BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Himrg W. Sage 1S91 Jft.HHklQ : 9LVv\a.V... Cornell University Library arV19554 Differential and integral calculus. 3 1924 031 480 894 I olin.anx 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/cu31924031480894 DIFFERENTIAL AND INTEGRAL CALCULUS. DIFFERENTIAL AND INTEGRAL CALCULUS, WITH APPLICATIONS. BY ALFRED GEORGE fiREENHILL, M.A., F.R.S., PROFESSOR OF MATHEMATICS TO THE SENIOR CLASS OF ARTILLERY OFFICERS, WOOLWICH. SECOND EDITION. IDxrnixm : MACMILLAN AND CO. AND NEW YORK. 1891. f [All rights reserved.} •^3 3 £^ /ccrIneilN ^university LIBRARY 4vJ PREFACE. The present Treatise is intended as an introduction to the study of the Differential and Integral Calculus, but will be found to contain what it is necessary to know in order to pass on to the subjects which presume a know- ledge of the Calculus. I have endeavoured to make this book suitable not only for the mathematical student, but also for men like engineers and electricians who require the subject for practical applications, to whom even a slight knowledge of the notation and methods of the Calculus is becoming more and more indispensable. Hitherto in this country the influence of Newton, although the inventor of Fluxions, has been employed to delay the study of this subject and make a know- ledge of it the privilege of a select few ; my object in writing this treatise has been mainly to present the subject in as simple a manner as possible, in order to encourage a larger number of students to cultivate it. With the object of keeping the size of the book within reasonable limits, it is assumed that the reader has already acquired a knowledge of the elements of Algebra, Trigonometry, and Coordinate Geometry, as given, for vi PREFACE. instance, in the treatises of Hall and Knight, J. B. Lock and 0. Smith ; accordingly I have at once proceeded to the operation and application of Differentiation with as little preliminary explanation as possible. I have followed the recent American treatises on this subject of Bice and Woolsey Johnson, Byerly and J. H. Taylor, in introducing the notion of Time as an inde- pendent variable, and the associated ideas of velocity and acceleration, in order to afford illustrations of the use of the Calculus ; this is after all only a return to the Method of Fluxions as invented by Newton, and carried out by Maclaurin and other writers in this country, until supplanted by the notation of the Differential Coefficients of the foreign mathematicians. The Doctrine of Fluxions is a useful and rigorous method of presenting the elementary ideas of the flow of varying quantities, and is employed in the treatises of Eice and Woolsey Johnson under the name of the Method of Eates; but the notation for a fluxion, for instance x the fluxion of x, though easily written is difficult to print, and has the inconvenience of not indicating the independent variable, so that the notation dob of Leibnitz, -^ instead of x, is now used almost uni- versally in printed books ; and to economise space, this notation it is now proposed to print in the form dxjdt. The chief novelties in the present work consist, first, in carrying on the subjects of the Differential and of the Integral Calculus together, instead of, as is usual, completing the Differential before passing on to the Integral Calculus ; secondly, in the use of the hyperbolic functions in conjunction with the ordinary circular PREFACE. vii trigonometrical functions, as thereby an exact analogy is preserved, which is not apparent when only the exponen- tial and logarithmic functions are employed. The notation of sinh, cosh, tanh, etc., to denote the hyperbolic sine, cosine, tangent, etc., has been employed, in accordance with what appears to be now the most universal custom. I have ventured also, on the grounds of symmetry, to introduce the inverse hyperbolic functions, and, fol- lowing Ferrers and Byerly, to denote them by sinh -1 , cosh -1 , tanh -1 , etc., by analogy with sin -1 , cos -1 , tan -1 , etc.; this idea of symmetrical symbolism will be found indicated in Bertrand's Integral Calculus, Chapter I., but has not been pursued apparently because of the lengthi- ness of the notation there employed, namely, sect, sin hyp., sect, cos hyp., sect, tang hyp., etc., instead of the above. By the use of the direct and inverse hyperbolic func- tions in conjunction with the direct and inverse circular functions, the Calculus is in my opinion considerably simplified; and the student is led on more naturally and readily to the consideration of the elliptic and other func- tions. The consideration of these last functions is how- ever beyond the scope of the present treatise. To exhibit more clearly the analogy and symmetry between the circular and hyperbolic functions, I have made a digression in Chapter I. on the formulas of the addition equation (as it may be called by analogy with elliptic functions) of ordinary trigonometry, showing how the formulas may all be deduced from a single figure, with the corresponding relations of the hyperbolic functions. viii PREFACE. Numerous collections of examples will be found throughout the book, introduced at each point to illus- trate what has immediately gone before, and chosen as having some bearing on subsequent stages of mathematics. The order of arrangement will be found in some respects different to what is customary ; for instance, the idea of tracing simple curves from their equations has been introduced into the first chapter, so far as is required for the ordinary applications of the Integral Calculus to finding the areas, etc., of these curves; the general theory of curves being resumed in Chapter VI. Maximums and Minimums also have been investigated without the aid of Taylor's Theorem. ChaDge of the Independent Variable has only been touched upon where necessary ; the general theory of Change of the Independent Variable, Lagrange's and Laplace's Theorems, and the Elimination of Constants and Functions, have been omitted as beyond the scope of an elementary practical treatise. I have to thank Mr. A. G. Hadcock, Inspector of Ordnance Machinery, Eoyal Artillery, for drawing the diagrams, and also for revising the proof sheets and preparing the index. Woolwich, December, 1885. PREFACE TO THE SECOND EDITION. In this second edition a rearrangement has been made, by which the general 1 theory of Integration has been relegated to a supplementary final Chapter ; only a slight preliminary sketch of Integration is now given in Chapter II., sufficient to carry on the student to some simple applications, and to enable him to follow the subsequent articles. In this rearrangement, and in many other details at the outset, I have received valuable advice and assistance from Mr. W. Gallatly, M.A. The theories of Change of the Variable, the Elimina- tion of. Constants and Functions, and the Theorems of Lagrange and Laplace have now been included, so that this new edition is increased in size. By following the custom of giving the examples in smaller print, this increase in size might have been apparently much reduced; but it was decided to use a uniform type throughout the book, as less trying to the eyesight. The solidus notation, for instance dx/dt for -77-, has now been used sparingly, in places where no confusion x PREFACE. or ambiguity could arise, and thereby some economy of space has been obtained. A few pages of explanation of the simplest Differen- tial Equations which occur in Dynamics and Electricity have been added; while the illustrations of Change of the Variable have been chosen from among the most typical differential equations; and it is hoped that this slight sketch will enable the student to solve an ordin- ary differential equation which may come in his way. The greater part of the diagrams have been redrawn by Mr. A. G. Hadcock, who has again helped me in the revision of the proof sheets and the preparation of the Index. Woolwtch, March, 1891. CONTENTS. I. — Differentiation, 1 § 1. Introduction. Constant and Variable Quantities. 2. Definition of a Function ; and of Independent and De- pendent Variables. 3. Definition of a Differential Coefficient. 4. Differentiation of Algebraical Functions. 5. Geometrical Interpretation. 6. Tangent and Normal. 10. Velocity ex- pressed by a Differential Coefficient. 11. Function of a Func- tion. 12. Differentiation of Sum, Product and Quotient. 13. Implicit and Explicit Relations between two Variables. 14. Curve Tracing. 15. Algebraical and Transcendental Func- tions. 16. Circular Functions. 21. The Cycloid. 22. Polar Coordinates. 25. Inverse Circular Functions. 28. The Ex- ponential Theorem and Logarithms. 30. Exponential or Logarithmic Curves and Coordinates. 32. Logarithmic Differ- entiation. 33. Hyperbolic Functions. 35. The Ellipse and Hyperbola compared. 36. The Catenary. 37. Inverse Hyper- bolic Functions. General Examples of Differentiation. II. — Integration, 83 § 38. Notation and Table of Integration. 40. Integration by Substitution. 41. Quadrature. 42. Fluxions and Fluents. 43. Corrected Integrals. 45. Integration between limits. Definite Integrals. 47. Centre of Gravity. 48. Quadrature of Parabola. 50. Quadrature of Circle and Ellipse. 52. Quadrature of Hyperbola and its conjugate. 55. Quadrature of the Cycloid. 56. Quadrature with Polar Coordinates. • 58. Rectification of Curves. 59. Volume and Surface of Solid of Revolution with Applications. 62. Theorems of Pappus or Guldin. 64. Moment of Inertia and Radius of Gyration. General Examples of Integration. xii CONTENTS. CHAP. PAGE III. — Successive Differentiation, 135 § 65. Notation of Successive Differential Coefficients. 66. Kational Algebraical Functions. 68. Circular and Hyper- bolic Functions. 69. Leibnitz's Theorem. 71. Maximum and Minimum Values of a Function. 74. Dynamical Applications. 75. Vertical Motion under Gravity. 77. Ex- perimental Determination of the Resistance of the Air. 78. Motion in a Plane. 79. Motion of a Projectile unresisted. 81. Dynamical Equations with Polar Coordinates. 82. Motion in a Circle. 83. Motion in a Field of Force. 84. Central Orbits. 85. Interchange of Independent and Dependent Variables. 86. Reciprocants. 87. Change of the Indepen- dent Variable. 88. Application to Differential Equations. 89. Geometrical Illustrations. Curvature. 95. The Evolute and Involute. 96. Circle of Curvature. 97. Evolute of Para- bola. 98. Evolute of Ellipse and Hyperbola, 99. Figure and Size of the Earth. 100. Dip and Distance of the Horizon. Variation in Length of a Degree and Minute of Latitude. 101. The Cycloid and its Evolute. Cycloidal Oscillations. 103. Harmonic Vibrations. 104. Intrinsic Equation of a Curve. 105. Envelopes. General Examples on Successive Differentiation. IV. — Expansion of Functions, ... . . 223 § 106. Taylor's and Maclaurin's Theorems. 107. Applica- tion to the Expansion of Functions. 111. Exponential values of the Sine and Cosine. 112. Resolution of Trigono- metrical Functions into Factors. 113. Bernoulli's and Euler's Numbers. 114. Remainder in Taylor's Series. 118. Geo- metrical Illustration. Contact of Different Orders. 119. Infinitesimals. 128. Indeterminate Forms or Singular Values. V.— Partial Differentiation and Integration, . . 257 §123. Functions of two Independent Variables. 124. Notation of Partial Differential Coefficients. 125. Derived Equations of Implicit Relations. 126. Expansion of a Func- tion of two or more Independent Variables. 127. Maximum and Minimum Values of a Function of two Independent Variables. 128. The Indicatrix. Normal Sections of a CONTENTS. x iii Surface. 129. Meunier's Theorem for Oblique Sections of a Surface. 130. Solid Geometry. 131. Functional and Differ- ential Equations of a Surface. 132. Change of Variables. 133. Conjugate Functions. 134. Double Integration. 137. Sign of an Area. 140. The Planimeter. 141. Functions of three or more Independent Variables. 143. Spherical Polar Coordinates. 144. Space, Surface and Line Integrals. 145. Green's Theorem. 146. Change of Variables in Space Inte- grals. 147. Confocal Quadrics. 148. Quantics. 149. Arbo- gast's Method of Derivation. 150. Theorems of Lagrange, Laplace and Burmann. - VI. — Curves in General, .... . 315 § 151. Curve Tracing in Cartesian Coordinates. Newton's Analytical Parallelogram. 152. Asymptotes. 153. Polar Coor- dinates. 154. Chord, Tangent, Asymptote, and Normal in Polar Coordinates. 155. Pedal Curves. 157. Orthoptic and Isoptic Curves. 158. Roulettes. 159. Centrodes. 160. Area of a Roulette. 161. Theorems of Holditch, Elliot, Leudesdorf and Kempe. 162. Euvelope of a Carried Line. 164. Epicycloids and Hypocycloids. 165. Teeth of Wheels. 166. Double Generation of Epi- and Hypo-cycloids. 167. Epi- and Hypo- Trochoids, or Bicircloids. 168. Inversion and Inverse Curves. 169. Mechanical Invertors. Parallel Motions. 170. Polar Reciprocals. 171. Orthogonal and Oblique Trajectories. 172. Critical Orbits. 173. Roulettes of Critical Orbits. 174. Vectors and Vectorial Equations. 175. Dipolar Circles. 176. Mercator's Chart. 177. Confocal Ellipses and Hyper- bolas. 178. Confocal Limacons. 179. Kepler's Laws of Planetary Motion. 180. Elliptic Planetary Motion. 181. Rectilinear Motion under Varying Gravity. 1 182. Kepler's Problem. 183. Fourier's Series. 184. Fourier's Series in Kepler's Problems. Examples on Fourier's Series. VII. — Integration in General, . . .387 § 186. General Integration of Algebraical Functions. 187. Integration of Rational Algebraical Functions. 191. Integra- tion of Irrational Algebraical Functions. 195. Integration by Rationalization. 196. Integration of Circular and Hyperbolic Functions. 202. Integration by Parts. 204. Formulas of xiv CONTENTS. PAGE Reduction. 205. Definite Integrals. 206. Bernoulli's and Euler's Numbers as Definite Integrals. 207. Differentiation of Definite Integrals. 210. Lagrange's Theorem. 211. Inte- gration considered as Summation. 212. Approximate Quad- rature. 213. Differential Equations. Appendix I. — Curves. II. — Equimomental Lines, . . 446 Appendix III.— Table for Hyperbolic Functions, . . .447 Articles in the earlier chapters marked with a * may be omitted at a first reading. ERRATA. Page 82, ex. 39, read— " with X = ax* + 4bx> + 6cx 2 + 4b'x + a'. " Page 125, line 7, read— " iwy 3 = }ir x PP' 3 ; ". Page 139, ex. 8, WBd -««g5? = ("-l)'{(l + »r-(-l + *)»}... 5 ' ' dx n (1-a; 2 )" Page 142, last line but one to be cancelled. Page 143, ex. 2. iv., read— "yV"+V = [ -l^rala" 2 "" 1 ." Page 151, ex. 18, the result as given is incorrect. Page 160, line 3 from bottom, read — "i{V+u)." Page 183, line 15, read— i 'Gu = u 1 fsu i dx~u. 2 /Su 1 dx." Page 231, line 4 from bottom, read — "partial fractions." DIFFERENTIAL AND INTEGRAL CALCULUS. CHAPTEE I. DIFFERENTIATION. 1. Introduction. Constant and Variable Quantities. The Calculus to be developed in the present treatise is the method of reasoning applicable to variable quantities in a state of continuous change. We call quantities variable when they change gradu- ally by increase or decrease ; on the contrary constant, when they remain unchanged while others change. Thus the abscissa and ordinate of points on a parabola vary, while the parameter remains constant. Again, the distance of a railway station from the terminus, or the latitude and longitude of a rock, of a lightship, or of an observatory, are constant quantities. But the distance of a railway train from the terminus, or the latitude and longitude of a steamer or traveller are variable quantities. Constant quantities are generally represented algebra- ically by the first letters of the alphabet, such as a, b, c, ..., or A, B, C, ..., or a, /3, y, ...; while variable quantities are represented by the last letters, ... t, u, v, w, x, y, z, or ... X, Y,Z, or ... £ r,, f 2 DIFFERENTIATION. 2. Definition of a Function; and of Independent and Dependent Variables. One variable quantity, denoted by y or i(x), is said to be a function of another variable quantity, denoted by x, when the value of y or f (x) depends on the value of x. The variable x, called the independent variable or argument, is one to which any value may be arbitrarily assigned. The variable y or f (x), called the dependent variable, is one of which the value is determined, when the arbitrary value of the independent variable x has been assigned. We may use the notation ix instead of f (x) when the in- dependent variable consists of a single term like x. Thus x 2 , x s , a; 4 , ..., x n , *Jx, £/x, sin a;, cos a;, tana;, cot a;, sec a;, coseca:, versa;, sm _1 a;, cos _1 a;, tan _1 a;, cot -1 *, sec _1 a;, cosec _1 a;, vers -1 a;, a x , expa;, cosh a;, sinha:, tanha;, ... log a;, cosh _1 a:, sinh _1 a;, tanh _1 a;, ..., all denoted generically by fee, and most of them presumably familiar to the student, are simple functions of x, which we shall require hereafter. We proceed to differentiate them, that is to find the differential doefficient, which is denned as follows : — 3. Definition of a Differential Coefficient. If fa; denotes any function of a variable quantity x, and if i(x + h) denotes the same function of x+h when x receives a small increment h, then the limiting value of i(x+h)-ix h when h is indefinitely diminished, is called the differential coefficient of ix with respect to x, and is denoted by dix ,., — r— or t x. dx DIFFERENTIATION. 3 This definition may be conveniently expressed as dix _, i(x + h) — ix dx~ h ' (It) being the abbreviation employed to denote the limit- ing value, as h is indefinitely diminished and ultimately becomes zero. Since i(x+h)—ix is the increment of ix corresponding to the increment h of x, therefore ^, i(x+h) — ix h is the ultimate ratio of the corresponding increments of ix and x, denoted by dix and dx, and called the differ- entials of ix and x ; and thus -y— measures the rate of increase or growth of ix ; while — ^ represents the average rate of increase of ix from x to x+h. The chief object at the outset of our subject is the determination of the differential coefficients of functions, and the application of them to the discussion of the geometrical and analytical properties of the functions. The algebraical difficulty in the determination of the differential coefficient lies in the reduction of its original indeterminate form, 77 to a determinate limit. J (Hall and Knight, Higher Algebra, chap, xx.) The name derivative or derived function is sometimes used instead of differential coefficient. The differential coefficient i'x of a function of ix may be supposed to derive its name from being the coefficient which turns the differential of x into the differential of ix. (Wicksteed, The Alphabet of Economic Science, p. 32.) 4 DIFFERENTIATION. 4. Differential Coefficient or Derivative of x n - From the definition of a differential coefficient in § 3, dx_-,,x+h — x_, dx h dx* _ lt (x+h) 2 -x* _0 dx hO u 2xh + h 2 _0 = lt— £ = lt(2z + A) = 2a3. dec 3 _ , (x + hf — X s _ da; A, u 3x*h + 3xh* + h s _0 ~ [t h = lt(3cc 2 +3a:/i+A. 2 ) = 3a: 2 . dx^^ jx + hy-x^ O dx~ h _ u 4x% + 6a3 2 /i 2 + toft, 3 + ft* _ ~ lt_ ~ A. _ = lt(4a 3 + 6x 2 h + to/i 2 + A 3 ) = 4ai 3 . And generally, when to is a positive integer, dx n _ , Jx + h) n -x n _0 dx~ h (and expanding by the Binomial Theorem) nx n -^h+ 7 ^—fix n -m+ ... +fc» ft _ u ii^ () = lt{m"- 1 +^y^^-%+ .- +h n -i\=nx n - 1 . By assuming the convergency of the Binomial Theorem we can make the same proof hold for showing that the differential coefficient of x n with respect to x is nap- 1 , DIFFERENTIATION. 5 where n represents any constant number, whether integral, fractional, positive or negative. But without using the Binomial Theorem or assuming its convergency; a proof may be given as follows : — (i.) Suppose n a positive integer, and denote x+h by x x ; then dx n _^ x x n -x n _0 dx x x —x (and then by Division) = \t(x x n " x + xx x n ~ 2 + . . . + x n - 2 x x + x n - a ) = nx n - \ when h = 0, and therefore x x = x. (ii.) Suppose n a positive fraction p/q, where p and q are positive integers ; and put x = zi, x x = z x q ; then dx n _T,oc£te—xPte zf-zH (dividing out z x — z from numerator and denominator) _ lt z/- 1 + z 1 P- 2 z+ ••• +z 1 zP- 2 +zp- 1 z 1 «- 1 +z 1 ?- 2 z+ ... +z 1 zV- 2 +zV- 1 qz?' 1 q (in.) Suppose n any negative number — m; then /7/y.ra /« -wi ™-m ™ m /v.m 1 lffa = — It- 1 - dx x x —x x x — x x x m x'' m™m X' dx n so that -=— = nx n ~ 1 , universally, where n denotes any constant quantity. DIFFERENTIA TION. Examples. — 1. Determine from the definition the d.c. (differential coefficient) with respect to x of Jx, x\ a* as*, a#, x\ -j£ -, ^, - 3 , -jp (x+a) n , x\ n 2a; + 3 ax+b (!)"■ a+bx+cx 2 +... x+1 Ax+B . 2. Give a rigorous proof that the differential coefficient of a: - ^ 2 with respect to x is ^/2 or^ 2-1 . 5. Geometrical Interpretation of Differentiation. y B dy / 1 •/ dt r ■ pIZ dt V y C N p E_+ R' A dx v dt / \ T O a l r> t Ik " G Fig.1 In Figure 1 the coordinates of a point P, referred to axes Ox and Oy at right angjes, are denoted hy x and y, where OM = x, MP = y; and in general, in the vertical elevation of an object, x is measured positive to the right, and negative to the left, y is measured positive upwards, and negative downwards; or, on a map or plan, Ox is drawn to the east, and Oy to the north. DIFFEREN TIA TION. 7 Then the equation y = ix represents some curve APB, the assemblage of points whose coordinates satisfy this equation; so that if OM=x, then MP = ix; the curve APB is now called the graph of the function ix (Chrystal, Algebra I, chap. xv.). If Mm = h, then Om = x+ h, and mp = f (x + h),Rp = i(x + h)-ix; so that *(* + h)-ix = Jg = tan Rp h PR r Now if the angle xTP is denoted by i/r, then since the direction of the tangent TP is the ultimate direction of the chord Pp when the point p has approached in- definitely near to P, therefore Rv tan i/r = It tan RPp = lt^yij ,,f(a: + A)— fa: cZfa; „, dv 7 ,, = lt j = ^, or ix, or ^ or dy/ds, since y = ix. By giving x a decrement h as well as an increment h, we may define a differential coefficient by the relation dix_, i(x+h)-i(x-h) dx~ 2h ' which is sometimes more elegant and useful than the definition in § 3, to which it is ultimately equivalent. Figure 1 shows that dy/dx or tan \p- is positive if y increases with x, but that dy/dx is negative if y dim- inishes as x increases; and thus dy/dx is zero at the turning points, such as B or G, where y from increasing begins to diminish, or from diminishing begins to increase. 8 DIFFERENTIATION. At B, y is said to have a maximum value, and at G a minimum value ; and in each case dy/dx = 0. Thus to discover the maximum or minimum value of y, a function of x, we must first find the values of x which make dy/dx = 0. Afterwards we must determine whether, as x increases continually, dy/dx changes from positive to negative, in which case y has a maximum value ; or whether dy/dx changes from negative to positive, in which case y has a minimum value ; we shall return to this subject here- after, although the present application to simple cases is easy. Thus ax — a? has a maximum value \a?, when x = \a (fig. 33); and 2sc 3 — 9a; 2 +12x — 3 has a maximum value 2 when x = l, and a minimum value 1 when x = 2 (fig. 29). Imagine the curve ABC to be a road, and dx a small horizontal forward step, dy the consequent vertical step ; then dy and therefore dy/dx is positive if the traveller is ascending, and dy and dy/dx is negative if the traveller is descending, supposing dx always positive. At the turning points,, such as the top of a hill or the bottom of a valley, where y is a maximum or minimum, dy/dx = 0. The road is steepest where dy/dx has its maximum numerical value', and then the differential coefficient of dy/dx will be zero. We may call tan yp- or dy/dx the gradient of the road, the ascent being dy/dx feet vertical for one foot hori- zontal ; or otherwise expressed, as on railway plans, the gradient is 1 in cot i/r or dx/dy, meaning 1 foot vertical for dx/dy feet horizontal. DIFFERENTIATION. 9 6. The Tangent and Normal of a Curve. If x', y' are the coordinates of any point P' on the tangent TPP' to the curve APB at P (fig.' 1), then fci = S or y'-y=i (x '- x) ' the equation of the tangent. TP. 2W is called the subtangent, and ilf (? the subnormal at P ; P6r being the normal at P, drawn perpendicular to the tangent through P. Therefore TM = y cot \js = ydx/dy ; MG = y tan \{r = ydy/dx. If T is to the right of M, tan \fr is negative; and MT= —ydx/dy, QM= -ydx/dy. Also the equation of the normal GP at P is y'-y=-o% (a! - x) - Let us apply these principles to some simple curves, the graphs of (i.) y = x 2 , (ii.) y = x$, (in.) y = %~\ (iv.) y 2 = x 3 . (i.) y=x 2 , a parabola (fig. 2 i.). Here dy/dx = 2x ; so that the equation of the tangent is y' — y — 2x(x' — x) ; which, since 2/ = cc 2 , may be written x' y' 2xx' = y'+y,ov^- y =l. Thus 0T= \x, TM= \x, and therefore T bisects 0#. The equation of the normal is aZfrz^ '7TTI ' (ii.) y = x^, another parabola (fig. 2 ii.). Here dy/dx=\x~^ = \y/x; so that the equations of the tangent and normal become il_^=i- x ' + y' -1- Jy a; ' x + i 2y 3 +y thus TO = OM, 0V= VN, and itfG=£. 10 DIFFERENTIA TION. N T M (?) Fig.2 (iii.) y = x~ 1 , a hyperbola (fig. 2 iii.) 1 Here dx x* x' and the equation of the tangent becomes 2x 2y ' so that 0T=2x, 0V=2y. The equation of the normal is xx' — yy' = x 2 — y 2 . (iv.) y 2 = x s , or y — x^, a curve called the semieubical parabola, and something like figure (i.). Here dy/dx = fas*, so that TM= f as, 0T= \x = \OM. (v.) Prove that in the curve y m = cx n , the equations of the tangent and normal are ^— = — — -, mxx' 4- nyy' = tux 2 + ny 2 . mx ny m n aa * 7. Supposing h again, as at first, to denote a finite increment of x, and supposing that neither ix nor i'x becomes infinite between P and p, and that they change gradually between these points ; then in the graph of fee * Articles which may be omitted at a first reading are marked with an asterisk* DIFFERENTIA TION. 11 the tangent TK at some point K between P and p (fig. 3) is parallel to the chord Pp ; and i( - X+ ^~ ix = tB,nRPp=t^nxTK=r(x+6h), where x+9h=0H, the abscissa of K; and then 6 = MB /Mm, a proper fraction, which is some un- known function of x and h. Therefore i(x + h) = ix + M'(x + Qh), a theorem required subsequently in Chapter IV. This leads to another definition of i'x, employed by Lagrange, according to which i'x is defined as the co- efficient of h in the algebraical expansion of i(x+h) in ascending positive integral powers of h. To illustrate geometrically exceptional cases of ix and i'x, consider f x = (x — b)/(x — a), or f x = (x — a) s , taking xa. Given that ix = a + bx+cx 2 , then 6 = i ; and generally, when h is small, we shall find 6™% ; the symbol ££ denoting approximate equality. 1 2 DIFFERENTIA TION. If k denotes a different increment of x, then f (x + k) = tx+ M'(x +k), where is a proper fraction, the same function of x and k which 6 is of x and h : so that i(x + h)-ix _h f'(x+6h) i(x + k) - ix k i'{x + k)' Ultimately, when h and k are sufficiently small, we may put i'(x+6h) _ 1 i\x + k) ' without sensible error ; and then i(x+h)-ix _h i(x+k) — ix Jc which is the rule of proportional parts, employed in Trigonometry ; equivalent to neglecting the curvature of the arc Pp, and replacing the arc by a straight line. Again supposing APp to represent the profile or elevation of a road, and i'x to represent the slope or gradient at P, ¥(x + h) at p, then the gradient i'(x+9h) at K is the average gradient of the chord Pp, repre- sented by {f (x + h) — ix}/h. 8. Instead of the letter h the symbol Ax is often em- ployed to denote the finite increment of x ; and Ay is then used to denote the corresponding increment of y, where y is a function of x, denoted by ix. Then, with this notation, y+Ay = i (x+Ax), and Ay = i{x+Ax) — ix\ so that Ay = Ep, if Ax = PR (fig. 1) ; and JUt^. ax Ax DIFFERENTIA TION. 13 In this notation, A (Delta) and d are not algebraical factors, but symbolic operations; so that Ax and dx must be considered as inseparable quantities, the x being quali- fied by the A or d, and not multiplied algebraically by it. Suppose for instance it is found that the difference of range of a gun over the sea at high and at low water is Ax, when Ay represents the rise and fall of the tide ; then the tangent of the angle of descent of the projectile into the water is Ay/ Ax, approximately. 9. Again, let the length of the arc AP of the curve APp, measured from any fixed point A to the variable point P, be denoted by s (fig. 1). Denoting by As the increment of s, corresponding to the increment Ax of x, then the arc APp = s+ As, and the arc Pp= As. Now, when the point p approaches to coincidence with P along the curve pP, it is assumed as axiomatic that , .chord P» , it = 1 ; arc .r-p ltx /(P^ + i^) =1 . arc Pp lta /(**+Ay') As \2. where Ax 2 means (Aas) or itf^+^Vi Therefore ~ 2 + 'jjL = 1 , or, as it is sometimes written, (d*\* + (dy\* = L \ds/ \dsJ 1 4 DIFFERENTIA TION. The assumption, which has been taken as axiomatic, that It (chord Pp)/(a,rc Pp) = l, is proved in Newton's Lemma VII. (Principia, Book I, § 1), as follows — P Fig.4 As the point p approaches to the point P, let the chord Pp, and the tangent Pv be produced to points Q, V, at a finite distance, where QV is parallel to pv, and the arc PQ is similar to the arc Pp; so that PQVm&j be con- sidered the magnified image of the similar microscopic figure Ppv. Now let the point p move on the curve pP close up to P, and let pv move parallel to itself up to P ; then the angle QPV always diminishes and ultimately vanishes : and the chord PQ, the arc PQ and the tangent PV are ultimately coincident; therefore also the chord Pp, the arc Pp and the tangent Pv, when vanishing together, will have a ratio of equality. PR Again (fig. 1) cos \/r = cos xTP = It cos RPp = It- — _i,Ax., arc Pp _dx . As chord Pp ds ' RP and sin • - *J(x 2 + a 2 ) , da; a; 2 +a 2 (a; 2 -|-a 2 ) (viii.) y = x /(2aa? - x 2 ), -£- = , ""^ «> • ^ eta /V /(2aa; — a; 2 ) (2) Differentiate with respect to a;, +a; 7 -faV+a 4 a; 3 -a 6 a:, xj(l-a?), x{\-2x 2 )J(l-x 2 ), a—x ja—x 2x Sx+x s 4a: — 4a; 3 5a;+10a; 3 +a; 6 _ a + x ^a+x'T^x 2 ' 1 + 3x 2 ' 1 - 6a; 2 +a: 4 ' 1 + 10a: 2 + 5a: 4 ' (3) Deduce the rule for the differentiation of a quotient from the rule for the differentiation of a product. 22 DIFFERENTIATION. 13. Implicit and Explicit Relations between two variables, x and y. When the variables x and y are connected together by any relation, for instance by ax 2 + 2hxy + by 2 + 2gx +2fy + c = 0, the general equation of a conic section, or by x s — 3axy +y 3 = 0, the equation of a cubic curve, this relation is called an implicit relation between x and y, and is denoted gener- ically by the notation F(»,y) = (A). When however it is possible by solution of the equation to obtain y in terms of x, or x in terms of y, then y is called an explicit function of x, or x of y, denoted generically as before by y = fee, or x = <$>y (in the notation of inverse functions, § 25, x = i~ 1 y). In the above implicit relations between x and y, we cannot obtain y explicitly in terms of x, or x in terms of y, except by the solution of a quadratic or cubic equa- tion ; but, for instance, from the implicit relation x 2 y 2 — a\x 2 — y 2 ) — 0, the equation of a quartic curve, we obtain explicitly, by mere transposition, 2 _ a 2 x 2 2 _ a?y 2 V ~a 2 +x 2 ' mX ~~a 2 =y~ 2 ' From an implicit relation (A) between x and y to find dy/dx, it is not necessary to express y as an explicit function of x ; but we differentiate (A) with respect to x, treating y as a function of x, and obtain what is called the first derived equation and thence determine dy/dx as a function of x and y, without solving the equation (A) ; DIFFERENTIA TION. 23 thus from the implicit relation or equation of a conic section we obtain the first derived equation. so that ty= _ax±hy±g dx hx+by+f For changing the coordinates x, y of a point on the conic section to x+Ax, y+Ay, the coordinates of an adjacent point on the curve, then ¥(x+Ax, y + Ay) = a(x + Ax) 2 + 2h(x + Ax)(y + Ay) + b{y + Ay) 2 + 2g(x + Ax) + 2f(y + Ay)+c = 0; and therefore, by subtracting F(x, y) = 0, a(2xAx + Ax 2 ) + 2h(Ax . y + xAy + Ax Ay) + b(2yAy + Ay 2 ) + 2gAx+2fAy = ; and dividing by Ax, a(2x+Ax) + 2h(y+x^+Ay) + b(2y+Ay)^ + 2 9+ 2f^ x = 0; which in the limit, when Ax and Ay are indefinitely small, becomes the first derived equation. Again, from the implicit relation x 3 — Saxy + y s = 0, we obtain the first derived equation 3x 2 ~Say-3ax^+3y 2 '^ c = 0, ; so that ^x^-ay ax ax — y' 24 DIFFERENTIATION. Examples. — Find dy/dx from the implicit relations (1) x 2 +y 2 = c 2 . (4) x$ + y* = a*. (2) ( x -a) 2 + (y~b) 2 = c 2 . (5.) x m y n = a m+n . ,~ oc" y 2 (6) x*+y°-5a?x 2 y = 0. {3) a 2± b 2 = l - (7) Given the implicit relation X s — Saxy + y 3 = 0, prove that dy/dx = 0, when x = a%/2, y = a%Jk; dx/dy = 0, when x = -2Xas-4)' "" (8) x 2 -4:x + 2y = 0,xy-2x-y = 0, (y-xf = \-x 2 . (9) Prove the properties of the following curves and sketch the curves : — (i.) In y 2 = p£C,aparabola, the subnormal If G is constant, (ii.) In x 2 + y 2 = a 2 , a circle, the normal PG is constant, (iii.) In x 2 + y 2 = 2ay, a circle, OT=TP. (iv.) In x^+y^-a^, the part of the tangent TV inter- cepted by the axes is of constant length, (v.) In by 2 = x 3 , a semicubical parabola, TM 2 = ^jb.MG; and generally in y m = cx n , MG varies as (TM)<- 2m - n ^ n . Ill x n+1 (vi.) In — H — - = — , 07= — — ; and determine p, the \ / x n yn a n> a n ' r< perpendicular from upon the tangent. (10) Prove that the equation of the tangent at (xy) of (i.) The circle (fig. 13) x 2 +y 2 = c 2 is xx'+yy'=c 2 ; or of the circle (x — af + (y — b) 2 = c 2 is (x-a)(x'-a) + (y-b)(y'-b) = c 2 ; (ii.) The ellipse (fig. 8) £+|J=l is ^-'+^ = 1 ; 26 DIFFERENTIATION. (Solution : — Forming the first derived equation, 2x , 2y dy ~ ,, , dy b 2 x . -a+TT -, =°< s0 tnat j = 2~ a 2 6 2 cfo da; and then s/ x s/V 0T+0V=a; and generally, the equation of the tangent of x n + y n = a n is x n - 1 x'+y n - 1 y' = a n . DIFFERENTIATION. 27 (12) Prove that if ( W =4*.,^( 1+ !) ; (13) Prove that the conic section ax? + 2hxy + by 2 +2gx + 2fy + c = can be described with component velocities dx/dt =hx+by +/, dy/dt =—ax—hy—g, or dxjdt= J ' {(h 2 -ab)x 2 + 2{fh-bg)x+p-bc}, dy/dt= - ,J{(h 2 -ab)y 2 + 2(gh-af)y+g 2 -ac}. Prove also that the equation of the tangent at (xy) is (ax +hy+ g)x' + (hx + by +f)y'+ gx +fy + c = 0. 15. Algebraical and Transcendental Functions. So far we have dealt only with algebraical functions of x, an algebraical function being denned as one which is composed of a finite number of the algebraical oper- ations of Addition, Subtraction, Multiplication, Division, Involution, and Evolution. An algebraical function of x may be denned in the most general manner as the root of an algebraical equa- tion of integral degree, the coefficients of which are rational integral algebraical functions of x; and a rational integral algebraical function of x is denned as the sum of a finite number of terms, multiples of integral powers of x, such as ax n +bx n - 1 +cx n - 2 + ... , where n is a positive integer ; and then n, the highest power of x, is called the degree of the function. 28 DIFFERENTIATION. But now we pass on to the functions of Trigonometry, such as sin a;, cos a;, ... , which are called transcendental functions; every function of x which is not an alge- braical function being called a transcendental function. Suppose we require to differentiate sin x ; then from the definition of a differential coefficient in § 3, d since , sin(x + h)— sing; . 2cos(x+$hjsm%h dx h h m i . 1 7 n sin hh = ltcos (x + $h) -^-r— = cosx; since It cos(cc + J/i) = cos x, and lt(sin %h)/^h = 1, when x and therefore h is expressed in circular measure. Or, more simply from the definition of § 5, cZsina; ,,sm(x + h)—sm(x — h) ,, sinh — , — = It — - — ■ — -ri = It cos x—r — = cos x ; dx 2h h and so on for the other functions of Trigonometry. But if x is given in degrees, or minutes, or seconds, then dsinx° _ it o = " dd • tttt- = cos a;. arc PQ Pi? arc PQ cZcosa; , OL-OM dx ~ arc PQ ,.uR chord PO = — sin x. = _n3^_= -H^ 2 chord PQ arc PQ PR arc PQ With the definition of § 3, dtanx _, tan (x + h)— tana; . fat da; A ~~ arc rP ,, Ou u to , ,» im -, arc fay 0£ since _=-_,lt — = 1 (§ 16), and d = ttt ; to OA arc fau arc rP CM. <2seca:_, wu _,,wu vu tv &rctw dx arc rP vu tv arc tw arc rP = tan x sec as : since .., wu , n .vu -Mm . ., to , avctw It — = 1. It — =lt7^-r = tana;, It — = 1, n = seca:. vu tv OA axctw arc rP 19. By means of the formula of § 11 for the differenti- ation of a function of a function, we can differentiate any power of the circular functions of x, and the circular functions of any function of x ; thus d(sin x) m /dx = wi(sin x) m ' 1 cos x ; ' d sin y/dx = cos ydy/dx ; d(sin y) m /dx = m(sin y) m ~ x cos ydy/dx. As an exercise, the student may in the same manner differentiate with respect to x the functions (cos a:)", (tana;)' 1 , (cot a:)", (sec a)", (coseca:)™, (versa;)", (fa;)™, sin fa;, {f(sina:)} ra , (cos fa;)",... 34 DIFFERENTIA TION. Examples. — (1) Prove the following differentiations : (i-) y = ix+lsm2x, dy/dx = cos?x, (ii.) y = \x — £ sin 2a;, dy/dx = sin 2 x. (iii.) y=\ cos 3 a; — cos x, dy/dx = sin 3 aj. (iv.) y = sin x — \ sin 3 a;, dy/dx = cos 3 a;. (v.) y = \x-\-\ sin 2x + ¥ \- sin ix, dy/dx = cos 4 a;, and write down y when dy/dx = sin 4 a;. (vi.) 2/ = tana; — x, dy/dx = ta,n 2 x. (vii.) 2/= —cot x— x, dy/dx = eot 2 x. (viii.) 2/ = i tan 3 a; — tan x-\-x, dy/dx = tan 4 a;. (ix.) y=\ tan 3 a; + tan x, dy/dx = sec 4 a;. (2) From the definition of § 3, differentiate with respect to x, sin 2a:, cos x/a, tan(ma; + 'n,), x tan x, tan x 2 , sin x n . (3) Differentiate x n , sin a;, tana;™, with respect to x m , 20. It will be useful also to draw the graphs of the circular functions ; thus the equations y/b = ± sin 2irx/a and ± cos 2-jrx/a represent in fig. 7, for two values of b, the plan of the spirals of a double-threaded screw of pitch a. Fig.7 DIFFERENTIA TION, 35 Examples. — Draw the curves: (1) y = sin x, cos a;, tana;, cot x, sec x, cosec x, vers x (on one diagram). (2) y = sm.