functions. A particular solution of a differential equation is a relation between the primitive variables which is consistent with the given differential equation, but which is less general than the general solution, although included in it. Theoretically, every particular solution can be obtained from the general solution by substituting in the general solu- tion particular values for the arbitrary constants or particular functions for the arbitrary functions ; but in practice it is often easy to obtain particular solutions directly from the differential equation when it would be difficult or impossible to obtain the general solution. (a) If a problem requiring for its solution the solving of a differential equation is determinate, there must always be given in addition to the differential equation enough outside condi- tions for the determination of all the arbitrary constants or arbitrary functions that enter into the general solution of the equation ; and in dealing with such a problem, if the differen- tial equation can be readily solved the natural method of pro- Art. 2.] homogeneous linear differential equations. 173 cedure is to obtain its general solution, and then to determine the constants or functions by the aid of the given conditions. It often happens, however, that the general solution of the differential equation in question cannot be obtained, and then, since the problem, if determinate, will be solved, if by any means a solution of the equation can be found which will also> satisfy the given outside conditions, it is worth while to try to get particular solutions and so to combine them as to form a result which shall satisfy the given conditions without ceasing to satisfy the differential equation. (b) A differential equation is linear when it would be of the first degree if the dependent variable and all its derivatives were regarded as algebraic unknown quantities. If it is linear and contains no term which does not involve the dependent variable or one of its derivatives, it is said to be linear and homogeneous. All the differential equations given in Art. i are linear and 1 homogeneous. (c) If a value of the dependent variable has been found which satisfies a given homogeneous, linear, differential equa- tion, the product formed by multiplying this value by any constant will also be a value of the dependent variable which, will satisfy the equation. For if all the terms of the given equation are transposed to the first member, the substitution of the first-named value must reduce that member to zero; substituting the second value is equivalent to multiplying each term of the result of the first substitution by the same constant factor, which there- fore may be taken out as a factor of the whole first member. The remaining factor being zero, the product is zero and the equation is satisfied. (d) If several values of the dependent variable have been found each of which satisfies the given differential equation, their sum will satisfy the equation ; for if the sum of the values in question is substituted in the equation, each term of the sum 174 HARMONIC FUNCTIONS. [CHAP. V. will give rise to a set of terms which must be equal to zero, and therefore the sum of these sets must be zero. (e) It is generally possible to get by some simple device particular solutions of such differential equations as those we have collected in Art. I. The object of this chapter is to find methods of so combining these particular solutions as to satisfy any given conditions which are consistent with the nature of the problem in question. This often requires us to be able to develop any given func- tion of the variables which enter into the expression of these conditions in terms of normal forms suited to the problem with which we happen to be dealing, and suggested by the form of particular solution that we are able to obtain for the differential equation. These normal forms are frequently sines and cosines, but they are often much more complicated functions known as Legendre's Coefficients, or Zonal Harmonics ; Laplace's Coef- ficients, or Spherical Harmonics ; Bessel's Functions, or Cylin- drical Harmonics; Lame's Functions, or Ellipsoidal Har- monics; etc. Art. 3. Problem in Trigonometric Series. As an illustration let us consider the following problem : A large iron plate n centimeters thick is heated throughout to a uniform temperature of ioo degrees centigrade ; its faces are then suddenly cooled to the temperature zero and are kept at that temperature for 5 seconds. What will be the tempera- ture of a point in the middle of the plate at the end of that time? Given a 2 =0.185 m C.G.S. units. Take the origin of coordinates in one face of the plate and the axis of X perpendicular to that face, and let u be the temperature of any point in the plate t seconds after the cool- ing begins. We shall suppose the flow of heat to be directly across the plate so that at any given time all points in any plane parallel Art. 3.] problem in trigonometric series. 175 to the faces of the plate will have the same temperature. Then u depends upon a single space-coordinate x ; — = o and — = o, and (4), Art. 1, reduces to ^ = a^t. (I) Obviously, tt = ioo° when t = o, (2) ?^ = o when .r = o, (3) and u = o when .*• = n ; (4) and we need to find a solution of (1) which satisfies the con- ditions (2), (3), and (4). We shall begin by getting a particular solution of (1), and we shall use a device which always succeeds when the equa- tion is linear and homogeneous and has constant coefficients. Assume* u = e? x+yt , where /J and y are constants; substi- tute in (1) and divide through by e px+yt and we get y = # 2 /f ; and if this condition is satisfied, u = e& x+yt is a solution of (1). u = eP x + a W is then a solution of (1) no matter what the value of /3. We can modify the form of this solution with advantage. Let /? = p.i,\ then u — e'"*^'^*' is a solution of (i\ as is also u = e-'We-**'. By (d), Art. 2, u = r"* 1 " ^ 33 ~t~ e ^ = g-" v " cos fXX (5) is a solution, as is also !i>v- xi p- »- x{ \ u _ e-^nKJL £ ) _ e -aw sin MX . ( 6 ) and n is entirely arbitrary. * This assumption must be regarded as purely tentative. It must be tested by substituting in the equation, and is justified if it leads to a solution, f The letter *' will be used to represent 4/ — 1. 1'76 HARMONIC FUNCTIONS. [CHAP. V. By giving different values to jx we get different particular solutions of (i) ; let us try to so combine them as to satisfy our conditions while continuing to satisfy equation (i). u = f-"V 2 < s i n px is zero when x = o for all values of jj. ; it is zero when x = n if fx is a whole number. If, then, we write u equal to a sum of terms of the form Ae' a '" lH sin mx, where m is an integer, we shall have a solution of (i) (see (d), Art. 2) which satisfies (3) and (4). Let this solution be u = A,e - a " sin x + A,e ~ wt sin 2x + A s e " 9a2 ' sin 3* -j- ... , (7) ^j, ^ a , ^ 3 , . . . being undetermined constants. When t = O, (7) reduces to u = A, sin x -\- A % sin 2jr -|- A, sin 3* -{- . . . . (8) If now it is possible to develop unity into a series of the form (8) we have only to substitute the coefficients of that series each multiplied by 100 for A, , A,, A, . . , in (7) to have a solution satisfying (1) and all the equations of condition (2), (3), and (4). We shall prove later (see Art. 6) that n 1 • I 1 sin x -\- - sin $x -\- — sin $x -\- . . . 3 '5 for all values of x between o and n. Hence our solution is 400 u = Tt e- aH sin x + -e'^ 1 sin 3* -] e-^ aH sin %x + . (9)' To get the answer of the numerical problem we have only 71 to compute the value of u when x = — and t = 5 seconds. As there is no object in going beyond tenths of a degree, four- place tables will more than suffice, and no term of (9) beyond the first will affect the result. Since sin — = 1, we have to 2 compute the numerical value of Art. 4.] problem in zonal harmonics. 177 -e where a' = 0.185 and * = 5. log a* = 9.2672 — 10 log 400 = 2.6061 log t = 0.6990 colog re = 9.5059 — 10 log aV = 9.9662 — 10 colog e a " = 9.5982 — 10 log log e — q.6378 — 10 , 1 1 an '~~k lo § u = 1.7102 log log e a " = 9.6040 — 10 ' log e"< = 0.4018 = 5I-3- If the breadth of the plate had been c centimeters instead of n centimeters it is easy to see that we should have needed the development of unity in a series of the form . nx . 2nx . -\nx A 1 sin — + A. sin 4- A. sin - — + c ' ' c ' s c ' Prob. 1. An iron slab 50 centimeters thick is heated to the tem- perature 100 degrees Centigrade throughout. The faces are then sud- denly cooled to zero degrees, and are kept at that temperature for 10 minutes. Find the temperature of a point in the middle of the slab, and of a point 10 centimeters from a face at the end of that time. Assume that ! = *- It ( . 7tX , I . T.71X , I . Z.7ZX , \ sin h - sin \- - sin • J — + ...) from x = o to x = c. V c 3 c 5 c ' J Ans. 84°.o; 49°.4. Art. 4. Problem in Zonal Harmonics. As a second example let us consider the following problem : Two equal thin hemispherical shells of radius unity placed together to form a spherical surface are separated by a thin layer of air. A charge of statical electricity is placed upon one hemisphere and the other hemisphere is connected with the ground, the first hemisphere is then found to be at poten- tial 1, the other hemisphere being of course at potential zero. At what potential is any point in the " field of force" due to the charge? We shall use spherical coordinates and shall let Vbe the potential required. Then f^must satisfy equation (5), Art. 1. 178 HARMONIC FUNCTIONS. [CHAP. V. But since from the symmetry of the problem V is obviously independent of 0, if we take the diameter perpendicular to the plane separating the two conductors as our polar axis, — - is zero, and our equation reduces to r^rV) i 9 ( Sing U) _ e 9r 2 ~*~ sin d& Vis given on the surface of our spherej hence V = f{(f) when r = I, (2) where /(#) = 1 if o < B < -, and /(0) = o if - < 6 < n. Equation (2) and the implied conditions that V is zero "at an infinite distance and is nowhere infinite are our conditions. To find particular solutions of (1) we shall use a method which is generally effective. Assume* that V = R@ where R is a function of r but not of 0, and @ is a function of 6 but not of r. Substitute in (1) and reduce, and we get 1 rd\rR) _r__ 4 sin T6) . (3) R dr* ~ & sin 6 dO Since the first member of (3) does not contain 6 and the second does not contain r and the two members are identically equal, each must be equal to a constant. Let us call this constant, which is wholly undetermined, m(m -f- 1) ; then rd\rR) 1 d \ Sln6 dff) . , . d\rR) whence r— — m(in -\- i)R = o, (4) / df)\ 1 d { 5[nd d») and shTe — Te — + *(* + i)® = o. ( S ) * See the first foot-note on page 175. Art. 4.] problem in zonal harmonics. 179 Equation (4) can be expanded into d'R , dR and can be solved by elementary methods. Its complete solution is R = Ar m + Br""- 1 . (6) ' Equation (5) can be simplified by changing the independ- ent variable to x where x = cos 0. It becomes dV d@l dx& ~ X ^Tx\ + m{ - m + ^® = °' (7) an equation which has been much studied and which is known as Legendre's Equation. We shall restrict m, which is wholly undetermined, to posi- tive whole values, and we can then get particular solutions of (7) by the following device : Assume* that © can be expressed as a sum or a series of terms involving whole powers of x multiplied by constant coefficients. Let & = 2a n x n and substitute in (7). We get 2[n(n — i)a„x"~' J — n{n + i)a„x n + m(m + i)a n x n ] = o, (8) where the symbol 2 indicates that we are to form all the terms we can by taking successive whole numbers for n. Since (8) must be true no matter what the value of x, the coefficient of any given power of x, as for instance x k , must vanish. Hence (k + 2){k -f i)a i+i — k{k -+- i)a k -+- m(m -f i)a k — o, m{m+i)-k{k+i) and a k+7 = {k+l){k + 2) «* (9) If now any set of coefficients satisfying the relation (9) be taken, & = '2a^c k will be a solution of (7). If k = m, then a k+ , = o, a k+t — o, etc. * See the first foot-note on page 175. 180 HARMONIC FUNCTIONS. [Chap. V. Since it will answer our purpose if we pick out the simplest set of coefficients that will obey the condition (9), we can take a set including a m . Let us rewrite (9) in the form (k + 2){k+l) a k (10) (m — k)(in -f- k — 1)' We get from (10), beginning with k — m — 2, m(tn — 1) U ' n - i - ~~ 2.(2;«- 1) "" _ m{m — i){m — 2){m — 3) a "'- 1 ~ 2. 4. (2»- i)(2;«- 3) a "" wz(/« — i)(*« — 2)(#z — 3)(ot — 4)(»/ — 5) a "'~° ~ 2.4.6. {2m — i)(2m — 3)(2;« — 5) If m is even we see that the set will end with #„; if m is odd, with a,. **(«* — 1) „ '-a m , etc. © = «. 2 . (2»2 — i) ;?z(w - i)Q - 2)m — 3>) x „_, ' 2 . 4 . (2w — i)(2m — 3) -...], where a m is entirely arbitrary, is, then, a solution of (7). It is found convenient to take a m equal to (2m — l)(2« — 3) ... I ?« ! ' and it will be shown later that with this value of a m , © — 1 when x = 1. © is a function of x and contains no higher powers of x than x m . It is usual to write it as P„,(x). We proceed to write out a few values of P m (x) from the formula p i x \ _ ( 2W ~ 0( 2 «* ~ 3) • ■ • 1 ;«(;« — 1) , 2 . (2»« — 1)' m(m - i)(m - 2)(m - 3) ^ m _, _ "1 2 .4.(2^ — l)(2W— 3) '"'J (») Art. 4.] problem in zonal harmonics. 181 We have : P.{x) = i or Plcos 0) = i, Plx) = x or P^cos 0) = cos 8, PI*) = *(3* ! - i) or Picas 0) = £(3 cos 2 - 1), PJ,*) = £(5* 3 - 3*) or P,(cos 0) = £(5 cos 3 - 3 cos 0), ^W = K35* 4 - 3<>*' + 3) or ^ (l2 > P 4 (cos 0) = i(35 cos'0 - 30 cos'0 + 3), ^.(*) = 1(63^ -70^ 3 +i5^) or P„(cos 8) = £(63 cos 6 — 70 cos 3 + 15 cos 0). We have obtained & = P m {x) as a particular solution of (7), and & = P„,(cos 0) as a particular solution of (5). PJx) or /^(cos 0) is a new function, known as a Legendre's Coefficient, or as a Surface Zonal Harmonic, and occurs as a normal form in many important problems. V= r m P m (cos 8) is a particular solution of (1), and r'"P m (cos 8) is sometimes called a Solid Zonal Harmonic. V = A,P a {cos 8) + A i rP l [cos 8) + A,r*P,(cos 8) + ASPlcos8)+... (13) satisfies (1), is not infinite at any point within the sphere, and reduces to V= A»P (cos 8) + A^cos 6) + A.P^cos 0) + A,P 3 (cos8)+... (14) when r = 1. F= A,P,(cosll) A,P,(co'ir> , A,P,(coie) + ^« + ... (.5) satisfies (1), is not infinite at any point without the sphere, is equal to zero when r = 00 , and reduces to (14) when r = I. If then we can develop /"(#) [see eq. (2)] into a series of the form (14), we have only to put the coefficients of this series in place of the A , A^ A^, ... in (13) to get the value of Ffor a point within the sphere, and in (15) to get the value of Fat a point without the sphere. 182 ' HARMONIC FUNCTIONS. [Chap. V. We shall see later (Art. 16, Prob. 22) that if /(#) = i for o < 6 < — and /(d) = o for — < < it, + _il. I _J-3p( cos m_... (^ 1 12 2.4 6V ' V ' Hence our required solution is V = j+ ^(cos A) - I • j ■ r*/>,(cos 0) -f- — • — 3 r\P,(cos 0)- . . . (17) T I2 2.4 6V . ' w/ at an internal point ; and V = — + -A-P i (cos6)-l---LpJcosff) 2r 4 r iK ' 8 2 r sV / _i_iL.illi/>( C osff)-... (18) 1 12 2.4^ v ?t an external point. If r = — and 8=0, (17) reduces to „ 1 . 3 1 7 1 1 . II 1 .3 1 . „ . . F =2-+f-4"8 2-7+r2- 2 Ti- 4 ^---' slnceP "' (l) = I - To two decimal places V = 0.68, and the point r = — , 6 = 4 is at potential 0.68. If r = 5 and 6 = — , (18) and Table I, at the end of this 4 chapter, give V =Trs + l'T'°- 707l + 7 s'y 3 4 -p-o.i^8+...=ai2, 71 and the point r = 5, = is at potential o. 12. 4 If the radius of the conductor is a instead of unity, we have only to replace r by — in (17) and (18). Art. 5.] PROBLEM IN BESSEL'S FUNCTIONS. 183 Prob. 2. One half the surface of a solid sphere 12 inches in di- ameter is kept at the temperature zero and the other half at 100 de- grees centigrade until there is no longer any change of temperature at any point within the sphere. Required the temperature of the center ; of any point in the diametral plane separating the hot and cold hemispheres ; of points 2 inches from the center and in the axis of symmetry ; and of points 3 inches from the center in a di- ameter inclined at an angle of 45° to the axis of symmetry. Ans. 50°; 50°; 73 . 9 ; 26°.i ; 77°.i ; 22°.o. Art. 5. Problem in Bessel's Functions. As a last example we shall take the following problem : The base and convex surface of a cylinder 2 feet in diameter and 2 feet high are kept at the temperature zero, and the upper base at 100 degrees centigrade. Find the temperature of a point in the axis one foot from the base, and of a point 6 inches from the axis and one foot from the base, after the permanent state of temperatures has been set up. If we use cylindrical coordinates and take the origin in the base we shall have to solve equation (6), Art. 1 ; or, represent- ing the temperature by u and observing that from the sym- metry of the problem u is independent of 0, 9r* ^ r dr ^ 3*" subject to the conditions u = o when z = o, (2) u = o " r--l, (3} u = 100 " z — 2. (4) Assume u = RZ where R is a function of r only and Z of z only; substitute in (1) and reduce. 1 d'R , 1 dR 1 d*Z , . WegCt R-dS + VR-Jr ~ ~ Z~d?- (5) The first member of (5) does not contain z ; therefore the second member cannot. The second member of (5) does not 184 HARMONIC FUNCTIONS. [CHAP. V. contain r ; therefore the first member cannot. Hence each member of (5) is a constant, and we can write (5) Rdr 1+ rRdr~ Z dz* ~ M ' ^ ' when ft is entirely undetermined. d'Z Hence — - — ftZ = o, (7) , d'R . idR . , D ... and — -L- jj.*R — o. (8) dr r dr Equation (7) is easily solved, and its general solution is Z = Ae* z -\- Be ' **, or the equivalent form Z = C cosh (/as) + D sinh (}jlz). (9) We can reduce (8) slightly by letting /xr = x, and it becomes d'R , idR . _ - r -4-R = o. (10) dx 1 ^ x dx ^ K ' Assume, as in Art. 4, that R can be expressed in terms of whole powers of x. Let R = ~2a n x" and substitute in (10). We get 2\n(n — l)a„x n " ' -f na n x" ~ 2 + a n x"~] = o, an equation which must be true, no matter what the value of x. The coefficient of any given power of x, as x k ~'', must, then, vanish, and k(k — i)a k + ka k -f a k _ , = O, or k'a k -\-a k - t = o, whence we obtain « i _,= — k*a k (n) as the only relation that need be satisfied by the coefficients in order that R = 2a k x h shall be a solution of (10). If k-=Q, « 4 _, = 0, a k _ t = o, etc. We can, then, begin with k = o as the lowest subscript. ART. 5.] PROBLEM IN BESSEL'S FUNCTIONS. 185 From (i i) a k =-^z2. Then «, = -§, «. = -*,. ^--f^,*,. Hence * = -.[, -*+-*., __£!_ + ...], where a may be taken at pleasure, is a solution of (10), pro- vided the series is convergent. Take a = I, and then R — J o (x) where /o(*)=I-2-, + 2 f ? - ? ^ 2 + ^^-... (12) is a solution of (10). J (x) is easily shown to be convergent for all values real or imaginary of x, it is a new and important form, and is called a Bessel's Function of the zero order, or a Cylindrical Har- monic. Equation (10) was obtained from (8) by the substitution of jr = iir ; therefore JAW 2 , "t- 2 . -4 . 2 \ 4 \&^ is a solution of (8), no matter what the value of /x ; and u =y o (/ir) sinh (pis) and u = JJ K pir) cosh (piz) are solutions of (i). u = /„(f* r ) s ' nn (y"^) satisfies condition (2) whatever the value of 11. In order that it should satisfy condition (3) /t must be so taken that 7oO) = o; (13) that is, li must be a root of the transcendental equation (13). It was shown by Fourier that ./„(/-<) = O has an infinite num- ber of real positive roots, any one of which can be obtained to any required degree of approximation without serious diffi- culty. Let yU,, jk,, yw 3 , . . . be these roots ; then u = A Jin/) sinh (n.z) + A J \(/A,r) sinh (/y?) + AJlus) sinh ( M ,z) + . . . (14) is a solution of (1) which satisfies (2) and (3). 186 HARMONIC FUNCTIONS. [Chap. V.. If now we can develop unity into a series of the form _ smh (2/t,) sinh (2/0 i = BJfaJ) + BJfar) + BJfar) 4 w = ioo satisfies (i) and the conditions (2), (3), and (4). We shall see later (Art. 21) that if /,(*) - ■ " ' ! (I5> rfjr I = 2 for values of r < 1. Hence J far) sinh (//,*) , J In/) sinh (/y?) + •• (16)' u = 200 "h (>,s) , /X/y) sinh (/ y?) , j , inh (2/*,) "** /*,/,()",) sinh (2/0 "*" ' - "J {7) ' -ft, -AC".) sil is our required solution. At the point r = O, 2 = 1 (17) reduces to sinh /i/j , sinh ju 2 « = 200 = 100 + -VjiiMd sinh (2//,) nJfa) sinh ( 2 /0 +...] + ...} -ft,/ fa) cosh A', /*,/,(/'„) cosh /i 2 since y„(o) = 1 and sinh (2jt) = 2 sinh jr cosh ^r. If we use a table of Hyperbolic functions* and Tables II and III, at the end of this chapter, the computation of the value of u is easy. We have ft, = 2 -405 /<,= 5.520 y.(^.) = - 0.3402 colog /(, = 9.6189 — 10 colog yu 2 = 9.2581 —10 " J fa) = 0.2848 " ./,(*0 = 0.4683/* " cosh /i, = 9.2530 — 10 " coshyu 3 = 7.9037 — 10 9.1567 10 7.6301 n — 10 See Chapter IV, pp. 162, 163, for a four-place table on hyperbolic funo Art. 5.] problem in bessel's functions. 187 0*i /i(/0 cosh /a,)- 1 = 0.1434 (/*»/.(/<.) cosh ju,) -1 = — 0.0058 0.1376; «=I3°.8 At the point r = \, z = 1, (17), reduces to « = 100 . /'./.(/O cosh /^ ^ M,/,(M,) cosh ^ /.(i/O = 0.6698 + ...} tog/ote/O =9.8259 - IO colog /*,/,(/<,) cosh //, = 9.1567 — 10 8.9826 - /,(*/*>) = - 0.1678 10 log /„(£//,) = 9.2248/z - colog //,/,(/*,) cosh yu, = 7.630 1 « — 10 10 6.8549 - /.(*".) _ oog6l 10 /*,/,(/0 cosh ju, /,(£aO _ 0.0007 . M* fi(}**) cos h /*a 0.0968 ' u = g°.y If the radius of the cylinder is a and the altitude b, we have only to replace fj. by /ua in (13) ; 2// ( , 2^, ... in the denomi- nators of (15) and (17) by /*,£, ///, . . . ; and //,,//,, /* 3 , . . in the denominators of (16) and (17) by /<,«, yu^, ft, a Prob. 3. One base and the convex surface of a cylinder 20 cen- timeters in diameter and 30 centimeters high are kept at zero tem- perature and the other base at 100 degrees Centigrade. Find the temperature of a point in the axis and 20 centimeters from the cold base, and of a point 5 centimeters from the axis and 20 centimeters from the cold base after the temperatures have ceased to change. Ans. 13°. 9; 9°,6. 188 HARMONIC FUNCTIONS. [CHAP, V. Art. 6. The Sine Series. As we have seen in Art. 3, it is sometimes important to be able to express a given function of a variable, x, in terms of sines of multiples of x. The problem in its general form was first solved by Fourier in his " Theorie Analytique de la Chaleur" {1822), and its solution plays an important part in most branches of Mathematical Physics. Let us endeavor to so develop a given function of x, fix), in terms of sin x, sin 2x, sin $x, etc., that the function and the series shall be equal for all values of x between o and n. We can of course determine the coefficients a l} a 2 , a„ . . . a n so that the equation f(x) = a x sin x -f- a, sin 2x -f- a, sin ^x -(-... -f- a n s ' n nx (0 shall hold good for any n arbitrarily chosen values of x between O and rt ; for we have only to substitute those values in turn in (1) to get n equations of the first degree, in which the n co- efficients are the only unknown quantities. For instance, we can take the n equidistant values Ax, 2Ax, %Ax, . . . nAx, where Ax = — ■ — , and substitute them ior x in n-\- 1 (i). We get /(Ax) = #, sin Ax + a, sin 2Ax -f- a 3 sin 3 Ax + . -f- a„ sin nAx, J[2Ax) = a x sin 2Ax -f- a, sin 4AX -f a, sin 6 Ax -\- . -\- a„ sin 2nAx, f{$Ax) — «j sin 3 Ax -f- «„ sin 6 Ax -f- «, sin 9^ + . + a n sin 3wz/^r, /(nAx) = a, sin «z/^ + «» s ' n 2«Jj; -j- «s s i n inAx -j- • -(- « M sin «Mjr, « equations of the first degree, to determine the n coefficients a,, a t , a,, . . . a„. Not only can equations (2) be solved in theory, but they can be actually solved in any given case by a very simple and r(2) Art, 6 ] the sine series. 189 ingenious method due to Lagrange,* and any coefficient a m can be expressed in the form 2 K=n a m = ^> /(kAx) sin (Km Ax). (3) If now n is indefinitely increased the values of x for which (1) holds good will come nearer and nearer to forming a con- tinuous set ; and the limiting value approached by a m will probably be the corresponding coefficient in the series required to represent/^) for all values of x between zero and n. Remembering that (n -\- \)Ax = n, the limiting value in question is easily seen to be IfM sin mxdx. (4) This value can be obtained from equations (2) by the fol- lowing device without first solving the equations : Let us multiply each equation in (2) by the product of Ax and the coefficient of a m in the equation in question, add the equations, and find the limiting form of the resulting equation as n increases indefinitely. The coefficient of any a, a K in the resulting equation is sin kAx sin mAx . Ax -\- sm 2k Ax sin zmAx . Ax -\- . . . -f- sin tiK Ax sin nmAx . Ax. Its limiting value, since (n + i)Ax = sr, is / sin kx sin mx . dx ; but f sin kx sin mx.dx = if [cos (m - k)x - cos(m + K)x]dx = o ° if m and k are not equal. * See Riemann's Partielle Differcntialgleichungen, or Byerly's Fourier's Series and Spherical Harmonics. 190 HARMONIC FUNCTIONS. [CHAP. V. The coefficient of a m is Ax{s\tf mAx -f- sin 2 2mAx + sin 2 imAx + . . . + sin 2 nmAx). Its limiting value is ./' 7 n sin 2 mx . ax = — . 2 The first member is /(Z/^r) sin mAx . Ax -\-f(2Ax) sin 2mAx . Ax -f- ■ +/(w//^r) sin mnAx . Ax, and its limiting value is / f(x) sin mx.dx. Hence the limiting form approached by the final equation as n is increased is n I f(x) sin mx . dx = — a n . "Whence — —J f(x) sin mx . dx (5) as before. This method is practically the same as multiplying the equation f(x) = «, sin x -j- a^ sin 2.x -)- # 3 sin $x -\- . . . (6) by sin »«r. akr and integrating both members from zero to n. It is important to realize that the considerations given in this article are in no sense a demonstration, but merely estab- lish a probability. An elaborate investigation * into the validity of the develop- ment, for which we have not space, entirely confirms the results formulated above, provided that between x = o and x = n the * See Art. 10 for a discussion of this question. Art. 6 ] THE SINE SERIES. 191 function is finite and single-valued, and has not an infinite num- ber of discontinuities or of maxima or minima. It is to be noted that the curve represented by y = f(x) need not follow the same mathematical law throughout its length, but may be made up of portions of entirely different curves. For example, a broken line or a locus consisting of finite parts of several different and disconnected straight lines can be represented perfectly well by_y = a sine series. As an example of the application of formula (5) let us take the development of unity. Here a m = IT 2 f = — /sin mx . dx sin mx . dx = cos mx s sin mx . dx = — (1 m m cos m-rt) = — |~i — (— i) w l m ' J o if m is even = — if m is odd. m 4 /sin x . sin %x . sin e>x . sin 7x . \ , . Hence 1 = ±(_ + -jL+ -A- +-J- + ...). (7) It is to be noticed that (7) gives at once a sine development for any constant c. It is, ac /sin x . sin $x sm ^x (8) Prob. 4. Show that for values of x between zero and it "sin x sin 2x . sin $x sin 4X (a) x = 2 + /ln ^ \ 4r sin * sin 3* sin 5* sin jx 192 HARMONIC FUNCTIONS. [CHAP. V. if f( x ) = x for o < x < -, and f(x) = n — x for - < * < n. 2 2 sin * , 2 sin 2X , sin 3* . sin 5a; . 2 sin 6a; sin jx 1 1 1 _, u._ . 1 2 3 5 6 ( f * J\ ) 2 2 V ' identically ; but '- '- is not changed by reversing the sign of x and is therefore an even function of x; and when fix) — fi— x) we reverse the sign of x, -^— * ^ is affected only to the extent of having its sign reversed, and is consequently an odd function of x. Therefore for all values of x between — n and n — - = -b„ + b l cos x + &, cos 2x -\- b, cos $x -f . . , . 2 //(*) +/(- ^) where b m = — I ^ '- cos mx . dx ; 71 d 2 o f( x \ — fi— x) and Tt-^-t — j-i = a x sin jr + a, sin 2^ + ^a sm 3-* + • • • , 2 n/(x)-f(-x) . where #„, = - / ^^^ — ^ i sin wjr . dx. 71 ..' 2 Art. 8.] Fourier's series. 195 b, n and a m can be simplified a little. * ._ 2 f A*) + A-*) . °m — ~z~J ~ cos mx . dx = -^ \J'A x ) cos mx • dx +ff{-x) cos mx . dx~\; but if we replace x by — x, we get ir -ir J A— x) cos mx . dx——Jf(x) cos mx.dx = ff(x)cos mx.dx, IT and we have b m = ~Jf(x) cos mx . dx. — IT In the same way we can reduce the value of a m to IT Hence A x ) = - ^0 + K cos x + b t cos 2* + b s cos 3* -f- . . . + a i sin x -\- a, sin 2* -|- a s sin 3* + . . . , (2) where # w = — / f(x) cos ^2^r . dx y (3) — JT IT and a m = — / /(^) sin mx . dx, (4) and this development holds for all values of x between — n and it. The second member of (2) is known as a Fourier's Series. The developments of Arts. 6 and 7 are special cases of development in Fourier's Series. Prob 6. Show that for all values of x from — n to n 2 sinh 7t\~j i 1 1 ,i , ^* = COS X H COS 2X COS XX-\ C0S4^ + ... n \_2 2 5 10 ° 17 196 HARMONIC FUNCTIONS. [CHAP. V. , 2 sinh n [~i . 2 . 3 . 4 . 1 A -sina sin 2X 4- — sin xx — — sin ax -+- . . . . 7t \_2 5 'iO ° 17 * J Prob. 7. Show that formula (2), Art. 8, can be written f(x) = -c e cos/J, + c, cos (* — /?,) + ^ 3 cos (2* — /? 2 ) + f, cos (3* — A,) + ... , where c m = (a M a + £,„*)* and /S„ = tan" 1 ~ Prob, 8. Show that formula (2), Art. 8, can be written fix) = ~c sin /?„ + c 1 sin (a: + /?,) + ', sin (2* + /? ) 2 + (4) y = fx " t = o. ( 5 ) Using the method of Art. 3, we easily get as particular solutions of (1) y = sin fix sin a fit, y — s in fix cos a fit, y = cos fix sin a fit, and y = cos/?* cos a fit. y = sin - r cos —j- satisfies (i), (2), (3), and (4). y = >a m sin — r - cos — — , (o) where 2 jffix) sin ^-f . ^ (7) is our required solution ; for it reduces to f{x) when t = o. See Art. 9. Prob. 13. Three edges of a square sheet of tinfoil are kept at potential zero, and the fourth at potential unity ; at what potential is the centre of the sheet ? Ans. 0.25. Prob. 14. Two opposite edges of a square sheet of tinfoil are kept at potential zero, and the other two at potential unity ; at Tvhat potential is the centre of the sheet ? Ans. 0.5. Prob. 15. Two adjacent edges of a square sheet of tinfoil are 202 HARMONIC FUNCTIONS. [Chap, V. kept at potential zero, and the other two at potential unity. At what potential is the centre of the sheet ? Ans. 0.5. Prob. 16. Show that if a point whose distance from the end of a harp-string is -th the length of the string is drawn aside by the player's finger to a distance b from its position of equilibrium and then released, the form of the vibrating string at any instant is given by the equation ion ,— ■ / 1 . mn . mnx mnat\ y = 7 r~i > I — sin — sin — — cos — — . (n— 1)71*^-, \m n I I I Show from this that all the harmonics of the fundamental note of the string which correspond to forms of vibration having nodes at the point drawn, aside by the finger will be wanting in the complex note actually sounded. Prob. 17.* An iron slab 10 centimeters thick is placed between and in contact with two other iron slabs each 10 centimeters thick. The temperature of the middle slab is at first 100 degrees Centigrade throughout, and of the outside slabs zero throughout. The outer faces of the outside slabs are kept at the temperature zero. Re- quired the temperature of a point in the middle of the middle slab fifteen minutes after the slabs have been placed in contact. Given a 2 = 0.185 in C.G.S. units. Ans. io°.3. Prob. 18.* Two iron slabs each 20 centimeters thick, one of which is at the temperature zero and the other at 100 degrees Centigrade throughout, are placed together face to face, and their outer faces are kept at the temperature zero. Find the temperature of a point in their common face and of points 10 centimeters from the com- mon face fifteen minutes after the slabs have been put together. Ans. 22 °.8; is°.i ; i7°.2. Art. 12. f Properties of Zonal Harmonics. In Art. 4, z — P m ix) was obtained as a particular solution of Legendre's Equation [(7), Art. 4] by the device of assuming that z could be expressed as a sum or a series of terms of the form a„x n and then determining the coefficients. We * See Art. 3. t The student should review Art. 4 before beginning this article. ART. 12.] PROPERTIES OF ZONAL HARMONICS. 203 can, however, obtain a particular solution of Legendre's equa- tion by an entirely different method. The potential function for any point (x, y, z) due to a unit of mass concentrated at a given point (x lt y iy #,) is ~ n* - *,)' + (^ 30' Tl^tF (I) and this must be a particular solution of Laplace's Equation [(3), Art. 1 J, as is easily verified by direct substitution. If we transform (i) to spherical coordinates we get V= 1 — (2) |V _ 2 rr,[cos cos 0, + sin 6 sin 1 cos (0—0,)] -\-r? as a solution of Laplace's Equation in Spherical Coordinates [(5), Art. I]. If the given point (jr,, y,, z t ) is taken on the axis of X, as it must be in order that (2) may be independent of 0, l =. o, and V= ,_ I (3) vr 1 — 2rr t cos -)- r, s is a solution of equation (1), Art. 4. Equation (3) can be written F=-(i-2-cos0 + ^)" i ; (4) / r r 2 \~i and if r is less than r, ( 1 — 2— cos 6 -\ J can be developed ««■ into a convergent power series. Let 2/> m — be this series, 1 r m p m being of course a function of 6. Then V=—2Sp M — is a 1 1 solution of (1), Art. 4. Substituting this value of V in the equation, and remem- bering that the result must be identically true, we get after a. slight reduction 204 HARMONIC FUNCTIONS. [CHAP. V. but, as we have seen, the substitution of x = cos B reduces this to Legendre's equation [(7), Art. 4]. Hence we infer that the coefficient of the wth power of z in the development of (1 — 2xz-\- 2 2 )~* io a function of x that will satisfy Legendre's equation. (1 — 2xz -\- z*)-* = [1 — z(2x — z)] -*, and can be developed by the Binomial Theorem ; the coefficient of z m is easily picked out, and proves to be precisely the func- tion of x which in Art. 4 we have represented by P„,(x), and have called a Surface Zonal Harmonic. We have, then, (i-m+*'}- , =^W+^W^+^)^+^(*).* , + • • • (5) if the absolute value of z is less than 1. If x = 1, (5) reduces to (l-2* + ^)-*=/' (l)+ J P l (l).*+/',(l).«»+^(l).^+...; but (1 -2z + z*)-i=(i —z)-'= I +£ + a + £ 3 + . . .; hence P m (i) = 1. (6) Any Surface Zonal Harmonic may be obtained from the two of next lower orders by the aid of the formula (» + l)P. +J (x) - (2» + l)xP„(x) + nP n _lx) = o, (7) which is easily obtained, and is convenient when the numerical value of x is given. Differentiate (5) with respect to z, and we get ( iit;+W = * <*> + 2p ^ ■ z + ^.w • *■ + • • • . whence ■**+•• •)- or by (5) (1 - 2xz + *■)(/>(*) + 2P,(*) . * 4- 3 P 3 (*) . g* . . •) + (* — *)(^,(*) 4- />,(*) . z + />,(*) . «• 4- • • • 1 = o. (8) Art. 13.] problems in zonal harmonics. 205' Now (8) is identically true, hence the coefficient of each power of z must vanish. Picking out the coefficient of z n and writing it equal to zero, we have formula (7) above. By the aid of (7) a table of Zonal Harmonics is easily com- puted since we have P a (x) — l > a °d P 3 (x) = x. Such a table for x — cos B is given at the end of this chapter. Art. 13. Problems in Zonal Harmonics. In any problem on Potential if Fis independent of so that we can use the form of Laplace's Equation employed in Art. 4, and if the value of Fon the axis of X \% known, and can be expressed as 2a m r m or as ^> -^j, we can write out our required solution as V=Za m r~P m{c os8) or V = ^^^; for since P m (i) = 1 each of these forms reduces to the proper value on the axis ; and as we have seen in Art. 4 each of them satisfies the reduced form of Laplace's Equation. As an example, let us suppose a statical charge of M units of electricity placed on a conductor in the form of a thin circu- lar disk, and let it be required to find the value of the Poten- tial Function at any point in the " field of force " due to the charge. The surface density at a point of the plate at a distance r from its centre is M a = 4* (3) Hence M F: -^(cos^ + f JP,(cos(?) -^> 6 (cos*)+. 7T ] (4) is our required solution if r < <2 and < — , as is 2 J"" |^(cos*) + I|> l( cos0) - l£/>(cos &) + ...] if r>a. (S) The series in (4) and (5) are convergent, since they may be obtained from the convergent series (2) and (3) by multiplying the terms by a set of quantities no one of which exceeds one in absolute value. For it will be shown in the next article that P m (cos 6) always lies between 1 and — 1. Prob. 19. Find the value of the Potential Function due to the attraction of a material circular ring of small cross-section. The value on the axis of the ring can be obtained by a simple M integration, and is , if M is the mass and c the radius of the Vc -f- r ring. At any point in space, if r < c V = M />,(cos 0) - I J>,(cos ff) + 1^ £p 4(cos 0) _ . . .1 and if r > <• Art. 14.] additional forms. 207 -P t {cos B) - x - i> s (cos 6) +Ll3 ^ (cos #) _ . . 1. _^ 2 r 2 . 4 r j Art. 14. Additional Forms. (a) We have seen in Art. 12 that P m {x) is the coefficient of s m in the development of (i — 2xz -f- ^)~ * in a power series. (l - 2xz + ^ 2 ) - * = [I — £(/>'' + * - «') + .s 2 ]-* - (i - ^«)" K 1 - ^~ 9 ~ 4 - If we develop (i — ze e ')~i and (i — ze~ ec )~iby the Bi- nomial Theorem their product will give a development for (i — 2.xz -\- z') ~ $. The coefficient of z m is easily picked out and reduced, and we get P„(cos 6) = 1.3.5... ( 2m — l) T a 1 r • ** ° D v — ' 2 cos ^(9 + 2 — 7 r cos (m — 2)6 2.4.6... 2m L I . (2W2 — 1) N ' \ .%. mim — 1) , , . , I + 2 r^ ^ — ,COS(»/-4)0+.. . (i) ^ I.2.(2«-l)(2«K-3) V ' ' J w If m is odd the parenthesis in (1) ends with the term con- taining cos 6 ; if m is even, with the term containing cos o, but in the latter case the term in question will not be multiplied by the factor 2, which is common to all the other terms. Since all the coefficients in the second member of (1) are positive, .P m (cos ff) has its maximum value when 6 = 0, and its value then has already been shown in Art. 12 to be unity. Obviously, then, its minimum value cannot be less than — 1. (b) If we integrate the value of PJx) given in (n), Art. 4, m times in succession with respect to x, the result will be j -y £ (2m 1^ found to differ from ' 3 , , , V — i) m by terms in " (2m) ! volving lower powers of x than the ;«th. T // m Hence PJx) = — , -^ - i>". (2) 208 HARMONIC FUNCTIONS. [CHAP. V. (c) Other forms for PJx), which we give without demon- stration, are It PJS) = \f\* + ^x^i ■ cos 0] ~d n (x) shall hold good for n -\- I equidistant values of x between — I and I, and taking its limiting value as n is indefinitely in- creased. Art. 16. Formulas for Development. We have seen in Art. 4 that z = P m (x) is a solution of d Legendre's Equation -5- (1 — x") — f \-\-m{m -f- i)z = o. (1) dx J d Hence -7- «;r L and < ' ~ **> ^r] + W(W + l )Pm{x) = °' (2) ( i -^)-irj + " (w + I)i> " (;r) = °- (3) Multiply (2) by />„(•*') and (3) by P m {x), subtract, transpose, and integrate. We have 1 [m{m + I) - »(« + l)-\f P m (x)P H {x)dx -1 210 HARMONIC FUNCTIONS. [Chap. V. - j'Pnkx) dx (I -**) dPJyX)- dx dP,„(x] dx (4) = \j> m {x){l -*)—— P n (x)(i - *■) — -1 by integration by parts, = O. 1 Hence J P m {x)P n (x)dx = o, (6) unless m = n. If in (4) we integrate from ^to 1 instead of from — 1 to \, we get an important formula. JPA*) J'P m (x)P n (x)dx: (I -,,[^4^ W ^>] m{m-\- 1) — n(n -\- 1) X and as a special case, since P (x) = I. , (7) 1 fP m {x)dx ' dx ■m{nt -\- 1) ' (8) unless m = o. 1 To get f[P m (x)ydx is not particularly difficult. By (2), Art. 14, fr-W^^kw/-^ I r d"'(x' — 1)" rf w (jr a - I)' a'x 7 " . dx (9) By successive integrations by parts, noting that .{• j~^*( x ' ~~ *)'" contains (x* — 1)" as a factor if k < ;«, and Art. 16.] formulas for development. 211 that — \^ = ( 2m V- we get afcr*' i f[P m {*)7dz = ^g^l /V - I)"*. (IO) -l ^ ■■' -l l l j\x i — i)'"dx = f(x — i)'"(x -f i)V# w + '* - f{x - i) n -\x+ i)'" +1 dx ■ \ 11! ! i Hence f\_PJxftdx = -^t-;- (n) -i i Prob. 20. Show that / P m (x)dx = o if wz is even and is not zero = (~ i)^ -7-V-^ ' 3 ' 5 ' 7 T^ , if m is odd. «^«+i) 2 . 4 . 4 . . . [m— i) i y> i [/^(^JV.* = -j— . Note that [P m (x)J' is an even function of x. Prob. 22. Show that if f(x) = o from x = — i to x = o, and /(*) = i from x = o to jc = i, /(*) = ; + p,w - J- ^.w + r 2 ■ rp-<*> — • fW = » Prob. 23. Show that F{9) = 2 B m P ,„(cos 0) where 2?„ = 2W+I fF{9)P n (cos 0) sin 5 <#. 212 HARMONIC FUNCTIONS. Prob. 24. Show that [Chap. V. 6 = See (1), Art. 14. Prob. 25. Show that 1 + ^j)^ (C ° S 6) + 9 &J F ^ C0S ff > + ~ J + {2 „ - 7) (»« +!)(«- I) ^ (x) + _ _ _ (*) 2.4 1 1 x H J'n(x)dx = -±- / *" £J£- I J~.dx, and use the 2 m m t/ tfx m -l -1 method of integration by parts freely. Prob. 26. Show that if Fis the value of the Potential Function at any point in a field of force, not imbedded in attracting or repel- ling matter; and if V =■ f{ff) when r = a, and where V - 2A m — P n (cos 6) if r a, 2m -\- 1 TT ff(d)P m {cos 6) sin 6d0. Prob. 27. Show that if V — c when r = a ; V — c if r < a, and V = — if r > a. r Art. 17. Formulas in Zonal Harmonics. The following formulas which we give without demonstra- tion may be found useful for reference : dPJx) ~-^={2n-i)P n _lx)+{ 2 n-$)P n _,{ x )+{2n- 9 )P n _ l {x)+...,{i) (2) fpix)dx = -_L_ [jP ^ |(ar) _ Pn+i{x)l (3) Art. 19.] spherical harmonics. 213 Art. 18. Spherical Harmonics. In problems in Potential where the value of f^is given on the surface of a sphere, but is not independent of the angle sin" /" P '"( M \ where ;U = cos 6, are particular solutions of (5), Art. 1. The factors multiplied by r m in these values are known as Tesseral Harmonics. They are functions of

) and V = ^ Y m (j*, ) (4) which is the result of differentiating (2) with respect to #. A table giving values of J,(x) and /,(*) will be found at the end of this chapter. If we write J t {x) for z in equation (2), then multiply through by xdx and integrate from zero to x, simplifying the resulting equation by integration by parts, we get xdj a {x) dx + JxJlx)dx = o, or, since J^x) dx X J xJlx)dx = xj^x). (5) If we write /„(*■) for z in equation (2), then multiply through by * 3 — ^ — , and integrate from zero to x, simplifying by inte- gration by parts, we get or [{ d ^) 2 + (M-r)y]-fx (/a{ x ) ydx = o, X' dx = - (/.(*))• +(/,(*))' (6) Art 20.] applications of bessel's functions. 215 If we replace x by px in (2) it becomes d*z , 1 dz , (See (8), Art. 5). Hence z = / (m x ) is a solution of (7). If we substitute in turn in (7) /„(>«*) and /J,Hi x ) for ,s, mul- tiply the first equation by xj^x), the second by x/ {fx K x), subtract the second from the first, simplify by integration by parts, and reduce, we get J xj l)x K x)J l^ L x)dx 1 Mk' — M \{wJAw)fti<* a )-ViaJlv*aVi(w)]. (8) Hence if /j k and /u, are different roots of /„(/<#) = O, or of JAw) = o, or of jxa/Xl-M) — XJ^jxa) = o, a J x J\l->« x )J {n<. x )dx — O. (9) We give without demonstration the following formulas, which are sometimes useful : 7T J by (5), Art. 19; and (3) reduces to A " = ^M ; (7> (4) reduces to ^ = o, (8> except for ,£ = 1, when ju k = o, and we have A, = i; (9) (5) reduces to J* = 7TJ - i ~ ,, T( r. (10)' Prob. 28. A cylinder of radius one meter and altitude one meter has its upper surface kept at the temperature ioo°, and its base and convex surface at the temperature 15°, until the stationary temper- atures are established. Find the temperature at points on the axis 25. 50, and 75 centimeters from the base, and also at a point 25, centimeters from the base and 50 centimeters from the axis. Ans. 29°.6; 47°. 6 ; 7i°.2 ; 25°. 8. ART. Si.] DEVELOPMENT IN TERMS OF BESSEL'S FUNCTIONS. 219" Prob. 29. An iron cylinder one meter long and 20 centimeters in diameter has its convex surface covered with a so-called non-con- ducting cement one centimeter thick. One end and the convex surface of the cylinder thus coated are kept at the temperature zero, the other end at the temperature of 100 degrees. Given that the con- ductivity of iron is 0.185 an d of cement 0.000162 in C. G. S. units. Find to the nearest tenth of a degree the temperature of the mid- dle point of the axis, and of the points of the axis 20 centimeters from each end after the temperatures have ceased to change. Find also the temperature of a point on the surface midway be- tween the ends, and of points of the surface 20 centimeters from each end. Find the temperatures of the three points of the axis, supposing the coating a perfect non-conductor, and again, suppos- ing the coating absent. Neglect the curvature of the coating. Ans.. O On On O O^^O O O O 15 .4 ; 40 .85 ; 72 .8 ; 15 .3 ; 40 .7 ; 72 .5 ; o .0 ; o .0 ; 1 .3. Prob. 30. If the temperature at any point in an infinitely long cylinder of radius c is initially a function of the distance of the point from the axis, the temperature at any time must satisfy the equation — = a (-ps H pT) ( see Art - I )> since ll ls clearly in- dependent of z and (p. Show that u = A ie -*wj.M + ^-° , "«V.(/v) where, if the surface of the cylinder is kept at the temperature zero, yu, , M, , Ms , • • • are roots of JS^ C ) = ° and Ak is tne vame given in (3) with c written in place of a ; if the surface of the cylin- der is adiabatic /*,,/«„ ;u„ ... are roots of /,(/#) = o and A k is ob- tained from (4); and if heat escapes at the surface into air at the tem- perature zero A*,, /*,, ;/„... are roots of pcjXvc) ~ ^/„(/^) = °> and A h is obtained from (5). Prob. 31. If the cylinder described in problem 29 is very long and is initially at the temperature 100° throughout, and the con- vex surface is kept at the temperature o c , find the temperature of a point 5 centimeters from the axis 15 minutes after cooling has begun ; first when the cylinder is coated, and second, when the coating is absent. Ans. 97°-2 ; o°.oi. Prob. 32. A circular drumhead of radius a is initially slightly distorted into a given form which is a surface of revolution about the axis of the drum, and is then allowed to vibrate, and z is the ordinate of any point of the membrane at any time. Assuming that 220 HARMONIC FUNCTIONS. [Chap. V. . , , . d'z Jd'z , i 3a\ L . ^ must satisfy the equation —5 = c ^^-5 _(- — —-. 1 subject to the con- dz ditions z = o when r = a, — = o when f = o, and 2 = /(r) when / = o, show that z — AJJ^n/) cos pjt + A t f a (j*,r) t cos t** ct + •■■ where /«,, /*,, /*,, = •• are roots of J t (ya) — o and A has the value given in (3). Prob. 33. Show that if a drumhead be initially distorted as in problem 32 it will not in general give a musical note ; that it may be initially distorted so as to give a musical note ; that in this case the vibration will be a steady vibration ; that the periods of the various musical notes that can be given are proportional to the roots of J a ( x ) = °> an d that the possible nodal lines for such vibrations are concentric circles whose radii are proportional to the roots of J.{x) = o. Art. 22. Problems in Bessel's Functions. If in a problem on the stationary temperatures of a cylinder u = o when z = o. u = o when z = b, and u = f{z) when r = a, the problem is easily solved. If in (2), Art. 20, and in the cor- responding solution 3 = cosh {j*z)JJijir) we replace fx by pit, we can readily obtain z = sin (/uz)/ (/^ri) and z — cos (l*z)J t ()xri) as particular solutions of (1), Art. 20; and /.(«) = 1+ £. + -^_ i + - i; £_ + ... (I) and is real. k = 00 7 ^— ., . RTtZ f(z) =^ A k sin — 2 f* K7ZZ ■where A k —j- I f( z ) sin -7- ds (2) by Art. 9. -/6 At-) Hence , = ^ ^ sin -^ — — - ( 3 ) k=i is the required solution. Art. 24.] lame's functions. 221 A table giving the values olj {xi) will be found at the end of this chapter. Prob. 34. A cylinder two feet long and two feet in diameter has its bases kept at the temperature zero and its convex surface at 100 degrees Centigrade until the internal temperatures have ceased to change. Find the temperature of a point on the axis half way between the bases, and of a point six inches from the axis, half way between the bases. Ans. 72.°!; 8o°.i. Art. 23. Bessel's Functions of Higher Order. If we are dealing with Laplace's Equation in Cylindrical Coordinates and the problem is not symmetrical about an axis, functions of the form /«(*) = 2"r(«+ 1) X* 2'(n + I) ' 2\ 2 \{n -f i)(« +2) play very much the same part as that played by ./„(.*) in the preceding articles. They are known as Bessel's Functions of the «th order. In problems concerning hollow cylinders much more complicated functions enter, known as Bessel's Functions of the second kind. For a very brief discussion of these functions the reader is referred to Byerly's Fourier's Series and Spherical Harmonics ; for a much more complete treatment to Gray and Matthews' admirable treatise on Bessel's Functions. Art. 24. Lame's Functions. Complicated problems in Potential and in allied subjects are usually handled by the aid of various forms of curvilinear co- ordinates, and each form has its appropriate Harmonic Func- tions, which are usually extremely complicated. For instance, Lame's Functions or Ellipsoidal Harmonics are used when solutions of Laplace's Equation in Ellipsoidal coordinates are required ; Toroidal Harmonics when solutions of Laplace's Equation in Toroidal coordinates are needed. For a brief introduction to the theory of these functions see Byerly's Fourier's Series and Spherical Harmonics. 222 HARMONIC FUNCTIONS. [Chap. V. Table I. Surface Zonal Harmonics. P, (cos 0) P, (cos 0) P 3 (cos 0) P, (cos 0) P 6 (cos 0) P 6 (cos 0) P^ (cos 0) 0° 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 .9998 .9995 .9991 .9985 .9977 .9967 .9955 2 .9994 .9982 .9963 .9939 .9909 .9872 .9829 3 .9986 .9959 .9918 .9863 .9795 .9713 .9617 4 9976 .9927 .9854 .9758 .9638 .9495 .9329 5 .9962 .9886 .9773 .9623 .9437 .9216 .8961 6 .9945 .9836 .9674 .9459 .9194 .8881 .8522 7 .9925 .9777 .9557 .9267 .8911 .8476 .7986 8 .9903 .9709 .9423 .9048 .8589 .8053 .7448 9 .9877 .9633 .9273 .8803 .8232 .7571 .6831 10 .9848 .9548 .9106 .8532 .7840 .7045 .6164 11 .9816 .9454 .8923 .8238 .7417 .6483 .o461 12 .9781 .9352 .8724 .7920 .6966 .5892 .4732 13 .9744 .9241 .8511 .7582 .6489 .5273 .3940 14 .9703 .9122 .8283 .7224 .5990 .4635 .3219 15 .9659 .8995 .8042 .6847 .5471 .3982 .2154 16 .9613 .8860 .7787 .6454 .4937 .3322 .1699 17 .9563 .8718 .7519 .6046 .4391 .2660 .0961 18 .9511 .8568 .7240 .5624 .3836 .2002 .0289 19 .9455 .8410 .6950 .5192 .3276 .1347 -.0443 20 .9397 .8245 .6649 .4750 .2715 .0719 -.1072 21 .9336 .8074 .6338 .4300 .2156 .0107 -.1662 22 .9272 .7895 .6019 .3845 .1602 -.0481 -.2201 23 .9205 .7710 .5692 .3386 .1057 -.1038 -.2681 24 .9135 .7518 .5357 .2926 .0525 -.1559 -.3095 25 .9063 .7321 .5016 .2465 .0009 -.2053 -.3463 26 .8988 .7117 .4670 .2007 -.0489 -.2478 -.3717 27 .8910 .6908 .4319 .1553 -.0964 -.2869 -.3921 28 .8829 .6694 .3964 .1105 -.1415 -.3211 -.4052 29 .8746 .6474 .3607 .0665 -.1839 -.3503 -.4114 30 .8660 .6250 .3248 .0234 -.2233 -.3740 — .4101 31 .8572 .6021 .2887 -.0185 -.2595 -.3924 -.4022 32 .8480 .5788 .2527 -.0591 -.2923 -.4052 -.3876 33 .8387 .5551 .2167 -.0982 -.3216 -.4126 -.3670 34 .8290 .5310 .1809 -.1357 -.3473 -.4148 -.3409 35 .8192 .5065 .1454 -.1714 -.3691 -.4115 -.3096 36 .8090 .4818 .1102 -.2052 -.3871 -.4031 — .2738 37 .7986 .4567 .0755 -.2370 -.4011 -.3898 -.2343 38 .7880 .4314 .0413 -.2666 -.4112 -.3719 —.1018 39 .7771 .4059 .0077 -.2940 -.4174 -.3497 -.1469 40 .7660 .3802 -.0252 -.3190 -.4197 -.3234 —.1003 41 .7547 .3544 -.0574 -.3416 -.4181 -.2938 -.0534 42 .7431 .3284 -.0887 -.3616 -.4128 -.2611 -.0065 43 .7314 .3023 -.1191 - 3791 -.4038 -.2255 .0398 44 .7193 .2762 -.1485 -.3940 -.3914 -.1878 .0846 45° .7071 .2500 -.1768 -.4062 -.3757 -.1485 .1270 TABLES. 233 Table I. Surface Zonal Harmonics. e Pi (cos «) P 2 cos fl) P 3 (cos 9) | P 4 (cos e> P 6 (cos 0) P (cos 9) P, (cos 8) 45° .7071 .2500 -.1768 -.4062 -.3757 -.1485 .1270 46 .6947 .2238 -.2040 -.4158 -.3568 -.1079 .1666 47 .6820 .1977 -.2300 -.4252 -.3350 -.0645 .2054 48 .6691 .1716 -.2547 -.4270 -.3105 -.0251 .2349 49 .6561 .1456 -.2781 -.4286 -.2836 .0161 .2627 50 .6428 .1198 -.3002 -.4275 -.2545 .0563 .2854 51 .6293 .0941 -.3209 -.4239 -.2235 .0954 .3031 52 .6157 .0686 -.3401 -.4178 -.1910 .1326 .3153 53 .6018 .0433 -.3578 -.4093 -.1571 .1677 .3221 54 .5878 .0182 -.3740 -.3984 -.1223 .2002 .3234 55 . 5733 -.0065 -.3886 -.3852 -.0868 .2297 .3191 56 .5592 -.0310 -.4016 -.3698 -.0510 .2559 .3095 57 .5446 -.0551 -.4131 -.3524 -.0150 .2787 .2949 58 .5299 -.07 -'8 -.4229 -.3331 .0206 .2976 .2752 59 .5150 -.1021 -.4310 -.3119 .0557 .3125 .2511 <50 .5000 -.1250 -.4375 -.2891 .0898 .3232 .2231 4 .4384 -.2117 -.4470 -.1841 .2123 .3240 .0818 «5 .4226 -.2321 -.4452 -.1552 .2381 .3138 .0422 <56 .4067 -.2518 -.4419 -.1256 .2615 .2996 .0021 67 .3907 -.2710 -.4370 -.0955 .2824 .2819 -.0375 68 .3746 -.2896 -.4305 -.0650 .3005 .2605 -.0763 69 .3584 -.3074 -.4225 -.0344 .3158 .2361 -.1135 70 .3420 -.3245 -.4130 -.0038 .3281 .2089 - 1485 71 .3256 -.3410 -.4021 .0267 .3373 .1786 -.1811 72 . 3090 -.3568 -.3898 .0568 .3434 .1472 -.2099 73 .2924 -.3718 -.3761 .0864 .3463 .1144 -.2347 74 .2750 -.3800 -.3611 .1153 .3461 .0795 -.2559 75 .2588 -.3995 -.3449 .1434 .3427 .0431 -.2730 76 .2419 -.4112 -.3275 .1705 .3362 .0076 -.2848 77 .2250 -.4241 -.3090 .1964 .3267 -.0284 -.2919 78 .?079 -.4352 -.2894 .2211 .3143 -.0644 -.2943 79 .1908 -.4454 -.2688 .2443 .2990 - .0989 -.2913 80 .1736 -.4548 -.2474 .2659 .2810 -.1321 -.2835 81 .1564 -.4633 -.2251 .2859 .2606 -.1635 -.2709 82 .1392 -.4709 -.2020 .3040 .2378 -.1926 -.2536 83 .1219 —.4777 -.1783 .3203 .2129 -.2193 -.2321 84 .1045 -.4836 -.1539 .3345 .1861 -.2431 -.2067 85 .0872 -.4886 -.1291 .3468 .1577 -.2638 -.1779 86 .0698 -.4927 -.1038 .3569 .1278 -.2811 -.1460 87 . 0523 -.4959 -.0781 .3648 .0969 -.2947 -.1117 88 .0349 -.4982 -.0522 .3704 .0651 -.3045 -.0735 89 .0175 -.4995 -.0262 .3739 .0327 -.3105 -.0381 90' .0000 -.5000 .0000 .3750 .0000 -.3125 .0000 224 HARMONIC FUNCTIONS. [Chap. V. Table II. Bessel's Functions. X Joix) JiW j X J»W) Jitx) X Jo'.xi Jt(X) 0.0 1.0000 0.0000 i 5.0 -.1776 -.3276 10.0 i -.2459 .0435 0.1 .9975 .0499 5.1 -.1443 -.3371 10.1 > -.2490 | .0184 0.2 .9900 .0995 1 5.2 -.1103 -.3432 10.2 -.2490 : .0066 0.3 .9776 .1483J 5.3 -.0758 -.3460 10.3 -.2477 -.0313 0.4 .9604 .1960 | 5.4 -.0412 -.3453 10.4 | -.2434 -.0555 0.5 .9385 .2423 1 5.5 -.0068 -.3414 10.5 i -.2366 -.0789 6 .9120 .2867 ' 5.6 .0270 -.3343 10.6 -.2276 -.1012 0.7 .8812 .3290 5.7 .0599 -.3241 10.7 -.2164 -.1224 0.8 .8463 .3688 5.8 .0917 -.3110 10.8 -.2032 -.1422 0.9 .8075 .4060 ' | 5.9 .1220 -.2951 10.9 -.1881 -.1604 1.0 .7652 .4401 i 6.0 .1506 -.2767 11.0 -.1712 -.1768 1.1 .7196 .4709 61 .1773 -.2559 11.1 -.1528 -.1913 1.2 .6711 .4983 6.2 .2017 -.2329 11.2 -.1330 -.2039 1.3 .6201 .5220 6.3 .2238 -.2081 11.3 -.1121 -.2143 1.4 .5669 .5419 6.4 .2433 -.1816 11.4 -.0902 -.2225 1.5 .5118 .5579 6.5 .2601 -.1538 11.5 -.0677 -.2284 1.6 .4554 5699 6.6 .2740 -.1250 11.6 -.0446 -.2320 1.7 .3980 .5778 6.7 .2851 -.0953 11.7 -.0213 -.2333 1.8 .3400 .5815 6.8 .2931 -.0652 11.8 .0020 -.2323 1.8 .2818 .5812 6.9 .2981 -.0349 11.9 .0250 -.2290 a.o .2239 .5767 7.0 .3001 -.0047 12.0 .0477 -.2234 2 1 .1666 .5683 7.1 .2991 .0252 12.1 .0697 -.2157 2 2 .1104 .5560 7.2 .2951 .0543 12.2 .0908 -.2160 2.S .0555 .5399 7.3 .2882 .0826 i 12 3 .1108 -.1943 2.4 .0025 .5202 7.4 .2786 .1096 12.4 .1296 -.1807 2.5 -.0484 .4971 7.5 .2663 .1352 | 12.5 .1469 -.1655 2.6 -.0968 .4708 7.6 .2516 .1592 ! 12.6 .1626 -.1487 2.7 - 1424 .4416 .2346 .1813 | 12.7 .1766 -.1307 2 ^ - 1 85(1 .4097 7.S .2154 .2014 , 12.8 .1887 -.1114 2 9 -.2243 .3754 7 9 .1944 .2192 1 12.9 .1988 -.0912 3.0 -.2601 .3391 8.0 .1717 .2346 13.0 .2069 -.0703 3.1 — 21121 .3009 8.1 .1475 .2476 13 1 .2129 -.0489 3.2 -.3202 .2613 , 82 .1222 .2580 13.2 .2167 -.0271 3.3 -.3443 .2207 1 8.3 .0960 .2657 13.3 .2183 —.0052 3.4 -.3643 .1792 8.4 .0692 .2708 13.4 .2177 .01B6 3.5 -.3801 .1374 8.5 .0419 .2731 13.5 .2150 .0380 3.6 -.3918 0935 8.6 .0146 .2728 13.6 .2101 .0590 3.7 -.3992 .0538 8.7 -.0125 .2697 13.7 .2032 .0791 3.8 -.4026 .0128 8 8 -.0392 .2641 13.8 .1943 .0984 3.9 -.4018 — .0272 8.9 -.0653 .2559 13.9 .1836 .1166 4.0 -.3972 -.0660 9.0 -.0903 .2453 14.0 .1711 .1334 4.1 -.3887 -.1033 9.1 -.1142 .2324 14.1 .1570 .1488 4.2 -.3766 -.1386 9.2 -.1367 .2174 14.2 .1414 .1626 4.3 -.3610 -.1719 9.3 -.1577 .2004 14.3 .1245 .1747 4.4 -.3423 -.202-! 9.4 -.1768 .1816 14.4 .1065 .1859 4.5 -.3205 -.2311 9.5 -.1939 .1613 14.5 .0875 .1934 4.6 -.2961 -.2566 ! 9.6 - .2090 .1395 14.6 .0679 1999 4.7 -.2693 f -.2791 9.7 -.2218 .1166 14.7 .0476 .2043 4.8 -.2404 -.2985 9.8 -.2323 .0928 14.8 .0271 .2066 4.9 -.2097 -.3147 9.9 -.2403 .0684 14.9 .0064 .2069 5.0 -.1776 -.3276 10.0 -.2459 .0435 15.0 -.0142 .2051 TABLES. 225 Table III. — Roots of Bessel's Functions. n *, for J„(x n ) = x n for J\(x n ) = 71 x n for J,(x„) = x n for J,(x„) = 1 2 3 4 5 2.4048 5.5301 8.6537 11.7915 14.9309 3.8317 7.0156 10.1735 13.3337 16.4706 6 7 8 9 10 18.0711 31.3116 24.3525 27.4935 30.6346 19.6159 22.7601 25 9037 29.0468 32.1897 Table IV.— Values of J„{xi). X J (xi) X J a (xi) X J (xi) 0.0 1.0000 2.0 2.2796 4.0 11.3019 0.1 1.0025 2.1 2.4463 4.1 12.3236 0.2 1.0100 2.2 2.6291 4.2 13.4425 0.3 1.0226 2.3 2.8296 4.3 14.6680 0.4 1.0404 2.4 3.0493 4.4 16.0104 0.5 1.0635 2.5 3.2898 4.5 17.4812 0.6 1.0920 3.6 3.5533 4.6 19.0926 0.7 1.1263 2.7 3.8417 4.7 20.8585 0.8 1 . 1665 2.8 4.1573 4.8 22.7937 0.9 1.2130 2.9 4.5027 4.9 24 9148 1.0 1.2661 3.0 4.8808 5.0 27.2399 1.1 1.3262 3.1 5.2945 5.1 29.7889 1.2 1.3937 3.2 5.7472 5.2 32.5836 1.3 1.4963 3.3 6.2426 5.3 35.6481 1.4 1.5534 3.4 6.7848 5.4 39.0088 1.0 1.6467 3.5 7.3782 5.5 42.6946 1.6 1.7500 3.6 8.0277 5.6 46.7376 1.7 1.8640 3.7 8.7386 5.7 51.1725 1.8 1.9896 3.8 9.5169 5.8 56.0381 1.9 2.1377 3.9 10.3690 5.9 61.3766 226 FUNCTIONS OF A COMPLEX VARIABLE. [CHAP. VI. Chapter VI. FUNCTIONS OF A COMPLEX VARIABLE. By Thomas S. Fiske, Adjunct Professor of Mathematics in Columbia University. Art. 1. Definition of Function. If two or more quantities are such that no one of them, when any values whatsoever are assigned to the others, suf- fers any restriction in regard to the values which it can assume the quantities are said to be " independent." If one quantity is so related to another quantity or to several independent quantities, that whenever particular values are assigned to the latter, the former is required to take one or another of a system of completely determined values, the for- mer is said to be a " function " of the latter. The quantity or quantities upon the values of which the value of the function depends, are said to be the " independent variables " of the function. A function is " one-valued " when to every set of values as- signed to the independent variables there corresponds but one value of the function. It is said to be " ^-valued " when to every set of values of the independent variables n values of the function correspond. The "Theory of Functions " has among its objects the study of the properties of functions, their classification accord- ing to their properties, the derivation of formulas which exhibit the relations of functions to one another or to their independ- ent variables, and the determination whether or not functions exist satisfying assigned conditions. Art. 2.] REPRESENTATION OF COMPLEX VARIABLE. 227 Art. 2. Representation of Complex Variable. A variable quantity is capable, in general, of assuming both real and imaginary values. In fact, unless it be otherwise specified, every quantity w is to be regarded as having the " complex " form u-\-v V — i, u and v being real. It is cus- tomary to denote V — i by i, and to write the preceding quan- tity thus : u -\- iv. If v is zero, w is real ; if u is zero, w is a " pure imaginary." A quantity z = x -\- iy is said to vary " continuously " when between every pair of values which it takes, c i = a 1 -+- ib^ and c i — a i "f" '^ > tne va l ue °f z varies in such a manner that x and y pass through all real values intermediate to a x and it follows that a' = e'^s" — exp(^ log a). 230 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. The following equations also are to be regarded as equations of definition : sin z tan 5* — COS.S cot z — , Kail <© — j cos z sin z I sec z = , i cosec Z = — : , cos.s sin z It may be shown that the formulas which are usually obtained on the supposition that the independent variables are real, and which express in that case properties of and relations between the preceding functions, still hold when the independent variables are complex. Prob. 2. Show that e m e" — e'" + ", m and n being complex. Prob. 3. Deduce cos z = W + e ~")> sin z = '~l e " — ''")• Prob. 4. Deduce cos (z J + 2 2 ) = cos z 1 cos z 2 — sin z, sin z 1 , sin (z, + ^) = cos z, sin z, + sin z, cos z a . Art. 5. Continuity of Functions. Let a function of a single independent variable have a determinate value for a given value c of the independent vari- able. If, when the independent variable is made to approach c, whatever supposition be made as to the method of approach, the function approaches as a limit its determinate value at c, the function is said to be " continuous " at c. This definition may be otherwise expressed as follows : A function of a single independent variable is continuous at the point c, when, being given any positive quantity e, it is possible to construct a circle, with center at c and radius equal to a determinate quantity S, so small that the modulus of the difference between the value of the function at the center and that at every other point within the circle is less than e. A function of several independent variables is said to be continuous for a particular set of values assigned to those vari- ables, when it takes for that set of values a determinate value c, and for every new set of values, obtained by altering the Art. 5.] CONTINUITY OF FUNCTIONS. 231 variables by quantities of moduli less than some determinate positive quantity ^ ff / F E D c \bW 1 j \y 6 / 5 i \ 3 SK 242 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. ^-plane serving to determine its image in the ze/-plane as an intersection of orthogonal curves. Case V. — Let w = z 3 . Writing w = u -\- iv, z = x -\- iy, the relations x — ixy ix y -y follow at once. If one of the variables x, y be eliminated from these two equations by means of the equation Ix -j- my -f- n = o, representing a straight line in the ^-plane, equations are ob- tained representing a unicursal cubic in the zf-plane. By putting w = p(cos