- 1 x, cos -1 *, tan" 1 ;?;, cot" 1 ^, sec" 1 **:, cosec -1 ^, vers _1 a; (on one diagram) ; and superpose them on the preceding diagram. (3) y = xsmx, (sincc)/x, (sin a) 2 , sin a; 2 , sin^/as, a;cot|-7ra;. (4) sin x = sin y, cos x = cosy, tan x — tan y, (sin x) 2 + (sin 2/) 2 = 1, (tan x) 2 + (tan j/) 2 = 1. (5) Prove that in the ellipse (a:/a) 2 + (y/b) 2 = 1, we may put x = a cos 6, y = b sin 6 (6 is then called the excentric angle); and A0p=6, and the equations of the tangent and normal are (fig. 8) (x/a)cos 6 + (y/b)sin 6=1; ax seed— by cosec 6 = a 2 — 6 2 . (Fig. 8 shows the elliptic trammel or compasses, the theory of which is obvious from this example.) 36 DIFFERENTIA TION. (6) Prove that in the curve x§ + y% = a$, we may put x = a{cosQf, y^a(sm6') 3 ; and then p = lj\axy); and the equations of the tangent and normal are asec0+2/cosec0 = a; xcos6—y smd=acos2d. 21. The Cycloid. As an illustration of the use of a variable angle or parameter for expressing the coordinates x and y of a point on a curve, consider the cycloid, the curve traced out by a point in the circumference of a circle which is rolling on a plane ; a curve often seen described by a piece of paper sticking to the rim of a carriage wheel. Starting from the origin where the point is origin- ally in contact with a horizontal plane, the point P describes the curve OP, and when the wheel has turned through an angle 6, the point P will have ascended a vertical height MP = a vers 9, while P will have advanced horizontally 0M=a6 — a sin 6, a denoting the radius of the wheel (fig. 9). y H A 1 p a n \ ^\£>' C~ N c \ P ' V V V TO A f c D Fig .9 B Thus x= OM =a(d- sin 0), y = MP = a vers 0; and the elimination of 6 gives x = a vers ~ 1 j//a — s /(2ay — y 2 ), the equation of the cycloid in x and y. DIFFERENTIA TION. 37 The complicated nature of this equation makes it pre- ferable to retain the angle d ; and now dx n dv . „ pavers 6=y,^ = asm0, so that PG, the normal to the curve, passes through G, the point of contact of the wheel with the ground. This is obvious- if we consider that the wheel is in- stantaneously turning about G, that is, G is the centre of instantaneous rotation ; and then GP is the normal, and TPH is the tangent. = a^/ (vex s 2 # + sin 2 (9) = 2a sin \ 6 ; ds ( 2a \l c% = C0SeC + = seC $ e= ka^) ■ Any other point P fixed on the wheel at a distance b from the centre will describe a curve in which x = a9 — b sin 6, y = a — bcos6; these curves are called trochoids ; and GP is still the normal at P. For a point on one of the spokes of a wheel, b < a, and the trochoid is called curtate. When b > a, the trochoid is called prolate ; a point on one of the floats of a paddle wheel describes a prolate trochoid. 22. Polar Coordinates. If OP is denoted by r and the angle xOP by 0, where P is any variable point on a curve AP (fig. 10), then r, 6 are called the polar coordinates of P. They are connected with the former coordinates (x, y) of § 5 by the relations x = r cos 6, y = r sin 6 ; or r = > /(x 2 +y 2 ), 6 = ta,n- 1 y/x. 38 DIFFERENTIATION. If r = £6 is the polar equation of the curve AP, and if xOP = d, then OP = r = id; so that the curve AP is, with polar co-ordinates, the graph of the function f 6. Such graphs are required, for instance, in plotting out the turning moment of an engine at any part of the revolution, or in drawing a cam in mechanism. When P moves to an adjacent point p on the curve, suppose the polar co-ordinates r, 6 to receive increments Ar, A6; so that r + Ar, 6 + A9 are the polar co-ordinates ofp; then xOp = 6+ Ad, P0p = A6; and 0p = £(d+A6) = r+Ar: Rp = f(6 + Ae)-W = Ar; also PR=rAd; if PR is the arc of a circle described with centre 0. Fig.lO Draw the straight line PR' perpendicular to Op ; then .PR , sinA0 , U R'R u versA0 _ PR = tr ~Ad~ = 1 ' b y§ 16 ^ andlt-p-£ = lt— ££— = 0,so that lt& Mp 1. DIFFERENTIATION, 39 (In the language of Infinitesimals, PR and PR' are called infinitesimals of the first order, and R'R an infinitesimal of the second order.) Now if the angle between OP produced and the curve Pp, or the tangent at P, is denoted by 0, then is called the radial angle, and + ^ u+ no iJ'-R' n-P-R' JBp Pi? ,,PR (since It-po = 1, and lt-p^= 1) _ u rA6_rd6 t , dr dlogr, and cot^=^=^|- anticipating the result of § 28. 23. If the arc AP, measured from any fixed point A up to the variable point P, is denoted by s, and if Pp, the increment of the arc, is denoted by As ; then since, as in § 9, u chord Pp _-t arc Pp r. i, ^ r, i.R'v i.Kp arcPp Rp therefore cos * = It cos OpP = It -^ =lt^ ^^ ^^ = lt^-(since lt^ = l, and m£"**E = 1) arc Pp v Pp arc Pp __. Ar_^r — As~~ds' PR' Similarly sin = It sin OpP = lt-p— — PR = It— = — , — arc Pp As ds ' Therefore g+^ = cos^+sinV = l. 40 DIFFERENTIATION. If Ot is drawn, at right angles to OP to meet the tangent at P in t, then Ot is called the polar swbtangent; and 0t = r tan = r 2 dd/dr. It is convenient for subsequent purposes to represent the reciprocal of r by u; and then r = -, -35^ 2.7j5! so that 0t= — dd/du. ■ If iO produced meets the normal at P~ in #, then Ogr is called the polar subnormal ; and , Og = r cot = dr/dO. Looking along OP from 0, t will be to the right and g to the left when dr/dQ is positive ; and vice versa if dr/d6 is negative. The tangent Pt = r sec = rds/dr; and the normal Pg = r cosec = ds/cW ; and denoting the perpendicular OY from the origin upon the tangent at P by p, then p = r- sin = r^dQjds. ' Again, from fig. 10, _!_-!+ 1 OY* Ot 2 VW 1 _fdu\ 2 p 2 a useful expression for p. Employing a dynamical interpretation as before in § 10, and supposing P to move in the curve AP (like a planet round the sun at 0) so that its polar co-ordinates r, 6 are given functions of the time t; then the component velocities of P in the direction OP and in the direction PS perpendicular to OP are, by resolution, ds dr ,ds : . rdd ■ ... , dt cos * = Tt and dt sm *=W' res P ectlvel y ; these are called the radial and transversal velocities of therefore V 2= \de) +V? DIFFERENTIATION. 41 P, the resultant velocity being ds/dt in the direction of the tangent tP. Therefore ™Jg ?*ap dt 2 dt 2 ^ dt 2 ' and by making t = 0, or r, we obtain the relations ds 2 _dr 2 2 de 2 ~dO~ 2+r ' cZs 2 rW , dr 2_ tir 2 ' while making £ = s gives, as before, ds 2 ds 2 It is important to have the power of plotting curves from the equations in polar co-ordinals, so the following examples should be worked, as preliminary practice. Examples. — -Draw the following curves whose equa- tions are given in polar co-ordinates : — (i) r=i;e, e\ s/e.e-i, e-\e-\..: (2) r = cos 0, cos 20, cos 30, cos 40. (3) r = sin 0, sin 20, sin 30, sin 40. (4) r = sec 9, cosec 0, tan 0, cot 0. (5) r = cos£0, sin£0, cos£0, cos|0, cos£0, cos|0, cosf0, cos f 0, sec \9, cosec £0. (6) r 2 =sin 20, cos 20, sec 20, cosec 20. (7) r = vers0, l+cos0, (versfl)" 1 , (1+COS0)- 1 , sec 2 £0, cosec 2 £0, (1 - cos a cos 0) _1 , (cos a - cos0)~\ cos — cos a, 1 — cos a cos (take a = \ir). (8) r = acos0 + 6sin0, asec0 + & cosec 0; l_cos0 , sin0 1 __ cos 2 . sin 2 ^ r a~ b ' r 2 a 2 ''' ¥ 42 DIFFERENTIATION. (9) Prove that in the curves (i.) r = a, 4> = %tt. (ii.) r = a sin 6, = 8. (iii.) r = 6 cos 6,

= ir — n6; and the curves (vi.) and (vii.) cut at right angles. (10) Prove that the curves r" = a n cos,(nQ — a) and r" = a n cos(n0 — j8) cut at an angle a — /3. (11) Prove that Ot is constant in r=a/6 (the reciprocal spiral) and Og — aQ; Og is constant in r=a6 (the spiral of Archi- medes, fig. 12, ii.) and Of = a0 2 ; Pg is constant in r = a sin (the circle). Find the locus of g and of t in the circle r = acos 6. (12) Prove that the locus of t is the straight line lu = ecos 6 in the curve fot= l/r = 1 + e cos 6 (a come section). Prove also that if 6rZ is drawn perpendicular to OP (fig. 10), then PL = l. (13) Prove that if x and 2 are the radial angles of the locus of t and of g, cot fa + cot 2 = 2 cot 0. (14) Prove that if, with given elevation of a gun to the horizon, the range of the projectile on the level is r yards, and if on a slight descending slope of one in m the range is increased hy Ar yards, the cotangent of the angle of descent is mAr/r, approximately. DIFFERENTIATION. 43 (15) Given that 5' extra elevation or depression of a gun increases or diminishes the range of r yards by Ar, prove that the cotangent of the angle of descent is about Ar cot 5'/r, and the slope of the descent is one in 688 Ar/r. (16) Prove that in (i.) r = a vers (the cardiod), ds/dd=2asm$Q. (ii.) r = 2a/(vers 6) (the parabola), ds/dO = a(cosec \&f <*•> *=^£=V( i+ s> <-or=«/e,:£=v( i+ £ 24. It has already been assumed in § 5 that dy/dx and dx/dy are reciprocal, or that their product is unity, i/ being any function of x and a; therefore a function of y. The proof, if any proof is required, may be given thus : if Ax is any increment of x and Ay the corresponding increment of y, then always ty X — = 1- Aa Ay ' and therefore proceeding to the limit, dx dy or dy/dx and dx/dy are reciprocal ; provided however the values of x and y are the same in each, in the case of many valued functions ; that is, functions y of x, which for any given value of x have more than one value of y, or vice versa; thus, if y = sin - 1 x, or x = sin y, then for any given value of x, y has any value comprised in the formula mr+(-l) n y. This theorem is required in the 44 DIFFERENTIATION. 25. Differentiation of the Inverse Circular Functions. If y is some function of x, denoted by ix, then x is some function of y, which it is convenient to denote by f -1 2/; so that f and f _1 denote functions inverse to each other, such that f ~ 1 (£x) = x, and i(i~ 1 y) = y. Thus in Algebra x n and Zjx are functions of x inverse to each other; for (£jx) n = x and 1!j{y n ) = y. In Trigonometry it is usual to employ the abbreviation sin 2 , . . . for (sin x) 2 , (sin x) 3 , . . . with positive powers ; but sin _1 a; is never used to mean 1/sin* or cosecaj, but to denote the cm. of an angle whose sine is the number x ; and so on for the other circular functions. To differentiate sin _1 a;, let sm~ 1 x = y, then x = siny, and dx/dy = cos y = ^/(l — sm 2 y) = ^/(l — x 2 ) ; ,, c dy 1 c£sin _1 a; 1 therefore -^- = — ^ s-, or ^ — = — ; -. dx y/{l-x 2 )' dx y/{l-x 2 ) Smnlarly — ^^=^^^- Let cos _1 a3 = 2/, a; = cosy, dx/dy = - sin y = - „y(l - cos 2 ?/) = - ^/(l - x 2 ), dy _dcos~ 1 x_ 1 dx dx ^(l—x 2 )' dcos~ 1 x/a_ _ 1 dx ,^/(a 2 — x 2 )' Since cos " x xfa + sin ~ ^/a = |7r, therefore dcoB-V + d8m-ya = () cue cfcc i dcos _1 a;/a__ _cZsm _1 x/a_ _ 1 dx ■ dx ' ,J{o? — x 2 )' thus illustrating the reason for this relation. and DIFFERENTIATION. 45 Again, let ta,n.- 1 x = y, then x = t&ny, dx/dy = sec 2 y = 1 + ta,n 2 y = 1 + a; 2 , and ^y_ ditan- 1 a! _ 1 dec dx 1 + x 2 ' ' also dtan-yo ^. a cte a 2 +a: 2 Let cot _1 a; = 2/, cc=cot2/, dx/dy = — cosec 2 2/ = — cot 2 i/ — 1 = — x 2 — 1, ' dcot _1 tc 1 and do; x 2 +l' dcot~ 1 x/a_ a dx x 2 + a 2 ' Since cot ~ x x/a, + tan " x x/a = \ic, ,, „ dcot'^/a . dta,n~ 1 x/a n theretore = — '— + , — '— = 0, dx dx ■, dcot~ 1 x/a_ _dta,n~ 1 x/a_ dx dx a 2 +x 2 ' Let sec~ 1 x = y,x= sec y, dx/dy = sec y tan y = x,J(x 2 — 1), dsec _1 a;_ 1 dx x s /(x 2 — Yj „,. ., , dcosec _1 a; 1 Similarly die = ~ x /(x 2 -l \ Let vers ~ h>/a = y,x = a vers y, dx/dy = a sin y = a^/(l — cos 2 y) = a x J(2 vers y — vers 2 y) = A J(2ax — x 2 ) ; d vers " x x/a 1 da; M J{2ax — x 2 )' 46 DIFFERENTIATION By the rule of § 11 for the differentiation of a function of a function, if cc is a function of t, dsin.~ 1 x_dsin~ 1 x dx _ 1 dx , ~llt = dx dt~ jQ.-x^dt' or dsm~ 1 x = dx/ > /(l—x 2 ), in the notation of differentials; and so on for the re- maining inverse functions. 26. The inverse circular functions are not much re- quired in Elementary Trigonometry, but are indispens- able in the Differential and Integral Calculus : so the principal formulas of Trigonometry, expressed by the direct functions, are given here on the opposite page, with the corresponding inverse notation. Examples. — (1) Construct a Table giving any direct circular function in terms of any other, and differ- entiate them as a verification. (2) Construct a similar Table for the inverse circular functions, and differentiate them. (3) Given 3/ = icos" 1 (2x 2 -l), J cos -\4a?- 3x), Icos-^Sxt-Sxt + l),... ,i L dy l prove that ^-^--^—^ (4) Given y = Jsin~ 1 2a; x /(1— a; 2 ), £sin -1 (3a; — 4a: 3 ), \ sin" 1 4a;(l- 2a= 2 ) x /(l -a; 2 ),... prove that g = -^L-^ 2tc 2a; 1 a; 2 (5) Given i/ = | tan "Jy— -g, ^sin" 1 ^-^, \cq^^— % , _ 1 3x — x s , , _ 1 4>x — 4ix s 1 _ x 5a; — 10a ; 3 + a; 5 * tan T=W' 4 tan I=6a?+^' T tan 1 - 10a 2 + 5af ' ' ,. , dv 1 prove that ^ = 1+? DIFFERENTIA TION. 47 W-, '-"-> N~"~ d d • r-l 02 03 O o • d o 03 U "d" h-=> n3 d 48 DIFFERENTIA T10N. *(6) Differentiate, f^tan-^x + l)/^, _,. , /l— cosa; . 1/N /(l+ce— a; 2 ) 2tan- 1 A / T - , sin- l VV , ,_ -, \l + cosa; J^/5 _ x // l—x+x 2 \ . _j 2 since C0S V\3 + 3a: + 3a;V' Bm ' v't.o-ecosaj+Scos 2 ^)' *(7) Differentiate f _1 a; with respect to x. 27. The derivative of some of these inverse functions can be obtained almost as simply by the direct process, from the definition of § 3 ; thus d tan ~ x x _ , ,tan ~ \x + h) — tan ~ he dx h ,.1 . . x+h — x = lt r tan - Y- — — - T r- = ldtaa-S A * A l+a?+xk 1+a; 2 ' because lt(tan" 1 z)jz = It 6>/(tan6>) = l, when and z = 0; and here tan 8 = z = 5 . \+x z + xh *Example. — Determine in the same way the d.c. with respect to x of sin _1 cc/a, cos _1 aj/a, tan _1 ai/a, cot _1 fl3/a, sec~ 1 x/a, cosec -1 a:/a, vers _1 cc/a. A™ rf sin ' tya _ . sin - \x + k)/a — sin ~ tya . V da; — A, — -■-A( i -S)'-iv{ i -e-?)> and 2 = 0, when A. = 0, so that lt(sin - V)/z = 1 ; while lt g -lt (a?+A) ^ a '~^>" a! ^ '-( a! + ft )^ ° h a 2 k (and rationalizing the numerator) DIFFERENTIATION. 49 ^ ,■ {x + h)\a?-x 2 )-x\a?-{x+K?} _0 a'hKx + ^J^-x^ + Xy/i^-ix + h) 2 }] 2x + h 1 = lt 7 (x + h)J(a?-x 2 ) + xJ{a 2 -{x + hf} ^(^-x 2 )' , ,, dsin-^/a 1 , „ \ and thus -5 — — = . 2 _ 2 . , as before ). 28. 2%e Exponential Theorem. Definition of the number e, and differentiation of logx and a x . Before d log x/dx or da x /dx can be found, the number e, called the base of the natural or NaperUm logarithms, must be defined. For, from the properties of logarithms, d log g x _ lt log g (a3 + h) — logqce dx h = l lt lo 8 .(l+^) = l ltl oga4 /(i +g ), x A/a; a; ° ^ when z = 0, on putting h/x = z. But now we must determine the value, when z = 0, of 4/(1+2:); or, on putting z = l/m, the value of (l + l/m) m = (l + 0) 1/0 , when.m is indefinitely great. When m is an indefinitely large number, we may suppose it to be an integer ; and now, by the Binomial Theorem, lt(l + l/m) m , when m = 00 , = (1 + 0) 1 / , ,, f, 1 , m(m— 1) 1 , m(m — l)(m — 2) 1 ) == It|i +m _ + ^^__ 2+ j-— - 8 +...| . i-i (1-IV1- •-lt11 + 1 1.2 1.2.3 ^1^1.2^1.2.3^ to! (denoting the product 1.2.3...%, called factorial n, by to !) 50 DIFFERENTIA TION. •16666667 •04166667 •00833333 •00138889 ■00019841 •00002480 •00000276 •00000028 = 1 = 1/1 = 1/2 = 1/3 = 1/4 = 1/5 = 1/6 = 1/7 = 1/8 = 1/9 = 1/10! 2-7182818... an incommensurable number, denoted generally by the letter e, and called the base of the natural logarithms. 29. Now replacing m by 1/z, so that z = when m = oo , then, when z = 0, ^/(l+z) = e ; and lt.log a 4/(l+z)=log a e, so that and d\0gqX = l l0gae> d log x _ 1 dx x and by the rule for the differentiation of a function of a function, ,f =- -^ (§ 20), when the base is e. Logarithms to base e are called natural logarithms ; natural logarithms are intended in this subject when no base is indicated, and not common logarithms to base 10, as in ordinary numerical calculations ; so also angles are always reckoned in circular measure, and not in degrees, minutes, or seconds : in this way extraneous factors are DIFFERENTIATION. 51 avoided, although we must return to the other measure- ment when we wish to employ mathematical tables for numerical calculations ; since logarithms are tabulated to base 10, and the circular functions are tabulated to degrees and minutes in the ordinary tables. and log 10 e = 0-43429448, called M, the modulus of the common logarithms. Again ^ = lt^^ = rf^i. dx h h Now let a h —l = z, therefore a h = l + z, and h = log a (l+z); also z = when h = 0. Therefore ^ = ^,--^ = 1 1 * ^ (fo l0g a (l + Z) l0g„4/(l + 2) a * si log a e Thus ^-=i^ = 10* x 2-30258509. 30. The Exponential or Logarithmic Curve, and the Equiangular or Logarithmic Spiral. Logarithmic Coordinates. The exponential or logarithmic curve is the graph of a x ; so that its equation is y = a x , or x = log a y. It is the curve showing the rate at which a quantity grows in geometrical progression, or at compound in- terest, and is approximately the curve seen passing through the top of a row of organ pipes (fig. 11) ; equi- distant ordinates being in geometrical progression. 52 DIFFERENTIA TION. Then dy/dx = a x \oga, and TM=ydx/dy = \og a e, so that the subtangent is constant in the logarithmic curve. The rate per cent, at which y = a x grows per unit of x is 100 dyfydx = 100 log a, which is constant ; and denoting this rate per cent, by c, then c = 100 log a, and y = e cx ' m - Fig-11 Suppose for instance x denotes years; then in one year, y will have increased to e c/10 ° times its value, which is at a rate of 100 (e c / 100 — 1) per cent, per annum. The equiangular or logarithmic spiral is the graph in polar coordinates of a , so that its equation is r = a e or 6 = log a r; it is called the equiangular spiral because the radial angle is constant. For cot = d log r/d6 = log a, a constant ; and thus r = ce e ° ota is the polar equation of an equiangular spiral, having a radial angle a (fig. 12 i.). When the radial angle a = ^ir, the equiangular spiral degenerates into a circle, r = c. Equispaced vectors of this spiral increase in geomet- rical progression; but in r = ad, the spiral of Archimedes (fig. 12-ii.) the vectors increase in arithmetical progression. DIFFERENTIA TION. 53 In plotting graphically an empirical relation, of the form y = ax m , between two variables x and y, say x the velocity of a projectile or ship and y the resistance of the air or water, a relation expressing the fact that dy/y = mdx/x, or that one per cent, increase in x gives m per cent. y (i) Fig- 12 increase in y, it is convenient sometimes to change to coordinates £ =log x and ^ = log y ; and now the empirical relation becomes }j = m^+c, so that the corresponding graph becomes a straight line. *31. Logarithms. The Exponential Theorem just employed forms part of Algebra, but for the sake of completeness the theorem has been established here. Again the theorems of logarithms employed can easily be established as follows : — Definition and principal properties of a Logarithm.— When a m = p, then m is called the logarithm of p to the base a : thus if p = 10 m , then m is the common logarithm of p, and if p = e m , then m is the natural logarithm of p. 54 DIFFERENTIATION. Now let a n = q; then by the Theory of Indices, as explained in Algebra, pq = a m xa n =a m+n (1) p/q = a m /a n =a m ~ n (2) p- = (a m ) r = a rm (3) i/p = (a m ) 1 ' r =a m ' r (4) Therefore the corresponding theorems in the logarith- mic notation are log a pq = m+n = log a p+\og a q (5) logp/q=m—n =log£>— logg (6) logp r =rm =rlogp (7) logZ/p = m/r =(logp)/r (8) Also if e b = a, then e = a 1 l h ; and therefore b = log e a, 1/6 = log a e, so that log e a x log a e = 1 ; and p — a m = e 6 ™, so that log a j» = log a e log e p- Thus, log 10 p = log 10 e log e p, and log 10 e = '4342945, the modulus M, while log.10 = 2-3025851. These theorems have already been employed in estab- lishing the preceding differentiations. The exponential function a? and the logarithmic function log^a; are functions inverse to each other; be- cause, if a m =p, then rn = log a p; and a loe a x = \og a a x = x. Starting with m = log a 23, then p is sometimes denoted by log _1 m, or by exp a m (called exponent of m to base a) instead of by a™, especially when m is a complicated mathematical expression, as thereby difficulties of print- ing are avoided. The letters exp must be considered an abbreviation of exponent; just as log, usually employed, is an abbreviation for logarithm. DIFFERENTIATION. 55 We have defined e as the lt(l + l/m) m , when m is in- definitely great ; and a similar expansion shows that e *=it(i+i/m;r==i+^+|?+ ... +2+ ... But suppose we reverse the procedure, and call this series exp x; the series obtained by changing x into y will be exp y ; and then a straightforward multiplication of the two series shows that the product is the series x + y (x + y) 2 , (x+y) n , T 1! ^ 2! ~*~ ••■ "'" n\ + "■' so that exTp(x + y) = exp x x exp y ; and thus, generally, exp(cc + 2/ + 2+ ... ) = exp x x exp y x expz x ... . Now suppose there are n of these factors, and x = y = z— ... =1; then since expl = e, therefore expn = e™ where n is a positive integer; and thence generally, by In- duction, exTpx = e x , for all values of x. (Cauchy, Oours d' Analyse; M. J. M. Hill, Proc. Cam. Phil. Soc. vol. v.) Writing bx for x, and putting e 6 = a, so that b = log a, a» ! =e te =e a!lo 8o = l+a3loga+^(loga) 2 +...H — ,(loga)»+... Changing x into h, then ^ = loga+A(loga) 2 + ... + ^ r 1 (lQ go)-+... > reducing to its first term, logos, when h=0, thus proving the lemma required in the differentiation of a x . Again a x+h = a x a h = a x (l + hloga + . . .), so that a^logct is the coefficient of h in the expansion of a x+h in ascending positive integral powers of h; and is therefore, according to Lagrange's definition (§ 7), the derivative of a x . 56 DIFFERENTIATION. Putting a=l+z, then (1 + zf = e h *>&+>) = l+Mog(l+z)+^{log(l+*)} 2 +...; so that log(l + z) is the coefficient of h in the expansion of (l+z) h by the Binomial Theorem; and similarly {log(l+2)} 2 is twice the coefficient of I?, and so on; therefore log(l+z) = ,-_ + _-....-L_J_+..., and l g(l-*)=-*-_-_-. ..-_-...; 1+2 Z 3 Z 5 z %n+l so that £log r -^=*+ T3 + 5 + ...+^ TT + .... Writing 1/% for 2;, then ' n — 1 to 3TO 3 ' 5n 5 the series employed in calculating the natural logarithms. Thus putting n = 2 gives log 3, n = 3 gives log 2, and thence log 4, log 6, log 8, and log 9 ; and putting n — 9 gives logclO, the reciprocal of which, log 10 e, is M, the modulus which converts natural into common logarithms. We calculate logarithms to base e, but tabulate them to base 10 ; because numbers with the same series of digits and differing only in the position of the decimal point have the same mantissa or decimal part in the logarithm to base 10, and differ only in the characteristic or integral part, the value of which is easily written down. Examples. (1) Calculate by logarithms the value of (1 + l/ni) m , when m is put equal to 10, 100, 1000, 10000, ... ; and show that these values tend to equality with e. , t n-f-t. 1,1.1, DIFFERENTIATION. 57 (2) Deduce the differentiation of e x from that of its inverse function log x ; and vice versa. *(3) Write down the series for {log(l+2)} 2 in ascending powers of z. *(4) Prove algebraically that ex V (x-^+ x *-...+(-l) n - 1 ^-..) = l+x; ( , X 2 . X s . , x n . \ 1 / . X 3 , X 5 . , x 2n + x , \ ll+x *(5) Draw the graphs of 2 X , e x , 10 s , e -a! cos£C, e' xi cosx. *(6) Calculate, to base e, log 2, log 3, log 7, log 10 ; and thence their logarithms to base 10, to seven decimals. 32. Logarithmic Differentiation. When the function to be differentiated is a single term consisting of factors raised to different powers, it is often simpler to take logarithms before differentiating; and then since (§ 29) d log y _ 1 dy dx y dx' therefore -, ~ = y — ~^-. dx ax rrim. . Thus, if y 2 -- (l+x) n then 2 log 1/ = m, log a; — wlog(l + a;) ; 58 DIFFERENTIATION. and differentiating with respect to x, 2dy_m_ n _m + (m — ri)x m y dx x \ + x x(l+x) dy_ , m+(m-^ .i . dx t (l+x)^ 1 this is called logarithmic differentiation. Let y = uvw...., the product of any number of functions u, v, w,.... of x ; then log 2/ = log u+ log v+logw+.... and differentiating this sum of functions, 1 dy _ 1 du 1 dw 1 dw y dx u dx v dx w dx dy _ n d log y 1 du , 1 dv , 1 c?iw da; da? /lw, low . 1 dw , \ = uvw....(- - 1 -+~ j-+— j-+—- h \u dx v dx w dx I the formula for the differentiation of a product (§ 12). Again, let y = u/v, log y = log u — log v, 1 dy _1 du_l dv y dx u dx v dx' j dy /du dv\ 1 9 and -f- = (—v — u i- M dx \dx ax/ j the formula for the differentiation of a quotient (§ 12). Let y = u v , where u and v are both functions of the independent variable x ; then log ?/ = 1' log w ; and differentiating this product with respect to x, 1 dy _dv-, ,vdu y dx dx u dx' du v dy dv , ,du -^— = ^r- = u v ^-\og:u+vu v - 1 ^ r . dx dx dx ° dx DIFFERENTIATION. 59 If u is constant, and v a function of x, then, by § 26, du? dv , — j— =u v -^- log u ; dx dx ° and if v is constant and u a function of x, by §§ 3 and 11, du v ,. ,du !)-!_ CLX (XX thus, when u and w are both functions of x, the dif- ferentiation of u v is performed by supposing the func- tions to vary one at a time, and then adding the results. Examples. — (1) Differentiate logarithmically with respect to x, jl-x (x + l)i(x+3)i 1 (sin map" 1+x' (a+2) 4 ' (x-a) m (x-b) n ' (cosnx)™' , ] +SmX , x*, *', Z/x, (1 + 1/flO- Z/{l+x), x l0BX , x. "~ * sin so (log a:)*, (ixY x ,\og x a. (2) Prove the following differentiations, (i.) y = log since, dy/dx = cot x. (ii.) y = log sec x, dy /dx = ton. x. (iii.) y = log tan(J-7T + \x) = log(sec x + tan x), =lo S/X l 1+sinx , dy / dx= sec x. ° \1 — sin a: (iv.) 1/ = log tan Ja: = log (cosec x — cot x), dy/dx = cosec a;. V 4 - Iog Vi+ cos a; -cos a; (v.) y = %sec 2 x— log sec a;, %/Ac = tan s a;. (vi.) y = \ sec x tan aj + J log (sec a; + tan x), dy/dx = sec 3 a;. (vii.) y = e?(x 2 -2x+2), dy/dx=e*x\ (viii.) 2/ = e !C (a; 3 -3a; 2 +6a:-6), dy/dx = e x x s . 60 DIFFERENTIATION. (ix.) y = e x \x n — nx n - 1 + n(n—l)x n - 2 —...}, dyjdx = e x x n . if n is a positive integer. (3) Prove that in the curve y/a = logsecx/a, or e^cos xja = 1 (the catenary of equal strength), ds x ds x , , -T- — sec -, -T- = cosec — , ana x = axis, dux a ay a *(4) Draw the graphs of x x , */x, (1 + 1 fa)*, 4/(1 + a:). 33. The Hyperbolic Functions. Corresponding to the trigonometrical functions of the circle, defined in § 16, there are certain functions as- sociated in a similar manner with the hyperhola, invented by Lambert (1768), called hyperbolic functions, which are of great use, and are defined here as follows : — Taking the exponential function e u and its reciprocal e~" of any variable quantity u, then their half sum, £(e" + e - *), is called the hyperbolic cosine of u, and is denoted by cosh u, and their half difference, \{e u — e~ u ) is called the hyperbolic sine of u, and is denoted by sinh u. The other hyperbolic functions of u are defined by analogy with the circular functions : the hyperbolic tangent of u, denoted by tanh u; „ coth u ; eoshu *.„v-.n.^ ^^g,^*, V*. cosh u sinh it cotangent 1/cosh u „ secant 1/sinh u „ cosecant cosh u— 1 „ versed sine „ sech u ; „ cech u ; „ versh u. The reason for these names will appear hereafter ; but the following formulas, which are easily established, show DIFFERENTIA TION. 6 1 that there is a Trigonometry of the hyperbolic functions exactly analogous to that of the circular functions ; so that modern Trigonometry must be considered to include the properties of the circular and hyperbolic functions, which mutually assist and illustrate each other. Thus cosh u + sinh u = e u or exp u, coshw — sinh u = e" u or exp( — u) ; cosh 2 w, — sinh 2 u = 1. Similarly tanh 2 « + sech 2 u = 1 , coth 2 u — cech 2 M = 1, sinh 2w = 2 sinh u cosh u, cosh 2u = cosh 2 u + sinh 2 u = 2 cosh 2 u —1 = 1 + 2 sinh%. . , „ 2 tanh it , „ l + tanh 2 w. smh2t(, — ^ — t — ,-s-. cosh 2 it =- 1 — tanh 2 ic — 1 — tarih 2 u ' sinh (u + v) = sinh ii cosh v + cosh u sinh v, cosh(u + v) = cosh u cosh v + sinh u sinh v, tanh w+ tanh i; ^ ' 1 + tanh ?a tanh?/ (These formulas are proved by noticing that coah(u+v) = $exp(u+v) + $ exp(«-w - v) = |exp it exp i>+£exp( — u) exp ( — v) = |(cosh u + sinh it)(cosh v + sinh v) + £(cosh u — sinh u)(coalo. v — sinh v) = cosh u cosh v+sinh u sinh v :. and so on.) sinh u + sinh v = 2 sinh £(w + -y)cosh £(it - 1>) ; sinh cosh ti — cosh v = 2 sinh £(«. + t;)sinh £(it — v) ; tanh it — tanh v = sinh(% — -y)sech u sech v. 62 DIFFERENTIA T10N. , d cosh u . , d sinh u Also , = smh«, — 9 = 00811%; d tanh u die = sech 2 u., d sech it dit = — sech u tanh u, du d coth it du d cech it dw = — cosech 2 i* = — cech u coth u ; and d versh u du = sinh u : which may be proved by straightforward differentiation of the exponential values of the hyperbolic functions; or else in the same manner as that employed for the circular functions, making use of the fact that, when h = 0, (sinh h)/h = 1, and (tanh h)jh = 1, by § 34. Fie.13 *34. Fig. 13 is drawn to illustrate geometrically the analogy between the circular and hyperbolic functions, and to justify the terminology given above. Denoting the cm. of the angle AOP by 6, and the radius OA by a, then from the definitions of Trigo- nometry, the arc AP=a6, the sector 0AP=\a?d, and DIFFERENTIATION. 63 OM = a cos 8, MP = a sin 8, AR or PT=atsxi8, OR or OT=a sec 0, and A.M = a vers 0. Now if the ordinate TQ is erected of length equal to the tangent TP or AR, then the coordinates x and y of Q are given by x = 0T= a sec 0, j/ = TQ = a tan ; so that by the elimination of 8 the equation of the locus of Q is x 2 — y 2 = a 2 , and Q therefore describes a rectangular hyperbola AQ, while P describes the circle A P. But since cosh 2 it — sinh 2 w, = 1 , we may also express the coordinates of Q more sym- metrically by putting OT=x = a cosh u, TQ = y = a sinh u ; so that OT and TQ represent the hyperbolic cosine and sine of u in the hyperbola, just as OM and MP represent the circular cosine and sine of in the circle ; hence the reason for these names. Now cosh u = sec 8, sinh u = tan 6 ; and while %a 2 6 is the area of the circular section OAP, it will be found that \a 2 u is the area of the hyperbolic sector OAQ ; so that the analogy between 6 and u is expressed through the areas of the circular and hyperbolic sectors, and not through the arcs or angles. When 6 and u axe connected by this relation, then 6 is called by Professor Cayley the Gudermannian of u, and sometimes also the hyperbolic amplitude of u, and is denoted by gd u, or amh u. Conversely exp u = cosh u + sinh u = sec 6 + tan 6, u=gd~ 1 9 = log(sec 9 + tan 8) = log tan(i7r + %6), by means of which u can be calculated as a function of 6, from the trigonometrical tables. 64 DIFFERENTIATION. A Table in the Appendix, taken from Legendre's Fonctions Mliptiques, t. ii., gives the value of u for every degree in the angle whose cm. is 6. When the hyperbolic functions for a certain value of u are required numerically, the corresponding value of 6 is found by proportional parts ; and then, by means of the tables, cosh u = sec 6, sinh u = tan 9, tanh u = sin 0,. . . It may also be noticed that when cosh u = sec 6, then l+tanh 2 |u> tanh u ; whence it follows, as in § 16, that when u = 0, (sinh u)ju = 1 , and (tanh u)/u = 1 . *35. The Ellipse and Hyperbola compared. If the ordinates in fig. 13 are all reduced (or enlarged) in a constant ratio, say b/a, by orthogonal projection, or by throwing the shadow of fig. 13 on a plane parallel to Ox by parallel rays of light, then we obtain fig. 14, in which the curve AP is an ellipse, and the curve AQ is a hyperbola, no longer rectangular. Now in the ellipse AP, OM=a cos 0, MP = 6 sin 6; while on the hyperbola A Q, OT=a cosh u, TQ = b sinh u, DIFFERENTIA TION. 65 so that a? b 2 is the equation of the ellipse, and of the hyperbola. b 2 V. < y b / B ~~^J K Q I ° M i' a; V ^ V s Fig.14 The angle 6 is called the excentric angle of the point P, or in Astronomical language the excentric anomaly ; while u may by analogy be called the hyperbolic excentric anomaly of the point Q on the hyperbola. The elliptic sector will be b/a of the circular sector OAp of which it is the projection, and its area will there- fore be %a 2 6xb/a = ^ab6; and similarly the hyperbolic sector AOQ will be \abvb in area, or ^ablog(x/a + y/b), since cosh u = x/a, sinh u = y/b ; and therefore xv , (x . y\ ex P tt = a + 6' U=l0g U + !> Carrying out the properties of the ellipse and hyper- bola on parallel lines by means of the excentric anomalies 6 and u, we find : — 66 DIFFERENTIA TION. Ellipse. Chord through (0, ) is - cos J(0 + ) + 1 sin J(0 + . 5 cosJ(0-0) ^ Hyperbola, Chord through (u, v) is - cosh Mu + v) - 1( sinh J(« + ») a b = cosh ^(m - v). Tangent at (u) is - cosh u - % sinh u = 1 . Normal at (u) is osx sech m + 6y cosech u - a 2 + ft 2 . Tangents at («, «) meet at g_ g coBh &(«+«) & sinh^(«+«) T coshi(M-'w)' coshJ(M-«) Normals at («, v) meet at ^+^QOBhi[u + v) coshucoshVi a cosh|(M-'y) _ tf+V rinhj(«+t;) sinh „ sinh „, b cosh J(« - v) Generally in the curve (x/a) n ± (yjb) n = 1, we may put (x/a) n = cos 2 or cosh 2 it, (y/b) n = sin 2 6 or smh 2 u; and then the equation of the tangent is x/a(eos Of ~ 2 ' n + (y/b)(sm 6) 2 - */» = 1, or (x/a)(eoshu) 2 - 2 l n -(y/b)(smhu) 2 - 2 l n =l. 36. The Catenary. To illustrate the practical use of the hyperbolic func- tions, an instructive curve to take is the catenary, the curve in which a uniform chain hangs ; the equation is y/a = cosh x/a ; so that the catenary is the graph of a cosh xja, and the ordinate of the catenary is the arithmetic mean of the ordinates of the two exponential curves y/a = e x ' a , and y\a=e,~ x ' a . DIFFERENTIA TION, 67 y \p' N X PS A \ \ y a x 2 yY\ T o M x Fig.15 Differentiating the equation of the catenary, we find tan i/r = dy/dx = sinh xja, so that sec \js = dsjdx = cosh xja = y/a or \/r = gd xja ; xja = log (sec \jr + tan i/r) , y/a = sec i/^. The equilibrium of the arc AP shows that if we denote the tension at the lowest point A by wa, then by resolv- ing horizontally, the tension at P = wasec\Jr = iuy. Also by resolving vertically, we see that wa and wy must represent weights of chain of length OA and MP ; and s = y sin i/r = ^(j/ 2 — a 2 ) = a sinh a:/a = a tan y]s. Again, if MQ is drawn perpendicular to the tangent at P, MQ = ycos\f, = a, and QP = /V /0 2 -2/ 2 )=s = arc^P. Hence to draw the axis Ox, called the directrix of the catenary, we measure off PQ in the tangent equal to the arc PA, and draw QM at right angles to QP to meet the vertical ordinate through P in M, 68 DIFFERENTIATION. The normal PG = y sec i/<- = y 2 /a. By a change of origin from to A the equation of the catenary becomes y + a = acoshx/a or y/2a = sinh 2 a;/2a ; or 1//6 = sinh 2 a;/6, if b — 2a ; and now (y + af = s 2 + a 2 , or s 2 = j/ 2 + 2cm/ = y 2 + ty- Suppose for example a steel telegraph wire 5000 feet long is stretched between two points P, P' at the same level, so that the versed sine or dip NA in the middle is 500 feet ; then y = 500, s = 2500, and therefore a = 6000 ; so that the tension at P or P' is the weight of 6500 feet of wire ; and taking the specific gravity of steel as 8, this would give a tensile stress of about 10 tons per sq. inch. Then 2x, the distance between P and P', will be 4865 feet. When the wire is screwed up tight so that the dip y is small, then s and x are very nearly equal, and a = isyy - \y SX \s*\y ^ \x*\y ; so that if telegraph wire sags y feet in the middle between posts I feet apart, the wire is screwed up to a tension equal to the weight of about \Pjy feet of wire. * If an endless chain, of length 21 or 4s, is suspended over smooth pullies at P and P' (fig. 15), then when P and P' are almost as far apart as possible, the two parts of the chain coincide in a single cateDaiy ; but when P and P' are moderately close, the chain hangs in two distinct festoons, catenaries having the same directrix, since the tension is unchanged in passing round the smooth pullies at P and P'. Also, if the festoons are partly supported by smooth inclined planes, the various catenaries will have the same directrix. DIFFERENTIATION. 69 Any catenary between the two festoons will have a higher directrix, and beyond them a lower directrix. Drawing the two common tangents to the two festoons, they will meet in 0, since the catenaries of the two festoons are similar curves, and is a centre of similitude. Therefore the tangents at P, P' of the upper festoon intersect above the directrix, and below the directrix for the tangents at P, P' of the lower festoon. Now, let P and P' be drawn apart ; the directrix of the two festoons will rise until they coalesce, when the tangents at P and P' will intersect in ; afterwards the directrix will descend again. In the separating case then, the tangents at P and P' intersect in ; and then V dv a . x . , x ta,n\b- = ~ = -^r-, or -cosn-=sinh- x dx' x a a' i J.-U / i x \ x + a or cosec ■»/<■ = coth x/a = x/a, - = ~Iog- , from which transcendental equation we find, by aid of the Table in the Appendix, the approximate root xja^l . 2 ; and then y/x = s/aZZl-5, s/xzz 1-25, x/s™S ; and yp- is the cm. of 56°30' about. The profile of the catenary is also seen in the surface of revolution, called the catenoid, formed by a soap bubble film, adhering to a circular wire which is raised gently in a horizontal position from the surface of soapy water. The same conditions give the stability of this capillary film (fig. 16) ; so long as the tangent cone along PP' has its vertex below the surface of the water, the film is stable; but the film always breaks when the vertex reaches the surface, and then x, the height of the wire 70 DIFFERENTIA TION. pC_ M *\ / ^^ y V Fig.16 PP' above the surface, and 2y its diameter are connected by 2y/x?z3, x/2y^i. (Maxwell, Capillary Action, Encyclopaedia Britannica.) If a thread, wrapped upon the catenary AP, is cut at A and then unwrapped, the end will describe the curve AQ, and QP will be the normal of this curve at P. Since QM, the tangent of this curve, is of constant length a, the curve Q is that which would be described by a body on a rough plane if drawn very slowly by the point M, moving in Ox, by a thread or chain MQ of con- stant length a ; hence the curve A Q is called the Tractrix. Examples. (1) Prove the following differentiations : (i.) y = \smh.2x+\x, dy/dx= cosh^x. (ii.) y = \ sinh 2x — \x, dy/dx = sinh?x. (iii.) y = % cosh 3 a; — cosh x, dy/dx = siah s x. (iv.) y = -fa sinh dx~ tf=&l x dx~ tf=tf>t x ? a )> ,., c?tan _1 a;/a_ _c?cot _1 a;/a_ a dx dx a 2 + x 2 ' It will be instructive for the student to make a table of the principal formulas of the circular and hyperbolic functions in parallel columns, to show the correspondence; this will be found to assist the memory in recollecting the formulas. DIFFERENTIATION. 