■Qv _ _ Qu dx ~ dy d* ~ dy At any point, therefore, where u and v admit second partial derivatives, one obtains 3*' i- a7 ' 9^97 ; that is, the functions u and v are solutions of Laplace's equa- tion for two dimensions. Any two real solutions p and q of this equation, such that p -f- iq is a monogenic function of x -f- iy, are called " conjugate functions." * Thus the examples of Art. 9 furnish the following pairs of conjugate functions: * Maxwell, Electricity and Magnetism, 1873, vo '- r > P- 227- 246 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. x -|- a, y -\- b ; r t r cos (0, -f- 0), r,r sin (0, -f 0) ; and q will be conjugate functions of x and y. Prob. ii. Show that if u and v are conjugate functions of x and y, x andjy are conjugate functions of u and v. Art. 12. Application to Fluid Motion. Consider an incompressible fluid, in which it is assumed that every element can move only parallel to the .sr-plane, and has a velocity of which the components parallel to the coordi- Art. 12.] APPLICATION TO FLUID MOTION. 247 nate axes are functions of x and y alone. The whole motion of the fluid is known as soon as the motion in the .s-plane is ascertained. When any curve in the £-plane is given, by the "flux across the curve"* will be meant the volume of fluid which in unit time crosses the right cylindrical surface having the curve as base and included between the .s-plane and a par- allel plane at a unit distance. The flux across any two curves joining the points z and z is the same, provided the curves enclose a region covered with the moving fluid. For, corresponding to the enclosed region, there must be neither a gain nor a loss of matter. Let z be fixed, and z be variable. Let ip denote the flux across any curve s z, reckoned from left to right for an observer stationed at z a and looking along the curve toward z. If /, m be the direction cosines of the normal (drawn to the right) at any point of the curve, and p, q be the components parallel to the axes of the velocity of any moving element, the value of ip will be ip = I (lp-\- mq)ds, where the path of integration is the curve joining z and z. The function ip is a one-valued function of z in any region within which every two curves joining z to z enclose a region covered with the moving fluid. If z moves in such a manner that the value of ip does not vary, it will trace a curve such that no fluid crosses it, i.e., a " stream-line." The curves ip = const, are all stream-lines, and ip is called the " stream-function." If p and q are continuous, and if z be given infinitesimal increments parallel to x and y respectively, one obtains dip dip &=-* Ty~ p - If now the motion of the fluid be characterized, as is the * Lamb's Hydrodynamics (1895), p. 69. 248 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. / = case in the so-called " irrotational" motion,* by the existence of a velocity-potential cj>, so that 90 _90 dx' q ~dy' the following equations result : d* ~ dy ' dx~ dy' Hence

-\-iip then becomes — *l> + i (s) are entire and without common factors.. This function is finite and admits an infinite number of suc- cessive derivatives for every finite value of z, except the roots of the equation (s) = o. Let a be such a root. Then the reciprocal of the given function is finite and admits an infinite nnmber of successive derivatives at the point a. Such a point * J. J. Thomson, Recent Researches in Electricity and Magnetism (1893), p. 208. f Love, Theory of Elasticity (1892), vol. 1, p. 331. Art. 13.] critical points. 251 is called a "pole." Any rational function having a pole at a can be put by the method of partial fractions in the form w = -A_ + . . . + A - +^( g ), s — a ' {z — ay ' T x ' where A lt . . ., A k are constants, A e being different from zero, and ip(z) is finite at the point a. The integer k is said to be the " order " of the pole, and the function is said to have for its value at a infinity of the £th order. In accordance with the definition of a derivative, w does not admit a derivative at a. From the character of the derivative in the immediate neighborhood of a, however, the derivative is sometimes said to become infinite at a. The trigonometric function cotz has a pole of the first order at every point z — run, m being zero or any integer posi- tive or negative. The function w = log (z — a) has for every finite value of z, except z = a, an infinite number of values. If z — a is writ- ten in the form Re i% , w = log R -f- i{® -)- 2tmi), where log R is real, and m is zero or any positive or negative integer. If z describes a straight line, beginning at a, © will remain fixed, but R will vary. The images in the ^r-plane will therefore be straight lines parallel to the axis of reals, dividing the plane into horizontal strips of width 2n. If now the .sr-plane is supposed to be divided along the straight line just drawn, and z varies along any continuous path, subject only to the restriction that it cannot cross this line of division, there will be a continuous curve as the image of the path of z in each strip of the w-plane. Each of these images is said to corre- spond to a " branch " of the function, or, expressed otherwise, the function is said to have a branch situated in each strip. The line of division in the .s-plane, which serves to separate the branches from one another is called a " cut." 252 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. At the point z = a no definite value is attached to the function. As z approaches that point the modulus of the real part of the function increases without limit, while the imagi- nary part is entirely indeterminate. Let z be an arbitrary point, distinct from a, and let log R„ -f- 2(9 -\- 2mni be any one of the corresponding values of the function. Sup- pose that z starts from z and describes a closed path around the point a, the values of the function being taken so as to give a continuous variation. Upon returning to the point z the value of the function will be lo g R * + i&o + 2 ( m + l ) ni > or log i? -j- t© -\-2{m — i)ni, according as the curve is described in a positive or negative direction. By repeating the curve a sufficient number of times it is evidently possible to pass from any value of the function at z to any other. When a point is such that a ^r-path en- closing it may lead in this manner from one value of a function to another value, it is called a " branch-point." In the case of the function here considered, the point z = a is called a "logarithmic branch-point," or a point of "logarithmic discontinuity." f(z) The function w = log -j\, where f(z) and +'* = c + id. If now a -f- ib is the reciprocal of / -|- iq, so that — f h - ~~ 1 a ~fT7' /+?' the preceding equation may be written i e a + ib — c _j_ j£ But whatever the value of the integer m, q -\- 2mit may be substituted for q without altering the value of c -\- id, and hence both a and b may be made less than any assignable quantity. The given function e^ therefore takes the value c -\- id at points a -\- ib indefinitely near to the origin. A point such that, when z approaches it, a function elsewhere one-valued tends toward an indeterminate limiting value is called an " essential sin- gularity." i Prob. 12. Show that for the function e z ~ a z — a is an essential singularity. i_ Prob. 13. The function e z% considered as a function of a real variable is continuous for every finite value of z, and the same is true of each of its successive derivatives. Show that when it is regarded as a function of a complex variable, z = o is an essential singularity. In order to illustrate still another class of special points take the function w = V{z — «,)(£ — «„)... (z — a„). 254 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. This function has at every finite point, except <*,,«,,..., a„, two distinct values differing in sign. At these points, however, it takes but a single value, zero. From each of the points a x , a, ,...,«„ let a straight line of indefinite extent be drawn in such a manner that no one of them intersects any other, and suppose the £-plane to be divided, or cut, along each of these lines. Along any continuous path in the .s-plane thus divided the values of the function form two distinct branches. For, writing z — a 1 = r^i, z — a, = r t e'\ . . . , z — a n = r n e'\ the function takes the form No closed path in the divided plane will enclose any of the points a lt a 1t . . . , a n , and the quantities 0, , ft,, . . . , ft,, after continuous variation along such a path, must resume at the initial point their original values. No such path, therefore, can lead from one value of the function at any point to a new value of the function at the same point. If, however, the cuts are disregarded and z traces in a positive direction, a closed curve including an odd number of the points a lt a,, . . . , a„, and not intersecting itself, then an odd number of the quantities ft, ft,, ... , ft, are each increased by 27r; and the value of the function is altered by a factor ^(S*+i)<»* i and so changed in sign. In the same way any closed path de- scribed about one of these points, and enwrapping it an odd number of times, leads from one value of the function to the other. On the other hand, a simple closed path enclosing an even number of these points, or a closed path which en- closes but one of the points, enwrapping it an even number of times, leads back to the initial value of the function. It fol- Art. 13.] critical points. 255 lows that each of the points a x , « 2 , . . . , a„ is a branch-point. Any point in the £-plane, closed paths about which lead from one to another of a set of different values of a function, the number of values in the set being finite, is called an "algebraic branch-point." As a further illustration, consider the function w = z* -\- [z — a)i, which is a root of the equation of the sixth degree, w° — $ziv* — 2(2 — a)w % -J- 3^w 2 — 6z(z — a)iv -\- (z—a)'—z*=o. The function has at every point, except s = o and z = a, six distinct values. Six branches are thereby formed which can be completely separated from one another by making cuts from the points z = O and z = a to infinity. Putting 00 for the cube root of unity, these six branches can be written w, = z -f (z — a) 1 , w,= — z + (z — a)', w 3 = z' + oo (z — a) 1 , w, =± — z -\-w{z — a)' , w t = z yi -\- oo'(z — {z) Prob. IS- Show that z = oo is an ordinary point for -rr~,, where $(z) th series of points occur. It may be shown that as the number of points in each of the series #,,... and z', ... is increased, the differences 5" — 5 and 5" — S' both approach zero, from which it follows that the difference S — S' has a limit equal to zero. For example, the difference S" — S has the value (*, - z )[f(r ) -/(A)] + (*,' - *tiA*d -/W] + (V-^')[/W-/(0] + -.. If M denotes the upper extreme of the quantities |/(O-/(0l- \A*J-M)\, W,)-M)l- •■ the modulus of S" — S will be less than dx dy . , . i any parameter t so that — — and — are continuous. For then the integral dt dt I \i ' dx 1 -\- dy 1 is finite. See, in this connection, Jordan, Cours d'Analyse, 2d Edition, Vol. I., p. too. Art. 15.] integral of a function. 259 But \z l — s \ is equal to the chord of the arc z^ lt and must therefore be less than or equal to this arc, and a similar result holds for each of the quantities 1 3/ — z 1 \ , \ z,' — z/ 1 , . . . Hence I S" - S\ " A*>) - A*,), /W - Ml • • • decrease indefinitely, ior f(z) is continuous. M acccordingly decreases indefinitely and the difference S" — S approaches zero. The limit, the existence of which has just been demon- strated, is called the integral of f{z) along the path L. It is written I f(z)ds. The definition here given is similar to that given for the integral of a function of a real variable. It is unnecessary to specify the path of integration when the inde- pendent variable is restricted to real values, since in that case it must be the portion of the axis of reals included between the limits of integration. The following well-known principles, applicable to the case of a real independent variable, may be readily extended to the general case : 1. The modulus of the integral cannot exceed the length of the path of integration multiplied by the upper extreme of the modulus of the function along that path. 2. The independent variable may be altered by any equa- tion of transformation, but L ', the path of integration in the transformed integral, must be such that it is described by the new variable while z describes L. 3. If F(z) is any one-valued function having everywhere f{z) for its derivative, the equation I J(z)dz = F{Z)-F{z;) must be true. 260 FUNCTIONS OF A COMPLEX VARIABLE. [CHAP. VI. To prove the third principle, write F(Z) — F(z a ) in the form F(Z)-F(z H )+F(z„)-F(z x . 1 )-\-. . . +F(z,)-F(z 1 )+F(z 1 )-F(z,). Since the derivative of F(z) isf(z), F(z m+1 ) - F{z m ) — [/(*„) + n„,](z m+1 - Zm), where rj m has zero for its limit when z m+l is made to approach z m . Hence F(Z) - F(z ) — limit 2f(z m )(z m+l - z, n ) + limit 2f/ m (z m+1 - z m ) ; or, since the second term of the right-hand member is equal to zero, F(Z)-F(z ) = jy(z)dz. If no function F{s) fulfilling the preceding conditions is known, the value of the integral requires further investi- gation. ■'dz ~? point z = — I to the point z = I, the path of integration being the upper half of the circumference of a unit circle described about the origin as a center. Writing z = exp (iff), z wilL describe the required path while 6 varies from JTto o. The equations — , = e' 2ie , dz = ie ie dd, z dz — = ur*dff = i cos B dff + sin B dd = id (sin B) — d (cos ff), follow at once. Hence for the path specified + ndz X X J —2 = t I d (sin ff)— id (cos ff) = — 2. The application of the direct and more familiar method gives the same result : /dz —i taken from the r — — -~l r i J z 2 ~ L z J a=1 L — z /:: i ! i = — 2. Art. 16.] reduction of complex integrals to real. 261 For a path along the axis of reals between the limits of integration this result is unintelligible. The discontinuity of dz the differential, — , at the point z = o, prevents the considera- tion of such a path ; and that the result should be negative when the differential is at every point of the path positive has no significance. The introduction of the complex variable furnishes a perfectly satisfactory explanation of the result. dz Prob. 16. Show that the integral of — along any semi-circum- ference described about the origin as a center is equal to ni. Art. 16. Reduction of Complex Integrals to Real. The integral f L Az)dz may be written in the form I {u -)- iv){dx -f- idy), or, separating the real and imaginary terms, / (iidx — vdy) -\- i {vdx -f- udy). Hence the calculation of the integral may be reduced to the calculation of two real curvilinear integrals. The equations dx~ dy du _ dy _ dv dx which express the condition that u -\-iv should be monogenic, express also that udx — vdy, vdx -\- udy are the exact differentials of two real functions of the variables x, y. Consider the case where these functions are one-valued. 262 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. Denoting them by P(x, y) and Q(x, y) respectively, the inte- gral may be written [P(X, Y) - P(x ,y )] + iiQ{X, Y) - Q(x ,y )], (x ,y a ) and (X, Y) being the initial and terminal points re- spectively of the path of integration. Art. 17. Cauchy's Theorem. Cauchy's Theorem furnishes the necessary and sufficient conditions that a one-valued function f(z), having a continuous derivative/'^), should yield a one-valued integral, that is, an integral the value of which, when the lower limit is fixed, de- pends simply on the upper limit, and not on the path of integration. It will be more convenient, before considering Cauchy's Theorem, to demonstrate the following lemma: Lemma. — Let^4 be a portion of the ^-plane, having a bound- ary S which consists of a closed curve not intersecting itself, or of several closed curves not intersecting themselves or one another. Denote by X the inclination to the axis of x of the exterior normal at any point of the boundary,* that is, the normal drawn to the right as the boundary is described in a positive direction. If at every point of the region A, including its boundary S, a function W ai the real variables x and y is one-valued and continuous and has continuous partial deriva- tives , — , the relations dx dy Wdy=J J^dxdy, (i) I Wdx = - I j d —dxdy (2) exist, the integrals in the first members being taken along the * It \i assumed that the boundary has a determinate tangent at every point. If the boundary of a given region is not of this sort, the theorem holds for any interior curve of which this assumption is true. Art. 17.] cauchy's theorem. 263 boundary in the positive direction, and those in the second members being taken over the enclosed area. If any straight line parallel to the axis of x be traced in the direction of increasing values of x, at each point where it passes into the area A, cos A is negative, and there- fore in the first member of (i) dy = cos\ds is negative. At each point where this straight line passes out of the area A, cos A, and there- fore dy, in the first member of equation (i), is positive. Hence in the first member of equation (i) the differ- entials Wdy corresponding to a given value of y, and taken in the order of increasing values of x, have signs which, compared with those of the corresponding values of W, first differ, then agree, and so on alternately. In order now to compare the integral in the first member of equation (i) with the integral in the second member, it is necessary to take dy as essentially positive. The sum of the differentials in the first member, correspond- ing to a fixed value of y, must therefore be written in the form dy{ -fv i+Wt -w t +w t - ...), where W l , W,,... are the corresponding values of W taken in the order of increasing values of x. But performing now in the second member of equation (i) an integration with respect to x, the same result is obtained, so that the two members of equation (i) become identical, and the equation is verified. To obtain equation (2) the same method is used. It is necessary in this case to observe that if a line parallel to the axis of y is traced in the direction of increasing values of y, at each point where it enters A, dx in the integral of the first 264 FUNCTIONS OF A COMPLEX VARIABLE. [CHAP. VI. member must be taken as positive; and at each point where this line passes out of A, dx in that integral must be taken as negative. By means of the preceding lemma, Cauchy's Theorem is easily proved. This theorem may be stated as follows : Theorem. — If, on the boundary of and within a given region A, a one-valued function w — f{z) is monogenic, and its deriv- ative f'{z) is continuous,* the integral I f{z)dz taken along the boundary S is equal to zero. For writing the integral in the form / wdz — I {udx — vdy) -f- i I udy -\- vdx), the preceding lemma gives {udx - vdy) = -JJJ^^dxdy, L {udy -\- vdx) = I I — — —-\dxdy; dx dy but since at every point of A csu ,dv _ 3« Qv _ dy~r~dx~°' dx~dy = C ' the given integral reduces to zero. Art. 18. Application of Cauchy's Theorem. From Cauchy's Theorem it follows that, if two different paths Z, and L, lead from the point z a to the point Z, and if along these paths and in the region inclosed between them a given function f{z) has no critical points, the integrals of the function taken along these two paths are equal. For two such paths taken together, one described directly, the other re- versed, constitute a closed curve, and the integral taken along *Otherwise expressed, the one-valued function j\z) has no critical 1 points on the boundary of or within A, ox f(z) is holomorphic in A. Art. 18.] application of cauchv's theorem. 265 it is equal to zero. But, since reversing the direction of the path of integration is equivalent to changing the sign of the integral, the equation JjW* - fjw, = o is obtained. The result just established may be stated in the following theorem : Theorem I. — If a function is holomorphic in any simply connected region bounded by a continuous closed curve, the integral of the function, from a fixed lower limit in that region to any point contained therein, is independent of the path of integration, and is a one-valued function of its upper limit. A region whose boundary is composed of disconnected curves is not necessarily characterized by the property stated in the theorem. Take, for example, the function w — V(z — a t ){2 — a t )...(z — a n ), and suppose that o < | Z ftZ+dZ £ fi*y* - j z Az)dz = j z f{z)dz =f(Z)j z dz + j z [A*)-AZ)~\d*- The first term is equal to f{Z)dZ. The modulus of second' term is equal to or less than M\ dZ\, where M is the upper ex- treme of \/{z) — f{Z) | along the line joining Z to Z + aZ. But since f(z) is continuous, the limit of M when Z-\-dZ approaches Z is zero. Hence J^ iZ f(z)dz - f z *A*y* = U{Z) + rj\dZ, where rj approaches zero with dZ. The integral therefore has f{Z)lor& derivative, and is holomorphic in 5. In the case of a region bounded by several disconnected: closed curves, of which one is exterior to all the others,, Cauchy's Theorem may be stated in the following form : Theorem III. — Let a function f\z) be holomorphic in a region A bounded by a closed curve C and one or more closed curves C lt Q, . . . interior to C. The integral of f[z) taken' along C will be equal to the sum of its integrals taken in the same direction along the curves C,, 6" , . . . For the integral of f(z) taken in a. positive direction completely around the boundary of A is equal to zero. But tne curves (7,, C„ . . . are then described in the direction oppo- Art. 19.] theorems on curvilinear integrals. 267 site to that in which C is described. Hence if all the curves are described in the same direction, the result may be written J' c f(z)dz =J c f{z)dz +f c A*)dz + ... If there is but one interior curve, so that the region A is included between two curves C and C lt the integral taken along every closed curve containing C, but interior to C has the same value, viz., the common value corresponding to the paths C and C x . Art. 19. Theorems on Curvilinear Integrals. Theorem I. — If f(z) be continuous in a given region except at the point a, the integral I f(z)dz, taken around a small circle c, having its center at a, will approach zero as a limit simulta- neously with the radius r of the circle c, provided only lim (z — d)J\z) = O when z = a. For let the upper extreme of the modulus of (z — d)f(z) on the circle c be denoted by M. Then at every point of c, v _ M _M mod f(z) — ■ 1 — — , and consequently c -M r mod J c f(z)dz ~ — J ds -^2nM. /dz -. r-, taken around any (z — ap closed curve C containing the point a, is equal to zero, except when n — i. When n = I, this integral is equal to 2ni. For the value of the integral will be the same if any circle described about a as a center be taken as the path of integration. Let then s — a = re' e , where r is a constant and B varies from o to zn. The integral becomes ~« - 1 j 268 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. which reduces to zero except when n = I. If n — I, its value is 2ni, whence / dz = 2711. Theorem III. — If f{z) is a function holomorphic in a given region S, C a closed curve the interior of which is wholly within S, and a a point situated within C, then f £&-ds = 27tif{a). Jc z— a J For describing about a as a center a small circle c of radius r, the equation PJVUp = fJ^Ldz v° z — a r. Let M be the upper extreme of the modulus/^) on C, and / the length of C. Then n fi 2 \ _ r M mod J c { g -a)* +t d * 27iiJ c (Z-aY + \Z~a-t) *" By taking n sufficiently great the modulus of R may be made less than any given positive quantity. Let M be the upper extreme of the modulus of f(z) on the circle C, p the modulus of t, and r the modulus of C — « or radius of C. Then < 27rJ c r" +1 (r—p) < r — p\r which, since p < r, has zero for its limit when n = oo . Writing now z for a-\-t, Taylor's Series becomes f(z)=f(a)+(z-a)f'(a)+^^f"{a)+. . .+ ( -£^f£/M( fl )+. . . The series is convergent and the equality is maintained for every point z included within a circle described about a as a center with a radius less than the distance from a to the nearest critical point oi f{z). When a is equal to zero, Taylor's Series takes the form f{z) = /(o) + zf{o) + ~f"{o) -f- . . . + Y7~^i f[nKo) + • ' • ' expressing/^) in terms of powers of z. This form is known as Maclaurin's Series. Art. 21.] Laurent's series. Art. 21. Laurent's Series. 271 Theorem. — Let S, a porticn of the ^-plane bounded by two concentric circles C x and £,, be situated in the interior of the region E, in which a given function f{z) is holomorphic. If a be the common center of the two circles, and a + t a point interior to S, /(a -\- t) can be expressed in a convergent double series of the form m = oo W/ = — CO With a -\- t as a center construct a circle <: sufficiently small to be contained within the region 5. If then C, be the greater of the two given circles, it follows from Article 18 that AZ)dZ AQdZ 2m " Ci Z — a — t 2ni u d Z — a — t But from Article 19, _L f MYZ O-rri Z — a-t' The two integrals of the right-hand member may be written _ r 2_ /- /(cyg r / (C-*)-J 1 (c-«)' 1 *»+» where # -A t*+'f(zyz R, 27tiJc*f ,+i ■ (Z-a-ty But |tf| < |£ — «| at every point of C,, and |*|> |C — «| at every point of C„ so that ^?, and i? 2 both have zero for a limit 272 FUNCTIONS OF A COMPLEX VARIABLE. [CHAP. VI. when n = oo . The value oif(a -j- t) can therefore be expressed in the form A_ t A., A^ ^ t ^ f ^ f ^•" - Since in the region S the function f{z)/(z — a) m+1 is holomor- phic for both positive and negative values of m, A m may be written 2mJ (C — a) + where C is any circle concentric with C, and C t and included between them. The series thus obtained is convergent at every point a A-t contained within the region S. It is important to notice, how- ever, that when the positive and negative powers of t are con- sidered separately, the two resulting series have different regions of convergence. The series containing the positive powers of t converges over the whole interior of the circle C, ; while the series of negative powers of t converges at every point exterior to the circle C v The region 5 can be regarded, therefore, as resulting from an overlapping of two other regions in which different parts of Laurent's Series converge. Writing z for a -f- t, Laurent's Series takes the form f{z) = A a + A,{z - a) + A& - df + . . . + A_ X (z- a )- l + A_ 2 (z _*)-■+... Consider as a special numerical example the fraction J = L_ i | i {Z - I) {Z — 2) (Z— 3) 2(3-1) Z — 2" 1 " 2{Z -3) If |^| < i, all three terms of the second member, when developed in powers of z, give only positive powers. If 1 < I -sr I < 2, the first term of the second member gives a series' of negative descending powers, but the others give the same series as before. If 2 <\z\< 3, the first and second terms both give negative powers. If \z\ > 3, all three terms give Art. 22.] Fourier's series. 273 negative powers, and the development of the given fraction can contain no positive powers. Thus a system of concentric annular regions is obtained in each of which the given frac- tion is expressed by a convergent power-series. Laurent's Series gives analogous results for every function which is holo- morphic except at isolated points of the .a-plane. Art. 22. Fourier's Series. Let w = f(z) be holomorphic in a region S , and let it be periodic, having a period equal to go, so that f(z -\- noo) = f{z), where n is any positive or negative integer. Denote by S n the region obtained from S a by the addition of noo to z ; and sup- pose that the regions . . . , S.„, . . . , 5., , S, , S, , . . . , S„ , . . . meet or overlap in such a manner as to form a continuous strip 5, in which, of course, the function w will be holomorphic. Draw two parallel straight lines, inclined to the axis of reals at an angle equal to the argument of go, and contained within the strip 5. The band T included between these parallels will be wholly interior to S. 2-niz By means of the transformation z' = e ™ the band T in the ^-plane becomes in the ^'-plane a ring T' bounded by two concentric circles described about the origin as a center, z and z -\- noo falling at the same point z'. Since w is holomorphic in a region including T, and d w dw dz go 217-iz dw dz' dz dz' 2ni " dz' w regarded as a function of z' will be holomorphic in T'. Hence, by Laurent's Theorem, w = ~2 A m z"», the quantity a in the general formula of the preceding article being in this case equal to zero. Substituting for z' its value, the preceding equation becomes ,«=» 2mni * w = 2 A m e " , 274 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. where i rwdz' i n z+ "' _ 2g ""'' Am = ^dJc^ = vJ s e " Wdz - In the latter integral the path is rectilinear. Denoting its independent variable by C for the purpose of avoiding confu- sion, the value of w becomes l »»=» />f-h" """•»' ,. ,,, »« = - co » = -J mx + l^r co S ^ ( ,-c)/(ck ^+tu 2JHTT* = - / /(QdQ + - ^ cos / cos M)dZ 2 *"°° . 2mnz /><+•» 2mnC J«=l Art. 23. Uniform Convergence. Let the series W= w -f- w, -f- w 2 +• • • + »» + . • . , each term of which is a function of z, be convergent at every point of a given region 5. Denote by W n the sum of the first n terms of W. If it is possible, whatever the value of the posi- tive quantity e, to determine an integer v, such that whenever n > v \W- W n \ I. If now this series be considered for the points within and upon a circle described about the origin as a center with an assigned radius less than unity, the remainder after n terms, z n or I — W n = can, by a suitable choice of n, be made I + z less in absolute value than any given quantity. In such a region, then, the series converges uniformly, and, by Theorem I, can have no point of discontinuity. A similar result holds for the region exterior to any circle described about the origin as a center with an assigned radius greater than unity. By means of Theorem II given above it can be shown that Laurent's Series is unique. For, assuming the notation used in the determination of the series, the series is uniformly con- vergent in the region included between any two given circles concentric with C, and Q , both being interior to C l and ex- terior to C t . Suppose, now, that two such series are possible : m = oo Hi = co /(a + t) = 2 Aj m = 2 A m 'r. m = — oo Divide by f + x , and integrate along any circle described about aasa center and included in the region of uniform converg. ence. The integral it" 1 ' K ~ x dt for such a path is zero, except when m — n\ the integral / t _1 dt = 2in. Hence for such a path, rga + W = 2inA = 2inA ,. 278 FUNCTIONS OF A COMPLEX VARIABLE. [CHAP. VI. from which it follows that A n = A„', and the two series are identical. Art. 24. One-valued Functions with Critical Points. Theorem I. — A function holomorphic in a region 5 and not equal to a constant, can take the same value only at iso- lated points of 5. For in the neighborhood of any point a interior to 5, by Taylor's theorem, f(z) -/0) = (* - «)/'(«) + { -^f /"(a) + ... Unless f{z) is constant over the entire circle of convergence of this series, the derivatives f'(a), f"{ a )> ■ • ■ cannot all be equal to zero. Let f-"\a) be the first which is not equal to zero. Then fiz) - /(a) = (z-a)"[ ^ ){d) 1 f" +1 \ a ) ( s - a )+ . . ." JK ' J v ' K ' \_\ . 2 . . . n ' i . 2 . . . (n -f- \y ' ' If \z — a\ be given a finite value sufficiently small, the modulus of the first term of the series within the brackets will exceed the sum of the moduli of all the other terms, and the same result will hold for every still smaller value of \z—a\. For values of z, then, distant from a by less than a certain finite amount, f(z) — f{d) is different from zero. If, on the other hand, the function is constant over the en- tire circle, described about a as a center, within which Taylor's series converges, it will be possible, by giving in succession new positions to the point a, to show that the value of the function is constant over the whole region S. Theorem II. — Two functions which are both holomorphic in a given region 5 and are equal to each other for a system of points which are not isolated from one another, are equal to each other at every point of 5. For let f(z) and (p(z) be two such functions. By the pre- ceding theorem, the difference/^) — /"(«) + • • • + ,%' , a \/ "V) + ■■■ But by Article 20, r being the radius of any arbitrary circle having its center at a, and M being the upper extreme of the modulus of f(z) on the circumference of this circle, , , w , x = I . 2 . . . nM mod f\a) < . But M is always finite, and r may be made indefinitely great. Hence /'"'(«) = o for all values of n, and /0)=/(4 Theorem IV. — If a function /(.s), holomorphic in a region 5, is equal to zero at the point a situated within 5, the function can be expressed in the form /0) = (* - a)'"" w = ^'\ where g(z) — h{z) -f log f(z ). Theorem X. — If two functions f{z) and {z)^\ where g{z) is holomorphic in every finite region of the 2-plane. For the ratio of the two functions has no zeros and no critical points in the finite portion of the .s-plane. Art. 25. Residues. If a one-valued function has an isolated critical point a, it is expressible by Laurent's series in the region comprised be- tween any two concentric circles described about a with radii less than the distance from a to the nearest critical point. Hence in the neighborhood of a f(z) = A,-\-A 1 (z-a) + A,(z-ay + ... + £,(*- *)-' + £,(* -«)- + ... The coefficient of (z — a)-' in this expansion is called the "residue" oi f(z) at the point a. If any closed curve C including the point a be drawn in the region of convergence of this series, and f{s) be integrated along C in a positive direction, the result will be /( (fz)dz = 2ntB 1 . ' c The following may be regarded as an extension of Cauchy's theorem : Theorem I. — If in a region 5 the only critical points of the one-valued function f{z) are the interior points a, a', . . , the Art. 25.] residues. 28S integral / f(z)dz taken around its boundary C in a positive direction is equal to f c f{x)dz = 2ni(B + B' + ...), where B, B' , . . . are the residues of f(z) at the critical points. For the integral taken along C is equal to the sum of the integrals whose paths are mutually exterior small circles de- scribed about the points a, «',... The following theorems are immediate consequences of the preceding : Theorem II. — If in a region having a given boundary C the only critical points of the one-valued function f(z) are poles interior to C, an equation flMds = 2in{M- N) JcA*) exists, M denoting the number of zeros and N the number of poles within C, each such point being taken a number of times equal to its order. For in the neighborhood of the point a f(z) = (* - a) m 4>(z) where {a v )/ij)'(a v ) = o. (See Prob. 18, Art. 25.) Art. 27. Weierstrass's Theorem. Any rational entire function of z, having its zeros at the points a lt a„ . . . , a m , can be put in the form A(z- «,)"■(* - O* 3 ...(«- «>, where yi is a constant and «,, n„ . . ., n m are positive integers. More generally, any function which has no critical point in the finite portion of the £-plane and has the points a x , . . ., a m as its zeros, is of the form <*«(* - «,)«. ...(z- a m f m , whereas') is holomorphic in every finite region. The extension of this result to the case where a function without finite critical points has an infinite number of zeros is due to Weierstrass. It is effected by means of the following theorem : Theorem. — Given an infinite number of isolated points a x , 288 FUNCTIONS OF A COMPLEX VARIABLE. [CHAP. VI. a it . . ., a,„ . . ., a function can be constructed holomorphic ex- cept at infinity and equal to zero at each of the given points only.* For the given points can be taken so that l«,i \a H \ increasing indefinitely with n. Consider the infinite product WW ^) = ? I 1 -£)'"- where P K (z) denotes the rational entire function Any factor may be written in the form But since ( z \- r dz z z " r z " dz log V~^J-~J« a n - z ~ ~ ^ ~' " ~ ^C ~J° a n n {a n -z) r the path of integration being arbitrary except that it avoids the points a lt a„ . . ., the product may be expressed as z*dz IJe^'K in which if>„(z) = —J In any given finite region of the ^-plane it will be possible to assume that | z\ = p < «,„ , since \a n \ increases indefinitely with n. Divide the product into two parts, n(i — \e p ^\ m^'\ 1 V ««/ m The second part is equal to m . e * The following proof is taken from Jordan, Cours d'Analyse, 2d edition,- Vol. II. Art. %l. J WEIERSTRASS'S THEOREM. 289 Consider the series 2ip n {z) and 2ip„'(z), each term of the sec- m m ond being the derivative of the corresponding term of the first. In the given region I0.'(*)l = \a,n\"{\a m \ — p)' a n \a n — z) oo Each term of 2ipJ(z) is accordingly less in absolute value than ill the corresponding term of a convergent geometrical progres- CO sion independent of z. The series 2$«'{s), therefore, converges ill oo uniformly. The series ^ip„{z) also converges, since I tp„(z) I = mod f",h K '(s)ds = - , ., P " 1 , r where / denotes the length of the path of integration. oo By Theorem IV, of Article 23, the series 2ip„(z) represents in the given region a holomorphic function. The exponential e also must be holomorphic. The other part of the product in - 1 a. 17J containing only a finite number of factors is everywhere holo- morphic, vanishing at all of the points a x , a„ . . ., which are situated within the given finite region. But this region may be extended arbitrarily. The product therefore fulfils the re- quired conditions. In the preceding demonstration it was tacitly assumed that none of the given points «,, «.,, . . . was situated at the origin. To introduce a zero at the origin it is necessary merely to mul- tiply the result by a power of z. The most general function without finite critical points 290 FUNCTIONS OF A COMPLEX VARIABLE. [Chap. VI. having its only zeros at the given points a x , a t , . . ., a„ . . ., can be expressed in the form f{z) = ^n(i - — V» w . where g{z) is holomorphic except at infinity; for the ratio of any two functions satisfying the required conditions is neither infinite nor zero at any finite point. By means of Weierstrass's theorem it is possible to express any function, F(z), whose only finite critical points are poles as the ratio of two functions holomorphic except at infinity. For, construct a function tp(z) having the poles of F(z) as its zeros. The product F{z). ip{z) = = I . Prob. 21. If ffii, and a>, be two quantities not having a real ratio, the doubly infinite series of which the general term is ; rr is absolutely convergent if p > 2. Hence show that the product \a n \ increasing without limit when n is increased indefinitely. Let, further, G„[ J be the series of negative powers of \Z — (X n i Art. 28.] mittag-leffler's theorem. 293 z — a n contained in the expansion oif{z) according to Laurent's Series in the neighborhood of a„. The function G„i j, having no critical point except at a„, may be developed by Maclaurin's series in the form G »{jzr^) = A ° w + A ^ z + ■■■ + A »" )z + • • • • and the series will converge uniformly within a circle described about the origin as a center with any determinate radius pn < |0«|- Within the same circle Maclaurin's series, applied to G„'( ), the derivative with respect to z of G„l \z — a„) \z — a„ converges uniformly. Hence, for any point within the circle \Z\ = Pn, G {z-±i) = w + *> G iz-^r) = *'<*> + *> F n (z) representing the first v -j- i terms of the development of G„( ] by Maclaurin's theorem, F n '{z) its derivative, and \z — a J R, R' remainders which by a suitable choice of v may be made less in absolute value than any given quantity. Choose the positive quantities £,,£,,. . .,£,,.. .so that the series E } -\- £, -\- . , . -\- £„ -\- . . . is convergent. Choose also in connection with each of the points a 1 , a lt . . ., a n , . . ., an integer v such that mod \g x [-L-^ _ Fiz)\ <£„ mod [^'(^) -*"/(*)] < ^ - if k! '{ih) -™] ' 'f I* I if kl in which the character of each critical point is exhibited. As an application of Mittag-Leffler's theorem consider cot z. Its critical points are z =0, ± n, ± 2n, .... In the neigh- borhood of z = O, cot 2 is holomorphic; and in the neigh- z borhood of z = nn, n being any positive or negative integer, is holomorphic. The series cot z — nn + » z— nn in which m is an arbitrary positive integer, is not convergent for finite values of z, even when \z\>'(cos x -\- i sin x (x^iy — nnf \(cos 2jr-)-2sin 2x)— &< When y = ± oo the first and last terms of the second member vanish. In regard to the series it can be proved that, Art. 28.] mittag-leffler's theorem. 29'? for any given region is which y is finite and different from zero, an integer r can be found such that the sum of the moduli of those terms for which |«| > y is less in absolute value than any previously assigned quantity e. As \y\ is increased the modulus of each of these terms is diminished. The modulus of their sum, therefore, cannot exceed e when y =±00. But whenj/=±oo the sum of any finite number of terms of the series is zero. Hence the limit of the whole series is zero. G'{z), therefore, never becomes infinite. Hence, by Theorem 111, Article 24, it is constant, and is equal to zero. It follows that G(s) is equal to zero. The expression for cot z is accordingly COt £=--[■' q z ^- Lz — nn nit — 00 The logarithmic derivative of the product expression for sin z, given in the preceding article as an example of Weier- strass's theorem, is 1 , 1 I + °° cot z=g'{z) + - + ," -Z — nn nn _ Hence g{s) in that expression is a constant. Making z = o, its value is seen to be unity. Prob. 22. From the expression for cot z deduce the equation + « COSeC 2 = ^> -, :j, ^ (z — nn)' where the summation does not exclude n = o. Prob. 23. Show that the doubly infinite series P(*) = 7.+ 1 _(z — a>y a? where 00= moo 1 -\- nw^ , defines a function whose only finite critical points are z = 00. This function is Weierstrass's ^-function. (Com- pare Problem 21.) Prob. 24. Prove that 72 *>(*) = - ^1 log '{z) — — 22- r- H where the summa- (z — 00) tion does not exclude oo = o. Art. 29. Critical Lines and Regions. The functions whose properties have been considered in the preceding articles have been assumed to have only isolated critical points. That an infinite number of critical points may be grouped together in the neighborhood of a single finite point is evident, however, from the consideration of such ex- amples as w = cot - , w = e cosec — . z In the former an infinite number of poles are grouped in the neighborhood of the origin. In the latter an infinite num- ber of essential singularities are situated in the vicinity of the point z = a. It is easy to illustrate by an example the occurrence of lines and regions of discontinuity. Take the series* *> = 7^+7=7 + 7^ +;£-, + - The sum of its first n terms is 1 which converges to unity if [ -sr | < 1, and to zero if \z\> 1. Hence the circle |#|= I is a line of discontinuity for this series. Consider now any two regions 5, and S,, the former situated within, the latter without, the unit circle. Let {£)t){z) + rp{z)\_l-0{z)-\ * This series is due to J. Tannery. See Weierstrass, Abhandlungen aus der Functionenlehre (1886), p. 102. Art. 29.] critical lines and regions. 299' will be equal to or dy = f{x)dx, (i) ■of which the general solution is written in the form y - j f(x)dx, (2) it is the object of the Integral Calculus to reduce the expres- sion in the second member of equation (2) to the form of a known function of x. When such reduction is not possible, the equation serves to define a new function of x. In the extension of the processes of integration of which the following pages give a sketch the given expression for the derivative may involve not only x, but the unknown function y ; or, to write the equation in a form analogous to equation (1), it may be Mdx + Ndy = o, (3) in which ikfand N are functions of x and y. This equation is in fact the general form of the differential equation of the first order and degree; either variable being taken as the independ- ent variable, it gives the first derivative of the other variable 304 DIFFERENTIAL EQUATIONS. [Chap. VII. in terms of x and y. So also the solution is not necessarily an expression of either variable as a function of the other, but is generally a relation between x and y which makes either an implicit function of the other. When we recognize the left member of equation (3) as an "exact differential," that is, the differential of some function of x and y, the solution is obvious. For example, given the equa- tion xdy -\-ydx = o, (4} the solution xy = C, (5), where C is an arbitrary constant, is obtained by " direct inte- gration." When a particular value is attributed to C, the result is a " particular integral ; " thusj = x~ x is a particular integral of equation (4), while the more general relation expressed by equation (5) is known as the " complete integral." In general, the given expression Mdx -j- Ndy is not an ex- act differential, and it is necessary to find some less direct method of solution. The most obvious method of solving a differential equation of the first order and degree is, when practicable, to '■ separate the variables," so that the coefficient of dx shall contain x only, and that of dy, y only. For example, given the equation (1 — y)dx-\- (1 -\- x)dy = o, (6) the variables are separated by dividing by (1 -j- x)(i — y). T , dx dy Thus — ■ — = o. 1 -j- x ' 1 — y Each term is now directly integrable, and hence log (i-\-x) -.log (1 — y) — c. The solution here presents itself in a transcendental form,, but it is readily reduced to an algebraic form. For, taking the exponential of each member, we find I -r- X _ = e* = C, whence 1 -1- x = C(l — y), (7) where C is put for the constant f. Art. 2.] geometrical representation. 305 To verify the result in this form we notice that differentia- tion gives dx— — Cdy, and substituting in equation (6) we find - C(i -y)+i+x = o, which is true by equation (7). Prob. 1. Solve the equation dy -\-y tan x dx = o. (Ans. y—C cos x.) Prob. 2. Solve $- + Pf = a\ (Ans. b l±± = ce ^\ dx J \ by — a J Prob. 3. Solve f- = ^±-\ f Ans. y = *±<-\ dx x +1 \ J 1 — C x } Prob. 4. Helmholtz's equation for the strength of an electric current C at the time t is C- — - - — R R dt' where E, R, and L are given constants. Find the value of C, de- termining the constant of integration by the condition that its initial value shall be zero. Art. 2. Geometrical Representation. The meaning of a differential equation may be graphically illustrated by supposing simultaneous values of x and y to be the rectangular coordinates of a variable point. It is conven- ient to put/ for the value of the ratio dy.dx. Then /'being the moving point (x,y) and

= Y, in which the constant of integration Y may be a function of y. The result of differentiating this is x'dx 4- 2xy dx -f- x*dy = dY, which should be identical with the given equation ; therefore, dY ■=■ y* dy, whence Y =\y* -\- C, and substituting, the com- plete integral may be written *■ + 3 x'y = / + C. The result is more readily obtained if we notice that all terms containing x and dx only, or y and dy only, are exact differentials ; hence it is only necessary to examine the terms containing both x and y. In the present case, these are . 2xy dx A- x'dy, which obviously form the differential of x*y ; whence, integrating and multiplying by 3, we obtain the result above. The complete integral of any equation, in whatever way it 310 DIFFERENTIAL EQUATIONS. [Chap. VII. was found, can be put in the form u = C, by solving for C. Hence an exact differential equation du = o can be obtained, which must be equivalent to the given equation Mdx + Ndy — o, (4) here supposed not to be exact. The exact equation du = must therefore be of the form jx(Mdx + Ndy) = o, (5) where /< is a factor containing at least one of the variables x andjj>. Such a factor is called an " integrating factor" of the given equation. For example, the result of differentiating equation (7), Art. I, when put in the form u = C, is (1 —y)dx-\-{i -\-x)dy _ n d-yf ' so that (1 — y*)~" is an integrating factor of equation (6). It is to be noticed that the factor by which we separated the variables, namely, (1 — y)~\i — x)~\ is also an integrating factor. It follows that if an integrating factor can be discovered, the given differential equation can at once be solved.* Such a factor is sometimes suggested by the form of the equation. Thus, given (y — x)dy -\- ydx — o, the terms ydx — xdy, which contain both x and y, are not ex- act, but become so when divided by either x' or y 1 ; and be- cause the remaining term contains y only, y~' is an integrating factor of the whole expression. The resulting integral is log.r + ^ = C. Prob. 10. Show from the integral equation in Prob. 9, Art. 3, that x~* is an integrating factor of the differential equation. Prob. 11. Solve the equation x{x' + 3y 2 )dx -\- y{y 1 -\- ^x^dy = 0. (Ans. x* + 6xY +/ = e.) * Since fiM and hJV in the exact equation (5) must satisfy the condition (2), we have a partial differential equation for fi; but as a general method of finding H this simply comes back to the solution of the original equation. Art. 5.] homogeneous equation. 311 Prob. 12. Solve the equation y dy -\-xdx-\ , , , . x +y (Ans. ?-±£ + unT' * = e.) 2 X Prob. 13. If u = c is a form of the complete integral and p. the corresponding integrating factor, show that /*/(#) is the general expression for the integrating factors. Prob. 14. Show that the expression x°yP(mydx + nxdy) has the integrating factor x km ' L ' a y k ""'^; and by means of such a factor solve the equation y(y' -\- 2x')dx + x(x* — 2y')dy — o. (Ans. 2x'y — y* = ex' .) Prob. 15. Solve (x* -\- y')dx — 2xydy — o. (Ans. x* — y % = «;.) Art. 5. Homogeneous Equation. The differential equation Mdx -\- Ndy = o is said to be homogeneous when M and N are homogeneous functions of x and j)/ of the same degree ; or, what is the same thing, when dy y -j- is expressible as a function of — . If in such an equation the variables are changed from x and y to x and v, where y v = — ; whence y = xv and dy = ;rafo -f- vdx, the variables # and v will be separable. For example, the equation {x — 2y)dx -\- ydy = o is homogeneous ; making the substitutions indicated and dividing by x, (1 — 2v)dx -f- v(xdv -\- vdx) = O, a^ir vdv whence - + (irzr ^ = o. Integrating, log x + log (v — 1) — — — - = C; and restoring _y, The equation Mdx -\- Ndy = o can always be solved when log (y-x)- j-- x = C. 312 DIFFERENTIAL EQUATIONS. [Chap. VII. M and N are functions of the first degree, that is, when it is of the form {ax -\-by-\- c)dx -f {a'x + b'y + c')dy = o. For, assuming x = x' + h, y = y' + k, it becomes («*'+*>'+ «A + W + c>&'+(aV+*y+fl , A+3 , ife+c')^=^ which, by properly determining h and /£, becomes {ax' + ^/')^' + {a'x' + £»r/, a homogeneous equation. This method fails when a\b = a':b', that is, when the equation takes the form {ax -f- by -\- c)dx -\- \m{ax -\- by) -j- c'~\dy = o ; but in this case if we put z = ax -\- by, and eliminate y, it will be found that the variables x and z can be separated. Prob. 1 6. Show that a homogeneous differential equation repre- sents a system of similar and similarly situated curves, the origin being the center of similitude, and hence that the complete integral may be written in a form homogeneous in x, y, and c. Prob. 17. Solve xdy — y dx — ^{x* -\- y*)dx = o. (Ans. x* — c l — 2cy.) Prob. 18. Solve {%y — jx -\- i)dx + {iy — J>x -f T,)dy = o. (Ans. (jc — x + i) 2 (j + x — i) 5 = c.) Prob. 19. Solve {x' -\-y')dx — 2xy dy = o. (Ans. x* — y* = ex.) Prob. 20. Solve (1 + xy)y dx + (1 — xy)x dy = o by introducing the new variable z = xy. (Ans. x = Cye*y.) dy Prob. 21. Solve -j-=ax-\-by-\-c. (Ans. abx-\-by j ra-\-bc=Ce bx .) Art. 6. The Linear Equation. A differential equation is said to be " linear " when (one of the variables, say x, being regarded as independent,) it is of the first degree with respect to y, and its derivatives. The linear equation of the first order may therefore be written in the form dy Art. 6.] the linear equation. 313 where P and Q are functions of x only. Since the second member is a function of x, an integrating factor of the first member will be an integrating factor of the equation provided it contains x only. To find such a factor, we solve the equation % + Py = o, ( 2 ) dy which is done by separating the variables ; thus, — = — Pdx ; whence log y = c — I Pdx or y = Ce-f™*. (3) Putting this equation in the form u = c, the corresponding exact equation is e^ Pix {dy -f- Pydx) = o, whence e' * is the integrating factor required. Using this factor, the general solution of equation (i) is efivy = f/ Pdx Qdx + C. (4) In a given example the integrating factor should of course be simplified in form if possible. Thus (i -f- x^dy — (m + xy)dx is a linear equation for y ; reduced to the form (i), it is dy x m 7 Separating the variables and integrating, Vx±\/y=±Vc, (2) and this equation rationalized becomes - y)' - 2<* + y) -f- c = o. (3). There is thus a single complete integral containing one arbi- trary constant and representing a single system of curves; namely, in this case, a system of parabolas touching each axis at the same distance c from the origin. The separate equa- tions given in the form (2) are merely branches of the same parabola. Recurring now to the geometrical interpretation of a differ- ential equation, as given in Art. 2, it was stated that an equa- tion of the first degree determines, in general, for assumed values of x and y, that is, at a selected point in the plane, a single value of p. The equation was, of course, then supposed 316 DIFFERENTIAL EQUATIONS. [Chap. VII. rational in x and y* The only exceptions occur at points for which the value of p takes the indeterminate form ; that is, the equation being Mdx + Ndy = o, at points (if any exist) for which M = o and N = o. It follows that, except at such points, no two curves of the system representing the complete integral intersect, while through such points an unlimited num- ber of the curves may pass, forming a "pencil of curves." f On the other hand, in the case of an equation of the second degree, there will in general be two values of p for any given point. Thus from equation (i) above we find for the point (4, 1), / = ± \; there are therefore two directions in which a point starting from the position (4, 1) may move while satis- fying the differential equation. The curves thus described represent two of the particular integrals. If the same values of x and y be substituted in the complete integral (3), the re- sult is a quadratic for c, giving c — 9 and c = 1, and these determine the two particular integral curves, \/~x -\- Vy = 3, and Vx — Vy = I. In like manner the general equation of the second degree, which may be written in the form Lp + Mp + N = o, where L, M, and N are one-valued functions of x and y, repre- sents a system of curves of which two intersect in any given point for which p is found to have two real values. For these points, therefore, the complete integral should generally give two real values of c. Accordingly we may assume, as the standard form of its equation, Pc* + Qc + R = o, * In fact f was supposed to be a one-valued function of x and y\ thus, p = sin~'x would not in this connection be regarded as an equation of the first degree. f In Prob. 6, Art. 3, the integral equation represents the pencil of circles pass- ing through the points (o, />) and (o, — b); accordingly^) in the differential equa- tion is indeterminate at these points. In some cases, however, such a point is merely a node of one particular integral. Thus in the illustration given in Art. 2, p is indeterminate at the origin, and this point is a node of the only particular integral, xy = o, which passes through it. Art - 8.] SINGULAR SOLUTIONS. 317 where P, Q, and R are also one-valued functions of x and y. If there are points which make p imaginary in the differential equation, they will also make c imaginary in the integral. Prob. 27. Solve the equation / +/ = 1 and reduce the inte- gral to the standard form. (Ans. (y + cos x)c* - 2c sin x + y — cos x = o.) Prob. 28. Solves/ + 2 xp - y = o, and show that the intersect- ing curves at any given point cut at right angles. Prob. 29. Solve {x* + 1)/ = 1. (Ans. cV* - zcxe* - 1.) Art. 8. Singular Solutions. A differential equation not of the first degree sometimes admits of what is called a " singular solution ; " that is to say, a solution which is not included in the complete integral. For suppose that the system of curves representing the complete integral has an envelope. Every point A of this envelope is a point of contact with a particular curve of the complete in- tegral system ; therefore a point moving in the envelope when passing through A has the same values of x, y, and p as if it were moving through A in the particular integral curve. Hence such a point satisfies the differential equation and will continue to satisfy it as long as it moves in the envelope. The equation of the envelope is therefore a solution of the equation. As an illustration, let us take the system of straight lines whose equation is y = cx + j, (1) where c is the arbitrary parameter. The differential equation derived from this primitive is y=P*+-p (2) of which therefore (1) is the complete integral. Now the lines represented by equation (1), for different values of c, are the tangents to the parabola f = Aax. (3) 318 DIFFERENTIAL EQUATIONS. [Chap. VII. A point moving in this parabola has the same value of p as if it were moving in one of the tan- gents, and accordingly equation (3) will be found to satisfy the differential equation (2). It will be noticed that for any point on the convex side of the parabola there are two real values of p ; for a point on the other side the values of p are imaginary, and for a point on the curve they are equal. Thus its equation (3) expresses the relation between x and y which must exist in order that (2) regarded as a quadratic for p may have equal roots, as will be seen on solving that equation. In general, writing the differential equation in the form Lp' + Mp+N=o, (4) the condition of equal roots is M * - 4LN = o. (5) The first member of this equation, which is the " discrimi- nant " of equation (4), frequently admits of separation into factors rational in x andjy. Hence, if there be a singular solu- tion, its equation will be found by putting the discriminant of the differential equation, or one of its factors, equal to zero. It does not follow that every such equation represents a solu- tion of the differential equation. It can only be inferred that it is a locus of points for which the two values of p become equal. Now suppose that two distinct particular integral curves touch each other. At the point of contact, the two values of/, usually distinct, become equal. The locus of such points is called a " tac-locus." Its equation plainly satisfies the discriminant, but does not satisfy the differential equation. An illustration is afforded by the equation Art. 8.] singular solutions. 319 of which the complete integral is f + (x — c)' = a\ and the discriminant, see equation (5), isy(y — a*) = o. This is satisfied by y = a, y = — a, and y = o, the first two of which satisfy the differential equation, while 2 = o does not. The complete integral represents in this case all circles of radius a with center on the axis of x. Two of these circles touch at every point of the axis of x, which is thus a tac-locus, while y = a and y = — a constitute the envelope. The discriminant is the quantity which appears under the radical sign when the general equation (4) is solved for/, and therefore it changes sign as we cross the envelope. But the values of p remain real as we cross the tac-locus, so that the discriminant cannot change sign. Accordingly the factor which indicates a tac-locus appears with an even exponent (as y 1 in the example above), whereas the factor indicating the singular solution appears as a simple factor, or with an odd exponent. A simple factor of the discriminant, or one with an odd ex- ponent, gives in fact always the boundary between a region of the plane in which/ is real and one in which/ is imaginary ; nevertheless it may not give a singular solution. For the two arcs of particular integral curves which intersect in a point on the real side of the boundary may, as the point is brought up to the boundary, become tangent to each other, but not to the boundary curve. In that case, since they cannot cross the boundary, they become branches of the same particular inte- gral forming a cusp. A boundary curve of this character is called a " cusp-locus " ; the value of / for a point moving in it is of course different from the equal values olp at the cusp, and therefore its equation does not satisfy the differential equation.* Prob. 30. To what curve is the line y = mx -\- a 4/(1 — n?) always tangent ? (Ans. y 1 — x* = a 2 .) Prob. 31. Show that the discriminant of a decomposable differ- * Since there is no reason why the values oip referred to should be identical, we conclude that the equation Lp % + Mp + JV= o has not in general a singular solution, its discriminant representing a cusp-locus except when a certain con- dition is fulfilled. 320 DIFFERENTIAL EQUATIONS. [Chap. VII. ential equation cannot be negative. Interpret the result of equating it to zero in the illustrative example at the beginning of Art. 7. Prob. 32. Show that the singular solutions of a homogeneous dif- ferential equation represent straight lines passing through the origin. Prob. 33. Solve the equation xp 2 — 2yp -\- ax — o. (Ans. x* — 2cy-\- ac 1 = o ; singular solution y* = ax'.) Prob. 34. Show that the equation p 1 + 2xp — y = o has no sin- gular solution, but has a cusp-locus, and that the tangent at every cusp passes through the origin. Art. 9. Singular Solution from Complete Integral. When the complete integral of a differential equation of the second degree has been found in the standard form Pt+Qc + R = o (1) (see the end of Art. 7), the substitution of special values of x and y in the functions P, Q, and R gives a quadratic for c whose roots determine the two particular curves of the system which pass through a given point. If there is a singular solution, that is, if the system of curves has an envelope, the two curves which usually intersect become identical when the given point is moved up to the envelope. Every point on the en- velope therefore satisfies the condition of equal roots for equa- tion (1), which is Q- A PR = o; (2) and, reasoning exactly as in Art. 8, we infer that the equation of the singular solution will be found by equating to zero the discriminant of the equation in c or one of its factors. Thus the discriminant of equation (1), Art. 8, or " c-discriminant," is the same as the "^-discriminant," namely, y 3 — ^ax, which equated to zero is the equation of the envelope of the system of straight lines. But, as in the case of the /"-discriminant, it must not be inferred that every factor gives a singular solution. For ex- ample, suppose a squared factor appears in the ^-discriminant. The locus on which this factor vanishes is not a curve in cross- ing which c and p become imaginary. At any point of it there Art. 9.] singular solution from complete integral. 321 will be two distinct values of p, corresponding to arcs of par- ticular integral curves passing through that point ; but, since there is but one value of c, these arcs belong to the same par- ticular integral, hence the point is a double point or node. The locus is therefore called a " node-locus." The factor repre- senting it does not appear in the /-discriminant, just as that representing a tac-locus does not appear in the ^-discriminant. Again, at any point of a cusp-locus, as shown at the end of Art. 8, the two branches of particular integrals become arcs of the same particular integral ; the values of c become equal, so that a cusp-locus also makes the c-discriminant vanish. The conclusions established above obviously apply also to equations of a degree higher than the second. In the case of the ^-equation the general method of obtaining the condition for equal roots, which is to eliminate c between the original and the derived equation, is the same as the process of finding the envelope or " locus of the ultimate intersections " of a system of curves in which c is the arbitrary parameter. Now suppose the system of curves to have for all values of c* a double point, it is obvious that among the intersections of two neighboring curves there are two in the neighborhood of the nodes, and that ultimately they coincide with the node, which accounts for the node-locus appearing twice in the dis- criminant or locus of ultimate intersections. In like manner, * It is noticed in the second foot-note to Art. 7 that for an equation of the first degree p takes the indeterminate form, not only at a point through which all curves of the system pass (where the value of c would also be found indeter- minate), but at a node of a particular integral. So also when the equation is of the «th degree, if there is a node for a particular value of c, the « values of c at the point (which is not on a node-locus where two values of c are equal) deter- mine n-\- 1 arcs of particular integrals passing through the point; and there- fore there are n + 1 distinct values of p at the point, which can only happen when/ takes the indeterminate form, that is to say, when all the coefficients of the ^-equation [which is of the «th degree} vanish. See Cayley on Singular So- lutions- in the Messenger of Mathematics, New Series, Vol. II, p. 10 (Collected Mathematical Works, Vol. VIII, p. 529). The present t-neory of Singular Solu- tions was established by Cayley in this paper and its continuation, Vol. VI, p. 23. See also a paper by Dr. Glaisher, Vol. XII, p. 1. 322 DIFFERENTIAL EQUATIONS. TChap. VIL if there is a cusp for all values of c, there are three intersections of neighboring curves (all of which may be real) which ulti- mately coincide with the cusp ; therefore a cusp-locus will appear as a cubed factor in the discriminant.* Prob. 35. Show that the singular solutions of a homogeneous equation must be straight lines passing through the origin. Prob. 36. Solve $p'y' — 2X )f + 4/ — x* = o, and show that there is a singular solution and a tac-focus. Prob. 37- Solve yp 1 + 2Xp — y = o, and show that there is an imaginary singular solution. (Ans. jc s = 2cx + c 2 .) Prob. 38. Show that the equation (1 — x 2 )p 2 = 1 — y represents a system of conies touching the four sides of a square. Prob. 39. Solve yp 2 — 4xp -\-y = o ; examine and interpret both discriminants. (Ans. c 1 + 2^(3/ — Sx 2 ) — ^y 1 -f-/ = o.) Art. 10. Solution by Differentiation. The result of differentiating a given differential equation of the first order is an equation of the second order, that is, it d % y contains the derivative t~t ; but, if it does not contain y ex- plicitly, it may be regarded as an equation of the first order for the variables x and/. If the integral of such an equation can be obtained it will be a relation between x, p, and a constant of integration c, by means of which p can be eliminated from the original equation, thus giving the relation between x, y, and c which constitutes the complete integral. For example, the equation f x + 2xy = *>+?, (I) * The discriminant of Pc l -\- Qc -f- R — o represents in general an envelope, no further condition requiring to be fulfilled as in the case of the discriminant of Lp* -)- Mp 4- N — o. Compare the foot-note to Art. 8. Therefore where there is an integral of this form there is generally a singular solution, although Lp? -\- Mp-\-JV= o has not in general a singular solution. We conclude, there- fore, that this equation (in which /., M, and N are one-valued functions of x and y) has not in general an integral of the above form in which P, Q, and R are one-valued functions of x and y. Cayley, Messenger of Mathematics, New Series, Vol. VI, p. 23. Art. 10.] solution by differentiation. 323 when solved for j/, becomes y = x + Yp; (2) whence by differentiation i dp The variables can be separated in this equation, and its inte- gral is Yp = C + e '* C- e* Substituting in equation (2), we find which is the complete integral of equation (1). This method sometimes succeeds with equations of a higher degree when the solution with respect to p is impossible or leads to a form which cannot be integrated. A differential equation between p and one of the two variables will be ob- tained by direct integration when only one of the variables is explicitly present in the equation, and also when the equation is of the first degree with respect to x and y. In the latter case after dividing by the coefficient of y, the result of differ- entiation will be a linear equation for x as a function of p, so that an expression for x in terms of p can be found, and then by substitution in the given equation an expression for y in terms of p. Hence, in this case, any number of simultaneous values of x and y can be found, although the elimination of p may be impracticable. In particular, a homogeneous equation which cannot be solved for p may be soluble for the ratio y : x, so as to assume the form y = x)%: in which the variables x and p can be separated. Another special case is of the form y = P* +AP), (I) 324 DIFFERENTIAL EQUATIONS. [Chap. VII. which is known as Clairaut's equation. The result of differ- entiation is which implies either dp * +/'(/) = o, or f- x = o. The elimination of p from equation (i) by means of the first of these equations * gives a solution containing no arbi- trary constant, that is, a singular solution. The second is a differential equation for p ; its integral is p = c, which in equation (i) gives the complete integral y = cx +/(<:). (2) This complete integral represents a system of straight lines, the singular solution representing the curve to which they are all tangent. An example has already been given in Art. 8. A differential equation is sometimes reducible to Clairaut's form by means of a more or less obvious transformation of the variables. It may be noticed in particular that an equation of the form y = nxp + {x, p) is sometimes so reducible by transformation to the independent variable z, where x = z* ; and an equation of the form y = nxp-\- u c 1 d x dy dz Prob. 53. Solve — , — = — j — — — ■ — . y -\- z z -j- x x -\-y (Ans. V{x +y + z) = -JL_ = _±_.) \ z ~- y x — z I r, . . o 1 dx dy dz Prob. 54. Solve ,-: r — = , —^ = -^ — . (b — c)yz (c — a)zx {a — b)xy (Ans. x> +y + z* = A, ax' 4- bf + c& = £.} Art. 13. Equations of the Second Order. A relation between two variables and the successive deriva- tives of one of them with respect to the other as independent variable is called a differential equation of the order indicated by the highest derivative that occurs. For example, o+*>£+4+~=o is an equation of the second order, in which x is the independent ART. 13.] EQUATIONS OF THE SECOND ORDER. 331 variable. Denoting as heretofore the first derivative by/, this equation may be written (i+*")^ + */ + ** = o, (i) and this, in connection with %=*• < 2 > which defines p, forms a pair of equations of the first order, connecting the variables x, y, and p. Thus any equation of the second order is equivalent to a pair of simultaneous equations of the first order. When, as in this example, the given equation does not con- tain y explicitly, the first of the pair of equations involves only the two variables x and/ ; and it is further to be noticed that, when the derivatives occur only in the first degree, it is a linear equation for/. Integrating equation (i) as such, we find >=~ w + V(I + ?) ; (3) and then using this value of/ in equation (2), its integral is y = c t — mx + c x log O + 4/(1 + *')], (4) in which, as in every case of two simultaneous equations of the first order, we have introduced two constants of integration. An equation of the first order is readily obtained also when the independent variable is not explicitly contained in the equation. The general equation of rectilinear motion in d's dynamics affords an illustration. This equation is — =/[s), where s denotes the distance measured from a fixed center of force upon the line of motion. It may be written — = f{s), in ds connection with •— = v, which defines the velocity. Eliminat- dt ing dt from these equations, we have vdv = f(s)ds, whose integral is %v* = f/(s)ds + c, the " equation of energy " for the unit mass. The substitution of the value found for v in the 332 DIFFERENTIAL EQUATIONS. [CHAP. VII. second equation gives an equation from which t is found in terms of s by direct integration. The result of the first integration, such as equation (3) above is called a "first integral" of the given equation of the second order ; it contains one constant of integration, and its complete integral, which contains a second constant, is also the "com- plete integral" of the given equation. A differential equation of the second order is " exact " when all its terms being transposed to the first member, that member is the derivative with respect to x of an expression of the first order, that is, a function of x, y and p. It is obvious that the terms containing the second derivative, in such an exact differ- ential, arise solely from the differentiation of the terms con- taining/ in the function of x, y and p. For example, let it be required to ascertain whether (-*?£- 4+'=° a is an exact equation. The terms in question are (1 — x 1 )— , dx which can arise only from the differentiation of (1 — x')p. Now subtract from the given expression the complete deriva- tive of (1 — x')p, which is (1 - s^y - 2x dy • {l Xi ^ 2X dx' civ the remainder is x^- ~\- y, which is an exact derivative, namely, dx that of xy. Hence the given expression is an exact differ- ential, and {i-^ + *y = c t (6) is the first integral of the given equation. Solving this linear equation for y, we find the complete integral y = c,x + c t 4/(1 - x'). (7) Prob. 55. Solve (1 - x')^X - xf- = 2. ax ax (Ans. y = (sin -1 x)' -f- c, sin" 1 x + O ART. 14.] THE TWO FIRST INTEGRALS. 333 Prob. S 6. Solve g = J . ( A ns. y = J + ,,**.) Prob. 57. Solve -^ = a'x — tfy. (Ans. a'x — d'y = A sin &x + £ cos A*.) Prob. 58. Solve y^ + (4j£j =1 . (Ans. / = *- + ,,* + ,,.) Art. 14. The Two First Integrals. We have seen in the preceding article that the complete integral of an equation of the second order is a relation be- tween x, y and two constants c, and c 2 . Conversely, any rela- tion between x, y and two arbitrary constants may be regarded as a primitive, from which a differential equation free from both arbitrary constants can be obtained. The process consists in first obtaining, as in Art. 3, a differential equation of the first order independent of one of the constants, say c, , that is, a rela- tion between x, y,p and c t , and then in like manner eliminating c, from the derivative of this equation. The result is the equa- tion of the second order or relation between x, y, p and q (q denoting the second derivative), of which the original equation is the complete primitive, the equation of the first order being the first integral in which c 1 is the constant of integration. It is obvious that we can, in like manner, obtain from the primi- tive a relation between x,y,p and c,, which will also be a first integral of the differential equation. Thus, to a given form of the primitive or complete integral there corresponds two first integrals. Geometrically the complete integral represents a doubly infinite system of curves, obtained by varying the values of c t and of c, independently. If we regard c l as fixed and c 3 as arbitrary, we select from that system a certain singly infinite system; the first integral containing c t is the differential equa- tion of this system, which, as explained in Art. 2, is a relation between the coordinates of a moving point and the direction of its motion common to all the curves of the system. But 334 DIFFERENTIAL EQUATIONS. [Chap. VII. the equation of the second order expresses a property involv- ing curvature as well as direction of path, and this property being independent of c x is common to all the systems corre- sponding to different values of c lt that is, to the entire doubly infinite system. A moving point, satisfying this equation, may have any position and move in any direction, provided its path has the proper curvature as determined by the value of q derived from the equation, when the selected values of x, y and/ have been substituted therein.* For example, equation (7) of the preceding article repre- sents an ellipse having its center at the origin and touching the lines x = ± 1, as in the diagram ; c, is the ordinate of the point of contact with x = 1, and c^ that of the point in which the ellipse cuts the axis of y. If we regard c, as fixed and c, as arbitrary, the equation represents the system of ellipses touching the two lines at fixed points, and equation (6) is the differential equation of this system. In like manner, if c 2 is fixed and be substituted in equation (1), the result will be the sum of the results of putting y = Fand of putting y = u ; the first of these results will be X, because Fis an integral of equation (1),. and the second will be zero because u is an integral of equa- tion (2). Hence equation (4) expresses an integral of (2); and since it contains the n arbitrary constants of equation (3), it is the complete integral of equation (1). With reference to this equation F is called " the particular integral," and u is called "the complementary function." The particular integral contains no arbitrary constant, and any two particular integrals may differ by any multiple of a term belonging to the comple- mentary function. If one term of the complementary function of a linear equation of the second order be known, the complete solution can be found. For let j/, be the known term ; then, if y = y,v be substituted in the first member, the coefficient of v in the result will be the same as if v were a constant : it will there- fore be zero, and v being absent, the result will be a linear equa- tion of the first order for v' , the first derivative o-f v. Under 338 DIFFERENTIAL EQUATIONS. [CHAP. VII. the same circumstances the order of any linear equation can in like manner be reduced by unity. A very simple relation exists between the coefficients of an exact linear equation. Taking, for example, the equation of the second order, and indicating derivatives by accents, if is exact, the first term of the integral will be P y' Subtracting the derivative of this from the first member, the remainder is (P, — P„')y' -\- P*y- The second term of the integral must therefore be (P 1 — P/)y ; subtracting the derivative of this ex- pression, the remainder, (P t — P/ -{- P„")y, must vanish. Hence P t — P/ + P„" = O is the criterion for the exactness of the given equation. A similar result obviously extends to equa- tions of higher orders. Prob. 61. Solve x— (3 + x)- — \- $y — o, noticing that e* is an integral. (Ans. y = c/ x + cj^x" + $s* -j- 6x -j- 6.) Prob. 62. Solve (x* — x)-p, + 2(2* -+- i)~ -f- 2y — o. (Ans. (x — i)"y = c^x* — 6x* -\- 2x — • | — 4X 3 log x) + c^x\) Prob. 63. Solve-3^3 + cos 6-jp — 2 sin 6-~ —y cos = sin 28. ( Ans. y = e~ sin e J e sin "{cfi + c,)dd + c,e ~ sin " sin p — 1 Art. 16. Linear Equations with Constant Coefficients. The linear equation with constant coefficients and second member zero may be written in the form A.D'y + A l D'-y + ... + A u y= (1) in which D stands for the operator -=-, D 1 for -7-5, etc., so that ax ax D" indicates that the operator is to be applied n times. Then, since De mx = me mx , D'e mx = m t e mx , etc., it is evident that if Art. 16.] linear equations, constant coefficients. 339 y _ e mx b e substituted in equation (i), the result after rejecting the factor e mx will be A t m'' + A l »f- i + ...+A n = o. (2) Hence, if m satisfies equation (2), e mx is an integral of equation (1) ; and if wz,, m t , . . .m n are n distinct roots of equation (2), the complete integral of equation (1) will be y = c^ x -\- c^e" 1 ** -(-...+ C*™"*. (3) For example, if the given equation is d'y dy - d s-T x - 2y = °' the equation to determine m is m 1 — m — 2 = o, of which the roots are m x = 2, m t = — 1 ; therefore the in- tegral is y = c,?'* + c t e-». The general equation (1) maybe written in the symbolic form/(Z>) .y = O, in which /"denotes a rational integral func- tion. Then equation (2) is f(m) = O, and, just as this last equation is equivalent to (in — m^[m — w,) . . . (m — m n ) — o, (4) so the symbolic equation /(D). y = o may be written (D — m t )(D — «0 ... (£> — m n )y = o. (5) This form of the equation shows that it is satisfied by each of the quantities which satisfy the separate equations (D — m^)y — o, (-D — m,)y = o . . . (D — m^)y — o; (6) that is to say, by the separate terms of the complete integral. If two of the roots of equation (2) are equal, say to m it two of the equations (6) become identical, and to obtain the full number of integrals we must find two terms corresponding to the equation [D - m,Yy = o ; (7) in other words, the complete integral of this equation of which y x == e m i* is known to be one integral. For this purpose we 340 DIFFERENTIAL EQUATIONS. [CHAP. VII. put, as explained in the preceding article, y =y{x _[_ c^o-im* — e °- x {c/V x -\- c 2 e-^ x ). Separating the real and imaginary parts of e'P" and e-'?", and changing the constants, the expression becomes e ax {A cos ftx-\-B sin fix). (11) For a multiple pair of imaginary roots the constants A and B must be replaced by polynomials as above shown in the case of real roots. When the second member of the equation with constant coefficients is a function of X, the particular integral can also be made to depend upon the solution of linear equations of the first order. In accordance with the symbolic notation introduced above, the solution of the equation dy j- x -ay = X, or (D - a)y = X (12) is denoted by y = (D — a)~'X, so that, solving equation (12),. we have 7^ X = e* x f e~ a *Xdx ( 1 3) D — a rJ as the value of the inverse symbol whose meaning is "that Art. 16.] LINEAR equations, constant coefficients. 341 function of x which is converted to X by the direct operation expressed by the symbol D — a." Taking the most convenient special value of the indefinite integral in equation (13), it gives the particular integral of equation (12). In like manner, the par- ticular integral of j\D)y = X is denoted by the inverse symbol -r-j—X. Now, with the notation employed above, the symbolic fraction may be decomposed into partial fractions with constant numerators thus : 1 N N N TTnyX = n ' X + n X + • ■ ■ + TT^ X >* ( x 4) f(D) D — m i D — m t D — m n in which each term is to be evaluated by equation (13), and may be regarded (by virtue of the constant involved in the indefinite integral) as containing one term of the complement- ary function. For example, the complete solution of the equation d"y dy v -— — 2 y = X dx dx is thus found to be y = ^Je-^Xdx — \e-*Je*Xdx. When X is a power of x the particular integral may be found as follows, more expeditiously than by the evaluation of the integrals in the general solution. For example, if X — x* the particular integral in this example may be evaluated by development of the inverse symbol, thus : _ i_ _ 2 _ _i \_ „» = - i[i - K^ - &) + xp - ny - . . .v = -i[i -$D + iD*- . . .]x* = -ix> + $x-i. * The validity of this equation depends upon the fact that the operations expressed in the second member of /(D) =(D- m,)(D - /»») + ...+(£> - m„) are commutative, hence ihe process of verification is the same as if the equation were an algebraic identity. This general solution was published by Boole in the Cambridge Math. Journal, First Series, vol. n, p. 114- It had, however, been previously published by Lobatto, Theorie des Characteristiques, Amster- dam, 1837. 342 DIFFERENTIAL EQUATIONS. [CHAP. VII. The form of the operand shows that, in this case, it is only- necessary to carry the development as far as the term contain- ing D\ For other symbolic methods applicable to special forms of X we must refer to the standard treatises on this subject. d 2 y dy Prob. 64. Solve 4 £i-3^+>=°. (Ans. y — ei*(Ax -f B) + ce~ x .) Prob. 65. Show that ^ i ^e ax = -j^V" D A-D) A a ) and that , ,. sin (ax -f /?) = — sin (ax + /?). A-ls I J \ a ) Prob. 66. Solve (Z> ! + \)y — e x -4- sin 2jc -4- sin a:. (Compare Prob. 59, Art. 14.) (Ans. y = A sin # -f- B cos # + £«* — \ sin 2* — \x cos *.) Art. 17. Homogeneous Linear Equations. The linear differential equation A ^% + A ^"& + • • ■ + ^ = °. (1) in which A , A iy etc., are constants, is called the "homogene- ous linear equation." It bears the same relation to x m that the equation with constant coefficients does to e mx . Thus, if y=x'" be substituted in this equation, the factor x m will divide out from the result, giving an equation for determining m, and the n roots of this equation will in general determine the n terms of the complete integral. For example, if in the equation „d*y , dy X 2 -4 + 2*-/- — 2y = O ax dx we put y = x m , the result is m(m — 1) -(- 2m — 2=0, or (m — \)(m -4- 2) = o. The roots of this equation are m^ = 1 and m, = — 2. Hence y = cjc -\- c t x~' is the complete integral. Equation (1) might in fact have been reduced to the form with constant coefficients by changing the independent van'- Art. 17.] homogeneous linear equations. 343 able to 6, where x = e e , or 6 — log x. We may therefore at once infer from the results established in the preceding article that the terms corresponding to a pair of equal roots are of the form {c l + c, log x)x"\ (2) and also that the terms corresponding to a pair of imaginary roots, a ± i/3, are x°-[A cos (/J log x)-\- B sin (/J log x)~\. (3) The analogy between the two classes of linear equations considered in this and the preceding article is more clearly seen when a single symbol $= xD is used for the operation of taking the derivative and then multiplying by x, so that ■8x m = mx m . It is to be noticed that the operation x*!) 1 is not the same as $' or xDxD, because the operations of taking the derivative and multiplying by a variable are not "commu- tative," that is, their order is not indifferent. We have, on the contrary, x'W = $($ — 1) ; then the equation given above, which is (x'D' + 2xD - 2)y = o, becomes [fl(0— i) + 2#— 2]j// = o, or (S — i)(S + 2> = o, the function of $ produced being the same as the function of m which is equated to O in finding the values of m. A linear equation of which the first member is homoge- neous and the second member a function of x may be reduced to the form f^).y = X. (4) The particular integral may, as in the preceding article (see eq. (14)), be separated into parts each of which depends upon the solution of a linear equation of the first order. Thus, solving the equation x^--ay = X, or (« - d)y = X, (5) ax we find tA- X = ** fx- a ~ l Xdx. (6) The more expeditious method which may be employed 344 DIFFERENTIAL EQUATIONS. [Chap. VII. when Xis a power of x is illustrated in the following example : Given x* -4 — 2-=- = ^ 2 . The first member becomes homo- dx ax geneous when multiplied by x, and the reduced equation is (S« _ 3^ = *•. The roots of _/[$) =0 are 3 and the double root zero, hence the complementary function is cj? -\- c, -j- c 3 log jr. Since in general f(d)x' = /(r)x r , we infer that in operating upon x 3 we may put -6 = 3. This gives for the particular integral 1 1 . 1 1 . -:* = - ~ X , « - 3 «' 9^-3 but fails with respect to the factor # — 3.* We therefore now fall back upon equation (6), which gives x 3 — x s I x~ l dx = x s log x. The complete integral therefore is y = <^ 3 + ^ + f s lo g •*■ + i* s J og #. d"*y dy Prob. 67. Solve 2X i -A i + 3*^ 3JC = x'K Prob. 68, Solve [x'D 3 + 3 ;cZ> 2 + Z>).y = - (Ans. y — c l + ^ log x + c s {\og x)'' + £(log x)\) (Ans. ^ = <:,.# + c, 00 ^ + |*"0 1 x Art. 18. Solutions in Infinite Series. We proceed in this article to illustrate the method by which the integrals of a linear equation whose coefficients are algebraic functions of x may be developed in series whose terms are powers of x. For this purpose let us take the equation ■ * The failure occurs because x" is a term of the complementary function having an indeterminate coefficient; accordingly the new term is of the same form as the second term necessary when 3 is a double root, but of course with a determinate coefficient. .Art. 18] SOLUTIONS IN INFINITE SERIES. 345 which is known as " Bessel's Equation," and serves to define the "Besselian Functions." If in the first member of this equation we substitute for y the single term Ax m the result is A(m' — ri l )x m + Ax'"+\ (2) the first term coming from the homogeneous terms of the equation and the second from the term x'y which is of higher degree. If this last term did not exist the equation would be satisfied by the assumed value of y, if m were determined so as to make the first term vanish, that is, in this case, by Ax n or Bx' n . Now these are the first terms of two series each of which satisfies the equation. For, if we add to the value of y a term containing x'"+ 2 , thus_y = A x m -\- A x x m ^ 2 , the new term will give rise, in the result of substitution, to terms containing x'" +2 and x m+i respectively, and it will be possible so to take A t that the entire coefficient of x'" +2 shall vanish. In like manner the proper determination of a third term makes the coefficient of x m+,i in the result of substitution vanish, and so on. We therefore at once assume y = 2 A r x m + 2r = A a x m -f- A t x m + 2 -f- A,x m + 4 -4- . . . , (3) in which r has all integral values from o to 00 . Substituting in equation (1) 2[{(m + 2rf— n , \A r x m+2r -\~ A r x m+2t - r+I) ] = o. (4) The coefficient of each power of x in this equation must sep- arately vanish ; hence, taking the coefficient of x m+2r , we have [(,„ + 2r) 1 -« , ]^ r + ^ r _,=o. (S) When r = O, this reduces to m* — n 2 = O, which determines the values of m, and for other values of r it gives Ar = _ (m + 2r + n)(m + 2r - n) Ar - i ' ^ the relation between any two successive coefficients. For the first value of m, namely n, this relation becomes A l A 346 DIFFERENTIAL EQUATIONS. [Chat>. VII. whence, determining the successive coefficients in equation (3), the first integral of the equation is ix' 1 x* "1 1 -7TT-,-> + 77, ,. T ^ , „s ^n--..J. (7) -V, = -4.*" 8-fl2' T (« + 1)0 + 2) 2*. 2 ! " In like manner, the other integral is found to be ^-^-"[ I + ^ rT ^ + (w _ l) I (w _ 2) i ^-, + ...],(8> and the complete integral is j/ = A y 1 -4- B^y v * This example illustrates a special case which may arise in this form of solution. If n is a positive integer, the second series will contain infinite coefficients. For example, if n ~ 2, the third coefficient, or £„ is infinite, unless we take B a = 0, in which case B 2 is indeterminate and we have a repetition of the solution jj/j. This will always occur when the same powers of x occur in the two series, including, of course, the case in which m has equal roots. For the mode of obtaining a new integral in such cases the complete treatises must be referred to.f It will be noticed that the simplicity of the relation between consecutive coefficients in this example is due to the fact that equation (1) contained but two groups of terms producing different powers of x, when Ax m is substituted for y as in ex- pression (2). The group containing the second derivative necessarily gives rise to a coefficient of the second degree in ;«, and from it we obtained two values of m. Moreover, be- cause the other group was of a degree higher by two units, the assumed series was an ascending one, proceeding by powers of x\ * The Besselian function of the «th order usually denoted by J n is the value of y\ above, divided by 2"»! if n is a positive integer, or generally by 2"F(«+i). For a complete discussion of these functions see Lommel's Studien iiber die Bessel'schen Functionen, Leipzig, 1868; Todhunter's Treatise on Laplace's, Lamp's and Bessel's Functions, London, 1875, etc - f A solution of the kind referred to contains as one term the product of the regular solution and log x, and is sometimes called a " logarithmic solution.'' See also American Journal of Mathematics, Vol. XI, p. 37. In the case of Bessel's equation, the logarithmic solution is the "Besselian Function of the cecond kind.'' Art. 18.] solutions in infinite series. 34? In the following example, d 2 y dy y -ds+ a T x - 2 x> = > <9> there are also two such groups of terms, and their difference of degree shows that the series must ascend by simple powers. We assume therefore at once y = 2A r x m + r . (io) The result of substitution is %\_{(m-\-r)(m+r-i)—2}A r x m+r - 2 + a(m+r)A r x m + r - t '}= o. (u, Equating to zero the coefficient of x'" +r ~ 2 , {m-\-r-\- i)(m + r — 2}A r -\-a{m-\-r — i)A r - l = o, (12) which, when r = o, gives {tn-\-\)(m — 2)A i =o, (13) and when r > o, tfi — r- T — I A r = — a-, j 1 — T-, j '-rA r . v (14) The roots of equation (13) are m — 2 and m = — I; taking m = 2, the relation (14) becomes r+i A (r+3)r whence the first integral is A^ = A a x{i- 2 -ax + ^a^-^ r6 a^ +...]. (i S > Taking the second value wz = — 1, equation (14) gives r — 2 , r(V - 3) r— I J whence 2?, = B , and 5, = o *; therefore the second inte- gral is the finite expression *.y, = b^-\i - \ ax ~\= B ^r x - \ ]• ( i6 ) *i? 3 would take the indeterminate form, and if we suppose it to have a finite value, the rest of the series is equivalent to £,)■,, reproducing the first integral. "34.8 DIFFERENTIAL EQUATIONS. [Chap. VII. When the coefficient of the term of highest degree in the result of substitution, such as equation (n), contains m, it is possible to obtain a solution in descending powers of x. In this case, m occurring only in the first degree, but one such solution can be found ; it would be identical with the finite integral (16). In the general case there will be two such solu- tions, and they will be convergent for values of x greater than unity, while the ascending series will converge for values less than unity.* When the second member of the equation is a power of x, the particular integral can be determined in the form of a series in a similar manner. For example, suppose the second mem- ber of equation (9) to have been x i . Then, making the sub- stitution as before, we have the same relation between consecu- tive coefficients; but when r = o, instead of equation (13) we have (in + \)(m — 2)A a x m - 2 — xi to determine the initial term of the series. This gives m = i\ and A a = -f ; hence, putting in = \ in equation (14), we find for the particular integral f 9-3 ' 9- n-3-5 A linear equation remains linear for two important classes of transformations ; first, when the independent variable is changed to any function of x, and second, when for y we put ■vflx). As an example of the latter, let y = e~ ax v be substituted in equation (9) above. After rejecting the factor e~", the result is - V ' df~ Z ' (I) where X, Y, Z are functions of x, y, and z, but not of t, we form the equation dx dx dy ,dz dz dz — d~ 4- ~d — - -4- -rd—- = Xdx + Ydy -4~ Zdz. dt dt ' dt dt ' dt dt ' J ' The first member is an exact differential, and we know that for a conservative field of force the second member is also exact, that is, it is the differential of a function U of x, y, and z. The integral is that first integral of the system (i) which is known as the equation of energy for the unit mass. Just as in Art. 13 an equation of the second order was re- garded as equivalent to two equations of the first order, so the system (l)in connection with the equation defining the resolved velocities forms a system of six equations of the first order, of which system equation (2) is an " integral " in the sense ex- plained in Art. 12. „ , . dx dy , Prob. 72. Solve the equations = — = dt as a system Iin- — my mx ear in /. (Ans. x = A cos mt-j- 2? sin mt, y ~A smmt—B costnt.) dz dy Prob. 73. Solve the system -. \- ny = e x , — -\- z = o. (Ans. y = Ae nx + £r" x + -f — , « = - nAe nx + nBe ~ nx - f -.) d'x dy Prob. 74. Find for the system —j— = x. / A d y ^x . 352 DIFFERENTIAL EQUATIONS. [Chap. VII. Prob. 75. The approximate equations for the horizontal motion of a pendulum, when the earth's rotation is taken into account, are d?x dy , gx d'y dx gy df- 2r dF+i=°' d7 + 2r ^+T = ' show that both x and y are of the form A cos nj + B sin nj -\- C cos nj ■+ D sin nj. Art. 20. First Order and Degree with Three Variables. The equation of the first order and degree between three variables x, y and z may be written Pdx + Qdy + Rdz = o, (1) where P, Q and R are functions of x, y and z. When this equation is exact, P, Q and R are the partial derivatives of some function u, of x, y and z ; and we derive, as in Art. 4, dP_d_Q dQ^dR_ dR^dP ( . dy ~ dx : dz dy' dx dz W for the conditions of exactness. In the case of two variables, when the equation is not exact integrating factors always exist; but in this case, there is not always a factor u such that /.iP, jaQ and pR (put in place of P, Q, and R) will satisfy all three of the conditions (2). It is easily shown, that for this purpose the relation \dz dy' \dx aW ^dy dxl u/ must exist between the given values of P, Q, and R. This is- therefore the " condition of integrability " of equation (1).* When this condition is fulfilled equation (1) may be inte- grated by first supposing one variable, say z, to be constant. Thus, integrating Pdx -(- Qdy = o, and supposing the constant of integration C to be a function of z, we obtain the integral, so * When there are more than three variables such a condition of integra- bility exists for each group of three variables, but these conditions are not alt independent. Thus with four variables there are but three independent con- ditions. Art. 20 ] first order and degree, three variables. 353 far as it depends upon x and y. Finally, by comparing the total differential of this result with the given equation we de- termine dC in terms of z and dz, and thence by integration the value of C. It may be noticed that when certain terms of an exact equation forms an exact differential, the remaining terms must also be exact. It follows that if one of the variables, say z can be completely separated from the other two (so that in equation (i) R becomes a function of z only and P and Q func- tions of x and_y, but not of z) the terms Pdx -\- Qdy must be thus rendered exact if the equation is integrable.* For example, zydx — zxdy — y 2 dz = o. is an integrable equation. Accordingly, dividing by y'z. which we notice separates the variable z from x and y, puts it in the exact form ydx — xdy dz i = o, y z of which the integral is x = y log cz. Regarding x, y and z as coordinates of a moving point, an integrable equation restricts the point to motion upon one of the surfaces belonging to the system of surfaces represented by the integral ; in other words, the point (x, y, z) moves in an arbitrary curve drawn on such a surface. Let us now consider in what way equation (i) restricts the motion of a point when it is not integrable. The direction cosines of a moving point are proportional to dx, dy, and dz; hence, denoting them by /, m and n, the direction of motion of the point satisfying equation (i) must satisfy the condition Pl+ Qm+ Rn=. o. (4) It is convenient to consider in this connection an auxiliary system of lines represented, as explained in Art. 12, by the simultaneous equations dx _ dy __dz ~p~~Q~~R' (5) * In fact for this case the condition (3) reduces to its last term, which ex- presses the exactness of Pdx-\- Qdy. 354 DIFFERENTIAL EQUATIONS. [Chap. VII. The direction cosines of a point moving in one of the lines of this system are proportional to P, Q and R. Hence, de- noting them by A, ju, v, equation (4) gives XI -j- fxm -\- vn = o (6) for the relation between the directions of two moving points, whose paths intersect, subject respectively to equation (1) and to equations (5). The paths in question therefore intersect at right angles; therefore equation (1) simply restricts a point to move in a path which cuts orthogonally the lines of the auxili- ary system. Now, if there be a system of surfaces which cut the auxiliary lines orthogonally, the restriction just mentioned is completely expressed by the requirement that the line shall lie on one of these surfaces, the line being otherwise entirely arbitrary. This is the case in which equation (1) is integrable.* On the other hand, when the equation is not integrable, the restriction can only be expressed by two equations involving an arbitrary function. Thus if we assume in advance one such relation, we know from Art. 12 that the given equation (1) together with the first derivative of the assumed relation forms a system admitting of solution in the form of two integrals- Both of these integrals will involve the assumed function. For any particular value of that function we have a system of lines satisfying equation (1), and the arbitrary character of the func- tion makes the solution sufficiently general to include all lines which satisfy the equation.f Prob. 76. Show that the equation (mz — ny)dx -f- (nx — lz)dy + (fy — mx)dz = o is integrable, and infer from the integral the character of the auxil- * It follows that, with respect to the system of lines represented by equations (5), equation (3) is the condition that the system shall admit of surfaces cutting them orthogonally. The lines of force in any field of conservative forces form such a system, the orthogonal surfaces being the equipotential surfaces. f So too there is an arbitrary element about the path of a point when the single equation to which it is subject is integrable, but this enters only into one of the two equations necessary to define the path. Art 21.] partial equations, first order, 355 iary lines. (Compare the illustrative example at the end of Art. 1 2!) (Ans. nx — Iz — C(ny — mz).) Prob. 77. Solve ysfdx — z'dy — e'dz = o. (Ans. yz =e x {i-\-cz).) Prob. 78. Find the equation which in connection with>» = /{x) forms the solution of dz = aydx -\- My. Prob. 79. Show that a general solution of ydx = (x — z){dy — dz) is given by the equations y — z=(p(x), y= (x- z)'(x). (This is an example of " Monge's Solution.") Art. 21. Partial Differential Equations of First Order and Degree. Let x denote an unknown function of the two independent variables x and y, and let _ df_ _dz p ~ dx' g ~dy denote its partial derivatives : a relation between one or both of these derivatives and the variables is called a " partial dif- ferential equation " of the first order.' A value of z in terms of .* and y which with its derivatives satisfies the equation, or a relation between x, y and z which makes z implicitly such a function, is a " particular integral." The most general equation of this kind is called the " general integral." If only one of the derivatives, say/, occurs, the equation may be solved as an ordinary differential equation. For if y is considered as a constant,/ becomes the ordinary derivative of z with respect to x\ therefore, if in the complete integral of the equation thus regarded we replace the constant of integra- tion by an arbitrary function of y, we shall have a relation which includes all particular integrals and has the greatest pos- sible generality. It will be found that, in like manner, when both p and q are present, the general integral involves an arbi- trary function. We proceed to give Lagrange's solution of the equation of 356 DIFFERENTIAL EQUATIONS. [CHAP. VII. the first order and degree, or " linear equation," which may be written in the form Pp+Qq = R, (!) P, Q and R denoting functions of x, y and z. Let u = a,in which u is a function of x, y and z, and a, a constant, be an integral of equation (i). Taking derivatives with respect to x and y respectively, we have 3« . 3« 3« . 3^ 3^+3^ = °' 37 + 3^ = °' and substitution of the values of / and q in equation (i) gives the symmetrical relation ^ + G 37 + ^ = a (2) Consider now the system of simultaneous ordinary differ- ential equations dx dy dz Let « = o be one of the integrals (see Art. 12) of this sys- tem. Taking its total differential, 3«, ,3", , 3k, —ax -+- —dy -+- —-dz = O : dx ^ dy J ^ dz and since by equations (3) dx, dy and dz are proportional to P, Q and i?, we obtain by substitution which is identical with equation (2). It follows that every integral of the system (3) satisfies equation (1), and conversely, so that the general expression for the integrals of (3) will be the general integral of equation (1). Now let v = b be another integral of equations (3), so that v is also a function which satisfies equation (2). As explained in Art. 12, each of the equations u = a, v = b is the equation of a surface passing through a singly infinite system of lines belonging to the doubly infinite system represented by equa- tions (3). What we require is the general expression for any Art. 21.] PARTIAL EQUATIONS, FIRST ORDER. 357 surface passing through lines of the system (and intersecting none of them). It is evident that f{u, v) =/(a, b) = C is such an equation,* and accordingly f(u, v), where / is an arbitrary function, will be found to satisfy equation (2). Therefore, to solve equation (1), we find two independent integrals u = a, v = b of the auxiliary system (3), (sometimes called Lagrange's equations,) and then put u — = *(*> " 9z, _u 9z, J 3« , 3« ., Adv dv "I and eliminating (x* -f- y 1 + z *)> ( I0 ) which represents any surface passing through the circles just mentioned, that is, any surface of revolution of which (9) is the axis.* Lagrange's solution extends to the linear equation contain- ing n independent variables. Thus the equation being the auxiliary equations are dx^ dx t _ _ dx n dz — ^ — '- — - = o is the condition that