73 Examples. (1) Construct a Table giving any direct hyperbolic func- tion in terms of any other ; and differentiate them as a verification. (2) Construct a similar Table for the inverse hyperbolic functions ; and differentiate them. (3) Prove that cosh~ 1 a!-|-cosh- 1 2/ = cosh- 1 {a;2/-|- x /(a; 2 — l)^/^ 2 — 1)} = Bwh- 1 {z l J(3f-l)+yJ(a?-l)} smh- 1 x+sinh- 1 y = smh.- 1 {x > /(l+y 2 )+y s /(l+x 2 )} = cosh- 1 { x /(« 2 +l)V(2/ 2 +l)+a;3/} tanh - he + tanh - hf = tanh - i£±!L 1+xy (4) Prove that if y = cosh ~ x x = \ cosh - 1 (2a; 2 — 1) = \ sinh ~ J 2x*/(x 2 — 1) = J cosh _1 (4a; 3 — 3a;), . . . dy_ l dx ^/(a; 2 — 1)' (5) Prove that if y = sinh - he = f sinh - *2as */0- + «*) = * cosh " K 1 + 2x *) = J sinh _ 1 (3a; + 4a; 3 ), . . . ^/ = L_ dx JQ.+X*)' %x %x (6) y = ts,nh- 1 x = ita,nh- 1 T - — - 2 = ^sinh- 1 z 1 , ,1+a; 2 ,, , ^a+a; 3 . da; 1— a; 2 ' 74 DIFFERENTIATION. (7) Prove that if = gdu = cos" 1 sechu = sin- 1 tanhtt = tan- 1 sinhw = 2 tan- 1 tanhitt = 2tan- 1 e"-^7r; 4^=sechw. du (8) Prove that if , u = gd.- 1 = cosh -1 sec = sinh- 1 tan = tanh _1 sin = 2tanh- 1 tan£0 ) du a (9) Prove that cosh log (sec 0+tan 0) = sec 6. sinh log(sec 6 + tan 6) = tan 6. tanh log(sec 6 + tan 6) = sin 6, etc. (10) In figure 13, prove that (i.) LA'AP+iAA'P = frr; (ii.) lAAQ-lAA'Q = %tt. (iii.) A'P and ^4P' intersect in Q, PMP' being the chord of the circle. Prove, geometrically, sec 6 ± tan 6 = tan (\ir ± £0). (11) 2/-cosh- ^ 3(a;+1) , S^(aj+l)V(a?+ y =Bmh i / R /«-^ » ;£= 1 i^5( ^W21og^g±^-W2tan-^ = W2tanh-^-W2tan-^g=^ I . (9) y = Alog{x t - l j2x+A)+B\og(x i +j2x+l) + Ctim- 1 ( s /2x-l)+Dt!m- 1 ( s /2x+l); , , . . „ ~ _ . dy x, oy x 2 , or x 3 , determine A, B, V, D, when ~ = ^—z. . (10) , = 2003-^ = 2^-^ = 2 tan-^^f = sin -iVV^_^^). dy_ 1 •J(a — /3) ' cfa: y/(a — x.x—f3)' (ll )2/ = 2sinh-^ = 2cosh- lA /^|=2tanh- lA /|^ = sinh- ^ (a i7 a ^-^ = 2 log{ V(« - «) + V(« - i8)> " lo g(« - 18) ; dse ^/(x — a.x — fi)' (12) 2/ = 2cosh- 1 /< /^ = 2sinh- 1 /V /^=2tanh-i/^ \a-p ya-p V a-X i(a-P) = 21og{ N /(a-a) + x /(/3-*)}-log(«-/3) -1 s /{a — x) — s /{^ — x)'dx ^/(a — x.ft — x)' DIFFERENTIATION. 77 ^ ' dx = J(ax* + 2bx + c)' 1 _ 1 . _ l s /{-a)J{ax i +2bx+c) y ~s/(-° 2 , x la — x or — t, rr cosJ:1 \\ 1 ' s/(a-b) Mb-x -J(a^F) Sinh '^x^b' or 78 DIFFERENTIA T10N. (16) ^ = - if ' dx {x-y) s /(x-a.x-^)' or 1 „j n]l -t x/(y-«-y-/3)V(^-«-a;-/3) x/(y-«-y-/3) K«-/3)(*-y) (17) ^= if dx (x — y) M J(a — x.x — f3)' v - 1 f ,j n -i s/(y-*-y-P)>/(a-y-x-l3) . J(a-y.y-l3) * i{a -/3)( X -y) Determine the degenerate forms when y = a or /3. (18) Denoting ax 2 + 2bx+c by i?, prove that dx (x—p) > /R (x—p) s /(ax 2 +2bx+c)' J(-ap 2 -2bp-c) *J(b 2 -ac){x-p) = 1 ,,„.,- i (ap+6)s+&P+c s /(-ap i -2bp-c) J(tf-ac)(x- V )' or 1 B j Tlh -i V(ap'+86j>+cWJZ s /(ap 2 +2bp + c) J{b 2 -ac){x~p) ■ l cosh " i ( a P+ b )®+ b P+c J(ap 2 +2bp+c) J{b 2 -ac){x-p)' or 1 „^-i s/(ap 2 +2bp+c)JR J(ap 2 +2bp+c) s/{ac-b 2 )(x-p) ■ 1 sinh - i ( a P+ b ) x + b P+c J{ap 2 +2bp+c) s/iac-VXx-p)' . DIFFERENTIA TION. 79 /io\ 1 i If G aoa 2 + a\ 1 x // az 2 +e\ or vcvK--4 dy _ cfa; (Aa?+C)J(aa?+cy (20) 2/ = e ax+6 cos(29a;+g'), -^ = a sec ae' K5+6 cos(pa; + g + a), where tan a = £>/a. (21) y = m sin mcc cosh nx+n cos mcc sinh was, -^ = (to, 2 + w 2 )cos ?na; cosh was, (22) y^lxjicfi-a^ + tahm-he/a, ^ = */(<*- i£ {ax+b)JR b 2 -ac 1 ■ _, »/(-«)*/* or */a JiW-acY 1 , , JaJR or —7- cosh" 1 7, "" , 2V (29) ?/ = sin - V( sin 2a; ) ± sinn " V( sin 2x )> j = s/(% * an #)> or ->y( 2 co * a3 )- (30) y = tan _ 1 /V /(tanh x) ± tanh _ 1 < /(tanh x), -f- = ^/(c ^ a; )» or V(*anh a;). (31) Prove that |^= -r4 , if v ' dux a+o cos x 1 .acosx+b 11= ,, , rorCOS 1 — — r V ( a " ) a+ocoscc 2 V(« 2 "^ 2 ) or tan_1 {(l4) itai1 ^} 1 , n acosa;+6 cosh ~ *■ — -j /^/(bP — a?) a+bcosx ) tan|a3j- J-^tanh-I^ V(6 2 -a 2 ) Wb + aJ _ 1 , y/(b+a)eos^x+y/(b— a)sin \x J(b 2 — a 2 ) ° ^/{b + a)cos \x — ,J{b — a)sin \x ( 32 ) 7?= ^ t, > if v ' ckc a + o cosh cc 1 _jacosha;+6 y = s /(b 2 -a 2 ) COS ~ a + b cosh a; V(& ! ra) tan_1 {(^) itanh ^} DIFFERENTIA TION. 8 1 or , 1 acosh» + b ^/(a 2 — 6 2 ) 2 + c 2 )P 1 , 1 aP-a 2 + 6 2 + c 2 OT V(-« 2 + & 2 + c 2 ) x/(^ + c 2 )-P " (35) Denoting a + bcoshx + csmhx by Q, prove that dx a+6cosha;+csinha; Q' 2/ "x/(-« 2 +^ 2 -c 2 ) cos V(^ 2 -c 2 )Q ' or V(« 2 -^ 2 +c 2 ) x/(^ 2 -c 2 )Q ' 1 . , 1 oQ-q» + y-c' 01 x /(a 2 -6 2 +c 2 ) smh */(c*-P)Q • 82 DIFFERENTIATION. (36) Prove that ^= ? —-i ,, if ax (l+ecosxy ,-, ox! . Jl- e 2 sina; 6*7(1 — e 2 )sm a; (1 — e 2 )y= sm- ] ^r- ^^ , a 1+ecosaj l+ecos* / o -ivi • t i«k/(e 2 — l)sinir . e^/(e 2 — l)sina: or (e 2 — 1) T «= — sinh- 1 ^^- ^ +^V • 1 + ecoscc 1+ecoscc 1)51 12 g V(^ 3 -l) + V3 cfa/_ 1 £B Write down the value of y when S = (^/W+i) (Euler) " (38) Prove that the equation (Riccati's) x-^- —ay+ by 2 = ex 2 " 1 is satisfied by or WH) tM1 {?V(- 6o )+4 (39) With X=aa;H46a: 3 +6ca! 2 +46 , a)+a', F the same function of y, and s =^('^r~f-Mx+y) 2 -b(x+y)-c; prove that, treating y as constant, dx/ /s /X = 2 -^' , > c?coth~Wa_ _ « tan _1 ^/a. f dx 1 ,_, , I — b = = — - cot L xla. Jaf+a* a = - sec i xja. /: cfe a; ^/(tf 2 - as 2 ) a ) x sj(x? - a?) a dx dx 1 _l ; ■ K = — - cosec '■x a. I JiZax-x 2 ) vers~ L x/a. fa x dx= logos' f^dx=^. J o /cosh mxdx — — sinh mx. m /sinh mxdx = — cosh mx. m /sech''mxdx= — tanhma;. J m /cosech 2 ma;o?a;= - — coth mx. I — =log«, or log( — x). J x I dx or log { *J(a? + x>) + x } f dx = cosh -1 #/a, or log{x+ s /(3t?-a?)}. [ f x =1 tanh-^/a (^a 2 ). ]x?- Jx 4/a' x ^F a?*' ^~ 2 ' af ' (ticc)^ 1 )"*' J.+5a;+Cx 2 +Da! 3 + ..., ax m +bx n +cxP+ .... (2) a;+a, (aj+a) 2 , (a?+o)» J(x+a), H x+a y ^p^> -j> (mx+nf, , ; fT -- (x+af V ;, a!-a (as-fe)* (3) a: 2 +a 2 , (£c 2 +a 2 ) 3 , a;(a 2 -a 2 )", xj(x*+a% ,, f , 2 , a; 1 1 a; x 2 — a 2 a; 2 -fa 2 x 2 — a 2 (a; 2 -fa 2 )* 2a: + 1 1 x + 2 Px+Q x + 2 ^ ' a> 2 + a; + l' x 2 +a;+l' a; 2 +a; + l' z 2 + a; + l x 2 + 2cc + 2' x-j2 2 1 Px+Q x 2 -J2x + V te 2 + 3' ax 2 +c' ax 2 + c' . . x — c x — c 1 Px + Q {(a5-c) 2 +a 2 }»(aj-c). . < 6 > VK 2 +2te+c) ' « a; 2 +2to+c > ^ +2to+c ) m ( ffla; + 6 )- 88 INTEGRATION. (7) —n — ■ — r— — 77 — — tti (rationalize the denominator), s /(x+a) + s /(x + b) v 7-o — TV (substitute a; 2 = — )> x 2 + l) K y/ (8) *V(* 2 + 1) ^"°^° " ~yj' xj(ax*+b) , (substitute s /(ax n +b) = y). (log x) n 1 x n ' x since f'ic i'x x xlogx ax n +b a+bcosx a+btx (a+ttx) p (ax n +b)- 1 - lln (substitute ax n +b = x n y n ). (9) cos 2 a3, cos 8 *, cos 4 a3, ... ; sin 2 a;, sin 3 a;, sin 4 a; . . . ; cos ma; cos nx, sin mx sin nx, cos mx sin nx. (Convert the powers and products of cos a; and sin a; into cosines and sines of multiples of x ; thus cos 2 a; = |(1 + cos 2 x), sin 2 a: = £(1 — cos 2 x) ; cos 3 a: = \ cos 3 x + f cos a;, sin 3 a: = f sin x — \ sin 3 x, ■ cos 4 a: = f + J cos 2 a; + £ cos 4 a;, sin 4 a: = § — \ cos 2 x + \ cos 4 a: ; cos ma; cos nx = \ cos(?n — w)a; + J cos(m + n)x ; sin ma; sin nx = \ cos(m — n)x — \ cos(m + n )x ; cos ma; sin nx = \ sin(m + n)x — J sin(m — m)a; ; and now the integrals with respect to x can be imme- diately written down). The answers of these examples are not given, because it is better practice in integration for the student to discover the results for himself; the correctness of a result can always be tested by differentiation. 41. Quadrature. The Integral Calculus was invented for the purpose of finding areas, or for quadrature, as it was formerly called; and the meaning of Integration is best illustrated by its application to finding the area of a curve. Let i/ = fa: (fig. 17) be the equation of a curve CPB; then if OM=x, MP = ix. INTEGRATION. 89 Let the area AMPG between the curve and the axis of x, bounded by an initial ordinate AG and the variable ordinate MP, be denoted by A; keeping AG fixed and varying MP, A is some function of x, which it is required to determine. Let the ordinate MP be moved into the adjacent position mp, by giving x the increment Ax ; and let AA be the corresponding increment of A, and Ay of y. Then Mm = Ax, mp = y + Ay ; and the area MmpP = A A. But MmpP lies between the rectangles Pm and Mp, and therefore A A lies between yAx and (y + Ay)Ax; or, AA/Aa; lies between y and y + Ay. Proceeding to the limit by making Ax, and therefore Ay and AA indefinitely small, dA ..AA „ T- = lt-r — = y = ix. ax Ax * Therefore A =Jydx + a constant =Jixdx + Sb constant ; so that to determine A we must know how to integrate ix. 90 INTEGRATION 42. Fluxions and Fluents. According to the Doctrine of Fluxions, the old-fashioned name of the Calculus in this country, the area AMPG, called the fluent, is supposed to be generated or to grow by the flow or motion of the variable ordinate MP ; and the rate of growth of the area, called its fluxion, is equal to the ordinate MP into the fluxion of x ; with Leibnitz's notation, dA _ olx dA. _ ~dt~ y ~dl ,0X ~dba~ y ' so that A=Jydx. Generally, "Mathematical quantity, particularly ex- tension, may be conceived as generated by continued local motion (as in the growth of a tree) ; and all quantities whatever, at least by analogy and accommodation, may be conceived as generated after a like manner." (Sir Isaac Newton, Fluxions, edited by Colson, 1736.) " The Differential Method of Leibnitz teaches us to consider Magnitudes as made up of an infinite Number of very small constituent parts, put together ; whereas the Fluxionary Method of Newton teaches us to consider Magnitudes as generated by Motion. A Line is described, and in describing is generated, not by an apposition of Points, or Differentials, but by the Motion or Flux of a Point." (Doctrine of Fluxions, by J. Hodgson, 1736.) Thus if the point M is supposed to move in the direc- tion MP, it will trace out the line MP. If the variable ordinate MP moves parallel to itself, it will trace out the area ABDG. INTEGRATION. 91 Similarly, any solid may be supposed generated by the motion of a variable plane area perpendicularly to itself; and then dV/dt = Adx/dt, if V denotes the volume, A the variable plane area, and dx/dt the velocity of the plane perpendicular to itself. For instance, if the volume Fis bounded by the surface formed by the revolution of the curve y = ix round the axis of x, then the fluent V may be supposed generated by the motion of an expanding (or contracting) circle of radius y, as in boring a hole of varying diameter, or turning a body in a lathe ; and therefore dV/dx = Try 2 , or V=irjy % dx. 43. Corrected Integrals. Denoting the indefinite integral J~ixdx by fjo:, then A=f 1 x+C, where G denotes a constant ; and denoting OA by a, then since A = when x = a, therefore C= —i-^a, and A = f 1 x — i 1 a. This is expressed by the notation A = /ixdx, or simply j ixdx = (i x x) x a = i x x — f x a, a & a being called the lower limit, and the integral is then called a corrected integral. Sometimes the fixed ordinate is taken to the right of the variable ordinate MP, at BD suppose, where OB = b ; and then the area MBDP is given by I ixdx, or simply lixdx = (f x aj)* = f x b — t-^x, x and b is called the upper limit. 92 INTEGRATION. Thus, the integrals of § 39, when, corrected, become f xmdx= (^ +1 Y = v m+1 -a m+ \ J \m + l)a m+1 a X m dx=[ -r = —, . xm+l/s m+1 7 1 X ~ mdx = {m-l)x™-* I cos ajcfoc = (sin xf = sin a?. / sin xdx = ( — cos xf = 1 — cos x = vers x. sin xdx = cos £C. / cosxdx = l — sinx. / sec 2 ccc&E = tan a. / cosecMa; = cot x. y"~ dx . _{x /*" c&c _ ^ V(a 2 -x 2 ) _Sm a y^F^) _C0S "a /~ cfa 1 , ,« /"°°c&c 1 , ,a; /-»-; — 9 = -tan x -- / -s-: 5 = -COt -1 -. J a?+x 2 a a J x 2 + a? a a y* 7 a x — l fdx . x a x ax = -, • / — = log-. log a J x °a ° a /cosh udu = sinh u. /sinh udii = cosh u - 1 = versh u, etc. 44. Another geometrical interpretation of integration is given here, adapted from Newton's Lemma II., Principia, Lib. I., § 1. Let the area ABDG (fig. 18) bounded by the curve y = ix, the axis of x, and the initial and final ordinates INTEGRATION. 93 AG and BD, be divided into a large number of narrow strips like PMmp by equidistant ordinates at distance Ax. Then the difference between the external rectangle Mp and the internal rectangle Pin is the rectangle Pp or Qq ; and therefore the difference between the sum of all the external rectangles and the sum of all the internal rectangles so described is the rectangle DE; also the area ABBG is intermediate to the sum of the external and the sum of the internal rectangles. Now in the limit when the breadth Ax of the rect- angles is indefinitely diminished and their number pro- portionately increased, the rectangle DE vanishes, and Fig.18 therefore the sum of all the external and the sum of all the internal rectangles each become equal to the area ABBG. If OM=x, MP=y, then the rectangle Pm=yAx; and denoting the sum of all the internal rectangles by XyAx, then the area ABDC=\t£yAx, = /ydx=/ixdx, replacing E byj", and Ax by dx in the limit, and supposing OA = a, OB=b. 94 INTEGRATION. With oblique coordinate axes, inclined at the angle w, the area ABBG '= sin wl ydx. a Any small part, such as AA or y Ax, is called a finite element of area, the symbol A (Delta) denoting as in § 8 a small finite increment ; and the symbol 2 (Sigma) is used, in conjunction with A, to denote the sum of a number of finite elements, such as "2,yAx, representing approximately the area ABDC ; and the transition to the Integral Calculus, where approximation ceases and the area is given exactly, is represented by replacing 2 by / and A by d. In practical problems of engineering and shipbuilding where we have recourse to the method of approximate quadrature, we divide the area ABBG into a finite number of elements MmpP, of which only one side Pp is curved, and then Pp may be made sufficiently small for it to be taken as straight without sensible error ; and now the area MmpP may be taken as the arithmetic mean of the outer rectangle Mp and the inner rectangle Pm. and the whole area ABDC as the arithmetic mean of the sum of the outer and of the inner rectangles. The student is recommended (by De Morgan) never to lose sight of the manner in which he would perform an integration, represented, graphically, by a quadrature, if a rough solution for practical purposes only was required. Rules for approximate quadrature, Simpson's, Weddle's, etc., will be given hereafter ; but meanwhile the arith- metic mean of a number of equidistant ordinates may be taken as the mean ordinate, which gives the height of the approximately equivalent rectangle. INTEGRATION, 95 45. Integration between Limits. Definite Integrals. Denoting the indefinite integral J~ixdx by i x x, then lixdx = (f 1 x)" a = fjb — f !«, a and this is now called a definite integral; a being called the lower limit and b the upper limit. (The term definite integral is however retained par excellence for integrals which can only be evaluated between certain definite limits, and of which the in- definite integrals cannot be found.) Examples. — Prove that the definite integrals (1) /x n dx = D - ?L_. V 'J n+1 a (2) /cos 6dd =J sin 0d8=l. o (3) fao&ddO =/2n 2 ddd = iir. (4) /(cos 0)W =/(sui 0) 3 cW = f . (5) /(cos 0) 4 cW = /(sin 0) 4 cW = t Vtt. ,a\ Hdx f"°dx j 1 m /^c&e ndx / "°dx 1 .„ 1 V 3+^y 3+^~ 2 y 3+^-18^^' 1 3 ^ V 'oJ+x 2 Z/ x 2 +a 2 4a' a < 9) /- (cc 2 +a 2 )(a; 2 + & 2 ) « + &' 96 INTEGRATION. 46. An interchange of limits changes the sign of a definite integral ; for lixdx = f j« — i-fi = — lixdx. b a Again, when we evaluate the integral by means of a substitution, say x = F, we must be careful to change the limits at the same time to the corresponding values of the new variable =/i(F$)F', equivalent" to f x b - ip = i^fi) - f /Fa). For instance, if we evaluate the definite integral / ^/(a 2 — x 2 )dx by means of the substitution x = a sin 0, then = makes x = 0, and = ^ir makes x = a, while dx = a cos d

d = \a 2 l (1 + cos 2)d = \-jra 2 . Considerations of symmetry and periodicity of the function fee to be integrated are useful. Thus if ix is an even function of x, so that f ( — x) = ix, then / ixdx = 2 / ixdx ; -a but if ix is an odd function, so that f( — x)= —ix, then 'ixdx = 0. 7 s Thus x 2 , x*, x 2n , cos x, sec x, vers x are even functions of x ; while x, x 3 , x 2n+1 , sin x, tan as, cot x, cosec a; are odd functions. INTEGRATION. 97 ( . /*3'r /"if For instance, / (cos x) 2n+1 dx = 2/ (cos x) 2n+1 dx, -iff but /(smx) 2n + 1 dx = 0; (sin a;) 2 "* 1 ^ = 2/ (sin a;) 2 ^ 1 ^, (cosx) 2n + 1 dx = 0. Again, if ix is a periodic function of period I, so that f (x + ril) = ix, where n is an integer, (for instance, sin 2-wxjl or cos 2-wxjl), then ixdx = n/ ixdx; and / f xdx = n I ixdx + / ixdx. Thus the period of sin x, cos x, sec x, cosec x, vers a?, or any odd power of these functions is 2x; the period of any even power, or of tan x, cot x, or an odd power of tan a; or cot a;, is tt; the period of any even power of tan a; or cot a; is \-w. It is advisable, in integration, to make use of these considerations in order to keep the limits of integration as close together as possible. Examples. — (1) Evaluate /(l, x, x 2 , x 3 ,... x 2n , x^+^dx. —a (2) /(cos 6) n dd and /(sin 6) n d6, for n = l, 2, 3, 4, ..., between the limits — \ir and |7r, and x, and f 7r, and 2ir. (3) Prove that (i.) lx,J(a— x)dx=-^ T a . (ii.) / x s /(a 2 —x 2 )dx = \a?. INTEGRATION. /•■• x f a dx , /• \ f a x 2 dx i _ 2 ( v -) /—. r, — - — s\- =t (substitute a? = asin 2 0). y J(ax-x % ) / - ; ----- J - O A M\ Fig.20 The mean effective pressure is obtained by dividing the work done by the distance of advance, so that the quadrature of the area ABDG leads to the value of the mean effective pressure ; and where the curve CPD is given by an Indicator, the method of approximate quadrature of § 44, or else a Planimeter is employed. When m = l, the pressure varies inversely as the volume, as in Boyle's law, and the above expression for the work done becomes illusory or indeterminate: But, by expanding (a/x)™- 1 in powers of m— 1 by §31, Pa l-Calx)™- 1 „ n x m-\ = Pa{\Qg- — ra—l l & a 2 reducing to Pa log xja, when m = 1. lo § D' ■+...}, INTEGRATION. 103 But, independently, in this case the work done is f Fa, „ , x —-ax = Pa log - x & a while the pressure curve GPJ) becomes a hyperbola; and thus the area AMPG is to the rectangle OAGE as log OM/OA to unity. Natural logarithms were formerly called hyperbolic logarithms for this reason, because they " square the hyperbolic areas " ; but the fact that all other systems of logarithms are equally connected with the hyperbola was not perceived when this name was given. The cross section of a gun has been drawn (fig. 20), taking OA as the length of the cartridge: but in the cylinder of a steam engine, the piston M starts from a point close to 0, and A must be taken to represent the point of cut-off of the steam, the pressure from to A being taken as the full boiler pressure P. The work done on the piston per square inch of cross section, in going from to M will now be represented by the area MPGE = Pa( 1 + log x/a) = Pa log ex ja ; and if we cut off the triangles OMP, OGE, each equal to \Pa, we are left with the sectorial area OPG=Pa\ogx/a = area AMPG, an important property of the hyperbola). (4) In the curve y = sin x, the area OMP = vers x, and in the curve y/b = sin x/a, the area OMP = ab vers x/a. (5) Prove that in the exponential curve y = a x (fig. 11) the area between the curve and the axis of x, cut off to the left by the ordinate MP = a x /log a = rectangle PM, MT; and in the catenary (fig. 15) y/a = cosh x/a, the area OMPA = a 2 sinhx/a. 104 INTEGRATION. 50, The Quadrature of ike Circle and Ellipse. In the circle (fig. 21), x 2 +y 2 = a 2 , or y = s /(a 2 —x 2 ); and therefore the area OMPB=/j(a 2 -x 2 )dx (L); and the area PM A = /,J(a 2 —x 2 )dx '. .(ii.). To find the first integral (i.), substitute x = a sin , then the angle BOP =

d 1) = | + f a 2 sin

, and the triangle OMP =£a z sin0cos0. INTEGRATION. 105 Expressed in terms of x, the area OMPB =_/ v /(a 2 — x 2 )dx = ^a 2 sha.- 1 x/a + \xj(a 2 — x 2 ). To find the second integral (ii.), substitute x = a cos 9, then the angle A OP = 6 ; and y = a sin 6, dx= —a sin 6W0. Therefore the area PMA = -/ahin 2 ed6 = \a 2 f(l - cos 2fl)dfl e ° = £ct 2 — £a 2 sin cos ; the difference between ^a 2 6, the area of the sector OAP, and \a 2 sm 6 cos 0, the area of the triangle OMP. Expressed in terms of as, the area PMA =J Jifl 2 — x 2 )dx = |a 2 cos _ hn/a — ^x^/ip? — x 2 ). Therefore the area of the quadrant OAB = J J {a? — x 2 )dx = \ira % ; and the area of the whole circle is -na 2 . . For the centroid of the area OMPB xA =fx*J(a?-x 2 )dx = %a?-l(a 2 -x 2 f, y A =J J(a 2 — x 2 )dx = \a 2 x — \x s . For the centroid of the area PMA xA =Jx > /(a 2 — x 2 )dx = \(a? — x 2 ) = ^y*, yA —/^(a 2 — x 2 )dx = ^a 2 (a — x) — \{a z — x 3 ). Therefore, for the quadrant OAB, x = y = \a\Zir. If we take A as the origin and AM = x, MP = y ; then y = ,/(2 ax — x 2 ) ; and the area AMP =/*/(% ax — x 2 )dx ; which is reduced to the above by putting x = a vers 6 = 2asin 2 £0, and then A0P=6. 106 INTEGRATION. 51. The equation of the ellipse (fig. 22) is - 2 +|- 2 =i )0 r 2/ =- x /(« 2 -^ 2 ); and therefore the area of a part of the ellipse is bjd of the corresponding part of the auxiliary circle cut off by the same ordinate, being called the excentric angle, and (j> the complementary excentric angle, so that Q +

/(x 2 — a 2 )dx. a To find this integral by the hyperbolic functions (§ 34), substitute x = acosh u ; then y — asinh u ; dx = a sinh udu. Therefore the area ATQ =fa 2 smh 2 udu = Jay (cosh 2u — l)du = |a 2 (£ sinh 2u — u) — Ja 2 cosh u sinh u — \a 2 u, of which the triangle OTQ = £a 2 sinh u cosh u, and there- fore the sector OAQ = \a 2 u (§ 34). Expressed in terms of x, the area ATQ =zJ~J(x 2 — a 2 )dx a — h x \Z( x ^ ~~ ffi2 ) — £a 2 cosh - x x\a = \xj(x 2 - a 2 ) - la 2 log {x + ^/(x 2 - a 2 )} /a. 108 INTEGRATION. To find the integral by the circular functions, sub- stitute x = a sec 6 ; then y = a tan 6, and the angle A OP = 6 ; and (§ 34) 6 = gd u, or amh u ; while u = log(sec 6 + tan 6). Then the area ^TQ = a, 2 fta,n 2 6 sec Odd, = £a 2 sec tan 6 — |a 2 log(sec 6+ tan 6) = $x s /(x 2 -a 2 )-%a 2 log{x+ > /(x 2 ^a 2 )}la, as before. If we take J. as the origin, and put AT=x, then TQ=y = J{2ax+x 2 ), and the area ATQ=J~ /s /(2ax+x 2 )dx, which is reduced by the substitution x = 2a sinh 2 £t&, 2a+x = 2a cosh 2 Jw ; then y=asinhu, and dx=2asioh %u cosh. ^udu> = a sink udu; so that the area A TQ = a 2 fsinh. 2 udu = %a 2 f(cosh 2u-l)du = £a 2 cosh u sinh u — \a % v J , as before. 53. In the conjugate rectangular hyperbola (fig, 24), 2/ 2 — x 2 = a 2 , or y = ^/(a 2 + x 2 ). Then the area OMRB=fj(a 2 +x 2 )dx, which is reduced to a 2 Jcosh 2 vdv bysubstituting x = asinhv, y = a cosh v, and then the sector OBR = |a% ; or the inte- gral is reduced to a 2 Js&c s (j>d(t> by substituting a;=atan0, o y = a sec . 55. The Quadrature of the Cycloid. Referring to fig. 9 of the cycloid (§ 21), and using 6 as the independent variable, the area OMP — lydx = /y^dd = /a 2 (vers 6) 2 dQ = . (5) a i y 2 = b 2 x 2 (a 2 — x 2 ), or sin~ 1 2;i//& = 2sin~ 1 a;/«, A = \ab, x = T \Tra, y = ±b. (6) x i -2axy 2 +y i =0, A = i > /2-7ra 2 , x=\a,y = ^ s /2aj-K. (7) In (y-x) 2 = a 2 -x 2 , A = ^a 2 , x = y = 4,a/STr; where A now denotes the area of the curve to the right of the axis of y. (8) In x i -2ax 2 y + a 2 (x 2 + y 2 )-a i = 0, A = \-ira 2 , x = 4a/37r, y = \a. (9) x* + 2x 2 y 2 + 4ax 2 y - 2a 2 (x 2 -y 2 +2ay) + a i =0, A = tto?{2 — % / J2), and determine x and y. *(10) Prove that the area of a loop of the curve (x 2 -na 2 ) 2 +(y 2 -a 2 ) 2 = a i is ^^/2a 2 {(n+l)cot- 1 ls /n — (n—l)coth.- 1 At /n}. 112 INTEGRATION. 56. Quadrature with Polar Coordinates. Let r = iff be the polar equation of a curve BPp (fig. 25) ; then if xOP = 0, OP = r = f 0. Let the sectorial area OBP enclosed by the curve BP, a fixed initial vector OB, and the variable vector OP be denoted by A. If AA and Ar denote the increments of A and r Fig. 25 corresponding to the increment A0 of 0; and if POp = A6, then x0p = 6 + Ad, Op = r+Ar = ?(6 + A6), and the area OPp = AA. Drawing the circular arcs PR, pr with centre 0, the sectorial area OPp is seen to be intermediate to the circular sectors OP R and Opr; or A A lies between \r 2 A0 and £(r + Ar) 2 A0, since the circular sector 0PR = ^r 2 A6, and the circular sector Opr = ^(r+Ar) 2 Ad; and therefore AA/A6 lies.between \r 2 and \{r-VAr)\ Proceeding to the limit, by making A9 indefinitely small, dA/d0 = ir 2 . Therefore A =J\r 2 dQ = ^J(i6) 2 dd, a a if the angle xOB = a ; this is the formula to be employed with polar coordinates for the quadrature of a sector such as OBP. INTEGRATION. 113 The revolving radius OP, starting from OB, sweeps out the area OBP (the fluent), and makes the area grow at the rate ^r 2 d0/dt per unit of time t. Also if x, y denote the coordinates of the centroid of the area OBP, then since g the centroid of the element OPp is ultimately in OP at a distance §r from 0, because OPp may he considered ultimately a triangle, therefore xA =f\r cos Q\r 2 dQ = \fr s cos Odd, a a yA =/%r sin 6 %r 2 d6 = ifr* sin 6dQ, a a where r = iO. Examples. (1) Prove geometrically that x zS~~y~J7 = r2 ju- (2) Find the area of a circle when its equation is given in the form r = 2a cos 6. (Here A =J\r 2 dQ = 2aycos 2 6d6 = af(l f(l + cos2e)dd = 7ra 2 .) -hir (3) Find the area and the centroid of a loop of the curve r = a cos 26. Answer : A = \ira 2 , x — 128, v /2a/105'7r. (4) Find the area and the centroid of a loop of the curve (i.) r = a cos n6, (ii.) r = bsinnQ, (iii.) r = a cos nd + b sin n6 (fig. 26). (Solution. — We must divide the right angle of the quadrants into n equal parts ; and now in (i.) r = a cos n9, the loop bounded by the lines 6 = ± \-w\n may be taken (fig. 26); and then 114 INTEGRATION, A =J %a 2 cos 2 n8dd = %<#/(! + cos 2nd)d6 so that the loop is half the circumscribing sector OEE' of the circle r=«, bounded by the radii 8= ±^w/n. Fig.26 This is geometrically obvious if we take vectors OP, OP' of the curve, such that E'OP'= AOP = 8; then, OP = a cos n8, 0P'= a sin n8, so that OP 2 +OP' 2 =a 2 ; and the sum of the areas swept out by OP and OP' is equal to the sector OAQ or OE'Q', Again, for the centroid of the loop, y = by symmetry, and x=OG = r; while INTEGRA TION. 115 8.4 =/$a 3 cos 3 n6 cos 0c?0 = £ay (3 cos «0 + cos 3n0)cos 0d0 hrln jn cos(?i-1)0+.3cos(w+1)0+cos(3m-1)0+cos(3tc-|-1)0}cZ0 , , ; j ■ i V t-l)fl + 3sin(-n, + l)9 + sin(3TO-l)9 sin(3w + l) 0\W» 3«-l 3w+l -J-a s cos —(—?- + — 1 _ 1 \_ 4a% 3 cos |tt/to 1J 2%W-1 to+1 3rc-l 3^+1/ (^-lX^-l)' - _ - _ 1 6an*cos \ir\n X ~ r ~Tr(n 2 -l)(yn*-l)' (ii.) In r = 6 sin nd, we may take the loop bounded by = and 0=71-/%; andnow,as before, A = \-Kb 2 \n; while by symmetry 6 = \irjn, and r has the above value with b written for a. (iii.) In r — a cos n6 + b sin nQ, it is convenient to in- troduce a subsidiary angle a, given by tan na = b/a ; and then r = a sec na cos w(0 — a). "We may take the loop bounded by the lines = a ± \-wfn; so that a £ a tH n o ,„ s,« 7ra 2 sec 2 na 7r(a 2 + 6 2 ) A =J £a 2 sec 2 na cos 2 %(0 — a)dd= j^ = ^ J a— lirln while = a, and r has the above value with A ^/(a 2 +b 2 ) or a sec na substituted for a, We thus see that the curves (i.), (ii.), and (iii.) are similar, but on a different scale, of a, b, and ^/(a 2 + b 2 ), and differently orientated with respect to the origin 0, which is the centre of similitude.) (5) Find the area and centroid of the car dioid r = avers 0. Answer : A = § rf, x = — fa. (6) Prove that in the curve r = a + b cos nd, where n is an integer, and b = -=- = s T (- e^ 2n— l\a/ (3) In x$ + y$ = a$, prove that s = %akx$. (4) Ina; 2 +2/ 2 = a 2 , s = asin _1 cc/a; and in y 2 = 2ax —> x 2 , s = a vers " x x]a. (5) In the catenary y = a cosh as/a, s = a sinh oj/a = ^(y 2 — a 2 ) = (area OMPA)/a. (6) In the tractrix J.Q {fig. 15), s = aloga/i/» (7) In the catenary of equal strength y = a log sec x/a> cosh s/a=secx/a, or x/a=gds/a. (8) In the cycloid (fig. 9), (rectified by "Wren, 1658) s = 8a vers |0 = 8a - 4 x /(4a 2 - 2a?/). (9) With polar coordinates, prove that in the curve r=acos0, or asin0, s = a6. (10) In r = a sec 0, s = a tan = ^/(j" 2 — a 2 ). (11) In the cardioid r = a vers 0, s = 4a Vers \ 6. (12) In the parabola r = 2a/(l + cos 0) = a sec 2 |0, s=a/sec s £0d0=asec £0tan|0+alog(sec£0+tan|0) = 7P+alog(sec £0+tan $0) (fig. 10) (Ex. 2. vi,§ 32). (13) In the equiangular spiral r = de 9cota (§ 30), s = r sec a = Pi, if measured from the origin 0. ■INTEGRATION. 119 59. The Volume and Surface of a Solid of Revolution. It has already been shown (§ 42) that for the volume V contained by the surface made by the revolution of the curve y = ix round the axis of x and by planes per- pendicular to the axis, d Vjdx = Try 2 = 7r(f x) 2 , and V= -Kjy 2 dx = -wfiixfdx ; so that the volume V may be supposed built up of elementary discs of radius ix and thickness dx. Also if x is the abscissa of the centroid of the solid V, the centroid being in the axis of revolution, xV= Tr/xy 2 dx =Jx{£x) 2 dx. Again, if S denotes the surface generated by the revolu- tion of the curve y=ix, then the fluent S may be sup- posed generated by the motion of an expanding (or contracting) circumference of radius y, and therefore dS ds dS ~ ds -dt = ^dt' 0V Tx =2 ^drx Therefore S = 2-TrJy-^-dx = 1-n-fyds, and for the centroid of the surface xS = 27rfxy-T-dx = lirfxyds. 60. Application to the Sphere, Spheroid, Paraboloid, and Gone. (i.) In the sphere, generated by the revolution of a circle (fig. 27) round the axis of x, y 2 =a 2 —x 2 , and therefore V= ir/{a 2 — x 2 )dx = 7r(a 2 x — \x 3 ), xV= irf(a 2 x — x 3 )dx — Tr(^a 2 x 2 — {x*). 120 INTEGRATION, For a hemisphere, therefore, F=f7ra 3 , and 5 = fa; and the volume of the complete sphere is f7ra 3 , § of the cir- cumscribing cylinder, or ^ird 3 , if d denotes the diameter. Again, since Qq ds _a dx y 8 = 2irjy-dx = 2-kwx ; o y _ t x xS=j2Traxdx = Traz 2 ,x=$x, (ii.) In the spheroid, gener- ated by the revolution of an ellipse round the axis of x, b 2 Fig.27 and therefore y 2 = -o(ct?-x*); and for the hemispheroid, V= %irdb 2 , x = fa. If the spheroid is prolate,- like a lemon, a is greater than b ; if oblate, like an orange, a is less than b. (iii.) In the paraboloid, generated by the revolution of the parabola y 2 = 2lx, V— irf2lxdx = irlx 2 = \ir£y 2 = | yolume of the circumscribing cylinder; and x = %x. Also S=2vfy^dy = 2Trfy^(^+l)dy M(^M INTEGRA TION. 121 (iv.) In the cone, generated by the revolution of the straight line y = xk&xia, where a is the semi- vertical angle of the cone, V= 7r/x 2 tan 2 acZx = %TrxHa,n 2 a = \irxy 1 = \ volume of the circumscribing cylinder ; and x — \x. Also 8 = 2irjy cosec ady = 7r2/ 2 cosec a ; and for the centroid of the surface S,x = fee. 61. If we refer to fig. 19, and compare the elements of volume swept out by the elements of area pn' and pm' when revolved about the axis Ox, we shall find these elements of volume are ultimately in the ratio of 2-irxydy _ xdy iry 2 dx ydx or twice the ratio of the elements of area pn' and pm'. Now, if the curve OP is given by the equation y n = p n - m x m , this ratio of areas of pn' and pm' becomes m/n, and the ratio of the elements of volume is therefore 2ml n; so that the area OMP is nj(m + n) of the rectangle OMPN (as before), while the volume swept out by the revolution of OMP about Ox is n/(2m+n) of the volume of the cylinder swept out by OMPN. When m = n, the curve OP is a straight line, and sweeps out a cone, and therefore the volume of the cone is one- third of the volume of the circumscribing cylinder ; and when 2m = n, the curve OP is a parabola with Ox for axis, and the volume of the paraboloid is one-half the volume of the circumscribing cylinder. When m = - n, the curve becomes a rectangular hyper- bola (fig. 20); and now the volume swept out by its 122 INTEGRA TlON> revolution round the asymptote Ox, to the right of the plane of MP, is equal to the cylinder swept out by the rectangle OMPN. Similarly the volume swept out by the revolution of y = a x (fig. 11) to the left of MP is half again as great as that of the cone swept out by TMP. In the sphere (fig. 27) the volumes swept out by the revolution round Ox of the elements of area Pn and Pq, in which NP=x, MP = y, rp=Ax, Pr= — Ay, are ulti- mately in the ratio 2-wxy(-dy) = ^J d V _r^^ = -a*(-=±)°Lh* -jixav- 2 ay n +1 pf 2 a Vl+W 2 a In a similar manner it may be shown that the area of the loop of x"+y i —5ax 2 y 2 =0 is -fa 2 , and that the volume generated by the revolution round Ox of the loop of x s + y B — 5ax 3 y = is -V-tto- 3 , 64. Moment of Inertia and Radius of Gyration. The moment of inertia of a body about an axis is defined to be, in the language of Indivisibles or Infinit- esimals, the sum of the products of the mass of each particle of the body and the square of its distance from the axis ; or in other words, in the language of Fluxions, the space integral throughout the body of the product of the density and the square of the distance from the axis. The distance k from the axis at which the body, a fly- wheel for instance, may be supposed concentrated without altering its moment of inertia is called the radius of gyration of the body about the axis ; and thus, if M denotes the mass of the body, then MJc 2 denotes its moment of inertia about the axis ; and k 2 is equal to the moment of inertia divided by the mass. When the body is homogeneous, we may suppose the density replaced by unity, and then the moment of inertia of the volume V is the space integral throughout the volume of the square of the distance from the axis, and may be denoted by Vic 2 , k denoting the radius of gyration about the axis. INTEGRATION, 127 : Similarly we may define the moment of inertia of an area A as the surface integral over the area of the square of the distance from the axis, and denote it by Ak 2 ; and we may define the moment of inertia of a line of length I about an axis as the line integral of the square of the distance from the axis, and denote it by Ik 2 , k denoting as before the radius of gyration. A knowledge of the M.I. (moment of inertia) and radius of gyration of a body is requisite in certain mechanical problems ; and as this quantity is difficult to find, except in some very elementary cases, unless we use the Integral Calculus, we shall discuss some simple cases here, in a manner analogous to that employed for deter- mining areas and their centroids, as an illustration of the power of the Integral Calculus. We shall first establish two fundamental theorems, which will simplify the subsequent operations. Theorem I. — If k denotes the radius of gyration about an axis through the centroid or centre of gravity of a body, and k x denotes the radius of gyration about a parallel axis at a distance h, then k 1 2 = k 2 +h 2 . Take the origin on the second axis and the plane of the paper perpendicular to the axis : then if m denotes the mass of a particle, and M the mass of the whole body, M = 2m, and Mk 2 = 2m(cc 2 + y 2 ) ; while Mk 2 =^m{(x-xf+{y -y) 2 }, if x, y, denote the coordinates of the centre of gravity ; the symbol 2 denoting summation or integration through- out the body for the separate particles represented by m. But Xmx=Mx, 2my=My, so that Mk 2 = tm(x 2 + y 2 ) - 2m(S 2 + y*) = Mk 2 - Mh 2 , or k 2 =k 2 -h 2 , k 2 = k 2 +h 2 . 128 INTEGRATION. From this theorem it follows that we need only calculate moments of inertia and radii of gyration about axes passing through the centroid or centre of gravity of a body ; as by Theorem I. we pass immediately from the k 2 or Mk 2 about an axis through the centre of gravity to that about a parallel axis. Theorem, II. — If Ak z 2 denotes the M.I. of a plane area A about an axis Oz perpendicular to the plane, and if Ak x 2 , Ak y 2 denote the M.I. about axes Ox, Oy at right angles to one another in the plane, then Ak 2 = Ak 2 + Ak 2 , or k z 2 = k*+k y \ For if dA denotes an element of the area at the point x, y, then Ak 2 =fyHA, Ak y 2 =fxHA, Ak 2 =f(x 2 +y 2 )dA so that Ak z 2 = Ak 2 + A h 2 , or k 2 = k 2 -\-k 2 . A similar proof will hold where the superficial density over the area is supposed to be variable. Examples. (1) Prove that k 2 = \a 2 for a circle (or cylinder) of radius a, about an axis through the centre perpendicular to the plane of the circle (or about the axis of the cylinder). (Divide the circle into concentric circular elements of radius r, and breadth dr ; then Ak 2 =f2Trrdrxr 2 = \-7ra i , and A = 7ra 2 , therefore k 2 = ^a 2 . The same method holds for the cylinder.) INTEGRATION. 129 (2) Prove that k 2 = \a 2 for a circle about a diameter. The k 2 about all diameters being the same, it is there- fore by Theorem II. half the k 2 about the axis through the centre perpendicular to the plane of the circle. Or independently, integrating with respect to x (fig. 21) Ak 2 =Jiydx xx 2 = k[x 2 J(a 2 - x 2 )dx ; -a and substituting x = a sin d = ay (sin20) 2 d0 o o fir = W/0- — cos *4>)d = iTra 4 , o and A = va 2 , so that k 2 = \a 2 . Similarly for an ellipse (fig. 22) about its axes, k 2 = \b 2 about OA, and k 2 = \a 2 about OB; and k 2 = \{a 2 + b 2 ) about an axis through perpendicular to the plane.) (3) Prove that k 2 = \a 2 + -^h 2 , for a cylinder of radius a and height h, about an axis through the centre and perpendicular to the axis of the cylinder. (By Example (2) and Theorem I., Vk 2 =j \a 2 dx(\a 2 + x 2 ) = ira 2 h{\a 2 + ^h 2 ) ; -JA and V=Tra 2 h, so that k 2 = \a 2 +^h 2 . When a = 0,k 2 = x^h 2 , the result for a thin rod or material line; and when h = 0, k 2 = \a 2 , the result for a thin disc.) (4) Prove that & 2 =fa 2 for a sphere of radius a about a diameter, or for a spheroid about its axis. (Here Vk 2 =fry 2 dx x \y 2 ^lyWx and y 2 = a 2 - i 130 INTEGRATION. Vk 2 = \ij[a 2 - x 2 ) 2 dx = Tr/ia* - 2a?x 2 + x*)dx = T B T Trq b ; —a and V=iTra s ,sotha,tk 2 = ia?.) (5) Prove that k 2 = ^a 2 , for a cone of height h and radius of base a, about the axis; and k 2 = ^h 2 +-^a 2 about an axis through the vertex and perpendicular to the axis of the cone. (About the axis of the cone Vk 2 = $Tr/y i dx —h^ri l&dx = TtrtraPh ; and V=^Tra 2 h, so that k 2 = ^a 2 . About the perpendicular axis through the vertex Vk 2 =py 2 dx{x 2 + \y 2 ) = iTra?h^h 2 + -&a 2 ), so that k 2 =%h 2 +^a 2 .) (6) Prove that k 2 = ^a 2 for a paraboloid of height h and radius of base a, about the axis; and that k 2 about a diameter of the base is %(a 2 +h 2 ). (7) Prove that k 2 =-g I d 2 for the ogival pointed head of an elongated projectile of diameter d about the axis, the ogival part being half a parabolic spindle. Prove also that, if the height of the ogival head is h, the distance of its centroid from the point is y-g-ft ; and that, about a diameter of the base of the head, h 2 =\h 2 + ^d 2 . (8) Prove that k 2 =^a 2 for a line of length a about its middle point, or for a rectangle of length a and breadth b about a line in its plane through its centre perpendicular to the sides of length a. INTEGRATION. 131 (9) Prove that 7c 2 = -Ma 2 +b 2 ) for this rectangle about an axis through its centre perpendicular to its plane. Prove also that k 2 = ^(a 2 + b 2 ) for a right solid of edges a, b, c about an axis through its centre perpendicular to the edges a and b; and that k 2 = %(a 2 + b 2 ) about an edge of length c. (10) Prove that k 2 =\h 2 for a triangle of height h, about an axis in its plane through the vertex, parallel to the base ; and thence h 2 = -^gh 2 for a parallel axis through the centroid of the triangle. (11) Prove that k 2 = ^h 2 +-J t a 2 for an isosceles triangle of height h and base a about an axis through its vertex perpendicular to its plane; and thence that k 2 has the same value about the axis of a right prism standing on a regular polygonal base, a denoting the length of a side and h the radius of the inscribed circle of the regular polygon. Deduce the value of k 2 about the axis of a circular cylinder. (12) Prove that if a solid is formed by the revolution through an angle 8 of a plane area about an axis Ox in its plane (fig. 28) the moment of the volume about Ox is equal to the moment of inertia of the plane area about Ox multiplied by 2 sinJ0. Prove also that the moment of the curved surface generated is equal to the moment of inertia of the perimeter of the plane curve about Ox multiplied by 2 sin£0. (Prof. H. Hart, Messenger of Mathe- matics, vol. xiv., p. 100.) 132 INTEGRATION. *General Examples of Integration. (1) Prove that the area of the curve y(l+x 2 ) = l — a; 3 , cut off by the axis of x, from x = to x = 1, is i7r+i log 2 -£ = -631972. (2) Prove that the area, between the curve and its asymptotes, of — — -» = -», or -s+ -» = -» is 4a 2 . J r x 2 y 2 a L x* y* cr (3) The area of r 2 =a 2 cos20 (the lemniscate) is a 2 ; and for a loop, sc = \tj2ira. (4) The area between the parabola y 2 = ax, and the circle y 2 =2ax—x 2 is. Jxa 2 — fa 2 , and a; = a {%tt — tt )/(!"" ~ f )• (5) Find the area in' the first quadrant contained by y 2 = 4aa5, a; 2 + y 2 = 2ax, y = 2(x — 2a). Express the area both when x is independent variable, and when y is independent variable. (6) A four-sided figure is formed by the three parabolas 2/ 2 -9ao;+81a 2 =0, y 2 -4ax+16a 2 =0, y 2 — ax+a 2 =0; and by y = 0. Prove that its area is 12a 2 , and is equal to the area enclosed by the chords of the arcs. (7) Prove that the area of each of the two equal and similar pieces bounded by the ellipse (x/a) 2 + (y/b) 2 = l, and by the hyperbola (as/a) 2 — (y/fi) 2 = 1, (n and 3& meaning the n tb derivative of y, must be carefully observed. If y = ix, and -j— is denoted by i'x, then tt-| is denoted d n y by fa;, and generally -j— by f"a;, or, more strictly, by f He. 135 (: 136 SUCCESSIVE DIFFERENTIATION. Sometimes also the notation y', y", y'", . . . and generally tfrt is employed to denote the successive derivatives of y with respect to x. The successive derivatives of a function are required, among other purposes, in the expansion of a function by Taylor's Theorem, as explained in the next chapter. 66. Successive Differentiation of Rational Functions. For instance (§ 4), dx m m i dx y-72/v.m therefore ^ r - s - = m(m — 1 )x m ~ 2 , ax 2 6?x m -£g =m(m-l)(m-2)x m - s , and generally —y-^ = m(m — 1 )(m — 2) . . . (m — n + 1 )xV l ~ ". If m is a positive integer d m x m -^ sr =m(m-l)(m-2)...2.1, denoted by m!, and all the higher derivatives vanish. Generally to differentiate successively any rational function of x with respect to x, the function should first be resolved into its partial fractions (Smith, or Hall and Knight, Algebra), and then the n th d.c. of a partial frac- tion A/(x—a), written in the form A(x — a) -1 will be Ai-lfnUx-a)-*- 1 or il /" 1 ^' ; \ y \ > (x—a) n+1 ' and of a partial fraction * w m be 3(-l)Mm+lMm + ^-l) (»-6) m (a;-6) m + n SUCCESSIVE DIFFERENTIATION. 137 67. To find the n^ d.c. of a partial fraction of the form Px+Q (x-af+fi 2 ' corresponding to a quadratic factor in the denominator, suppose it resolved into its conjugate imaginary partial fractions of the form, A+iB A-iB x — a — i(3 x — a+ifi' and then the % th d.c. will be y ' \(x-a-il3) n + 1 ^(x-a + i/3) n + 1 ) -( 1Y JA+iBXx-a+if3)^+(A-iB)(x-a-il3y+ l K ' {(x-a) 2 + ^} n + 1 which is easily thrown into a real form. Also (§29) d\og(x-a) = _l_ t ax x—a and therefore d-logix-^J-iy-Kn-iy. dx n (x — a) n ' 68. Successive Differentiation of Circular and Hyper- bolic Functions. By § 17, — ^ — = cos x = sin(a; + %ir), ct m n / y therefore — =-3— =sin(a3+i27r), dx 2 and generally , = sm(a; + §nir). Similarly, d -^^±M} = p«sm(px + q + imr), dnc0s ^ + ti=p«cos(px + q + in?r). 138 SUCCESSIVE DIFFERENTIATION. Since ^ = a*loga(§29), ax d n CL x -i d n e x therefore generally -=-^ = a x (log a) n ; and -=-^ = e* a i /o oo\ ^ 2 *sinh cc . , cP" +1 sinh cc , A1S0 (§ 33) -^^ = Smh!r ' ***» =ZC ° ShX - d*xxhx = CQsh ^ +1 coshx = ginh da; 2 " cfe 2M+1 To differentiate any powers or products of the sine or cosine, circular or hyperbolic, we must express them as sines or cosines of the multiples of the argument, as in Integration, § 40. Denoting by Fa; any rational algebraical function of x, then F © eto = ebX¥b ' F (S a " = aX¥ Q°Z a) ; and supposing ~Fx divided into its even part fa? 2 , and its odd part xtpx 2 (§ 46), then (^ \ sin sin cos -T-) mx = i(~ ™ 2 ) mx+m F™ that £ -^'-O; and given z=(ay + b)j(Ay+B), prove that z' -IW/ y 2\ This function of y or z is called by Professor Cayley the Schwartzian derivative, and is denoted by him by (y, x) or (z, x). 69. Leibnitz's Theorem. We have already proved the rule for the differentiation of uv, the product of u and v , two given functions of x ; namely (§12) _=_,+^ ; .and Leibnitz's Theorem enables us to generalize this •differentiation for any number of repetitions of the operation. SUCCESSIVE DIFFERENTIATION. 141 For differentiating again, each term on the right-hand side, being a product, gives rise to two terms ; and taking care not to invert the order of u and v, d 2 uv_d 2 u dw dv ,du dv d 2 v dx 2 dx 2 dx dx dx dx dx 2 _d?u ,qdu dv d 2 v dx 2 dx dx dx 2 ' . • d 3 uv_d 3 u ~d 2 u dv ~du d 2 v d 3 v # dx 3 dx 3 dx 2 dx dx dx 2 dx 3 ' , d*uv _ d 4 u .d B u dv d 2r w d 2 v du d s v d*v dx 4, dx 4, dx 3 dx dx 2 dx 2 dx dx s dx*' We now perceive the law for any number of differen- tiations, by analogy with the Binomial Theorem ; and the law can be proved by Mathematical Induction. For assume that d n uv d n u , d n ~ x u dv , n(n — I) d n ~ 2 u d 2 v . = v-\-n + — h dx n dx n dx 71 ' 1 dx 1.2 dx n ~ 2 dx 2 n{n—\) d 2 u d n ~ 2 v du d n ~h) d n v . + 1.2 do? dx^ Z2+n dba d^ zl+U dx^ W Differentiating again, each term on the right-hand side of (1) gives rise to two terms, of which the second of one term coalesces with the first of the next term ; so that d n+1 uv_d n+1 u . ^.d n wdv , (n+l)nd n - 1 wd 2 v cfe»+i ~~ do^+ lV +{n+ >~dx~ n dx 1.2 -^^ rr (fo2 + "" - J _ (n-^\)n d 2 u d n - x v .dudPv d n+1 v + 1.2 dtfdx n - 1+{n+ \tfe» + V i,,,,( ' If therefore the law expressed by (1) holds for n, it holds when n is changed into n+1, as expressed in (2). But the law holds when n is 1, 2, 3, 4, and therefore it holds when n is 5, 6, ... and generally when n is any positive integer. This law is called Leibnitz 's Theorem. 142. SUCCESSIVE DIFFERENTIATION. Leibnitz's Theorem can be established by a symbolical proof, which is easily extended to the case of the differ- entiation of any number of factors ; for if y = uvw... where u, v, w,.-. are functions of x, then dy du dv dw i z =T-w,..+M-rW...+w - 7 — ... + ... dux, dx dx dx = (D 1 +D 2 +D s +...)uvw..., where D 1 represents the operation of differentiation on u only, i) 2 on v, D 8 on w,.... Then, since these operators represented by D obey the same laws as algebraical quantities, g=(Z) 1 +D,+D 8 +...)"ut W ..., so that the coefficient of -*— -5— -j-f. . . is equal to the coefficient of x p y q z r ... in the expansion of (x+y+z+...) n by the Multinomial Theorem. . 70. , By Leibnitz's Theorem, = e-{a« + na^ x+ ^a^ + ...}y = e ax C a+ £) n y- with the symbolical notation. Therefore, if Fx denotes a rational function of x, / d 2 \ ( d 2 \ and F \dxV y sin &* + 2) = sin (p* + ?) F ^ - p 2 )y> theorems of great use in Differential Equations. SUCCESSIVE DIFFERENTIATION. 143 Examples on Leibnitz's Theorem. (1) Differentiate n times, a; sin re, o; 2 cosh2a3, (xsmxf, x m e ax , x 2 -r^ 2 + x-^-+y. (2) Prove that, if (in d 1 j (i.) y = sin.- 1 x, ( 1 ~ a;2 )^- a; ^ = () > and - when x = 0, 2/ (2») = 0, y = 0, j/W = 2 . 2 2 . 4 2 . . . . (2n - 2) 2 . (iii.) i/ = sin(msin- 1 a;+a) ) (l-x 2 )^-x-^+m 2 y = 0, Determine 2/ (m) when a; = 0. (Newton.) (iv.) 2/ = tan-^. ^ 2 +^8 + 2 4l = ' and (tf+^^ + 2(» + l>^+»(» + l)g = 0. When a = 0, y=(-l) n 2n\ a,- 2 * 1 .. ( d \ n (3) Prove that the differential equation (-5 — a) y = is satisfied by y = (C +C 1 x+... + C n - 1 x n - 1 )e™; ■d 2 / d 2 \ n and (^- 2 ±P 2 ) y = 0by ^ = (C' +C' 1 a ; +... + C' n . ia; -^ c C o >+O a; )- 144 SUCCESSIVE DIFFERENTIATION. 71. TheMaximum andMinimum Values of a Function. One of the most useful applications of the Calculus is to the determination of the maximums or minimums of a function y or ix of a variable quantity x. The subject has been already touched upon in § 5, where it has been shown that since dy/dx is positive if y increases with x, but dy/dx is negative if y diminishes as x increases, therefore dy/dx = at the turning points, such as A, where y from increasing begins to diminish, or as B, where y from diminishing begins to increase again. y Fig.29. x At A, y is said to have a maximum value ; and dy/dx is diminishing in passing through the value zero from a posi- tive to a negative value, and therefore d 2 y/dx 2 is negative. At B, y is said to have a minimum value ; and dy/dx is increasing through the value zero from a negative to a positive value, and d 2 y/dx 2 is positive. Thus to discover the maximum or minimum value of y, a function of x, we must first find the values of x which make dy/dx = 0. If one of these values makes d 2 y/dx 2 negative, the corresponding value of y is a maximum ; but if it makes d 2 y/dx 2 positive, the value of y is a minimum. SUCCESSIVE DIFFERENTIATION. 145 But if d 2 y/dx 2 is also zero, then a closer examination is required ; and it may happen, as at C (fig. 31) that y has neither a maximum or minimum value, although dy/dx is zero. Generally d 2 y/dx 2 is zero when dy/dx is a maximum or minimum, that is where the road ABC is steepest, as at D ; such a point D is called a point of inflexion on the curve ABC, the curve crossing the tangent at the point, and changing its curvature from one way to the other ; as seen in railway lines, on an S curve. As x increases continuously, the maximums and mini- mums of y must occur alternately, because y after reaching a maximum must diminish again to a minimum before increasing again to a maximum. It is advisable in the following examples to sketch the graph of the function whose maximums or minimums are required. Examples. (1) If y = 2x s -9x 2 +12x-S, (§ 5), the equation of the curve of fig. 29 ; dy/dx = 6x 2 - 18a: + 12 = 6(x - 1) (x - 2), which vanishes, when x = l, or 2 ; and d 2 y/dx 2 = 6(x - 2) + 6(x - 1). (i.) When x = l, d 2 y/dx 2 = — 6, and y = 2, a maximum ;• (ii) when x=2, d 2 y/dx 2 = 6, and y = 1, a minimum. (2) y=ax—x 2 , is a maximum \a 2 , when x=\a (§ 5). (3) y = x- x 3 , is a maximum and equal f^/3, when x = J^/3 ; a minimum — f-,,/3, when x= —\sj^- (4) Prove that y — x 3 — 3x 2 + 6x has no max. or min. value. (5) If y = (1 + x 2 )(J — x 3 ) 2 ; when x = 0, y — 49, a minimum ; £B=1, y = 72, a max.; aj= {/7, 2/ = 0, a minimum. 146 SUCCESSIVE DIFFERENTIATION, (We find dy/dx = 2x(l-x)(7-x s )(7+4>x+4x + 4cc 2 ) + . . . . = — 180 ; and therefore y is a maximum ; (iii.) when x = 4/7, d 2 y/dx 2 = - 6a; 2 (l - x){7 + 4a; + 4a; 2 ) + . . . . = a positive number, and y is a minimum.) (6) y = (x-l)(x-2)/(x-3) is a max., 3-2^/2, when x = '6-J2; a min., 3 + 2^/2, when 35 = 3 + ^/2. Explain how the min. is greater than the max. x 2 — x+l (7) If y=-YT T> wn en x = 0, y=—l, a maximum; when a; = 2, y=%, a minimum. (8) Determine the maximum and minimum of _ ax 2 +2bx + c y ~Ax 2 +2Bx+C (This can be done algebraically, by solving this equation as a quadratic in x, and determining the limits of y from a consideration of the expression under the radical. Then (Ay-a)x 2 +2(By-b)x+Cy-c=0; and solving this quadratic in ,x, -(By-b)±J\(By-bf-(Ay-aXCy-c)} _ Ay-a SUCCESSIVE DIFFERENTIATION, 147 The turning points of y are given by (Ay-a)(Gy-e)-(By-bf = 0, or(AG-B 2 )y 2 -(Ac+aG-2Bb)y+ac-b 2 = (1) and then x = — {By — b)/(Ay — a), ax+b bx+c ,„. y= AxTB = B^+G (2) so that (Ab-aB)x 2 +{Ac-aG)x+Bc-bC=Q (3) the transformation of (1) into (3) by the homographic substitution (2) ; obtainable also by putting dy/dx = Q. The roots of (1) and (3) will be imaginary only when the roots of Ax 2 -j-2Bx + C=0 separate the roots of ax 2 +2bx + c = 0, as will be seen on drawing the graph of y = (ax 2 + 2bx + c)j{Ax 2 + 2Bx + G) ; for instance, the graphs of y = (x-l)(x-3)/(x-2)(x-4<), or y=(x-l)(x-2)/(x-3)(x-4<). In all other cases the roots of (1) and (3) are real ; and denoting them by y v y 2 , and x v x 2 , then q . v _ ( A yi- a )( x - x i) 2 .... _ (a-Ay 2 )(x-x 2 ) 2 Bx U- Ax 2 +2Bx+G y y2 Ax 2 +2Bx+G and y 1 being the max., y 2 will be the mm. value of y. We now find ax 2 +2bx+c = p(x—x 1 ) 2 -i-q(x — x 2 ) 2 , Ax 2 +2Bx+C=P( l x-x 1 ) 2 +Q(x-x 2 ) 2 ; where P= 4*1=*, Q= ^Al 2 , 2/1-2/2 3/1 — 2/2 2/1-2/2 2/1-2/2 ' so that p=Py 2 , q=Qy v 148 SUCCESSIVE DIFFERENTIA TION. In constructing numerical examples, we may assign arbitrary integral values to x v a; 2 , P, Q, p, q ; and then integral values of A, B, 0, a, b, c result, which make equations (1) and (3) have rational roots. Thus y = (x 2 +x+l)j(x 2 -^x+\) may be written (x-iy+s(x+iy ,, y 3(x-i) 2 + (x+iy so that 3^=1, x 2 = — 1, P=3, Q = l,p=l, q=3 ; and ^i = 3, y 2 = i. Again y in Ex. 7 may be written y = {Sx 2 +(x-2) 2 }/{5x 2 -(x-2) 2 }). (9) Determine the maximum and minimum of r=a cos 6+b sin 0. (Writing it r = x /(a 2 +& 2 )cos(0-tan- 1 6/a), then r = „J (a 2 + b 2 ) a maximum, when = tan " 1 6/a ; r= —y/(a 2 +b 2 ), a minimum, when # = 7r+ tan - ^/a.) (10) Supposing given currents C and C produce de- flexions a and a in a tangent galvanometer, so that tan a/tan a'=C/C ; show how to make a — a a maximum. C—C (Here sin(a — a') = jrrTv sul (a + «') J and therefore a — a' is a maximum when a+a' = Jx.) (11) Determine the maximum and minimum of y, when x and y are connected by the implicit relation (§ 13) X s — 3axy + y s = 0. (Forming the first derived equation 3x 2 -3ay-3ax d £+3yf x = 0; then dy/dx = 0, if x 2 —ay = 0. Combining this with the implicit relation we obtain x = a£/2,y = a£/4!. SUCCESSIVE DIFFERENTIATION. 149 To find the corresponding value of d 2 y/dx 2 , form the second derived equation by differentiating the first derived equation ; but omitting terms involving dy/dx, since they vanish ; therefore or d 2 y _ 2x dx 2 ax — y 2 a so that the corresponding value of y is a maximum. Similarly, by differentiating with respect to y, equating dx/dy to zero, and examining the sign of d 2 x/dy 2 , we find that when y = a 4/2, x has the maximum value of a£/4<. These considerations are sometimes useful in drawing a curve whose equation is given as above.) (12) If xyix — y) = 2a 3 , determine the maximum and mini- mum values of x and y ; also if x i +y i — 4a 2 xy = 0. (13) Determine the greatest rectangle which can be in- scribed in a given isosceles (or scalene) triangle. Fig.30. (Let ABE be the given isosceles triangle (fig. 30) and let a denote the altitude OA and 26 the base BB' ; and x the height and-2y the breadth of the inscribed rectangle PNN'P'; or PNN"F' in the scalene triangle ABB". 150 SUCCESSIVE DIFFERENTIATION. Then x/a + y/b = l, since P lies on the straight line AB. Also, if u denotes the area of the rectangle PNN'P 1 , u = 2xy = 2xb( 1 — J = 2b( x ), and £= 26 ( i - 2 I) =o - d?u b when x = \a, y = ^b; and then -=-^=— 4-; so that « = £a&, a maximum. This is the problem of cutting the greatest rectangular log from a triangular log ; at least half the material is wasted.) (14) Determine the greatest cylinder which can be cut from a given cone. (Suppose the preceding figure to be made to revolve round the axis of x, and to describe a cone of altitude a and radius of base b ; also a cylinder of altitude x, and radius of base y. Then the volume of the cylinder V= Trxy 2 = Trb 2 x(l — x/a) 2 ; *r =rf j(i_5y_i?(i_5)i =rf (i_?Yi_s5) Ba o, ax l\ aJ a\ a/) \ a/\ a/ when x = a, or x = ^a. d 2 V b 2 (i.) When x = a, -j-j = 2tt— ; and V= 0, a minimum : d 2 V b 2 (ii.) When x=$a, -^p= -2-7T-; and F=/ T 7ra& 2 = £ volume of the cone, a maximum.) (15) Determine the cylinder of greatest curved surface which can be inscribed in a given cone. (With the same notation as before, the surface £ = 2-irxy, and therefore, as in Ex. 13, 8 is a maximum when x = %a,y = \b.) SUCCESSIVE DIFFERENTIA TION. 151 (16) Determine the greatest cylinder which can be cut from a given sphere. (Here the volume V=2-wxy 2 , where y 2 = a 2 — x 2 ; so that V=2 v (a 2 x-x s ); and dV/dx = 2Tr(a?-2x 2 ) = 0, when x 2 = ^a 2 , x = % s /?>a; y/x = ^/2 ; and then V=$ s /3Tra s = % / J3 volume of the sphere.) (17) Determine the cylinder of greatest curved surface which can be inscribed in a given sphere. (Here the surface 8=&Trxy = 4 when the surface is a minimum. (The diameter is double the height. This is the problem of the gasholder of given capacity and minimum weight ; the gasholder being a cylindrical vessel closed at the top and open at the bottom, where it sinks into water. Thus a gasholder of 8 million cubic feet capacity should be about 137 feet high and 273 feet in diameter.) (22) Determine the proportions of a cylinder of given volume, closed at both ends, in order that the whole surface should be a minimum. (The diameter equal to the length or height. This is the problem required in the design of a cylin- drical boiler of given volume, and minimum weight.) (23) Determine the proportions of a cylindrical tin canister to require minimum material for given volume, sup- posing the ends doubled down to overlap cylindri- cally (i.) a given distance, (ii.) a given fraction of the length of the cylinder. (The diameter equal to (i.) the difference, (ii.) the sum of the lengths of the cylinder and of the ends.) (24) Prove that a conical tent of given capacity will re- quire the least amount of canvas when the height is */2 times the radius of the base ; and that the canvas will then, when laid out flat, form a sector of a circle of angle %^/Stt radians, or about 208°. (25) Prove that, accordingto the regulations of the Parcel Post, which require the sum of the length and girth of a parcel not to exceed 6 feet — (i.) The greatest sphere allowed is about 17| inches in diameter, and a little over 1£ cubic feet in volume; SUGCESSI VE DIFFERENTIA TION. 153 (ii.) The greatest cube is 14f inches long, and nearly If cubic feet in volume ; (iii.) The greatest rectangular box is 2 feet long and 1 foot square, and 2 cubic feet in volume ; (iv.) The greatest parcel of any shape is a cylinder, 2 feet long, and 4 feet in girth, and over 2£ cubic feet in volume. (Rev. W. A. Whitworth.) (26) Determine the speed most economical in fuel to steam against a tide, supposing the resistance to vary as the 71 th power of the velocity through the water. (Solution.' — Let a denote the velocity of the tide, x the velocity of the steamer through the water; then x — a will be the velocity of the steamer relatively to the bank. The power required and therefore the coal burnt per hour will vary as the product of the resistance and the speed, that is, as x n+1 , and therefore the coal burnt per mile will vary as x n+1 /(x — a). This is a min. when x/a = l + l/n, or (x — a)/a — l/n. Thus if the resistance is taken to vary as the square of the velocity, the speed past the bank should be half the velocity of the current.) (27) Determine the length of cartridge which will give maximum velocity to a projectile in a gun, suppos- ing the powder instantaneously ignited, and that in expanding its pressure varies inversely as the m th power of its volume (ex. 3, p. 101). (Denoting by I the length of the bore, we must make \Trd?Pa{l - (a/Z)™" 1 }/^ - 1) a maximum by variation of a ; so that l-m(a/l) m - 1 = 0, or a/Z=(l/m) 1 * B - 1 >. When m = 1, this gives a/.l = 1/e — -368). 154 SUCCESSIVE DIFFERENTIA TION. *72. In finding a maximum or minimum, exceptional cases sometimes occur, where for a certain value of x, not oDi yS= o ' butaiso S =o '§=° In such cases it is generally simpler to notice that as x increases continuously, dy/dx changes sign from posi- tive to negative as y passes through a maximum value ; and dy/dx changes sign from negative to positive as y passes through a minimum value ; but if dy/dx does not change sign, y is neither a maximum or minimum. Sometimes also y has a maximum or minimum value when dy/dx changes sign by passing through the value infinity; but these cases require special investigation, and are conveniently solved by tracing the curve whose equation is y = ix. These cases are represented graphically in fig. 31. At A, y has a maximum, and at B, a minimum value. At G, dy/dx = 0, but does not change sign, so that y is neither a maximum or minimum, and G is called a point of inflexion on the curve. At D, dy/dx = oo , and changes sign from positive to negative, so that y is a maximum, and D is called a cusp. At E, y = co , and dy/dx = oo , but does not change sign, and y changes sign from — oo to +°° on crossing the asymptote. At F, y = oo , and dy/dx — oo , and changes sign from positive to negative, and y has an infinite max. value. At G, dy/dx is discontinuous, and changes abruptly from a negative value to a positive value, and y is a minimum. At H, y has a maximum value, but dy/dx does not vanish. (De Morgan, Biff, and Int. Calculus, p. 45.) SUCCESSIVE DIFFERENTIATION. 155 Fig.31 Supposing the curve to represent the plan of a railway, then we may compare D to a station like Cannon Street station ; while G may represent a turntable, and H an engine house on a siding. Examples. (1) If y = x*-?>x i +hx i , thendylda = 5x i -20x 3 +15x 2 = 5xXx-l)(x-3) = when x = 0, or 1, or 3. When x = 0, d 2 y/dx 2 = 0, and dy/dx does not change sign, so that y is neither a maximum or minimum, as at C. When x = 1, d 2 y/dx 2 = — 10, and y = 1, a maximum ; and when a; = 3, d 2 y/dx 2 — 90, and y= — 27, a minimum. (2) If (&//<& = (a; -l)(a:-2) 2 (a;-3) 3 , then dy/dx = 0, when a; = 1, or 2, or 3. When 03=1, dy/dx changes from positive to negative, and y is a maximum ; when x = 2, dy/dx does not change sign, and 3/ is neither a maximum or minimum ; similarly when x = 3, y is a minimum. 156 SUCCESSIVE DIFFERENTIATION. (3) If y = x m (a-x) n , then dy/dx=x m - 1 (a-x) n - 1 {ma-( / m+n)x} = 0, when x = 0, or ma/(m+ri), or a. When a; = 0, dyjdx changes sign from negative t6 positive, if m is even, and y is then a minimum; but dy/dx does not change sign if m is odd, and then y is neither a maximum or minimum. Similarly when x = a, y is a minimum if n is even; 2/ is neither a maximum or minimum if n is odd. Therefore the intermediate value x = ma/(m+n) makes y a maximum. (4) If y = a-b(x-cf, then dy/dx =x i +7). When a; = 0, dyjdx=, and changes from negative to positive, so that y is a minimum. When x = 1, y is a max. ; when x = 7, y is a. minimum. (6) Discuss the maximums and minimums of y when (i.)y = (x-\f{x-2f{x-2,f; (ii.) dyjdx = {x- l)\x - 2)\x - 3) " 3 ; (iii.) dyjdx =(x— o)(x — /S) 2 (x — y)(x — S) ; (iv.) dyjdx =(x-a)(x-/3) s (x-y)(x-S); a=KV+v)t, also, from (3), %v 2 = \ V 2 — gy. If the body N rises to the height h, and if T denotes the whole time of going up and coming down again, then \T is the time of rising and also of falling, and y= hgT, hV 2 =9K and h = %gT\ Thus with a foot and second as units, and <7 = 32, then h = (2T) 2 , or the height attained in feet is the square of twice the number of seconds the body is in the air. Also v=g(lT-f), and y = izgt(T-t) = \gtt',iW = T-t; %v 2 =g(h-y). 162 SUCCESSIVE DIFFERENTIATION. *76. Vertical motion of a body in a resisting medium. When a body like a sphere is projected vertically in the air or any other resisting medium, in which it is assumed that the resistance varies as the nth power of the velocity ; then if the weight of the body is W lb., and if w denotes its ternvmal velocity, that is the velocity with which it descends uniformly when the upward resistance of the medium balances the down- ward attraction of gravity, the resistance of the air at any other velocity v will be a force of W(v/w) n pouDds, and the retardation due to this resistance will be g(vjw) n The terminal velocity is observable in falling rain- drops, hailstones, or meteorites ; also in a train or steamer at full speed. In a balloon or parachute we seek to make the terminal velocity as small as possible, but in projec- tiles we increase the terminal velocity and thereby the ranging power by giving them an elongated form. When the body is moving downwards with velocity v, the resistance of the medium acts upwards, and the equation of motion is dv (v\ n ,, N dC g ~ g U (1); and when the body is moving upwards, with velocity v, both , gravity and the resistance act downwards, and the equation of motion is dv /v\ n ... irr-v-Aw) < 2 >- We shall take the resistance to vary as the square of the velocity, so that n = 2 ; and now equation (1) may be dt 1 1 written - T -= - ^ — , , .„ > dv g 1 — (v/wy SUCCESSIVE DIFFERENTIATION, 163 f _ I /~ dv so that t= - A — . .„ #/l-(-u/w) 2 so that t' = - /= — "/",, ,„ = -tan' 1 - = ^-loe > or — tanh A — , by(y),p. 65. Z# °M)-» g W • /w/ ' t or v = dy/dt = wta,nhgt/w (3) if 1/ denotes the depth of the body below the highest point after falling a time t (seconds). Therefore, integrating again, y = w /tanh — dt = — log cosh "- (4). a J w g & w w In the ascending motion, suppose the body was at a depth y below the highest point t' seconds before reaching it, and that its velocity was v' ; then v' = dy/dt', and from(2) |:=^(l+S) (5), dv' w _ 1 v / . + (v/wf g w or v' = w tern gt'/w (6), and y = w/taxi ^-dif = — log sec — (7). Z/ w g b w v From these equations (3) to (7) we find exp gy/w 2 = sec gt'/w = cosh gt/w = v'/v, so that gt'/w = gd gt/w, and (w/v') 2 — (iv/v) 2 = 1 . If in fig. 13 the circular sector OAP is taken pro- portional to if, the time of ascent to a height y, then the hyperbolic sector OAQ will bear the same ratio to t, the time of falling back again; and the velocities v' of projec- tion and v of fall will be represented by AR and ATJ, while the height y will be proportional to the logarithm of the secant OT. Also at half the times of ascent and descent the velocities will be equal, and represented by At. (Newton, Principia, lib. ii., Prop, ix.) 164 SUGCESSIVE DIFFERENTIATION. The upward velocity is therefore infinite at an infinite depth below the highest point, but the time of ascent is finite, namely \-ww\g. In the downward motion the velocity gradually grows from zero to w the terminal velocity, which is reached asymptotically at an infinite depth, and in an infinite time. The terminal velocity of a rifle bullet may be taken as 400 f.s. ; and therefore, if fired vertically upwards with velocity 1200 f.s., instead of attaining in the absence of resistance a height of 22500 feet in 37£ seconds, it will ascend only 5750 feet in 15 seconds ; returning to the ground again in 23 seconds with velocity 380 f.s. To attain within one per cent, of the terminal velocity, when vjw is equal to or less than - 99, occupies a time £=-^log e 199 = ^x 5-2933, 2g * 2g during which the body will have fallen iy 2 , 100 Aw 2 orvnh71 Thus if a man falls in a parachute 800 yards in 2 minutes, we may suppose w = 20 ; and with g = 32, we find t=V t 7, and 2/ = 24 - 4, the time in seconds and the distance fallen in feet to attain within one per cent, of the terminal velocity. In these equations g represents the acceleration of gravity, corrected for buoyancy ; so that it may happen, as with a balloon in the air, or a buoyant body rising to the surface in water, that the sign of g must be changed. In the motion of a steamer through the water against a resistance varying as the square of the velocity, or of a railway train moving against a resistance consisting of a constant part and a part proportional to the square of the SUCCESSIVE DIFFERENTIATION. 165 velocity, we must replace g by /, the acceleration with which the steamer or train starts or stops, and take w to represent the full speed. Thus if a steamer of W tons displacement is propelled at a velocity w (feet per second) with a uniform thrust of T tons, f—gT/W, and the horse- power at full speed is 22402 , io/550 ; and to stop by rever- sing the engines takes ^ttw// seconds, during which the steamer will have gone (|ty 2 //)log e 2 = {%w 2 /f) x 06931 feet. *(77) Experimental Determination of the Resistance of the Air. The velocity of projectiles and the resistance of the air is now inferred experimentally from the instants of time recorded by a chronograph at which a series of equi- distant screens are passed, the record being made by the projectile cutting wires carrying an electric current ; the projectile is supposed to travel so fast that it may be taken to fly in a horizontal straight line. If the shot takes t seconds to go s feet, then the velocity v and the acceleration / are given by (§§ 10, 11, 74) _ds ._dv _d 2 s _ dv v ~di't~d~t~~de~ v ds' But the chronograph gives t as a function of s, not s as Idt a function of i, so that we must write v=l ~r ', while / as , dv d f, ldt\ ( dH\ l/dt\ 2 dH , f= v drs =v A l ld S r\-dsVl\ds) =-^ 3 - Then if the shot weighs W lb., the resistance of the air is W^v 3 poundals, or W%-^u 3 /g pounds Mr. Bashforth in his experiments found that d 2 t/ds 2 was a slowly varying quantity ; and if we take it as constant, 166 SUCCESSIVE DIFFERENTIATION. we assume that the resistance varies as the cube of the velocity ; and now if d 2 t/ds 2 = c, and we integrate, suppos- ing Fis the initial velocity, where s = and t = 0, 1 dt 1 , , . si 2 1/1 1\ The average velocity £7" over s is given by U= 8 It, and i_*_i,i _V 1 ^ 1 U~'s~V + 2 CS ~2\V + v. so that the average velocity U is the actual velocity at the distance Js of the mid point, and is the harmonic mean of the initial velocity V and the final velocity v. Suppose that three equidistant screens I feet apart are cut at instants of time by chronograph t v t 2 , t s seconds ; then, with dH/ds 2 = c, we assume that t = a+bs + ^cs 2 . Measuring s backwards and forwards from the middle screen, we find a = t 2 , 2bl = t s — t v cl 2 = t 1 — 2t 2 -\-t s ; and at the middle screen V— 1/6 = 2lj(t s — t^), the average velocity from the first to the third screen ; while the resistance of the air at this velocity is WcV s /g pounds. For instance if W= 70,1 = 150 and t x = 2-3439, t 2 = 2"4325, £ 3 =2'5221; then 1^=1684 f.s., and the resistance of the air is 462 pounds. (The Bashforth Chronograph, 1890.) When a projectile, whose terminal velocity is w, is flying horizontally against a resistance of the air varying as the nth power of the velocity, the equations of motion dv vdv _ (v\ n . dt~ ds ~ ~^\w/ ' so that ^r V -^= * {(T- (T 1 } * w J if 1 n — l\\vJ \VJ ) V w 2 ~J v"- 1 ~~n-2\\v) \VJ )' S UGGESSIVE DIFFERENTIA TION. 167 But when n = 1 , gtjw — log V/v, gs/w 2 = V— v ; j i o a£ w iu as . F" and when n = 2, ?- = -==, ^= = loo- — . w v V w A ° v 78. Equations of Motion in a Plane. If x, y are the coordinates at the time t of a point P moving in a plane (fig. 1), then (§ 10), -^ and -^ are the component velocities of P parallel to Ox and Oy. Similarly, -^ and -^f- are the component accelerations parallel to Ox and Oy ; since the acceleration in any direction is defined to be the rate of change of velocity in that direction, reckoned by the growth per unit of time. 79. Motion of a Projectile when Unresisted. Suppose, for instance, that a body P is projected from (fig. 33) with velocity V at an angle a to the horizon, and that the resistance of the air is left out of account. Then, if the axis Ox is horizontal, and the axis Oy drawn vertically upwards, the equations of motion are dt 2 ' dt 2 ~ g - Integrating with respect to t : -jr = a const. = Fcos a, -^ = a const. — gt = Vain a— gt; and integrating again, supposing t = at 0, x = Vt cos a, y=Vt sin a — \gt 2 . Therefore t = x/(Vcosa), and substituting this value of qx 2 t my, V = x tan a — ^p4 9~> a a 2K 2 cos 2 a the equation of ,the trajectory; which will be found to be a parabola. 168 SUCCESSIVE DIFFERENTIATION. y H N K X L r ^^> fi \ \ \ 1 \ \ 1 \ \ 1 \ \ V / -* T C Fig.33. B * For treating the equation as a quadratic in x, la; sinacosa) = cos 2 a \y — = \ g / g V 2 V 2 . ^-sin a a 2 9 and comparing this with the equation (x-hf=-p(y-k), which is the equation of a parabola as in fig. 33, of which the vertex is at (h, h), the latus rectum is p, and the concavity downwards ; then & = (F 2 sinaCOSa)/sr, k = %(V*sm*a)lg, p = 2(V 2 cos 2 a)/g. If HK is the directrix of the parabola, then The height iV 2 /g a body must fall to acquire the velocity V was called the impetus of the velocity v; so that the impetus at is OH. SUCCESSIVE DIFFERENTIA TION. 169 . . dx d 2 x dy d 2 y dy Again dtdf 2 + dtW~~ 9 dt ; and integrating, v denoting the velocity at P, The velocity v at any point P is therefore the velocity which would be acquired in falling freely from the level of the directrix; for \v 2 = g . OH-g .MP=g . PK ; or the impetus of the velocity at any point P is PK, the depth below the directrix. Produce MP to meet the tangent at in IT; then since OM = x = Vt cos a, therefore OT=Vt; and since MP = y = Vt sin a - \gt 2 , and M T= Vt sin a, then TP = \gt 2 . Thus the parabola OP may be supposed described by a body which is carried in t seconds from to T by the original velocity of projection V, without being influenced by gravity or resistance (the motus violentus of ancient writers), and which afterwards falls from T to P in t seconds under gravity without resistance (the motus naturalis), the combination of these two motions being called the motus rrdxtus; this is the method of Galileo employed in Elementary Dynamics. Then with 0T= Vt, TP = %gt 2 , the elimination of t gives OT*/TP = VH 2 l\gV = 2 V 2 /g = WH, so that OP =WH. TP, or P V 2 = 4?H0 . V, (fig. 33), whence it is inferred, by a theorem of Geometri- cal Conies, that the trajectory OP is a parabola. But we can easily show that the curve OP satisfies the definition of a parabola as " the locus of a point which moves so that its distance from a fixed point is equal to its distance from a fixed straight line" by making the angle TOS equal to the angle TOH, and 08 equal to OH ; 170 SUCCESS! VE DIFFERENTIA TION. now if OT and the parallel VP meet SH in F and Z; then since P V 2 = 4>H . V, therefore, by similar triangles, PN* = 4HY. YZ=(HY+YZ) 2 -(HY-YZf = HZ 2 - SZ 2 = HP 2 - SP 2 , so that SP 2 = HP 2 - PN 2 = PK 2 , or SP = PK, and P therefore describes a parabola, according to this last elementary definition. A jet of water issuing from in the direction OT with velocity V (or a stream of bullets from a Maxim gun), will form a continuous parabola, and since the horizontal component of the velocity is constant, equi- distant vertical ordinates will cut off equal volumes of water, or the line density of the jet is proportional to cos \p- ; so that the height of the centre of gravity of the jet is the average height of the ordinates or § the height of the vertex A , since the area OAB is f of the circum- scribing rectangle. The jet if frozen will stand as an arch, even when cracked across by normal planes ; and when inverted will hang as a catenary, without change of form if flexible. In a suspension bridge the weight may be supposed concentrated in a uniform roadway, and the chains will then assume the parabolic curve. 80. To find the range OB, and the time of flight on the horizontal plane through 0, put 2/ = ; then = x tan a — gx 2 /2 F 2 cos 2 a, = Vt am a- \gt 2 . Therefore, if the range is denoted by R, R = 2 V 2 sin a cos ajg = V 2 sin 2a/ y Y^T*-^ JV p\ rdtVdt? I \ffs dx P > dt' B V M X (ii) dx dr . . Q dd dy dr . . , n d6 di = dt COsd - rsme dt> dt = dt sm6+rcose d-f Therefore the component velocity in the direction OP, called the radial velocity, as before (§ 23), dx , , dy . „ dr = dt COs6+ l sm6 =M> and the component velocity perpendicular to OP, called the transversal velocity, dx . - dy n dQ ■dt sme+ dt cose = r W Differentiating again, d 2 x (d?r d%\ fd 2 r dG*\ a /_„. ,„, .„ . , dt*= {-aW- r dT*) C0S 6 -\^dt di +r w) Sm 6 > , ~drdd d*e\ . w) sn d*y (d*r dG 2 \ . a , f a dr dd , d 2 d\ a w = (w- r dr*) smd+ ( 2 dt dt +T w) C0Sd - SUCCESSIVE DIFFERENTIATION. 