= f (-)) dz dz f ' " ' ins. xy— z~ = t \yi Prob. 8i. Solve (y -f- z)p -f- (z -f jc)^ = jc -f j\ Prob. 82. Solve (# + j)(/ — q) = z. (Ans. {x^ r y)\ogz—x—f(x+y).) Prob. 83. Solve jc(^ — z)p +y(z — x)q = z(x —y). (Ans. x -\-y -\- z = f{xyz).) Art. 22. Complete and General Integrals. We have seen in the preceding article that an equation be- tween three variables containing an arbitrary function gives rise to a partial differential equation of the linear form. It follows that, when tbe equation is not linear in p and g, the general integral cannot be expressed by a single equation of the form = k\ a and b being regarded as arbitrary, be taken as the primitive, the derived equations are x — a -f- zp = o, y — b -f- zq = o, and the elimination of a and b gives the differential equation *(/ + {a) ) = o, ^ a A x , y, z, a, (a)) = o. (3) These two equations together determine a line, namely, the " ultimate intersection of two consecutive surfaces." Such lines are called the " characteristics " of the differential equa- tion. They are independent of any particular form of the complete integral, being in fact lines along which all integral surfaces which pass through them touch one another. In the illustrative exr.mple above they are equal circles with centers in the plane of xy and planes perpendicular to it* The example also furnishes an instance of a "singular so- lution " analogous to those of ordinary differential equations. *The characteristics are to be regarded not merely as lines, but as " linear elements of surface," since they determine at each of their points the direction of the surfaces passing through them. Thus, in the illustration, they are cir- cles regarded as great-circle elements of a sphere, or as elements of a right ■cylinder, and may be likened to narrow hoops. They constitute in all cases a triply infinite system. The surfaces of a complete integral system contain them all, but they are differently grouped in different integral surfaces. If we arbitrarily select a curve in space there will in general be at each of its points but one characteristic through which the selected curve passes; that is, whose tangent plane contains the tangent to the selected curve. These char- acteristics (for all points of the curve) form an integral surface passing through the selected curve ; and it is the only one which passes through it unless it be itself a characteristic. Integral surfaces of a special kind result when the se- lected curve is reduced to a point. In the illustration these are the results of rotating the circle about a line parallel to the axis of z. '66'Z DIFFERENTIAL EQUATIONS. [Chap. VII. For the planes z = ± k envelop the whole system of spheres, represented by the complete integral, and indeed all the sur- faces included in the general integral. When a singular solu- tion exists it is included in the result of eliminating a and b- from equation (2) and its derivatives with respect to a and b, that is, from 3/3/ f=°> da = °' db=°' (4) but, as in the case of ordinary equations, this result may in- clude relations which are not solutions. Prob. 84. Derive a differential equation from the primitive Ix + my -\- nz — a, where /, m, n are connected by the relation f + m* + n* = 1. Prob. 85. Show that the singular solution of the equation, found in Prob. 84 represents a sphere, that the characteristics con- sist of all the straight lines which touch this sphere, and that the general integral therefore represents all developable surfaces which touch the sphere. Prob. 86. Find the integral which results from taking in the general integral above /' -{-m 2 = cos 2 (a constant) for the arbitrary- relation between the parameters. Art. 23. Complete Integral for Special Forms. A complete integral of the partial differential equation F(x, y, z, p, q) = O (i)> contains two constants, a and b. If a be regarded as fixed and b as an arbitrary parameter, it is the equation of a singly in- finite system of surfaces, of which one can be found passing through any given point. The ordinary differential equation of this system, which will be independent of b, may be put in the form dz = pdx -\- qdy, (2)' in which the coefficients/' and q are functions of the variables and the constant a. Now the form of equation (2) shows that these quantities are the partial derivatives of z, in an integral of equation (1); therefore they are values of p and q which Art. 33.] complete integral for special forms. 363 satisfy equation (i). Conversely, if values oip and q in terms of the variables and a constant a which satisfy equation (i) are such as to make equation (2) the differential equation of a sys- tem of surfaces, these surfaces will be integrals. In other words, if we can find values of p and q containing a constant a which satisfy equation (1) and make dz = pdx -\- qdy inte- grate, we can obtain by direct integration a complete inte- gral, the integration introducing a second constant. There are certain forms of equations for which such values of p and q are easily found. In particular there are forms in which p and q admit of constant values, and these, obviously make equation (2) integrable. Thus, if the equation contains p and q only, being of the form F(p, q) = o, (3) we may put/ = a and q = b, provided F{a, b) = o. ( 4 > Equation (2) thus becomes dz = adx -\- bdy, whence we have the complete integral z = ax -\- by -f- c, (5} in which a and b are connected by the relation (4) so that a, b and c are equivalent to two arbitrary constants. In the next place, if the equation is of the form B=px + qy+f(p,q), (6) which is analogous to Clairaut's form, Art. 10, constant values of p and q are again admissible if they satisfy z = ax + by+/(a, b), (7) and this is itself the complete integral. For this equation is of the form z =ax-\- by -\- c, and expresses in itself the rela- tions between the three constants. Problem 84 of the preced- ing article is an example of this form. In the third place, suppose the equation to be of the form F(z,p,q) = o, (8) 364 DIFFERENTIAL EQUATIONS. [Chap. VII. in which neither x nor y appears explicitly. If we assume q = ap, p will be a function of z determined from F(z, p, ap) — o, say p = a Y x + f .{y> a ¥y + b. (13) It is frequently possible to reduce a given equation by trans- formation of the variables to one of the four forms considered in this article.* For example, the equation x*p' -\- f q' = £ may be written (xdzV lydz\_ Kzdxl^ \zdyl ~ *' *The general method, due to Charpit, of finding a proper value of^ consists of establishing, by means of the condition of integrability, a linear partial dif- ferential equation for/, of which we need only a particular integral. This may be any value of p taken from the auxiliary equations employed in Lagrange's process. See Boole, Differential Equations (London 1865), p. 336 ; also For- syth, Differential Equations (London 1885), p. 316, in which the auxiliary equa- tions are deduced in a more general and symmetrical form, involving both / and q. These equations are in fact the equations of the characteristics regarded as in the concluding note to the preceding article. Denoting the partial deriva- tives of F(x, y, z, p, q) by X, Y, Z, P, Q, they are dx dy dz dp dq ~P~~~ Z ~Q~ Pp+Qq = ~X+Zp = ~^+Zq' See Jordan's Course d'Analyse (Paris, 1887), vol. m, p. 318 ; Johnson's Differ- ential Equations (New York, 1889), p. 300. Any relation involving one or both the quantities p and q, combined with p=o, will furnish proper values of/ Art. 24.] partial equations, second order. 365 whence, putting x' = log x, y' = log y, z' = log z, it becomes /" + t" = i» which is of the form F{p', q') = o, equation (3). Hence the integral is given by equation (5) when a" + 6* = 1; it may therefore be written z' = x' cos a -\-y' sin a -\- c, and restoring x, y, and z, that of the given equation is z = cx cos a y sin a - Prob. 87. Find a complete integral for/ 5 — q* = 1. (Ans. a = x sec a +jy tan a -\- i.) Prob. 88. Find the singular solution of z — p x -f- qy + pq. (Ans. z = — jy. ) Prob. 89. Solve by transformation q — 2yp'. (Ans. 2 = ax + « 2 y + 2>) Prob. 90. Solve z{p*—g 2 ) — x — y. (Ans. s* = (* + «)» + (y -f <*)* + £.) Prob. 91. Show that the solution given for the form F{z,p, q) = o represents cylindrical surfaces, and that F{z, o, o) = o is a singular solution. Prob. 92. Deduce by the method quoted in the foot-note two complete integrals of pq = px -f- gy. (Ans. 22 = (-J- + ay J +/J, and x = xy + y */(*' + a) + b.) Art. 24. Partial Equations of Second Order. We have seen in the preceding articles that the general solution of a partial differential equation of the first order de- pends upon an arbitrary function ; although it is only when the equation is linear in p and q that it is expressible by a single equation. But in the case of higher orders no general account can be given of the nature of a solution. Moreover, when we consider the equations derivable from a primitive con- taining arbitrary functions, there is no correspondence between their number and the order of the equation. For example, if and q. Sometimes several such relations are readily found ; for example, for the equation e=pg we thus obtain the two complete integrals z = (y-\-a)(x + b) and 42 =(- + ay + /Sj . x -^ -3&6 DIFFERENTIAL EQUATIONS. [CHAP. VII. the primitive with two independent variables contains two ar- bitrary functions, it is not generally possible to eliminate them and their derivatives from the primitive and its two derived equations of the first and three of the second order. Instead of a primitive containing two arbitrary functions, let us take an equation of the first order containing a single arbitrary function. This may be put in the form u = ' s ~ dxdy' ~ ay the result is found to be of the form Rr + Ss-\- 7V+ U(rt- s°) = V, (i) in which R, S, T, U, and V are functions of x, y, z, p, and q. With reference to the differential equation of the second order the equation u = 0(w) is called an " intermediate equation of the first order " : it is analogous to the first integral of an ordi- nary equation of the second order. It follows that an inter- mediate equation cannot exist unless the equation is of the form (i); moreover, there are two other conditions which must exist between the functions R, S, T, and U. In some simple cases an intermediate equation can be ob- tained by direct integration. Thus, if the equation contains derivatives with respect to one only of the variables, it may be treated as an ordinary differential equation of the second order, the constants being replaced by arbitrary functions of the other variable. Given, for example, the equation xr — p = xy, which may be written xdp — pdx = xy dx. This becomes exact with reference to x when divided by x', and gives the intermediate equation p =yx\og x -f- x(y) + f(y). ART. 24.] PARTIAL EQUATIONS, SECOND ORDER. 367 Again, the equation p-\- r -\-s= i is already exact, and gives the intermediate equation which is of Lagrange's form. The auxiliary equations* are dx = dy = "- — , x _ B _j_ 0(j,y of which the first gives x — y = a, and eliminating x from the second, its integral is of the form z = a-\- (y).) Prob. 95. Solve y*(s — t) = x. (Ans. z=(x +y) logy + ) + 'p(z).) Prob. 97. Show that Monge's equations (see foot-note) give for Prob. 96 the intermediate integral / = {z), and thence the general integral. (Ans. y + x, D')z = o (2) will constitute an equally general solution of equation (1). It is, however, only when f(D, D') is a homogeneous function of D and D' that we can obtain a solution of equation (2) containing n arbitrary functions,* which solution is also the "comple- mentary function" for equation (1). Suppose then the equation to be of the form d"z , d"z , , . d"z . . ^s=+^s=^+---+^ = * (3) and let us assume z = '>, etc. Substituting in equation (3) and rejecting the factor D'"z or

+ *«,#) + . . . + 0„O + w„^) (6) is the general integral of equation (3). d'z d'z For example, the general integral of -j-j — -^-, = o is thus found to be z = 0(j/ -\- x) -\- tp(y — x). Any expression of the form Axy -\- Bx -\- Cy -\- D'\s a particular integral ; accordingly it is expressible as the sum of certain functions of x -\- y and x — y respectively. The homogeneous equation (3) may now be written sym- bolically in the form (D - m,D')(D - mJD') ...(£>- m n D')z = o, (7) in which the several factors correspond to the several terms of the general integral. If two of the roots of equation (5) are equal, say, to *«,, the corresponding terms in equation (6) are equivalent to a single arbitrary function. To form the general integral we need an integral of (D — m x DJz = o (8) in addition to 0(y -\- m,x). This will in fact be the solution of (D — m^D')z = , D')F{ax + by) = f(a, b)F*\ax + by). (13) That is, if t = ax -f- £jj/, the operation of /(Z>, Z)') on F{t) is equivalent to multiplication by/" («, £) and taking the rath de- rivative, the final result being still a function of t. It follows that, conversely, the operation of the inverse symbol upon a function of t is equivalent to dividing by f{a, b) and integrating n times. Thus, jijhr) F{ax + h) = 7+dx-dy - 2 d? = Sin {X -A + Sm <* +» or (Z? — Z7) (Z? + 2Z?> = sin (x — y) -\- sin (* + j/). The complementary function is 0(j + x) -f- ^-(;/ — 2;r). The part of the particular integral arising from sin (x —y), in which « = 1, b = — 1, is / /sin tdf — - sin (jt — y). That aris- * The symbolic form of this theorem is D -m/y F (*' yS > = emXD ' I '' mxD ' F ^ x ' y} dx corresponding to equation (13), Art. 16. The symbol *«*D' here indicates the addition of mx to y in the operand. Accordingly, using the expanded form of the symbol, e mxD'F{y) = (i + mx — + —f- J\ + ' • ') ^ W = F ^ + mx ">> the symbolic expression of Taylor's Theorem. 372 DIFFERENTIAL EQUATIONS. [CHAP. VII. ing from sin [x -\-y) which is of the form of a term in the com- plementary function is jz jr, cos (x -\-y), which by equa- tion (10) is — \x cos (x-\-y). Hence the general integral of the given equation is z = (p(y + x) + f(y - 2x) + | sin (x —y)- \x cos ( x -\-y). If in the equation /{£>, D')z — o the symbol f(D, D'), though not homogeneous with respect to D and D ', can be separated into factors, the integral is still the sum of those corresponding to the several symbolic factors. The integral of a factor of the first degree is found by Lagrange's process ; thus that of (D — mD' — a)z = o (15} is z = e ax cf){y -f- mx). (16) But in the general case it is not possible to express the solution in a form involving arbitrary functions. Let us, how- ever, assume 3=a* x + k y, (17) where c, h, and k are constants. Since De kx+iy = ke'' x+l * and D'e hx+ky — ke hx + h , substitution in f'{D, D')z = o gives cf{h, k)e hx + ky = O. Hence we have a solution of the form (17) whenever h and k satisfy the relation f(k, k) = o, (18) c being altogether arbitrary. It is obvious that we may also write the more general solution z=2ce hx+my , (19) where k = F(h) is derived from equation (18), and c and h admit of an infinite variety of arbitrary values. Again, since the difference of any two terms of the form e kx + F <- li >y with different values of h is included in expression (19), we infer that the derivative of this expression with respect to h is also an integral, and in like manner the second and higher derivatives are integrals. For example, if the equation is d^z dz dx'' dy ~ ' Art. 25.] linear partial equations. 373 for which equation (18) is A" — k = o, we have classes of in- tegrals of the forms e hx + h *y, e hx + h Xx + 2/iy), e" x + h \{x + 2hyY + 2jj/)], e" x + h \(x + 2hy) a + 6y(x + ihy)\ In particular, putting A = owe obtain tfu. algebraic integrals c x x, c 1 (x 1 -\-2y), c t (x'-\-6xy), etc. The solution of a linear partial differential equation with variable coefficients may sometimes be effected by a change of the independent variables as illustrated in some of the exam- ples below. Prob. 99. Show that if m l is a triple root the corresponding terms of the integral are x'(p(y -f- m^x) -f- xip(y-{- m^x)-\- xiy+n^x)- pi c 1 d " z ^ z ^ z Prob. 100. Solve 2— — 3—— 2— - a = o. c 1 9'* , 3"* . 3 3 * 1 Prob. 101. Solve , + 2 4- — - = -j-. 3^ 3y 3x3/ ' 9/ * (Ans. 2 = 0(x) + ip(x -f j) -f xj(* +jc) — ^ log #.) Prob. 102. Solve (Z> 2 + 5Z>Z>' + 6D n )z = ( v - 2a:)" 1 . (Ans. a = 0(j> — 2x) + ^(_v — 3X) -|- x log (jy — 2a:).) Prob. 10?. Solve — — ; — . - + ^ 2=0. •* 3a: 2 dxdy ^ 3je Prob. 104. Show that for an equation of the form (15) the solu- tion given by equation (19) is equivalent to equation (16). v , „. l3*2 I 32 I 3 2 2 I 32 Prob. 105. Solve — r~j s ^ = ~i ^ ; 5" b y transposi- a: 3a: jc' 9.x jy 3/ /3/ don to the independent variables x* and /. ■a u c c 1 a3 3 ' sr 1 3* z 1 2^ z Prob. 106. Solve * — + «*_ +y^r = o. 374 GRASSmann's space analysis. [Chap. VIIL Chapter VIIL GRASSMANN'S SPACE ANALYSIS. By Edward W. Hyde, Professor of Mathematics in the University of Cincinnati. Art. 1. Explanations and Definitions. The algebra with which the student is already familiar deals directly with only one quality of the various geometric and mechanical entities, such as lines, forces, etc., namely, with their magnitude. Such questions as How much? How far? How long? are answered by an algebraic operation or series of operations. Questions of direction and position are dealt with indirectly by means of systems of coordinates of various kinds. In this chapter an algebra* will be developed which deals directly with the three qualities of geometric and mechanical quantities, viz., magnitude, position, and direction. A geomet- ric quantity may possess one, two, or all three of these prop- erties simultaneously ; thus a straight line of given length has all three, while a point has only one. The geometric quantities with which we are to be concerned are the point, the straight line, the plane, the vector, and the plane-vector. When the word "line" is used by itself, a "straight line" will be always intended. A portion of a given straight line of definite length will be called a " sect " ; though when the length * The algebra of this chapter is a particular case of the very general and comprehensive theory developed by Hermann Grassmann, and published by him in 1844 under the title "Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik." He published also a second treatise on the subject in 1862. Art. 2.J explanations and definitions. 375 of the sect is a matter of indifference, the word line will fre- quently be used instead. Similarly, a definite area of a given plane will be called a " plane-sect." If a point recede to infinity, it has no longer any significance as regards position, but still indicates a direction, since all lines passing through finite points, and also through this point at infinity, are parallel. Similarly, a line wholly at infinity fixes a plane direction, that is, all planes passing through finite points, and also through this line at infinity, are parallel. Thus a point and line at infinity are respectively equivalent to a line direction and a plane direction. A quantity possessing magnitude only will be termed a "scalar " quantity. Such are the ordinary subjects of algebraic analysis, a, x, sin 0, log z, etc., and they may evidently be in- trinsically either positive or negative. The letter T prefixed to a letter denoting some geometric quantity will be used to designate its absolute or numerical magnitude, always positive. Thus, if L be a sect, and Pa. plane- sect, then TL is the length of L, and TP is the area of P. That portion of a geometric quantity whose magnitude is unity will be called its " unit," and will be indicated by prefixing the letter U; thus UL = unit of L = sect one unit long on line L* Hence we have TL . UL = L. Art. 2. Sum and Difference of Two Points. In geometric addition and subtraction we shall use the or. dinary symbols -(-, — , =, but with modified significance, as will appear in the development of the subject. Every mathematical, or other, theory rests on certain fun- damental assumptions, the justification for these assumptions * The word "scalar ''and the use of the letters T and U, as above, were introduced by Hamilton in his Quaternions, ^stands for tensor, i.e., stretcher, and TL is the factor that stretches UL into L. The notation \ L \ for absolute magnitude is not used, because the sign | has been appropriated by Grassmann to another use. 376 grassmann's space analysis. [Chap. VIII. lying in the harmony and reasonableness of the resulting theory, and its accordance with the ascertained facts of nature. Our first assumption, then, will be that the associative and commutative laws hold for geometric addition and subtrac- tion, that is, whatever A, B, C may represent, we have A + B + C = (A + £) + C = A + (B + C) = A + C-{-B = (A + C)+£. We shall also assume that we always have A — A = o, and that the same quantity may be added to or subtracted from both sides of an equation without affecting the equality. Now let/; , /, be two points, and consider the equation A+A-A=A + (A~A)=A- (i) In this form we have an identity. Write it, however, in the form A-A+A = (A-A)+A=A- (2) and it appears that/, — /, is an operator that changes/, into /,by being added to it. Conceive this change of/, into/, to take place along the straight line through /, and />, ; then the operation is that of moving a point through a definite length or distance in a definite direction, namely, from/, to/,. This operator has been called by Hamilton " a vector," * that is, a carrier, because it carries/, rectilinearly to/,. Grassmann gives to it the name Strecke, and some writers now use the word " stroke " in the same sense. Again,/,—/, is the difference of two points, and the only difference that can exist between them is that of position, i.e. a certain distance in a certain direction. Hence we may regard /, — /, as a directed length, and also as the operator which moves /, over this length in this direc- tion. Writing/, — /, = e, equation (2) becomes A + e = A- (3) * See the first of Hamilton's Lectures on Quaternions, where a very full discussion of equation (2) will be found. Also Grassmann (1862), Art. 227. Art. 2.] sum and difference of two points. 377 Thus the sum of a point and a vector is a point distant from the first by the length of the vector and in its direction. Since A — A — — (p t — A)> it appears that the negative of a vector is a vector of the same length in the opposite direction. If A — A = o, or A = A> A must coincide with A because there is now no difference between the two points. The question arises as to what, if any, effect the operator A— A should have on any other point/,, that is, what is the value of the expression p, —A +A ? We will assume that it is some point A > so that we have A— A+A=A> or A-A=A-A (4) This implies that the transference from A to A * s the same in amount and direction as that from A to A> that is» that A> A' A> A are the four corners of a parallelogram taken in order. Thus equal vectors have the same length and direction, and, conversely, vectors having the same length and direction are equal. Note that parallel vectors of equal length are not neces- sarily equal, for their directions may be opposite. Equation (4) may also be written A+A=A+A, (5) so that, whatever meaning may be assigned to the sum of two points, if we are to be consistent with assumptions already made, we must have the sum of either pair of opposite corner- points of a parallelogram equal to the sum of the other pair. The sum cannot therefore depend on the actual distances apart of the points forming the pairs, for the ratio of these two distances may be made as large or as small as we please. If n be a scalar quantity, ne will denote that the operation € is to be performed n times on a point to which ne is added, that is, the point will be moved n times the length of e ; hence 378 grassmann's space analysis. [Chap. VIII. ne is a vector n times as long as e, and having the same or the opposite direction according to the sign of n. In the figure above, let A— A = e i> A— A = e .> A-A = e i» A— A = e 4 . Then e. + e, = A —A + A —A =A —A +A —A = A— A = e„ (5) since, by Art. 4, A — A =A — A- Also, e 3 - e, = A — A = • • -A De any n points whatever taken as corners of a closed polygon, we shall have (A-A)+(A-A)+(A-A)+---+(A-A- 1 )+(A-A) = o; that is, the sum of vectors represented by the sides taken in order about the polygon is zero. By " taken in order " is not meant that any particular order of the points must be observed in forming the polygon, which is evidently unnecessary, but simply that, when the polygon is formed, the vectors will be the operators that will move a point from the starting position along the successive sides back to this position again, so that the final distance from the starting-point will be nothing. Art. 3. Sum of Two Weighted Points, f Consider the sum w,A + ^ a A> in which m { and m, arescalars, that is, numbers, positive or negative, and A> A are P°i nts - * Grassmann (1844), § 15. t Grassmann (1844), § 95, and (1862), Art. 227. Art. 3.] sum of two weighted points. 37& The scalars m x and »z a will be regarded as values or weights assigned to the points/, and/,. When any weight is of unit value the figure i will be omitted, so that p means ip, and is called a unit point. Occasionally, however, a letter may be used to denote a point whose weight is not unity. To assist his thinking, the reader may consider the weights initially as like or unlike parallel forces acting at the points. In order to arrive at a meaning for the above expression we shall make two reasonable assumptions, which will prove to be consistent with those already made, viz., first, that the sum is a point, and second, that its weight is the sum of the weights of the two given points. Denoting this sum-point by p, we write »*. A + *» S A = C'«i + *«.)A (7) Transposing, we have *«,(/, — p) = m,(p — /,), or P^zIJ^rJi, (8) Both members of (8) are vectors, and, being equal, they must, by Art. 4, be parallel. This requires that/ shall be collinear with /, and p v Also, since /, — p and p — p, are vectors whose lengths are respectively the distances from/, to p and (romp to p v it follows that these distances are in the ratio of m, to in,. Hence, p is a point on the line p,p, whose distances from/, and/, are inversely proportional to the weights of these points. We shall call p the mean point of the two weighted points. If ;/z, and #z 2 are both positive, (8) shows that p must lie be- tween /, and/,; but if one, say m t , is negative, let m t ——m^. Thus «,(A - /) = <(A — P\ (9) and / is on the same side of each point, that is, its direction from each point is the same. Also, since its distances from the two points are inversely as their weights, p must be nearest the point whose weight is greatest. 380 grassmann's space analysis. [Chap. VIII. Case when «, -|- m^ = o, or m t = —m .* — With this con- dition equations (7) and (8) become **.A + m J* = m X A — A) = o . J, (10) and /-A=/-A- (11) Thus ^> is in the same direction from each point, that is, not between them, and yet is equidistant from them. This re- quires either that the two points shall coincide, that is, p t =p v which evidently satisfies (10) and (11); or else, p 1 and/,, being ■different points, that p shall be at an infinite distance. Thus the sum is in this case a point of zero weight at infinity.! Eq. (10) shows that a zero point at infinity is equivalent to a vector, or directed quantity, as stated in Art. 1. It has been shown in Art. 2 that p^ =p s is the condition that p^ and p, coincide ; let us consider the equality of weighted points in general, say m 1 p 1 =m,p 2 . Hence, by (7), there is found m l p 1 — w*,A = (^, — w,)/ = o; hence, since p cannot be zero, m 3 — m^ = O, or «, = m 2 ; and therefore m l (p 1 —p,) = 0, or, since «k,^0, /, — p^ =0, that is, p^ =p v Therefore, if any two points are equal, their weights must be the same and their positions identical, that is, they are the same point. Exercise 1. — To find the sum and difference of the two weighted points 3/, and p^ : 3A+A = 4A 3A - A = 2p', and the mean points are as shown in *| the figure. The reciprocals of the L_ 1 i _ ' Sp ' 3p ' ip ft distances of p, p v and/ from/,, viz., \, \, \, are in arithmetical progression, hence the points form a harmonic range. Exercise 2. — Given a circular disk with a circular disk of *Grassmann (1862), Art. 222. t Compare the case of the resultant of unlike parallel forces of equal magnitude. 2— ^_i_^ JL-.- Art. 4.] sum of two weighted points. 381 half its radius removed, as in the figure ; to find the centroid of the remaining portion. Take /j at center of large circle, /, at center /^ />M\ of small circle, and /, at the point of contact ; f p,( p \ then p 3 = £( />, -+- A)- The areas of the two cir- xX\^£/ cles are as i 14; call them 1 and 4. Then it is as if there were a weight 4 at p x , and a weight — 1 at/, ; hence P = [4A - KA + A)] + 3 = (7A - A) - 6 - Prob. 1. Show that p s ,p„ «,/, + »*,A> an d w iA ~~ w jA are four points forming a harmonic range. Prob. 2. An inscribed right-angled triangle is cut from a circular disk ; show that the centroid of the remainder of the disk is at the point (37T — 2 sin 2«) /, — /, sin 2a 3(7? — sin 2a) \lp x is the center of the circle, p, the opposite vertex of the triangle, and a one of its angles. Art. 4.. Sum of any Number of Points. As in the last article we assume the sum to be a point whose weight is equal to the sum of the weights of the given points ; thus, n — n 2mp=p2m. (12) 1 1 M Let e be some fixed point, and subtract e2m from both 1 sides of (12) ; thus we have 2m{p — e)={p — e)Sm, (13) an equation which gives a simple construction for/. If 2m = o, then m l = — 2m, and 1 2 / 2mp\ 2mp = m^pt + 2mp = m\ p 1 — —„ — , (14) 382 grassmann's space analysis. [Chap. VIII. so that the sum becomes the difference of two unit points, or a vector whose direction is parallel to the line joining p 1 with the mean of all the other points of the system, and whose length is m i times the distance between these points. Since any point of the system may be designated as A> it follows that the line joining any point of the system to the mean of « all the others is parallel to any other such line. If 2mp — o, i equation (14) shows that/, is the mean of all the other points of the system, and, since any one of the points may be taken as />,, any point of the system is the mean of all the others. Let n — 3 in (12) and (13); then »',A + »*, A + '«, A = (■>»! + »*. + tn,jp, (15) «,(/, — ')+ mlP*—e) + nh{p 3 — e)={m 1 -\-m / ~A = 4(A -A) + 5(A -A) - 2(A -A) = A -A+A -A +7- A- Art. 4.] SUM OF ANY NUMBER OF POINTS. 383 When any number of geometric quantities can be connected with each other by an equation of the form 2mp = o, in which the m's are finite and different from zero, then they are said to be mutually dependent, that is, any one can be expressed in terms of the others. If no such relation can exist between the =:5P, / W.P, / / / / „- -U*< -2P 3 / / 4 4p„ p. S> quantities, they are independent. We obtain from what has preceded the following conditions; That two points shall concide, m l p 1 ^ r m i p i = 0. (17) That three points shall be collinear, m l p l -\-m i p l -\-m s p t = o. (18) That four points shall be coplanar, **,A + »*»A + w sA + m *P* = °- (*9) It follows that three non-collinear points cannot be con- nected by an equation like (18) unless each coefficient is separately zero. Similarly four non-coplanar points cannot be connected by an equation like (19) unless each coefficient is separately zero. The significance of these statements will be presently illus- trated. The following are corresponding equations of condition for vectors : That two vectors shall be parallel, «,e, + w 5 e, = o. (20) 384 GRASSMANN S SPACE ANALYSIS. [Chap. VIII. That three vectors shall be parallel to one plane, «, 6 , + «,e. + « s e 3 = o. (2r> These conditions follow from the results of Art. 2, or from equations (17) and (18) by regarding the e's as points at infinity. If in addition to (21) we have »i + »» + n > = °> (22) the extremities of the three vectors, if radiating from a point, will be collinear : for, let e„ . . . e % be four points so taken that if, — e a = e, , e, — e, = e, , e, — e a = e a ; then (21) becomes «,(>■ - '0) + «.(*> - O + »s(^s - *.) = o, or by (22) n 1 e 1 + «/, + » s * s = o, which by (18) requires e,, e,, e, to be collinear. It may be shown similarly that 2ne = 2n = o (23) are the conditions that four vectors radiating from a point shall have their extremities coplanar. Exercise 3. — Given a triangle e a e,e t and a point p in its plane; pe, cuts e/, in q t , pe 1 cuts e^e t in q,,pe a cuts e ejn q„ q,q, cuts„ -f (x, - yjifo - n,(f a -\-y,)e, = o. Now the e's are not collinear, and yet are connected by a ART. 4.] SUM OF ANY NUMBER OF POINTS. 385 relation of the form of equation (18); hence, as was there shown, each coefficient must be zero ; accordingly ■*. — .?i». = *—y**i =y* +7. = o, whence we find x : x 1 = n a : — n v hence (ra — «,)/, = n,e — n l e l , and similarly (*i—» a )A = «A — n,e t , («„ -»„)/>, = n,e 2 -n e a . Adding, we have («. - ».)A. + («. - «»)A + (», - »,)A = o ; therefore, by (18), A> A> A are collinear. 2 2 Exercise 4. — Let / = 2ne -f- 2 n be any point in the plane of the triangle *„*,-?,: show that lines through the middle points of the sides e t e t , e t e t , and e t e 1 of the triangle parallel to ej>, e x p, and ^ 3 /> meet in a point A - [(», + «>o + (». + ».)', + (», + «,k] -*- 2^». By the conditions the vector from the middle point of e^ to p' is a multiple of the vector e — p ; hence A - £0i + '.) = •*( — *)*, + [(4 - ^» + njj — x)]e, = o ; therefore, as in the previous exercise, each coefficient must be zero, whence x =y = f, and substituting we find p' as above. It follows also that the distances of p' from the middle points of the sides are the halves of the distances of p from the oppo- site vertices. 2 Prob. 3. Show that e = \2e is collinear with p and p' of Exer- 386 GRASSMANN S SPACE ANALYSIS. [Chap. VIII. cise 4. Also that, by properly choosing/, it follows that e is col- linear with the common point of the perpendiculars from the vertices on the opposite sides, and the common point of the perpendiculars to the sides at their middle points. Prob. 4. Given two circles and an ellipse, as in the figure, with centers at ( 22 7 e , + 175*3 + 239^, + *,) ). Art. 5. Reference Systems. Let p be any unit point, e a , i?, , e t three fixed unit points, and w, x, y scalars ; then, writing p = we, + xe x + ye t , (24) we must have also, because p is a unit point, w-\-x-\-y=i, (25) and p is the mean of the weighted points we , xe^, ye t . The point p may occupy any position whatever in the plane e l e l e i ; for it is on the line joining we, -\- xe 1 with e t , and by varying IS) y and w-\-x, — remaining constant, / may be moved along Art. 5.] REFERENCE SYSTEMS. 387 W this line from — °o to -\- ; while by varying the ratio — the point we„ -\- xe^ may be moved from — oo to -f- °° along „ + (£,/, + k.m, + *,»,)*, + (A>4 + £,>«, + &,«,), to p„ or generated by a point moving rectilinearly from^ to/ a . The student should note carefully the difference between p^pi and/ a — p, ; they have the same length and direction, but the sect p^p, is confined to the line through these two points, while the vector p, — p i is not. The sect has position in addi- tion to the direction and length possessed by the vector. Again, in plane space, two sects determine a point, the intersection of the lines in which they lie, and also an area, as will appear later, so that AA = A m which p is not in general a unit point. In solid space, however, two lines do not, in general, meet, and hence cannot fix a point ; but two sects, in this case, determine a tetrahedron of which they are opposite edges. It appears, therefore, that a product may have different interpretations in spaces of different dimensions. Hence we will consider separately products in plane space, or planimetric products, and those in solid space, or stereometric products. Products of the kind here considered are termed " com- binatory," because two or more factors combine to form a new quantity different from any one of them. This is the fundamental difference between this algebra and the linear associative algebras of Peirce, of which quaternions are a special case. Before discussing in detail the various products that may arise, we will give a table which will serve as a sort of bird's-eye view of the subject. * Grassmann (1844), Chap. 2 ; (1862), Chap. 2. f See Art. 1. Art. 6.] nature of geometric multiplication. 391 In this table and generally throughout the chapter we shall use/, p lt p„ etc., for points; e, e,, e 2 , etc., for vectors; L, L lt etc., for sects, or lines ; rj, rf it etc,, for plane-vectors ; and P, P lt etc., for plane-sects, or planes. Also/,/,, etc., as used in this table will not generally be unit points. The products are arranged in two columns, so as to bring out the geometric principle of duality. Planimetric Products. AA = L. £/-,=/• PiAA — area (scalar). L^L^L % = (area) a (scalar). pL = area (scalar). Lp = area (scalar). A . L t L t = L. A-AA = A PJi • AA = A L,L t . L,L. = L. AA-AA .AA=(area)'(scalar). L l L,.L,L l .L l L,= (area) 4 (scalar) e,e 2 = area (scalar). Stereometric Products. AA = L. PJ>* = L. P,P,P, = P- P,P,P a =p. PiPiPsP* — volume (scalar). P,P,P s P t = (volume) 3 (scalar). pP = volume (scalar). Pp = volume (scalar). L^ = volume (scalar). L t L t = volume (scalar). pL = Lp = P. PL = LP = p. p.PA = P- P-P,P*=P- p . P,P a P s = L. P-P,P,P, = L. -£-AAA=A L . PfJ> t = P e,e 2 = rj. VSh = e - €,6,6, = volume (scalar). r/jisi, = (volume) 2 (scalar). e,e,- e a e. = e. V,V* ■ V,V, = V- 392 grassmann's space analysis. [Chap, vili Laws of Combinatory Multiplication. — All combinatory products are assumed to be subject to the distributive law ex- pressed by the equation A{B + C) = AB -f- AC. The planimetric product of three points or of three lines and the stereometric product of three points or planes, or of four points or planes, are subject to the associative law. That is, In Plane Space : AAA = AA • A = A • AA ; AAA = AA • A = A • AA- In Solid Space : AAA = A ■ A A = A A A ; AAA = A • A A = AA • A AAAA=A-AAA = AA-AA; P P P P = P . P PP = P P pp The commutative law of scalar algebra does not, in general, hold. Instead of this, in the products just given as being asso- ciative, a law prevails which may be expressed by the equation AB=- BA, from which it follows that the interchange of any two single factors of those products changes the sign of the product.* Since vectors are equivalent to points at oo , the associative law holds for e^e, and Tf^fj,. Art. 7. Planimetric Products. Product of Two Points.f — This has been fully defined in Art. 6, and it is evident from its nature as there given that AA=-AA- (3 2 ) If p^ =p lt this becomes p i p 1 = o, which must evidently be true, since the sect is now of no length. Also, A(A -A) =AA -AA = AA- (33) * Grassmann (1862), Chap. 3. f Grassmann (1862). Arts. 245, 246, 247. Art. 7.] PLANIMETRIC PRODUCTS. 393 But/, — p v is a vector, say, e ; hence A e = A A : (34) or the product of a point and a vector is a sect having the di- rection and magnitude of the vector ; or, again, multiplying a vector by a point fixes its position by making it pass through the point. To find under what conditions pp' will be equal to /,/,. Take any other point p % in the plane space under consideration, and write p - x,p, + xj>, -\-xJ>„ p' =y,p 1 +AA, +J> / a A> with the conditions for unit points 2x = ~2y = o. Th e „ //=|,;,: >■*+ ,:; '■>•+ ,.,. **■ If this is to reduce to AA> we must have the third condition x,?, — *>y* — x *y* ~ x ^= — °' which requires that x 3 = y, = O, unless the coefficient of /,/., is to vanish also. Thus //' must be in the same straight line with /,/,. If, moreover, in addition x t y 2 — x t y t — i, we shall have pp' = pp,. Hence pp' is equal to p,p., when, and only when, the four points are collinear, and p' is distant from/ by the same amount and in the same direc- tion that/, is from/,. Product of Three Points. — By Art. 6 the product is what is determined by the three points. In solid space they would fix a plane, but, as we are now confined to plane space, this is not the case. The points evidently fix either a triangle or a parallelogram of twice its area, and the product p,pj>, will be taken as the area of this, or an equivalent, parallelogram. This area is taken rather than that of the triangle, because it is what is generated by/,/, as it is moved parallel to its initial position till it passes through /,. We have pj>J> % — p x .pj>, = - A -AA = -AAA- so that if we go around the triangle in the opposite sense the sign is changed. As this product possesses only the properties of mag- nitude and sign it is scalar. Write/ = 2xp,p' = 2yp,p" = 2zp; then 394 grassmann's space analysis. [Chap. VIII. X 'j x^ x % PP'P" - i\ y* y* AAA; (35> •^i z i z * that is, any triple point product in plane space differs from any other only by a scalar factor.* Finally, AAA = A(A - A)(A - A) =Aee', (36) if e = A —A and e' =/, — A- Product of Two Vectors. — -Using the values of e and e' just given, we see that e and e' determine the same paral- lelogram that A> A> an d A do; hence the meaning of the product is the same in all respects in two-dimensional space. We shall have ee' = — e'e, for ee' = (A - A)(A - A) = - (A - A)(A ~A) = -e'e; since we have shown that inverting the order changes the sign: in a product of points. The result may be obtained also by regarding e and e' as points at infinity, or by consideration of a figure. As we have seen that ee' has, in plane space, precisely the same meaning as AAA we mav write AAA = A ee ' = ee ' = (A - A)(A -A) = AA + AA+AA- (37) Thus the sum of three sects which form the sides of a triangle, all taken in the same sense as looked at from outside the triangle, is equal to the area of the triangle. Product of Two Sects. — Any two sects in plane space,. p L v L v determine a point, the intersec- /' tion of the lines in which they lie, and y an area, that of a parallelogram as in. the figure. Let A be the intersection, and take A and A so that L x ^p^p, and Z a =AA- The area. * Grassmann (1862), Art. 255. /i 2 Art. 7.] planimetric products. 395 determined by Z, and Z 2 is then the same that we have given as the value of AAA- We write therefore 44 = Pop, -P.P. = AAA • A- (38) The third member of (38) is not to be regarded as derived from the second by ordinary transposition and reassociation of the points, for the associative law does not hold for the four points taken together, since AAA -A — °- The third member simply results from the definition of L^L^* It may be taken as a model form which will be found to apply to several other cases, for instance to (38) when points and lines are inter- changed throughout. Thus, if />, = L L, and A = L L^ we have A A = 44 • 44 = 444 • 4- (39) For take p/ and A' so that/,// = Z, and/,^'= Z„; AA is. evidently some multiple of Z , say «Z ; hence AA = «Z„ = Jr( A A • AA') • (AA • AA') = ^(AAA'-A)-(AAA' -A), by (38), = -* ■ PJ>*Px-PiP*P* ■ AA» because AAA' and AAA' are scalar, = ^ (/.A • AA' -AA') • 4- by (38), = L^L.L^ . Z , which was to be proved. Product of Three Sects. — The method has just been indi- cated, but we may also proceed thus : Let the lines be Z„, L„ Z 2 , and let A- A- A be their common points. Take scalars n„ «, n s so that Z = «,/,/„ etc., then Z Z,Z, = »,»,», . AA . AA • AA = - *.».». • AAAA-AA = - «,»,», . AAA • AAA = ».*.».( AAA)'. (4°) * Grassmann applies the terms "eingewandt" and "regressiv " to a prod- uct of this kind, the first term being used in the Ausdehnungslehre of 1844. and the second in that of 1862. See Chapter 3 of the first, and Chapter 3, Art. 94, of the second. 396 grassmann's space analysis. [Chap. viii. Product of a Point and Two Sects. — Let/ be any point and let L x and Z, be as in (38) ; then pL,L t =p.pj> % .pj* =P-Pop i p,-Po =P<,P>P,-PPo- (40 It has been here assumed that pL x L^ —p . L x L r The prod- uct is not associative, for pL x . Z 2 is the line Z 2 times the scalar pL x , a different meaning from that assigned in (41). As a rule, to avoid ambiguity, the grouping of such products will be indicated by dots. Product of Two Parallel Sects. — Let them be/,e and np t e; then, as in (38), p,e . np^e = n.p 1 e .p,e = n . ep,. ep, = n . ep,p t . e, (42) that is, a scalar times the common point at 00 . Addition and Subtraction of Sects. — Let Z, and Z, be two sects, p their common point, and p x and p a so taken that A = AA > L 2 = P„P* ; then Z, + Z 3 =p t p l +p p, =p,(p t +A) = 2p p, (43) p being the mean of p x and/ 2 ; hence the sum is that diagonal of the parallelogram which passes through p Q . Also Z.-Z, = A(A~A)> (44) so that the difference of the two passes also through/,, and is parallel to the other diagonal of the parallelogram determined by Z, and Z a . If the two sects are parallel let them be «,/,e and n,p,e\ then «,A e + nj,e = {nj>, + n,p,)e = (ft, + n,)pe, , (45) so that the sum is a sect parallel to each of them, having a length equal to the sum of their lengths, and at distances from them inversely proportional to their lengths. If «, = — «, the two sects are oppositely directed and of equal length, and the sum is «.(/.« -A<0 = »,(A -AH (4 6 ) which, being the product of two vectors, is a scalar area. Art. 7.] Consider next n sects p i e l , p e , . arbitrarily chosen point ; then PLANIMETRIC PRODUCTS. 397 p„e n , and let e„ be some 2pe = e,Se - e 2e + 2pe = e 2e + 2{p - e a )e. (47) The second term of the third member of this equation, being a sum of double vector products, that is, a sum of areas, is itself an area, and is equal to the product of any two non-parallel vec- tors of suitable lengths. Therefore, a and yS being such vec- tors, write 2e = a and 2(J> — >.*. + {K + £.k'„- Multiply by e e 1 to find where the resultant cuts this line ; then (£,+&,>„*, . ,=eje x e % . [(£,+£,>,— (k t -\-k,)e t ], or e t e t cuts the resultant at the point [(*, + *>,-(*. + *.)'.]-(*,-*.)• Similarly the resultant cuts the other sides of the reference triangle at [(*, + £,>, - (k, + i,)e,] -»- (k t + k % -k a - k) and at [(£. + k t )e, - (k x + £,)*,] - (k a - k,). Suppose k = k, = k, = k,; then each of the three points just found recedes to infinity ; but in this case 2pe reduces to 2£oO.*\ + and the system is equivalent to a couple. Prob. n. Construct the resultant of Exercise 8 when k — i, *.= 2 > K~ 3- K = 4! when k = i, ^= - 2, k = 3, £,= - 4: when i„ = 3, -4, = k, = 2, k^ = 1 ; and when £, = &, = 1, /£„ = &, = — 2- Art. 8.] the complement. ' 399 Prob. 12. There are given n points p, . . .p„; to find a point e such that forces represented by the sects ep l , ep t , etc., shall be in equilibrium. (The equation of equilibrium is 2ep~e2p = —ep = o. n Hence e coincides with the mean point of the/'s.) Prob. 13. If a harmonic range e it p, e it p' be given, together with some point e x = — e » e > ■ e » e x = — W, • e, = *„ by (a) and (38) ; I VA= ko • ki • k, = 'A • *A • V> = ('.-V.)' = 1 = VA> which agrees with {c) ; *„ | *, = e Q e 1 e = o = V.+W l = W,(j i ~ j)(l ~ j) = A- (50) Thus the complement of a point is a line,* which may be easily constructed by the fourth member of (50); which ex- presses this line as the product of the points in which it cuts the sides e/^ and e a e t of the reference triangle. Comparing this equation with Ex. 3 in Art. 4, it appears that |/, above is 2 e related to the point 2-,as the line/ / a of Ex. 3 is to the point 2ne. Hence \p t may be found by constructing this line cor- responding to 2 -. as shown in the figure of Ex. 3, Art. 4. Again, the line \p t may be shown to be the anti-polar of p With respect to an ellipse of such dimensions, and so placed upon *//, that, with reference to it, each side of the reference triangle is the anti-polar of the opposite vertex.* From this it appears that complementary relations are polar reciprocal relations. Take any point p, — ~2me, and we have A lA = (4a + I A + l * e *)( m /A + **A*. +*W0 = 2lm = 2me.2l\e=p,\p lt (51} *See Hyde's Directional Calculus, Arts. 41-43 and 121-123. Art. 8.] THE COMPLEMENT. 401 so that this product is commutative about the complement sign, and scalar. This is true of all such products when the quantities on each side of the complement sign are of the same order in the reference units. Take for instance the product AAlAA- This is scalar because \p a p t is a point, so that the whole quantity is equivalent to a triple-point product ; and we have/A \pj k = \pj l .pj t = | (pj A \pj\ - A p t \pj v by ( fl) and (c). If, however, such a quantity be taken as pj> t . \p, it is neither scalar nor commutative about the sign | ; for, \p % being a line, the product is that of two lines, that is, a point, and AA • I A = - 1 A • AA = - I (A • IAA)- (52) Such products as we have just been considering are called by Grassmann " inner products," * and he regards the sign | as a multiplication sign for this sort of product. Inasmuch, however, as these products do not differ in nature from those heretofore considered, it appears to the author to conduce to simplicity not to introduce a nomenclature which implies a new species of multiplication. For instance,/^ will be treated as the combinatory product of p into the complement of q, and not as a different kind of product of/ into q. The term co-product may be applied to such expressions, regarded as an abbreviation merely, after the analogy of cosine for complement of the sine. Consider next a unit normal vector system, tion we have I »,='.. I*i= 1(10= - because i 1 \i 1 = i^i, = I, Also, z, | z 2 = Next iet e, = m^ + mj, and e 2 = n l i l -f- n^ ; * Grassmann (1862), Chapter 4. By the defini- n It ft / 1 fr-wCf" h / 1\ I «\ 1 . \ I 1 1= i Co 1, ; * — -- ~W,U H 402 grassmann's space analysis. [Chap. VIII. then, by (b) and (c), I e, = *», | *,+ **, | *, = »*,*„ — **,*!• (S3) By the figure it is evident that | e, is a vector of the same length as e, and perpendicular to it, or, in other words, taking the complement of a vector in plane space rotates it positively through 90°. The co-product e, | e„ is the area of the parallelogram, two of whose sides are e, and | e, drawn outwards from a point ; if e, is parallel to | e 2 , this area vanishes, or e, | e 2 = o ; but, since | e 2 is perpendicular to e„ , e, must in this case be perpendicular to e a ; hence the equation e, | e 2 = O (54) is the condition that two vectors e, and e 2 shall be perpendicu. lar to each other. The co-product e, | e 1 , which will usually be written e,-, and called the co-square of e, , is the area of a square each of whose sides has the length 7e, ; hence Te 1 =V^W 1 =V^?. (SS) Let a x and a t be the angles between i, and e, and between i l and e 2 respectively, as in the figure. Then e,e 2 = ;«,«, - mji, = 7e, 7e 2 sin (a, - a,), (56) the third member being the ordinary expression for the area of the parallelogram 6,6,. Also = ;«,», -f- mji^ = Te x Te, cos (a, — a,), (57) the last member being found as before, remembering that sin (90 -[- a, - «,) = cos (a, - a x ). If in (57) we let e 2 = e,, whence «, = »«, and n^ = m % , we have 7c, = e* = ^/< 4 «,'. (58) ^ 7\ = TV, = 1, then w, = cos a,, m t = sin a,,n x = cos «„ «, = sin a, , and equations (56) and (57) give the ordinary trigo- nometrical formulas sin(o- 2 - a,) = sin a, cos a, — cos a, sin «„ Art. 8.] the complement. 403 and cos (or, — a,) = cos «, cos a, + sin a, sin a, . Squaring and adding (56) and (57), there results T\ .T% = e*e* = (e.e,) 3 + (e, | e 2 )\ (59) Attention is called to the fact, which the student may have already noticed, that such an equation as AB = AC, in which AB and A Care combinatory products, does not, in general, imply that B=C, for the reason that the equation A(B—C)=o can usually be satisfied without either factor being itself zero. Thus pL r = pL, means simply that the two quantities which are equated have the same magnitude and sign, which permits Z„ to have an infinity of lengths and positions, when p and L, are given. The equation p,p 2 —p,p,, or p^p 2 — p t ) = o,p, and /, being unit points, implies, however, that/ 2 =p„ unless p l is at 00 , that is, a vector. Exercise 9. — A triangle whose sides are of constant length moves so that two of its vertices remain on two fixed lines : find the locus of the other vertex. Let * e, and £ e 2 be the two fixed lines, and //'/" the triangle. Let pe be per- pendicular to p'p", p' — e„ = xe, and P" — e„= ye* ; then /' — p' = ye, — xe x , 7\ye a — xe, ) = c = constant, by the con- ditions. Also, Tp'e = constant = mc, say, and Tep = constant = nc, say. Hence VB ~—~ X 6 e -p' = Tp'e . U{e - p') = mc . ^ _ ' = m(ye 2 - #e,), and similarly p — e = n \ (ys, — xe t ). Therefore p-e, = p = xe l + m(ye, - xe,) + n \ (ye, - xej, an equation which, with the condition T(ye 2 — xe,) = c, or y'e* — 2xye x | e 3 + x'e^ = c\ determines the locus to be a second-degree curve, which must in fact be an ellipse, since it can have no points at infinity. Let us rearrange the equation in p thus : p = x[( 1 — m)e l — n | e, J + y[me, + n \ e J = xe + ye', say, 404 GRASSMANN S SPACE ANALYSIS. [Chap. VIIL so that e = (i — m)e l — « | e, and e' = «e, -\-n\e^; then multi- ply successively into e and e'; therefore pe = ye'e and pe' = xee'. Substituting these values of x andjj/ in the equa- tion of condition, we have el . (pey + 2e, | e 2 . pe . pe' + e^pej = c\ee')\ a scalar equation of the second degree in p. Exercise io. — There is given an irregular polygon of n sides : show that if forces act at the middle points of these sides, proportional to them in magnitude, and directed all out- ward or else all inward, these forces will be in equilibrium. Let e a be a vertex of the polygon, and let 2e„ 2e 2) . .. 2e n represent its sides in magnitude and direction. Then the mid- dle points will be ^-f-e,, e a -4- 2e, -4- e 2 , etc., and, using the complement in a vector system, we have 2pe = 0,+eO | e 1 +(^ +26 I +e.) | 6 2 +(^ +2e 1 +2e 2 +e 3 ) | e 3 + . . . . + ('„ + 2e, + . . . + 2e„_ a + e„)|e s . = | c, . e s e, = O, an equation of the second degree in p, and hence representing a conic. Prob. 15. If a, b, c are the lengths of the sides of a triangle, prove the formula a* = f 1 -f- c* — zbc cos A, by taking vectors e,, e a , and e a — ei equal to the respective sides. Prob. 16. If e„e 1 and e e* are two unit lines, show that the vec- tor perpendicular from e a on the line (e + # e i)(*o + &O is (bet — aej, of which the length is — r: — r-. From (be % -aey-'" "' b T(be* - ae x ) this derive the Cartesian expression for the perpendicular from the origin upon a straight line in oblique coordinates, ab sin w -4- (a 1 + b* — 2tzb cos w)^, 00 being angle between the axes. Prob. 17. If three points, me a + «*,, wz*?, + w 2 , ot or ».A H""»A = °> the two points coincide ; AAA = o,| or 2np = o, j the three points are collinear ; 6,6, = o, or n.e, + n,e, — o, (62) the two vectors are parallel (points at infinity coincide); (63) AA = o, or n,L t + « 3 A = o, the two lines coincide; AAA = o, a or ~2nL = o, J the three lines are confluent. 406 grassmann's space analysis. [Chap. VIII. the two vectors are perpendicular ; either point lies on the com- plementary line of the other. Z a | L, = o, (64) either line passes through the complementary point of the other. If we write the equation P = *A + *,e„ j? 1 e 1 is the projection of p on ei parallel to e,, and x^e, is the projection of p on e a parallel to e,. Multiply both sides of the equation into e 3 ; therefore pe, = x^fi*, or x 1 = pe, ^- 6,6,. Similarly, multiplying into e,, we have pe, = x,e,e lt or x, = pe, -~ e 2 e,, whence p = + . (65} e,e 2 e 2 e, v » The two terms of the second member of (65) are therefore the projections of p on e, parallel to e a , and on e 2 parallel to e„ respectively.* Let e, and e 2 be unit normal vectors, say, 1 and |z; then (65) becomes p = 1. p\i — 1 1 . pi = 1. p\i-\- ip . 1 1 ; (66) or, if z, and z 2 be used instead of z and | z, P= VPK + h-P\h- ( 6 7) Again, in (65) let p = e 3 , clear of fractions, and transpose; therefore e,e 2 • e 3 + e 2 e s . e, + 6,6, . e, = O, (68) a symmetrical relation between any three directions in plane space. Let 7e, = 7e 2 = Te 3 =.- 1, and multiply (68) into |e„ thus 6 1 e 3 + €,e,.e I |e, + 6,e I .e a |6, = 0, (69) which is equivalent to sin (or ± /J) = sin a cos /? ± cos a sin /?, the upper or lower sign corresponding to the case when e s is * Grassmann (1844), Chapter 5 (1862), Art. 129. Hyde's Directional Calcu- lus, Arts. 46 and 47. Art. 9.] EQUATIONS OF CONDITION, AND FORMULAS. 407 between e J and e„ or outside, respectively. Writing in (69) instead of e 2 , we have 6,|e 2 — e,|e,.e,|e 1 +e s 6 1 .e a e, = 0, (70) which gives the cos (a ± /J). These formulas being for any three directions in plane space, are independent of the magni- tude of the angles involved. There is given below a set of formulas for points and lines, arranged in complementary pairs, and all placed together for convenient reference, the derivation of them following after. /=(AAA)"'[A AAA + A AAA + A -PPJX \ L^L^LytL, . LL X L, + Z, . LL,L, + A • LLJL& A=(AAA)"'[IAA-/IA +1 AA-AA + I AA-A A]. Z^AA^-'HAA-AA+IAA-AA+IAA-^IA] AA-AA = — A -AAA +A -AAA = A -AAA — A -AAA. (71) (72) (73) AA- l?.= - Al?i& = A A I ?i?. = A Aki A Aki ki Al?. l& Al& Ak Alf. Ak. A I?. AA|^,= LX\MM, a aw A AW 1 \Af t L,\M, \M t L t \M t » AW AW l AW 1 r ^t K (74) (75) (76) AAA ■ 1&3* = (77) PM>PMi P.\s% Ak. Ak. Alft Ako At?, Al?. The complementary formula to (77) is not given, but may be obtained by putting L's and M's for p's and ^'s. Derivation of Equations (7i)-(77)- — Equation (71). Write / = *„/„ + *,/, -4- x,p„ and multiply this equation by p t p, ; then A A/ = *oAAA> or *. = A*. A -s- AAA- Multiplying similarly by ^ A and by p p„ we find *. =/AA -^-AAA and *» =/AA -^AAA' The substitu- 408 GRASSMANN S SPACE ANALYSIS. [CKAP. VIII. tion of these values gives the first of (71), and the second is similarly obtained or may be found by simply putting Z's for p's in the first. Equation (72). Write/ = x \p x p % + *, | / 2 / -f x, \ p t p v and multiply into |/ ; thus/|/ = -*\,AAA- Find in the same way values of x x and x„ and substitute. Equation (73). Write /,/, .p 3 p t — xp, + yp v and multiply by PA : therefore//, -AA • AA = xpp t p v or . b y Art. 23, A/A -AAA = x PPiP, =~xp,pp:, or, * = — AAA- Multiply- ing by//, we find _y = AAA- an ^ on substituting obtain the first of (73). For the second put /,/, .p s p t — xp a + yp t , and proceed in a similar way. Equation (74). In the first of (73) put p s p t =\g t . Equation (75). In the fourth of (73) put AA, =A> A = k.> A = !&• Equation (76). Multiply (75) by/,. Equation (77). In the first of (72) put q. t for/, and multiply by AAA, •?» A ko = m o X o + W A + '».*« etc. Sub- stituting these values in (yy), we have the required result. A solution may also be obtained directly without the use of (yj). 2 Let the ^'s be as above, but write/, = 2/q,p l ~ 2mq,p^ = ~2nq. Then AAA= 2 ^-^^-^=[y„, «*,» «J? n ?,&= (7„> m v v *\- * Grassmann (1862), Art. 173. Art. 9.] equations of condition, and formulas. 