173 Therefore the component radial acceleration d?x a , d 2 y . . d 2 r d6 2 W eosd+ oW sm6= W- r W and the component transversal acceleration d 2 x . n , d 2 y n a dr dO , d 2 9 -W 2Sme+ W cose = 2 Ttdt +r df 2 ' 1 d / df)\ which is usually written in the form - ji\ T ' i ^i)> to which it is equivalent, as may he verified by differentiation. 82. Motion in a Circle. Suppose, for instance, that P describes a circle round as centre ; then r is constant and dr/dt = ; so that the radial acceleration is — rdd 2 /dt 2 and the transversal or tangential acceleration is rd 2 6/dt 2 . Also, since s = r6, if v denotes the velocity in the circle, v = ds/dt = rd6/dt — nr ; where n = d6/dt, called the angular velocity round ; so that the tangential acceleration is -jr or -^ or v-*-, the same as for rectilinear motion ; and the central or normal acceleration in the direction PO is r(dd/dt) 2 = v 2 /r = n 2 r = nv. To realize this circular motion practically, suppose a plummet, suspended from a fixed point by a fine thread of length I, to be projected with velocity v so as to describe a horizontal circle of radius r = I sin a under gravity. Denoting by Q the tension of the thread, then v 2 /r '. g '.'. Q sin a '. Q cos a, or v 2 /r = g tan a. Now if T denotes the period, or time of revolution in the circle, T=2Trr/v; therefore gta,na = v 2 /r=4:Tr 2 r/T 2 , or T=2TT,J(r cot alg) = 2tt /V /(£ cos *aK)-* V Now r 2 d0/dt is generally denoted by h, and then h = 2dA/dt (§ 56), twice the rate the sectorial area A is swept out by the vector OP, revolving about 0. Change the independent variable in the equations of motion from t to 0, and denote r by 1/u (§ 23). Then since r 2 dd/dt = h, therefore d6/dt = hv? ; and equa- tion (2) becomes dh T dh T dlh? T — . r\y — r\v A — dt u' dO hu s ' dd w 3 . . dr dr dO 1 du, „ du, A ^ am dt = TQdt= ~u 2 dO hu= ~dO h '' +*, + d 2 r _ d 2 u dO h du dh _ d 2 u, 2 2 du T so that -^--3^^ A -5gr^--3gs*«-50-- Then equation (1) becomes, since r(d0/dt) 2 = h 2 u s , Oj U-. 9 « OjU JL 7 o Q T» — 'job' 1 u j7\ n?u 3 = R, dO 2 dO u d 2 u _ ^_ _7^du or d6 2 + U ~ h?u 2 ~h 2 u 3 dQ {6) ' the differential equation of the orbit of P. SUCCESSIVE DIFFERENTIATION. 175 *84. Central Orbits. For a central field of force in which the attraction is always directed to the origin 0, T=0, and h is constant. Denoting the central acceleration due to the attraction towards by P, so that P — —R, then d?u _ P , . d6 2+U -hW W ' or P=h 2 u 2 (jj^ + u whence the required value of P is found when the equation of the orbit is given. Examples. (1) The polar equation of a conic section with a focus at the origin being l/r or lu= 1 +e cos 9, then l (w +u ) = 1 ' and P = h?vJ i /l = n 2 a s r~ i , which varies as u 2 or r~ 2 ; a denoting the mean distance, and n the mean motion or mean angular velocity round ; and therefore 27r/n the period of revolution. Thus Newton's Law of Gravitation, by which the Sun attracts the planets with intensity inversely proportional to the square of the distance, is deduced from Kepler's Law, that the planets describe ellipses of which the Sun occupies a focus. (2) Prove that, for a conic section with the centre at the origin, P varies as r. (3) Prove that P varies as v? in the curves (Cotes's Spirals) (i.) aw = cosh. n6, (ii.) cwt = exp n6 (the equiangular spiral, p. 53), (iii.) wvu = sinh n6, (iv.) au = nd (the reciprocal spiral), (v.) au = sinnO, or cos nO, or cos n(9 — a). 176 SUCCESSIVE DIFFERENTIATION. (4) Prove that P varies as r~ 2 "~ 3 in r n = a n cos n6. (5) (i.) P varies as at 5 in a-w, = tanh %y/W or coth \^JW ; .... -r, . ,. cosh 0-2 cosh 0+2 (li.) F varies as u* in cw = — , n , 1 or — ,— ^ — ? ; v ' cosh 0+1 cosh — 1 ..... „ . ,. , „ cosh20-l cosh 20 + 1 (m.) P varies as^m a %^ cosh20 + 2 or cosh2e _ 2 . (In these curves au=l or r = a when = oo ; so that after an infinite number of revolutions round the origin, the curves approach to coincidence with the circle r = a, which is therefore called an asymptotic circle. A body describing this asymptotic circle freely under the central attractions of (i), (ii), (iii), would be unstable, and would be found ultimately describing one of the corresponding curves.) (6) Prove that the radial and transversal P and T in the orbit (r, 0) become changed to pP + p(q 2 — l)h 2 u s and pqT in the orbit (pr, q6). (7) Prove that the orbit au = gd m0 or cos au cosh m0 = 1 can be described with P = h 2 u 3 , T = mh 2 u 3 sin au; and the orbit m0 = gdcra or cosh au cos m0 = 1 with P = h 2 u B , T= —mh 2 u 3 sinh.au; and that dr/dt is constant. The inverse problem of the determination of the orbit when the central force is given as a function of u is more complicated; we multiply equation (4) by du/dO, and integrate both sides with respect to ; when whence is found as a function of u by transposition, and a single quadrature or integration. SUCCESSIVE DIFFERENTIATION. 177 85. Interchange of the Independent and Dependent Variable. In §§ 77 and 83, the practical need in Dynamics of expressing differential coefficients, such as dx/dt, d 2 x/dt 2 , in terms of differential coefficients of other variables, independent and dependent, has been illustrated; and the general operation required is called change of the variable, which we shall now investigate in its generality, and illustrate still further. In § 77, we found -^ = — j^/ij') > thus inverting the independent and dependent variable ; and with variables x and y, the general theory is established as follows : — d 2 y _ d / /dx\_ d /_ jdx\dy _ _d 2 x //dx\ s dx 2 dx\ / dy) dy\ / dy/dx dy 2 l \dy) ' d s y_ __{dx x, then n\dx n ^f =1 (m ' s) s! dt*' where 0(», S ) is the coefficient of h n in the expansion of e 8 or {(x+h) — )*-S3 ' -{a-t- 1 )}-^^- 1 )* a' n 1/ We must distinguish between a?S-f , the product of as" and -5-^, and ( x-j- j y, obtained by operating n times by x dx on y-> iov ( x d-Jy = i$' iix=e - *88. Application to Differential Equations. The theory of change of the variable is of great practical use in the solution of differential equations. The general linear differential equation of the second order may be written tf + 2Py' + Qy = 0, or R, (A), where P, Q, R are given functions of x. By putting y = uv, then \v I \V V J V and the coefficient of u' may be removed by making v'\v = — P, or log v = —JPdx, v — exp_/( — Pdx) ; and now v"/v+2Pv'/v+Q = Q-P 2 -P', a quantity, denoted -by /, and called the differential in- variant; so that, in its canonical form, u" +Iu = R exTpfPdx = 8, suppose (B). With R = 0, and u v u 2 denoting two particular solu- tions of the differential equation u"+lu = 0, so that u = Au 1 +Bu 2 is the most general solution ; then 182 SUCCESSIVE DIFFERENTIATION. u{' + Jttj = 0, « 2 " + Iu 2 = 0, so that, eliminating /, u-[' u 2 — u t u 2 " = 0, and integrating, u( u 2 — u x u 2 = G, a constant. Denoting by s the quotient uju 2 , then s"/s'= — 2u 2 '/u 2 ; and ?-l© 2 = 27 ; <°>' a non-linear differential equation of the third order, in which the left hand side is the Schwartzian derivative, (s, x) ; and of which the most general solution is thence s = (clu-l + bu 2 }l(A.u x + Bu 2 ) ; since {(as+b)/(As+B), cc}=(s, x), by ex. 16, p. 140. Knowing any solution s of this equation (C), then from u 2 /u 2 = —%s"/s', we obtain, by integration, log u 2 = constant — \ log s', or u 2 = C$s'~i ; and then u^dss'^i; so that equation (0) may be written in the same canonical form as (B), Again, denoting the product u x u 2 by z, then z' = u/ug + ^6 1 ^^ 2 / , z" = u{u 2 + 2u-[u 2 + lijtt/ = 2{u{u 2 ' — Iz), z"' = 2(u{u 2 ' + u 1 'u 2 "-Iz'-rz) = -2I(u 1 u 2 '+u 1 'u 2 )-2(Iz'+rz)=-4Iz'+2rz = (D), a linear differential equation of the third order, also satisfied by u* and u 2 , and therefore generally by z = au^ + 2bu{w 2 + cu 2 2 . A first integral of this equation is zz"-%z' 2 + 2Iz 2 + %C 2 = 0, a non-linear equation of the second order, one solution of which is z = u x u 2 = Cs/s'. SUCCESSIVE DIFFERENTIATION. 183 Supposing a solution z of this differential equation (D) is known ; then since u l' u 2 ~ ^1*2' = C» therefore <_< = ^ and integrating, log i^/i^ = log s =fCdx\z ; or u 1 /u 2 = s = expjCdx/z; while UjU 2 = z; so that i^ = ^(23) = ^/as exp \f(Cdx\z), u 2 = s/( z l s ) = J z ex P hf( ~ Gdx/z). The solution of the more general equation (B) can now be found, and thence the solution of (A) ; for if u" + Iu = 8, while u{' + lu x = 0, u 2 " + Iv 2 = ; then v!'vb x — uu{ = 8u v u"u 2 — uu 2 " = Su 2 ; and integrating w'u-l — im-[ =J~8vb 1 dx, vfu 2 — uu 2 =J~8u 2 dx ; whence u = Cv^jSu^dx — Gu 2 j8u 2 dx : to which the complementary function Avj x -\-Bu 2 may be added, to obtain the most general solution. For instance, the solution of y" ± n 2 y = ix is a cos u = A , cosh nx + B 2* nx + \f x 2X® - i)Wbc _ « . dr 2 \ p Jrdr p 2 and on changing the dependent variable from u to w, where u = wr a lP-l, to the form employed by Laplace, 1 d 2 w _ n(n + l) ± o w dr 2 r 2 where n replaces a\p—\, and ±q 2 is written for ±4 when ^ = sinf ; (ii.) ( a *+x*)^ + 2x^ = to § = 0, when a; = atan t; (iiL) ^aJ* +a! cfce +7lSy=0 to dl +M22/ = ' when x=eK (iv.) a; 2 "f^±a«2/ = to b n< ^-±u = 0, *■ y da" ^ dz™ ' when se/a = e' = 6/0 and y=x n ~ lr w. (2) Given 1/ = sin(m(9 + a), and a; = sin 6, prove that (1 — x 2 )y" — xy'+m 2 y = ; and that, when cc = 0, 2/( 2«-i) = m (l2_ m 2)(32_ m 2)...{(27i-3) 2 -m 2 }cosa, 2/(2") = _ m 2 (2 2 - m 2 ) (4 2 ~ m 2 ) . . . { (2t? - 2) 2 - m 2 } sin a. *(3) Denoting the Schwartzian derivative by (s, x), .... . fds\ 2 , \ / 1 9 •£ ^ logs' (ii.) (s,x) = {^{(s,y)-(x,y)}; a\\\ ( es+ f ax+b )-(c ™+b \_ (Ax+B? (m - } \Es~+F> Ax+B) ~ V s ' Ax+BJ ~ (aB-Ab) ^ x) ' 186 SUCCESSIVE DIFFERENTIATION. *(4) Prove that the linear differential equation of the third order R a y'"+3R 1 y" + 2>R 2 y'+R z y=0, where the R's are functions of x, is reduced to the form u'"+3P 2 u'+P 3 u = 0, by the substitution u = y exp_/ifi! 1 cfoc/.R ; and this again to the canonical form v dz 3 by the substitution v = uz', where z 1 is given by the differential equation (z, £C) = |P 2> and z' 3 9 = P 8 -fP,'. *(5) Verify that y = AFx+EF{-x) satisfies (i-) - ^ = 2cosec 2 x + cot 2 a, or 2 cosech 2 a?+coth 2 a, if Fa; = (cotfl3+cota)e- :,;cota , or (cotha; + coth can ^e ma( ie to pass through the points P v P 2 , P 3 , ...., or else touch the chords P 1 P 2 , P 2 P 3 , ..... on being wound on or unwound. . In the limit when the points P v P 2 , P 3 , are taken close together on the curve, the polygon Q X Q 2 Q 3 .... becomes a continuous curve, the locus of the centres of curvature, called the evolute of the curve P 1 P 2 P 3 ...., and touched by all the normals of the curve P. 91. Any small arc Pp may be considered as ultimately coincident with the arc of the circle of curvature, and therefore Q, the point of intersection of the normal at p with the normal at P (fig. 35), is ultimately the centre of curvature at P; and then QP is called the radius of curvature; and denoting it by p, then p = ltAs/A\fr = ds/d-ft. The curvature at any point is therefore 1/p or d\Jr/ds. Now (§ 5) tan yfr = J or yf, = tan" 1 ^ ; ^ 9 > =-V( i+ £) ! SUCCESSIVE DIFFERENTIATION. 189 ds _ ds /dyp-_ A. ,dy*\i ld 2 y " V dxV/dx 2 ' Fig.35 and therefore, with x for independent variable, _ ds d\fs dxl dx the expression for p in terms of y, ^ and , 92. Taking t as the independent variable, then dx d?y _ d 2 x dy ^ to tn\ d\lr dt dt 2 dt 2 dt and (§12) J= ; dP + dP , , » /da; 2 , cfoAf //das d 2 v cf 2 a; di/\ therefore P = (^ + JfJ / (&&-&!£} We have now changed the independent variable from cc to t in the expression for p. 190 SUCCESSIVE DIFFERENTIATION. Since tfj^+ d jt , dx , dy . , and -tl=vcos \}s, -^=vsm\fr, therefore p = v 2 ( cos ty-Tar — sin ^ j^)> T = cos^-sm^, , d 2 i/ . , d 2 x the normal component of the acceleration of P (fig. 34 ii.). The tangential component of the acceleration . d 2 x . . , d 2 y C0S *d¥ +sm *W _ 1 (dx d 2 x , dy d 2 y\ _dv_ dv_ d\v 2 v\dt dt 2 dt dt 2 ) dt ds ds the same as for rectilinear motion ; and v 2 /p, the normal acceleration, is the same as for motion in the circle of curvature (§ 82). For instance, in discussing the motion of a projectile in a resisting medium, in which the resistance acts in the direction opposite to motion, we begin by resolving normally, so as to eliminate the resistance ; and then v 2 /p = g cos \Js. Now v = ds/dt, and p = — ds/d\fs, the negative sign appearing because \Js is always diminishing as s increases ; so that v 2 d\fr/ds = vd^/dt = — g cos t//-. Put tam/r=2>, and v cos \Jr = u = dx/dt , the horizontal component of the velocity ; then dt u -, dx u 2 -j-= , and -y-= , dp g dp g fundamental equations in the theory of the motion of a projectile in a resisting medium. SUCCESSIVE DIFFERENTIATION. 191 93. Changing to polar coordinates by putting £C = rcosO, y = r sin6>, then by § 81, dx d?y d 2 x dy _dr / dr dd d 2 6\ fd 2 r cW 2 \ dt~d!>~W dT~^\dtTt +r W)~ r \df~ r WJ' and ^ 2 4 .# 2 _^ 2 + r 2 — ■ dt 2 + dt 2 ~dt 2 ^ dt 2 ' whence we find the value of the radius of curvature in terras of the polar coordinates, and their derivatives with respect to t. Making 6 the independent variable by putting t=6, then d6/dt=l,d 2 6ldt* = 0; and p = (g +r »)y(^ + 2g-rg) ; so that, at the origin 0, where r = 0, p = ^dr/d6. On replacing r by 1/u, fdu 2 , 2 V fd 2 u , \ „ 1 du 2 and since -^ = ^ + % 2 (§ 23), therefore the chord of the circle of curvature through is 2/j sin = 2p£>/r, =2 (S +u2 )/© +u ) m2; and therefore -7™ + ^ = cZ(9 2 p sin 3 9> „. c£ 2 w , 1 cfo Since — +u= --^ therefore the chord of curvature through is — 2pdu/(u 2 dp) = 2pdr/dp ; and the diameter of curvature is 2rdr/dp. 192 SUCCESSIVE DIFFERENTIATION. 94. Taking s, the arc of the curve, for the independent variable, then since (§ 9) dx . dy ■ , therefore differentiating with respect to s, d 2 x_ . ,d\Js_ dy 1 ds 2 ~ ^ ds ~ ds p d 2 y _ . dyfs _ dxl ds 2 ~ *as ds p Squaring and adding, \ds 2 ) + \dsV p 2 ' , dy ld 2 x _ dx ld 2 y , ds ds 2 ds/ ds 2 ' also 1 _ d sin if r __d cos i/r _ p dx dy ' j , d 2 y . , d 2 x and cos^ = p^, sm^-p^. If a, /3 denote the coordinates of Q, the centre of curvature at P, a = x - p sin yj, = x + p 2 -^, = y + p cos ^ = y + p 2 ^|. With x for independent variable, the expressions are not so symmetrical ; for x - a= psin ^= fr^-=fl^= d A^+i^l% r T d\[s ds dxl dx dx\ dx 1 ) / dxr ,_„__,„,__* *-,/g~(i + g)/£ We ™ y write /.-i*S£fl/£ 3 (oc -\- r u\ /el/ cc and by symmetry, a = $ dy i J ^ SUCCESSIVE DIFFERENTIATION. 193 95. The Evolute and Involute. Since a = x — psim/r, P = y + pcos\js; then differentiating with respect to s, da dx dp . , , d\ls dp . d^ = d^~d^ sm ^~ pC0S ^d^ = ~d9 Hm ^' UsO tvo U>o Ct/O CvO d3 dy . dp , , d\ls dp , ds ds ds r r r ds ds r and therefore ^- = — cot i/r = tan(^x + \b). da The locus of (a, /3) the centre of curvature Q, is called the evolute of the curve J.P (§ 90) ; and the preceding equation shows that the tangent to the evolute at Q is QP, the normal to the curve at P (fig. 35). Also <*+& = %<„&& ds* ds i ds* ds 2 ds 2, if o- denotes the length of the arc of the evolute, measured from a fixed point. Therefore -^ = ±-r> as as and o- = a constant ± p. (i.) Suppose , y= &sin0; dx de — — a sin 9, dy d6 = b cos 6 ; d 2 y , . n Thence p = (a 2 $m 2 6+b 2 cos 2 9)*/ab = a 2 b 2 (°^+t) i =^^TP.PV/OY. Denoting the angle J. OF by », then tan w = — t- = T tan : ct2/ b a sin 6 cos 6 sma> = „/(a%in , + &W0)' cos w = J(a 2 siD. 2 6+b 2 cos 2 dy SUCCESSIVE DIFFERENTIATION. 199 Therefore a 2 sin 2 + 6'-cos 2 # n x — a =? p cos w = ! cos a = a cos — °^ — cos 3 #, a or a« = (a 2 — 6 2 )cos 3 = (a 2 — b 2 )x s /a? ; o • a 2 sin 2 + 6 2 cos 2 # . n y — p = p sin w = J sin a 2 — 6 2 = 6 sin + — = — sin 3 (9, b and 6/3 = - {a 2 - 6>in 8 = - (a 2 - 6% 3 /6 3 . Therefore, eliminating 0, (aa)% + (b/3)% = (a 2 -b 2 f, the equation of the evolute of the ellipse. Similarly the equation of the evolute of the hyperbola (x/ a y-(y/bf=i, ■ is (aaf-(b0 = (a 2 + b 2 )%; obtained by putting x = a cosh u, y=b sinh u ; when da = (a 2 + 6 2 )cosh 3 u, 6/3 = — (a 2 + 6 2 )sinh 8 it. Examples. (1) Prove that the radius of curvature PQ of a parabola at P is 2PP, where R is the point where the normal at P meets the directrix of the parabola. Prove also that the evolute cuts the parabola at points which are the centres of curvature at x = I of the parabola, at an angle cosec _1 3. (2) Prove that in the rectangular hj'perbola xy = c 2 , a = %x + \y z \c 2 , /3 = -fi/ + \x z \c 2 ; and that the equation of the evolute is (a+j8)*-(o-j8)* = (4o)». (3) Prove that in the hyperbola xy + Ax + By = 0, the radius of curvature p = \TP .PV\OY. 200 SUCCESSIVE DIFFERENTIATION. 99. The Figure and Size of the Earth. i The size and figure of the Earth is inferred from the / measured length of a degree or minute of latitude at sea level ; the length is found to vary slightly with the , latitude, showing that the Earth is not exactly spherical ; but the length of a mean sexagesimal minute of latitude is i taken as one geographical, nautical, or sea mile (French mille), so that the length of a quadrant of a meridian is 90 X 60 = 5400 sea miles; and the length of the Admiralty measured mile being taken as 6080 feet, the circumference of the Earth is 360 X 60 x 6080 feet. On the Metric System the quadrant is divided cen- tesimally, and the mean centesimal minute of latitude is taken as the kilometre ; so that the quadrant is 10000 = 10 4 kilometres, or 10 7 metres, or 10 9 centimetres. This would make the metre = 3'2832 feet; but as the metre is more nearly 3"2809 feet, this shows that the quadrant is about 10007 kilometres; but for practical purposes it is sufficient to take the round numbers. The globe terrestre au millionieme at the Paris Exhib- ition of 1889 was a sphere 40 metres in circumference and therefore 12 - 732 metres in diameter, representing the Earth, 40,000 kilometres in circumference, and 12,732 kilometres in diameter, on a scale of one-millionth. In navigation the speed of a vessel is always measured in knots (French nceuds, German knoten, Dutch knoopen, Spanish nudos, Italian nodi), one knot being a speed of one sea mile or mean sexagesimal minute of latitude per hour ; when the knots on the log line are spaced so that the number of knots which pass over the taffrail in half a minute give the speed in knots, the knots must then be 6080-=- 120 = 50-7 feet apart. SUCCESSIVE DIFFERENTIATION. 201 100. Dip and Distance of the Horizon. Variation in Length of a Degree and Minute of Latitvde. When we ascend vertically from sea level at J. to a point T, the horizon in consequence of the curvature of the Earth is bounded by the tangent lines TP (fig. 5) and the angle TPM is called the dip of the horizon, while TP is called the distance of the horizon, being the distance at which a light at T is visible from sea level, in the absence of atmospheric refraction. Supposing the Earth is exactly spherical, then the dip TPM=AOP=d = sec- 1 (l+h/R), if It denotes the radius of the Earth, and h the height of T above sea level ; while the distance of the horizon TP = R tan 6 = J(2hR + h?). The quantity h/R is so small in all places accessible to us that we may neglect (h/R) 2 ; and then TP= /s /(2hR) ; while tan d = ^/(2h/R) ; so that we may put d the dip in . . , 180x60x60 I2h sexagesimal seconds = -V~P~' With the metre as unit of length, _R = 10 7 -4-^71-; so that, for a height of h metres, , 180x60x60 I2hxl-7r aAO ,,,,,„ x d = y 1Q7 * = 648VCVKM = log - \\ log h+ 2-063); while 2 , P = x /(2^xl0 7 H-^7r) = 20000 x /(/i/107r) (metres), = iOJih/lOw) = log" XJ log h + -5525) (kilometres). With i2 = 10 4 -=-^7r (kilometres), the surface of sky covered with a uniform canopy of cloud at a height of x kilometres will be (§ 60) 2tt(R + x)x = 2ttRx(1 + x/R) = 40,000 x, (square kilometres) neglecting the fraction x/R, which is practically very small. 202 SUCCESSIVE DIFFERENTIA TION. Defining the latitude of a place as the angular altitude of the celestial pole, measured from the horizontal plane of the sea level or surface of mercury at the place, a plane perpendicular to the plumb line (Sir W. Thomson, Navigation), then it is found by geodetic measurements that the length of a minute of arc of latitude on the Earth's surface in latitude \Js may be taken as 1 — c cos %$r , sea miles or kilometres, according as the minute is sexa- gesimal, or centesimal ; so that in latitude i/r the radius of curvature of the meridian is (1 — c cos 2^)5400 -±\-w sea miles, or (1 — c cos 2i/r)10000 -t-^tt kilometres. Denoting by B the mean radius of the Earth, and by a, b its equatorial and polar radii, then ds/dyjr = B(l — c cos 2i/r), dx/difr = — -R(l — c cos 2i/r)sin \fs, dyjd-^r = B(l — c cos 2i/r)cos \Js ; x/B =/(l - ccos2\fr)sin-ftd\fr = (1 + £c)cos^ — £ccos S\fr, ft yjB =/(l - ccos2^)cosi/<-cfyr = (1 - £c)sim/r- £csin 3i/r ; o and a/B=l+§c, b/B=l-\c; so that (a — b)/B — ^c, and §c is called the compression of the Earth ; and it is found by geodetic measurements that c = 1/200, about ; so that the compression is 1/300. Thus with B = 6366 kilometres, we find a — 6 = 6366-7-300 = 21 kilometres, about; on the Globe au ruillionieme this would amount to only 21 millimetres, and would be quite insensible. The length of the corresponding minute of- longitude is x/B = (1 + £c)cos \p--\c cos 3^, miles or kilometres. SUCCESSIVE DIFFERENTIATION. 203 Examples on Curvature: (1) Denoting the length of the normal PO by n, and the radius of curvature by p, prove that in the (i.) Rectangular hyperbola, 2p = normal chord, as in the circle, (ii.) Catenary, p = n, as in the circle, (iii.) Tractrix, p = a 2 /n, or pn = a 2 . (iv.) Cycloid, p = 2n, as in the parabola, (v.) Conic, p = n 3 /l 2 . (vi.) Sinusoid, y/b = sin x/a, p = (a 2 + b 2 — y 2 )*/ay. (vii.) In x% + y$ = a$, p = S(axy)i = 3p(Ex. 6, p. 36). (viii.) In the curves (Sumner lines), sinh y/a = tan a sin x/a, or coshy/a = sec/3cosa;/a; p = a cosec a cosec x/a, or a sin j3 sec ai/a. (2) Prove that in y/a = logsecx/a, the chord of curvature parallel to the axis of y is 2a, and p/a = cosh s/a. (3) Prove that, if x = a + 2bt+ct 2 , y = A+2Bt + Ot 2 , the point describes a parabola; and that p varies as sec 8 T/f, where \p- is the angle the tangent makes with the line cx + Cy = 0. (4) Determine p as a function of 9 in the curve, an ellipse, described by the point x = a+b cos 8 + csin 6, y = A+Bcos6+C'sm6. (5) Prove that, with polar coordinates, in the curve (i.) r = ad, p = (a 2 +r 2 )l/(2a 2 +r 2 ). (ii.) r = a/8, P = r(r 2 +a 2 //a s . (iii.) r = a e , p = Pg- (iv.) r = a sin d, or a cos 6, p = \a. (v.) r = a sec 0, or cosec 6, p = . (vi.) r=avers6, p = % s /2(ar)%. (vii.) r = a V( cos 2 #)> or a s/( s ^ x 2 #)> P = & a2 / ? '' (viii.) r = ay/(sec26), or a^cosec 20), p = r s /a 2 . 204 SUCCESSIVE DIFFERENTIATION. (6) Prove that in the curve r = a+b cos 6, p = (a 2 +2ab cos 6 + b 2 fl(a 2 + Sab cos 0.+ 26 2 ) ; and that in the conic l/r = lu = 1 + e cos 0, p = 1(1 + 2e cos + e 2 )*/(l + e cos 6f = I cosec 3 0, denoting the radial angle. (7) Prove that the chord of curvature through of r n = a n cosnd is 2r/(n+l), and p = Pg/(n+l). *(8) Prove that the normal chord which divides the area of an ellipse most unequally is equally inclined to the axes. *(9) Prove that in the curves (x/a) n ±(y/b) n =l, and (x/a) m =(y/b) n , p is respectively 2/(1 — n) times and twice the radius of curvature of the tangent hyperbola xy + Ax + By = 0, at the point of contact. *(10) Prove that the locus of the centre (a, /3) of a rect- angular hyperbola, touching the ellipse (§ 98), and having the same curvature, the axes of the ellipse and hyperbola being parallel, is given by aa = (a 2 + b 2 ) cos 3 0, b/3 = (a 2 + b 2 ) sin 3 ; or (aaf + (bfif = (a 2 + b 2 )l *(11) Prove that in the lemniscate (x 2 +y 2 ) 2 = a 2 (x 2 — y 2 ), or r 2 = a 2 cos 26, we may put x = a cos B/J (cos 26), y = asin 6^(cos2d) ; and then a = fa cos^VOec 26), f3 = - fa sin^^sec 26) ; and (J+p%) 2 (J-ffi = ia 2 . *(12) Theevolute of the equiangular spiral r = c exp(0cota) is the same curve turned about through an angle ^7r+tan a log tan a (radians). SUCCESSIVE DIFFERENTIATION. A H 205 101. The Cycloid and its Evolute. Gycloidal Oscilla- tions. The cycloid, defined in § 21, is now drawn inverted, for purposes of dynamical illustration, being generated by the rolling of a circle on the under side of the horizontal line B'B (fig. 40). If the circle were to change to rolling on the upper side of Ox, then a cycloid as in fig. 9 would be described, cutting the other cycloid at right angles; thus the orthogonal trajectories of equal cycloids with base qn Ox are equal cycloids with base on BB'. With the new coordinate axes of fig. 40, x = a(d + sm.Q), y = a vers 6; so that x = a vexs' 1 y/a+ s /(2ay— y 2 ) (1) The curve described by P' when NP' is made equal to the arc OR is called Koberval's companion of the cycloid, but then x — a6, y = avers6; or y = a vers x/a ; a curve of the same form as the sinusoids (§ 20) y = a cos x/a or y = a sin x/a. 206 SUCCESSIVE DIFFERENTIATION. IP is the normal and TP is the tangent at P, I being the centre of instantaneous rotation of the rolling circle ; and now, as in § 21, \}r = |0, while ds/dd = 2a cos ^6 ; so that s = 4a sin J0 = 4a sin'i/r (2), s denoting the arc OP, which is therefore equal to 2TP. The value of p, the radius of curvature at P, can be obtained from the formula (§ 92) /da? dy 2 \§ l/dx d 2 y d 2 x dy\ \dd 2+ ddV l\Wde 2 ~Wde)' and will be found to be 4a cos £0; but this can be obtained immediately from equation (2) by § 91 ; for p = ds/d\}r = 4a cos i^ = 4a cos £0. Therefore PQ = 2PI, if Q is the centre of curvature at P, and this shows that the evolute A Q is an equal cycloid. For a, j8 denoting the coordinates of Q, then (§ 91) a = x — p sin i/r = a(6 + sin 9) — 4a cos £0 sin %d = a(6 — sin 6), /3 = y + p cos i/r = a(l — cos 6) + 4a cos 2 |0 = 4a — avers 6; which proves that, as P describes the cycloid OPB, Q describes an equal cycloid AQJB, produced by rolling a circle of radius a on the horizontal straight line through A. This, the first problem of an evolute, was invented by Huyghens (1673), with the object of making a particle oscillate in a cycloid ; for let the cycloid BOB' be cut out of some material and divided at BO, and B'O be placed in the position AB, and OB in the position B'A, in a vertical plane with BB' horizontal ; a particle at hanging vertically from A by a fine thread of length OA will, if made to oscillate in the vertical plane of the figure, describe an arc of the cycloid BOB'. SUCCESSIVE DIFFERENTIATION. 207 102. Isochronism of Cycloidal Oscillations. The peculiarity of these oscillations is that the period is the same for all arcs of oscillation; this property is called the isochronism of the cycloid. For resolving tangentially at P, dv/dt = d^v 2 /ds = — g sin \js = — gs/l, ii 1 = 4a, the length of the thread. Integrating with respect to s, and denoting gjl by n 2 , \v 2 = \n 2 {s 2 -s 2 ), supposing P to be drawn aside from through an arc s v and then let go. Therefore v = ds/dt = — % x /(s 1 2 — s 2 ), the negative sign being taken because P begins moving towards 0; and dt 1 __, ± s fh ds ds 1 ,. N fh ds ,s or s = s 1 cosn(t — r), supposing t to denote the instant of time when P is let go. If \T denotes the time of oscillation from rest to rest, then n(t— t) increases by v, while t — r increases by \T, so that nT= 2tt, and s = s 1 cos{2'7r(i — t)/T}, and T=2Tr > /(l/g), which isindependent of s 1; and therefore the same f orallarcs of vibration; which proves the isochronism of the cycloid. When s v the amplitude of vibration, is small, the arc of vibration in the cycloid may be considered coincident with the arc of the circle of curvature at 0, a circle of radius I and centre A ; so that the period of a small plane oscillation of a simple pendulum of length I is thus proved to be ^Try/il/g), the same as the period of revolution in a small horizontal circle (§ 82). 208 SUCCESSIVE DIFFERENTIATION. 103. Harmonic Vibrations. If a point P describes the circle (fig. 21) with constant velocity in the periodic time T, then 6/2ir=(t-T)IT, or 6 = n(t-r), supposing t is an instant of time when P is at A ; and then 0M= x = acosd = a cos{27r(i - t)/T}. The point M oscillates between A and A' in the time \T, and M is said to perform a simple harmonic motion (S.H.M.) ; and a is called the amplitude, T the period, and t is called the phase, being one of the instants at which M is at A ; while « is variously called the speed, or angular velocity, (or mean motion, in Astronomy). In a steam engine the piston Jf performs a simple harmonic motion very approximately while the crank P moves with constant velocity, the slight discrepancy being due to the obliquity of the connecting rod; and in § 102, P makes a harmonic vibration on the cycloid. If x = acos{2Tr(t- T )/T}, then d 2 x/dt 2 = — ^x/T 2 = — n 2 x, so that the point M vibrates as if attracted to with intensity proportional to the distance from 0. The small vibrations of elastic bodies producing musical notes are of this nature, whence these vibrations are called harmonic. In the plane oscillations of a simple circular pendulum composed of a small plummet at the end of a fine thread of length I feet, the plummet oscillates in a circle ; and since s = 19, resolving tangentially, dt 2 ■a ■ s = — #sin#= —gsmj- SUCCESSIVE DIFFERENTIATION. 209 The complete integration of this equation of motion requires the Elliptic Functions ; but when the oscillations are so small that we may replace smsjl by its cm. s/l, then d 2 s/dt 2 = — gs/l = — nh, so that the pendulum has a S.H.M. of period Sir^il/g). Again, if in the motion of P on the ellipse (§ 51, figs. 8 and 22), we put = 2Tr(t-T)IT=n(t- T ), then x = a cos n(t — r),y = b sin n(t — t) ; so that the motion of P is compounded of two s.H.M.'s in directions parallel to Ox and Oy ; and since d 2 xjdt 2 = — n 2 x, d 2 y/dt 2 = — n 2 y, therefore P moves as if attracted to with intensity proportional to the distance from 0. Here, in the most general case of projection, x = a cos n(t — T),y = b cos n(t — r'), with different phases t and t' ; and thence sinnt=( - eosnr' — T cos nr) /sin n(i — t'), — sin nr' + r sin nr) sin n(j — t) ; so that, squaring and adding, g-2^cos % (T-T') + | 2 = sinMT-T') ) the equation of a system of ellipses; having the same orthoptic circle x 2 +y 2 = a?+b 2 , and all inscribed in the rectangle bounded by x=±a, y=±b; the points of contact forming a parallelogram of constant perimeter 2 N /(a 2 +6 2 ), the sides of which are parallel to the dia- gonals of the rectangle. 210 SUCCESSIVE DIFFERENTIATION. We may for brevity write x and y for x/a and y/b, and put cos n(r — t')=c; and now cos _1 x± cos _1 y — cos ~ x c, the ambiguity of sign corresponding with the two directions in which P can move round the ellipse ; and this is equivalent to xy + ^/O-x^il-y^c, or xj(l-y 2 )±yj(\-x 2 ) = j{\-c 2 ), or x 2 — 2cxy + y 2 =l—c 2 . In the corresponding case for hyperbolas we have cosh _ Y x ± cosh " x y = cosh ~ a c, or x 2 — 2cxy + y 2 =l — c 2 , giving hyperbolas circumscribing a rectangle ; while sinh ~ 1 x± sinh ~ l y = cosh ~ Vj, represents the conjugate hyperbolas ; and cosh _ x x± sinh ~ 1 y, or sinh " he ± cosh ~ x y = sinh _1 c, represent rectangular hyperbolas and their conjugates; all described under a repulsion from proportional to the distance, when we put xja = cosh or sinh n(t — t), y/b = cosh or sinh n(t — t') ; so that d 2 x/dt 2 = n 2 x, d 2 yjdt 2 = n 2 y. With the component velocities of Ex. 13, p. 27, the general conic is described with component accelerations d 2 xjdt 2 = (h 2 — ab)x — bg +fh = (h? — ab)(x — x), d 2 y/dt 2 = (h 2 - ab)y - af+gh = (h 2 - ab)(y - y), where x, y are the coordinates of the centre ; and if X = (h 2 - ab)x 2 -2(bg -fh)x +f - be, Y=(h 2 -ab)y 2 -2(*f-gh)y+g 2 -ac, then dt = dxj^/X = — dy\,J Y ; so that the integral of this differential relation is ax 2 + 2hxy + by 2 + 2gx + 2fy+c = 0. SUCCESSIVE DIFFERENTIATION. 211 With two component s.h.m's of different periods, x = a cos n(t — r), y = b cos m(t — t') ; and eliminating t, m cos " x xja — n cos ~ 1 y/b = mn(j' —r) = a, a constant; curves exhibited practically, in Lissajous's method, by a spot of light on a screen, reflected from small mirrors on two tuning forks of corresponding periods, vibrating in perpendicular planes. (Ganot, Physics; George M. Hopkins, Experimental Science.) The student may exercise himself in drawing simple curves of this nature : for instance, with a = 0, \-w, . . . and with m/n = 1, 2, 3, 4, . . . 2/3, 3/4,. . .. (Clifford, Kinematic.) Draw also the curves given by dt = JXdx = *J Ydy, and dt = dx = dyj,J Y, or l, the curve lies inside the circle of cusps BBB ..., and is a hypocycloid — that is, the curve described by a point on the circumference of a circle rolling inside a fixed circle (fig. 41, iii.). The points A, A, ... on these curves midway between the cusps are called the apses ; and in the cycloid the apses lie on a straight line, in an epicycloid the apses lie on a circle greater than the circle of cusps, in the hypo- cycloid on a lesser circle. To describe a curve from its intrinsic equation, s = fi/r, we find its radius of curvature p = dsjd\}r = i'\}r ; and start- ing from a point A when ^ = 0, we describe a series of successive short circular arcs of appropriate radius p and curvature A\Js, and thus build up a close approximation to the shape of a curve. In this way the centering of arches is described prac- tically (called in French anse de panier), and curves of section of complicated surfaces; such as capillary sur- faces, of which the equations cannot be integrated (Sir "W. Thomson, Capillary Attraction). In drawing the curves s = Isinmty, where p = toZcostoi/t, then as ty increases from to |x/to, p diminishes from ml to 0, and half a branch is described; and the remainder of the curve is formed of symmetrical repetitions. SUCCESSIVE DIFFERENTIATION. 213 Fig.41 For practice the student may draw the epicycloids for m = £, \, |, \, f , . . . ; and the hypocycloids for m = 2, 3, 4, §, f , . . . ; also their evolutes. If s = fi/r is the intrinsic equation of a curve, then o- = ds/d\lr = i'\[r is the intrinsic equation of the evolute (§ 92), measuring \-k, . . . . (3) Prove that the equation of the epicycloid s = I sin \ty can be written 4(x 2 +y 2 ) — 3a*a#— a 2 = 0, if a = f£. 105. Envelopes. Since the normal of a curve is a tangent to the evolute (§§ 90, 95), the evolute is called the envelope of the normals, and the equation of the evolute is readily determined from this consideration; because the point of intersection of the normal at P with a consecutive normal is ultimately Q, the point of contact of QP on the evolute (fig. 35), and also the centre of curvature at P. For instance, the equation of the normal to the ellipse at a point whose excentric angle is 6 is (ex. 5, p. 35) axsecO — by cosec 6 — a 2 +b 2 = (i.) SUCCESSIVE DIFFERENTIATION, 215 Denoting this equation by F0 = O, then, as explained below, to find the point of ultimate intersection of this normal with the consecutive normal, we must determine x and y from the equations F0 = O, andF'0 = O; and the equation of the evolute is obtained by eliminating between these equations. Here Y 6 = ax sec t&n 6+ by cosec 6 cot 0=0 (ii.); and from (i.) and (ii.), ax = (a 2 — b 2 ) cos 3 0, by= — (a 2 — b 2 ) sin 3 ; giving the coordinates of the centre of curvature ; and eliminating 0, (ax)* + (by)* = (a 2 - b 2 f, the equation of the evolute (§ 98). Similarly for the envelope of the normals of the hyper- bola and of the parabola, in the forms ax sech u+by cosech u = a 2 + b 2 , and y = m(x — l) — ^m s l, for all values of u and m (§§ 97, 98). Generally if F0 = denotes the equation of any curve, involving x and y and a parameter 0, as it is called; keeping 6 constant, a particular curve of the series is obtained; but by varying a series of curves is produced. To find the points of ultimate intersection of the curve F# = with a consecutive curve of the series, suppose 6 to receive a small increment Ad ; then we must find x and y from the equations F0 = O,andF(0+A0) = O, where A0 = 0, ultimately ; that is, from the equations F0 = O, and lt{F(0 + A0)-F0}/A0 = O, or F'0 = O. The locus of these points is called the envelope of the series of curves, and its equation is found by eliminating between the equations F0 = 0, and F'0 = 0. 216 SUCCESSIVE DIFFERENTIA TION. This curve is called the envelope of the series, because each curve touches the envelope where it intersects the consecutive curve of the series. For, take three consecutive curves of the series defined by the parameters 6 + A8, 6, and 6 — Ad; and suppose the first and second to intersect in P, and the second and third in P' ; then P, P' are two consecutive points on the envelope, and also on the curve F# = 0, and therefore ultimately the envelope and a curve of the series have the same tangent where they meet. Familiar instances of envelopes of straight lines, besides evolutes, are discussed in Optics with caustic curves, the envelopes of rays reflected or refracted at given curves or surfaces ; as seen by reflexion on the surface of the water, or refracted through a glass of water. Thus the caustic of rays reflected by a circle, (i.) emanating from a point in the circumference of the circle, is a cardioid (ex. 11, p. 118) ; (ii.) of parallel rays is the curve of ex. 3, p. 214 ; (iii.) the caustic of rays emanating from a point, and refracted by a plane (§ 73), is the surface formed by the revolution of the evolute of an ellipse or hyperbola. Examples. (1) Find the envelope of a straight line of given length c, which moves with its ends on the coordinate axes. (If the straight line makes an angle 6 with the axis of x, it makes intercepts c cos 8, c sin 6 on the axes, and x sec 6 + y cosec 6 = c. Differentiating with respect to c, x sec 6 tan Q — y cosec 6 cot = 0; and therefore x=c cos 3 0, y = c sin s 6, and eliminating 6 (ex. 9 iv., § 14), x$+y$ = c$.) SUCCESSIVE DIFFERENTIATION. 