409 Also /„ = l 2\e + l^jxe -f- l t 2ve with similar values for p 2 and/ 2 , which on being substituted in AAA §' ve ^ e result - Equation (77), however, exhibits the product in a very compact, symmetrical, and easily remembered form.* Exercise 13. — Show that the sides AA> AA> AA of the tri- angle /,/>,/,, cut the corresponding sides \p s , \p lt \p t of the com- plementary triangle in three collinear points. The three points of intersection are, using (74), AA • I A = - A -A I A+ A • A I A. AA • I A = ~A -A I A +A -A I A. AA-lA = -A-AlA+A-AlA> of which the sum is zer °> showing that the points are collinear. It may be shown in the same way that the lines joining corresponding vertices are confluent. Exercise 14. — If the sides of a triangle pass through three fixed points, and two of the vertices slide on fixed lines, find the locus of the other vertex. Let the fixed points and lines be p v A, A, A. A, and /, p', p" the vertices of the triangle, as in the ' ' \ p figure. Then p'p s p" = o ; p' coin- cides with //, . L l and p" with pp, . Z, ; hence substituting {pp x . L^p % {L t . p,p) = O, the equation of the locus, which, being of the second degree in p, is that of a conic. Prob. 18. Show that if the three fixed points of the last exercise are collinear, then the locus of / breaks up into two straight lines. Use equation (73). Prob. 19. If the vertices of a triangle slide on three fixed lines, and two of the sides pass through fixed points, find the envelope of the other side. (This statement is reciprocally related to that of Exercise 14, that is, lines and points are replaced by points and * These methods may be applied to determinants of any order by using a space of corresponding order. 410 grassmann's space analysis. [Chap. viii. lines respectively, and the resulting equation will be an equation of the second order in L, a variable line.) Prob. 20. Show that if the three fixed lines of Exercise 5 are confluent, then the envelope of L reduces to two points and the line joining them. Art. 10. Stereometric Products. The product of two points in solid space is the same as in plane space. See Art. 7. Product of Three Points. — Any three points determine a plane, and also, as in Art. 7, an area ; hence p,p,p, is a plane-sect or a portion of the plane fixed by the three points whose area is double that of the triangle pj>^p s . It may be shown, in the manner used in Art. 7 for the sect, that no plane-sect, not in this plane, can be equal to AAA> and that any plane-sect in this plane having the same area and sign will be equal topjji,* Of course AAA is not now scalar. Product of Four Points. — Any four non-coplanar points determine a tetrahedron, say AAAA> an d six times the vol- ume of this tetrahedron is taken for the value of the product, because this is the volume of the parallelepiped generated by the product AAA» — i-e. the parallelogram/,,^,— when it moves parallel to its initial position from/, to /,. Let A -A = £ .A ~ A = e'.A - A = e ". then AAAA = AAAe" = AP^'e" = p.ee'e". (78) 333 3 If p, =2ke,ps = ~2le, p, = 2me,p t = 2ne, then 00 AAAA = 2ke2le2me2ne = [k„, /,, m„ n s ] . e a e,e,e s ; (79) from which it appears that any two quadruple products of points differ from each other only by a scalar factor, that is, they differ only in magnitude, or sign, or both ; hence such products are themselves scalar.f If AAAA = °> the volume of the tetrahedron vanishes, so that the four points are coplanar. * Grassmann (1862), Art. 255. f Grassmann (1862), Art. 263. *>4 Art. 10.] STEREOMETRIC PRODUCTS. 411 Product of Two Vectors. — The two vectors determine an area as in Art. 7, but they also determine now a plane direc- tion, so that the product e,e, is a plane-vector, and is not scalar as in plane space. Also, 6,6, differs from />,e,e, now just as e differs from pe ; namely, 6,6, has a definite area and plane direction, that is, toward a certain line at infinity, while ^,6,6, is fixed in position by passing through /,. Equation (37) there- fore does not hold in solid space. Product of Three Vectors. — Three vectors determine a parallelepiped as in the figure above, and ee'e" is therefore the volume of this parallelepiped. Any other triple vector product can differ from this only in magnitude and sign. For let 6,6,6, be such a product, and write 8 3 3 e = x^e x -f- x,e t -\- x s e 3 = 2xe, e' — 2 ye, e" = 2ze ; then •*'l x , x, a y. y* 2, 2 i *. ee'e" = 2xe2ye2ze = y, y t y, e,e a e s , (80) z \ z i Z i so that the two products only differ by the scalar determinant factor. Hence the product of three vectors must be itself a scalar, by Art. 1. Since, then, the product of four points has precisely the same signification as that of three vectors, we may write AAAA =f 1 ee'e" = ee'e" = f>, - p,)(A ~P,)(p t -p t ) = P*P>P< ~ AAA + AAA - AAA- (81) Thus the sum of the plane-sects forming the doubles of the faces of a tetrahedron, all taken positively in the same sense as looked at from outside the tetrahedron, is equal to the volume of the tetrahedron. Compare equation (37). If ee'e" = O, the volume of the parallelepiped vanishes, and the three vectors must be parallel to one plane. Product of Two Sects. — In solid space two sects determine a tetrahedron of which they are opposite edges. Thus AAAA = AA • AA = 44 = AA • AA = 44. (82) so that the stereometric product of two sects is commutative, and has the same meaning as that of four points. •412 grassmann's space analysis. [Chap. VIII. Product of a Sect and a Plane-Sect. — Let them be L and P, and let A be their common point; take A.A.A so that Z=AA and f=AAA- L and p evidently determine the point A> and also the parallelepiped of which one edge is L and one face is P, so that the product should be made up of these two factors. Hence we write LP = AA ■ AAA = AAAA • P.; PL = AAA • A/, = AAAA -A = LP. j (8s) If L is parallel to P, A is at infinity, and, replacing it by e, (83) becomes PL = LP= ep x . ej>J, = e^AA ■ * (84) Product of Two Plane-Sects. — Let them be P 1 and P v and let L be their intersection, while/, and/, are such points that P, = Lp l and P t = Lp^, then P 1 and P, determine the line L and also a parallelepiped of which they are two adjacent faces, and PA, = Lp x . Lp % = Lp iP > . L = - />,/>,. (85) If Z 3 , and P 2 are parallel, L is at infinity, and is equivalent to a plane-vector, say to r/ ; hence, substituting in (84), PA = nPi ■ nP* = vpJ>* -v=- PA- (86) Product of Three Plane-Sects.— By (85) and (83) this must be the square of a volume times the common point of the three planes ; or, if A>A> A» A De taken in such manner that P, = AAA. P> =AAA> P, = AAA. then PAA* = 023 . 031 . 012 = 023 . 0123 . 01 = (AAAA) 1 -A ; (87) the suffixes being used instead of the corresponding points. If A be at infinity, the three planes are parallel to a single line, and may be written P i = n l ep^p„ etc., and then treated as above. Product of Four Plane-Sects.* — Let the planes be P t . . . P,, and let A • • • A De the four common points of the planes taken three by three. n . . . w 3 may be so taken that P„ = n e pj>,p v etc. ; then PAAA, = ».»,«,,«»■ 123. 230. 301. 012 = «."i« a ra a (AAAA)'- ( 88 ) * Grassmann (1862), Art. 300. ART. 10.] STEREOMETRIC PRODUCTS. 413 Product of Two Plane-Vectors. — Let ^ and % be two plane- vectors or lines at infinity ; let e be parallel to each of them, and e, and e a so taken that r/ 1 = ee i; tj, — ee s , then VST, = ee . • £ e a = ee s e > .e— — r)^, (89) because ^, and % determine a common direction e, and a paral- lelepiped of which three conterminous edges are equal to e, e,, e„ respectively. Product of Three Plane-Vectors. — Take e,, e s , e 3 so that ViV*V, = n . e 2 e 3 . e 3 e, . e,e 2 = ra(e,e a e 3 ) s . (90). The directions e, . . . e 3 are common to the plane-vectors ?j 1 . . . r/ s taken two by two. Several conditions are given here together which follow from the results of this article. AA = o, AA = o, (91) Two points coincide. Two planes coincide. AAA = o, AAA = o, (92) Three points collinear. Three planes collinear. AAAA = AA • AA A A A A = P.P. ■ P.P. = AA = 0, = AA = o, (93) Four points coplanar; two Four planes confluent; two lines intersect. lines intersect. e.e a = o, w, - o, (94) Vectors parallel. Plane-vectors parallel. e.e.e, = o, tjms), = o, (95) Three vectors parallel to Three plane-vectors parallel to one plane. one line. Sum of Two Planes. — Let them be A and P„ let Z be a sect in their common line, and take p 1 and/, so that P x = Z/,, P, = Lp, ; then f 1 + J P, = Z(/ 1 +A) = 2Zj, ( 9 6> p being the mean of/, and />,,. Also ^_p 3 = z(a-a); (97) whence the sum and difference are the diagonal plane through Z, and a plane through Z parallel to the diagonal plane which is itself parallel to Z, of the parallelepiped determined by P x 414 grassmann's space analysis. [Chap. VIII. and P, If TP X = TP„ P 1 ± P, will evidently be the two bisecting planes of the angle between them. The bisecting planes may also be written -^P±Y^ or P,Tp±prp. ( 9 g) If the two planes are parallel, let tf be a plane-vector parallel to each of them, that is, their common line at infinity, and let/, and p t be points in the respective planes; then we may write P, = «,/",??, Pi = n iP*V> whence P* + P* = (».A + n *P*)v = K + n Spv- (99) If n i -\- n t = o, this becomes P, + P, = «.(A -/.)?. (100) the product of a vector into a plane-vector and therefore a scalar, by (80). Two plane-vectors may be added similarly, since they will have a common direction, namely, that of the vector parallel to both of them. Exercise 15. — If two tetrahedra e^e^e^e, and e/e/e/ej are so situated that the right lines through the pairs of corresponding vertices all meet in one point, then will the corresponding faces cut each other in four coplanar lines. The given conditions are equivalent to e t e t ' . e t e^' = = '„'„' • e,e t ' = \hh = 1(10 = h | k = v,> Im, = 1(10 = h r- (ioi) U. = m„ k*,= 1(10 = Z 3 ) 3 Let e = .27z ; then -~-^j,,-' Wa I e = 4v, + A v, + 4v. = j(4*. - '.Oft 1 . - £0. (102) so that | e is a plane-vector. The figure, which is drawn in isometric projection, shows that the two vectors /^ — / 2 z, and /jZ 3 — / 3 z,, whose prod- uct is /, . | e, are both perpen- dicular to e ; for the first is perpendicular to /,i, -f- Z^, which is the orthogonal pro- jection of eupon ZjZ^andto z s , and therefore is also per- pendicular to e, while the second is perpendicular to /,z, + / 3 z 3 and to t t , and therefore to e. Hence | e is a plane-vector perpendicular to e ; and, since | ( | e) = e, the converse is also true, i.e. the complement of a plane-vector is a line-vector normal to it. The figure shows that e is equal to the vector diagonal of the rectangular parallelepiped whose edges have the lengths l s , 4 A i hence Te = VIS + i: + /,". (103) Multiply equation (102) by e; therefore e I « = (A*. + / ,«, + '.0(4 v. + A 1 ,*! + foO = /,'+/,' + /,•= T'e =e»-, (104) so that the co-square of a vector is equal to the square of its tensor. The product e|e is that of a vector e into a plane- vector perpendicular to it, as has just been shown ; it is there- Art. 11.] THE COMPLEMENT IN SOLID SPACE. 417 fore a volume which is equivalent to Te. T\e; hence, by (104), e|e= Te . T\e= T*e, or Te = T\e. Hence, the complement of a vector in solid space is a plane-vector perpendicular to it and having the same tensor, or numerical measure of magni- tude.* s Let a second vector be e' =2mi ; then 1 e I e' = / 1 m l -+- / 2 w a + z>z 3 = e' | e. (105) Now e|e', being the product of e into the plane-vector | e', is the volume of the parallelepiped in the fig- ure, that is, TeTe' sin (angle between e and |e') = TeTe' cos l'. Hence e\e' = e' | e—l,m 1 +l i m 2 +l 3 m 3 = TeTe' cos *'. (106) If Te = Te' = 1, /, . . . l % , m x . . . m 3 are di- rection cosines, and (105) gives a proof of the formula for the cosine of the angle between two lines in terms of the direction cosines of the lines. We have also in this case ee' = (l,m x - l,m,) \ z 3 + (l 2 m a - l,m,) | z, + (/,»*, - /,«,) | z„ and, taking the co-square, (ee'y= (sin f)' = (/,«,- / 1 * 1 ) , + (/,*«.-/,«,)'+(A« l -'.'«,)'- ( io 7) If e|e'=o, (108) e is parallel to the plane-vector perpendicular to e', that is, e is perpendicular to e', as is also shown by (106). Let r) = \e,rf= \ e' ; then v \ v ' = I e . e' = e> \ e = e\ e' = 7e7V cos f = T V T V ' cos ^', (109) and v \r}' = is the condition of perpendicularity of two plane-vectors, either e\r)' = 0, or rf\e = o, is the condition that a vector shall be perpendicular to a plane- vector, for the first means that e is parallel to a vector which is (no) Also (in) * Grassmann (1862), Art. 335. 418 GRASSMAnn's space analysis. [Chap. VIII. perpendicular to r/', and the second that rf is parallel to a plane- vector which is perpendicular to e. Equations (71)— (77) of Art. 9 become stereometric vector formulae if e,, e,, etc., be substituted for /,,/,, etc., and 77,, 77,, etc., for Z„ L 2 , etc. For instance, (76) gives the vector formulas "l w 2 I 1 3 e. I e, e, I e, e.k,' e.le.' >7i% I 7. V = (112) For lack of space no treatment of the complement in a point system in solid space is given. Exercise 17. — To prove the formulas of spherical trigo- nometry cos a = cos b cos c -j- sin b sin c cos A, and sin a sin b sin c sin A sin 5 sin C Take three unit vectorsCe, , e„, e 3 parallel to the radii to the vertices of the spherical triangle, then «=(angle bet. e, and e,), ^=(angle bet. e,e, and e,e s ), etc. In eq. (112) put 6,6, for e/e,'; hence e,e a | e,e 3 = sin (5 sin whence derive 2 7(C/e,e, + c7e,e 3 ) t , ,. . /sin .$ sin (s — a) the ordinary value * / -. y sin b sin c Expanding, the numerator becomes 1 -\- Ue^lUe^, and the denominator 4/2(1 -\- £/e,e 2 | £/e,e 3 ). Also there is obtained £/e,eJ £76,6,, = ' ' ,=M-. The remainder is left to the stu- 76,6,76,6, dent. Prob. 24. If e l( e 2 , e 3 , drawn outward from a point, are taken as three edges of a tetrahedron, show that the six planes perpen- ART. 12.] ADDITION OF SECTS IN SOLID SPACE. 419 dicular to the edges at their middle points all pass through the end of the vector p = — — -( | e,e, . e^+ 1 ^e, . e/ + 1 e,e 3 . e 3 s ). (Sug- ze i e a e s gestion. We must have (p — ^e t ) | e, = o, with two other similar expressions.) Prob. 25. Show that e, \ ee' and ee'. | e are three mutually per- pendicular vectors, no matter what the directions of e and e' may be. Prob. 26. Let e,, e a , e 3 be taken as in Prob. 24 ; let A be the area of the face of the tetrahedron formed by joining the ends of these vectors, and 2A, — Te^e % , etc.; also 6 1 , = Angle between e^ and e^j, etc.: then show that we have the relation, analogous to that of Prob. 15, Art. 8, A*= A'+A^+A^ — 2A^A S cos 0,— 2A< t A 1 cos 6,— 2A 1 A i cos 6 t . It 0, ... 6, are right angles, this becomes the space-analog of the proposition regarding the hypotenuse and sides of a right-angled triangle. (Suggestion. 2A a = T(e, - e 1 )(e i - e,).) Prob. 27. There are given three non-coplanar lines e a e l , e e^, i? e 3 ; planes cut these lines at right angles, the sum of the squares of their distances from e being constant. Show that the locus of the common point of these three planes is (p\ e 1 ) 2 +(p| e a )"+(p| e 3 ) 2 =, + *J>*)("x £ x + u ^) -I- O^, + JsAM^e, + V.) = C*i«i +J>W)Ae> + (*.«> +y, v ^P^ + (x.tt, + y,v,)p 1 e % -f { Xi u, +y,v i )p t t 1 . This will be equal to the given pair if we have #,«,+ y,v, = x t u*-\-y l v l —i, and x,u, +y 1 v 1 = *,?*, +_y,w 1 = o). Since there are eight arbitrary quantities with only four equations of condition, the desired result can evidently be ac- complished in an infinite number of ways. Let p l e 1 , / 2 e 2 .... p n e„ be ii sects, and let 5 be their sum,, and e any point, then S=2pe = e 2e - e,2e + 2pe = e,2e + 2{j> - e a )e (i 13) 1 the sum of a sect and a plane-vector as before. If 2(p — e )e is parallel to 2e it may be written as the prod- uct of some vector e' into 2e, that is, e'2e, when the sum be- comes 5 = e 2e -j- e'2e = (/ -+- e')2e, a sect, because e a -f- e' is a point. In no other case does 5 reduce to a single sect. If 2e = o S becomes a plane-vector. Of the two parts compos- ing 5, the sect will be unchanged in magnitude and direction if e c be moved to a new position, while the plane-vector will in general be altered. It is proposed to show that a point q may be substituted for e a such that the plane-vector will be perpen- dicular to ^e. Writing S = 9 2e-(g-e t )2e+ 2(p - e,)e, and, for brevity, putting q — e = p, 2e — a, 2(p — e^e — \fir so that S = qa — pa + |/S, ("4> ART. 12.] ADDITION OF SECTS IN SOLID SPACE. 421 we must have for perpendicularity, by (in), ( | ft — pa) | a = o = | ft a — pa . \ a, or pa .\a = a.p\a — p. a- = \fta. ("S) The second member is obtained from the first by substitut- ing in eq. (74) p for^, and a for/, and g lt in accordance with the statement at the end of Art. 1 1. If in (i 1 5) we make p | a = o, p will be the vector from e to ^ taken perpendicularly to a, say p, — \aft-^o?-=q^ — e„. (116) Since a and /? are known, the required point has been found. Multiply (115) by or; then, using (75), — ap . a- = pa . a 2 - = a \ fta — \ft.a-— | a . a \ ft, whence, substituting in (1 14), c \ a \P \ v- , ^e2(p — e\e S=g<*+-±T-\« = qZe + . ^-iL.ie. („;) This may be called the normal form of S* The sects of this article represent completely the geometric properties of forces, hence all that has been shown applies immediately to a system of forces in solid space. We have only to substitute the words force and couple for sect and plane- vector. The resultant action of any system of forces is 5", called by Ball in his Theory of Screws " a wrench." The con- dition for equilibrium is S = 0, which gives at once 2e = o and 2(p — e )e = o; (118) since otherwise we must have e 2e = — 2(J> — e )e, which is an impossibility. The line g2e is the central axis of the sys- tem of forces 6". Lack of space forbids a further development of the subject, but what has been given in this article will indicate the perfect adaptability of this method to the requirements of mechanics. Exercise 19. — Reduce p^e^ -\- / 2 e 3 = 5 to its normal form. •S - e l e i + e .) + (A - O 6 . + (A - ji + a,6J/-\- ajjj+ajbji. If we assume, as suggested by ordinary algebra, that the square of a sign of direction is -\-, and further that the product of two directions at right angles to one another is the direction normal to both, then the above reduces to AB = a 1 b 1 -f- aj?^ + (ajb^ — aj>^)k. Thus the complete product breaks up into two partial products, namely, a 1 b 1 -j- aj}^ which is independent of direc- tion, and (a 1 b^ — <*J>?)k which has the axis of the plane for direction.* * A common explanation which is given of ij = k is that i is an operator,y'an operand, and k the result. The kind of operator which i is supposed to denote is a quadrant of turning round the axis i ; it is supposed not to be an axis, but a quadrant of rotation round an axis. This explains the result ij = k, but unfortunately it does not explain ii = -)- ; for it would give ii = i. Art. 3.] PRODUCTS OF COPLANAR VECTORS. 433 Scalar Product of two Vectors.— By a scalar quantity is meant a quantity which has magnitude and may be positive or negative but is destitute of direction. The former partial product is so called because it is of such a nature. It is denoted by SAB where the symbol S, being in Roman type, denotes, not a vector, but a function of the vectors A and B. The geometrical mean- ing of SAB is the product of A and the orthogonal projection of B upon A. Let OP and OQ represent the vectors A and B; draw QM and NL perpendicular to OP. Then (OP)(OM) = (OP)(OL) + (OP)(LM), ■=•{«■?+ ^}- = «A + <*A- Corollary I. — SB A = SAB. For instance, let A denote a force and B the velocity of its point of application ; then SAB denotes the rate of working of the force. The result is the same whether the force is projected on the velocity or the velocity on the force. Example I. — A force of 2 pounds East -)- 3 pounds North is moved with a velocity of 4 feet East per second -j- 5 feet North per second ; find the rate at which work is done. 2X4+3X5=23 foot-pounds per second. Corollary 2. — A' =± a' -\- a* = a*. The square of any vector is independent of direction ; it is an essentially positive or signless quantity ; for whatever the direction of A, the direction of the others must be the same; hence the scalar product cannot be negative. Example 2. — A stone of 10 pounds mass is moving with a velocity 64 feet down per second -(- 100 feet horizontal per second. Its kinetic energy then is — (64' + ioo") foot-poundals, 434 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX. a quantity which has no direction. The kinetic energy due to 6 4 2 the downward velocity is 10 X — and that due to the hori- zontal velocity is — X I00 2 ; the whole kinetic energy is ob- tained, not by vector, but by simple addition, when the com- ponents are rectangular. Vector Product of two Vectors. — The other partial product from its nature is called the vector product, and is denoted by VAB. Its geometrical meaning is the product of A and the projection of B which is perpendicular to A, that is, the area of the parallelogram formed upon A and B. Let OP and OQ represent the vectors A —6, — > ~~i and B, and draw the lines indicated by the figure. It is then evident that the area of the triangle OPQ = aj, — \a,a % — %bj> t — £(«, - b,){K ~ «,), = U a A — a A)- Thus (#,#, — a,b^)k denotes the magnitude of the parallelo- gram formed by A and B and also the axis of the plane in which it lies. It follows that VBA = — VAB. It is to be observed that the coordinates of A and B are mere component vectors, whereas A and B themselves are taken in a real order. Example. — Let A = (io? -4- 117) inches and B = (S«'+ l2 J) inches, then VAB = (120— $5)k square inches; that is, 65 square inches in the plane which has the direction k for axis. If A is expressed as aa and B as b/3, then SAB = ab cos afi, where a/3 denotes the angle between the directions a and ft. Example. — The effective electromotive force of 100 volts per inch /go° along a conductor 8 inch /45° is SAB = 8 X 100 cos /45 u /90 volts, that is, 800 cos 45 volts. Here /45 indicates the direction a and /o,o° the direction /?, and /45 /90 means the angle between the direction of 45° and the direction of 90°. Also VAB = ab sin a/3 . ~a/3, where a/3 denotes the direction which is normal to both a and /?, that is, their pole. ART. 3.] PRODUCTS OF COPLANAR VECTORS. 435 Example. — At a distance of 10 feet /30 there is a force of 100 pounds /6o° . The moment is NAB — 10 X 100 sin /jo? /6o° pound- feet 907 /90 . = 1000 sin 30 pound-feet 90 / /go". Here 90 / specifies the plane of the angle and /90 the angle. The two together written as above specify the normal k. Reciprocal of a Vector. — By the reciprocal of a vector is meant the vector which combined with the original vector pro- duces the product -j- 1. The reciprocal of A is denoted by A' 1 . Since AB — ab (cos a/3 -f- sin a/3 . a/7), b must equal a~ l and /3 must be identical with a in order that the product may be 1. It follows that _ 1 «a aj + aj /1 — ~ a — ~r~ — i — 1 r* a a a x -\- « 2 The reciprocal and opposite vector is — A~\ In the figure let OP = 2/3 be the given vector ; then OQ = \fl is its recipro- cal, and OR = f( — /3) is its reciprocal and opposite.* k Q ? Example.— If A = 10 feet East + 5 feet North, A~ l = — feet East + — feet North and — A" = feet 125 ~ 125 125 East — — feet North. 125 Product of the reciprocal of a vector and another vector. — A-B = \AB, a = ^ !«A + a A + ( a A — <*A)<*P}> = - (cos «/? + sin a/3 . aft). * Writers who identify a vector with a quadrantal versor are logically led to define the reciprocal of a vector as being opposite in direction as well as recip- rocal in magnitude. 436 VECTOR ANALYSIS AND QUATERNIONS. [Chap. IX b b __ Hence SA~ l B = -cos a6 and VA~'B = -sin a/3. a6. a a r r Product of three Coplanar Vectors. — Let A = aj. -f a j, B — b x i -f bj, C — cj -\- cj denote any three vectors in a common plane. Then (AB)C = \{aA + aA) + (a A - «AWfo* + ej) = («A + «A)fo* + C J) + ( a A - *A)(— cj + cj). The former partial product means the vector C multiplied R\ by the scalar product of A and B ; while the latter partial product means the comple- /c mentary vector of C multiplied by the mag- / s' nitude of the vector product of A and B. (> J If these partial products (represented by OP and OQ) unite to form a total product, the total product will be represented by OR, the resultant of OP and OQ. The former product is also expressed by SAB . C, where the point separates the vectors to which the S refers ; and more analytically by abc cos aft . y. The latter product is also expressed by (VAB)C, which is equivalent to V(VAB)C, because VAB is at right angles to C. It is also expressed by abc sin aft . afty, where afty de- notes the direction which is perpendicular to the perpendicular to a and ft and y. If the product is formed after the other mode of association we have A(BC) = (a,f + a J) Ac, + V.) + («,* + **/)(V. - V>)* = (V, + b,c,)(a 1 i + a,j) + Ac, — b^(a t i - a J) = SBC.A -\-VA(VBQ. The vector a,i — a^j is the opposite of the complementary vector of aj + a t j. Hence the latter partial product differs with the mode of association. Example.— Let A = i/o + 2 /90° , B = 3^0°+ 4/90°, C = 5/0 + 6/go°. The fourth proportional to A, B, Cis Art. 3.] products of coplanar vectors. 437 (A-*B)C = I X I 3 3 + ^ X4 { 5/2! + 6 M 1 -' J : 4 ~ ' : 3 -l -6/0° + 5/90°} 1 I 2 + 2° = 13.4/0° + II.2/90°. Square of a Binomial of Vectors. — If A -\- B denotes a sum of non-successive vectors, it is entirely equivalent to the resultant vector C. But the square of any vector is a positive scalar, hence the square of A -\- B must be a positive scalar. Since A and B are in reality components of one vector, the square must be formed after the rules for the products of rect- angular components (p. 432). Hence {A +B)> = (A + B){A + B), = A' + AB + BA + B\ = A 3 + B> + SAB + SBA + NAB + NBA, = A* + B 1 + 2S^^. This may also be written in the form a% ~t~ V "+" 2a b cos a fi° But when A -\- B denotes a sum of successive vectors, there is no third vector C which is the complete equivalent ; and con- sequently we need not expect the square to be a scalar quan- tity. We observe that there is a real order, not of the factors, but of the terms in the binomial ; this causes both product terms to be AB, giving {A + B)*= A' + 2AB + B- = A' + B* + 2S^^ + 2NAB. The scalar part gives the square of the length of the third side, while the vector part gives four times the area included between the path and the third side. Square of a Trinomial of Coplanar Vectors. — Let A -f- B -(- C denote a sum of successive vectors. The product terms must be formed so as to preserve the order of the vectors in the tri- nomial ; that is, A is prior to B and C, and B is prior to C. 438 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX.. Hence (A + B 4- Cf = A' +B* + C + 2AB+2AC + 2BC, = A> + B' + C + 2{SAB + SAC+ SBC), (1) + 2(VAB + VAC+ VBC). (2) Hence S{A + B + Cf = (1) = a 3 + F -\- c? -\- 2ab cos aft -\- 2ac cos ay + 2bc cos fty and V(A + B + C)* = (2) = j 2ab sin ayff + 2ac sm a Y + 2 ^ c s ' n Py\- a ft The scalar part gives the square of the vector from the be- c ginning of A to the end of C and is all that exists when the vectors are non-successive. The vector B part is four times the area included between the successive sides and the resultant side of the A polygon. Note that it is here assumed that V(A + B)C — WAC-\- VBC, which is the theorem of moments. Also that the prod- uct terms are not formed in cyclical order, but in accordance with the order of the vectors in the trinomial. Example.— Let A = 3/0^ B = 5/30 , C = 7 /45° ; find the area of the polygon. iV(AB+AC + BC), = i{i5sin/o/30° + 2i sin/o/45° + 35 sin/30 /45 }, = 3-75 + 742 + 4-53 = 15-7- Prob. 10. At a distance of 25 centimeters /20 there is a force of 1000 dynes /8o°; find the moment. Prob. n. A conductor in an armature has a velocity of 240 inches per second /3oo° and the magnetic flux is 50,000 lines per square inch /o; find the vector product. (Ans. 1.04 X io 7 lines per inch per second.) Prob. 12. Find the sine and cosine of the angle between the directions 0.8141 E. -f- 0.5807 N., and 0.5060 E. + 0.8625 N. Prob. 13. When a force of 200 pounds /270 is displaced by 10 feet /30 , what is the work done (scalar product) ? What is the meaning of the negative sign in the scalar product ? Art. 4.] coaxial quaternions. 439 Prob. 14. A mass of ioo pounds is moving with a velocity of 30 feet E. per second + 5 o feet SE. per second; find its kinetic energy. Prob. 15. A force of 10 pounds /45J is acting at the end of 8 feet /2oo° ; find the torque, or vector product. Prob. 16. The radius of curvature of a curve is 2/0 + 5/90 ; find the curvature. (Ans. ■oz/o^ + ^j'j/ 9 o°J Prob. 17. Find the fourth proportional to 10/0 -f- 2/90" 8/0° - 3 /9o_°, and 6/0° + 5/90°. Prob. 18. Find the area of the polygon whose successive sides are 10 /30° , 9/100°, 8/180° , 7/225°. Art. 4. Coaxial Quaternions. By a "quaternion " is meant the operator which changes one vector into another. It is composed of a magnitude and a turning factor. The magnitude may or may not be a mere ratio, that is, a quantity destitute of physical dimensions ; for the two vectors may or may not be of the same physical kind. The turning is in a plane, that is to say, it is not conical. For the present all the vectors considered lie in a common plane ; hence all the quaternions considered have a common axis.* Let A and R be two coinitial vectors ; the direction normal to the plane may be denoted by /?. The operator which changes A into R consists of a scalar multiplier and a turning round the axis fi. Let the former be denoted by r and the latter by 6 e , where denotes the angle in radians. Thus R = r/3 e A and recip- rocally A = -p- e R. Also ~R = r/3 e and ±A = -6~ e - : r A ' R y The turning factor /J 9 may be expressed as the sum of two component operators, one of which has a zero angle and the other an angle of a quadrant. Thus /3» = cos . /3° + sin . /?»/». * The idea of the "quaternion " is due tn Hamilton. Its importance may be judged from the fact that it has made solid trigonometrical analysis possible. It is the most important key to the extension of analysis to space. Etymologi- cally "quaternion" means defined by four elements; which is true in space ■ in plane analysis it is defined by two. 440 VECTOR ANALYSIS AND QUATERNIONS. [Chap. IX. When the angle is naught, the turning-factor may be omitted ; but the above form shows that the equation is homogeneous, and expresses nothing but the equivalence of a given quaternion to two component quaternions.* Hence rfP = r cos 9 -\- r sin 6 . (F^ and rfiPA — pA + g/3"/*A = pa . a-\-qa . /3 n/t a. The relations between r and 0, and / and q, are given by r = Vf + q\ = tan ""£ Example. — Let E denote a sine alternating electromotive force in magnitude and phase, and / the alternating current in magitude and phase, then E= (r + 2nnl . p"/*)I, where r is the resistance, / the self-induction, n the alternations per unit of time, and /? denotes the axis of the plane of repre- sentation. It follows that E = rl -\- 2nnl . /J'/ 2 /; also that .I~ l E = r-\-2nnl . fi«l*, that is, the operator which changes the current into the elec- tromotive force is a quaternion. The resistance is the scalar part of the quaternion, and the inductance is the vector part. Components of the Reciprocal of a Quaternion. — Given R = (p + g.j3">)A, then A = —. -^ R P -q.fi*'* R ~(P + q.F /% ){p-g./W*) '-P +9 P +q J * In the method of complex numbers /J*/2 is expressed by i, which stands for |/ — I. The advantages of using the above notation are that it is capable of being applied to space, and that it also serves to specify the general turning factor /3 s as well as the quadrantal turning factor f}*/2. Art. 4.] coaxial quaternions. 441 « Example.— Take the same application as above. It is im- portant to obtain / in terms of E. By the above we deduce that from E — (r + 27ml. p*/*)I j _ ( r 2nnl \ " : \r*-\-(27m/y r'-\-{27tn/y- pn ] E ' Addition of Coaxial Quaternions. — If the ratio of each of several vectors to a constant vector A is given, the ratio of their resultant to the same constant vector is obtained by tak- ing the sum of the ratios. Thus, if ^ = (A + / is introduced to specify the plane in which the angle from the initial line lies. If we are given R in the form rcp//6, then we deduce the other form thus : R = r cos 6 . i -j- r sin cos .j-\-r sin 6 sin (p . k. If R is given in the form xi -\- yj -\- zk, we deduce R- Vx* -4-/ -\-z' tan-' y , For example, B = io 30°/ /45° = io cos 45°. i-\- io sin 45 cos 30° .j-\- 10 sin 45° sin 30° . k. Again, from C ■= 32" — f— 47' -f- $k we deduce ~ 5 // V41 C = Vg + 16 + 25 tan- - jj tan" — = 7.07 Sl°.4/ / 6 4"-9 - To find the resultant of any number of component vectors applied at a common point, let R x , R 2 , . . . R n represent the n vectors or, 444 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX. R 1 =x 1 i+yj+e l k, K = *j+yJ+\ k > R n — x n i + y„j + z n k ; then 2R = (2x)i + {2y)f+ {2z)k and r=V{2xy+i?yf + VS*y, tan = 2z , , a V{2y)' + {2zy - -— - and tan — ^ — . 2y ^x Successive Addition. — When the successive vectors do not lie in one plane, the several elements of the area enclosed will lie in different planes, but these add by vector addition into a resultant directed area. Prob. 23. Express A — 41' — 5/ ' + 6 & and B — 5/ + 6/ — ik in the form rff/ '/#. (Ans. 8.8 1307/63° and 10.5 3 ii°7 /6i°.5. ) Prob. 24. Express C = 123 577/142° and Z> = 456 657 /200° in the form xi + yj + z& 7T //7T n 1 1 n . Prob. 2<. Express is = 100 - // - and F = 1000 - // 3- in 4// 3 6// 4 the form #*' -f-jy' + zk. Prob. 26. Find the resultant of 10 2Q° //3o° , 20 307 /40° , and 30 4^7/5^°- _ Prob. 27. Express in the form rcf>/ /B the resultant vector of 1/ + 2/ — $k, 4* — 5/ + 6/5, and — 7/ + 8/ + 9^. Art. 6. Product of two Vectors. Rules of Signs for Vectors in Space. — By the rules z" = -+-, J' = -|-, z)'= £, and/z = — £ we obtained (p. 432) a product of two vectors containing two partial products, each of which has the highest importance in mathematical and physical analysis. Accordingly, from the symmetry of space we assume that the following rules are true for the product of two vectors in space : i° = +, ./* = +, k> = +, if = k, jk = i, ki = j, ji = — k, kj = — i, ik = —j. The square combinations give results which are indepen- ART. 6.] PRODUCT OF TWO VECTORS. 445 dent of direction, and consequently are summed by simple addition. The area vector determined by * and/ can be represented in direction by k, because k is in tri-dimensional space the axis which is complementary to i and/. We also observe that the three rules if = k, jk = i, ki =/ are derived from one another by cyc- lical permutation ; likewise the three rules ji ■=. — k, kj — — i, ik = —J. The figure shows that these rules are made to represent the relation of the advance to the rotation in the right-handed screw. The physical meaning of these rules is made clearer by an application to the dynamo and the electric motor. In the dynamo three principal vectors have to be considered : the velocity of the conductor at any instant, the intensity of magnetic flux, and the vector of electromotive force. Frequently all that is demanded is, given two of these directions to determine the third. Suppose that the direction of the velocity is i, and that of the flux/, then the direction of the electromotive force is k. The formula ij '= k becomes velocity flux = electromotive-force, from which we deduce flux electromotive-force = velocity, and electromotive-force velocity = flux. The corresponding formula for the electric motor is current flux = mechanical-force, from which we derive by cyclical permutation flux force = current, and force current = flux. The formula velocity flux = electromotive-force is much handier than any thumb-and-finger rule ; for it compares the three directions directly with the right-handed screw. Example. — Suppose that the conductor is normal to the plane of the paper, that its velocity is towards the bottom, and that the magnetic flux is towards the left ; corresponding to the rotation from the velocity to the flux in the right-handed screw we have advance into the paper: that then is the direc- tion of the electromotive force. Again, suppose that in a motor the direction of the current 446 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX. along the conductor is up from the paper, and that the mag- netic flux is to the left ; corresponding to current flux we have advance towards the bottom of the page, which therefore must be the direction of the mechanical force which is applied to the conductor. Complete Product of two Vectors. — Let A = aj + a J '-(- a t k and B = bj.-\-bJ -\- b,k be any two vectors, not necessarily of the same kind physically, Their product, according to the rules (p. 444), is AB = («,*'+ aj+ aJt)(bj. + bJ-\-bJi), = aJ>ji-\-aJ>ijj -\-a.b s kk, + aj>jk + a s b t kj + aJ>JH + ajbjk + afijj + a,bji = a A + a A + a A, + {a A - «A)* + W, - aA)J+ ( a A — <*A)k = a A + a A + a A + «. a a a, *, K K i J k Thus the product breaks up into two partial products, namely, «,£,+ a A~\~ a A > which is independent of direction, and « x a, « 3 b, b, b, , which has the direction normal to the plane of i j k A and B. The former is called the scalar product, and the latter the vector profluct. In a sum of vectors, the vectors are necessarily homogene- ous, but in a product the vectors may be heterogeneous. By making a, = b s = o, we deduce the results already obtained for a plane. Scalar Product of two Vectors. — The scalar product is de- noted as before by SAB. Its geometrical meaning is the product of A and the orthog- onal projection of B upon A. Let OP rep- resent A, and OQ represent B, and let OL, LM, and MN be the orthogonal projections upon OP of the coordinates bj, bj\ b 2 k re. spectively. Then ON is the orthogonal pro- jection of OQ, and Art. 6.] product of two vectors. 447 OP X ON = OP X (OL + LM + MN), \ a a a — ajb x -\- aj}^ -\- a % b % = SAB. Example. — Let the intensity of a magnetic flux be B— bj-\-bJ ' -\- b 3 k, and let the area be S = sj-\- sj -{- s t k; then the flux through the area is SSB = b^, -\- b 2 s, -+- b 3 s,. Corollary i. — Hence SB A — SAB. For h a i + ^A + b,a, - a,b, + a,b, -f a,b, . The product of B and the orthogonal projection on it of A is equal to the product of A and the orthogonal projection on it of B. The product is positive when the vector and the pro- jection have the same direction, and negative when they have opposite directions. Corollary 2. — Hence A ' = a^-\-a^ -\-a^=a". The square of A must be positive ; for the two factors have the same direction. Vector Product of two Vectors. — The vector product as before is denoted by NAB. It means the product of A and the component of B which is perpendicular to A, and is rep- resented by the area of the parallelogram formed by A and B. The orthogonal projections of this area upon the planes of jk, ki, and ij represent the respective components of the product. For, let OP and OQ (see second figure of Art. 3) be the or- thogonal projections of A and B on the plane of i and/; then the triangle OPQ is the projection of half of the parallelogram formed by A and B. But it is there shown that the area of the triangle OPQ is %( a A ~ a A)- Thus ( a A ~ a A)k denotes the magnitude and direction of the parallelogram formed by the projections of A and B on the plane of i and/ Similarly (aj, — a,b t )i denotes in magnitude and direction the projec- tion on the plane of /' and k. and [ajb, — afi^j that on the plane of k and i. Corollary 1.— Hence NBA = - NAB. Example.— Given two lines A = ji — 10/ '+ 3^ and B = — gi -4- 4/ — 6k; to find the rectangular projections of the par- allelogram which they define : 448 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX. NAB = (60 - i2> + (- 27 + 42)/+ (28 - 90)* = 482' + 1 SJ — 62k. Corollary 2. — If A is expressed as aa and B as £/?, then SAB = ab cos aft and \Mi? = a3 sin aft . aft, where aft de- notes the direction which is normal to both a and ft, and drawn in the sense given by the right-handed screw. Example.— Given A - rep) ' [B_ and B = r'^~//&. Then SAB = rr' cos ^//6_^~//_^_ = 7t'|cos 6 cos 0' + sin d sin 0' cos (0' — cp)\. Product of two Sums of non-successive Vectors. — Let A and B be two component vectors, giving the resultant A -\- B, and let C denote any other vector having the same point of appli- cation. Let A = aj -j- a J -\- a s k, ■A+B B = bj + bj + b a k, C = c,i + cj + c,k. Since A and B are independent of order,, A + B = (a, + *,)* + (a, + *,)/ + (*. + W consequently by the principle already established S(A + B)C = (a, + £>, + (*, + *,)'. + («. + *.K = a/, + a/, + a s t, + b t c, + V» + Vt = S^C"+Sj5C. Similarly V(^ + 5)C = { (a, + *,K - (a. + *X } f + etc. = (a,<:, - «,c>' + ( V. - V>" + • • • = V^C + V5C Hence (^ + 5)C = ^4 C+ 5C In the same way it may be shown that if the second factor consists of two components, C and D, which are non-successive in their nature, then (A+B)(C+D) = AC+AD + BC + BD. Art. 7.] product of three vectors. 449 When A -\- B is a sum of component vectors (A + B? = A" + B' + AB + BA = A* + B"-\-2SAB, Prob. 28. The relative velocity of a conductor is S.W., and the magnetic flux is N.W.; what is the direction of the electromotive force in the conductor ? Prob. 29. The direction of the current is vertically downward, that of the magnetic flux is West; find the direction of the mechani- cal force on the conductor. Prob. 30. A body to which a force of 2/ + 3/' + 4^ pounds is applied moves with a velocity of 5/+ 6/ + "jk feet per second; find the rate at which work is done. Prob. 31. A conductor 8/+ 9/ + io ^ inches long is subject to an electromotive force of nz'+ 12/ + 13^' volts per inch; find the difference of potential at the ends. (Ans. 326 volts.) Prob. 32. Find the rectangular projections of the area of the parallelogram defined by the vectors A = 12/— 23/'— 34^ and B = - 45* - 5 6 / + 6 1 k - Prob. 33. Show that the moment of the velocity of a body with respect to a point is equal to the sum of the moments of its com- ponent velocities with respect to the same point. Prob. 34. The arm is gi + ii/'-f- 13/J feet, and the force applied at either end is 17* + 19/ -f- 23/J pounds weight; find the torque. Prob. 35. A body of 1000 pounds mass has linear velocities of 50 feet per second 3o°//45°> and 60 feet per second 6o°//2 2°.5; find its kinetic energy. Prob. 36. Show that if a system of area-vectors can be repre- sented by the faces of a polyhedron, their resultant vanishes. Prob. 37. Show that work done by the resultant velocity is equal to the sum of the works done by its components. Art. 7. Product of Three Vectors. Complete Product. — Let us take A = aj. -)- a^j -\- a s k, B = bj -\- b,J '-(- b s k, and C = c x i -j- c t j-\- c,k. By the product of A, B, and C is meant the product of the product of A and B with C, according to the rules p. 444). Hence ABC = OA -f a A + a s b,){cj + cj+ cjt) + {(*A — «A>"+ ( a A — aA)J+ ( a A ~ <*A)&\(c l i + cJ+ 'J?) = («/, + a A + a A) {*, i + C J + C J*) ( T ) 450 VECTOR ANALYSIS AND QUATERNIONS. [Chap. IX. + a, a, «„ «, a x a, K K hh K K '. c % c * i J k (2) + a, a, «, K K £, e, c* c* (3) Example.— Let ^4 = \i ' + 2/*+ 3/6, £ = 4*' + 5/ -f- 6k, and C = 7? + 87' -+- 9/6. Then (1) = (4 + 10 + i8)(7* + 87 + 9*) = 32(7*'+ 8/+ 9*). 782" + 6/ — 66& (2) = -3 6 -3 7 8 9 *' 7 k (3) = 1 2 3 4 5 6 7 8 9 = 0. If we write A = aa, B = b/3, C = c^, then ABC — «fc cos a/? . y (1) -(- abc sin a/3 sin ar/J^ . a/3y (2) -)- afo sin a/3 cos a/3/, (3) where cos a(3y denotes the cosine of the angle between the directions a/3 and y, and afiy denotes the direction which is normal to both a/3 and y. We may also write ABC=SAB.C+V(yAB)C+S(VAB)C. (1) (2) (3) First Partial Product. — It is merely the third vector multi- plied by the scalar product of the other two, or weighted by that product as an ordinary algebraic quantity. If the direc- tions are kept constant, each of the three partial products is proportional to each of the three magnitudes. Second Partial Product. — The second partial product may be expressed as the difference of two products similar to the first. For V(VAB)C = 1 - (V, + VsK + ('A + c A )W + 1 - (V, + , + ('A + c,a^b,\k. Art. 7.] PRODUCT OF THREE VECTORS. 451 By adding to the first of these components the null term {b l c 1 a l — c^afi^i we get — SBC . , in the usual for- mula for the volume of a parallelepiped. Example. — Let the velocity of a straight wire parallel to itself be V— 1000/30° centimeters per second, let the intensity of the magnetic flux be B — 6000 790° lines per square cen- timeter, and let the straight wire L = 15 centimeters 60°/ /45 . Then V VB = 6000000 sin 6o° 90 / /9Q° lines per centimeter per second. Hence S(VVB)L = 15 X 6000000 sin 6o° cos lines per second where cos0 = sin 45° sin 60°. Sum of the Partial Vector Products. — By adding the first and second partial products we obtain the total vector product of ABC, which is denoted by V(ABC). By decomposing the second product we obtain V{ABC) = SAB. C-SBC.A + SCA .B. By removing the common multiplier abc, we get Y(a/3y) = cos a/3 . y — cos fly . a -(- cos ya . fi. Similarly V(/3ya) = cos /3y . a — cos ya . /3 -\- cos a/3 . y and V(yafj) — cos ya . /3 — cos a/3 . y -\- cos /3y . a. These three vectors have the same magnitude, for the square of each is cos 2 a/3 + cos" /3y -f cos a ya — 2 cos a/3 cos /3y cos ya, r ' that is, I -{S{a/3y)}\ They have the directions respectively of a',. /3', y' , which are the corners of the triangle whose sides are bisected by the corners a, /3, y of the given triangle. Prob. 38. Find the second partial product of 9 20°/ /30 , 10 30°/ 740° , 11 45"/ /45° . Also the third partial product. Prob. 39. Find the cosine of the angle between the plane of / t i+m l J+ n x k and /,«+/«,/'+«,* and the plane of /,*" + «,/+«,& and I J, -\- m i j -\- nfi. Prob. 40. Find the volume of the parallelepiped determined by the vectors 100/ + 50/ + 25^, 50/+ io/-|-8o£, and — 752'+ 407 — 80&- ART. 8.] COMPOSITION OF QUANTITIES. 453 Prob. 41. Find the volume of the tetrahedron determined by the ■extremities of the following vectors : 32 — 2/ + 1^, — 4* + 5/ — lk, y — V — 2k > 8 * + 4/ — 3& Prob. 42. Find the voltage at the terminals of a conductor when its velocity is 1500 centimeters per second, the intensity of the mag- netic flux is 7000 lines per square centimeter, and the length of the •conductor is 20 centimeters, the angle between the first and second being 30 , and that between the plane of the first two and the direc- tion of the third 6o°. (Ans- .91 volts.) Prob. 43- Let a = 2^7/10°, /? = 3^7/25°, Y = 4?7/35°- Find Vafiy, and deduce Yfiya and Yya/3. Art. 8. Composition of Quantities. A number of homogeneous quantities are simultaneously located at different points ; it is required to find how to add or compound them. Addition of a Located Scalar Quantity. — Let m A denote a mass m situated at the extremity of the radius- vector A. A mass m — m may be introduced at the extremity of any radius-vector R, so that m A = (m — ni) R -f- m A = m R + m A — m R = m R -+- m(A — i?). Here A — R is a simultaneous sum, and denotes the radius- vector from the extremity of R to the extremity of A. The product m(A — R) is what Clerk Maxwell called a mass-vector, and means the directed moment of m with respect to the ex- tremity of R. The equation states that the mass m at the extremity of the vector A is equivalent to the equal mass at the extremity of R, together with the said mass-vector applied at the extremity of R. The equation expresses a physical or mechanical principle. Hence for any number of masses, m l at the extremity of A lt m 7 at the extremity of A^, etc., 2m A = 2% + 2{m(A — R)\, 454 VECTOR ANALYSIS AND QUATERNIONS. [Chap IX. where the latter term denotes the sum of the mass-vectors treated as simultaneous vectors applied at a common point. Since 2{m(A — R)\ = 2mA — 2mR =■ 2mA — R2m, the resultant moment will vanish if R = — ^ — , or R2m = 2mA 2 m Corollary. — Let R = xi ' + yj ' + zk, and A =a,t + £,_/ -f- c t k ; then the above condition may be written as _ 2 (ma) . «' (2mb) .j . 2(mc) . k 2m 2m 2m ' 2(ma) 2(mb) 2mc therefore x = — ^ — , y = -^ — . z = — — 2m 2m 2m Example. — Given 5 pounds at 10 feet 45°/ /30 and 8 pounds at 7 feet 6o //45° ; find the moment when both masses are transferred to 12 feet 75°//6o°. m^A x = 5o(cos 30°« + sin 30 cos 45°/+ sin 30 sin 45°£), m *A* = 56(cos45 t Y4-sin45°cos6o _/+sin45°sin6o /£), (m l -\-m^R= i26(cos6o°« + sin6o°cos 75y+ s i n 6o°sin 75 °£), moment = m l A l -f- m,A t — (w, + m,)R. Composition of a Located Vector Quantity. — Let F A de- note a force applied at the extremity of the radius-vector A. As a force F — F may introduced at the ex- tremity of any radius-vector R, we have F A = {F-F) R + F A = F R + V(A - R)F. This equation asserts that a force F applied at the extremity of A is equivalent to an equal force applied at the extremity of R together with a couple whose magnitude AR.T. 8.] COMPOSITION OF QUANTITIES. 455 and direction are given by the vector product of the radius- vector from the extremity of R to the extremity of A and the force. Hence for a system of forces applied at different points, such as F, at A lt F^ at A„ etc., we obtain 2{F A ) = 2(F R ) + 2V(A - R)F = (2F) R + 2V(A - R)F. Since 2V(A - R)F = 2VAF - 2VRF = 2VAF-VR2F the condition for no resultant couple is VR2F = 2VAF, which requires 2F to be normal to 2VAF. Example. — Given a force \i -\- 2j -\- ik pounds weight at 4* + Sj-\~6k feet, and a force of 7* + Q/ + ilk pounds weight at loi -\- I2j -\- 14k feet; find the torque which must be sup- plied when both are transferred to 21 -f- 5/ '+ 3^, so that the effect may be the same as before. VAf^y-ej+ik, VA,F 2 = 6t- \2j + 6k, 2VAF= gz — iSj + gk, 2F= 8z+ii/-f 14/fc, VR2F=tfi-4j- 18*, Torque = — 281 — i\j + 27k. By taking the vector product of the above equal vectors with the reciprocal of 2F we obtain v\(VR2F)^\ = v{(2VAF)^}. By the principle previously established the left member resolves into — R -\- SR-yTp.. 2F; and the right member is equivalent to the complete product on account of the two factors being normal to one another ; hence - R + SR-^ . 2F = 2{VAF)-±p; 456 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX. that is, R = 2p2{VAF) + SR^j? • 2F - (0 (2) The extremity of R lies on a straight line whose perpen- dicular is the vector (i) and whose direction is that of the resultant force. The term (2) means the projection of R upon that line. The condition for the central axis is that the resultant force and the resultant couple should have the same direction ; hence it is given by V j 2VAF- VR2F\ 2F=o; that is, V(VR2F)2F = V(2AF)2F. By expanding the left member according to the same prin- ciple as above, we obtain — (2FyR 4- SR2F. 2F = V{2AF)2F; therefore R = ^V2F(V2AF) + |g^ . 2F v(-^j(V2AF) + SR*.2F. This is the same straight line as before, only no relation is now imposed on the directions of 2F and 2VAF; hence there always is a central axis. Example. — Find the central axis for the system of forces in the previous example. Since 2 F— St-\- nj-\- 14k, the direction of the line is 81 -f- 11/+ 14^ V64 — |— 121 — f— 196' Since ' = ? "{ and 2VAF = gi- 1 87 4- ok, the 2F 381 v J perpendicular to the line is ., 8?'4-ii/'+i4^ . „. , , 1 , . . . ,, V—^ST^-W - 187 + 9^ = 33^351* + 54/ - 243*}- Prob. 44. Find the moment at qoV / 2 7o° of 10 pounds at 4 feet io o //20° and 20 pounds at 5 feet 3o°//i2o°. Art. 9.] SPHERICAL TRIGONOMETRY. 457 Prob. 45. Find the torque for 4/ +3/+ 2k pounds weight at 21 — 3/' + i/e feet, and 21 — 1/— impounds weight at — 3/+ 47 + 5& feet when transferred to — 3/ -+- 2/ — &,k feet. Prob. 46. Find the central axis in the above case. Prob. 47. Prove that the mass-vector drawn from any origin to a mass equal to that of the whole system placed at the center of mass of the system is equal to the sum of the mass-vectors drawn from the same origin to all the particles of the system. Art. 9. Spherical Trigonometry. Let i,j, k denote three mutually perpendicular axes. In order to distinguish clearly between an axis and a quadrantal version round it, let i v,i t f l1 , k" /2 denote fc quadrantal versions in the positive sense about the axes i,j, k respectively. The directions of positive version are indicated -j| 'by the arrows. By f^i"/* is meant the product of two quadrantal versions round i; it is equiv- alent to a semicircular version round i ; hence 2 Similarly f^f 1 * means the product of two quadrantal versions round./, and/V'* =f = -. Similarly V 1 **'* = k" = -. By {"^J"'* is meant a quadrant round i followed by a quad- rant round/; it is equivalent to the quadrant from j to 2, that is, feo — k"^. But f^'i"/' is equivalent to the quadrant from — i to — j, that is, to k"'''. Similarly for the other two pairs of products. Hence we obtain the following Rules for Versors. j£ lift" 1 1 _ f/sV"/! rV A "A.-'/a *Ufl* __ £"■/» fUfU _ jfl*. t ir/i jit/, -it/j ir/, £v/i ;*/i __ fly The meaning of these rules will be seen from the follow ing application. Let li + mj + nk denote any axis, then 458 VECTOR ANALYSIS AND QUATERNIONS. [Chap. IX. (/?' + ittf-\- n£)"/* denotes a quadrant of angle round that axis. This quadrantal version can be decomposed into the three rectangular components h , /«/ , n£"'* ; and these components are not successive versions, but the parts of one version. Sim- ilarly any other quadrantal version (l'i-\-m'j -\-n'kf ,% can be resolved into /'z , «/V , n'tf 1 '*. By applying the above rules, we obtain (It + mj + nk)"'\l'i -f m'j + n'k) w/ * = — (//' -(- ww' -(- ««') = — (//' -f- /WW*' -f- ««') - {(«»' - **'»>" + («/' - »7)/ + (/*«' - l'm)k\"'\ Product of Two Spherical Versors. — Let fi denote the axis and b the ratio of the spherical versor PA, then the versor itself is expressed by /? s . Similarly let y denote the axis and c the ratio of the spherical versor AQ, then the versor itself is expressed by y c . Now /?* = cos b + sin b . /T /2 , and y c = cos c -\- sin c . y"^ ; therefore /?V = (cos b + sin 3 . /T A )(cos c + sin <: . y* 7 ') = cos b cos t -f- cos bsinc . y"'' -\- cos c sin b . /J* 7 ' -\- sin bsinc. fT'*y''** But from the preceding paragraph pUyU _ _ cos ^ _ s ; n ^ _ ^A . therefore /3 b y c — cos £ cos c — sin £ sin c cos /?/ (i) -f- 1 cos 3 sin c . y -f- cos c sin b . fi — sin b sin ^ sin fly . flyY 1 *- (2) The first term gives the cosine of the product versor ; it is equivalent to the fundamental theorem of spherical trigonom- etry, namely, cos a = cos b cos c -\- sin b sin c cos A, Art. 9.] spherical trigonometry. 459 where A denotes the external angle instead of the angle in- cluded by the sides. The second term is the directed sine of the angle; for the square of (2) is equal to 1 minus the square of (1), and its di- rection is normal to the plane of the product angle.* Example.— Let /J = 307 /45 ° and y = 607 /30°. Then cos /3y = cos 45° cos 30° -f- sin 45" sin 30 cos 30°, and sin fJy . fiy = V 'fly ; but /? = cos 45° i -f- sin 45° cos 30°/+ sin 45° sin 30 k, and ;/ = cos 30 i -J- sin 30° cos 60°/ + sin 30° sin 60° k ; therefore \/3y = {sin 45° cos 30° sin 30° sin 60° — sin 45 sin 30°sin 30° cos 60°}/ -4- {sin 45° sin 30° cos 30° — cos 45° sin 30° sin 60° £/ -f- {cos 45 sin 30°cos6o° — sin 45° cos 30 cos 30° \k. Quotient of Two Spherical Versors. — The reciprocal of a given versor is derived by changing the sign of the index ; y~ c is the reciprocal of y c . As /?* = cos ^ -j- sin b . /3" /il , and y~ c = cos c — sin c . y" 2 , /3 h y~ c = cos b cos c -\- sin b sin c cos fly -f-jcos c sin b . fl — cosb sin c . y -\- sin b sin c sin fly . fJy (''''• s Product of Three Spherical Versors. — Let a" denote the versor PQ, fl b the versor QR, and y- the versor RS ; then a a fl b y- denotes K PS. Now a'fjy = (cos a-\- sin a . a ! *)(zos b -\- s\xvb . /T /2 )(cos c -f- sin c . y *) = cos a cos b cos c (1) -\- cos a cos b sin c. ?/ + cos a cos £ sin b . fT -j- cos $ cos c sin « . a (2) -{- cos a sin £ sin c . fF' % y"'* -f- cos £ sin asmc. ay -\- cos c sin « sin $ . a ft (3) * Principles of Elliptic and Hyperbolic Analysis, p. 2. 