217 (2) Find the envelope of the parabolas of § 79, described by bodies projected from a fixed point with given velocity V, but different elevation. (Denoting by 6 the variable elevation, and putting the impetus of projection ^V 2 /g = h, then the equation of the parabola is y = x tan 6 — lx 2 sec 2 d/h ; and therefore the envelope is (fig. 33) y = h — $x 2 /h, or x 2 = 4 +h coa{'px + q). (ii.) g-2(a2-^)g+(^+^ = by y = A cosh (ax + 6) cos(pcc + g) or 5sinh(oa;+6)sin(px+g'). (iii.) 5 +2rlsinh "S"^^ 0, 8 " 2n Sinh a S + ^ = °' by x = a cos 7ie a (£ — f J + b cos -n-e _ a (i - 1 2 ), y = a sin %e a (£ — i x ) — b sin we - a (i — * 2 ) ; and then -^ + 2w 2 cosh 2a-j7g + n*x = 0, . . . . (4) Given the differential equations -^ = w(aj cos 2cat + 2/ sin 2a,t ), ^ = tt(a5 sin 2coi - ?/ cos 2(d), ,, c£ 2 * , -, dy „ A d 2 i/ „ cfo 2 n then^ + 2coj-^ = 0, J-2 WWt -n*y = 0, and ^ + 2(2o, 2 -7i 2 )^+^x = ) .... Putting to = « cosh a or n cos a, according as co < W, these equations are satisfied by x = ae ' * a cos n(fe a — t) + ae* a sin n(te - a + t), y = ae~ * a sin 7i(£e a — t) — ae* a cos w(fe " " + t), when co>5i ; and give the solution when co< n. 220 SUCCESSIVE DIFFERENTIATION. s=m— 1 (5) Prove ilaty = e^^(A t ^j)x+B t ^ b px), s=0 satisfies the differential equation {(lr«)V}Vo. /r\ r \ _ acosmf fcicosTii p cosh mt ,q sinh. nt ' ^ '' m 2 —n 2 2n m 2 +n 2 2n 2 _ a cos mt b sin wi p cosh mf gi cosh nt m 2 +n 2 2n 2 m 2 —n 2 2n satisfy the differential equation -j~2 ± n 2 x — a cos wit + 6 sin nt +p cosh.mf+ q sinh nt. cue bt m \ , ce*" (ii.) x = e-(c o+ C 1 t + ... + G m .^ + ^) ,^ -ji — a ) x = be at + ceP t ; and write down the solution of (d? \ m -j-^±n 2 ) x — a cos mt+b sin nt+p cosh mf+gsinhwi. (7) Prove that x^-t^ = y is satisfied by 2/ = J.e 2 V*(l-2 x /:c) + £e -V"{(1 + JxynBijSjx + a) + s /(3x)sm( > /3 s /x+ a) }. (8) Determine the most economical speed of a troopship, costing £400 a day for provisions and wages ; given that the speed is 8 knots on a consumption of 50 tons of coal a day, costing 10s. a ton ; and given that the consumption of coal per day varies as the cube of the speed. (9) A cylindrical boiler is to be constructed of sheet iron of uniform thickness, with a longitudinal cylin- SUCCESSIVE DIFFERENTIATION. 221 drical flue of one-third of the external diameter. Prove that for given volume the weight of the boiler will be least when the length is two-thirds of the diameter. Prove that with any given number of flues or tubes of given thickness, the diameters of which are proportional to the external diameter of the boiler, the weight for given volume will be a minimum when the weight of the cylindrical part and of the tubes is double the weight of the ends. (10) A number n of incandescent lamps each of internal resistance r ohms, are inserted in a single circuit of resistance R ohms. Show that in the most economical arrangement the number n should be the nearest integer to R/r, and that only about fifty per cent, of the energy can be utilized. (11) The strength of a rectangular beam of breadth b and depth d being proportional to bd?, and its stiffness to bd 3 , prove that the strongest and stiffest beams which can be cut from a circular log are such that the perpendicular from the corner of the cross section on the opposite diagonal will cut off a third or a fourth part of the diagonal. (12) Determine where it is economical to change from a cutting to a tunnel in constructing a level railway, crossing a ridge I yards broad and sloping on each side at an angle a ; taking the cost of tunnelling as £A per linear yard, and of the cutting as £B per cubic yard, the breadth of the railway being b yards, (i.) when the sides of the cutting are vertical, (ii.) when they slope at an angle /3. 222 SUCCESSIVE DIFFERENTIATION. (13) Determine geometrically where to stand in the road so as to throw a ball over a house with the least velocity. Determine also where a ship comes within range of a fort on a cliff; and where the ship can return the fire; and where the ship can best batter the fort. (14) Prove that if w denotes the terminal velocity of a projectile and the resistance of the air is taken to vary as the velocity (§ 77), the trajectory for initial velocity V and elevation a is given in terms of the time of flight t by Vw w x = cos a(l — e-M™), y = (tan a+y sec a)x r- wt . Prove also that, in a maximum range on an inclined plane through the point of projection, the angles of ascent and descent are complementary. (15) Show that the path of a rocket, if the velocity is maintained constant, is an inverted catenary of equal strength (ex. 2, p. 71) ; and that the range is proportional to the elevation. (16) Prove that, in epi- or hypo-cycloids, ds s ds ds 2 (17) Prove that if c denotes the curvature 1/p, I 2 rpo /-pit i(a+x)=ia+xfa+j ] f'a+^f"a+... + ~^a+... > fa, fa, f"a, . . . i n a, denoting the successive derivatives of ia with respect to a. For assume that £(a+x) = A +xA 1 +x 2 A 2 +x 3 A s +...+x n A n +... where the A's are functions of a only, and not of a;. 223 224 EXPANSION OF First put x = 0, then f a = A . Next differentiate successively with respect to x, and put x = after each differentiation ; then i'(a+x) = A 1 + 2xA 2 + ,f'a = A 1 ; f'(a+x) = 2\A z +2 . SxA 3 + , Fa = 2!4 2 ; f"(a+x) = S\A i + ,r"a = 3\A 3 ; i n (a+x) = n\A n + , i n a =n\A n . Therefore A = f a, A x = i'a, A 2 = ^f "a, A 3 = ^f '"a, and generally A n = -fi n a. We have assumed here that ^ a+x)== dx i{a + x) ' and this is evident, if we consider that it is immaterial whether we change the function by increasing a or x. Therefore, as Taylor's Theorem asserts, i{a+x) = ia+xi'a+%-fa+^i'"a+ +~i n a+ As a simple illustration, consider the example of ia = a m ; theni'a=ma m - 1 ,¥'a=m(m — l)a m - 2 , ; and therefore f(a+x) = (a+x) m = a m +ma m - 1 x+ m ( m ~ 1 ' a m - 2 x 2 + , a verification of the Binomial Theorem. Symbolical Form of Taylor's Theorem. x n d Employing the abbreviations of x n for — and D for -j-, w. da then Taylor's expansion can be written i(a+x) = (1 +xD+x 2 D 2 + . . . +x n D n + . . Ma ; FUNCTIONS. 225 and treating the operator D as an algebraical quantity (§ 69), this is equivalent to f(a+x) = e xl> ia, or exp(xD)fa, which is called the symbolical form of Taylor's Theorem. Maclaurin's Theorem. Suppose a = in Taylor's Theorem ; then fa=fO+rf'0+ft"0+'S"'0+ +^f"0+ 2! o! n\ which is called Maclaurin's Theorem ; but this theorem was first given by Stirling in 1717. The meaning of f™0 is that ix must be differentiated n times with respect to x, and then x put equal to after the differentiation. In Taylor's Theorem i(a+x) is expanded in ascending powers of x, a part of the whole argument a+x of the function i(a+x) ; in Maclaurin's Theorem fee is expanded in powers of x, the whole argument of the function of x. We might have proved Maclaurin's Theorem first, in the manner above in which Taylor's Theorem was obtained, and then have derived Taylor's Theorem by putting ix = ¥(a+x); and then, differentiating n times with respect to x, i n x = F n (a+x) ; and putting x = 0, f"0 = F w a. Substituting in Maclaurin's Theorem, ¥(a+x) = 'Fa+xF'a+~¥"a+ +^F»a+ , which is Taylor's Theorem. In fact the two theorems of Taylor and Maclaurin are the same when considered geometrically as the equation of a curve with different origins at a distance a apart on the axis of x. p 226 EXPANSION OF 107. Application to the Expansion of Fimctions. (1 ) Let ix = sin x, then £ = ; also (§ 68) i n x = sin (x + n\if), f "0 = sin \n-K ; so that f 2n = sin n-w = 0, f 2B+1 = sin(«7r + \-k) = ( - l) m - Therefore, by Maclaurin's Theorem, x s , x 6 , (-1)'V V + 1 , amB^-gj+gj- + V+1)! + (2) Let ix — cos a;, then f = 1 ; and £"a3 = cos(a3 + %nir), f "0 = cos \nir ; so that P n = ( - l) m , P n+1 = 0. Therefore, by Maclaurin, /j.2 ^.4 / 1 \rin3m, 2! 4! 2n\ (3) Let fx = e*, thenf0=l; also f "a; = e x , and f "0 = 1 . Therefore e x = l+x+~ + +^+ = 2av 2! tc! „ = o Changing x into ex, C2/yi2 srtl/yin 2! n\ Again, suppose fx = a x , thenf0 = l; f n x = ^(log a) m , and f "0 = (log a) n . Therefore «■ -i i i , aMoera) 2 , , x n (loga) n , a !B =l+a;logaH — v ° y + + — v ° y — h This expansion is the same as the preceding, if c = log a, since a x =e x ^ a (% 31). (4) sinha;=J(e :c — e _!C ) a; 3 a; 6 aj 8 " - *" 1 = a; + 3! + 5! + + (2^+l)! + FUNCTIONS. 227 (5) coshx = J(e a! + e- a; ) = i+ i + :i+ + fe. + These expansions might have been obtained independ- ently by Maclaurin's Theorem. (6) Let f(l+a:)=log(l+a;), then f 1=0; fCi+»)= ^ n = i; f"(l+ai) = ( ~ 1 / ) "" 1(r ''~ 1)! . M = (-l)»- 1 (w-l)!. 7 (l+x) w \ t \ I Therefore by Taylor's Theorem (here a=l) log(l+a ; ) = a; --+---+ +± L + We cannot expand logo; in ascending powers of x, because if fee = log x, then fO, f'0, f"0, ... are all infinite; the same applies to vers _1 a;, exp( — 1/aj), exp( — 1/x 2 ), .... (7) From the preceding expansion, tanh-^logl±! = , + J + ! 5 + + ££ + (8) A 2 . h* \ , /, h 3 . ¥ 8in(x+h) =sinx\l — ^ + j- r .. J +cos x[h — ^ + 5!' eos(x+h) = cosx[l-^, + -^...J-smx[h-^+-^ J; whence the expansions of sin A and cos h in ascending powers of h are inferred, from the formulas sin(iz+A) = sina3cosA+cosa;sinA, cos(ce+A) = 228 EXPANSION OF 108. Some expansions can be derived from others by differentiation or integration ; thus d sin x , d sinh x cos x = — 5 — , cosh x = 1 , dx ax giving the expansion of cos x or cosh x when that of sin x or sinh x is known, and vice versa. Again, by integration, \og(l+x)=J^ x =f{\+x)-Hx o o - /Tl ,2 3 , \,7 X 2 X S X* = /(l — x+x i — x i +...)dx=x — -a-+-~—-T-+... . -. , * *x / aax And tanh ~ - = / -^ , a J a i — x i X 5 , = f( l +t 3+ t 5+ ) dx ^+f 3+ . J \a a 3 a, the series becomes divergent, and then ,x , ,a , a , a 8 a 5 , tan 1 - = i 7 r — tan 1 - = *'jr h-s ■=-«+.... a J x ^ oj 3a; 5a; 5 Similarly, ^~\=J J{a ,_ x ,y =-J [}-&) dx _/7 ,lg' 1.3 a} 4 , 1.3.5 ai 8 \,ai V V + 2 a 2 2.4 a* 2.4.6 a 6 " 1 " /a FUNCTIONS. 229 _x,\ x 3 1.3 x 5 ■ 1-3.5 x> a 23a 3 2.4 5a 6+ 2.4.6 7a 7+ ' j . , -,x f dx x 1 x 3 , 1 . 3 x 5 . and sinh" 1 -=/— 7--g „=- — ~ ^— „+^ — rr— i + --.; a J J{a?+x 2 ) a 2 3a 3 2.4 5a 5 ' "V or, without integration, we may assume that sin- 1 x= A 1 x+A^> 2 +A^c 3 +A i x i +A & x 5 + ; and then by differentiation A 1 +2A 2 x+3A z x 2 +4A i x 3 +5A 5 x i + = (l-^ ) -i = 1 + | a .2 + li| a ,4 + > by the Binomial Theorem ; and equating coefficients of like powers of x, 1 13 J. 1 = l, A 2 = 0, J. 3 =2-g, J. 4 = 0, ^5 = 2 4 5' > as before. Put x = 1, then sia " 11 = ^ = 1+ + 2 i 4^5 + ' another series for the calculation of tt ; or put x= \, then ™- 1 i = fcr = 2 + 2^8 + 2.4.*5.32 + ' a more rapidly convergent series for it. To expand any rational algebraical function of x, we resolve it, as for Integration, into its quotient and partial fractions, and then expand each partial fraction in powers of x, by the Binomial Theorem. Any powers or products of sines or cosines of multiples of x, circular or hyperbolic, are expanded immediately, when we resolve them into sines or cosines of other multiples of x, as in Integration (§ 40). 230 EXPANSION OF 109. Many expansions of functions are readily obtained, in Newton's manner, by forming the differential equation satisfied by the function, and then deducing by successive differentiation a recurring relation between the coefficients. For instance, if y = ix = exp(a sin - 1 x), then (1-^-^-^ = 0; and differentiating n times, by Leibnitz's Theorem, >dx 2 (1 _ a; 2 )?5 - (2« + l)asSS ~ (« 2 + W?S = ; v / dx n+2 v -* J dx n+1 v c and now putting x = 0, f m + 2 0-(a 2 + Of»0 = 0, a recurring formula; whence, since f0=l, f'0 = a, we deduce f "0 = a 2 , f "0 = a(a 2 + 1 2 ), f ""0 = a 2 (a 2 + 2 2 ), . . . ; so that fee = exp(cs sin ~ 1 x)=l+ax + a 2 x 2 + a(a 2 + l 2 )x s + a\a 2 + 2 2 )x± + a(a 2 + l 2 )(a 2 + &)x & +.... By expanding exp(asin _1 a;) in powers of asin _1 x, and equating coefficients of a, a 2 , a 3 ,... we deduce the expan- sions in powers of x of sin _1 a;, %(aia.~ 1 xf,$(gm~ 1 xf,...; thus sin -1 a; = a;+a;3+3 2 x 5 +3 2 . 5 2 .cc 7 +..., %(sin- 1 x) 2 = x 2 + 2 2 .x i + 2 2 . 4 2 . x 6 +2 2 . 4 2 . 6 2 .x 8 + ..., Ksin- 1 £B) 3 = a ; 3+(l 2 + 3X + (l 2 + 3 2 + 5 2 K+...; and so on; the expansion of sin"" 1 ^ having been given already in § 108; and putting x = sm6, we obtain the expansions of 6, |0 2 , %6 3 ,. . • in powers of sin 6. The versed sine, or rather the half versed sine, denoted by havers, is much used in Navigation ; and havers 6 = |(1 — cos 6) = sin 2 |# ; so that the preceding expansions give 6, |0 2 , %6 3 ,... in a series of powers of havers 26. FUNCTIONS. 231 Then putting 26 = , we obtain , , havers f 2 2 .4 2 (havers<£) 3 , ^ = — 2! + 4! + 6! + " ; t o , (vers ) 3 , or ^ 2 = vers0 + v 6 +35 3 +" ; and differentiating both sides with respect to vers (j>, 12 12 3 <{> cos = 1 + 1 vers + ^(vers 0) 2 + ' ' w (vers tj>f + . . . In a similar manner the expansion of sin(msin -1 a: + a) can be established, the differential equation and the recurring formula being (ex. 2 iii, p. 143 ; ex. 2, p. 185) and f™+ 2 0-(% 2 -m 2 )f"0 = 0. By differentiation we obtain the expansion of (1 — * 2 )"^sin(m sin-^+a) ; and by putting cc = sinO, we obtain the expansion of sin(m(9 + a) in powers of sin 6. Examples. (1) Expand, in ascending (and descending) powers of x, J-j (A' j it ■, • • * (x— a)(x— b){x— c) also in powers of x — a, or x — b, or x — c, (Resolve into partial functions, of the form a? — a a; — x — c then a; — a a\ a/ \a a? a 8 a n+1 / or = ^(l-^)- 1 - ^(1+-+^+...+^!+...). x\ x/ \X X i ■ X 6 x n / 232 EXPANSION OF To expand in powers of x—b, put x— b = z; then A A = —v , which is expanded in powers of z, ascend- x— a z — a+b r r ing or descending, as above.) (2) Expand tx/(a — x) m in ascending powers of x, fee denoting a rational integral function of x. Determine the remainder when fx is divided by x—a, (x—a)(x—b), {x — af, (x—af (3) Prove that e x oos " cos(a; sin a) x 2 _ x n = l+*cosa+ijTCOS 2a+...H — r cos%a+.... z! n\ (4) e^cos^a; a 2 cc 2 a n x n = 1 +ax-\ — ^- cos 2a(sec a) 2 + . . . -\ j-cos na(seca) n + ..., where tan a =p/a ; and determine the expansion of e^sin^as. (5) cosh ax cos px = 1 H — ~- cos 2a(sec a) 2 +...+ ~ , cos 2i2a(sec a) 2 ™+ .... ; and write down the expansions of sinh okc cos pc, cosh ax sin px, sinh ax sin px. (6) Expand sin(mx + nh), cos(mx + nh), i(mx + nh) by- Taylor's Theorem in powers of h, n, m, or x. a -J-) mx ctx/ cos = (sinh, or tanh, or sinh -1 , or tanh _1 )am_ TOS moj. (ii.) tanh|/i ^{f (cc + &) + f x} = f (a; + h) - ix. (Hi.) exp(rf^)&-f^). *(8) From the relation log(l + a;) 2 = 2 log(l+a;), deduce the expansion of log(l+a;). FUNCTIONS. 233 110. In many cases we require only a few terms, three or four, at the beginning of the series which is the expan- sion of a function ; and when the function is composed of constituent functions, of which the expansions are known, then the first three or four terms are readily obtained by a combination of the expansions of the constituents of the function. Thus e^sec x = e x -h cos x (-. , . x 2 . x 3 , \ /, x 2 , a; 4 \ and performing the division as far as five terms of the quotients, using the methods of detached coefficients and contracted division, /yi2 /yi4 \ sy,2 ™3 ™4 1 -2- + 24-) 1+a;+ 2- + 6 + 24- + l+l+£ + +1+1+0 + f + £ so that e x secx=l+x+x 2 +%x s +$x i +. Similarly, by the method of detached coefficients in multiplication, the * denoting a missing term, e a! cosa; = l+a; + * — %x s — \x^ e x sinx = x + x 2 + $x s + * — ^x s ... + * +^+- 40 Examples. (1) (i.) tanx=x+f+^+~ + ■ (ii.) (iii.) 15 315 _l_a;_fl3 3 _ 2x 5 _ COtX ~x 3 45 945 ' x secx = 1 + 2 + 2i 2 . 5x i . 61» 6 720 234 EXPANSION OF (iv.) CO secx=-+Q+ m+iEm +... . . i» . , * a; 5 a; 7 a; 9 (v.) e^sinx= a3 + -i8o-2835-90720~- , . . j - .. *, U!- tt' J- \-JU (vi.) e i»cosa ; = l+*- I 2- s -33gQ-... (vii.) (cos a)™ m 2 , 3tc 2 -2to 4 157t s -30ro 2 +16TO 6 _1 2~ + 24 * - 720 - a3 +" .... /sinaiV 1 m 2 5% 2 — 2% 4 Son 3 — 4>2n 2 +16n 6 = 1_ ~6~ + 360 X 45360 X+ - Write down the corresponding expansions of tanh x, coth x, sech x, cosech x, .... ,o, ,-, , , ■ s ,x 3 x a 107a; 7 (2) (i.) tea(smx) = x+j--jfi--£QJQ... ,.., . , L v , a; 3 a; 5 275a; 7 (n.) Hm (taaiaO = fl!+g-- i o-gQ i o-.- (3) 4/(l+x) = e(l-|a;+iia ; 2 -J ir a' 3 + f^a;*-...). (This is readily effected by writing 4/(l+a3) = explog^/(l + a;) = exp(l-Ja;+Ja; 2 -ia; 3 + ...) = e . e~i x . e& 2 . e-i* 3 . e** 4 . . . , and then calculating by multiplication the coefficients of X, X 2 , X 3 , 03*...) (4) (tan- 1 a;)/(l+a; 2 ) = a;-(l + ^a; 3 +(l + i+*K--" + ( _l)»-i(l + i+t+ ...+_J_y»-i... . Deduce the expansion of |(tan _1 a;) 2 in powers of x, or of |0 2 in powers of tan 0. (5) log sec x = £(sin a:) 2 + \ (sin a;)* + . . . + ^— (sin xf n + FUNCTIONS. 235 *111. The Exponential Values of the Sine and Cosine. Comparing the expansions in § 107 of cos x and cosh x, sin x and sinh x, we notice that, if i denotes y/(— 1), the circular and hyperbolic functions are connected by cos ix = cosh x, sin ix = i sinh x, tan ix = i tanh x ; so that sin(u + iv) = sin u cosh v + i cos u sinh v, cos(u + iv) = cos u cosh v — i sin u sinh v. Then cos cc = cosh ix = £(e ir,: + e - ix ), sin a; = — i sinh ia; = \{e™ — e~ ix )/i ; the exponential values of the circular cosine and sine ; and cosh(t; + iu) = cos u cosh v + i sin u sinh v, sinh(i; + iu) = cos u sinh i> + i sin u cosh v. Also cos 6+i sin = e l8 , so that cos n6+i sin n6 = e m = (cos + i sin 0) w , De Moivre's Theorem, analogous to cosh u + sinh u = e u , and cosh nu + sinh tcw = e nu = (cosh if. + sinh u) n . * 112. The Resolution of the Trigonometric Functions i/nto Factors, A rigorous proof is given in treatises on Trigonometry of the resolution into factors of sin and cos 6, in the form e 2 ^e=e{i-^){i-~r 2 ){i-^) (i.) C08 e = (i_^)(i_^)(i__|L) ( ii.) the factors being originally inferred from the values of 6 which make sin 6 and cos 6 zero. Therefore also, changing 6 2 into — u 2 , sinhu = u(n-j)(l+^)(l+^- 2 ) (iii.) C0S hu=(l + ^)(l+^ ? )(l + ^ ? ) (iv.) 236 EXPANSION OF Again, l-cos0=2sin 2 |0 -H^-m-^-m' <:•> l+cos0=2cos 2 £0 -*-S)T>-£Ki-&)' <*> and 1+8tae= ( 1+ ^)*( 1 _ s |_) , ( 1+ _«.)' (viL) ^-(t-m+^-S)' ^ so that l+sin0 . . . 1 ,.. //l + sin0\ .KX-^O .(ix.). (^KiX 1 -^)- Taking logarithms of (i.) and expanding, we find — -sS)"^— •• w where the series 1 -p + 2 -*> + 3 -■*>+... or 2m. "* is denoted by $ p ; the symbol 2 denoting summation for all succes- sive positive integral values of n from 1 to infinity. Also, from (ii.), denoting 2(2m— l) - ? by T p , logsec, = (^)\l + ^ + ^ + ...) + ... + ^)\ lv ...( x i.) FUNCTIONS. 237 But S p -T p = 2 ~pS p ; so that ^ = (1-2-*)^, S p = &T p /(2p-1). Now differentiating (x.) and (xi.) we obtain ^ x =l~l^S ^ (*"•) 4, / jj. \2n-l tana;= ~^,[r~ ) T ^> (xiii.). Similarly taking logarithms of equation (ix.) and expanding, gives log(see x + tana;) = log tan(^7r + \x) = gd _1 £C +K£)"('-W-^-)+- 2 / 3; \ 2»+l + (2^+1) feJ ^ +1 + (xiv -> where l-p-3--P+5--P-7 _ -P+... = 2(-l)*- 1 (2'n,-l)- 1 ' is denoted by U p ; and differentiating 4 /a; \ 2 ™ seca;=H — X(r~) ^W 1 ( xv -); the first term reducing to unity, because (§ 108) i-m-++-=iT. The series for cosec 2 a) and sec 2 a; or cot 2 a; and tan 2 £ are obtained by the differentiation of the series (xii.) and (xiii.) for cot x and tan x. sin(m+w)o; n sin (m + n)x Also - — : — — and '— sin rax sin nx cos mx cos nx can be expanded by writing them in the form cot mx + cot nx and tan mx + tan nx. 238 EXPANSION OF *113. Bernoulli's Numbers. In the general expansion of tan x, cot x, cosec x, tanh x, coth x, cosech x, the coefficients are certain rational num- bers called Bernoulli's numbers, which are thus denned. e x -\- 1 x Suppose £ + +( - i ) ra " i £^ + • then B v B 2 , ..., B n , ... are called Bernoulli's Numbers. With our notation (§ 33) |a^t- = |a;eoth£a;, so that, changing \x into x, x coth x= 1 + ¥ VB 1 --*JB i + ... + (_i ) »-i|- ! 2^ B +..(i.). Again, changing a; into ■iai, then (§ 110) ix coth ix = x cot a; /y»2 ~»4 ~,2?l = i-|2^-fr 2 ^--^^-... or cotas=±-£.2«B 1 -f r 2*S a -...-^ r - r 2*»fl B - (ii.). a; 2! 1 4! 2 2to! v ' Now tan x = cot x — 2 cot 2a;, so that tana; = |2^-l)5 1 +^(2*-l)5 2 +... +^j-2«-(2«--l)5 n + (hi.); FUNCTIONS. 239 and therefore changing x into ix, tanha; = |2^-l)5 1 -|2^-l)5 2 +... nftn-l + (-l)»- 1 ~2 2 »(2 2 »-l)5 n + (iv.). Again cosec x = tan \x + cot x, so that cosecx=-+|25 1+ |2(23-l)5 2 +|2(2«-l)£3+...(v.) ; and therefore cosech a; = --| ! 2 J B 1 + | I 2(23-l) J B 2 -| [ 2(2«-l)£3 + ...(vi.). The first nine numbers of Bernoulli are T> _£ TJ jj^ TJ J^ TJ _jj^ T> _J>^ p _ 0"1 1 ~ 6' 2 ~ 30' s _ 42' * ~ 30' 5 _ 66' 6 ~ 2730' _7 R _3617 ^T- 6 . -°8- 510 '^9- Comparing these expansions with those of the last article (§ 112), we notice that o _(2tt) 2 "p ^ n_ 2W! so that S2n/'7! 3n is a rational number. If the expansion of sec x is written /via /Y»4: /yii?l seca ; = l + |^ 1 +|^+...+^+..., then -Ej, E % , ..., E n , ... are called Euler's numbers, and # 1 = 1, ^ 2 = 5, ^ 3 = 61, # 4 =13S5,.... Comparing this expansion of sec a; with that of the last article, we find 2n- 1, x n j 2 2n_1 — 1 cosech x = — 22( — 1 )" " 1 B n x 2n - i ; x n sec x = 1 + 1,E n X2 n , sech x = l— 2( — 1 )" - 1 E n X2n- versa? = 2( — l)* -1 ;^, versha;=2a;2 n . Since 1 = cos a; sec a; = sin x cosec *, and tan x = sinajseca;, the B's and E's are connected by relations easily written down ; namely, writing E Q for unity, 2«! 2!(2»-2)! ■»" — ^ 2n! 2(2»-^-l) H 2(2 2 "-^-l) , (-1)" _ n l!2m! " 3!(2m -2)! "' * ~ l ~ - " " 1 "(2»+1)I ' K-i _ ^- 2 , , (-l)"- 1 ^ _ 2 2 "(2 2 "-l) ll(2m-2)! 8!(2ro-4)! (2»-l)! 2«I "' FUNCTIONS. 241 Suppose 6 = gd u, or u = gd ~ 1 6 = log(sec 6 + tan 6) ; then cos 6 cosh u = l, or cosh iQ cos iu = l, so that m = gdi$, or i0=log(seciu+tanm). Now u = log(sec + tan 6) =/(sec 0d0 = 6 + 2 j&VWi. so that, conversely, = u — 2( — l)™ - 1 £ , „u 2m+ i, a curious instance of the reversion of a series. We may compare the reversions y = e * - 1 = 2x n , and x = log(l + y) = S( - 1)™ " y/ 71 ; also the reversions of i/= (l+x) m , sinx, Examples. (1) Calculate to seven decimals from these series the value of tan 18° = tan T 1 T5 -Tr ; also of sin 18° and cos 18°. (2) Prove that cosec 1° = 573, cotl° = 57-29; cosec 1'= 3437-7 = cot 1'; cosec 1" = 206265 = cot 1"; cosec 8"'76 = 23546; and thence, taking the sun's parallax as 8"'76, prove that the distance of the sun is about 150 million kilometers, or 81 million nautical miles. (3) In the expansion (l-cc)"^exp(^x + ^a; 2 ) = SJ.„cc, l , prove that A n is always an integer. (4) Given exp(e a; — \) = Y + 1 l L n x n , prove that, for rc = l, 2,3,...,Z„ = 1, 2, 5, 15, 52,... (5) Given sinlog(l + x) = 1,A n x n , coslog(l +x) = 1 + ^B^c n , calculate the first eight A's and B's. (6) Prove that the coefficient of x 2n in the expansion of x/(e*-l) is 2£ 2n /(27r) 2m ; and that , cosh x — cos x , .„ x 2n 2 m _B TC cos 1%tt . log « h 2, 77-r — = 0. 8 x 2 2nl n (7) Prove that S s = 1-2020569 ; and that 2to- 2 (to+1)- 2 = £7t 2 -3; 2rc- 3 (% + l)- 3 = 10-7r 2 ; 2w- 4 (%+ 1)" 4 = A^ + ^tt 2 - 35. 242 EXPANSION OF 114. The Remainder im Taylor's Series. The previous expansions extend to an infinite number of terms, and are therefore only true when convergent. But some functions, for instance sec _1 a;, cosh" 1 ^, or coth _1 a?, cannot be expanded in an infinite series in ascending powers of x, because x must be greater than unity, and the expansion by Taylor's or Maclaurin's Theorem would be divergent, and the theorem is then said to fail. This difficulty will be avoided if we can make the series terminate after a finite number of terms ; we shall proceed to explain how this can be done. Suppose i(a + h) expanded, in Taylor's Series, i(a+h) =ia+M'a+~i"a+ +^f n a+R, h n+i Since all the terms of R involve -. — — r^-, as a factor, we (n+l)l may put J R = __^P ) and seek to determine an expression for P. We shall prove that P = i n+1 (a+9h), where 6 is a proper fraction, some unknown function of a and h; then and R is called Lagrange's Form of the Remainder in Taylor's Series ; so that Taylor's Theorem is now f(a+A)=fa+M'«+|f''a+...+^f M «+^ I y,f»+ 1 («+^), thus avoiding the use of an infinite series ; and incidentally establishing Taylor's Theorem in a rigorous manner. FUNCTIONS. 243 To prove that P = f n+1 (a + Oh), we write down a function ¥x, such that Fx = fx+(a+h-x)£'x+ ( - a+h 2 ~ x) * i"x+... {a+h-xY (a+h-xy+i "*" ' n\ XX+ (to+1)! ^' then Fas = ^±^)!(f »+^ - P) ; also Fa = f (a + A,) and F(a + h) = f(q, + h). If we draw a curve BQK, whose equation is y = ¥x, with ordinates AB, MQ, HK, at B, Q, K, then if OA = a, AB = ~Fa = i(a+h); andif OS = a+/i, fflr=F(a+/i) = f(a+A); and the chord BK is therefore parallel to the axis of x. Now if ix, fx, i"x,... i n x are all finite and change gradually between x = a and x = a+h, then Fx and Yx are also finite and continuous; and therefore at some point Q of the curve BQK between B and K the tangent is parallel to the axis of x. Exceptional cases where Fa; is not continuous are seen represented in fig. 31. If a+dh is the abscissa of this point Q, 8 is a proper fraction, and AM = 6h; then Y(a + 6h) = 0, or {h ~Q h)n {£ n +\a+dh)-P} = 0, and therefore P = i n+1 (a+6h). *115. The actual value of 6 is seldom assignable, and is not a matter of practical interest. We can, however, assign an approximate value, and a few terms of a series in powers of h, giving a closer ap- proximate value. 244 EXPANSION OF For expanding 11 in two series, writing h n for h n /n\, R = h n+1 (i n+1 a + 6M K + 2 a + 6 2 h 2 f. n+3 a +...) M = h n+1 i »+% + h n+ d n+2 a + h n+s i n + s a +...; an equation for determining 6 by the method of suc- cessive approximation or reversion of series (§85). Thus, as a first approximation, 6 = l/(n + 2) ; and to a second approximation, fl _ 1 n + 1 f"+ 3 q , , n + 2 2(n + 2f{n + S)i n+2 a ' and so on ; the next term being (TO+l)(5^+12)f m + 2 c t f"+ 4 a-^3(w+l)(TO+4)(f"+ 3 a) 2 /t2 6(n + 2) 3 (n + 3)(n + 4)(f n + 2 a) 2 For instance, if n = Q, or as in § 7, f(a+h)=ia+M'(a+6h), then = H _ _+ ^-A-2 _+...; and if f (a +h) = ia+hfa+ h 2 i"(a + dh), ., fl , , fa A , 17f 3 af B a-15(f 4 a) 2 A 2 then = £ + _- + ^o ,-55; f 3 a 36 (f 3 a) 2 " 1620 *116. If we had put B = hP, and ^x = ix+{a+h-x)i'x+^ a+h ~ x ^i"x+... 2! + ( a+h ~ X)n f n x+(a + h-x)P, so that Fa = F(a + A) = f(a + A) ; then Yx = <*+ A =*£i "+ 1 ® - P, nl and F(o + 0/0 = 0, when P = ^(1 - 0)«f »+i( + 0&) ; and then JR = {L_(l _ 0)»f»+i(a + 0A), Cauchy's Form of the Remainder in Taylor's Series. FUNCTIONS. 245 More generally, if we had put R = hP +1 P, and Fx = ix+(a+h-x)fx+(a+h-x) 2 i"x+ + (a+h- x)J. n x +{a+h- x) v+1 P, so that Fa = F(a + h) = f (a + h), then F'x ^^'^ i^x-ip+l^a+h-xyP; and F'(a+6h) = 0, when P = - !f" V ^ (l - e>-^f"+ 1 (ffl+ 0A), w!(p+l) v and then R = ,^ + V , (1 - 0)»-*f»+\a + 0A) ; Schlomilch and Roche's Form of the Remainder. When p = n, this becomes Lagrange's Remainder, and Cauchy's when p = 0. 117. Put a = 0, and change h into x ; then f x = f + as£'0 + ffljfO + . . . + asJE "0 + « n+1 f »+ I (ftB), Maclaurin's Theorem with Lagrange's Remainder; so that now we may write P = i n+1 (6x). Thus P has the following values for the corresponding ' functions, when expanded in powers of x. (i.) (a + x) m , P = m(m-l)...(m — n)(a+6x) m - n - 1 > (ii.) sin(cc+a), P = sin{6x + a+(n + l)%Tr}. (iii.) sinh x, or cosh x, P = \{(? x ± ( - l) n e-° x } ; (iv.) e x , or a* P = A or a e "(log a) n+1 ; (v.) log(l+a), P = (-l) m n!(l + te)-"- 1 ; (vl) tanh"^, P=«P» 2 (l-e 2 a 2 )"+i ; (vii.) tan -1 a;, P = n\(l + 6W)-» l + 1 hm{(n + l)(lTr+ta,n- 1 dx)} ; (viii.) exp(a; cos a)cos(a; sin a), P = exp(te cos a)cos{ &c sin a + (w + l)a }. 246 EXPANSION OF Fig. 42 118. Geometrical Illustration of Taylor's Theorem. Contact of Different Orders. If y = ix is the equation of a curve Pp, and if OM = x, then MP=ix (fig. 42). If Mm = h, then Om = x+h, and mp = i(x+h). Draw the tangent TPV at P, cutting mp in V, and draw PR parallel to Ox. Then ta,n RPV = i'x, so that i?F=M'a3, and mV=ix+hi'x. But, by Taylor's Theorem, as far as three terms, f (x + h) = ix + M'x + |ft 2 f "(x + 6h) ; so that Vp = $h 2 i"(x+dh)„ Describe a circle touching the curve at P and passing through p, and let Vp produced meet the circle in U. Then, since PV 2 = VU. Vp, therefore VU=PV 2 /Vp = (PR 2 +RV 2 )/Vp _ h 2 {l + (i'xf} _ l + (i'x) 2 %Wi"{x+dh) %i"(x+6h> FUNCTIONS. 247 Now make p approach to coincidence with P by- diminishing h to zero ; the circle becomes the circle of curvature at P, and VU becomes the chord of curvature parallel to the axis of y, and as before (§ 94), _ l + (rxf _ J dy*\/d*y ii"x V^dxV/ dx 2 ' Thus, at a maximum or minimum y, dy/dx = 0, and d 2 y/dx 2 = l/p, the curvature, estimated positive upwards. Suppose x = a makes dy/dx = 0, then i'a — ; and i{a±h) = ia+%h 2 i"(a±dh). When h is small fa may be written for f"(a ± Bh); and, when ffa = 0, the ordinate fa is less than the adjacent ordinates f (a ± h) if f "a is positive, and fa has therefore a minimum, value; but if i"a is negative, then fa is greater than f (a ± h), and fa is therefore a maximum; the Theory of Maximum and Minimum is often discussed in this manner by the aid of Taylor's Theorem. Let f (x + h) = ix+ hi'x + h 2 f'x + h z i'"(x + Bh) , when expanded by Taylor's Theorem as far as four terms; and let Pq be the arc of the parabola, which has its axis parallel to the axis of y, and which osculates the curve Pp at P, that is, which has the same circle of curvature at P ; then Vq = \m'x, so that qp = \M"\x + eh). The geometrical interpretation of i(x+h) = ix+M'(x+dh) has been given in Chapter I., § 7. The graph of the equation y = f x + M'x + h 2 i "x + h & i'"x will for different values of h, keeping x constant, be a curve of the third degree, touching the curve Pp at P, and having the same curvature, and in addition the same value of d & yjdx % at P; it is then said to have a contact of the third order with the curve y = ix. 248 EXPANSION OF Generally, the graph of the equation y = fx+ M'x + h 2 Fx +...+ h n i n x will represent for variable values of A. a curve of the n th degree, and it is said to have contact of the n th order at P with y = ix; two curves being said to have a contact of the n th order at a point of intersection when the first n d.c.'s of y with respect to x (and therefore of x with respect to y) are equal in the two curves at this point. *119. Infinitesimals. In discussing the properties of a curve OP in the neighbourhood of a point 0, it is convenient to take as origin, and the tangent Ox and normal Oy as coordinate axes, as in § 96; but now to take the arc s measured from as the independent variable. The student can easily supply the figure. Then by Taylor's or Maclaurin's Theorem, x = x's+x\+x'"s a +..., y = y's + y"s 2 + y'"s s +... ; the accents denoting differentiation with respect to s and that s is afterwards made zero, so as to refer to 0. Now -^-= cos^, ^ = sin^; as as d?x . , d^b- d % y , d\lr _=- sl nV^ ( ^ = cos^; d s x , d\ls 2 . , d 2 \Is -^-cos^-sm^, d 3 y . .dxl/t ,d 2 \lr d*x ./dxlS d B \l,\ .dylrdSl, ^__ 3sin ^^_cos^^-^V )>-- FUNCTIONS. 249 Denoting by c the curvature i// or \jp at 0, then, by- putting s = and i^=0, we find x' = l, y'=0; x " = 0,y" = c; x'" = -c 2 , y'" = c'; x""=-3cc',y""=-c* + c"; ...; so that, to three terms of the expansion, x = s — c 2 s 3 — Scc's i + ..., y = cs 2 + c's a — (c 3 — c")s v . . ; 1 , p' „ 2p' 2 p" or, since c = -, c = — *-, c ' = -^- — ^, ..., P P P P Then cos ^=_ = l-_ + ^..., so that tan\/r = sini^/cosi/r = — ^ri+o - ^ 1 +p' 2 — Ipp")---- p ^/o op Along Op, the circle of curvature at 0, we take c or p as constant, so that c', />', c", p", . . . vanish ; and then _ s 3 _ s 2 s 4 and therefore, if the arcs OP and 0^ are each equal to s, we find that It Pp/i? = $cp' Let the normal at P meet the normal at in Q, so that <2 is ultimately the centre of curvature at ; then OQ=y + xcoti, = p + hs^ + ^(pp"-p' 2 )...; and lt(0 Q — p)/s = ^dp/ds. Similarly, if It denotes the radius of the circle which touches OP at and passes through P, 2R = x*/y + y = 2p + %sp'..., so that \t(B— p)/s=$dp/ds. 250 EXPANSION OF Any magnitude o on the figure is called an infinitesimal of the n th order, with respect to s, if It o/s™ is a finite quantity; thus x and -fy are infinitesimals of the first order, y of the second order, Pp of the third order, .... In d n y/dx n , d n y is of the n tlL order compared with dx. As exercises in Infinitesimals, the student may prove that, if the chord PNP' is parallel to the tangent at 0. (i.) lt( OT-PT )/s 2 =^c'/c; (ii.) lt( PQ-OQ )/s 3 = tW; (iii.) lt( TQ-OQ )/s 2 =£c; (iv.) lt( OT+TP- arc OP )/s 3 = -&c 2 ; (v.) lt(arc OP - chord 0P)/s 3 = ¥ \c 2 ; (vi.) lt( PG-PM )/s 4 = £c 3 ; (vii.) lt( arc OP -arc OP' )/s 2 = lc'/c; (viii.) lt( NP-NP' )/s 2 = lc'/c. The equation of the conic which touches the curve OP at is of the form y = l(ax 2 + 2hxy + by 2 ) ; and if in addition the conic osculates the curve at 0, that is has the same curvature, then a = c = 1/p, the curvature (§ 91). We can make the values of y'" in the curve and the conic the same, by taking \=\c'\o, and now the conic has a contact of the third order, b being still arbitrary ; and the locus of the centre of the conic is the straight line ax+hy = 0. If in addition we determine b so that y"" is the same in the curve and the conic, then the conic has a contact of the fourth order with the curve, and it is called the conic of closest contact; it may be considered the ulti- mate form of the conic which passes through five con- secutive points near ; and we shall find ,_lc"_4c' 2 , lc~ 2 9^ + C ' FUNCTIONS. 251 120. Indeterminate Forms, or Singular Values. In general the value of a function of x is determinate, and is obtained by substitution for any particular value of the independent variable x. But when for a particular value of x, say x = a, the function assumes one of the indeterminate forms, or singular values, 0/0, oo /oo , oo x 0, oo — oo , 1", oo °, 0°, the real value must be obtained by the method of limits; that is, the value of the function must be found for x = a+h, where h is small, and reduced and simplified as much as possible by cancelling factors, etc., and after- wards h must be made to vanish. We have seen this exemplified in Chapter I. in finding the derivatives of the simple functions ; for dfx _ ,J(x + h) — ix dx h assumes at first the indeterminate form 0/0, before re- duction; and in the evaluation of these indeterminate forms we have employed lemmas which enable us to write down the values, when x = 0, of (sin mx)/nx. {(tan mx)/nx}P, (sin- :i mx)/nx, {(t&n' 1 mx)/nx}P, ^/(1 + x), (l+x/afl*, {\og a (l+x)}/x, etc. By ordinary algebraical and trigonometrical reductions the indeterminate form may in general be evaluated ; but the Differential Calculus affords a general method. Thus, suppose that when x = a, the function ix/Fx assumes the indeterminate form 0/0, because fa = and Fa = 0. Then, when x = a, ix _,, f(a+h) _, ia+M'a+h 2 fa+ ... Fx~ ~ F(a +h)~ Fa + KE'a + h 2 ¥"a +... i'a+^hi"a+... fa ~ F'a+%hF"a+...~F'a 252 EXPANSION OF fa If however fa = and Ya — 0, the true value is ™-; s a and so on, till the value is obtained. Thus, when x = a, = - = It — = — =«a n - 1 - x — a 1 Or, putting x = a + h, as in § 4, It = It- r = na n \ x—a h If to is a positive integer, then x — a divides x n — a n , and the quotient, as"- 1 + ax n ~ 2 + aV 1-3 + + a*~ 2 a: + a*~ 1 , becomes na n ~\ when 33 = a. Again, as another example, when x = 0, sinh a: — since _0 _, coshce — cossc ~~ a? ~0' M = 0' __, ,sinhce+sina3_0 _., cosh aj+ cos aj_l _lt to 0' -lt 6 ~3 ; and thus sinh ce — since is an infinitesimal of the third order with respect to x. But by using the expansions of sin x and sinh x, then this function can be evaluated as follows : — when as = 0, (sinh x — sin x)/«? =lt(a5+a; 3 +a? 5 +a; 7 + — x+x s — cc 6 +ce 7 — )ja? — 2 lt(a!g+a! y +a! 11 + )jx 3 = 21t (3! + fl + ri! + ) = 3! = 3' aS be£ ° ra 121. The indeterminate form oo /oo can be thrown into the form 0/0 by interchanging numerator and denomin- ator, and is evaluated in the same manner. FUNCTIONS. 253 -n, . , , , sec(2n + l)x oo I or instance, when as = W, h, rr = — 5 J sec(2m+l)a: oo' but, when numerator and denominator are interchanged, _ cos (2m + l)sc _ _,,(2m + 1) sin (2m + l)x ~ cos(2ti+l)a;~ 0~ (2n + l)sm(2n+l)x = 2m + l 2n + V } The indeterminate form oo x assumes the form 0/0 by throwing the oo into the denominator. For instance, whena3=oo,2 a! sm^=oo xO = a— ^ r = ^ = « (§ 16). The indeterminate form oo — oo can also be made to assume the form 0/0 by reduction of the terms to a single fraction. Thus, for example, when x = Jx, sec x — tan x = oo — oo 1 — sina: 1 — sinai /l— sina; rVr cos a; ,^(1— sin 2 a;) \l+sina; and, again, |7r sec a; — a: tan a; = oo — co \tv — ccsina; . — sina: — a:cosa: _ = 0; cosa; — sina; 122. To evaluate 1°°, oo°, 0°, take the logarithm ; this will be found to assume the form 0/0 or oo/oo , and can then be evaluated by the preceding rules. For instance, when a? = 0, (cosma;) ,l / ;!:2 =l°°; and . ,„ lop; cos ma; log (cos mxp^ = n— 2 — % = h ■-nit x 2 — m tan ma; 2a; n , — m 2 sec 2 ma; , „ = n\t g = -¥ ein > and therefore (cos mxf 1 ^ = exp ( — £m 2 «). 254 EXPANSION OF Examples. — Prove that, when n\ -9 a; 3 - 19a; + 30 _7, W 8- ^ cc 3 -2a; 2 -9a;+18 _ 5' 8=3, 4 : 3' s/a — ^Jx + ^ja — x) (2) * ~ "' ' V(« 2 "* 2 ) V(2«)' .„. „ tan x + sec a; — 1 ., (3) a; = 0, t t = 1 - v ' tana: — seca: + 1 ... , cos x + 1 — sin as , (4) x = \-w, -.. -. — =1. w * cos x — 1 + sin a; ._. , sina; — cosa; , /0 (5) x = ^' sin2*-cos2as-l = K/2 - .... , 1— sina; — 2sin 2 as _ v ' b 1 — 3-sm a; + 2 sin 2 a; /it\ n (2 sin a; — sin 2a;) 2 nnn . (7) aj = 0, ^ 5-^- = 0.064. v ' (sec a; — cos 2as) d (8). = 1, fel- (9)B=0l r^h= L v ' log (1 + as) (10) «=— 1, — nr = log- v 7 %+l ° a (In this way / x~ x dx is deduced from tx n dx.) a a (ll)as = a,^-^=l. (12) 33 = 0, .F~x = F^'' ix — ia i'a Fx-Fa F'a' ix — fa — {x — a)i'a _i"a . Fx-Fa-(x-a)F'a~Wa'' {x—£a — (x — a)£'a — %(x — a) 2 £"a _ i'"a Fx - Fa - (x - a)F'a -\{x- a) % F"a ~ F^a ' and so on. .,„, ax + b a < 17 > a;=0O 'Z^T5 = Z ; ax 2 + 2bx + c a . Ax 2 + 2Bx+G A' ax m + bx m - 1 + cx m - % + ...+k n a Ax n + Bx n ~ x + Gx n - % +... + K 'A' according as m is < , = , or > n. Write down the values when x = 0. 256 EXPANSION OF FUNCTIONS. (18)x = Q, lo % COtx = 0. coseea? (19) x = 0, o; n log x = or — oo , according as n is positive or negative. (20) x = a, (a — x)ta,n^Trx/a = 2a/TT. (21) x=,x(Z/a-l) = \oga. (22) x = , z/(x n + a n )-x = 0. (23) a, = l _±_- ,j-L- = l. v 7 logic log a; (24) a; = 0, — cota; = 0; -* — cot 2 a; = #, cosec 2 * — s = 4. v ' x ' x 2 s x 2 3 (25) aj = l, a; 1 /a-^) = l/e. (26) x = x,(l + l/xy c =e. (27) %=oo, (cos «/%)* = 1; (cosa/%)" 2 = exp(-£a 2 ); (cos a/n) nS = ; /sinaM* («J^Mr 2 = exp( _ ia2) . (^f = . V a/n J \ a/n / rv b \ a/% / , oc . _ /sinccY 00860 *)'" . . ,. (28) as = 0, {— ) = 1, exp( - ft, or : (sec axy- mUx ?= 1, exp J a 2 /6 2 ; or « '5 according as r is <, =, or > 2. (29) a;=0, (cosa;) cota; = l. (30) £C = |tt, (sinaj) tanfl; =l; (sin a;) taA = 1/^/e. (31) x = \ir, (tana;) tari2a! = l/e. (32) tc = 0, (l + l/cc) a; =l; (cota;) 8ina; =l. (33) x = ao,*/x = l, 4/(l+as) = l. (34) sc = 0, os* = l; 4/a? = 0; (sina;) tana; =l. CHAPTEE V. PARTIAL DIFFERENTIATION AND INTEGRATION. 123. Functions of Two Independent Variables. When y is a function ix of a single variable x, the relation y = ix is exhibited graphically by means of a plane curve, in which the abscissa is x and the corresponding ordinate y is ix (fig. 1), the curve being called the graph of the function ix (§ 5). But when a variable quantity z is a function of two independent variables x and y, expressed by the notation s = ifay) ■.....(A), then x and y may be supposed to be the coordinates of any point on a datum (horizontal) plane, and z to be the height of a surface above this point on the datum plane, e.g. the surface of the land ; so that the relation (A) is the equation of the surface, and the graph of a function f (x, y) of two independent variables x and y is a surface. For instance, the relation pv = JRd, connecting p the pressure, v the volume, and 6 the absolute temperature of a given quantity of a perfect gas, where R is a constant, may be represented graphically by means of the surface, cz=xy, where x represents the volume v, y the pressure p, and z the temperature 6. 258 PARTIAL DIFFERENTIATION. With coordinate axes Ox, Oy, Oz, at right angles, the curve y = ix will, by revolution round Ox, sweep out the surface whose equation is /J(y 2 + z 2 ) = ix, or y 2 + z 2 = (ix) 2 ; and, by revolution round Oy, the surface _ y = i{J(x 2 + z 2 )}; while the curve given by the implicit relation F(a?, y) = will sweep out the surfaces F(x, s /y T +z 2 )=0, and F^x^Tz^, y) = 0. We may also write the relation (A) in the form ■F(x,y,z) = (B), when an implicit relation connects x, y, z (§ 13). Thus (x/af + (y/bf + (z/c) 2 = 1 is the equation of an ellipsoid (fig. 43) ; but (x/a) 2 + (y/b) 2 = l is now in space the equation of a cylinder standing on an elliptic base, the axis being parallel to Oz. And generally, in space, the equations y = ix or ~F(x, y) = will represent cylindrical surfaces, the cross section being the graphs of the corresponding plane curve. 124. Notation of Partial Differential Coefficients. Now returning to equation (A), where, to fix the ideas, Oz is supposed vertical, and making a section of the surface by a vertical plane parallel to the axis of x (fig. 43) ; then the tangent of the slope to the horizon of the curve of section will be the d.c. of z with respect to x, keeping y constant ; this tangent of slope is expressed by ^- or — ' ^ (generally abbreviated to — J, and this is called the partial d.c. of z with respect to x, 3 being used when the differentiation is partial. PARTIAL DIFFERENTIATION. 259 Again, if a section of the surface is made by a vertical plane parallel to the axis of y (fig. 43), then the tangent of the slope of the curve of section will be the d.c. of z with respect to y, keeping x constant ; and is expressed by or — ' * (abbreviated to — ). dy dy \ dyJ Now, to find dzjdt, the rate of increase of z, when x and y increase at given rates dx/dt and dy/dt, let Ax and Ay denote small finite increments of x and y, and Az the corresponding increment of z. Then z + Az =i:(x + Ax, y + Ay), where z = f (x, y), Az=i(x + Ax, y + Ay)-i(x, y), Az _ f (x+ Ax, y+Ay)- i(x, y+Ay)+i(x, y+Ay)- £(x, y) At~ At _ i(x + Ax,y + Ay) - i(x, y + Ay) Ax Ax At i(x,y + Ay)-i(x,y) Ay Ay At' Proceeding to the limit, when At, Ax, and Ay are made indefinitely small, then lt f (x, y + Ay)- i(x, y ) _ 3f (x, y) _ dz . Ay ~ dy dy' and lt f (x + Ax,y + Ay) - f (x, y + Ay) Ax _ lt df(x, y + Ay) _ di(x, y) _dz , dx dx dx' so that dz_ dz dx.dz dy n . dl~dlc dt ) ty dl ( '' or dz= — dx+— dy dx dy a in the notation of Differentials (§ 11). 260 PARTIAL DIFFERENTIATION. Thus we deduce from the relation z=(JX- s /Y)/(x-y), where X = ax 2 +2bx+c, Y=ay 2 +2by+c, 2dz _ dx dy ~a~^z~ 2 ~^/X + JY' and more generally, from the results of Ex. 39, p. 82, ds _ dx , dy J{**-g#-gj JX ' JY where X = ax i + 4>bx 3 + 6cx 2 + 4 ?=°< ox dy but z is neither a maximum or minimum. These geometrical considerations are useful in the pro- blem of finding the maximum or minimum of a function of two independent variables. (Cayley, Contour and Slope Lines, Phil. Mag., 1859 ; Maxwell, Hills and Dales, Phil. Mag., 1870; Rev. E. Hill, Messenger of Mathematics, vol. v.). *128. The Indicatrix. Dwpin's Theorems for Normal Sections of a Surface. By changing the origin to a point on the surface (A) where dz/dx = and dz/dy = 0, we may write the equation of the surface in the form (§ 126) z = \{ax 2 + 2hxy + by 2 ) + higher powers, etc., of x and y ; dx 2 ' ~Gxdy dy 2 This is now the general form of the equation of a surface in the neighbourhood of a point, when the tangent plane is taken as the plane xOy, and the normal Oz as the axis of z ; analogous to the method of § 96, where the tangent and normal at any point of a plane curve are taken as coordinate axes. Neglecting terms in x and y of a degree higher than the second, then the equation z = ^(ax 2 +2hxy + by 2 ) ■ represents a paraboloid, called the osculating paraboloid, or paraboloid of curvature at the point. where a, h, b, are the values of ^-p ^-57,9 -^- 2 a * ^- 266 PARTIAL DIFFERENTIATION. Sections of this paraboloid by parallel planes z = c close to the tangent plane, which are very approximately sec- tions of the surface, will be similar conic sections ax 2 + 2hxy +by 2 = 2c; and in the tangent plane the similar conic section ax 2 + 2hxy + by 2 =l is called the Indicatrix (Dupin). If ab — h 2 is positive, the indicatrix is an ellipse, and a plane z — c, close to the tangent plane, will, if it meets the surface, cut off a small cup from the surface ; the point on the surface is then called a synclastic or cup point ; as at the top of a hill, or bottom of a valley ; so that the contour lines in the neighbourhood are approximately similar ellipses ; and the osculating paraboloid is elliptic. But if ab — h 2 is negative, the indicatrix is a hyperbola, and the tangent plane cuts the surface in two lines crossing at the point of contact; such a point is called an anticlastic or saddle point, such as a pass or bar ; so that the contour lines in the neighbourhood are approximately similar hyperbolas; and the osculating paraboloid is hyperbolic. Making a normal section of the surface and its osculat- ing paraboloid by a plane in which x = r cos 0, y = r sin Q, then It 2z/r 2 = a cos 2 + 2h sin 6 cos 6 + b sin 2 0. But It 2z/r 2 is the curvature 1/p of the normal section of the surface (§ 96), so that p the radius of curvature of a normal section is equal to the square of the corre- sponding radius vector of the indicatrix, since l/p = a cos 2 + 2h sin 6 cos 6 + b sin 2 #. If p' is the radius of curvature of the normal section at right angles to the first, then PARTIAL DIFFERENTIATION. 267 1/p = a sin 2 — 2h sin cos 8 + b cos 2 0, so that l/p + l/p' = a + b, & constant : a theorem due to Euler; and if = 0, 1/p — a, l/p' = b. We may revolve the axes Ox and Oy round Oz so as to make h disappear; and now the equation is of the form ' z=lx 2 /R + ly 2 /R+...; and R, R' the radii of curvature of the normal sections xOz, yOz, are called the principal radii of curvature of the surface, and Ox and Oy are called the directions of the lines of curvature at ; and now l/p = cos 2 6/R+sin 2 6/R' for a normal section making an angle 6 with Ox. The quantity a +b, or 1/p + 1/p, or 1/R + l/R' is called the curvature of the surface (Sir W. Thomson, Capillary Attraction) ; and to measure this curvature a spherometer is employed, consisting of a small plate or table, resting on the surface by three feet at the corners of an equi- lateral triangle, with a fourth foot at the centre which can be screwed down to touch the surface by means of a graduated micrometer screw. Now if c denotes the radius of the spherometer, and z v z 2 , z 3 the distances of the three feet from the tangent plane at the central foot, then for any angle of orienta- tion 6, 2zJc 2 = acos 2 6 -\-2hsm6eos9+bsin 2 6 2z 2 /c 2 = a cos 2 (6 + f tt) + 2z 3 /c 2 = a cos 2 (0 + fir) + so that the distance of the central foot from the plane of the three feet = K% + z 2 + z s) = & 2 (a+ b) ; which is independent of the orientation 6, and the same as for a sphere of equal curvature. 268 PARTIAL DIFFERENTIATION. *129. Meunier's Theorem, for oblique sections oj a Surface. When we draw a normal plane to a sphere at any point, cutting the sphere in a great circle of radius R, the radius of the sphere, and when we draw through the same tangent line an oblique plane, inclined at an angle 6 to the normal plane, cutting the sphere in a small circle, then the radius of this small circle is obviously R cos 8. A similar theorem, called Meunier's Theorem, connects the radius of curvature of any oblique section of a surface with the normal section having the same tangent line: for if the oblique plane z = xtan 6 cuts the surface z = i(ax 2 + 2hxy + by 2 ) + . . . , then the curvature of the section of the surface is It 2z sec 0/a; 2 , with y — 0, which is equal to a sec 6; so that the radius of curvature is cos 6/a. *130. Solid Geometry. Further geometrical applications would lead us beyond the scope of this book; we may however enunciate a number of useful definitions and propositions in Solid Geometry, for the student either to establish himself, or to refer for their demonstration to the treatises of Smith, Frost, or Salmon. A line of curvature on a surface is one whose tangent is an axis of the indicatrix at every point. Two lines of curvature pass at right angles through an ordinary point on a surface. A point where the principal radii of curvature R and R' are equal is called an umbilicus. Lines of curvature converge to an umbilicus. In the neighbourhood of a point on a surface the normals along a line of curvature ultimately intersect, PARTIAL DIFFERENTIATION. 269 but other normals pass through two (focal) lines at right angles to one another, one at each centre of principal curvature. These focal lines are seen experimentally when a narrow beam of light is received directly on a screen at a variable distance ; rays of light being always capable of being cut orthogonally by a surface. When normals are. drawn round a small closed contour of surface A, described on a surface round a point, and parallel normals are drawn on a sphere of radius c, then the corresponding contour on the sphere has an area c 2 A/RR' ; this is easily established if the contour is bounded by the neighbouring lines of curvature. The quantity \jRR' is called Gauss's measure of cur- vature; if the surface is bent in any manner without stretching, then 1/RR' is unaltered. Tortuous Curves. Two surfaces intersect in a line, called a tortuous curve if it does not lie in a plane. The two tangent planes of the surfaces intersect in the tangent line of the curve, while the two normal lines of the surfaces lie in the normal plane of the curve. The properties of a tortuous curve are investigated, as in § 90, by considering the curve as the limit of a twisted polygon of short links or chords. The plane through two consecutive links is called the osculating plane ; the normal line in this plane is called the principal normal, and the normal line perpendicular to this plane is called the binormal. The centre of curvature in the osculating plane is called the centre of principal curvature ; and p = ds/d\[s is called the radius of principal curvature, Ai/r denoting the angle between consecutive normal planes. 270 PARTIAL DIFFERENTIATION. Accents now denoting differentiation with respect to the arc s, we shall find, as in § 94, a = x+ p 2 x", /3=y+ P 2 y", y=z+ P V, a, /3, y denoting the coordinates of the centre of principal curvature ; and this point will be the foot of the perpen- dicular drawn from the point P on the line AB, joining A and B the centres of curvature of the normal sections of any two surfaces intersecting in the curve. The normal planes of the curve carve out a surface called the polar developable. Three consecutive normal planes intersect in a point which is called the centre of spherical curvature. For if normal planes are drawn through P v P 2 , P 3 , the middle points of three consecutive links of a tortuous chain (fig. 35), then their point of intersection Q 2 will be equidistant from the four ends of the three links, and therefore Q 2 will be the centre of a sphere passing through these points, and R, the radius of this sphere, is called the radius of spherical curvature. The locus of these points Q is called the edge of regres- sion of the polar developable. The angle between two consecutive osculating planes being denoted by At, then dr/ds is called the torsion; and denoting it by 1/cr, then 2 — b 2 ), the generating lines become perpen- dicular to the axis of the cylinder. But if the radius of the circle AA' is changed from a to b, the surface will become the hyperboloid z 2 _x 2 +y 2 _■. F 2 ~a^¥ ' Generally, one inextensible surface of revolution can be wrapped upon another surface of revolution, when the meridian curve AP of the first surface is a plane section of a cylinder, and the meridian curve AP' of the second is the curve which AP becomes when the cylinder is flattened into a plane. For if the equal arcs AP, AP' are denoted by s, and if y, y' denote the corresponding ordinates of P, P' with respect to Ox, the line of intersection of the plane of AP with a plane base or cross section of the cylinder, with which it makes an angle a, then y' = ysina; and the first surface can be applied to the second, so that the meridian arc AP becomes AP' without change of length, but the circular cross section of the surface at P becomes contracted into a circle, smaller in the ratio of sin a to 1. In the case above, the base of the cylinder is a common catenary, and the curve AP is a section of the cylinder made by a plane inclined at an angle cos _1 6/a to the base, and is therefore a modified catenary ; and when the PARTIAL DIFFERENTIATION. 275 cylinder is developed into a plane, it is readily seen that the curve AP becomes changed into a hyperbola AP'. As another illustration, we may take the base of the cylinder a circle ; and now the curve A P is an ellipse of excentricity e = sin a, which by revolution round Ox, the minor axis in which it cuts a circular cross section of the cylinder, sweeps out an oblate spheroid ; and this oblate spheroid is applicable on a surface of which the meridian curve AP' is a sinusoid (§ 20). We may therefore cut away 1 — e of the surface of the prolate spheroid by two meridian cuts, and join the edges together, when the new surface will be formed by the revolution of a sinusoid. Examples on Partial Differentiation. (1) Deduce the differential relation dz dx dy yQ.-cs? + *) = J(l-ai?+aty jQ.-cyi'+t) from the integral relation z = {x V(l - cy 2 + y*) + yj(l - ex* + -mW =^; and thence A^ + ^+^i-^HO, and |{( 1 -^f l } + ^ +1 ) P »= a (P K is called the 2cma£ surface harmonic of the m th degree, and h n P n the zonal solid harmonic.) Prove also that 1 dV • 7 ,„ 1 3F , T . -^3^ = ^-/07^^ = ^; and deduce (i.) (w+l)P„ +1 -(2w+l) M P„+wP, l _ 1 = ; (ii.) PUi-aMP'^+P'n-^P™. the accent denoting differentiation with respect to /*; (iii.) A iP'„-P'„_ 1 = 'n,P„; (iv.) P'„+i-P'»-i = (2^ + l)P„; (v.) ( M 2 -l)P' Jl =^P„-P„_ 1 ) = (7 l + l)(P n+i - M P„) = «(%+ l)(P n+1 - P n _!)/(2 TO + 1). PARTIAL DIFFERENTIATION. 271 *(7) Putting fih = z, h 2 =r*+z 2 , and denoting A"P„ by Z n , prove that ^§=nZ n . h ^=n(n-l)...(n-p+l)Z n . p ; and thence that writing z+ c for z changes Z n into Z,+w»Z tt -i+ TO ^~ 1 W ,-,+ ... +w*»'- 1 Z 1 +_ x _cos0. 3a; a; 2 + 2/ 2 r ' dy x' 2 + y 2 r and we should have found immediately, by differentiation with respect to the old variables, dz dz dr , dz d6 dz n dz . . dx dr dx d9 dx dr rdd dz dz dr , dz 30 dz . . , dz a ^ = d-rdy + Wdy = dr Sme+ rd9 C ° Se - Differentiating again, with respect to x and y, will determine |J, J^, §^ 2 , •••; and we shall find that d 2 z 3 2 0_3 2 s , 3z , 3 2 z 2 3?/ 2 3r 2 r3r r 2 30 2 ' d 2 z ^_f^3^\ 2 == d^zfdz^ ^z_\_(_^z__dz\ . 3^3^ \3a%/ 3r 2 \r3r r 2 30V Vr3r30 rW ' so that these expressions are unaltered by orthogonal transformation, that is, by a change to any other system of rectangular axes. 280 PARTIAL DIFFERENTIATION. When the old and new variables are connected by implicit relations, it is immaterial whether we differentiate with respect to the old or the new system ; but it is ad- visable to keep entirely to the system first chosen. Now, with the implicit relations, i(x, y, u, v) = 0, ¥(x, y, u, v) = 0, and differentiating for instance with respect to the old variables, 3f , 3f du , di ~dv_ft dx ~du dx dv dx dF , dFdu ,3F cto =0 . dx du dx dv dx so that ^-g^M^, dx d{v, x)l d(u, v) 3^_ 3(f, F) / 3(f, F) dx d(x, u)/ d(u, v)' dx du dx dv dx = (dz_ 3(f,F) dz 3(f, F) \ / 3(f, F) Xdw d(v, x) dv d(x, u)j/ d(u, v) Similarly for — and the higher partial derivatives. It is sometimes necessary to permute the variables x, y, z, and to make, say x the dependent, and y, z the in- dependent variables in a relation of the form (B) (§ 123), -F{x,y,z) = 0. w 3F dx . dF n 3F dx , 3F „ -Now — — +;^- = 0, — — + — = 0; dxdy dy dx dz dz while, with the former use oip,q,r,..., dx dz^ dy dz 1 PARTIAL DIFFERENTIATION. 281 so that ^=1, 5?= -2. "bz p dy p , Similarly we find, with the notation of p. 273, 3 2 o;_ _ r ?) 2 x _qr—ps d 2 x_ q 2 r — 2pqs+p 2 t_ dz 2 p s ' dydz p 3 ' dy 2 p s Analogous to the Keciprocants of § 86, we obtain Ternary Reciprocants, functions of the partial derivatives p, q, r, s, t, ..., the form of which is unaltered when the variables x, y, z are permuted in cyclical order. (Elliott, Proc. London Math. Society, 1886-1889 ; Forsyth, Phil. Trans., 1889.) Examples on Change of the Variables. (1) Given x = u+v, y = uv, prove that dz / 3a dz\ I, N dz fdz dz\ /, N (2) With x — r cosh u, y = r sinh u, prove that d 2 z 3 2 «_3 2 s jte_ d 2 z "dx 2 dy 2 dr 2 rdr r 2 du 2 (3) Prove that, on changing to oblique coordinates £ and r\, the axes being inclined at an angle w, TPz.-dH fd 2 z „ &z ,d 2 z\ / . , and write down the transformation of 3% 3 2 s_/ d 2 z \ 2 dx 2 dy 2 Xdxdy) ' x- — 2/~-J changes i(x, y) into f (x cos a — ysin a, xsina+y cos a) ; that is, turns the axes through an angle a. (5) From f (x, y, z) = 0, F(oj, y, z) = 0, prove that dy = d(i, F) /3(f, F^ , of which two are in- dependent, and E a function of v, constant). *133. Conjugate Functions. Two quantities u and v are called conjugate functions of x and y, when the complex quantities u + iv and x+iy are functions of each other, where i denotes ^( — 1) (Maxwell, Electricity, vol. I., Chap. xii.). Conjugate functions are useful in physical problems on the plane flow of liquid and electrical currents. We denote u + iv by w and x+iy by z, for brevity ; and now, if w = iz, or u + iv = £(x + iy), ,, dw du , ,dv „, then ■^- — s, — H;s- = i2 : , ax dx dx dw du , .dv .„, dy dy dy ,, , du , .dv .du dv so that - — H^- = *~ — ^-; dy dy dx dx ■ PARTIAL DIFFERENTIATION. 283 and equating the real and imaginary parts, du __dv , du _ dv dx dy' dy dx' n tl d 2 u 3 2 v d 2 u d 2 v Consequently ^2 = 3a; 2 dxdy' dy 2 ~ dxdy' d 2 v_ _ 3% 32?;_ 3% 3cc 2_ dxdy'dy 2 ~ dxdy' ., , 3% , 3% . 3 2 w , 3 2 v A SOthat 3^ + 3F 2 = °'3¥ 2+ 3y = °- Denoting by J" the Jacobian du dv du dv d(u, v) dx "by dy dx. d(x, y)' T (du\ 2 , /duV (dv\ 2 [dv\ 2 J ^) + \dy)' 0X \dx) + \dy) ; and J=f(x+iy)i'(x — iy). Conversely, x and y are conjugate functions of u and v. given by x + iy = f - 1 (u + iv) ; since i* + w = f (x + iy), or w = iz ; and differentiating now with respect to u and v as inde- pendent variables, , „ /dx .dy\ . „, /3» .32/\ . sothat ^ = &f?=-|* ; 3it 3v 3i> 3tt S 2 ^ 3%_ <% , 3^/ 3u 2+ 3i; 2 'Sw 2 "^ 2 ' and the Jacobian d(x, y) _dxdy 3a; dy d(u, v)~du dv dv du \2 /Tli a 2 W/ " t "\3w/ \3u/ \dvJ 1 1 ■f^+iy^aj-^-J-' 284 PARTIAL DIFFERENTIATION. or d(u,v) x d(x, y) =1 d(x, y) d(u, v) ' provided u and v are conjugate functions of x and y. If (j> and \Jr denote conjugate functions of u and v, it stands to reason that and yfr are also conjugate functions of x and y ; but if

, ^tanh- 1 r2m + ffl2m ■ (3) Determine the conjugate functions u, i; of cc, y from w = z 2 , Jz, z n , z n - c", 1/z, (az + b)/(A» + B), sin z, tans, sees, sin _1 z, tan _1 s, sec _1 s, logs, exp z, cosh %, sinh 2, tanh z, ... ; and sketch the corresponding curves. (4) Prove that log J" is a conjugate function. (5) Prove that F(£)ysin( P x +q ) = sin(pa; + q)i (^ p)y + cos(px + q){^ p)y, where, resolved into its conjugate functions F(x+iy) = i{x, y)+icj>(x, y). (The statement at the foot of p. 142 is incorrect, and must be altered to this result.) 286 PARTIAL INTEGRATION. 134. Double Integration. Denote by V the volume of the solid which is bounded by the surface z=i(x, y) and a cylinder standing on any base A in the plane of z = 0, with generating lines parallel to the axis of z (fig. 43). Then V, considered as a fluent, may be supposed gener- ated by the motion of the fluxion dVjdy, an area moving with its plane perpendicular to the axis of y ; as seen for instance in gradually rilling up the volume V with water, when the axis Oy is held vertical. Again the area dV/dy, considered as a fluent, may be supposed generated by the motion of the ordinate z, moving parallel to the axis of x ; so that z is the fluxion of dV/dy with respect to x, or dxdy " ' PARTIAL INTEGRATION. 287 and integrating, doubly, Y=ffzdxdy, the integration extending over the area of the base A. In the Infinitesimal method, the volume V may be supposed built up of filaments of length % and cross section dxdy. Also if x, y, z denote the coordinates of the centroid of the volume V, xV=ffxzdxdy, y V=ffyzdxdy, zV ' =ff\z 2 dxdy : Applying this method to the determination of the volume of an octant of the ellipsoid (fig. 43) (x/a) 2 + (y/b) 2 + (z/c) 2 = l, and integrating, first with respect to x, the limits are and a^Q. — y 2 /b 2 ) ; and integrating afterwards with respect to y, the limits are and b ; so that J cj{l-{xlaf-(ylbf}dxdy, X 2 ( "u\ which, on substituting — 2=( 1 — ts) sin 2 ^, makes V= /ac(l — ¥^)dy/cos 2 as independent variables, 13j9 13v = l i§P.l£^ =0 . pd9 vdd 9' pd$ v 30 ndp , mdv_ () n dp m dv _ , pd9 'vdd - ' pd$ ~v d$~ ' ,, , m— w dp rn m—ndp -, so that, -£= -^, ^= - 1 ; p dd 9 p dcp m—n dv n m—n dv 30 9' v d~ 1; and 3(P.«) _ 1 jw_ 1 3(0, ,6) m — ncd Therefore the area on the p, v diagram of the Garnot- cycle bounded by the two isothermals 9{, 9 2 and by the two adiabatics

- and therefore, taken round the curve, Jy^jjdt, oxJydx= -A ; Therefore, taken round the curve, J \ y ~dl +ay dt) dt ' OT J(y dx + xd y) =0 '> and ydx-\-xdy = d{xy) is called a perfect differential ; its integral between two limits is independent of the intermediate values of x and y and of the path described between the limits; so that, taken round any closed path, the integral is zero. and xdy—ydx is not a perfect differential, so that its integrated value depends on the path taken between two limits ; and in a closed path it represents twice the area enclosed by the path. Similar theorems hold for the integrals ffzdxdy and Jjzdydx. 138. Changing to polar coordinates by putting x=rcos 6, y = rsinO, ,, dy dx 9 d0 then ^-y^Tt'' so that A ==/i^dt =J\rHQ, taken round the curve. 292 PARTIAL INTEGRATION. For an origin inside a closed oval curve, the limits of 6 may be taken as and 2-tt. But if the origin is outside the area, draw the tangents OF, OH to the curve from ; then along FPH, ddjdt is positive, and /* \r^dt= axenOFPCHO; but along ERF, ddjdt is negative, and / \r^dt= -area OFQHO ; so that, taken round the curve, A =J\f^dt =/lr 2 d9 =/%x 2 dv, if y/x = v= tan 6. Although not a perfect differential, xdy—ydx can be made one by dividing by x 2 +y 2 , and then xd y-y?*=dtan-iy=de; x^+y 2, x so that f xd V-y?* =/d9 = 2*, or 0, J x 2 +y z J according as the origin is inside or outside the curve. Consequently if each point of the contour is displaced through small distances c 2 x/r 2 , c 2 ylr 2 , parallel to the axes, the change in the area will be 2ttc 2 or zero, according as the origin is inside or outside the contour. 139. A convenient independent variable to take is s, the length of the arc of the perimeter measured from a fixed point; so that the point P may be supposed to move with unit velocity ; and now, with p denoting the length of the perpendicular from on the tangent at P (§ 9), dy dx x ^-y^=p> PARTIAL INTEGRATION. 293 so that A = /%pds =/ip-r-dr = /h'P sec d T =/h'P r d /r liJ( 1 * ~ P*)- When fig. 44 represents an indicator diagram, and KL the reduced stroke of the piston, while the ordinate y represents the pressure of the steam, the pencil will de- scribe the contour with the area to the left, when the steam pressure is urging the piston from L to K. The diagram taken on the return stroke from the other end of the cylinder will be described in the opposite sense, with the area on the right hand of the describing pencil. Sometimes a loop is found on the diagram, described in the reverse sense ; this loop shows that a cushioning effect takes place, which requires attention, as the nega- tive area of this loop represents so much lost work. In the general case, where the perimeter cuts itself a number of times, then the area obtained by integrating once round a loop will be positive or negative with the above formulas, according as the area is on the left or right of the describing point, as it travels round the curve. (Clifford, Common Sense of the Exact Sciences; Cremona, Graphical Statics.) Familiar instances of such looped contours are seen in Lissajous's figures (Ganot, Physics, § 281), whose general equation may be written as (§ 103) m sin - tya —nsm.~hjjb=& constant, or oc/a = sin(nt+e),ylb = sin(mt + e). Also with the polar curves r = a cos (m6/n), where rn and n are integers, curves seen on the back of an engine-turned watch. 294 PARTIAL INTEGRATION. 140. The Planimeter. This instrument in the most usual form, that invented by Amsler of Schaffhausen, consists of two bars OA, AP, pivoted at and jointed at A, and carrying in PA produced a small graduated roller B, with axis fixed parallel to PA (fig. 45). Fig. 45 The instrument is used to measure areas ; the pointer P is carried round the perimeter of the curve whose area is required, an indicator diagram, or the section of a ship for instance ; the roller B, which rolls and slides on the plane of the paper, then registers the area. If precision is required, the point P may be carried say ten times round the perimeter, and the reading of the roller be divided by ten. PARTIAL INTEGRATION. 295 To explain the theory of the instrument, we shall suppose the pointer P to travel round a finite area PP 1 PJP 3 (something like a Carnot cycle in Thermo- dynamics), in which PP V P 2 P S are circular arcs described round as centre, and P X P Z , P 3 P, are arcs round A v A as centres ; in this way we analyse the motion due to the joints and A separately. Let OA=a, AP=b, AR=c; and let the direction of a positive motion of the roller, as marked by the gradua- tions, be that which on a right-handed screw would give a motion in the direction AR. » Drop the perpendicular 01 from on AR. (i.) Fix the joint A, and move P to P x by rotation round on the circle PP X through an angle 6; the angular velocity dd/dt will give to the roller R the component velocities 01. dd/dt in the direction IR and IR . d6/dt perpendicular to IR ; the first component drags the roller over the paper, and the second com- ponent makes the roller turn with circumferential velocity IR . dd/dt ; and therefore the whole travel of the roller, or its graduations, will be IR . 6. (ii.) Fix the joint 0, and move P x to P 2 by rotation round the joint A 1 through an angle tp ; the angular velocity d$\dt of AP will communicate a circumferential velocity — cd$\dt to the roller ; and the travel of the roller will therefore be c, backwards. (iii.) Fix the joint A, and move P 2 to P 3 by rotation round through an angle 6 ; the roller will now move backwards, and its travel will be I t R . 6. (iv.) Fix the joint 0, and move P 3 to P by rotation round A through an angle 0; the travel of the roller will be C0, forwards ; this cancels the travel of (ii.). 296 PARTIAL INTEGRATION. In completing the finite circuit PP-JP^P^ the total forward travel of the roller will then be (IR — I^Q. But the area PP 1 P 2 P S = area PP&Q = sector OPP-y — sector OQQ± = l(0P*-0Pi)6 = \{OA^ + AF i + 2AI.AP-OA i -AP i -2AI 1 .AP)6 ^(AI-AIje^IR-IjKfl = b times the travel of the roller. Thus by altering the length of b by an adjustment of the instrument, which allows the arm AP to slide in the sleeve AR and be clamped, the area can be read off in any required unit, say the square inch or square centimetre. Any irregular area, such as for instance an indicator diagram or the cross section of a ship, must be supposed built up of infinitesimal elements formed in the same manner as PPJP^P^; and will be read off when the pointer P completes a circuit of the perimeter, both joints being now free to turn simultaneously. When I coincides with R, the roller will not turn, and then P describes a circle called the zero circle, represented by the middle dotted circular line (fig. 45) of radius JiORt+RP 2 )^ s /{a*-c*+(b+cf} = s /(a?+b 2 +2bc). When I lies on the same side of R as A, the travel of the roller is reversed, but in a complete circuit the reading is unaltered. If however the origin is taken inside the area to be measured, the area of the zero circle must be added to the reading of the roller. Prof. G. B. Mathews explains the theory of the Plani- meter in a slightly different manner, by PARTIAL INTEGRATION. 297 (i.) moving OA to 0A V and P to P v keeping AP parallel to itself; and then the curvilinear area APP 1 A 1 = b times the travel of the roller ; (ii.) moving A 1 P 1 to A 1 P 2 by rotation round A 1 ; (iii.) moving 0A X to OA and P 2 to P 3 , keeping A-^P^ parallel to itself; and then the curvilinear area A A 1 P 2 P 3 = b times the travel of the roller, backwards ; (iv.) completing the circuit by moving AP S to AP by rotation round A, when the sector APP 3 = sector A-yP-^P^, and the travel of the roller cancels the travel in step (ii.). Therefore the area PP 1 P 2 P 3 = area APP 1 A 1 — area A^P^P^A = b times the travel of the roller. The end A may be guided in a slot of any form and the area will be read off as before ; a straight slot is often employed, with the advantage that the pencil P can then cover a greater area ; and with appropriate mechanism can be made to register the moment of the area, and its moment of inertia about the straight line of the slot. (J. F. Branrwell, British Association, 1872 ; H. S. Hele Shaw, Proc. I.G.E., 1885.) Examples. (1) Prove that the area is ir(Bc — bC) of the curve, an ellipse (Ex. 4, p. 203), given by x = a + b cos + esin 6, y = A+Bcos6+Osm6. (2) Trace and find the area of the curve (r — a cos 6) 2 = a 2 cos 26. (3) Prove that the volume cut off from the surface z n = Ax 2 + %Hxy + By 2 by the plane z=c is n/(n+l) of the cylinder on the same base. 298 PARTIAL INTEGRATION. 141. Functions of three or more Independent Variables. A function of three independent variables, x, y, z, de- noted by f(x, y, z), may be supposed to represent some function of the position of a point in space whose co- ordinates are x,y,z; for instance, the density or tempera- ture or pressure at the point. Then i{x, y, z) = G,& constant, would imply a relation connecting x, y, z, and would be the equation of a surface; for instance, a surface of equal density, or pressure. If V denotes the volume contained in a closed surface S, then V=fffdxdydz, the integration including all the infinitesimal brick shaped elements of volume dxdydz (fig. 43) contained in S. When the density p within the surface S is variable and a given function of x, y, z, then the mass M contained by the surface 8 is given by the triple integration M=fffpdxdydz, so that ^-^=P, and the mass is the space integral of the density p throughout the volume V; while x, y, z, the coordinates of the centre of mass, and k X) k y , h z , the radii of gyration about Ox, Oy, Oz, are given by xM=fffxpdxdydz, yM=fffypdxdydz, zM=fffzpdxdydz ; M h 2 =fff(y i + z 2 )pdxdydz, Mk* =///{z 2 + x z )pdxdydz, Mh z 2 =///(??■ + y 2 )pdxdydz. Functions of more than three independent variables cannot be interpreted geometrically without the introduc- tion of the fiction of space of more than three dimensions, a thing which is inconceivable. PARTIAL INTEGRATION. 299 A function F(t, x, y, z) of four independent variables t, x, y, z, may however be interpreted, as in Hydrodynamics, as representing the velocity, or density, or pressure, at the time t at a point in space whose coordinates are x, y, z. 142. We have used the words mass and density; the mass of a body is the quantity measured by the balance against certain standard lumps of metal, called weights in the Acts of Parliament (French, poids, German, Gewichte), the standard in this country being the Pound Weight, and in the Metric System the Kilogramme of 1000 grammes. The density of a body is defined as the number of units of mass in the unit of volume ; with British units, the density is the number of lbs. in a cubic foot of the substance, and with Metric units is the number of grammes in a cubic centimetre, or of tonnes of 1000 kilo- grammes per cubic metre. The units of length are thus the foot in the British System, and the metre or centimetre in the Metric System; while the unit of time in universal use for theoretical investigations is the second, the mean solar sexagesimal second. When therefore we speak of a time t, we mean t seconds; and coordinates x, y, z are measured in the unit of length, which is either the foot or else the metre or centimetre. For practical purposes there are only three systems of fundamental units which need be considered, (i.) the British foot-pound-second (f.p.s.) system, ; (ii.) the C.6.S. (centimetre-gramme-second) system ; (iii.) the metre-kilogramme-second (m.k.s.) system; and from these fundamental units of length, mass, and time, all other units, of area, volume, density, velocity, accelera- tion, momentum, energy, force, etc., may be derived. 300 PARTIAL INTEGRATION. *143. Spherical Polar Poordvnates. In this system of coordinates the position of a point P on a sphere with centre at is defined by 6, its angular distance from a fixed pole If on the sphere, and by tp the angle which the plane ONP makes with a fixed initial plane ; so that on the terrestrial sphere,