460 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX. 4- sin a sin b sin c . a n ' 3 /3 y . (4) The versors in (3) are expanded by the rule already ob- tained, namely, p"/y"A _ _ cos p y _ sin py . Jf>\ The versor of the fourth term is a U f? h y U = - (cos a/3-\- sin a/3 . ~a~f? u ~)y l * — — cos a/3 . ^" /a +sin a/3 cos or/3^+sin a/3 sin a/?;/ . a/3y" /:l . Now sin a/3 sin a/S;-' . afiy — cos ay . (3 — cos /3j/ . a (p. 45 1), hence the last term of the product, when expanded, is sin a sin b sin c\ — cos a§ . y ~r cos a Y • ft" 3 — cos fiy . a^" + cos~a/3y}. Hence cos a a /3 b y c = cos a cos £ cos c — cos a sin b sin c cos ySy — cos b sin # sin c cos ay — cos c sin « sin b cos a/3 -|- sin a sin $ sin c sin a/3 cos a/3;j/, and, letting Sin denote the directed sine, Sin a" fry' = cos a cos b sin c . y -f- cos « cos t sin 3 . /3 4- cos 3 cos c sin a . a — cos « sin £ sin c sin /3j/ . /3y — cos (5 sin a sin c sin ay . ay — cos <: sin a sin # sin a-/? . a/? — sin a sin (5 sin tj cos a/3 . y— cos ay . /3-|-cos /3y . <*}•* Extension of the Exponential Theorem to Spherical Trigo- nometry. — It has been shown (p. 458) that cos (3 b y c = cos b cos c — sin b sin c cos f3y and (sin /?y) V2 = cos c sin 3 . /3" /a + cos * sin c . y wh — sin b sin c sin /3y . f3y • Now cos b = 1 - + —-77 + etc. 2 ! 4! 6! * In the above case the three axes of the successive angles are not perfectly independent, for the third angle must begin where the second leaves off. But the theorem remains true when the axes are independent ; the factors are then quaternions in the most geneial sense. Art. 9.] SPHERICAL TRIGONOMETRY. 461 and sin b = b r -4- — r — etc. Substitute these series for cos b, sin b, cos c, and sin c in the above equations, multiply out, and group the homogeneous terms together. It will be found that cos PY — 1 — — |0 3 + 2bc cos Py + c*\ + -\{ b" + 40V cos fiy + 60V + 46? cos fly + c*\ — gj!^ + 60V cos /3y + 150V + 2O0V cos /?;/ + 1 50V + 6&r' cos J3y + <:" | -f . . ., where the coefficients are those of the binomial theorem, the only difference being that cos fly occurs in all the odd terms, as a factor. Similarly, by expanding the terms of the sine, we obtain (Sin py)** = b.ff /2 + c. y 1 - be sin /3y . Jf' % -~{b s . /T /2 + 30V . y" /s + 3bc\ /3 n/ * + c° . /'* \ + — r f^ 3 + b ' c \ sin ®y ■ @y 3 JL { y . p* + S b'c . y /% + io0V . /f /s + IO0V ■ y"'* + 5fc\ /J V2 + c 6 . Y w/ * \ - i\ { h ' c +rrl^ v + fc6 1 sin ^- ^ ~ • ■ • ■ By adding these two expansions together we get the ex- pansion for fi h y c , namely, - -\ \ a + 2bc (cos /Jy + sin fly . Jy' % ) + ^ \ - -L j 0\ /S" /2 + 3 0V • y" + $bc\ p" % + e . y" /% \ + -L\ b' + 4 0V(cos /? r + sin /J r . /?/ A ) + 60 V + 4 0£ s (cos /Jy + sin /? y . T) "'*) + **} + ■■- 4! 462 VECTOR ANALYSIS AND QUATERNIONS. [CHAP. IX. By restoring the minus, we find that the terms on the second line can be thrown into the form and this is equal to ■^\b.fr*+c.y*\\ where we have the square of a sum of successive terms. In a similar manner the terms on the third line can be restored to tf . tf*l* 4- 3 p c . p'y w/t + 36c' ; /J V V + c° . y iW!s \ that is, ^{b.f + c.y*^. Hence _J>-£ + c -y * Extension of the Binomial Theorem. — We have proved above that e 6 ?"'* e*"'* = ^* /! + ^ ! provided that the powers of the binomial are expanded as due to a successive sum, that is, the order of the terms in the binomial must be preserved. Hence the expansion for a power of a successive binomial is given by {b.fT^ + c. y h \ n = b" . /J""' 2 + nF-'e . /?" l *'*y'' 1 n(n — 1) + -^"-V.^-'^/sy-l-etc. * At page 386 of his Elements of Quaternions, Hamilton says: "In the present theory of diplanar quaternions we cannot expect to find that the sum of the logarithms of any two proposed factors shall be generally equal to the logarithm of the product ; but for the simpler and earlier case of coplanar quaternions, that algebraic property may be considered to exist, with due modification for multiplicity of value." He was led to this view by not dis- tinguishing between vectors and quadrantal quaternions and between simul- taneous and successive addition. The above demonstration was first given in my paper on "The Fundamantal Theorems of Analysis generalized for Space." It forms the key to the higher development of space analysis. Art. 10.] composition of rotations. 463 Example.— Let b = ^ and c=\, /3 = 307/45", Y = 607/30°. (b . f? 1 * + c . y^y = -{& + R = (cos 6 + sin B . /3 w/2 )rp 464 VECTOR ANALYSIS AND QUATERNIONS. [Chap. IX. = r(cos 6 + sin B . y6" /2 )(cos ftp. ft -\- sin ftp . ftpft) — r{cos ftp . /S-fcos sin ftp . ftp ft -4- sin sin /Jp. ftp~\. When cos ftp =o, this reduces to ft e R — cos OR + sin OV(ftR). The general result may be written ft e R = S/J72 . /?+ cos 6{VftR)(J + sin 0V/W?. Note that (VftR)ft is equal to V{VftR)ft because Sy3i?/3 is O, for it involves two coincident directions. Example. — Let ft = li -\- mj -\- nk, where P -j- in' -{- if = i and R = xi -\- yj -\- zk ; then SftR = Ix -f- »y -4- «,s V(/^)/J = and Hence w^ — «y «^r - Iz ly — mx / m « z / k VftR = / m 11 x y z i j k • Ix -j- ^7 + nz)(li -\- mj "-j- #£) -{- cos # mz — ny nx — Iz ly — mx I m n / ■/' k 4- sin I m n x y z i j k To prove that ft i p coincides with the axis of ft- b / a p'/ s ftV s . Take the more general versor p e . Let OP represent the axis ft, AB the versor ft~ h ''\ BC the versor p". Then (AB)(BC) = AC = DA, therefore >d (AB)(BC)(AE) = (DA)(AE) = DE. Now DE has the same angle as BC, but its axis has been rotated round P by the angle b. Hence if 6 — nJ2, the axis of /J-*/»p*/»/j*/» will coincide with ft b p* The exponential expression for * This theorem was discovered by Cayley. It indicates that quaternion, multiplication in the most general sense has its physical meaning in the compo- sition of rotations. ART. 10.] COMPOSITION OF ROTATIONS. 465 fl-t/tp/ip/i i s ,_ w p/»+w r /» + W r / , > which may be expanded according to the exponential theorem, the successive powers of the trinomial being formed according to the multinomial theorem. Composition of Finite Rotations round Axes which Inter- sect. — Let /3 and y denote the two axes in space round which the successive rotations take place, and let /3 b denote the first and y* the second. Let fi b X y 1 " denote the single rotation which is equivalent to the two given rotations applied in succession ; the sign X is introduced to distinguish from the product of versors. It has been shown in the preceding para- graph that and as the result is a line, the same principle applies to the subsequent rotation. Hence y c ((3 h p) — y-'/\ft- b /*p"/zfi"/*)y c /* because the factors in a product of versors can be associated in any manner. Hence, reasoning backwards, /?» x y c = (p/y^Y- Let m denote the cosine of /JVyA, namely, cos b/2 cos c/2 — sin b/2 sin c/2, and n. v their directed sine, namely, cos b/2 sin c/2.y J r cos c/2 sin b/2 . /?— sin b/2 sin c/2 sin /3y . §y; then /J d X y c = m* — n* + 2mn ■ v - Observation. — The expression (/S'/^A)* is not, as might be supposed, identical with fi b y c . The former reduces to the lat- ter only when /J and y are the same or opposite. In the figure /S* is represented by PQ, y c by QR, /?V by PR, pl*y«* by ST, and (/S'/yA)* by SU, which is twice ST. The cosine of SU differs from the cosine of PR by the term —(sin b/2 sin c/2 sin §y)\ evident from the figure that their axes are also different. 466 VECTOK ANALYSIS AND QUATERNIONS. [CHAP. IX. Corollary. — When b and c are infinitesimals, cos /3 h Xy c =i, and Sin /S 4 x y c ~ b. /S + c. y, which is the parallelogram rule for the composition of infinitesimal rotations. Prob. 54. Let fi = 307/45°, = n/ 3 , and R = 21 - 3/ + 4J ; calculate p B £. Prob. 55. Let /J = 9^7/90°, = w/ 4j J? = - /+ 2/- 3*; calculate f?R. Prob. 56. Prove by multiplying out that /3- b /zp*Mfi t fr = \fi b p\*h- Prob. 57. Prove by means of the exponential theorem that y- c P b y c has an angle b, and that its axis is y^/3. Prob. 58. Prove that the cosine of (/SWy*/*)' differs from the cosine of fi b y c by — (sin - sin - sin /3y) . Prob. 59. Compare the axes of (fi b /*y c fcy and (Py*. Prob. 60. Find the value of /3 b X y c when /3 =~o7/ 9 o c and y = oo //9Q°- Prob. 61. Find the single rotation equivalent to j"'/ 8 Xj w/2 X >4"A Prob, 62. Prove that successive rotations about radii to two corners of a spherical triangle and through angles double of those of the triangle are equivalent to a single rotation about the radius to the third corner, and through an angle double of the external angle of the triangle. Art. 1.] INTRODUCTION. 467 Chapter X. PROBABILITY AND THEORY OF ERRORS. By Robert S. Woodward, Professor of Mechanics in Columbia University. Art. 1. Introduction. It is a curious circumstance that a science so profoundly mathematical as the theory of probability should have origi- nated in the games of chance which occupy the thoughtless and profligate.* That such is the case is sufficiently attested by the fact that much of the terminology of the science and many of its familiar illustrations are drawn directly from the vocabulary and the paraphernalia of the gambler and the trick- ster. It is somewhat surprising, also, considering the antiquity of games of chance, that formal reasoning on the simpler questions in probability did not begin before the time of Pascal and Fermat. Pascal was led to consider the subject during the year 1654 through a problem proposed to him by the Chevalier de Mere, a reputed gambler.f The problem in question is known as the problem of points and may be stated as follows : two players need each a given number of points to win at a certain stage of their game ; if they stop at this stage, how should the stakes be divided ? Pascal corresponded with his friend Fermat on this question ; and it appears that the letters which passed between them contained the earliest distinct formulation of principles falling within the theory of probability. These * The historical facts referred to in this article are drawn mostly from Tod- hunter's History of the Mathematical Theory of Probability from the time of Pascal to that of Laplace (Cambridge and London, 1865). t " Un probleme relatif aux jeux de hasard, propose^ aun austere janseniste par un homme du monde, a et6 l'origine du calcul des probabilitfes.'' Poisson, Recherches surla Probability des Jugements (Paris, 1837). 468 PROBABILITY AND THEORY OF ERRORS. [Chap. X acute thinkers, however, accomplished little more than a correct start in the science. Each seemed to rest content at the time- with the approbation of the other. Pascal soon renounced such mundane studies altogether ; Fermat had only the scant leisure of a life busy with affairs to devote to mathematics; and both died soon after the epoch in question, — Pascal in 1662, and Fermat in 1665. A subject which had attracted the attention of such dis- tinguished mathematicians could not fail to excite the interest of their contemporaries and successors. Amongst the former Huygens is the most noted. He has the honor of publishing the first treatise* on the subject. It contains only fourteen propositions and is devoted entirely to games of chance, but it gave the best account of the theory down to the beginning of the eighteenth century, when it was superseded by the more elab- orate works of James Bernoulli, f Montmort,^: and De Moivre.§ Through the labors of the latter authors the mathematical theory of probability was greatly extended. They attacked, quite successfully in the main, the most difficult problems; and great credit is due them for the energy and ability dis- played in developing a science which seemed at the time to- have no higher aim than intellectual diversion. | Their names,, undoubtedly, with one exception, that of Laplace, are the most important in the history of probability. Since the beginning of the eighteenth century almost every mathematician of note has been a contributor to or an expos- itor of the theory of probability. Nicolas, Daniel, and John Bernoulli, Simpson, Euler, dAlembert, Bayes, Lagrange, Lam- bert, Condorcet, and Laplace are the principal names which figure in the history of the subject during the hundred years *De Ratiociniis in Ludo Aleae, 1657. fArs Conjectandi, 1713. JEssai d'Analyse sur les Jeux de Hazards, 1708. § The Doctrine of Chances, 1718. I! Todhunter says of Montmort, for example, " In 1708 he published his work on Chances, where with the courage of Columbus he revealed a new world, to Mathematicians." Art. 1.] INTRODUCTION. 469 ■ending with the first quarter of the present century. Of the contributions from this brilliant array of mathematical talent, the Th6orie Analytique des Probabilites of Laplace is by far the most profound and comprehensive. It is, like his Me- ■canique Celeste in dynamical astronomy, still the most elabo- rate treatise on the subject. An idea of the grand scale of the work in its present form* maybe gained by the facts that the non-mathematical introduction! covers about one hundred and fifty quarto pages ; and that, in spite of the extraordinary brevity of mathematical language, the pure theory and its ac- cessories and applications require about six hundred and fifty pages. From the epoch of Laplace down to the present time the extensions of the science have been most noteworthy in the fields of practical applications, as in the adjustment of obser- vations, and in problems of insurance, statistics, etc. Amongst the most important of the pioneers in these fields should be mentioned Poisson, Gauss, Bessel, and De Morgan. Nu- merous authors, also, have done much to simplify one or an- other branch of the subject and thus bring it within the range of elementary presentation. The fundamental principles of the theory are, indeed, now accessible in the best text-books on algebra : and there are many excellent treatises on the pure theory and its various applications. Of all the applications of the doctrine of probability none is of greater utility than the theory of errors. In astronomy, geodesy, physics, and chemistry, as in every science which at- tains precision in measuring, weighing, and computing, a knowledge of the theory of errors is indispensable. By the aid of this theory the exact sciences have made great progress dur- *The form of the third edition published in 1820, and of Vol. VII of the complete works of Laplace recently republished under the auspices of the Academie des Sciences by Gauthier-Villars. This Vol. VII bears the date 1886. f " Cette Introduction," writes Laplace, "est le developpement d'une Lecon ■sur les Probabilites, que je donnai en 1795, aux Ecoles Normales, ou je fus ap- pele comme professeur de Mathematiques avec Lagrange, par un decret de la Convention nationale." 470 PROBABILITY AND THEORY OF ERRORS. [Chap. X. ing the present century, not only in the actual determination of the constants of nature, but also in the fixation of clear ideas as to the possibilities of future conquests in the same di- rection. Nothing, for example, is more satisfactory and in- structive in the history of science than the success with which the unique method of least squares has been applied to the problems presented by the earth and the other members of the solar system. So great, in fact, are the practical value and theoretical importance of the method of least squares, that it is frequently mistaken for the whole theory of errors, and is sometimes regarded as embodying the major part of the doc- trine of probability itself. As may be inferred from this brief sketch, the theory of probability and its more important applications now constitute an extensive body of mathematical principles and precepts. Obviously, therefore, it will be impossible within the limits of a single chapter of this volume to do more than give an out- line of the salient features of the subject. It is hoped, how- ever, in accordance with the general plan of the volume, that such outline will prove suggestive and helpful to those who' may come to the science for the first time, and also to those who, while somewhat familiar with the difficulties to be over- tome, have not acquired a working knowledge of the subject. Effort has been made especially to clear up the difficulties of the theory of errors by presenting a somewhat broader view of the elements of the subject than is found in the standard treatises, which confine attention almost exclusively to the method of least squares. This chapter stops short of that method, and seeks to supply those phases of the theory which are either notably lacking or notably erroneous in works hitherto published. It is believed, also, that the elements here presented are essential to an adequate understanding of the well-worked domain of least squares.* *The auihor has given a brief but comprehensive statement of the method of least squares in the volume of Geographical Tables published by the Smith- sonion Institution, 1894. Art. 2.] permutations. 471 Art. 2. Permutations. The formulas and results of the theory of permutations and combinations are often needed for the statement and so- lution of problems in probabilities. This theory is now to be found in most works on algebra, and it will therefore suffice here to state the principal formulas and illustrate their mean- ing by a few numerical examples. The number of permutations of n things taken r in a group is expressed by the formula (n) r = n{n — i)(« - 2) ...(«- r + i). (i) Thus, to illustrate, the number of ways the four letters a, b, c, d can be arranged in groups of two is 4 . 3 = 1 2, and the groups are ab, ba, ac, ca, ad, da, be, cb, bd, db, cd, dc. Similarly, the formula gives for n = 3 and r = 2, (3), = 3.2 = 6, n= 7 " r = 3, (7)3 = 7-6.5 =210, n — 10 " r = 6, (10), = 10.9.8.7.6. 5 = 151200. The results which follow from equation (1) when n and r do not exceed 10 each are embodied in the following table : Values of Permutations. 10 9 8 7 6 s 4 3 3 2 2 I I I 10 9 8 7 6 5 4 2 90 72 56 42 30 20 12 6 2 3 720 504 33 6 210 120 60 24 6 4 5040 3024 1680 840 360 120 24 S 30240 15120 6720 2520 720 120 6 151200 60480 20160 5040 720 7 604800 181440 40320 5040 8 18:4400 362880 40320 9 3628800 362880 10 3628800 4 I S t 9864100 986409 109600 13699 1956 325 64 15 The use of this table is obvious. Thus, the number of per- mutations of eight things in groups of five each is found in the fifth line of the column headed with the number 8. It will be 472 PROBABILITY AND THEORY OF ERRORS. [Chap. X. noticed that the last two numbers in each column (excepting that headed with i) are the same. This accords with the for- mula, which gives for the number of permutations of n things in groups of n the same value as for n things in groups of (n — i). It will also be remarked that the last number in each column of the table is the factorial, n\, of the number n at the head of the column. For example, in the column under 7, the last number is 5040 = 1.2.3.4.5.6.7 = 7!. The total number of permutations of n things taken singly, in groups of two, three, etc., is found by summing the numbers given by equation (1) for all values of r from 1 to «.. Calling this total or sum S p , it will be given by S t = Z(n\. (2) To illustrate, suppose n = 3, and, to fix the ideas, let the three things be the three digits 1, 2, 3. Then from the above table it is seen that S t = 3 + 6 -\- 6 = 15 ; or, that the number of numbers (all different) which can be formed from those dig- its is fifteen. These numbers are 1, 2, 3 ; 12, 13, 21, 23, 31, 32; 123, 132, 213, 231, 312, 321. The values of S t for n = 1, 2, ... 10 are given under the corresponding columns of the above table. But when n is large the direct summation indicated by (2) is tedious, if not impracticable. Hence a more convenient formula is desirable. To get this, observe that (1) may be written ( ^=(^)!' (I) ' if r is restricted to integer values between 1 and (n — 1), both inclusive. This suffices to give all terms which appear in the right-hand member of (2), since the number of permutations for r — (n — 1) is the same as for r = n. Hence it appears that S t —n\-\-- 1 ^1.2^' ••{n - l)\ .Art. 3.] combinations. 473 But as n increases, the series by which n ! is here multiplied approximates rapidly towards the base of natural logarithms ; that is, towards * = 2.7182818 -h log e = 0.4342945. Hence for large values of n Sp = n\e, approximately.* (3) To get an idea of the degree of approximation of (3), sup- pose n = 9. Then the computation runs thus (see values in the above table) : log 91 = 362880 5-5597630 e 0.4342945 9!* = 986410 5-9940575 S p = 986409 by equation (2). The error in this case is thus seen to be only one unit, or about one-millionth of Sp.j- Prob. 1. Tabulate a list of the numbers of three figures each which can be formed from the first five digits 1, . . . 5. How many numbers can be formed from the nine digits ? Prob. 2. Is Sj, always an odd number for n odd ? Observe values of Sp in the table above. Art. 3. Combinations. In permutations attention is given to the order of arrange- ment of the things considered. In combinations no regard is paid to the order of arrangement. Thus, the permutations of the letters a, b, c, d in groups of three are (abc) (abd) bac bad acb {acd) cab cad adb adc dab dac bca {bed) cba cbd bda bde dba dbc cda cdb dca deb * See Art. 6 for a formula for computing »! when n is a large number. f When large numbers are to be dealt with, equations (1)' and (3) are easily managed by logarithms, especially if a table of values of log («!) is available. Such tables are given to six places in De Morgan's treatise on Probability in the Encyclopaedia Metropolitana, and to five places in Shortrede's Tables {Vol. I, 1849). 474 PROBABILITY AND THEORY OF ERRORS. [Chap. X. But if the order of arrangement is ignored all of these are seen to be repetitions of the groups enclosed in parentheses, namely, {abc), (add), (acd), (bed). Hence in this case out of twenty-four permutations there are only four combinations. A general formula for computing the number of combina- tions of n things taken in groups of r things is easily derived. For the number of permutations of n things in groups of r is by (i) of Art. 2 (n) r = n(n — i)(n — 2) ...(« — r -f- 1) ; and since each group of r things gives 1.2.3.. ■ f — rl per- mutations, the number of combinations must be the quotient of («) r by r\. Denote this number by C(n) r . Then the gen- eral formula is »(»- i)(»-2)...(»-r+i) 4")r = y\ W This formula gives, for example, in the case of the four let- ters a, b, c, d taken in groups of three, as considered above, 4.3.2 Multiply both numerator and denominator of the right-hand member of (1) by (n — r)\ The result is n\ , , , v ,r r\ (n — r) ! which shows that the number of combinations of n things in groups of r is the same as the number of combinations of n things in groups of (« — r). Thus, the number of combina- tions of the first ten letters a, b, c . . J in groups of three or seven is 10! — : — : = 120. 3 '• 7 1 The following table gives the values C(n) r for all values of n and r from 1 to 10. The mode of using this table is evident. For example, the number of combinations of eight things in sets of five each is found on the fifth line of the column headed 8 to be 56. Art. 3.] combinations. Values of Combinations. 475 10 9 8 7 6 5 4 3 3 2 2 I 1 I 10 9 8 7 6 s 4 2 45 36 28 21 15 10 6 3 1 3 120 84 5b 35 20 10 4 1 4 210 126 70 35 15 5 1 5 252 126 56 21 b 1 6 210 84 28 7 1 7 I20 36 8 1 8 45 9 1 9 10 1 10 [ s c 1023 5" 255 127 63 31 15 7 3 It will be observed that the numbers in any column show a maximum value when n is even and two equal maximum values when n is odd. That this should be so is easily seen from (1)', which shows that C{n) r will be a maximum for any value of n when r ! (it — r) ! is a minimum. For n even this is a minimum for r = ^n ; while for n odd it has equal minimum values for r = \(n — 1) and r = J(« + 1). Thus, maximum of C(ti) r = n \ for » even, (2) Ill^il for « odd. 2 2 The total number of combinations of n things taken singly,, in groups of two, three, etc., is found by summing the numbers given by (1) for all values of r from 1 to n both inclusive. Calling this total or sum S c , S e = 2C{n) r . The same sum will also come from (i)'by giving to rail values from 1 to (n — 1), both inclusive, summing the results, and in- creasing their aggregate by unity. Thus by either process c_ ; ,, v Thus, suppose there are two urns C/j and U v the first of which contains «, white and b 1 black balls, and the second a t white and b, black balls. Then the probability of drawing a white ball from U 1 is/, = «,/(«, + £,), while that of drawing a white ball from cT, is A = a,/(a a -\- b,). The total number of different pairs of balls which can be formed from the entire number of balls is («, + £,)(«, + b,). Of these pairs «,« a are favorable to the concurrence of white in simul- taneous or successive drawings from the two urns. Hence the probability of a concurrence of white with white = «,«,/(«, -(- <$,)(# 2 + £,), white with black = («A + «A)/(«. + b,)(a % + &,), black with black = bfij{a x -f- £,)(«, -f- &,), and the sum of these is unity, as required by equations (2) of Article 4. In general, if A, A, A . . . denote the probabilities of several independent events, and P denote the probability of their concurrence, ^ = AAA- (I) To illustrate this formula, suppose there is required the probability of getting three aces with three dice thrown simul- taneously. In this case/, =/, =/, = 1/6 and P=(i/6) 3 = 1/216. Similarly, if two dice are thrown simultaneously the proba- bility that the sum of the numbers shown will be 11 is 2/36; and the probability that this sum 1 1 will appear in two succes- sive throws of the same pair of dice is 4/36.36. The probability that the alternatives of a series of events will concur is evidently given by Q = q&q, . . . = (I -AX I - A)(i -A) (2) Thus, in the case of the three dice mentioned above, the probability that each will show something other than an ace is 480 PROBABILITY AND THEORY OF ERRORS. [CHAP. X.. g, — q^ = g a = 5/6, and the probability that they will concur in. this is Q = 125/216. Many cases of interest occur for the application of (1) and (2). One of the most important of these is furnished by suc- cessive trials of the same event. Consider, for example, what may happen in n trials of an event for which the probability is p and against which the probability is g. The probability that the event will occur every time is/". The probability that the event will occur (n — 1) times in succession and then fail is p n ' 1 g. But if the ; order of occurrence is disregarded this last combination may arrive in n different ways ; so that the prob- ability that the event will occur (n — 1) times and fail once is np n ~ l q. Similarly, the probability that the event will happen (n — 2) times and fail twice is \n(ii — l)p~"*g* ; etc. That is, the probabilities of the several possible occurrences are given by the corresponding terms in the development of (/ -f- q)". By the same reasoning used to get equations (2) of Art. 3 it may be shown that the maximum term in the expansion of {p -\- q) n is that in which the exponent m, say, of q is the whole number lying between (n-\- i)q — 1 and (n -\- i)g. In other words, the most probable result in n trials is the occurrence of the event (n — 111) times and its failure m times. When n is large this means that the most probable of all possible results is that in which the event occurs n — nq = n{\ — q) = np times and fails nq times. Thus, if the event be that of throwing an ace with a single die the most probable of the possible results in 600 throws is that of 100 aces and 500 failures. Since q" is the probability that the event will fail every time in n trials, the probability that it will occur at least once in n trials is I — q n - Calling this probability r* r= 1 —g"= 1 -(1 —pf. (3) If r in this equation be replaced by 1/2, the corresponding value of n is the number of trials essential to render the * See Poisson's Probability des Jugements, pp. 40, 41. Art. 5.] PROBABILITY OF CONCURRENT EVENTS. 481 chances even that the event whose probability is/ will occur at least once. Thus, in this case, the value of n is given by log 2 n — ° log(i-/) - This shows, for example, if the event be the throwing of double sixes with two dice, for which / = 1/36, that the chances are even {r = 1/2) that in 25 throws (u — 24.614 by the formula) double sixes will appear at least once. Equation (3) shows that however small/ may be, so long as it is finite, n may be taken so large as to make r approach in- definitely near to unity ; that is, n may be so large as to render it practically certain that the event will occur at least once. When n is large (1 -/)" = 1 - »/ + \ 2 V — —rrkj-y + • • • {npy («/)' , = \-np-\- = e~ np approximately. Thus an approximate value of r is r = 1 - e - nt , log e = 0.4342495. (4) This formula gives, for example, for the probability of drawing the ace of spades from a pack of fifty-two cards at least once in 104 trials r — 1 — e~'' = 0.865, while the exact formula (3) gives 0.867. Similarly, the probability of the occurrence of the event at least t times in n trials will be given by the sum of the terms of (p -j- q) n from p" up to that in fig"' 1 inclusive. This proba- bility must be carefully distinguished from the probability that the event will occur t times only in the n trials, the latter being expressed by the single term in fiq"- 1 . Prob. 9. Compare the probability of holding exactly four aces in five hands of whist with the probability cf holding at least four aces in the same number of hands. Prob. 10. What is the probability of an event if the chances are even that it occurs at least once in a million trials ? See equation (4). 482 PROBABILITY AND THEORY OF ERRORS. [Chap. X. Art. 6. Bernoulli's Theorem. Denote the exponents of/ and q in the maximum term of {p _j_ qY by fx and m respectively, and denote this term by T. Then „ n(n — i)(n — 2) . . . O + 1) m n\ As shown in Art. 5, M in this formula is the greatest whole number in (n -f- i)A and sw the greatest whole number in (w 4- 1)^ ; so that when n is large, jj. and in are sensibly equal to np and ft£ respectively. The direct calculation of T by (1) is impracticable when n is large. To overcome this difficulty the following expression is used : * n\ = n n e- n \/~2~nn[l + — +-^+. . •)• (2) \ ' \2n ' 288» ' / v ' log e = 0.4342495, log 2tz = O.7981799. This expression approaches n n e~" V27tn as a limit with the increase of n, and in this approximate form is known as Stir- ling's theorem. Although a rude approximation to n ! for small values of n this theorem suffices in nearly all cases wherein such probabilities as T are desired. Making use of the theorem in (1) it becomes V 2nnpq That this formula affords a fair approximation even when n is small is seen from the case of a die thrown 12 times. The probability that any particular face will appear in one throw is p = 1/6, whence q = 5/6; and the most probable result in 12 throws is that in which the particular face appears twice and fails to appear ten times. The probability of this result com- puted from (3) is 0.309, while the exact formula (1) gives 0.296. The probability that the event will occur a number of times * This expression is due to Laplace, Thfiorie Analytique des Probabilites. See also De Morgan's Calculus, pp. 600-604. Art. 6.] Bernoulli's theorem. 483 comprised between (jji — a) and (/.< + a) in n trials is evidently expressed by the sum of the terms in {p -\- qf for which the exponent of/) has the specified range of values. Calling this probability R, putting A* — n P + "> an d m = nq — u, and using Stirling's theorem (which implies that n is a large number),* Vznnpq^ np I \ nq I very nearly ; and the summation is with respect to u from u = — a to u — -{- a. But expansion shows that the natural logarithm of the product of the two binomial factors in this equation is approximately — ic'/mpq. Hence R = 2 — e-«*/*«pq . S/2nnpq and, since n is supposed large, this may be replaced by a definite integral, putting Thus dz = l/V2tipq, and z' = u'/lnpq. -+- a/ \Jlnpq a/ VZnp'q R = 4= I e-**dz = - 2 - I e~*dz. (4) This equation expresses the theorem of James Bernoulli, given in his Ars Conjectandi, published in 171 3. The value of the right-hand member of (4) varies, as it should, between o and 1, and approaches the latter limit rap- idly as z increases. Thus, writing for brevity z * See Bertrand, Calcul des Probabilities, Paris, 1889, for an extended discus- sion of the questions considered in this Article. 484 PROBABILITY AND THEORY OF ERRORS. [Chap. X. the following table shows the march of the integral : s / / z / o.oo 0.000 0.75 0.711 1.50 O.966 .25 .276 1. 00 ■843 1-75 .987 .50 .520 1-25 •923 2.00 •995 To illustrate the use of (4), suppose there is required the probability that in 6000 throws of a die the ace will appear a number of times which shall be greater than 1/6 X 6000 — 10 and less than 1/6 X 6000 -f- 10, or a number of times lying between 990 and 1010. In this case a = 10, n = 6000,/ = 1/6, q = 5/6. Thus, a/V2npq = 10/V2 . 6000 . 1/6 . 5/6 = 0.245. Hence, by (4) and the table, R = O.27. Prob. 11. If the ratio of males to females at birth is 105 to 100,. what is the probability that in the next 10,000 births the number of males will fall within two per cent of the most probable number ? Prob. 12. If the chance is even for head and tail in tossing a coin, what is the probability that in a million throws the difference between heads and tails will exceed 1500? Art. 7. Inverse Probabilities.* If an observed event can be attributed to any one of several causes, what is the probability that any particular one of these causes produced the event ? To put the question in a concrete form, suppose a white ball has been drawn from one of two urns, Ui containing 3 white and 5 black balls, and lT, contain- ing 2 white and 4 black balls ; and that the probability in favor of each urn is required. If U, is as likely to have been chosen as U v the probability that U x was chosen is 1/2. After such choice the probability of drawing a white ball from U l is 3/8. Before drawing, therefore, the probability of getting a white ball from U 1 was 1/2 X 3/8 = 3/16, by Art. 5. Similarly, before drawing the probability of getting a white ball from U, was 1/2 X 2/6 = 1/6. These probabilities will remain un- changed if the number of balls in either urn be increased or * See Poisson, Probability des Jugements, pp. 81-83. Art. 7.] inverse probabilities. • 485 •diminished so long as the ratio of white to black balls is kept constant. Make these numbers the same for the two urns. Thus let the first contain 9 white and 15 black, and the second 8 white and 16 black; whence the above probabilities may be written 1/2 x 9/24 and 1/2 X 8/24. It is now seen that there are (9 -f- 8) cases favorable to the production of a white ball, each of which has the same antecedent probability, namely, 1/2. Since the fact that a white ball was drawn excludes considera- tion of the black balls, the probability that the white ball came from U x is 9/17 and that it came from cT, is 8/17 ; and the sum of these is unity, as it should be. To generalize this result, let there be m causes, C,, C„ . . . C m . Denote their direct probabilities by q v q„ ■ . .q m \ their antecedent probabilities by r lf r a , . . . r m \ and their resultant probabilities on the supposition of separate existence by A>A> • • ■ p m . That is, Pi = ?i r » A = 9S*> • • • A» = ?«*■»• (0 Let D be the common denominator of the right-hand mem- bers in (1), and denote the corresponding numerators of the several fractions by s,, s„ . . . s m . Then p, = sJD, p, = sJD, . . .p m = s m /D ; and it is seen that there are in all (s t -f- j a -\- . . . s m ) equally possible cases, and that of these s x are favorable to C lt s, to C„ . . . Hence, if P„ P„ . . . P m denote the probabilities of the several causes on the supposition of their coexistence, Pi = sj& + *. + • • • J ™) = A/(A + A + • • ■ A,)- Thus in general P 1= pj2p, P, = pJ?.p,...P m =p m /2p. (2) To illustrate the meaning of these formulas by the above concrete case of the urns it suffices to observe that for U lt q, = 3/8 and r, = 1/2, for U„ q, = 1/3 and r, = 1/2 ; whence p i = 3/16, /, = 1/6, /,+/,= 17/48 ; and P,=9/i7, /", = 8/17. As a second illustration, suppose it is known that a white 486 • PROBABILITY AND THEORY OF ERRORS. [CHAP. X. ball has been drawn from an urn which originally contained m balls, some of them being black, if all are not white. What is the probability that the urn contained exactly n white balls? The facts are consistent with m different and equally probable hypotheses (or causes), namely, that there were i white and (m — i) black balls, 2 white and (m — 2) black balls, etc. Hence in (1), q l = q t = . . . = I, and p l — i/m, p, = 2/m, . . . p„ — n/m, . . . p m = m/m. Thus 2p = (i/2)(m+ 1), 2n and P„ = p„/2p = — ~, , — \. This shows, as it evidently should, that n = m is the most probable number of white balls in the urn. The probability for this number is P m = 2/{m -\- 1), which reduces, as it ought, to I for m=\. Formulas (1) and (2) may also be applied to the problem of estimating the probability of the occurrence of an event from the concurrent testimony of several witnesses, X n X„ . . . Denote the probabilities that the witnesses tell the truth by x t x„ . . . Then, supposing them to testify independently, the probability that they will concur in the truth concerning the event is x^x, . . . ; while the probability that they will con- cur in the only other alternative, falsehood, is (1 — x^i —x t ) . . . The two alternatives are equally possible. Hence by equations (1) and (2) p, = x,x, ..., p t — {l - 0(1 — *,) . . ., X \X '„ . . . p.= x,x, . . . + (I — *,)(i - x t ) . . .' X t X^ . . .+(1 - JT,)(l - X,) . . .' Pj being the probability for and P^ that against the event. To illustrate (3), if the chances are 3 to 1 that X, tells the truth and 5 to 1 that X, tells the truth, jtr, = 3/4, x, — 5/6, and P l = 15/16; or, the chances are 15 to 1 that an event occurred if they agree in asserting that it did.* * For some interesting applications of equations (3) see note E of Appendix to the Ninth Bridgewater Treatise by Charles Babbage (London, 1838). Art. 8.] probabilities of future events. 487 It is of theoretical interest to observe that if Xl , x , . . . ir (3) are each greater than 1/2, P l approaches unity as the number of witnesses is indefinitely increased. Prob. 13. The groups of numbers of one figure each, two figures each, three figures each, etc., which it is possible to form from the nine digits 1, 2, ... 9 are printed on cards and placed severally in nine similar urns. What is the probability that the number 777 will be drawn in a single trial by a person unaware of the contents of the urns ? Prob. 14. How many witnesses whose credibilities are each 3/4 are essential to make P t = 0.999 m equation (3) ? Art. 8. Probabilities of Future Events. Equations (2) of Art. 7 may be written in the following manner: 5 = 5- 3l = _L A A Pm~W (I) If A> A. • • • Pm are found by observation, P„ P t , . . . P m will ex- press the probabilities of the corresponding causes or their effects. When, as in the case of most physical facts, the num- ber of causes and events is indefinitely great, the value of any p or P in (1) becomes indefinitely small, and the value of 2p must be expressed by means of a definite integral. Let x de- note the probability of any particular cause, or of the event to which it gives rise. Then, supposing this and all the other causes mutually exclusive, (1 — x) will be the probability against the event. Now suppose it has been observed that in (m -\- n) cases the event in question has occurred m times and failed n times. The probability of such a concurrence is, by Art. $,cx m (i — x) n , where c is a constant. Since x is unknown, it may be assumed to have any value within the limits o and 1 ; and all such values are a priori equally possible. Put y = cx'"(i — x) n . Then evidently the probability that x will fall within any as- signed possible limits a and b is expressed by the fraction o 1 I ydx I I ydx ; 488 PROBABILITY AND THEORY OF ERRORS. [CHAP. X. so that the probability of any particular x is given by x m (l — xfdx P = ~7 S " • ( 2 ) /V"(i — xfdx This may be regarded as the antecedent probability of the cause or event in question. What then is the probability that in the next (r -f- s) trials the event will occur r times and fail s times, if no regard is had of the order of occurrence ? If x were known, the answer. would be by Arts. 2 and 5 (r-\-s)\ , But since x is restricted only by the condition (2), the required probability will be found by taking the product of (2) and (3) and integrating throughout the range of x. Thus, calling the required probability Q, fx m+r (i — x) n+s dx 6=^^; . ( 4 ) r\s\ fx m {\ — x) n dx The definite integrals which appear here are known as Gamma functions. They are discussed in all of the higher treatises on the Integral Calculus. Applying the rules derived in such treatises there results * (r -\- s) ! (m -\- r) ! {n -\- s) ! (m -\- n -\- 1 ) ! ^ ~ r\s\m\n\(m -{■ n -\- r -\- s +1)! ' ^' If regard is had to the order of occurrence of the event ; that is, if the probability required is that of the event happen- ing r times in succession and then failing s times in succession, * It is a remarkable fact that formula (5) is true without restriction as to values of m, n, r, s. The formula may be established by elementary considera- tions, as was done by Prevost and Lhuilier, 1795. See Todhunter's History of he Theory of Probability, pp. 453-457. Art. 8.] probabilities of future events. 489 the factor (r -+- s)\/r\s\ in (3), (4), (5) must be replaced by unity. To illustrate these formulas, suppose first that the event has happened m times and failed no times. What is the prob- ability that it will occur at the next trial ? In this case (4) gives Q = fx m+1 dx J Jx m dx = (m+ i)/{m + 2). When m is large this probability is nearly unity. Thus, the sun has risen without failure a great number of times m ; the probability that it will rise to-morrow is ( I+ i)( I+ i.r =I+ i_i+... which is practically 1. Secondly, suppose an urn contains white and black balls in an unknown ratio. If in ten trials 7 white and 3 black balls are drawn, what is the probability that in the next five trials 2 white and 3 black balls will be drawn? The application of {5) supposes the ratio of the white and black balls in the urn to remain constant. This will follow if the balls are replaced after each drawing, or if the number of balls in the urn is sup- posed infinite. The data give m = j, n = 3, r = 2, s = 3, m-{-r = g, n -\- s = 6, r -j- s = $, m-\- n-\- 1 = 11, m-\-n-\-r-\-s-\-i = 16. Thus by (5) S!c.!6!n! Q = 2 ! 3 ! 7 ! 3 !i6! = 3o/9i. Suppose there are two mutually exclusive events, the first of which has happened m times and the second n times in m-\-n trials. What is the probability that the chance of the ■occurrence of the first exceeds 1/2 ? The answer to this ques- tion is given directly by equation (2) by integrating the nume- rator between the specified limits of x. That is, 490 PROBABILITY AND THEORY OF ERRORS. [CHAP X. i /V*(i — xfdx _ 0.5 P =— • (6) fx m {\ —xfdx Thus, if tn = i and n = o, P = 3/4 ; or the odds are three to one that the event is more likely to happen than not. Simi- larly, if the event has occurred m times in succession, P= 1 -(1/2)™+', which approaches unity rapidly with increase of n. Art. 9. Theory of Errors. The theory of errors may be defined as that branch of math- ematics which is concerned, first, with the expression of the re- sultant effect of one or more sources of error to which com- puted and observed quantities are subject ; and, secondly, with the determination of the relation between the magnitude of an error and the probability of its occurrence. In the case of computed quantities which depend on numerical data, such as tables of logarithms, trigonometric functions, etc., it is usually possible to ascertain the actual values of the resultant errors. In the case of observed quantities, on the other hand, it is not generally possible to evaluate the resultant actual error, since the actual errors of observation are usually unknown. In either case, however, it is always possible to write down a symbolical expression which will show how different sources of error enter and affect the aggregate error ; and the statement of such an expression is of fundamental importance in the theory of errors. To fix the ideas, suppose a quantity Q to be a function of several independent quantities x, y, z . . .; that is, Q=/(x,y, z...), and let it be required to determine the error in Q due to errors in x, y , z . . . Denote such errors by AQ, Ax, Ay, As . ■ ■ Then, supposing the errors so small that their squares, prod- ucts, and higher powers may be neglected, Taylor's series gives Art. 10.] laws of error. 491 ^ = t**+f/>+f^+-- <■> This equation may be said to express the resultant actual error of the function in terms of the component actual errors, since the actual value of AQ is known when the actual errors of x, y, z . . . are known. It should be carefully noted that the quantities x, y, z . . . are supposed subject to errors which are independent of one another. The discovery of the independent sources of error is sometimes a matter of difficulty, and in general requires close attention on the part of the student if he would avoid blunders and misconceptions. Every investigator in work of precision should have a clear notion of the error-equation of the type (i) appertaining to his work; for it is thus only that he can distinguish between the important and unimportant sources of error. Prob. 15. Write out the error-equation in accordance with (1) for the function Q — xyz -+- x 3 log (y/z). Prob. 16. In a plane triangle a/b = sin^/sin B. Find the error in a due to errors in b, A, and B. Prob. 17. Suppose in place of the data of problem 16 that the angles used in computation are given by the following equations : ^ = ^+$(180°-^- B- Q, B = B, + |(i8o° - A-B t - C, ), where A u B l , C, are observed values. What then is Aa ? Prob. 18. If w denote the weight of a body and r the radius of the earth, show that for small changes in altitude, Aw/w = — Ar/r\ whence, if a precision of one part in 500000000 is attainable in com- paring two nearly equal masses, the effect of a difference in altitude of one centimeter in the scale-pans of a balance will be noticeable.* Art. 10. Laws of Error. A law of error is a function which expresses the relative frequency of occurrence of errors in terms of their magnitudes. Thus, using the customary notation, let e denote the magni- * This problem arose with the International Bureau of Weights and Measures, Whose work of intercomparison of the Prototype Kilogrammes attained a pre- cision indicated by a probable error of 1/500 000 oooth part of a kilogramme. 492 PROBABILITY AND THEORY OF ERRORS. [Chap. X. tude o. any error in a system of possible errors. Then the law of such system may be expressed by an equation of the form y = 0(e)- (i) Representing e as abscissa and y as ordinate, this equation gives a curve called the curve of frequency, the nature of which, as is evident, depends on the form of the function 0. This equation gives the relative frequency of occurrence of errors in the system ; so that if e is continuous the probability of the occurrence of any particular error is expressed by yde = cp(e)de; which is infinitesimal, as it plainly should be, since in any con- tinuous system the number of different values of e is infinite. Consider the simplest form of 0(e), namely, that in which 0(e) = c, a constant. This form of 0(e) obtains in the case of the errors of tabular logarithms, natural trigonometric func- tions, etc. In this case all errors between minus a half-unit and plus a half-unit of the last tabular place are equally likely to occur. Suppose, to cover the class of cases to which that just cited belongs, all errors between the limits — and -fa are equally likely to occur. The probability of any individual error will then be (e)de = I. (3) -a • Art. 11. Typical Errors of a System. Certain typical errors of a system have received special designations and are of constant use in the literature of the theory of errors. These special errors are the probable error, the mean error, and the average error. The first is that error of the system of errors which is as likely to be exceeded as not ; the second is the square root of the mean of the squares of all the errors ; and the third is the mean of all the errors regardless of their signs. Confining attention to systems in which positive and negative errors of the same magnitude are equally probable, these typical errors are defined mathe- matically as follows. Let e p = the probable error, e m = the mean error, e a = the average error. 494 PROBABILITY AND THEORY OF ERRORS. [CHAP. X. Then, observing (2), of Art. 10, f(p(e)de =J'(p(e)de = fcp(e)de = J (e) = c(±a^e), 4>(e)=c cos'(7te/2a); c being a constant to be determined in each case and e having any value between — a and + a. Art. 12. Laws of Resultant Error. When several independent sources of error conspire to pro- duce a resultant error, as specified by equation (1) of Art. 9, there is presented the problem of determining the law of the resultant error by means of the laws of the component errors. The algebraic statement of this problem is obtained as follows for the case of continuous errors : In the equation (1), Art. 9, write for brevity e=AQ, e,=^z/*, e^-^Ay,.' . . ; dx dy Art. 12.] laws of resultant error. 495 and let the laws of error of e, e t , <=„ . . . be denoted by 0(e), 0,(e,), 0,(e a ) . . . Then the value of e is given by «=*, + *,+•.. (i) The probabilities of the occurrence of any particular values of e„ e„, . . . are given by 0,( e .K e .. 0,O»K e a> • • • ; and the probability of their concurrence is the probability of the cor- responding value of e. But since this value may arise in an infinite number of ways through the variations of e,, e 2 , . . . over their ranges, the probability of e, or 1 (e l )de l i (e^de l . . . subject to the restriction (i). This latter gives e, = e— e 2 — e 3 . . ., and de 1 = de for the multiple integration with respect to e 3 , e„ . . . Hence there results (P(e)de = rfey 0,(6 - e, - e s - . . .)(f> t (e,)de, ..., or 0(0 = yW* -e, - e, - . . .) t {e t )de,f t (e,)de t ... (2) It is readily seen that this formula will increase rapidly in complexity \vrith the number of independent sources of error.* For some of the most important practical applications, how- ever, it suffices to limit equation (2) to the case of two inde- pendent sources of error, each of constant probability within assigned limits. Thus, to consider this case, let e, vary over the range — a to -\- a, and e 2 vary over the range — b to -\- b. Then by equation (2), Art. 10, 0,(6.) = i/(2«), 2 (e 3 ) = \/{2b). Hence equation (2) becomes ^ = ibf de >- In evaluating this integral e, must not surpass ± b and e, = e — e, must not surpass ± a. Assuming ay b, the limits of the integral for any value of e = e, -f- e, lying between — (a-\- b) and — (a — b) are — b and -(- (e + «)• This fact is * The reader desirous of pursuing this phase of the subject should consult Bessel's Untersuchungen ueber die Wahrscheinlichkeit der Beobachtungsfehler; Abhandlungen von Bessel (Leipzig, 1876), Vol. II. 496 PROBABILITY AND THEORY OF ERRORS. [CHAP. X. made plain by a numerical example. For instance, suppose a = 5 and b = 3. Then — (a + b) — — 8 and — (« — b) = - 3. Take e = — 6, a number intermediate to — 8 and — 3. Then the following are the possible integer values of e, and e which will produce e = — 6 : e e, e, limits of e a -6= -5 -1, -1 =+(e + «), = - 4 - 2, = -3-3. - 3 = - b. Similarly, the limits of e, for values of e lying between — (a — b) and + (a — b) are — b and + b ; and the limits of e, for values of e between -f- (a — b) and + {a -f- b) are + (e — a) and + b. Hence 0(e) = i/^ = f= S? f ° r -<«+*> < 6< ~ ('"'J' -b hf de <=3> i0r - {a -* )<6<+(a - d) ' -b ^bf^^r 1 for +<«-*><«<+<«+*>• *«>-£* 0(e) I (3) Thus it appears that in this case the graph of the resultant law of error is represented by the upper base and the two sides- of a trapezoid, the lower base being the noil , , axis of e and the line joining the middle mom . points of the bases being' the axis of d>(e). iiiioiiii f & w (See the first figure in Art. 13. ) The prop- iimoimi v & , J ' r r erties of u), including the determination imiioimii & of the limits, are also illustrated by the iimiiomim J adjacent trapezoid of numerals arranged imiiiioniiim J , . b to represent the case wherein a = 0.5 and b = 0.3. The vertical scale, or that for 0(e), does not, how- ever, conform exactly to that for e. Art. 13.] errors of interpolated values. 497 Prob. 20. Prove that the values of 0(e) as given by equation (3) satisfy the condition specified in equation (3), Art. 10. Prob. 21. Examine equations (3) for the case wherein a — b and b = o; and interpret for the latter case the first and last of (3). Prob. 22. Find from (3), and (1) of Art. 11, the probable error of the sum of two tabular logarithms. Art. 13. Errors of Interpolated Values. Case I. — One of the most instructive cases to which formulas (3) of Art. 12 are applicable is that of interpolated logarithms, trigonometric functions, etc., dependent on first differences. Thus, suppose that v x and w 2 are two tabular logarithms, and that it is required to get a value v lying t tenths of the interval from v x towards v v Evidently v =. v x + (y, — v x ) t = (1 - f)v x + tv t ; and hence if e, e lf e^ denote the actual errors of v, v lt z> 2 , re- spectively, e={i-t)e l + te t . (1) It is to be carefully noted here that e as given by (1) re- quires the retention in v of at least one decimal place be- yond the last tabular place. For example, let v = log (24373) from a 5-place table. Then ^=4.38686, z', = 4.38703, v t — v t — -j-0.00017, t = 0.3, and v =4.38691.1. Likewise, as found from a 7-place table, e x = — 0.45, e^ = -f- 0.37 in units of the fifth place; and hence by (1) *= —0.20. That is, the actual error of v = 4.38691. 1 is = 0.20, and this is verified by reference to a 7-place table. The reader is also cautioned against mistaking the species of interpolated values here considered for the species common- ly used by computers, namely, that in which the interpolated value is rounded to the nearest unit of the last tabular place. The latter species is discussed under Case II below. Confining attention now to the class of errors specified by equation (1), there result in the notation of the preceding article . e, = (1 — t)e x , e 2 = te v and e = e = e x + e 2 ; and since e and e^ each vary continuously between the limits 498 PROBABILITY AND THEORY OF ERRORS. [CHAP. X. ± 0.5 of a unit of the last tabular place, a and b in equations (3) of that article have the values a = 0.5(1 — t), b =0.5/. Hence the law of error of the interpolated values is ex- pressed as follows : cp(e) = — for values of e betw. —0.5 and —(0.5—/), for values of ebetw. — (0.5—^) and +(0.5—/), 1 - t — for values of e betw. +(0.5— () and +0.;. (1 - t)t r\ a ; t 3 (2) The graph of 0(e) for t = 1/3 is shown by the trapezoid AB, BC, CD in the figure on page 500. Evidently the equa- tions (2) are in general represented by a trapezoid, which degen- erates to an isosceles triangle when t = 1/2. The probable, mean, and average errors of an interpolated value of the kind in question are readily found from (2), and from equations (1) of Art. 11, to be e t = (l/4)(i ~t) for o < /? < 1/3/ = 1/2 - (i/2)V2*(i — t) for 1/3 < t < 2/3, = i/4t for 2/3 < t < 1. e -=i 96(1-/)/ \ - y (3) 1 - (1 - 2ty *a 7 a7~ for o < t < 1/2, 24(1 — t)t I - (2t - I) 3 for 1/2 < t < 1. 24(1 — f)t It is thus seen that the probable error of the interpolated value here considered decreases from 0.25 to 0.15 of a unit of the last tabular place as t increases from o to 0.5. Hence such values are more precise than tabular values ; and the computer who desires to secure the highest attainable precision with a given table of logarithms should retain one additional figure beyond the last tabular place in interpolated values. Art. 13.] errors of interpolated values. 499 Case II. — Recurring to the equation v = v t + t(v t — v,) for an interpolated value v in terms of two consecutive tabular values v t and v„ it will be observed that if the quantity t(v, — v t ) is rounded to the nearest unit of the last tabular place, a new error is introduced. For example, if v x — log 1633 = 3.21299, and v t = log 1634= 3.21325 from a 5-place table, z>, — », = + 26 units of the last tabular place ; and if t = 1/3, t(v 1 — v t ) = 8f ; so that by the method of interpolation in question there results v = 3.21299 -f 9 = 3.21308. Now the actual errors of z\ and ■v, are, as found from a 7-place table, — 0.38 and +0.21 in units of the fifth place. Hence the actual error of v is by equation 0)» I X — 0.38 + \ X + 0.21 — \ = — 0.52, as is shown di- rectly by a 7-place table. It appears, then, that in this case the error-equation cor- responding to (1) is e=(i-t)e l + te t +e t , (4) wherein e x and e^ are the same as in (1) and e,'\s the actual error that comes from rounding t(v t — &,) to the nearest unit of the last tabular place. The error e t , however, differs radically in kind from e x and j = 1 for - 1/6 < e < + 1/6, I (6) = (1/4X5 -6e) for + i/6« a Actual Error. I O.250 0.289 0.250 1/2 1/2 .292 .408 •333 I i/3 .256 •347 .287 5/6 i/4 .276 .382 ■313 1 M 5 .268 ■3/0 •303 9/10 1/6 .277 .385 •315 1 i/7 .274 .380 •3" 13/14 1/8 .279 •389 • 3>8 1 1/9 .278 .386 .316 17/18 i/io .281 •392 .320 1 When the interpolating factor t has the more general form m/n, wherein m and n are integers with no common factor, the possible values of e 3 are the same as for t = \/n. But equa- tions (3) of Art. 12 are not the same for t = m/n as for^ = \/n, and hence for the more general form of t, = (s/6)(l3/io ± e) for e between :p 1/10 and ^ 3/10, = (5/3)(4/5 ± e ) f° r e between =F 3/'° and T 7/ l °< = (5/6)(9/iO ± e) for e between =F 7/ 10 a °d T 9/iO. The graph of the right-hand half of A ^J> this function is shown in the accompany- ing diagram, the whole graph being symmetrical with respect to OA, or the axis of (e) — 2(1 — e), no regard being paid to the signs of the errors. The probable error is then found from 2 f(l ~ e)^e = h whence e P = 1 — \ V2 = 0.29. Similarly, the percentage of the whole number of errors which may be expected to lie, for example, between 0.0 and 0.2 in this system is 0.2 2 / (1 — e)de = 0.36. Using the same five hundred interpolated values cited above, but rounding them to the nearest unit of the last tabu- lar place and computing their actual errors by means of a 7-place table, the following comparison is afforded : T . . , _ Actual Theoretical Limits of Errors. Percentage. Percentage. 0.0 and 0.2 35-8 3 6 0.2 and 0.4 27.8 28 0.4 and 0.6 18.6 20 0.6 and 0.8 12.2 12 0.8 and 1.0 5- 6 4 O.o and 0.29 49-8 5° 506 PROBABILITY AND THEORY OF ERRORS. [Chap. X. The agreement shown here between the actual and theoretical percentages is somewhat closer than in the preceding case, the maximum discrepancy being only 1.6 and the average only 0.6 per cent. Finally, the following data derived from one thousand act- ual errors may be cited. The errors of one hundred inter- polated values rounded to the nearest unit of the last tabular place were computed * for each of the interpolating factors 0.1, 0.2, . . . 0.9. The averages of these several groups of act- ual errors are given along with the corresponding theoretical errors in the parallel columns below : Interpolating Actual Theoretical Factor. Average Error. Average Error. O. I O.338 O.32O 0.2 O.288 O.303 O.3 O.32I O.304 O.4 O.268 O.29O 0.5 O.324 O.333 0.6 0.276 0.290 0.7 0.321 0.304 0.8 0.289 0.303 0.9 0.347 0.320 The average discrepancy between the actual and theoret- ical values shown here is 0.017. It is, perhaps, somewhat smaller than should be expected, since the computation of the actual errors to three places of decimals is hardly warranted by the assumption of dependence on first differences only. The average of the whole number of actual errors in this case is 0.308, which agrees to the same number of decimals with the average of the theoretical errors, f * By Prof. H. A. Howe. See Annals of Mathematics, Vol. Ill, p. 74- The theoretical averages were furnished to Prof. Howe by the author, \ The reader who is acquainted with the elements of the method of least squares will find it instructive to apply that method to equation (1), Art. 13, and derive the probable error of e. This is frequently done without reserve by Art. 14.] statistical test of theory. 507 Prob. 25. Apply formulas (3) of Art. 12 to the case of the sum or difference of two tabular logarithms and derive the correspond- ing values of the probable, mean, and average errors. The graph of 0(e) is in this case an isosceles triangle whose base, or axis of e, is 2, and whose altitude, or axis of 0(e), is 1. those familiar with least squares. Thus, the probable error of e t or ei being 0.25, tne probable error of e is found to be 0.25 Vl — 2/ -\- 21 1 . This varies between 0.25 for /= o and 0.1S for t = £ ; while the true value of the probable error, as shown by equations (3), Art. 13, varies from 0.25 to 0.15 for the same values of t. It is, indeed, remarkable that the method of least squares, which admits infinite values for the actual errors ei and e^, should give so close an approximate formula as the above for the probable error of e. Similarly, one accustomed to the method of least squares would be inclined to apply it to equation (4), Art. 13, to determine the probable error of e. The natural blunder in this case is to consider e u ei , and e s independent, and e s like ei and ei continuous betweer the limits 0.0 and 0.5 ; and to assign a probable error of 0.25 to each. In t'..is manner the value 0.25^2(1 - t + t*) is derived. But this is absurd, since it gives 0.25 V2 instead of 0.25 for t = o. The formula fails then to give even approximate results except for values of t near 0.5. 5U8 HISTORY OF MODERN MATHEMATICS. [CHAP. XI Chapter XI. HISTORY OF MODERN MATHEMATICS. By David Eugene Smith, Professor of Mathematics in the Michigan State Normal School. Art. 1. Introduction. Modern Mathematics is a term by no means well denned. Algebra cannot be called modern, and yet the theory of equa- tions has received some of its most important additions during the nineteenth century, while the theory of forms is a recent creation. Similarly with elementary geometry; the labors of Lobachevsky and Bolyai during the second quarter of the century threw a new light upon the whole subject, and more recently the study of the triangle has added another chapter to the theory. Thus the history of modern mathematics must also be the modern history of ancient branches, while subjects which seem the product of late generations have root in other centuries than the present. How unsatisfactory must be so brief a sketch may be in- ferred from a glance at the Index du Repertoire Bibliographique des Sciences Mathematiques (Paris,' 1893), whose seventy-one pages contain the mere enumeration of subjects in large part modern, or from a consideration of the twenty-six volumes of the Jahrbuch fiber die Fortschritte der Mathematik, which now devotes over a thousand pages a year to a record of the pro- gress of the science.* The seventeenth and eighteenth centuries laid the founda- * The foot-notes give only a few of the authorities which might easily be cited. They are thought to include those from which considerable extracts have been made, the necessary condensation of these extracts making any other form of acknowledgment impossible. Art. 1.] INTRODUCTION. 509' tions of much of the subject as known to-day. The discovery of the analytic geometry by Descartes, the contributions to the theory of numbers by Fermat, to algebra by Harriot, to geometry and to mathematical physics by Pascal, and the discovery of the differential calculus by Newton and Leibniz, all contributed to make the seventeenth century memorable. The eighteenth century was naturally one of great activity. Euler and the Bernoulli family in Switzerland, d'Alembert, Lagrange, and Laplace in Paris, and Lambert in Germany, popularized Newton's great discovery, and extended both its theory and its applications. Accompanying this activity, how- ever, was a too implicit faith in the calculus and in the in- herited principles of mathematics, which left the foundations, insecure and necessitated their strengthening by the succeed- ing generation. The nineteenth century has been a period of intense study of first principles, of the recognition of necessary limitations of various branches, of a great spread of mathematical knowl- edge, and of the opening of extensive fields for applied mathe- matics. Especially influential has been the establishment of scientific schools and journals and university chairs. The great renaissance of geometry is not a little due to the founda- tion of the Ecole Polytechnique in Paris (1794-5), and the simi- lar schools in Prague (1806), Vienna (181 5), Berlin (1820), Karlsruhe (1825), and numerous other cities. About the mid- dle of the century these schools began to exert a still a greater influence through the custom of calling to them mathemati- cians of high repute, thus making Zurich, Karlsruhe, Munich, Dresden, and other cities well known as mathematical centers. In 1796 appeared the first number of the Journal de 1'Ecole Polytechnique. Crelle's Journal fur die reine und angewandte- Mathematik appeared in 1826, and ten years later Liouville began the publication of the Journal de Mathematiques pures et appliquees, which has been continued by Resal and Jordan.. The Cambridge Mathematical Journal was established in 1839,, and merged into the Cambridge and Dublin Mathematical!' 510 HISTORY OF MODERN MATHEMATICS. [Chap. XI. Journal in 1846. Of the other periodicals which have contrib- uted to the spread of mathematical knowledge, only a few can be mentioned : the Nouvelles Annales de Mathematiques (1842), Grunert's Archiv der Mathematik (1843), Tortolini's Annali di Scienze Matematiche e Fisiche (1850), Schlomilch's Zcitschrift fur Mathematik und Physik (1856), the Quarterly Journal of Mathematics (1857), Battaglini's Giornale di Mate- matiche (1863), the Mathematische Annalen (1869), the Bulle- tin des Sciences Mathematiques (1870), the American Jour- nal of Mathematics (1878), the Acta Mathematica (1882), and the Annals of Mathematics (1884).* To this list should be added a recent venture, unique in its aims, namely, L'lnter- mediaire des Mathematiciens (1894), and two annual publica- tions of great value, the Jahrbuch already mentioned (1868), and the Jahresbericht der deutschen Mathematiker-Vereini- gung (1892). To the influence of the schools and the journals must be added that of the various learned societies f whose published proceedings are widely known, together with the increasing liberality of such societies in the preparation of complete works of a monumental character. The study of first principles, already mentioned, was a nat- ural consequence of the reckless application of the new cal- culus and the Cartesian geometry during the eighteenth century. This development is seen in theorems relating to in- finite series, in the fundamental principles of number, rational, * For a list of current mathematical journals see the Jahrbuch iiber die Fort- schritte der Mathematik. A small but convenient list of standard periodicals is given in Carr's Synopsis of Pure Mathematics, p. 843 ; Mackay, J. S., Notice sur le journalisme mathematique en Angleterre, Association francaise pour l'Avancement des Sciences, 1893, II, 303 ; Cajori, F., Teaching and History of Mathematics in the United States, pp. 94, 277 ; Hart, D. S., History of Ameri- can Mathematical Periodicals, The Analyst, Vol. II, p. 131. f For a list of such societies consult any recent number of the Philosophical Transactions of Royal Society of London. Dyck, W., Einleitung zu dem fur den mathematischen Teil der deutschen Universitatsausstellung ausgegebenen Specialkatalog, Mathematical Papers Chicago Congress (New York, 1896), p. 44. Art. 2.] theory of numbers. 5] 1 irrational, and complex, and in the concepts of limit, conti- unity, function, the infinite, and the infinitesimal. But the nineteenth century has done more than this. It has created new and extensive branches of an importance which promises much for pure and applied mathematics. Foremost among these branches stands the theory of functions founded by Cauchy, Riemann, and Weierstrass, followed by the descrip- tive and projective geometries, and the theories of groups, of forms, and of determinants. The nineteenth century has naturally been one of specializ- ation. At its opening one might have hoped to fairly compass the mathematical, physical, and astronomical sciences, as did Lagrange, Laplace, and Gauss. But the advent of the new generation, with Monge and Carnot, Poncelet and Steiner, Galois, Abel, and Jacobi, tended to split mathematics into branches between which the relations were long to remain ob- scure. In this respect recent years have seen a reaction, the unifying tendency again becoming prominent through the theories of functions and groups.* Art. 2. Theory of Numbers. The Theory of Numbers, f a favorite study among the Greeks, had its renaissance in the sixteenth and seventeenth centuries in the labors of Viete, Bachet de Meziriac, and es- pecially Fermat. In the eighteenth century Euler and Lagrange contributed to the theory, and at its close the sub- ject began to take scientific form through the great labors of Legendre (1798), and Gauss (1801). With the latter's Disquisi- tiones Arithmetical (1801) may be said to begin the modern theory of numbers. This theory separates into two branches, the one dealing with integers, and concerning itself especially * Klein, F., The Present State of Mathematics, Mathematical Papers of Chicago Congress (New York, 1896), p. 133- f Cantor, M., Geschichte der Mathematik, Vol. Ill, p. 94; Smith, H. J. S., Report on the theory of numbers; Collected Papers, Vol. I; Stolz, O., Gros- sen und Zahlen, Leipzig, 1891. 512 HISTORY OF MODERN MATHEMATICS. [Chap. XL with (i) the study of primes, of congruences, and of residues, and in particular with the law of reciprocity, and (2) the theory of forms, and the other dealing with complex numbers. The Theory of Primes* has attracted many investigators during the nineteenth century, but the results have been de- tailed rather than general. Tchebichef (1850) was the first to reach any valuable conclusions in the way of ascertaining the number of primes between two given limits. Riemann (1859) also gave a well-known formula for the limit of the number of primes not exceeding a given number. The Theory of Congruences may be said to start with Gauss's Disquisitiones. He introduced the symbolism a=b (mod c), and explored most of the field. Tchebichef published in 1847 a work in Russian upon the subject, and in France Serret has done much to make the theory known. Besides summarizing the labors of his predecessors in the theory of numbers, and adding many original and noteworthy contributions, to Legendre may be assigned the fundamental theorem which bears his name, the Law of Reciprocity of Quad- ratic Residues. This law, discovered by induction by Euler, was enunciated by Legendre and first proved in his Theorie des Nombres (1798) for special cases. Independently of Euler and Legendre, Gauss discovered the law about 1795, and was the first to give a general proof. To the subject have also contributed Cauchy, perhaps the most versatile of French mathematicians of the century; Dirichlet, whose Vorlesungen uber Zahlentheorie, edited by Dedekind, is a classic ; Jacobi,. who introduced the generalized symbol which bears his name ; Liouville, Zeller, Eisenstein, Kummer, and Kronecker. The theory has been extended to include cubic and biquadratic reciprocity, notably by Gauss, by Jacobi, who first proved the law of cubic reciprocity, and by Kummer. * Brocard, H., Sur la frequence et la totalite des nombres premiers; Nou- velle Correspondence de Mathematiques, Vols. Vand VI; gives ecent history to- 1879. Art. 3.] IRRATIONAL AND TRANSCENDENT NUMBERS. 513 To Gauss is also due the representation of numbers by binary quadratic forms. Cauchy, Poinsot (1845), Lebesgues (1859, 1868), and notably Hermite have added to the subject. In the theory of ternary forms Eisenstein has been a leader, and to him and H. J. S. Smith is also due a noteworthy ad- vance in the theory of forms in general. Smith gave a com- plete classification of ternary quadratic forms, and extended Gauss's researches concerning real quadratic forms to complex forms. The investigations concerning the representation of numbers by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and the theory was completed by Smith. In Germany, Dirichlet was one of the most zealous workers in the theory of numbers, and was the first to lecture upon the subject in a German university. Among his contributions is the extension of Fermat's theorem on x n -\-y n = z", which Euler and Legendre had proved for n = 3, 4, Dirichlet showing that x h -\- y" ^£ ag" . Among the later French writers are Borel ; Poincare, whose memoirs are numerous and valuable ; Tannery, and Stieltjes. Among the leading contributors in Germany are Kronecker, Kummer, Schering. Bachmann, and Dedekind. In Austria Stolz's Vorlesungen uber allgemeine Arithmetik (1885-86), and in England Mathews' Theory of Numbers (Part I, 1892) are among the most scholarly of general works. Genocchi, Sylvester, and J. W. L. Glaisher have also added to the theory. Art. 3. Irrational and Transcendent Numbers. The sixteenth century saw the final acceptance of negative numbers, integral and fractional. The seventeenth century saw decimal fractions with the modern notation quite generally used by mathematicians. The next hundred years saw the imaginary become a powerful tool in the hands of De Moivre, and especially of Euler. For the nineteenth century it re- mained to complete the theory of complex numbers, to separate irrationals into algebraic and transcendent, to prove the exist- ence of transcendent numbers, and to make a scientific study 514 HISTORY OF MODERN MATHEMATICS. [Chap. XI. of a subject which had remained almost dormant since Euclid, the theory of irrationals. The year 1872 saw the publication of the theories of Weierstrass (by his pupil Kossak), Heine (Crelle, 74), G. Cantor (Annalen, 5), and Dedekind. Merayhad taken in 1869 the same point of departure 'as Heine, but the theory is generally referred to the year 1872. Weierstrass's method has been completely set forth by Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and the recent indorsement by Tan- nery (1894). Weierstrass, Cantor, and Heine base their the- ories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Meray. Continued Fractions, closely related to irrational numbers (and due to Cataldi, 16 13),* received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Lagrange. Other noteworthy contributions have been made by Drucken- miiller (1837), Kunze (1857), Lemke (1870), and Gunther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Mobius, and Gunther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject. Transcendent Numbers f were first distinguished from alge- braic irrationals by Kronecker. Lambert proved (1761) that n cannot be rational, and that e w (n being zero or rational) is irrational, a proof, however, which left much to be desired. *But see Favaro, A'., Notizie storiche sulle frazioni continue dal secolo deci- moterzo ai decimosettimo, Boncompagni's Bulletino, Vol. VII, 1874, pp. 451, 533- f Klein, F., Vortrage tibtr ausgewahlte Fragen der Elementargeometrie, 1895, p. 38 ; Bachman, P., Vorlesungen iiber die Natur der Irrationalzahlen, l8q2. -Art. 4.] complex numbers. 515 Legendre (1794) completed Lambert's proof, and showed that n is not the square root of a rational number. Liouville (1840) showed that neither e nor e 1 can be a root of an integral quadratic equation. But the existence of transcendent numbers was first established by Liouville (1844, 185 1), the proof being subse- quently displaced by G. Cantor's (1873). Hermite (1873) first proved e transcendent, and Lindemann (1882), starting from Hermite's conclusions, showed the same for n. Lindemann's proof was much simplified by Weierstrass (1885), still further by Hilbert (1893), and has finally been made elementary by Hurwitz and Gordan. Art. 4. Complex Numbers. The Theory of Complex Numbers* may be said to have attracted attention as early as the sixteenth century in the recognition, by the Italian algebraists, of imaginary or impos- sible roots. In the seventeenth century Descartes distin- guished between real and imaginary roots, and the eighteenth saw the labors of De Moivre and Euler. To De Moivre is due (1730) the well-known formula which bears h is name, (cos -f- i sin -f- i sin «0, and to Euler (1748) the formula cos -(- z'sin = **'. The geometric notion of complex quantity now arose, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus. In the eighteenth century Kuhn (1750) and Wessel (about 1795) made decided advances towards the present theory. Wessel's memoir appeared in the Proceed- ings of the Copenhagen Academy for 1799, and is exceedingly *Riecke, F., Die Rechnung mit Richtungszahlen, 1856, p. 161 ; Hankel, H., Theorie derkomplexen Zahlensysteme, Leipzig, 1867 ; Holzmuller, G., Theorie der isogonalen Verwandtschaften, 1882, p. 21; Macfarlane, A., The Imaginary of Algebra, Proceedings of American Association 1892, p. 33 ; Baitzer, R., Einfiihrung der komplexen Zahlen, Crelle, 1882 ; Stolz, O., Vorlesungen uber allgemeine Arithmetik, 2. Theil, Leipzig, 1886. 516 HISTORY OF MODERN MATHEMATICS. [Chap. XI, clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbe Buee independently came upon the same idea which Wallis had suggested, that ± \/ — 1 should repre- sent a unit line, and its negative, perpendicular to the real axis. Buee's paper was not published until 1 806, in which year Ar- gand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Men- tion should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of di- rectional numbers are scientifically laid. The general accept- ance of the theory is not a little due to the labors of Cauchy and Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known. The common terms used in the theory are chiefly due to the founders. Argand called cos

-{- i s\n cp the " reduced form"(l'expression reduite); Gauss used i for V — 1, introduced the term " complex number " for a -\- bi, and called a* + V the " norm." The expression "direction coefficient ", often used for cos

a real angle, i, J, or k should be any directed unit line such that i' =_/ 3 = k" = — i. It also necessitated a withdrawal of the commutative law of multiplication, the adherence to which obstructed earlier dis- covery. It was not until 1853 that Hamilton's Lectures on Quarternions appeared, followed (1866) by his Elements of Quaternions. In the same year in which Hamilton published his discov- ery (1844), Grassmann gave to the world his famous work, Die lineale Ausdehnungslehre, although he seems to have been in possession of the theory as early as 1840. Differing from Hamilton's Quaternions in many features, there are several essential principles held in common which each writer discovered independently of the other.* Following Hamilton, there have appeared in Great Britain numerous papers and works by Tait (1867), Kelland and Tait (1873), Sylvester, and McAulay (1893). On the Continent Hankel (1867), Hoiiel (1874), and Laisant (1877, 1881) have written on the theory, but it has attracted relatively little attention. In America, Benjamin Peirce (1870) has been especially prominent in developing the quaternion theory, and Hardy (1881) Macfarlane, and Hathaway (1896) have con- tributed to the subject. The difficulties have been largely in the notation. In attempting to improve this symbolism Macfar- lane has aimed at showing how a space analysis can be de- veloped embracing algebra, trigonometry, complex numbers, Grassmann's method, and quaternions, and has considered the general principles, of vector and versor analysis, the versor being circular, elliptic logarithmic, or hyperbolic. Other recent contributors to the algebra of vectors are Gibbs (from 1881) and Heaviside (from 1885). The followers of Grassmann f have not been much more * These are set forth in a paper by J. W. Gibbs, Nature, Vol. XLIV, p. 79. \ For bibliography see Schlegel, V., Die Grassmann'sche Ausdehnungs- lehre, SchlSmilch's Zeitschrift, Vol. XLI. Art. 6.] theory of equations. 519 numerous than those of Hamilton. Schlegel has been one of the chief contributors in Germany, and Peano in Italy. In America, Hyde (Directional Calculus, 1890) has made a plea for the Grassmann theory.* Along lines analogous to those of Hamilton and Grassmann have been the contributions of Scheffler. While the two former sacrificed the commutative law, Scheffler (1846, 1851, 1880) sacrificed the distributive. This sacrifice of fundamental laws has led to an investigation of the field in which these laws are valid, an investigation to which Grassmann (1872), Cayley, Ellis, Boole, Schroder (1890-91), and Kraft (1893) have contributed. Another great contribution of Cayley's along similar lines is the theory of matrices (1858). Art. 6. Theory of Equations. The Theory of Numerical Equations f concerns itself first with the location of the roots, and then with their approxi- mation. Neither problem is new, but the first noteworthy contribution to the former in the nineteenth century was Budan's (1807). Fourier's work was undertaken at about the same time, but appeared posthumously in 1831. All processes were, however, exceedingly cumbersome until Sturm (1829) communicated to the French Academy the famous theorem which bears his name and which constitutes one of the most brilliant discoveries of algebraic analysis. The Approximation of the Roots, once they are located, can be made by several processes. Newton (171 1), for example, gave a method which Fourier perfected; and Lagrange (1767) discovered an ingenious way of expressing the root as a con- tinued fraction, a process which Vincent (1836) elaborated. It * For Macfarlane's Digest of views of English and American writers, see Proceedings American Association for Advancement of Science, 1891. f Cayley, A., Equations, and Kelland. P., Algebra, in Encyclopaedia Bri- tannica; Favaro, A., Notizie storico-critiche sulla costruzione delle equazioni. Modena, 1878; Cantor, M., Geschichte der Mathematik, Vol. Ill, p. 375. 5JW HISTORY OF MODERN MATHEMATICS. [CHAP. XI. was, however, reserved for Horner (1819) to suggest the most practical method yet known, the one now commonly used. With Horner and Sturm this branch practically closes. The calculation of the imaginary roots by approximation is still an open field. The Fundamental Theorem* that every numerical equation has a root was generally assumed until the latter part of the eighteenth century. D'AIembert (1746) gave a demonstration, as did Lagrange (1772), Laplace (1795), Gauss (1799) and Argand (1806). The general theorem that every algebraic equation of the nth degree has exactly n roots and no more follows as a special case of Cauchy's proposition (1831) as to the number of roots within a given contour. Proofs are also due to Gauss, Serret, Clifford (1876), Malet (1878), and many others. The Impossibility of Expressing the Roots of an equation as algebraic functions of the coefficients when the degree ex- ceeds 4 was anticipated by Gauss and announced by Ruffini, and the belief in the fact became strengthened by the failure of Lagrange's methods for these cases. But the first strict proof is due to Abel, whose early death cut short his labors in this and other fields. The Quintic Equation has naturally been an object of special study. Lagrange showed that its solution depends on that of a sextic, " Lagrange's resolvent sextic," and Malfatti and Vandermonde investigated the construction of resolvents. The resolvent sextic was somewhat simplified by Cockle and Harley (1858-59) and by Cayley (1861), but Kronecker (1858) was the first to establish a resolvent by which a real simplifi- cation was effected. The transformation of the general quintic into the trinomial form x 6 -\- ax -\- b = o by the extraction of square and cube roots only, was first shown to be possible by * Loria, Gino, Esame di alcune ricerche concernenti l'esistenza di radici nelle equazioni algebriche; Bibliotheca Mathematica, 1891, p. 99; bibliography on p. 107. Pierpont, J., On the Ruffini-Abelian theorem, Bulletin of American Mathematical Society, Vol. II, p. 200. -Art. 6.J theory of equations. 521 Bring (1786) and independently by Jerrard * (1834). Hermite (1858) actually effected this reduction, by means of Tschirn- hausen's theorem, in connection with his solution by elliptic functions. The Modern Theory of Equations may be said to date from Abel and Galois. The latter's special memoir on the subject, not published until 1846, fifteen years after his death, placed the theory on a definite base. To him is due the discovery that to each equation corresponds a group of substitutions (the "group of the equation ") in which are reflected its essen- tial characteristics.! Galois's untimely death left without suffi- cient _ demonstration several important propositions, a gap which Betti (1852) has filled. Jordan, Hermite, and Kronecker were also among the earlier ones to add to the theory. Just prior to Galois's researches Abel (1824), proceeding from the fact that a rational function of five letters having less than five values cannot have more than two, showed that the roots of a general quintic equation cannot be expressed in terms of its coefficients by means of radicals. He then investigated special forms of quintic equations which admit of solution by the extraction of a finite number of roots. Hermite, Sylves- ter, and Brioschi have applied the invariant theory of binary forms to the same subject. From the point of view of the group the solution by radi- cals, formerly the goal of the algebraist, now appears as a single link in a long chain of questions relative to the transfor- mation of irrationals and to their classification. Klein (1884) has handled the whole subject of the quintic equation in a simple manner by introducing the icosahedron equation as the normal form, and has shown that the method can be general- ized so as to embrace the whole theory of higher equations.^: He and Gordan (from 1879) have attacked those equations of * Harley, R., A contribution of the history ... of the general equation •of the fifth degree, Quarterly Journal of Mathematics, Vol. VI, p. 38. f See Art. 7. t Klein, F., Vorlesungen iiber das Ikosaeder, 1884. 522 HISTORY OF MODERN MATHEMATICS. [Chap. XI. the sixth and seventh degrees which have a Galois group of 168 substitutions, Gordan performing the reduction of the equation of the seventh degree to the ternary problem. Klein (1888) has shown that the equation of the twenty-seventh degree occurring in the theory of cubic surfaces can be re- duced to a normal problem in four variables, and Burkhardt (1893) has performed the reduction, the quaternary groups in- volved having been discussed by Maschke (from 1887). Thus the attempts to solve the quintic equation by means of radicals has given place to their treatment by transcendents. Hermite (1858) has shown the possibility of the solution, by the- use of elliptic functions, of any Bring quintic, and hence of any equation of the fifth degree. Kronecker (1858), working from a different standpoint, has reached the same results, and his method has since been simplified by Brioschi. More recently Kronecker, Gordan, Kiepert, and Klein, have contributed to the same subject, and the sextic equation has been attacked by Maschke and Brioschi through the medium of hyperelliptic functions. Binomial Equations, reducible to the form x n — 1 = o r admit of ready solution by the familiar trigonometric formula 2kn . 2kn . x = cos + ?sin — ; but it was reserved for Gauss (1801Y n ■ ' n to show that an algebraic solution is possible. Lagrange (1808) extended the theory, and its application to geometry is- one of the leading additions of the century. Abel, generaliz- ing Gauss's results, contributed the important theorem that if two roots of an irreducible equation are so connected that the one can be expressed rationally in terms of the other, the equa- tion yields to radicals if the degree is prime and otherwise depends on the solution of lower equations. The binomial n—i equation, or rather the equation ^2 x m = o, is one of this class considered by Abel, and hence called (by Kronecker) Abelian Equations. The binomial equation has been treated notably by Richelot (1832), Jacobi (1837), Eisenstein (1844, 1850), Cay- Art. 6.] theory of equations. 523 ley (185 1), and Kronecker (1854), and is the subject of a treatise by Bachmann (1872). Among the most recent writers on Abelian equations is Pellet (1891). Certain special equations of importance in geometry have been the subject of study by Hesse, Steiner, Cayley, Clebsch, Salmon, and Kummer. Such are equations of the ninth degree determining the points of inflection of a curve of the third de- gree, and of the twenty-seventh degree determining the points in which a curve of the third degree can have contact of the fifth order with a conic. Symmetric Functions of the coefficients, and those which re- main unchanged through some or all of the permutations of the roots, are subjects of great importance in the present theory. The first formulas for the computation of the symmetric func- tions of the roots of an equation seem to have been worked out by Newton, although Girard (1629) had given, without proof, a formula for the power sum. In the eighteenth century Lagrange (1768) and Waring (1770, 1782) contributed to the theory, but the first tables, reaching to the tenth degree, appeared in 1809 in the Meyer- Hirsch Aufgabensammlung. In Cauchy's cele- brated memoir on determinants (1812) the subject began to assume new prominence, and both he and Gauss (1816) made numerous and valuable contributions to the theory. It is, how- ever, since the discoveries by Galois that the subject has be- come one of great importance. Cayley (1857) nas given sim- ple rules for the degree and weight of symmetric functions, and he and Brioschi have simplified the computation of tables. Methods of Elimination and of finding the resultant (Bezout) or eliminant (De Morgan) occupied a number of eighteenth-century algebraists, prominent among them being Euler (1748), whose method, based on symmetric functions, was improved by Cramer (1750) and Bezout (1764). The leading steps in the development are represented by Lagrange (1770-71), Jacobi, Sylvester (1840), Cayley (1848, 1857), Hesse (1843, 1859), Bruno (1859), and Katter (1876). Sylvester's dialytic method appeared in 1841, and to him is also due (1851) the 524 HISTORY OF MODERN MATHEMATICS. [CHAP. XI. name and a portion of the theory of the discriminant. Among recent writers on the general theory may be mentioned Burn- side and Pellet (from 1887). Art. 7. Substitutions and Groups. The Theories of Substitutions and Groups* are among the most important in the whole mathematical field, the study of groups and the search for invariants now occupying the atten- tion of all mathematicians. The first recognition of the im- portance of the combinatory analysis occurs in the problem of forming an ?«th-degree equation having for roots in of the roots of a given rath-degree equation (in < n). For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a bi- quadratic expression necessarily leads to a sextic equation, and Le Sceur (1748) and Waring (1762 to 1782) still further elabo- rated the idea. Lagrangef first undertook a scientific treatment of the the- ory of substitutions. Prior to his time the various methods of solving lower equations had existed rather as isolated artifices than as a unified theory.^; Through the great power of analy- sis possessed by Lagrange (1770, 1771) a common foundation was discovered, and on this was built the theory of substitu- tions. He undertook to examine the methods then known, and to show a priori why these succeeded below the quintic, but otherwise failed. In his investigation he discovered the important fact that the roots of all resolvents (r£solvantes, re- duites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a " Calcul des Combinaisons," the first * Netto, E., Theory of Substitutions, translated by Cole; Cayley, A., Equa- tions, Encyclopaedia Britannica, gth edition. f Pierpont, James, Lagrange's Place in the Theory of Substitutions, Bulletin of American Mathematical Society, Vol. I, p. 196. J Matthiessen, L., Grundzuge der antiken und modernen Algebra der Iittera- len Gleichungen, Leipzig, 1878. Art. 7.] substitutions and groups. 525 important step towards a theory of substitutions. Mention should also be made of the contemporary labors of Vander- monde (1770) as foreshadowing the coming theory. The next great step was taken by Ruffini* (1799). Begin- ning like Lagrange with a discussion of the methods of solving lower equations, he attempted the proof of the impossibility of solving the quintic and higher equations. While the attempt failed, it is noteworthy in that it opens with the classification of the various "permutations" of the coefficients, using the word to mean what Cauchy calls a "systeme des substitutions conjuguees," or simply a " systeme conjugue," and Galois calls a " group of substitutions." Ruffini distinguishes what are now called intransitive, transitive and imprimitive, and transitive and primitive groups, and (1801) freely uses the group of an equation under the name "l'assieme della permutazioni." He also publishes a letter from Abbati to himself, in which the group idea is prominent. To Galois, however, the honor of establishing the theory of groups is generally awarded. He found that if r lt r it . . . r„ are the n roots of an equation, there is always a group of permuta- tions of the r's such that (1) every function of the roots invari- able by the substitutions of the group is rationally known, and (2), reciprocally, every rationally determinable function of the roots is invariable by the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on the group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI). Cayley and Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The popularizing of the sub- ject is largely due to Serret, who has devoted section IV of his *Burkhardt, H., Die Anfange der Gruppentheorie und Paolo Ruffini, Ab- handlungen zur Geschichte der Mathematik, VI, 1892, p. 119. Italian by E. Pas- cal, Brioschi's Annali di Matematici, 1894. 526 - HISTORY OF MODERN MATHEMATICS. |ChAP. XI. algebra to the theory ; to Camille Jordan, whose Traite des Substitutions is a classic ; and to Netto (1882), whose work has been translated into English by Cole (1892). Bertrand, Her- mite, Frobenius, Kronecker, and Mathieu have added to the theory. The general problem to determine the number of groups of n given letters still awaits solution. But overshadowing all others in recent years in carrying on the labors of Galois and his followers in the study of discontin- uous groups stand Klein, Lie, Poincare, and Picard. Besides these discontinuous groups there are other classes, one of which, that of finite continuous groups, is especially important in the theory of differential equations. It is this class which Lie (from 1884) has studied, creating the most important of the recent departments of mathematics, the theory of trans- formation groups. Of value, too, have been the labors of Killing on the structure of groups, Study's application of the group theory to complex numbers, and the work of Schur and Maurer. Art. 8. Determinants. The Theory of Determinants* may be said to take its origin with Leibniz (1693), following whom Cramer (1750) added slightly to the theory, treating the subject, as did his predecessor, wholly in relation to sets of equations. The re- current law was first announced by Bezout (1764). But it was Vandermonde (1771) who first recognized determinants as inde- pendent functions. To him is due the first connected exposi- tion of the theory, and he may be called its formal founder. Laplace (1772) gave the general method of expanding a deter- minant in terms of its complementary minors, although Van- dermonde had already given a special case. Immediately fol- lowing, Lagrange (1773) treated determinants of the second * Muir, T., Theory of Determinants in the Historical Order of its Develop- ment, Part I, 1890; Baltzer, R., Theorie und Anwendung der Determinanten, 1881. The writer is under obligations to Professor Weld, who contributes Chap. II, for valuable assistance in compiling this article. ART. fci.] DETERMINANTS. 527 and third order, possibly stopping here because the idea of hyperspace was not then in vogue. Although contributing nothing to the general theory, Lagrange was the first to apply determinants to questions foreign to eliminations, and to him are due many special identities which have since been brought under well-known theorems. During the next quarter of a century little of importance was done. Hindenburg (1784) and Rothe (1800) kept the subject open, but Gauss (1S01) made the next advance. Like Lagrange, he made much use of de- terminants in the theory of numbers. He introduced the word "determinants" (Laplace had used "resultant "), though not in the present signification,* but rather as applied to the dis- criminant of a quantic. Gauss also arrived at the notion of reciprocal determinants, and came very near the multiplication theorem. The next contributor of importance is Binet (181 1, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of m = n reduces to the multiplication theorem. On the same day (Nov. 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. In this he used the word " determinant " in its present sense, summa- rized and simplified what was then known on the subject, im- proved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's. He was the first to grasp the subject as a whole ; before him there were determi- nants, with him begins their theory in its generality. The next great contributor, and the greatest save Cauchy, was Jacobi (from 1827). With him the word "determinant" received its final acceptance. He early used the functional determinant which Sylvester has called the " Jacobian," and in his famous memoirs in Crelle for 1841 he specially treats this subject, as well as that class of alternating functions which Sylvester has called "Alternants." But about the time of Jacobi's closing memoirs, Sylvester (1839) an d Cayley began * "Numerum bb-ac, cuius indole proprietates formae (a, b, c) imprimis pen- dere in sequenti^is docebimus, determinantem huius uocabimus." 528 HISTORY OF MODERN MATHEMATICS. [Chap. XI. their great work, a work which it is impossible to briefly sum- marize, but which represents the development of the theory to the present time. The study of special forms of determinants has been the natural result of the completion of the general theory. Axi- symmetric determinants have been studied by Lebesgue, Hesse, and Sylvester ; per-symmetric determinants by Sylvester and Hankel ; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley ; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius ; compound determinants by Sylvester, Reiss, and Picquet ; Jacobians and Hessians by Sylvester ; and sym- metric gauche determinants by Trudi. Of the text-books on the subject Spottiswoode's was the first. In America, Hanus (1886) and Weld (1893) have published treatises. Art. 9. Quantics. The Theory of Quantics or Forms * appeared in embryo in the Berlin memoirs of Lagrange (1773, 1775). who considered binary quadratic forms of the type ax* -\- bxy -\- cy 2 , and estab- lished the invariance of the discriminant of that type when x -\- Xy is put for x. He classified forms of that type accord- ing to the sign of b 2 — \ac, and introduced the ideas of trans- formation and equivalence. Gaussf (1801) next took up the subject, proved the invariance of the discriminants of binary and ternary quadratic forms, and systematized the theory of binary quadratic forms, a subject elaborated by H. J. S. Smith, Eisenstein, Dirichlet, Lipschitz, Poincare, and Cayley. Galois also entered the field, in his theory of groups (1829), and * Meyer, W. F., Bericht Uber den gegenwSrtigen Stand der Invarianten- theorie. jahresbericht der deutschen Mathematiker-Vereinigung, Vol. I, 1890-91; Berlin 1892, p. 97. See also the review by Franklin in Bulletin New York Mathematical Society, Vol. Ill, p. 187 ; Biography of Cayley, Collected Papers, VIII, p. ix, and Proceedings of Royal Society, 1895. f See Art. 2. Art. 9.] quantics. 529 the first step towards the establishment of the distinct theory is sometimes attributed to Hesse in his investigations of the plane curve of the third order. It is, however, to Boole (1841) that the real foundation of the theory of invariants is generally ascribed. He first showed the generality of the invariant property of the discriminant, which Lagrange and Gauss had found for special forms. Inspired by Boole's discovery Cayley took up the study in a memoir "On the Theory of Linear Transformations " (1845), which was followed (1846) by investigations concerning co- variants and by the discovery of the symbolic method of find- ing invariants. By reason of these discoveries concerning invariants and covariants (which at first he called " hyperdeter- minants ") he is regarded as the founder of what is variously called Modern Algebra, Theory of Forms, Theory of Quantics, and the Theory of Invariants and Covariants. His ten memoirs on the subject began in 1854, and rank among the greatest which have ever been produced upon a single theory. Syl- vester soon joined Cayley in this work, and his originality and vigor in discovery soon made both himself and the subject prominent. To him are due (1851-54) the foundations of the general theory, upon which later writers have largely built, as well as most of the terminology of the subject. Meanwhile in Germany Eisenstein (1843) had become aware of the simplest invariants and covariants of a cubic and bi- quadratic form, and Hesse and Grassmann had both (1844) touched upon the subject. But it was Aronhold (1849) wri ° first made the new theory known. He devised the symbolic method now common in Germany, discovered the invariants of a ternary cubic and their relations to the discriminant, and, with Cayley and Sylvester, studied those differential equations which are satisfied by invariants and covariants of binary quantics. His symbolic method has been carried on by Clebsch, Gordan, and more recently by Study (1889) and Stroh (1890), in lines quite different from those of the English school. In France Hermite early took up the work (185 1). He 530 HISTORY OF MODERN MATHEMATICS. [CHAP. XI. discovered (1854) the law of reciprocity that to every covariant or invariant of degree p and order r of a form of the mth. order corresponds also a covariant or invariant of degree m and of order r of a form of the pth order. At the same time (1854) Brioschi joined the movement, and his contributions have been among the most valuable. Salmon's Higher Plane Curves (1852) and Higher Algebra (1859) should also be men- tioned as marking an epoch in the theory. Gordan entered the field, as a critic of Cayley, in 1868. He added greatly to the theory, especially by his theorem on the Endlichkeit des Formensystems, the proof for which has since been simplified. This theory of the finiteness of the number of invariants and covariants of a binary form has since been extended by Peano (1882), Hilbert (1884), and Mertens (1886). Hilbert (1890) succeeded in showing the finiteness of the com- plete systems for forms in n variables, a proof which Story has simplified. Clebsch * did more than any other to introduce into Ger- many the work of Cayley and Sylvester, interpreting the pro- jective geometry by their theory of invariants, and correlating it with Riemann's theory of functions. Especially since the publication of his work on forms (1871) the subject has at- tracted such scholars as Weierstrass, Kronecker, Mansion, Noether, Hilbert, Klein, Lie, Beltrami, Burkhardt, and many others. On binary forms Faa di Bruno's work is well known, as is Study's (1889) on ternary forms. De Toledo (1889) and Elliott (1895) have published treatises on the subject. Dublin University has also furnished a considerable corps of contributors, among whom MacCullagh, Hamilton, Salmon, Michael and Ralph Roberts, and Burnside may be especially mentioned. Burnside, who wrote the latter part of Burnside and Panton's Theory of Equations, has set forth a method of transformation which is fertile in geometric interpretation and binds together binary and certain ternary forms. * Klein's Evanston Lectures, Lect. I. Art. 10.] calculus. 531 The equivalence problem of quadratic and bilinear forms has attracted the attention of Weierstrass, Kronecker, Chris- toffel, Frobenius, Lie, and more recently of Rosenow (Crelle, 1 08), Werner (1889), Killing (1890), and Scheffers (1891). The equivalence problem of non-quadratic forms has been studied by ChristofTel. Schwarz (1872), Fuchs (1875-76), Klein (1877, 1884), Brioschi (1877), and Maschke (1887) have contributed to the theory of forms with linear transformations into them- selves. Cayley (especially from 1870) and Sylvester (1877) have worked out the methods of denumeration by means of generating functions. Differential invariants have been studied by Sylvester, MacMahon, and Hammond. Starting from the differential invariant, which Cayley has termed the Schwarzian derivative, Sylvester (1885) has founded the theory of recipro- cants, to which MacMahon, Hammond, Leudesdorf, Elliott, Forsyth, and Halphen have contributed. Canonical forms have been studied by Sylvester (185 1), Cayley, and Hermite (to whom the term " canonical form " is due), and more recently by Rosanes (1873), Brill (1882), Gundelfinger (1883), and Hil- bert (1886). The Geometric Theory of Binary Forms may be traced to Poncelet and his followers. But the modern treatment has its origin in connection with the theory of elliptic modular func- tions, and dates from Dedekind's letter to Borchardt (Crelle, 1877). The names of Klein and Hurwitz are prominent in this connection. On the method of nets (reseaux), another geometric treatment of binary quadratic forms Gauss (1831), Dirichlet (1850), and Poincare (1880) have written. Art. 10. Calculus. The Differential and Integral Calculus,* dating from New- ton and Leibniz, was quite complete in its general range at * Williamson, B., Infinitesimal Calculus, Encyclopaedia Britannica, 9th edi- tion; Cantor, M., Geschichte der Mathematik, Vol. Ill, pp. 150-316; Vivanti, G., Note sur l'histoire de 1'infiniment petit, Bibliotheca Mathematica, 1894, p. 1 ; Mansion, P., Esquisse de l'histoire du calcul infinitesimal, Ghent, 1887. Le 532 HISTORY OF MODERN MATHEMATICS. [Chap. XI. the close of the eighteenth century. Aside from the study of first principles, to which Gauss, Cauchy, Jordan, Picard, Meray, l and those whose names are mentioned in connection with the theory of functions, have contributed, there must be men- tioned the development of symbolic methods, the theory of definite integrals, the calculus of variations, the theory of dif- ferential equations, and the numerous applications of the Newtonian calculus to physical problems. Among those who have prepared noteworthy general treatises are Cauchy (1821), Raabe (1839-47), Duhamel (1856), Sturm (1857-59), Bertrand (1864), Serret(i868), Jordan (2d ed., 1893), and Picard (1891-93). A recent contribution to analysis which promises to be valu- able is Oltramare's Calcul de Generalization (1893). Abel seems to have been the first to consider in a general way the question as to what differential expressions can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the century. Frullani's theorem (1821), Bierens de Haan's work on the theory (1862) and his elaborate tables (1867), Dirichlet's lectures (1858) era- bodied in Meyer's treatise (1871), and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlomilch, Elliott, Leudesdorf, and Kronecker are among the noteworthy con- tributions. Eulerian Integrals were first studied by Euler and after- wards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows: x"-'(i — xf~ l dx, J e- x x"-'dx, although these were not the exact forms of Euler's study. If n is integral, it follows that / e~*x n l dx = n !, but if n is fractional it is a transcendent function. To it Legendre assigned the symbol F, and it is deux centime anniversaire de l'invention du calcul diffferentiel ; Mathesis, Vol. IV, p. 163. Art. 10.] calculus. 533 now called the gamma function. To the subject Dirichlet has •contributed an important theorem (Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others. On the evaluation of Fx and log /jr Raabe (1843-44), Bauer {1859), and Gudermann (1845) nave written. Legendre's great table appeared in 18 16. Symbolic Methods may be traced back to Taylor, and the analogy between successive differentiation and ordinary ex- ponentials had been observed by numerous writers before the nineteenth century. Arbogast (1800) was the first, however, to separate the symbol of operation from that of quantity in a differential equation. Francois (1812) and Servois (1814) seem to have been the first to give correct rules on the subject. Hargreave (1848) applied these methods in his memoir on dif- ferential equations, and Boole freely employed them. Grass- mann and Hankel made great use of the theory, the former in studying equations, the latter in his theory of complex num- bers. The Calculus of Variations* may be said to begin with a problem of Johann Bernoulli's (1696). It immediately occu- pied the attention of Jakob Bernoulli and the Marquis de 1'Hopital, but Euler first elaborated the subject. His contri- butions began in 1733, and his Elementa Calculi Variationum gave to the science its name. Lagrange contributed extensively to the theory, and Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of max- ima and minima. To this discrimination Brunacci (1810), Gauss (1829), Poisson (1831), Ostrogradsky (1834), and Jacobi (1837) have been among the contributors. An important gen- eral work is that of Sarrus (1842) which was condensed and im- proved by Cauchy (1844). Other valuable treatises and me- moirs have been written by Strauch (1849), Jellett (1850), Hesse (1857), Clebsch (1858), and Carll (1885), but perhaps the most * Carll, L. B., Calculus of Variations, New York, 1885, Chap, V; Tod- hunter, I., History of the Progress of the Calculus of Variations, London, 1861 ; Reiff, R., Die AnfSnge der Variationsrechnung, Mathematisch-natur- wissenschaftliche Mittheilungen, Tubingen, 1887, p. go. 534 HISTORY OF MODERN MATHEMATICS. [Chap. XI. important work of the century is that of Weierstrass. His. celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and un- questionable foundation. The Application of the Infinitesimal Calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the eighteenth century these appli- cations were multiplied, until at its close Laplace and Lagrange had brought the whole range of the study of forces into the realm of analysis. To Lagrange (1773) we owe the introduc- tion of the theory of the potential* into dynamics, although the name " potential function " and the fundamental memoir of the subject are due to Green (1827, printed in 1828). The name "potential" is due to Gauss (1840), and the distinction between potential and potential function to Clausius. With its development are connected the names of Dirichlet, Rie- mann, Neumann, Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami, and many of the leading physicists of the century. It is impossible in this place to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations of Euler on vibrating chords; Sophie Germain on elastic membranes ; Poisson, Lam6, Saint- Venant, and Clebsch on the elasticity of three-dimensional bod- ies; Fourier on heat diffusion; Fresnel on light; Maxwell, Helm- holtz, and Hertz on electricity; Hansen, Hill, and Gylddn on astronomy; Maxwell on spherical harmonics; Lord Rayleigh on acoustics; and the contributions of Dirichlet, Weber, Kirchhoff, F. Neumann, Lord Kelvin, Clausius, Bjerknes, MacCullagh, and Fuhrmann to physics in general. The labors of Helm- holtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of me- chanics as well as on those of pure mathematics. * Bacharach, M., Abriss der Geschichte der Potentialtheorie, 1883. ThiS' contains an extensive bibliography. ART. 11.] DIFFERENTIAL EQUATIONS. 535 Art. 11. Differential Equations. The Theory of Differential Equations * has been called by Lief the most important of modern mathematics. The influ- ence of geometry, physics, and astronomy, starting with New- ton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients. The first method of integrating linear ordinary differential equations with constant coefficients is due to Euler, who made d n y d"~'y the solution of his type, ^„ + A, -—^ -f ... -f A„ y = o, de- pend on that of the algebraic equation of the «th degree, F(s) =z*-\-A 1 zt"-'-\-. . .-\-A n = o, in which z h takes the place of d"y -r- k (k — I, 2, . . . n). This equation F(z) = O, is the " char- acteristic " equation considered later by Monge and Cauchy. The theory of linear partial differential equations may be said to begin with Lagrange (1779 to 1785). Monge (1809) treated ordinary and partial differential equations of the first and second order, uniting the theory to geometry, and intro- ducing the notion of the " characteristic," the curve represented by F(z) = o, which has recently been investigated by Darboux, * Cantor, M., Geschichte der Mathematik, Vol. Ill, p. 429 ; Schlesinger, L., Handbuch der Theorie der linearen Differentialgleichungen, Vol. I, 1895, an ex- cellent historical view ; review by Mathews in Nature, Vol. LI I, p. 313; Lie, S., Zur allgemeinen Theorie der pariiellen Differentialgleichungen, Berichte iiber die Verhandlungen der Gesellschaft der Wissenschaften zu Leipzig, 1895;, Mansion, P., Theorie der partiellen Differentialgleichungen ier Ordnung, Ger- man by Maser, Leipzig, 1892, excellent on history ; Craig, T., Some of the De- velopments in the Theory of Ordinary Differential Equations, 1878-1893, Bul- letin New York Mathematical Society, Vol. II, p. 119 ; Goursat, E., Leconssur. l'intSgration des Equations aux derivees partielles du premier ordre, Paris, 1891; Burkhardt, H., and Hefter, L., in Mathematical Papers of Chicago Congress, p. 13 and p. 96. f " In der ganzen modernem. Mathematik ist die Theorie der Differentialr gleichungen die wichtigste Discipiin '536 HISTORY OF MODERN MATHEMATICS. [Chap. XI. Levy, and Lie. Pfaff (1814, i8i5)gave the first general method of integrating partial differential equations of the first order, a method of which Gauss (181 5) at once recognized the value and of which he gave an analysis. Soon after, Cauchy (18 19) gave a simpler method, attacking the subject from the analyt- ical standpoint, but using the Monge characteristic. To him is also due the theorem, corresponding to the fundamental theorem of algebra, that every differential equation defines a function expressible by means of a convergent series, a propo- sition more simply proved by Briot and Bouquet, and also by Picard (1891). Jacobi (1827) also gave an analysis of Pfaff's method, besides developing an original one (1836) which Clebsch published (1862). Clebsch's own method appeared in 1866, and others are due to Boole (1859), Korkine (1869), and A. Mayer (1872). Pfaff's problem has been a prominent sub- ject of investigation, and with it are connected the names of Natani (1859), Clebsch (1861, 1862), DuBois-Reymond (1869), Cayley, Baltzer, Frobenius, Morera, Darboux, and Lie. The next great improvement in the theory of partial differential equations of the first order is due to Lie (1872), by whom the whole subject has been placed on a rigid foundation. Since about 1870, Darboux, Kovalevsky, Meray, Mansion, Grain- dorge, and Imschenetsky have been prominent in this line. The theory of partial differential equations of the second and higher orders, beginning with Laplace and Monge, was notably advanced by Ampere (1840). Imschenetsky* has summarized the contributions to 1873, but the theory remains in an imperfect state. The integration of partial differential equations with three or more variables was the object of elaborate investigations by Lagrange, and his name is still connected with certain subsid- iary equations. To him and to Charpit, who did much to develop the theory, is due one of the methods for integrating the general equation with two variables, a method which now bears Charpit's name. * Grunert's Archiv fur Mathematik, Vol. LIV. ART. 11.] DIFFERENTIAL EQUATIONS. ifyfTy The theory of singular solutions of ordinary and partial differential equations has been a subject of research from the time of Leibniz, but only since the middle of the present cen- tury has it received especial attention. A valuable but little- known work on the subject is that of Houtain (1854). Dar- boux (from 1873) has been a leader in the theory, and in the geometric interpretation of these solutions he has opened a field which has been worked by various writers, notably Caso- rati and Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as at present accepted. The primitive attempt in dealing with differential equations had in view a reduction to quadratures. As it had been the hope ■of eighteenth-century algebraists to find a method for solving the' general equation of the nth. degree, so it was the hope of analysts to find a general method for integrating any differen- tial equation. Gauss (1799) showed, however, that the dif- ferential equation meets its limitations very soon unless complex numbers are introduced. Hence analysts began to study these equations as functions, thus opening a new and fer- tile field. Cauchy was the first to appreciate the importance of this view, and the modern theory may be said to begin with him. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their in- tegrals, but whether a given differential equation suffices for the definition of a function of the independent variable or "variables, and if so, what are the characteristic properties of this function. Within a half-century the theory of ordinary differential equations has come to be one of the most important branches of analysis, the theory of partial differential equations remain- ing as one still to be perfected. The difficulties of the general problem of integration are so manifest that all classes of inves- tigators have confined themselves to the properties of the in- tegrals in the neighborhood of certain given points. The new departure took its greatest inspiration from two memoirs by 538 HISTORY OF MODERN MATHEMATICS. [CHAP. XL Fuchs (Crelle, 1866, 1868), a work elaborated by Thome and Frobenius. Collet has been a prominent contributor since 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch * (1873) at- tacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f=0 under rational one-to-one transformations. Since 1870 Lie's f labors have put the entire theory of dif- ferential equations on a more satisfactory foundation. He has shown that the integration theories of the older mathema- ticians, which had been looked upon as isolated, can by the introduction of the concept of continuous groups of transfor- mations be referred to a common source, and that ordinary differential equations which admit the same infinitesimal trans- formations present like difficulties of integration. He has also emphasized the subject of transformations of contact (Beriih- rungstransformationen) which underlies so much of the recent theory. The modern school has also turned its attention to the theory of differential invariants, one of fundamental im- portance and one which Lie has made prominent. With this- theory are associated the names of Cayley, Cockle, Sylvester, Forsyth, Laguerre, and Halphen. Recent writers have shown the same tendency noticeable in the work of Monge and Cauchy, the tendency to separate into two schools, the one inclining to use the geometric diagram, and represented by Schwarz, Klein, and Goursat, the other adhering to pure anal- ysis, of which Weierstrass, Fuchs, and Frobenius are types. The work of Fuchs and the theory of elementary divisors has formed the basis of a late work by Sauvage (1895). Poincare's- * Klein's Evanston Lectures, Lect. I. f Klein's Evanston Lectures, Lect. II, III. Art. 12.] infinite series. 539' recent contributions are also very notable. His theory of Fuchsian equations (also investigated by Klein) is connected with the general theory. He has also brought the whole sub- ject into close relations with the theory of functions. Appell has recently contributed to the theory of linear differential equations transformable into themselves by change of the func- tion and the variable. Helge von Koch has written on infinite determinants and linear differential equations. Picard has un- dertaken the generalization of the work of Fuchs and Poincare in the case of differential equations of the second order. Fabry (i885)has generalized the normal integrals of Thom6, integrals which Poincare has called "integrals anormales," and which Picard has recently studied. Riquier has treated the question of the existence of integrals in any differential system and given a brief summary of the history to 1895.* The number of contributors in recent times is very great, and includes, besides those already mentioned, the names of Brioschi, Konigsberger, Peano, Graf, Hamburger, Graindorge, Schlafii, Glaisher, Lom- mel, Gilbert, Fabry, Craig, and Autonne. Art. 12. Infinite Series. The Theory of Infinite Series f in its historical develop- ment has been divided by Reiff into three periods: (1) the period of Newton and Leibniz, that of its introduction; (2) that of Euler, the formal period ; (3) the modern, that of the scientific investigation of the validity of infinite series, a period beginning with Gauss. This critical period begins with the publication of Gauss's celebrated memoir on the series I +^ + "-("+ I >- /? - (/? + I W....in 1812. Euler ~ 1 .y ^ 1 . 2. y.{y+ 1) * Riquier, C, Memoire sur l'existence des integrates dans un systeme dif- ferentiel quelconque, etc. Memoires des Savants Strangers, Vol. XXXII, No. 3. f Cantor, M., Geschichte der Mathematik, Vol. Ill, pp. 53- 7' : Reiff. R., Geschichte der unendlichen Reihen, Tubingen, 1889 ; Cajori, F., Bulletin New York Mathematical Society, Vol. I, p. 184; History of Teaching of Mathe- matics in United States, p. 361. 540 HISTORY OF MODERN MATHEMATICS. [Chap. XI. had already considered this series, but Gauss was the first to master it, and under the name " hypergeometric series " (due to Pfaff) it has since occupied the attention of Jacobi, Kummer, Schwarz, Cayley, Goursat, and numerous others. The partic- ular series is not so important as is the standard of criticism which Gauss set up, embodying the simpler criteria of con- vergence and the questions of remainders and the range of convergence. Gauss's contributions were not at once appreciated, and the next to call attention to the subject was Cauchy (1821), who may be considered the founder of the theory of con- vergence and divergence of series. He was one of the first to insist on strict tests of convergence ; he showed that if two series are convergent their product is not necessarily so; and with him begins the discovery of effective criteria of converg- ence and divergence. It should be mentioned, however, that these terms had been introduced long before by Gregory (1668), that Euler and Gauss had given various criteria, and that Maclaurin had anticipated a few of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form. His test for convergence is still one of the most satisfactory when the integration in- volved is possible. Abel was the next important contributor. In his memoir 7H DZ\Wl I ) (1826) on the series 1 -) x -\ j -x' + ... he corrected certain of Cauchy's conclusions, and gave a completely scien- tific summation of the series for complex values of m and x. He was emphatic against the reckless use of series, and showed the necessity of considering the subject of continuity in ques- tions of convergence. Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; Art. 12.] INFINITE SERIES. 541 of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Tchebichef (1852), and Arndt 1853). General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory. The Theory of Uniform Convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Stokes and Seidel (1847-48). Cauchy took up the problem again (1853), acknowl- edging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thome used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform converg- ence, in spite of the demands of the theory of functions. Semi-Convergent Series were studied by Poisson (1823), who also gave a general form for the remainder of the Mac- laurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a differ- ent formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlomilch (Zeitschrift, Vol. I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function P{ x ) — i» _J_ 2«-(- . . . -\-{x— 1)". Genocchi (1852) has further contributed to the theory. Among the early writers was Wronski, whose " loi supreme " (181 5) was hardly recognized until Cayley (1873) brought it into prominence. Transon (1874), Ch. Lagrange (1884), Echols, and Dickstein* have published of late various memoirs on the subject. Interpolation Formulas have been given by various writers * Bibliotheca Mathematica, 1892-94; historical. 542 HISTORY OF MODERN MATHEMATICS. [CHAP. XI. from Newton to the present time. Lagrange's theorem is well known, although Euler had already given an analogous form, as are also Olivier's formula (1827), and those of Minding (1830), Cauchy (1837), Jacobi (1845), Grunert (1850, 1853), Christoffel (1858), and Mehler (1864). Fourier's Series* were being investigated as the result of physical considerations at the same time that Gauss, Abel, and Cauchy were working out the theory of infinite series. Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann (1701) and still earlier by Viete. Euler and Lagrange had simplified the subject, as have, more recently, Poinsot, Schroter, Glaisher, and Kummer. Fourier (1807) set for himself a different prob- lem, to expand a given function of x in terms of the sines or cosines of multiples of x, a problem which he embodied in his Theorie analytique de la Chaleur (1822). Euler had already given the formulas for determining the coefficients in th& series ; and Lagrange had passed over them without recog- nizing their value, but Fourier was the first to assert and at- tempt to prove the general theorem. Poisson (1820-23) also attacked the problem from a different standpoint. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner. Dirichlet's treatment (Crelle, 1829), while bringing the theory of trigonometric series to a temporary conclusion, has been the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schlafli, and DuBois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series have been Dini, Hermite, Halphen, Krause, Byerly and Appell. * Historical Summary by Bocher, Chap. IX of Byerly's Fourier's Series and Spherical Harmonics, Boston, 1893; Sachse, A., Essai historique sur la representation d'une fonction .... par une serie trigonomfetrique. Bulletin des Sciences mathematiques, Part I, 18S0, pp. 43, 83. Art. 13. 1 theory of functions. 543 Art. 13. Theory of Functions. The Theory of Functions * may be said to have its first development in Newton's works, although algebraists had al- ready become familiar with irrational functions in considering cubic and quartic equations. Newton seems first to have grasped the idea of such expressions in his consideration of symmetric functions of the roots of an equation. The word was employed by Leibniz (1694), but in connection with the Cartesian geometry. In its modern sense it seems to have been first used by Johann Bernoulli, who distinguished between algebraic and transcendent functions. He also used (1718) the function symbol d the deter- mination of self-corresponding points for algebraic correspond- ences of a simple kind are fundamental in the present theory, subjects to which Bacharach, Brill, and Noether have also con- tributed extensively. Cayley added much to the theories of rational transformation and correspondence, showing the distinc- tion between the theory of transformation of spaces and that of correspondence of loci. His investigations on the bitangents of plane curves, and in particular on the twenty-eight bitangents of a non-singular quartic, his developments of Pliicker's conception of foci, his discussion of the osculating conies of curves and of the sextactic points on a plane curve, the geometric theory of the invariants and covariants of plane curves, are all note- worthy. He was the first to announce (1849) t ne twenty-seven lines which lie on a cubic surface, he extended Salmon's theory of reciprocal surfaces, and treated (1869) the classification of cubic surfaces, a subject already discussed by Schlafli. He also con- tributed to the theory of scrolls (skew-ruled surfaces), orthog- onal systems of surfaces, the wave surface, etc., and was the first to reach (1845) an Y very general results in the theory of curves of double curvature, a theory in which the next great advance was made (1882) by Halphen and Noether. Among Cayley's other contributions to geometry is his theory of the Absolute, a figure in connection with which all metrical prop- erties of a figure are considered. * Biographical Notice in Cayley's Collected papers, Vol. VIII. ART. 15.] ANALYTIC GEOMETRY. 557 Clebsch* was also prominent in the study of curves and surfaces. He first applied the algebra of linear transformation to geometry. He emphasized the idea of deficiency (Geschlecht) of a curve, a notion which dates back to Abel, and applied the theory of elliptic and Abelian functions to geometry, using it for the study of curves. Clebsch (1872) investigated the shapes of surfaces of the third order. Following him, Klein attacked the problem of determining all possible forms of such surfaces, and established the fact that by the principle of continuity all forms of real surfaces of the third order can be derived from the particular surface having four real conical points. Zeuthen (1874) has discussed the various forms of plane curves of the fourth order, showing the relation between his results and those of Klein on cubic surfaces. Attempts have been made to extend the subject to curves of the nth order, but no gen- eral classification has been made. Quartic surfaces have been studied by Rohn (1887) but without a complete enumeration, and the same writer has contributed (1881) to the theory of Kummer surfaces. Lie has adopted Plucker's generalized space element and ex- tended the theory. His sphere geometry treats the subject from the higher standpoint of six homogeneous coordinates, as distinguished from the elementary sphere geometry with but five and characterized by the conformal group, a geometry studied by Darboux. Lie's theory of contact transformations, with its application to differential equations, his line and sphere complexes, and his work on minimum surfaces are all prominent. Of great help in the study of curves and surfaces and of the theory of functions are the models prepared by Dyck, Brill, O. Henrici, Schwarz, Klein, Schonflies, Kummer, and others.f The Theory of Minimum Surfaces has been developed along * Klein, Evanston Lectures, Lect. I. f Dyck, W., Katalog mathematischer und mathematisch-physikalischer Modelle, Mtinchen, 1892 ; Deutsche Universitatsausstellung, Mathematical Papers of Chicago Congress, p. 49. 55S HISTORY OF MODERN MATHEMATICS. [CHAP. XI. with the analytic geometry in general. Lagrange (1760-61) gave the equation of the minimum surface through a given contour, and Meusnier (1776, published in 1785) also studied the question. But from this time on for half a century little that is noteworthy was done, save by Poisson (18 13) as to cer- tain imaginary surfaces. Monge (1784) and Legendre (1787) connected the study of surfaces with that of differential equa- tions, but this did not immediately affect this question. Scherk (1835) added a number of important results, and first applied the labors of Monge and Legendre to the theory. Catalan (1842), Bjorling (1844), and Dini (1865) have added to the subject. But the most prominent contributors have been Bonnet, Schwarz, Darboux, and Weierstrass. Bonnet (from 1853) has set forth a new system of formulas relative to the general theory of surfaces, and completely solved the problem of determining the minimum surface through any curve and admitting in each point of this curve a given tangent plane. Weierstrass (1866) has contributed several fundamental theo- rems, has shown how to find all of the real algebraic minimum surfaces, and has shown the connection between the theory of functions of an imaginaay variable and the theory of minimum surfaces. Art. 16. Modern Geometry. Descriptive,* Projective, and Modern Synthetic Geometry are so interwoven in their historic development that it is even more difficult to separate them from one another than from the analytic geometry just mentioned. Monge had been in possession of his theory for over thirty years before the publi- cation of his Geometrie Descriptive (1800), a delay due to the jealous desire of the military authorities to keep the valuable secret. It is true that certain of its features can be traced back to Desargues, Taylor, Lambert, and Frezier, but it was Monge who worked it out in detail as a science, although * Wiener, Chr. , Lehrbuch der darstellenden Geometrie, Leipzig, 1884-87; Geschichte der darstellenden Geometrie, 1884. ■ART. 16.] MODERN GEOMETRY. 559 Lacroix (1795), inspired by Monge's lectures in the Ecole Polytechnique, published the first work on the subject. After Monge's work appeared, Hachette (1812, 1818, 1821) added materially to its symmetry, subsequent French contributors being Leroy (1842), Olivier (from 1845), de la Gournerie (from i860), Vallee, de Fourcy, Adhemar, and others. In Germany leading contributors have been Ziegler (1843), Anger (1858), and especially Fiedler (3d edn. 1883-88) and Wiener (1884-87). At this period Monge by no means confined himself to the de- scriptive geometry. So marked were his labors in the analytic geometry that he has been called the father of the modern theory. He also set forth the fundamental theorem of recip- rocal polars, though not i.n modern language, gave some treat- ment of ruled surfaces, and extended the theory of polars to quadrics.* Monge and his school concerned themselves especially with the relations of form, and particularly with those of surfaces and curves in a space of three dimensions. Inspired by the general activity of the period, but following rather the steps of Desargues and Pascal, Carnot treated chiefly the metrical rela- tions of figures. In particular he investigated these relations as connected with the theory of transversals, a theory whose fundamental property of a four-rayed pencil goes back to Pappos, and which, though revived by Desargues, was set forth for the first time in its general form in Carnot's Geometric de Position (1803), and supplemented in his Theorie des Trans- versales (1806). In these works he introduced negative mag- nitudes, the general quadrilateral and quadrangle, and numer- ous other generalizations of value to the elementary geometry of to-day. But although Carnot's work was important and many details are now commonplace, neither the name of the theory nor the method employed have endured. The present Geometry of Position (Geometrie der Lage) has little in com- mon with Carnot's Geometrie de Position. *On recent development of graphic methods and the influence of Monge upon this branch of mathematics, see Eddy, H. T., Modem Graphical Develop- ments, Mathematical Papers of Chicago Congress (New York, 1896), p. 58. 560 HISTORY OF MODERN MATHEMATICS. [CHAP. XL Projective Geometry had its origin somewhat later than the period of Monge and Carnot. Newton had discovered that all curves of the third order can be derived by central projection from five fundamental types. But in spite of this fact the theory attracted so little attention for over a century that its origin is generally ascribed to Poncelet. A prisoner in the Russian campaign, confined at Saratoff on the Volga (1812-14), "prive," as he says, " de toute espece de livres et de secours, surtout distrait par les malheurs de ma patrie et les. miens propres," he still had the vigor of spirit and the leisure to conceive the great work which he published (1822) eight years later. In this work was first made prominent the power of central projection in demonstration and the power of the principle of continuity in research. His leading idea was the study of projective properties, and as a foundation principle he introduced the anharmonic ratio, a concept, however, which dates back to Pappos and which Desargues (1639) had also used. Mobius, following Poncelet, made much use of the an- harmonic ratio in his Barycentrische Calciil (1827), but under the name " Doppelschnitt-Verhaltniss " (ratio bisectionalis), a term now in common use under Steiner's abbreviated form " Doppelverhaltniss." The name " anharmonic ratio " or " function " (rapport anharmonique, or fonction anharmonique) is due to Chasles, and " cross-ratio " was coined by Clifford. The anharmonic point and line properties of conies have been further elaborated by Brianchon, Chasles, Steiner, and von Staudt. To Poncelet is also due the theory of " figures homo- logiques," the perspective axis and perspective center (called by Chasles the axis and center of homology), an extension of Carnot's theory of transversals, and the " cordes id^ales " of conies which Pliicker applied to curves of all orders. He also discovered what Salmon has called " the circular points at in- finity," thus completing and establishing the first great principle of modern geometry, the principle of continuity. Brianchon (1806), through his application of Desargues's theory of polars> Art. 16.] modern geometry. 561 completed the foundation which Monge had begun for Ponce- let's (1829) theory of reciprocal polars. Among the most prominent geometers contemporary with Poncelet was Gergonne, who with more propriety might be ranked as an analytic geometer. He first (1813) used the term " polar" in its modern geometric sense, although Servois(i8n) had used the expression " pole." He was also the first (1825- 26) to grasp the idea that the parallelism which Maurolycus,, Snell, and Viete had noticed is a fundamental principle. This principle he stated and to it he gave the name which it now bears, the Principle of Duality, the most important, after that of continuity, in modern geometry. This principle of geomet- ric reciprocation, the discovery of which was also claimed by Poncelet, has been greatly elaborated and has found its way into modern algebra and elementary geometry, and has recently been extended to mechanics by Genese. Gergonne was the first to use the word "class" in describing a curve, explicitly defining class and degree (order) and showing the duality between the two. He and Chasles were among the first to study scientifically surfaces of higher order. Steiner (1832) gave the first complete discussion of the pro- jective relations between rows, pencils, etc., and laid the foun- dation for the subsequent development of pure geometry. He practically closed the theory of conic sections, of the corre- sponding figures in three-dimensional space and of surfaces of the second order, and hence with him opens the period of special study of curves and surfaces of higher order. His treat- ment of duality and his application of the theory of projective pencils to the generation of conies are masterpieces. The theory of polars of a point in regard to a curve had been studied by Bobillier and by Grassmann, but Steiner (1848) showed that this theory can serve as the foundation for the study of plane curves independently of the use of coordinates,, and introduced those noteworthy curves covariant to a given curve which now bear the names of himself, Hesse, and Cayley, This whole subject has been extended by Grassmann, Chasles. 5G2 HISTORY OF MODERN MATHEMATICS. [Chap. XI. Cremona, and Jonquieres. Steiner was the first to make prom- inent (1832) an example of correspondence of a more com- plicated nature than that of Poncelet, Mobius, Magnus, and Chasles. His contributions, and those of Gudermann, to the geometry of the sphere were also noteworthy. While Mobius, Plucker, and Steiner were at work in Germany, Chasles was closing the geometric era opened in France by Monge. His Apercu Historique (1837) is a classic, and did for France what Salmon's works did for algebra and ' geometry in England, popularizing the researches of earlier writers and contributing both to the theory and the nomen- clature of the subject. To him is due the name "homo- graphic" and the complete exposition of the principle as applied to plane and solid figures, a subject which has received attention in England at the hands of Salmon, Townsend, and H. J. S. Smith. Von Staudt began his labors after Plckuer, Steiner, and Chasles had made their greatest contributions, but in spite of this seeming disadvantage he surpassed them all. Joining the Steiner school, as opposed to that of Plucker, he became the greatest exponent of pure synthetic geometry of modern times. He set forth (1847, 1856-60) a complete, pure geometric system in which metrical geometry finds no place. Projective proper- ties foreign to measurements are established independently of number relations, number being drawn from geometry instead of conversely, and imaginary elements being systematically introduced from the geometric side. A projective geometry based on the group containing all the real projective and dual- istic transformations, is developed, imaginary transformations being also introduced. Largely through his influence pure geometry again became a fruitful field. Since his time the distinction between the metrical and projective theories has been to a great extent obliterated,* the metrical properties * Klein, F., Erlangen Programme of 1872, Haskell's translation, Bulletin of New York Mathematical Society, Vol. II, p. 215. Art. 17.] elementary geometry. 563 being considered as projective relations to a fundamental con- figuration, the circle at infinity common for all spheres. Un- fortunately von Staudt wrote in an unattractive style, and to Reye is due much of the popularity which now attends the subject. Cremona began his publications in 1862. His elementary work on projective geometry (1875) in Leudesdorf's translation is familiar to English readers. His contributions to the theory of geometric transformations are valuable, as also his works on plane curves, surfaces, etc. In England Mulcahy, but especially Townsend (1863), and Hirst, a pupil of Steiner's, opened the subject of modern geometry. Clifford did much to make known the German theories, besides himself contributing to the study of polars and the general theory of curves. Art. 17. Elementary Geometry. Trigonometry and Elementary Geometry have also been affected by the general mathematical spirit of the century. In trigonometry the general substitution of ratios for lines in the definitions of functions has simplified the treatment, and certain formulas have been improved and others added.* The convergence of trigonometric series, the introduction of the Fourier series, and the free use of the imaginary have already been mentioned. The definition of the sine and cosine by series, and the systematic development of the theory on this basis, have been set forth by Cauchy (1821), Lobachevsky (1833), and others. The hyperbolic trigonometry,! already founded by Mayer and Lambert, has been popularized and further, developed by Gudermann (1830), Hoiiel, and Laisant (1871), and projective formulas and generalized figures have * Todhunter, I., History of certain formulas of spherical trigonometry, Philosophical Magazine, 1873. t Giinther, S., Die Lehre von den gewohnlichen und verallgemeinerten Hyperbelfunktionen, Halle, 1881; Chrystal, G., Algebra, Vol. II, p. 288. 564 HISTORY OF MODERN MATHEMATICS. [CHAP. XI. been introduced, notably by Gudermann, Mobius, Poncelet, and Steiner. Recently Study has investigated the formulas of spherical trigonometry from the standpoint of the modern theory of functions and theory of groups, and Macfarlane has generalized the fundamental theorem of trigonometry for three-dimensional space. Elementary Geometry has been even more affected. Among the many contributions to the theory may be men- tioned the following: That of Mobius on the opposite senses of lines, angles, surfaces, and solids ; the principle of duality as given by Gergonne and Poncelet ; the contributions of De Morgan to the logic of the subject ; the theory of transversals as worked out by Monge, Brianchon, Servois, Carnot, Chasles, and others ; the theory of the radical axis, a property dis- covered by the Arabs, but introduced as a definite concept by Gaultier (1813) and used by Steiner under the name of " line of equal power " ; the researches of Gauss concerning inscrip- tible polygons, adding the 17- and 257-gon to the list below the 1000-gon ; the theory of stellar polyhedra as worked out by Cauchy, Jacobi, Bertrand, Cayley, Mobius, Wiener, Hess, Hersel, and others, so that a whole series of bodies have been added to the four Kepler-Poinsot regular solids ; and the re- searches of Muir on stellar polygons. These and many other improvements now find more or less place in the text-books of the day. To these must be added the recent Geometry of the Tri- angle, now a prominent chapter in elementary mathematics.. Crelle (18 16) made some investigations in this line, Feuerbach (1822) soon after discovered the properties of the Nine-Point Circle, and Steiner also came across some of the properties of the triangle, but none of these followed up the investigation. Lemoine * (1873) was the first to take up the subject in a sys- * Smith, D. E., Biography of Lemoine, American Mathematical Monthly, Vol. Ill, p. 29; Mackay, J. S. , various articles on modern geometry in Proceed- ings Edinburgh Mathematical Society, various years; Vigarie, £., Gfeomfetrie du triangle. Articles in recent numbers of Journal de Math§matiques speciales, Mathesis, and Proceedings of the Association francaise pour l'avancement des sciences. Art. 18.] non-euclidean geometry. 565 tematic way, and he has contributed extensively to its de- velopment. His theory of " transformation continue" and his " geometrographie " should also be mentioned. Brocard's con- tributions to the geometry of the triangle began in 1877. Other prominent writers have been Tucker, Neuberg, Vigarie, Emmerich, M'Cay, Longchamps, and H. M. Taylor. The theory is also greatly indebted to Miller's work in The Educa- tional Times, and to Hoffmann's Zeitschrift. The study of linkages was opened by Peaucellier (1864), who gave the first theoretically exact method for drawing a straight line. Kempe and Sylvester have elaborated the subject. In recent years the ancient problems of trisecting an angle, doubling the cube, and squaring the circle have all been settled by the proof of their insolubility through the use of compasses and straight edge.* Art. 18. Non-Euclidean Geometry. The Non-Euclidean Geometry t is a natural result of the futile attempts which had been made from the time of Proklos to the opening of the nineteenth century to prove the fifth postulate (also called the twelfth axiom, and sometimes the * Klein, F., Vortrage iiber ausgewahlten Fragen; Rudio, F., Das Problem von der Quadratur des Zirkels. Naturforschende Gesellschaft Vierteljahr- schrift, 1890; Archimedes, Huygens, Lambert, Legendre (Leipzig, 1892). f Stackel and Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss, Leipzig, 1895; Halsted, G. B., various contributions: Bibliography of Hyperspace and Non-Euclidean Geometry, American Journal of Mathematics, Vols. I, II; The American Mathematical Monthly, Vol. I; translations of Loba- chevsky's Geometry, Vasiliev's address on Lobachevsky, Saccheri's Geome- try, Bolyai's work and his life; Non-Euclidean and Hyperspaces. Mathe- matical Papers of Chicago Congress, p. 92. Loria, G., Die hauptsachlichsten Theorien der Geometrie, p. 106 ; Karagiannides, A., Die Nichteuklidische Geometrie vom Alterthum bis zur Gegenwart, Berlin, 1893; McClintock, E., On the early history of Non-Euclidean Geometry, Bulletin of New York Mathe- matical Society, Vol. II, p. 144; PoincarS, Non-Euclidean Geom., Nature, 45 : 404 ; Articles on Parallels and Measurement in Encyclopaedia Britaneica, 9th edition; Vasiliev's address (German by Engel) also appears in the Abhand- lungen zur Geschichte der Mathematik, 1895. 566 HISTORY OF MODERN MATHEMATICS. [CHAP. XL eleventh or thirteenth) of Euclid. The first scientific investi- gation of this part of the foundation of geometry was made by Saccheri (1733), a work which was not looked upon as a pre- cursor of Lobachevsky, however, until Beltrami (1889) called attention to the fact. Lambert was the next to question the validity of Euclid's postulate, in his Theorie der Parallellinien (posthumous, 1786), the most important of many treatises on the subject between the publication of Saccheri's work and those of Lobachevsky and Bolyai. Legendre also worked in the field, but failed to bring himself to view the matter outside the Euclidean limitations. During the closing years of the eighteenth century Kant's* doctrine of absolute space, and his assertion of the necessary postulates of geometry, were the object of much scrutiny and attack. At the same time Gauss was giving attention to the fifth postulate, though on the side of proving it. It was at one time surmised that Gauss was the real founder of the non- Euclidean geometry, his influence being exerted on Loba- chevsky through his friend Bartels, and on Johann Bolyai through the father Wolfgang, who was a fellow student of Gauss's. But it is now certain that Gauss can lay no claim to priority of discovery, although the influence of himself and of Kant, in a general way, must have had its effect. Bartels went to Kasan in 1807, and Lobachevsky was his pupil. The latter's lecture notes show that Bartels never mentioned the subject of the fifth postulate to him, so that his investigations, begun even before 1823, were made on his own motion and his results were wholly original. Early in 1826 he sent forth the principles of his famous doctrine of parallels, based on the assumption that through a given point more than one line can be drawn which shall never meet a given line coplanar with it. The theory was published in full in 1829-30, and he contributed to the subject, as well as to other branches of mathematics, until his death. * Fink, E., Kant als Mathematiker, Leipzig, 1889. AKT. 18.] NON-EUCLIDEAN GEOMETRY. 567 Johann Bolyai received through his father, Wolfgang, some of the inspiration to original research which the latter had received from Gauss. When only twenty-one he discovered, at about the same time as Lobachevsky, the principles of non- Euclidean geometry, and refers to them in a letter of Novem- ber, 1823. They were committed to writing in 1825 and published in 1832. Gauss asserts in his correspondence with Schumacher (1831-32) that he had brought out a theory along the same lines as Lobachevsky and Bolyai, but the publi- cation of their works seems to have put an end to his investi- gations. Schweikart was also an independent discoverer of the non-Euclidean geometry, as his recently recovered letters show, but he never published anything on the subject, his work on the theory of parallels (1807), like that of his nephew Taurinus (1825), showing no trace of the Lobachevsky-Bolyai idea. The hypothesis was slowly accepted by the mathematical world. Indeed it was about forty years after its publication that it began to attract any considerable attention. Houel (1866) and Flye St. Marie (1871) in France, Riemann (1868), Helmholtz (1868), Frischauf (1872), and Baltzer (1877) in Ger- many, Beltrami (1872) in Italy, de Tilly (1879) in Belgium, Clifford in England, and Halsted (1878) in America, have been among the most active in making the subject popular. Since 1880 the theory may be said to have become generally understood and accepted as legitimate.* Of all these contributions the most noteworthy from the scientific standpoint is that of Riemann. In his Habilitations- schrift (1854) he applied the methods of analytic geometry to the theory, and suggested a surface of negative curvature, which Beltrami calls "pseudo-spherical," thus leaving Euclid's geometry on a surface of zero curvature midway between his own and Lobachevsky's. He thus set forth three kinds of *For an excellent summary of the results of the hypothesis, see an article by McClintock, The Non-Euclidian Geometry, Bulletin of New York Mathematical Society, Vol. II, p. 1. 568 HISTORY OF MODERN MATHEMATICS. [Chap. XI. geometry, Bolyai having noted only two. These Klein (1871) has called the elliptic (Riemann's), parabolic (Euclid's), and hyperbolic (Lobachevsky's). Starting from this broader point of view* there have con- tributed to the subject many of thedeading mathematicians of the last quarter of a century, including, besides those already named, Cayley, Lie, Klein, Newcomb, Pasch, C. S. Peirce, Killing, Fiedler, Mansion, and McClintock. Cayley 's contribu- tion of his " metrical geometry " was not at once seen to be identical with that of Lobachevsky and Bolyai. It remained for Klein (1871) to show this, thus simplifying Cayley's treat- ment and adding one of the most important results of the entire theory. Cayley's metrical formulas are, when the Absolute is real, identical with those of the hyperbolic geome- try ; when it is imaginary, with the elliptic ; the limiting case between the two gives the parabolic (Euclidean) geometry. The question raised by Cayley's memoir as to how far pro- jective geometry can be defined in terms of space without the introduction of distance had already been discussed by von Staudt (1857) and has since been treated by Klein (1873) and by Lindemann (1876). Art. 19. Bibliography. The following are a few of the general works on the history of mathematics in the nineteenth century, not already men- tioned in the foot-notes. For a complete bibliography of recent works the reader should consult the Jahrbuch uber die Fort- schritte der Mathematik, the Bibliotheca Mathematica, or the, Revue Semestrielle, mentioned below. Abhandlungen zur Geschichte der Mathematik (Leipzig). Ball, W. W. R., A short account of the history of mathematics (London, 1893). Ball, W. W. R., History of the study of mathematics at Cam- bridge (London, 1889). Ball, W. W. R., Primer of the history of mathematics (London, 1895). * Klein, Evanston Lectures, Lect. IX. Art. 19.] bibliography. 569 Bibliotheca Mathematica, G. Enestrom, Stockholm. Quarterly. Should be consulted for bibliography of current articles and works on history of mathematics. Bulletin des Sciences Mathematiques (Paris, H i6me Partie). Cajori, F., History of Mathematics (New "York, 1894). Cayley, A., Inaugural address before the British Association, 1883. Nature, Vol. XXVIII, p. 491. Dictionary of National Biography. London, not completed. Valuable on biographies of British mathematicians. D'Ovidio, Enrico, Uno sguardo alle origini ed alio sviluppo della Matematica Pura (Torino, 1889). Dupin, Ch., Coup d'ceil sur quelques progres des Sciences mathe- matiques, en France, 1830-35. Comptes Rendus, 1835. Encyclopaedia Britannica. Valuable biographical articles by Cayley, Chrystal, Clerke, and others. Fink, K., Geschichte der Mathematik (Tubingen, 1890). Bib- liography on p. 255. Gerhardt, C. J., Geschichte der Mathematik in Deutschland (Munich, 1877). Graf, J. H., Geschichte der Mathematik und der Naturwissen- schaften in bernischen Landen (Bern, 1890). Also numerous bio- graphical articles. Giinther, S., Vermischte Untersuchungen zur Geschichte der mathematischen Wissenschaften (Leipzig, 1876). Giinther, S., Ziele und Resultate der neueren mathematisch- historischen Forschung (Erlangen, 1876). Hagen, J. G., Synopsis der hoheren Mathematik. Two volumes (Berlin, 1891-93). Hankel, H., Die Entwickelung der Mathematik in dem letzten Jahrhundert (Tubingen, 1884). Hermite, Ch., Discours prononce devant le president de la republique le 5 aout 1889 a l'inauguration de la nouvelle Sorbonne. Bulletin des Sciences mathematiques, 1890 ; also Nature, Vol. XLI, p. 597. (History of nineteenth-century mathematics in France.) Hoefer, F., Histoire des mathematiques (Paris, 1879). Isely, L., Essai sur l'histoire des mathematiques dans la Suisse francaise (Neuchatel, 1884). Tahrbuch iiber die Fortschritte der Mathematik (Berlin, annu- ally, 1868 to date). Marie, M., Histoire des sciences math6matiques et physiques. Vols. X, XI, XII (Paris, 1887-88). Matthiessen, L., Grundziige der antiken und modernen Algebra der litteralen Gleichungen (Leipzig, 1878). 570 HISTORY OF MODERN MATHEMATICS. [CHAP. XI.. Newcomb, S., Modern mathematical thought. Bulletin New- York Mathematical Society, Vol. Ill, p. 95; Nature, Vol. XLIX,. P- 325- Poggendorff, J. C, Biographisch-literarisches Handworterbuch zur Geschichte derexacten Wissenschaften. Two volumes (Leipzig, 1863). Quetelet, A., Sciences mathematiques et physiques chez les Beiges au commencement du XIX e siecle (Brussels, 1866). Revue semestrielle des publications mathematiques redigee sous les auspices de la Societe mathematique d' Amsterdam. 1893 to date. (Current periodical literature.) Roberts, R. A., Modern mathematics. Proceedings of the Irish Academy, 1888. Smith, H. J. S., On the present state and prospects of some branches of pure mathematics. Proceedingsof London Mathemat- ical Society, 1876; Nature, Vol. XV, p. 79. Sylvester, J. J., Address before the British Association. Nature,. Vol. I, pp. 237, 261. Wolf, R., Handbuch der Mathematik. Two volumes (Zurich, 1872). Zeitschrift fur Mathematik und Physik. Historisch-literarische Abtheilung. Leipzig. The Abhandlungen zur Geschichte der Mathematik are supplements. For a biographical table of mathematicians see Fink's Ge- schichte der Mathematik, p. 240. For the names and positions- of living mathematicians see the Jahrbuch der gelehrten Welt, published at Strassburg. INDEX. 571 INDEX. Abelian functions, page 545. Abel's quintic demonstration, 22, 520. Absolute convergence, 228. Absolute, the, 556. Addition of vectors, 426, 443. Adjustment of observations, 469. Algebraic equations, 1-32. Alternants, 527. Alternating current, 153 electromotiveforce, 440, 442. Analytic geometry, 552. Anharmonic ratio, 560. Anti-hyperbolic functions, 116. Approximation of roots, 3, 12, 519. Arched catenary, 146. Array of determinant, 36. Associative law, 392. Ausdehnungslehre, 374-424, 517. Automorphic functions, 549. Average error, 493. Axial projection, 72. Bernoulli's theorem, 482. Bibliography, 508, 568. Binary forms, 513, 531. Binomial equations, 16, 522. factors, 46. theorem, 462. Bessel's functions, 169, 183, 213, 215, 220, 221, 224, 345, 549- Branch of a function, 251. point, 252. Canonical forms, 531. Calculus, 531. of variations, 533. Catenary, 14, 145, 327. of uniform strength, 147. Cauchy's expansion, 52. theorem, 262, 264, Center and diameter, 94. Central axis, 456. Centroids, 381, 386. Chance, games of, 467. Characteristic ratios in conies, no. Clairaut's equation, 324. Coaxial quaternions, 439. Cofactors in determinants, 47, 62. Combinations, 473. Combinatory multiplication, 392. Commutative law, 392. Complete integral, 309, 333, 359, 362. Complement, the, 399, 416. Complementary function, 399,416. Composition of quantities, 453. of rotations, 463. of vectors, 428. Complex hyperbolic functions, 140. integrals, 261. numbers 138, 515. roots, 31. variable, 226-302. Concurrent events, 479. Conduction of heat, 183, 219. Conformal representation, 236, 238, 244. Congruences, 512. Conic ranges, 91. Conies, points on, 107. 572 HIGHER MATHEMATICS. Conies, projective generation of, 86, 561. sectors of, no. triangles of, no. Conjugate functions, 245. Cosine series, 192. Consistence of equations, 58. Continued fractions, 515. Continuity of functions, 230. Convergence, of series, 540. uniform, 274. Conversion hyperbolic formulas, 118. Cooling of iron plate, 174. Coplanar vectors, 426, 432. Correlation, 74. Correspondence in conies, 107. Cosine series, 192. Covariants, 529. Critical lines, 298. points, 250, 278. regions, 298. Cross ratio, 104. Cubic equations, 17. surfaces, 556. Curves of second degree, 82, Curvilinear integrals, 267. Cusp locus, 319. Cut, 251. Cutting, 72. Cylinder, floating, 13. Deficiency curves, 548. De Moivre's formula, 515. quintic, 23. Derivative equation, 9. Derivatives of complex functions, 233- gudermanians, 130. hyperbolic func- tions, 120, 122. Descriptive geometry, 554. Determinant array, 36. Determinants, 33-69, 408, 526. Development of determinants, 49, 52. Differential calculus, 531. equations, 172, 303-373, 535- Differentiation, 54, 120, 322. Direct probabilities, 476. Dirichlet's conditions, 198. Discontinuous groups, 526. Discriminant, 319. Distributive law, 392. Doubly ruled surfaces, 102. Duality of plane and point, 96. principle of, 74. Dynamo, 445. Elastic catenary, 148. Elementary functions, 229. geometry, 564. Electric charges, 177. currents, 153, 305, 440, 452. motor, 445. Elements at infinity, 72. of a group, 34. of roots, 15. Eliminant, 59, 61. Elimination, 523. Ellipse, functional relations, III. Elliptic functions, 23, 107, 291, 522, 544- modular functions, 549. Equation of energy, 331. Equations, differential, 169, 172, 303-373. 535- solution of, 1-32. theory of, 519. Equilibrium of forces, 404. Equipotential curves, 248. Energy, equation of, 332. Envelopes, 317. Error function, 492. Errors, theory of, 467-507, 550. Essential singularity, 253. Eulerian integrals, 532. Exact differential equations, 308. Expansion of hyperbolic functions, 123, 125. Exponential expressions, 124. functions, 229. theorem, 460. First integrals, 332, 333. Flexure and tension, 151. Flotation of bodies, 13. INDEX. 573 Fluid motion, 246. Flux across a curve, 247. magnetic, 445, 452. Forces, equilibrium of, 404. resultant of, 398, 430. Forms, 530. Fourier's series, 194, 196, 273, 542. Fractions, continued, 514. Functions, Bessel's, 169-220, 549. determinantal, 37. elliptic, 107, 522, 544. harmonic, 169-225. hyperbolic, 107-168. of a complex variable, 226-302. symbols for, 543. theory of, 543. with n values, 300. Galois's group theory, 525. Games of chance, 467. Gamma functions, 488, 533. General integrals, 359. Geometric applications, 325. elements, 7°- multiplication, 390. representation, 227, 305. Geometry, analytic, 552. descriptive, 558. elementary, 563. modern, 558. non-Euclidean, 104, 565. projective, 76-106, 560. Graphic representations, 232. solution of equations, 3. Graphs of equations, 9. of hyperbolic functions, 132. of laws of errors, 492, 496, 500, 503- Grassmann's space analysis, 374- 424. Groups, 34, 524. Gudermanian angle, 129. function, 128. tables of, 168. Harmonic analysis, 169. elements, 77. functions, 169-225. Heat conduction, 174, 183, 215, 219. Hessian curve, 555. History of modern mathematics, 508-570. Holomorphic function, 256. Homogeneous differential equations, 172, 311, 342. linear equations, 60. Horner's method, 2, 12, 519. Howe truss strut problem, 21. Hyperbola, functional relations, in. Hyperbolic functions, 107-168. paraboloid. 99. Hyperboloid of one nappe, 100. Hyperdeterminants, 525. Hyperelliptic functions, 546. Hypergeometric series, 348, 540. Icosahedron equation, 521. Imaginaries, 141, 227, 516. Infinitesimal calculus, 534. Infinite series, 539. Infinity, elements at, 73. point at, 256. Inner products, 401. Integral calculus, 531. Integral, complex, 261. curvilinear, 267. hyperbolic, 135. one-valued function, 284. probability, 483. Integrals, 257, 303-373. Intermediate differential equation, 366. Interpolated values, 497. Interpolation formulas, 541. Invariants, 531. Inverse probabilities, 484. Inversions in permutations, 35. Involution, 88. Irrational numbers, 513. Irrotational motion, 248. Jacobians, 357. Journals, mathematical, 509. Kern curve, 88. 574 HIGHER MATHEMATICS. Lagrange's equation, 356. lines, 358. resolvent, 15, 520. series, 549. Lam 's functions, 221. Laplace's equation, 170, 203. Laurent's series, 271. Laws of error, 491. Least squares, 470, 550. Legendre's equation, 179. Limiting values, 119. Linear algebraic equations, 56. differential equations, 172, 312, 336, 338, 368. Literature, 436, 569. Locus, 317, 319, 321. Logarithmic branch point, 252. discontinuity, 252. expressions, 127. functions, 229. solution of equations, 32. tables, 497, 499, 503. Loxodrome, 150. Maclaurin's configuration, 35. series, 271. Magnetic flux, 445, 449. Magnification, 237. Map drawing, 237. Mathematical bibliography, 568. history, 508-570. periodicals, 509. Mathematicians, living, 570. Matrix, 60. McClintock's method for equations, 29. Mean error, 41,3. Mercator's projection, 151, 245. Mereomorphic function, 256. Metrical geometry, 104, 568. Minimum surfaces, 557. Minor determinants, 47. Mittag-Leffler's theorem, 292, 548. Models, 557. • Modern geometry, 558. mathematics, 508-570. Modulus of complex variable, 227. Modulus of integral, 259. Monge's equations, 367. Monogenic function, 233, 235. Multiplication, geometric, 390, 457. theorem, 65. Newton's approximation rule, 6. Node locus, 321. Non-Euclidean geometry, 104, 565. Notations for determinants, 39. for functions, 543. for vectors, 427. Numbers, projective definition, 104. theory of, 511, 513. Numerical equations, 10, 519. Observations, errors of, 550. Omega functions, 547. One-valued functions, 226, 278. Orders of determinants, 40. Orthogonal trajectories, 326. Parallel lines, 43, 566. Partial derivatives, 245. differential equations, 355, 365, 368, 535- Particular integral, 337. Pencils, 71. Periodicals, 509. Periodic functions, 169-225, 547. Permutations, 34, 471. Physics, 534. Plane and point duality, 96. Plane perspective, 76. sects, 412. vectors, 413. Planes, sum of, 413. Planimetric products, 390, 392. Pliicker's equations, 554, 556. Point analysis, 374-424. at infinity, 256. Points, sum of, 375. Pole and polar, 87, 559, 561. Polygons, 564. Polygrams, 74. Polyhedra, 564. Polystims, 74. Potential, 177, 534. Primal forms, 70. INDEX. 575 Prime numbers, 512. Primitive of differential, 307. Probable error, 493. Probabilities, 467-507, 550. Product of arrays, 68. of determinants, 67. of points, 392, 410. of sects, 394, 411. of vectors, 394, 432, 444, 449. of versors, 459. Products, planimetric, 390, 392. stereometric, 390, 410. Projecting, 72. Projective conic ranges, 91. geometry, 70-106, 560. Projectivity, 80. Quadrantal versors, 457. Quadratic equations, 16. Quadric surfaces, 98. Quantics, 528. Quantity, complex, 515. Quartic equations, 19. Quaternions, 425-466, 517. Quintic equations, 21, 520. Raising the order, 55. Reciprocal of vectors, 435. determinants, 69. Reciprocity, 512. Rectangle inscribed in rectangle, 20, 25. Reference systems, 386, Regions, critical, 298. Regula falsi, 5. Removal of terms, 11, 22, 520. Representation, conformal, 236,238, 244. geometric, 305. graphic, 232. Residues, 382, 512. Resolvent equations, 17. sextic, 23, 520. Resultant error, 494. Resultant of equations, 59, 61, 523. of forces, 398. of vectors, 429. Roots of Bessel's functions, 225. of equations, 1-32, 520. of unity, 15. Rotations, 463. Rows and columns, 42. Ruled quadric surfaces, 98. Rules for versors, 457. Sarrus's rule, 40. Scalar products, 433, 446. quantities, 375, 433. Schools of mathematics, 509. Second-degree curves, 82. Semi-convergent series, 534, Separation of roots, 8. of variables, 304. Series, convergence of, 274. for roots, 27, 30. Fourier's, 273. infinite, 228, 342, 537. Laurent's, 271 ■ Maclaurin's, 271. Taylor's, 269. trigonometric, 174, 542. Sextic resolvent, 17, 520. Simultaneous differential equations, 327- Sine series, 188, 542. Singular solutions, 317, 320, 537. Singularity, essential, 253. Solution of equations. 1-32, 519. linear equations, 56. Space analysis, 374-424, 425. Sphere, conformal representation, 244. depth of immersion, 13, 15. Spherical harmonics, 169, 213. trigonometry, 418, 457. versors, 458. Stereometric products, 391, 410. Stirling's theorem, 482. Stream function, 247. Sturm's theorem, 2, 8, 12, 519. Substitutions, 524. Sum and difference formulas, 116. of points, 375. Surfaces, ruled, 98. Surveying problems, 80. 576 HIGHER MATHEMATICS. Sylvester's method of elimination, 63- Symbolic methods, 533. Symmetric functions, 523. Synectic function, 256. Synthetic geometry, 552. Systems of curves, 306, 325, 333. of differential equations, 349- Tables of Bessel's functions, 224. of combinations, 475. of gudermanians, 168. of hyperbolic functions, 160- 168. of permutations, 471. of probability integral, 484. of roots of Bessel's functions, 225. of surface zonal harmonics, 222. of values of Jo(xi), 225. Tabular values, 494, 503. Tac locus, 318. Taylor's series, 269. Tension and flexure, 151. in catenary, 14, 146. Ternary forms, 509. Tetrahedra, 412, 422. Theory of errors, 467-507. of functions, 226, 543. of numbers, 511. Theta functions, 546. Torque, 455. Tractory, 149. Trajectories, 325. Transcendent equations, 1-15. functions, 538. numbers, 513. Triangle, geometry of, 564. Trigonometric series, 174, 200. solution of equations r 24. Trigonometry, 559. Typical errors, 493. Uniform convergence, 274, 541. Uniform strength, catenary of, 147, Variations, calculus of, 533. Vector analysis, 425-466. products, 434, 447. quantities, 138, 374-466, 518- Versors, 457. Water pipe, 13. Weierstrass's ^-function, 297. theorem, 287. Weighted points, 378. Whist, game of, 477. Zero determinants, 62. formulas, 51. Zonal harmonics, 169, 177, 202, 205, 208, 212, 222. SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OF JOHN WILEY & SONS, New York. London: CHAPMAN & HALL, Limited. ARRANGED UNDER SUBJECTS. Descriptive circulars sent on application. Books marked with an asterisk are sold at net prices only. 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