, called the spherical polar coordinates in space. With ON coincident with Oz, and with the plane xOz as the prime meridian, these coordinates are connected with the orthogonal coordinates x, y, z, by means of the relations x = r sin 6 cos , y = r sin 6 sin $, z=r cos 6. . If we replace rsinfl by w, then ©, + ^d, xM=///'pr s sm 2 d cos drd6d, zM=fffpr z sva. 6 cos Bdrddd^. PARTIAL INTEGRATION. 301 *144. Space, Surface, and Line Integrals. Consider a fixed closed surface S, and a function X of the coordinates x, y, z of a point in space. Then in the triple integration extending throughout the volume enclosed by the surface S, called a space integral, Jjff^*dyte=Jf( - X 1 +Z 2 -X 3 + . ..)dydz, where X v X 2 , X s , ... denote the values of X where a point moving from -oo tow parallel to the axis Ox succes- sively enters and leaves the interior of the surface S. Denoting by l v l v l z ,... the cosines of the angles which the outward drawn normals of the surface S at these points make with Ox, then dydz = — l 1 dS 1 = l % dS z = — l s dS 3 =..., supposing the infinitesimal prism on the base dydz parallel to Ox to cut out the elements of surface dS v dS 2 , dS s , ..., on entering and leaving the surface S. Therefore, denoting the element of volume by dV, JJf^ y=Jjf(h^xd8 t + kX^dS, + ...) =fflXdS. . .(i), the double integration extending over the surface S, and this is called a surface integral; so that a volume integral can always be expressed as a surface integral. Similarly, with Y, Z other given functions of x, y, z, where m, n denote the cosines of the angles the outward drawn normal of the surface S makes with Oy, Oz. Therefore, by addition .#(S+f+!^-#< !X+ " F+ "^ <"■>■ Thus V=fffdxdyz = lff(lx+my+nz)dS. 302 PARTIAL INTEGRATION. For instance, integrating over the surface of the ellip- soid, p denoting the perpendicular from the centre upon the tangent plane, JpdS = 4nrabc, and JdS/p = §Tr(bc/a + ca/b + ab/c) ; and, as an exercise, the student may calculate JdS/p and JdS/p 3 for the hyperboloids, as -well as for the ellipsoid. Again, suppose X, Y, Z are the component forces per unit of volume acting throughout a fluid at rest, in which the pressure at any point is represented by p ; then the equation of equilibrium of the fluid within the closed surface 8, obtained by resolving parallel to Ox, is fffXdV=fflpdS. But changing the surface integral into a space integral, Jf^-MJ^r- so that -~ = X; and similarly ^=Y,^ = Z: dx J dy dz or dp = Xdx + Ydy + Zdz, so that the space variation of the pressure of a fluid at rest in any direction is equal to the component force per unit of volume in that direction ; and surfaces of equal pressure are cut orthogonally by the lines of force. As another illustration, we may suppose X, Y, Z to represent the components of flux (estimated with British units in lb. per square foot per second) of a fluid in motion ; then Jf(lX +mY+nZ)dS represents the number of lb. which is flowing out across the surface S per second; while if p denotes the density, in lb. per cubic foot, of the PARTIAL INTEGRATION. 303 fluid at any point of the interior of S, so that the mass M within 8 ia JffpdV \b., then dM/dt represents the rate of increase, in lb. per second, of the quantity of fluid inside S. Equating this gain and loss, ■dMldt+J/(lX+mY+nZ)dS=0; and, replacing by space integrals, leading to the equation of continuity in Hydrodynamics, when we replace X, Y, Z by pu, pv, pw, where u, v, w ■denote the components of velocity of the fluid. In a plane, dA denoting an element of the area A, of which ds represents an element of length of the closed •contour s, equation (ii.) becomes M& + %y u= f« z+m7y *' ; « thus expressing a surface integral by a line integral; and I = dy/ds, m = — dxjds ; jot, if F(cc, y) = is the equation of the contour, «-i/V{(D ,+ (f)> »=f/v(©'+(sn- A similar theorem connects the surface integral on a •curved surface S, which is a portion of a closed surface, with a line integral round the edge of S ; #Kf-S)+KS-SMf-f)}^ =/(^4+4)* <*•> •(Maxwell, Electricity, i., p. 25), this surface integral vanishing, by equation (ii.), for a closed surface. 304 PARTIAL INTEGRATION. For a plane surface, we may take 1 = 0, m = 0, n=l ; and now, as in (iii.), *145. Green's Theorem. ' Now suppose TJ and V are given functions of x,y,z; then from equation (i.), integrating by parts, and therefore, denoting by — y 2 the operator M ox 2+ oy 2+ dz 2 ' w_ viy + -du ?w + vu ?E)dv ox dx oy oy dz dz J =ff u '^ d8+ Jlf u ' A2UdV ' (vL) ' and therefore, by symmetry, =JT u ^ ds +ffl™ u ' dv > (viL) ' , jdU , dV , dU dU where I— + m—~+ n — =—, ox ay az ov representing the rate of growth of U in the direction of the outward drawn normal of 8. Equations (vi.) and (vii.) constitute Green's Theorem, a theorem of great use in the mathematical theories of Electricity and Magnetism. (An Essay on Electricity and Magnetism, by G. Green ; edited by N. M. Ferrers.) PA R TIA L INTEGRA TION. 305 *146. Change of the Variables in Space Integrals. Generally in changing from x, y, z to any new. inde- pendent variables u, v, w, we may consider that space is divided up into elements of volume bounded by the surfaces for which u, v, w are constant; and now the element of volume will be changed from dxdydz to \ ' V' — (dudvdiv, a(u, v, w) ,}' ' *' — ( denoting the determinant o{u, v, w) dx dy dz du dii,' du, dx' dy dz dv' dv' dv dx dy dz diu "dw' dw called the Jacobian of x, y, z with respect to u, v, w. Denoting by ds, the element of length, then ds 2 becomes changed from dx 2 + dy"- + dz 2 to -dw) +(^dw+ ... ) +(~du + . dx -, dx -, dx , \ 2 (dy 7 . \' 2 . (dz . —du + —dv + „ du dv dw = A 2 du 2 + B 2 dv 2 + G 2 di, \du " ' ' V ' \du" ■ 2Ddvdw + lEdwdu + 2Fdudv, a 2 _ 5^! + ^y 2 + f^i b—^ x ^ + "^y ^y + ^ z ^ z du 2 du 2 du 2 ' " dv dw dv dw dvdw' It is convenient to choose the new variables u, v, w, so that the corresponding surfaces cut at right angles, and then D, E, F vanish ; while 1 Su 1 dy 1 dz A du A du A du represent the cosines of the angles the normal to the surface u makes with the axes Ox, Oy, Oz. 306 PA RTIA L INTEGRA T10N. But with x, y, z as independent variables, and denoting 3m- 2 dv? du 2 , , „ dv 2 ,79 dw 2 , , , 3^ + oy + a?" b y V. ^ + - by V. ^ + - by V; ,, 1 3m, 1 3tt 1 3k then 7- =— , ,- =— , t- ^r- /ij 3a: /ij 3y Aj 3z are also the cosines of the angles which the normal to the surface u makes with Ox, Oy, Oz; so that 1 3ti_ 1 dx 1 3u_ 1 dy 1 3m, _ 1 dz Aj dx A du' h x "dy A du' h x 3s A du' and similarly, 1 dv _ 1 dx 1 dv _ 1 3y 1 3v _ 1 3z '~0 dv' .J. 3s. A 2 3a; £ 3v' h 2 dy B dv' h 2 dz 1 3tw _ 1 dx 1 dw_ 1 dy 1 3w_ h s dx~Vdw' A 8 3y G dw' h s dz Also, since cZu, = du , , 3it , , 3m- 7 rf-V = —(foe + ^ <% + rr- 7 , dw 7 , 3w 7 w = — cte 4- — - dy + — - cfe ; 3a; 3i/ 3s therefore, multiplying these equations by 1 du 1 dv 1 3ttf A7 2 3a? V^®' A/3^' ,,,. , 1 3u 7 , 1 3w, , 1 3w, and adding, ds^ ^dn + ^ 2 ^dv+^ 2 -div; so that^r— = r-9 ^-> •■• J or Ah,=Bh 9 = Ch„ = l. 3u hf dx *■ i i Now denoting by ds v ds 2 , ds 3 , the elements of the normals, intersections of the surfaces u, v, w, dSi = Adu, ds 2 = Bdv, ds s = Cdw ; so that g^ = ABC, and ^hJ£l hhA=x/ABQ 3(w, v, w) d(x, y, z) 1 2 s ' PARTIAL INTEGRATION. 307 Now if V denotes any function of x, y, z, or u, v, w, then 372 -QY2 dV 2 _ 2 3T^ 2^0. & 2^! 3x 2 + dy 2 + dz 2 ~ * 3u 2 + 2 3v 2 + fts 3w 2 ' also, which are readily proved by taking the axes parallel to the normals to u, v, w at the point. With spherical polar coordinates r, 0,

2 , so that A = \, B — r, C=r%mQ ; v r 2 3rV 3r / Vsinfl 30Y a 30/ r 2 sin 2 a^ *147. Confocal Quadrics. A familiar instance occurs with confocal quadric sur- faces, where, with X, /*, v for new variables, « 2 + X 6 2 + X c 2 + X 'a 2 + / u 'a« + v + ""~ ' whence, by solution, ( a 2 + A)(a 2 + M )(q 2 + „) 2 _ . (a 2 -6 2 )(a 2 -c 2 ) ■■ ^ -•••» *-—, and j« (\-/i)(X-y) £2= ™_ and ^ _ 4(a 2 + X)(6 2 + X)(c 2 + X)' jD ■••.0-.... Denoting by p the perpendicular from the centre on the tangent plane of the surface X, then we can show that ± _x 2 y 2 z 2 3X_ 2p 2 x ,_ p 2 (a 2 + X) 2 (6 2 + X) 2 (c 2 + X) 2 ' 3a; a 2 + X' "■' V-*P - 3 2 X . 3 2 X , 3 2 X a J 1,1,1 and 3^ + V + 3^~ 2p A^+x + & 2 +x + c 2 +x; and thence that F=fX will satisfy the condition y 2 V=0, provided fX-q/"{(a 2 + X)(& 2 + X)(c 2 + X)HdX. 308 PARTIA L INTEGRA TION. Examples. (1) Interpret geometrically the equations fr = 0, W = 0, i

; and that the generating lines of (ii.) are given by cc-csecusechwcosw _ 2/-csecu.seclm$imc; _ 2-ctanwtanhw sechvsin(u + w) -secht;cos(?4±w) ~ Itanhv ' the angle between them being 2cos~ 1 (cosusechi>). Write down the corresponding equations for a system of confocal prolate spheroids and hyper- boloids of two sheets. (3) Prove that the equations c^fc + S TnPP = 8a(aCOsll^0 - a, ) <*■>• — ^j . 91 =8a(acosh'y — x) (ii.), cos^ sin^v v ' \ " sTnh^ + coi^ =8a(aCoshw + a: ) ^ represents a system of confocal paraboloids inter- secting at right angles in the point PARTIAL INTEGRATION. 309 x = a(cosh u + cos v — cosh w), y = 4a cosh Jw, cos £u sinh Jty, = 4a sinh £u sin \v cosh £ta Determine A,B,G for this system, and the equa- tions of the generating lines of (ii). Prove also that y 2 (u, v,orw) = 0; as also in Ex. 2. (4) Verify that 1/r, , log tan \d, and their products are annihilated hy the operator y 2 - Also r _2 cos 6, r~ 2 sin 6cos(

where x x = r sin ^sindj... ,sin 0»- -1, M.j = r 2 , x 2 = r sin ^sinfr,... cos &n -1, «2 = r 2 sin %• a; 3 = 7' sin ^sin 2 ... cos $« -2, M 3 = r 2 siu 2 1 sin 2 2 , cc n _i = ?'sin 6-fios 6 2 , ii n _i = r 2 sin 2 1 sin 2 2 ...sin 2 n _ 2 , x n = rcos6 v W r =r'"'- 1 sin' l - 2 1 sin m - 3 2 ...sin0 n _ 2 . Examine the case of n = 3. Prove also that 1 dx 1 2 3* 2 ' ' "dx n dr ' 310 PARTIAL INTEGRATION. 148. Quantics. A rational integral homogeneous algebraical function of the n th degree in m variables x,y,z, ... is defined to be " a function in which the sum of the indices of the variables in each term is constant and equal to n, the indices being positive integers" ; such a function is called a quantic, and is denoted by (x, y, z, ...)". Thus (x, yf, or (a, b, c, .. .){x, y) n represents the binary quantic in x, y of the 71 th degree ax n + nbx n ~ hj + ^n(n — 1 )cx n ~ 2 y 2 + Denoting the general quantic by u, then ~du du , du , This is proved for each single term of the quantic, say AxVyQz r ..., where p + q + r+ ... =n ; for, denoting this term by v, then dv dv dv . dv . dv , dv , and x-- — \-y—+z—+...=nv. dx "dy dz More generally d . 3.3 2 X 1Tx +y o\ +Z a\ + ~) « = *(«" l)(»-2)u, and so on; in which it is important to notice that the expression in brackets on the left hand side must be ex- panded by the Multinomial Theorem as if x, — , y, — ,... were independent algebraical quantities, and then u sup- plied in the partial derivatives. PARTIAL INTEGRATION. 311 / 3 3 \ k For if (x- — \-y---\- „\ * and thence generally d n Fy 7s "?\ 7\ \ Therefore, expanded in powers of a; by Maclaurin's Theorem, so that the coefficients of the powers of x are derived by the repeated operation of this process is called Arbogast's Method of Derivation. In Arbogast's second method of Derivation ft/ = f (a + bx + ex 2 + dx s + ex* + . . . ), is expanded in powers of x, in the form ><(^^ + «| + ^ + ...)V = A + Bx+Cx 2 +Dx s +Ex i + ..., suppose ; and now we find PARTIAL INTEGRATION. 313 A=fa B = f'a.b C=?a.c+f"a.~ D = ?a.d+i"a.bc+i'"a.% E=fa.e+ i"a(bd + £c 2 ) + f ' "a . \Wc + i""a . % and so on, by a simpler mode of Derivation. For example, applying this method to Ex. 3, p. 234, tf(l + x) = exp(l - \x + \v? - \x* + ix* . . . ) ; here a = l, b= - £, c = £, d= —$, e = \, ... , and fa = fa = f "a = =e; thence A = l, B=~h C=H, D= - T V, E=$m *150. Theorems of Lagrange, Laplace, and Burmann. Put y = a + bx+cx 2 + dx 3 +..., and y. Then !_„+.,,£ !_, + .,£; and thus -^ = Ay-^-. 3x ^3a. Then if « = %, du du dy , „, cty , 3u SjTIQ. S1I1C6 -dx\ y daJ y dxda^~ y dxda da\ y 'dx/' therefore 3a; 2 ^(""D-K^-D-aM}' 314 PARTIAL INTEGRATION. and generally, by Induction, r dx n 3a»- 1 \^ 2/ ^ 3a J ' Now make ce = 0, therefore ?/ = a, and u = ia; and this is called Lagrange's Theorem,. Thus, for instance, putting iy = y, we find , 1 db 2 , 1 1 ; and their equation will then be When e=l, the conic is a parabola, and the asymptote is given by m = 0, c = qo ; so that it lies at an infinite distance, and does not therefore satisfy the definition of an asymptote. If u n has a factor x, then to determine the corre- sponding asymptote, we must expand x in descending powers of y, in the form x = c'+p'y- 1 + q'y 2 +... • or we may put, in general, x = m'y + c+py- 1 +..., and determine m', c', p\ ... as before. 320 CURVES IN GENERAL. Sometimes the expansion of y in descending powers of x or x in powers of y must be written y= Ax 2 + mx + c+ px~ 1 + ..., x = A'y 2 +m'y + c'+p'y- 1 + ..., and then y= Ax 2 + mx + c, or x — A'y 2 +m'y + c is called a parabolic asymptote ; and so on. When it is possible to obtain y explicitly in terms of x, or x in terms of y (§ 13) from the implicit relation f(x, y) = 0, the asymptotes are then readily determined by expanding by the Binomial Theorem and other algebraical operations in descending powers. Thus if x 3 +y 3 = a 3 , then y = £/(a s — x 3 )= — x(l — a 3 x~ 3 ft = — x + \a 3 x~ 2 +....; or x = ^/(a 3 -y 3 )= -y{l-a 3 y- 3 )^ = -y + \a 3 y- 2 +.... Also if x = a makes y = od , or y = b makes x = oo , then x — a 4 +Z> 4 , (ii.) a i +¥>c i >a i , (iii.) c^a*, (iv.) a 4 > c 4 > b\ (v.) c 4 = 6 4 , (vi.) 6 4 > c 4 . (6) Draw the curves (i.) tan " 1 x/a + tan ' 1 y/b = tan _1 o; (ii.) tanh" 1 a;/a + tanh" 1 3//6 = tanh'' 1 c; (iii.) (tana;) 2 + (tan2/) 2 =4, or 3. 322 CURVES IN-GENERAL. 153. Polar Coordinates. In discussing properties of a curve connected with straight lines radiating from any origin 0, it is convenient to change to polar coordinates (r, 6) by putting (§ 22) x = r cos 6, y = r sin d. Substituting in the rational integral equation (A), p. 315, u r- n + (A cos + £sin 0)r-"+ 1 + (a coa 2 6+2k sin d cos + b sin 2 0)r-"+ 2 + ... = ; an equation of the n th degree in r _1 ; so that a straight line cuts a curve of the n th , degree in n points, some pairs of which however may be imaginary. ■ Denoting by r v r 2 , r 3 , ... r n the roots in r of this equa- tion and their harmonic mean by r, then r- 1 = (r 1 - 1 + r 2 - 1 +...+r n - 1 )/n=-(Acos6 + Bsm6)/nv , or Ar cos 8 + Br sin 6 + mt = 0, Ax + By + nu = 0, or u x + nu = 0, the equation of a straight line, the locus of a point P, such that OP is the harmonic mean of 0P V OP 2 , ... OP m where P v P 2 , ... P n are the n points in which the straight line cuts the curve ; this straight line is called the polar line of 0, by analogy with the polar line of a conic section, with which it coincides when n = 2. The straight line ^ + ^,, = is the locus of P when r- 1 = r 1 - 1 + r s - 1 +...+r n ~\ In the same way the polar conic of 0, . i 1X . n(n — 1) n u 2 + (n - 1 )u x + K 1 2 -\ = {), is the locus of P when s Vop opJkop op J 0; and so on. CURVES IN GENERAL. 323 Writing, as usual, u ibr 1/r, then since for the points of intersection with the curve, 2it = — (A cos 6 + B sin 6)/nu , therefore ■ 2(d 2 u/de 2 + u) = 0. But d 2 u/d6 2 +u = c cosec 3 0, by § 93, where c denotes the curvature, and the radial angle at which the vector OP cuts the curve ; so that 2c cosec 3 <£ = 0, at the points of intersection of OP with the curve. (Dr. Kouth, Quarterly J. of Math., xxiv., p. 257.) We have found that in a central field of force (§ 84), the acceleration to the centre P = h 2 u\d 2 u/dd 2 + u), so that the orbit is a straight line, and d 2 u/d6 2 +u = 0, when P = 0; but the orbit is concave to the origin when P and therefore d 2 u/d6 2 + u is positive; and convex when they are negative; and at a point of inflexion, where the curve changes from concavity to convexity, or vice versa, d 2 u/dd 2 +u vanishes and changes sign. Definition. — A curve is said to be concave with respect to a point or line when it lies on the same side of its tangent as the point or line in the neighbourhood of the point of contact ; and convex when it lies on the opposite side of its tangent ; and at a point of inflexion the curve crosses its tangent. In interpreting the above, some of the p's must be negative, and we shall take p as positive or negative according as the curve is concave or convex to the origin. Incidentally we deduce that the three points of inflexion on a cubic lie in a straight line. 824 CURVES IN GENERAL. 154. Equation of the Chord, Tangent, Asymptote, and Normal of a Curve in Polar Coordinates. It is convenient to employ u, the reciprocal of r (§ 23), and now the equation of a straight line can be written u= Acqs 9+B sin 6; or, more generally, u = Acos(6 — a) + Bsm ($ — a), equivalent in Cartesian coordinates to 1—A(x cosa + j/sin a)+ B(y cos a — a; sin a), an equation of the first degree is x and y, and therefore the equation of a straight line. To find the equation of the chord of the curve, whose equation is u = id, which passes through the two points whose vectorial angles are a±/3, we must determine A and B from the equations f( a + /3) = A cos /3 + B sin fi, f (a - /3) = A cos /3 - B sin /3 ; and therefore , f(q + /3) + f(«-/3) f(q + /3)-f(a-/3) , 2co Sj 8 ' 2sin/3 so that the equation of the chord is f(a+/3)+f(a-/3) . a x,fi a+j8)-f + sin(0 — a)}. sin a Examples. (]) Find the asymptotes of (i.) r = asec0, 6cosec0, «sec0 + 6cosec6), asec(8 — a), b cosec (8-/3), a sec (8 — a) +b cosec (8-/3). (ii.) r = a tan 8, a cot 8. (iii.) r = a(sec 8 — cos 8), a(sec + tan 8) ; and deter- mine their equations in x and y. (iv.) r = sec 28, cosec 28, sec 38,... sec n8, cosec nd. (v.) r 2 =a 2 sec 28, (§ 22) at correspond- ing points. Therefore, denoting Y by p, and the angle AO Y by w, cot 7 — sin &>7o- doo duS to ctar The relation p = fa), connecting p and w for the curve .4P, is called its tangential polar equation ; and p = fw is the polar equation of the pedal curve A Y, with p and to as polar coordinates; and then q = i\w+\-7r) will be the polar equation of the pedal of the evolute BQ. The original curve AP is the envelope (§ 105) of x cos to + 1/ sin co = fto ; and with reference to the curve A Y, the curve AP is called the first negative pedal of A Y with respect to 0. Thus the first negative pedal of a circle is an ellipse or hyperbola, according as the pole is inside or outside the circle; the first negative pedal of a straightlineisaparabola. For instance .the directions of motion of the parts of a rope in contact with a moving pulley are at any instant tangents to a conic, of which a focus is at the instantaneous centre of rotation of the pulley. We may take the pedal of the pedal, positive or negative, with respect to the pole 0, any number of times n, and then we obtain a curve called the n th pedal, positive or negative. Suppose rays of light proceeding from are incident on a reflecting curve AP; the reflected ray will pass through Z on OY, produced so that 0Z=20Y, and will be parallel to YI, and therefore normal to the locus of Z ; the envelope of the reflected rays, called the katakaustic (§ 105), is therefore the evolute of the locus of Z, a curve double the size of the pedal curve A Y. CURVES IN GENERAL. 329 Fig. 47 156. Limacons, the Pedals of a Circle. Consider the pedal of a circle, centre C and radius a, with respect to any point 0, where OC=b. Then p = 07=17+01 =a + b cos oo, the equation connecting the polar coordinates p and w of the locus of 7, and this curve is called a limacon. The limacon is called a conchoid of a circle, because it is described by producing the vector of the circle r = &cos0 a constant distance a. The equation of the limacon may be written p= ±a + b cos w, corresponding to parallel tangents of the circle ; so that the chord of the limacon through is of constant length 2a. If b < a, is inside the circle, and the pedal consists of 'a single 'oval curve (fig. 47, i.). This oval has points of inflexion, if 6 > \a. If b > a, is outside the circle, and the pedal is looped, having a double point (fig. 47, ii.). If 6 = a, is on the circumference, and p = o(l + cos to), the equation of a cardioid (fig. 47, iii.). If in fig. 8 we fix the bar PQR, and move the cross OQR, then any point fixed in OQ or OR will describe a limacon. 330 CURVES IN GENERAL. 157. Orthoptic and Isoptic Curves. If two tangents PR, QR to the curve APQ intersect at a constant angle a, in the point R, the locus of R is called an isoptic curve of the curve APQ; and if the angle a is a right angle, the locus of R is called the orthoptic curve. Thus the isoptic curve of a circle is a concentric circle the orthoptic locus of an ellipse or hyperbola is a circle sometimes called the director circle ; the orthoptic locus of a parabola is its directrix, while the isoptic locus of a parabola is a confocal hyperbola. The equation of the isoptic locus for tangents inclined at an angle a is obtained by eliminating w between x cos(w + Jot) + y sin(w -(- |a) = f(a> + £a), x cos(w — Ja) + y sin(w — |a) = f(a> — |a) ; and a = \-w gives the orthoptic locus. The normal RI of the isoptic locus at R will pass through I the point of intersection of the normals at the points P, Q; since I is the centre of instantaneous rotation of the constant angle PRQ. Examples. (1) Prove that the equation of the pedal of an ellipse with respect to the centre is £> 2 = (X 2 cos 2 a> + & 2 sin 2 a), or [x 2 -\-y 2 ) 2 = a 2 x 2 + b 2 y i - Show that the pedal has points of inflexion when b 2 /a 2 < J, or e 2 > | ; and that, when b = 0, the pedal reduces to two circles. (2) Prove that the pedal of a parabola with respect to the vertex is the cissoid p = a(seca>—cos w), or y 2 = x 3 /(a — x). (3) Prove that the isoptic locus of a parabola is a hyper- bola, of excentricity sec a ; and explain how the two branches of the hyperbola are formed. CURVES IN GENERAL. 331 (4) The isoptic locus of an ellipse is given by (x 2 + 2/2 _ a 2 - b 2 ) 2 = 4 cot 2 a(a?y 2 b 2 + x 2 - a 2 b 2 ). (5) The isoptic locus of a cycloid is a trochoid. (6) The orthoptic locus of x% + y$ = a% is \J*L a cos 20. (7) The orthoptic locus of the cardioid is composed of a circle and a limacon (Wolstenholme). (8) Prove that if p v p 2 , p & , ... are the radii of curvature of the envelopes of the sides of a polygon, whose sides are a, b, c, ... then ap 1 + bp 2 +cp s + ... is equal to twice the area of the polygon. (9) Prove that the polar equation of the curve OP of § 119, in the neighbourhood of 0, is r = 2 p e + ipp'6 2 +.... (10) Prove that the pedal of an involute of a circle, with respect to the centre of the circle, is a spiral of Archimedes (fig. 36). Apply this to the theory of a weighing machine, show- ing the weight on a dial provided with equal graduations, when the body is weighed against a fixed weight, sus- pended by a rope which wraps on the involute of a circle. 158. Roulettes. When a curve, carrying a point P fixed to it, rolls on a straight line (or any given curve), the path traced out by the poiut P is called the roulette of P with respect to the straight line (or given curve). Thus, when a circle rolls on a straight line, the roulette of a point on the circumference is a cycloid, and the roulette of any other point fixed in the plane of the circle is a trochoid (§ 21). An involute of a curve (§95) is thus the roulette of a point on a straight line which rolls on the curve. 332 CURVES IN GENERAL. Fig. 48 A remarkable analogy, pointed out by Steiner, exists between the roulette of a point with respect to a straight line and the pedal of the rolling curve with respect to the point as pole. Steiner's Theorems assert that (i.) The length of the arc of the roulette is equal to the length of the corresponding arc of the pedal ; (ii.) The area bounded by an arc of the roulette, the ordinates at the ends of the arc, and the straight line on which the curve rolls is twice the area bounded by the corresponding arc of the pedal and the vectors from the origin to the ends of the arc. For, if AP is the roulette of the point P when the curve is rolled on the straight line Ox (fig. 48), and if PM is the perpendicular from P on Ox, the tangent at I to the rolling curve, then relatively to P the locus of M is the pedal of the rolling curve with respect to P ; and therefore relatively to M the locus of P is the same curve, and the subnormal IM of the roulette is the q or dpjdw of the rolling curve, or the polar subnormal of its pedal. CURVES IN GENERAL. 333 We may suppose the pedal A'P rolled on the roulette AP, so that M is always vertically over P if Ox is hori- zontal ; and the pedal, if loaded so that the centre of gravity is at M, will rest in neutral equilibrium on the roulette, provided the friction is sufficient, or else that teeth are cut, to prevent slipping. The arc AP of the roulette will then be equal to the corresponding arc A'P of the pedal, which is Steiner'a first theorem. The arc s in the roulette is therefore the same function of y as in the pedal of p. Also if the pedal is rolled into a consecutive position so that M comes to M', and the point p of the pedal comes into contact with the point P' of the roulette, then the element MM'P'P, which is the increment of area of the roulette, is ultimately double the element MPp, which is the increment of area of the pedal, or lt(area MM'P'P)jaxe& MPp= 2 ; and therefore, by integration, the area OMPA of the roulette is double the area A' MP of the pedal, which is Steiner's second theorem. Examples. (1) Draw the figures and compare the arcs and areas of the following pairs of curves, the first curve of a pair being fixed, and the second rolling on it, so that its pole describes a straight line Ox (or Oy) ; thus illustrating Steiner's Theorems, (i.) The parabola y 2 = 2lx, and the spiral r = IB. (ii.) The circle x 2 + y 2 = a 2 , and the circle r = a cos 6. (This principle is employed in the parallel motion of Deleuil's air pump. Deschanel, Physios.) 334 CUR VES IN GENERAL. (iii.) The -ellipse (x/a) 2 + (y/b) 2 = l, and r = b cos(b6/a) or a cos(a6/b). (iv.) The hyperbola (x/a) 2 -(y/b) 2 =l, and r = a cosh (ad/b) or 6 sinh (b6/a). (v.) The exponential curve 1/ = fce^ , and the reciprocal spiral r = c/9. (vi.) The cycloid and the cardioid. (vii.) The trochoid and the limacon. (viii.) The straight line y = x tan a, and the equiangular spiral r = ae" tana . (ix.) The catenary y = a cosh x/a, and the straight line ?* = /dp, where is the radial angle and p the perpendicular on the tangent from the pole on the rolling curve. (4) Prove that if the centre of an ellipse is fixed at a distance b from a plane, and the ellipse is rolled on the plane, the point of contact will describe the Cotes's spiral r cosh (aeO/b) = ae. 159. Centrodes. When a moving plane figure slides or turns on another plane, which may be considered fixed, then a point I in the moving plane can always be found which has no velocity; this point / is called the centre of instantaneous rotation (§ 21), and the relative motion of the two planes is assigned by the position of / and by the angular velocity n of the moving plane round /. The point / will in the general case describe a curve in the moving plane and a curve in the fixed plane, and the motion of the moving plane will be given by rolling the first curve on the second ; so that any point carried by the rolling plane will describe a roulette of the first curve with respect to the second curve ; these curves described by / are now called the centrodes of the relative motion of the two planes. Another point J can always be found of which the acceleration is zero ; and now the acceleration of any other carried point P at a distance r from J will be composed of component accelerations n 2 r towards J and nr perpendicular to JP, n denoting the angular acceler- ation dn/dt. 336 CURVES IN GENERAL. Fig.49 Therefore the resultant acceleration of any point, such as P, makes the same angle a = cot~ 1 (n,'n 2 ) with JP; and the lines of resultant acceleration at any instant are equiangular spirals of radial angle a, the magnitude of the acceleration at distance r from J being «V cosec a. Denoting by R the radius of curvature PP' of the trajectory of P, by v the velocity of P, and by 9 the angle IP J, then (§ 92) the normal acceleration of P, v 2 /R = n 2 r cos 9 — nr sin 6 = nhr cosec a sin (a — 9). CURVES IN GENERAL. 337 The points whose trajectories have zero curvature, and which are therefore at this instant describing straight paths (as M in fig. 48), or rather are passing through points of inflexion on their trajectories, are obtained by putting 6 = a, and therefore lie on a circle IQJ passing through / and J, called the circle of inflexions (the circle . PMI in fig. 48) ; and this circle will touch the centrodes at /, since the acceleration at M is in the normal. Now if IP meets this circle in Q, then the curvature of the trajectory of P is 1/iJ = n 2 r sin(a — 6)/v 2 sin a = nKPQIv 2 = PQ/PI\ Thus PP' = PI 2 /PQ, IP' = ppipq -pi= pi . iq/pq, 1 PQ 1^ J^ IP' PI.IQ IQ PI' or 1 , ! _ ! _sec0 PriP' IQ ID' where IB is the diameter of the circle of inflexions, and where now denotes the angle between IP and the common normal of the centrodes at J. A point P inside the circle of inflexions will thus describe a trajectory convex with respect to I, but concave if the point is outside this circle; so that the rolling centrode, if loaded so that its centre of gravity is at P, will be in stable or unstable equilibrium with IP vertical according as P is inside or outside this circle. The diameter ID of the circle of inflexions is inferred by placing the carried point P for a moment at C, the centre of curvature of the rolling centrode at /, when it is easily seen that the centre of curvature of the roulette 338 CURVES IN GENERAL. through G will be at C, the corresponding centre of curvature at 1 of the fixed centrode ; and now, with = 0, iD C/ IC P + p" or ii) = pp'jip + p), where p, p denote the radii of curva- ture CI, IC of the rolling and fixed centrodes, reckoned . positive when the centrodes are convex to each other. Now ^.L^ec^ + A p p, and the symmetry of this relation shows that P is the centre of curvature of the roulette of P' with respect to the former moving centrode ; also that GP, UP' intersect in a point T on IT the perpendicular to IP. When a curve rolls symmetrically on an equal curve, the roulette of any point will be similar to the pedal of the curve with respect to P, but enlarged to twice the scale ; the reason being that the reflexion of the carried point P in the tangent at / is a fixed point. Suppose for instance, as drawn in fig. 49, that the fixed and rolling centrodes are equal ellipses, and that the carried point P is at one of the foci ; the roulette of P will be a circle, and the centre of curvature P' will lie at a focus of the fixed ellipse. We can pivot these ellipses at the other foci and 0', and now revolve them in contact with each other ; teeth may be cut to prevent slipping, and the revolving foci P and P' may be connected by a link to prevent separation. This mechanism is sometimes employed, and the ellipses roll on each other as if connected with a crossed parallelogram of bars OP, PP', P'O', O'O. CURVES IN GENERAL. 339 160. The Area of a Roulette. As / moves along the rolling and fixed centrodes with the same velocity ds/dt, the moving plane will turn in the time dt through an angle dm, which is the sum of the curvatures ds/p and ds/p' of the equal arcs ds of the rolling and fixed centrodes ; so that 4_/l M dt \p p')di The normal PI of the roulette of P now sweeps out an area, denoted hy {PI), at a rate which, by Guldin's Theorem generalized (§ 62), will be measured by the product of PI and of the component velocity of the middle point of PI perpendicular to PI. This component velocity, being the arithmetic mean of the velocities of / and P in the same direction, is equal to | cos or (b + c)(A)-b(R)-c(P)=mrbc(b+c); this is called Holditch's Theorem. If the closed contours described by A and P lie entirely outside each other, the bar AP can only oscillate between two extreme positions, and n = 0. CURVES IN GENERAL, 341 In using the planimeter, the pivot is generally fixed outside the contour described by P, and the joint A oscillates on the arc of a circle of radius a, so that (A) = and rc, = 0; and thus b(R) + c(P) = 0, attending to the sign of the area. Thus if P describes a circle of radius r not enclosing 0, the motion of the bars OA, AP is similar to that of the beam and connecting rod of a steam engine; and now (R) = — 7rr 2 c/6; this is independent of the length of the beam, and is therefore the same for the ordinary direct action steam engine, as applied in the locomotive engine, where A oscillates in a straight line. A new form of Planimeter has been lately brought out, in which the end A of the bar AP is constrained to move in a straight or curved slot, while the pencil P is carried round the area to be measured ; but now the area is registered by the motion of a wheel W which can slide and turn on a graduated round bar CD, projecting at right angles to AP from a point C, which we may take to be the middle point of AP. The rate at which AP sweeps out area is the product of AP into the component velocity of C perpendicular to AP ; and this component velocity is equal to the velocity with which the wheel W slides along CD ; the wheel being supposed to roll, but not to slide on the paper ; in this manner the sliding motion of the roller R on the paper of Amsler's Planimeter is obviated. Mr. Elliott has generalized Holditch's theorem by showing that the bar PAR may be replaced by an elastic thread, which stretches, but keeps the ratio b/c constant, when a similar Theorem holds. 342 CURVES IN GENERAL. Mr. Leudesdorf has extended the theorem to the relation connecting the areas of the roulettes (A), (B), (C) of any three points A, B, C with the area (P) of the roulette of any fourth point P ; and he finds that (P) — x(A) — y(B) — z(C) = mr times the rectangle on the segments of a chord through P of the circle circum- scribing the triangle ABC, when x, y, z denote the ratios of the triangles PBG, PGA, PAB to the triangle ABG. For (P)-(A) = mr(GP 2 -GA 2 ), ...; so that (P)-x(A)-y(B)-z(C) = mr(GP 2 -x . GA 2 -y . GW-z.GG 2 ) =mr{x . BC 2 +y . CA 2 +z . AB 2 ) which can be ghown to lead to Mr. Leudesdorf's result; and reduces to Holditch's Theorem when P lies in a side BC. Mr. Kempe points out that the locus of P, for which (P) is zero, is a circle ; and the locus of P, for which (P) is constant, is a concentric circle, exceeding in area the first circle by an amount proportional to (P) ; but that if n = 0, this system of concentric circles must be replaced by a system of parallel straight lines. {Messenger of Mathematics, vol. vii., 1878.) 162. The Envelope of a Carried Line. Similar theorems hold for the envelope of a carried line Nx, fixed in the moving centrode and carried round by it. The point of contact N on the envelope is the foot of the perpendicular IN drawn from I on the carried line ; and now if we denote by y the ordinate IN of the point / of the rolling centrode with respect to the line Nx, by x the corresponding abscissa on the line Nx, and by S the length of the arc described by N ; then dS dx , du> CORVES IN GENERAL. 343 so that S—x=Jyd(o=J\(c + c')yds; or S the length of the arc of the envelope exceeds x the projection of the corresponding arc of the rolling centrode on the carried line hy the moment about the carried line of the same distribution of density \{c + c') as before. Carried lines which have envelopes of the same length will therefore touch a circle, with the centre of gravity G of the perimeter as centre. Again, if $ denotes the angle between IN and the common normal of the centrodes at I, the radius of curva- ture R' of the envelope of the carried line at N is given by ,,, dS dx , ,,ds , dw> dai a r dm a = Q'I+W=Q'N, supposing NI meets in Q' the circle IQ'D', which is the reflexion of the circle of inflexions in the common tangent of the centrodes at /. Therefore, for all carried lines passing through D', R' = 0, and the envelope has a cusp lying on this circle, called for this reason the circle of cusps ; while in addition the circle is the locus of the centres of curvature of all carried lines, the centre of curvature Q' being obtained by dropping the perpendicular D'Q' on IN. The normal NI of the envelope will now sweep out area (N~I) at a rate d{Nl)_ t fdx dS\_ dx , 2 dw ~dT '- iy \dt + 7TtJ- y di + ^ 'It'' so that the area swept out by the normal JS T I exceeds the area swept out on the rolling centrode by the ordinate NI by the M.i. of the perimeter, of variable density %(c + c'), round the carried line. 344 CURVES IN GENERAL. Thus from Theorem II, p. 128, it follows that if two carried lines intersect at right angles in P, the sum of the areas of their envelopes exceeds the area of the roulette of P by the area of the rolling centrode. Carried lines which sweep out equal areas are such that the m.i. of the perimeter round them is the same; they are therefore equimomental lines, and therefore tangents of an equimomental ellipse; and by varying the area we obtain a system of equimomental confocal ellipses, by well known theorems. (Besant, Roulettes and Olissettes ; Kempe, Messenger of Mathematics ; Nature, 1878 ; Darboux, Bulletin des Sciences Mathematiques, 1878; Larmor, Nature, 1881 ; Proc. Gam. Phil. Soc, 1890; Minchin, Uniplanar Kinematics, 1882.) 163. When the fixed centrode becomes a straight line, c' = ; and the area swept out by the normal PI (fig. 48) in a complete revolution of the rolling curve is the same as that swept out by the ordinates MP, and is therefore double the area of the pedal of the rolling curve with respect to P. Now, if the rolling curve is coated with matter of linear density \c per unit of length, the mass will be l\cds = l\d 2ff w = tt; and therefore the area of the roulette of P will exceed the area of the roulette of G, the centre of gravity of this wire, by tt . GP 2 ; and consequently the area of the pedal of the rolling curve with respect to P will exceed the area of the pedal with respect to G by ^tt.GP 2 . CURVES IN GENERAL. 345 Thus, for instance, the pedal of a circle with respect to its centre being the circle itself, will have an area -n-a? ; and consequently the area of the lima§on p = a+b cos w will be 7ra 2 + |7r& 2 , while the area of the corresponding trochoid will be 2tto, 2 + irb z ; and with a = b, the area of the cardioid i p = a{\+eo&w) will be |7ra 2 , and of the cycloid 3ira 2 . In a roulette with respect to a straight line, 1 1 _sec0 R~^N + N~~p ' or NR-N i = pRcos(j>, or 11 p cos < /> R N iV 2 ' If the rolling curve is an ellipse and the carried point at a focus, then (fig. 48) p cos = N(2a - N)/a = 2N- N 2 /a, so that _i_-_. R^N a If this roulette sweeps out a surface by revolution round the fixed straight line, the curvature of this surface will be everywhere 1/a, a constant; a soap-bubble film of revolution will assume the shape of this surface, or a surface similarly generated by the roulette of the focus of a hyperbola or parabola. For instance, the roulette of the focus of a parabola is a catenary and we obtain the catenoid of fig. 16. Again, the roulette of the pole of the involute of a circle with respect to a straight iine is a parabola; and the roulette of the centre of the rectangular hyperbola r 2 cos20 = a 2 is a curve in which R = $N, or Ry — \(P, an elastica or lintearia. ■ ■ 346 CURVES IN GENERAL. 164. Epicycloids and Hypocycloids. These curves are the roulettes of a point P on the cir- cumference of a circle which rolls on the outside or inside of a fixed circle. Let denote the centre and a the radius of the fixed circle, C the centre and c the radius of the rolling circle, and / the point of contact of the circles ; then IP is the normal of the roulette of P, because I is the centre of instantaneous rotation of the rolling circle (fig. 50). y Fig.50. Draw the diameter PCQ of the rolling circle, and sup- pose Q originally in contact with the fixed circle at B, and that P is then at A ; A is then an apse and B a cusp of an epicycloid (§ 104). CURVES IA T GENERAL. 347 Denoting the angle xOI by 6 (radians), then the arc JB = a8 = &xcIQ, so that the angle ICQ = ad/c; and the coordinates of P in terms of 9, for the epicycloid, are x = (a + c)cos d+c cos(l + a/c)6, y = (a + c)sin 6 + c sin(l + a/c)6 ; and for a hypocycloid, change c into — c. Thence we find, for the epicycloid, ^ = 2(a+ C )cos^, and integrating, the arc AP .c. . . ad = a = 4-(a+c) a m^; while the ar c 50 = 4- (a + c) vers 1- . a 2c With the notation of p and to of § 154, p = (a + 2c)cos(a0/2c), and o> = (1 + a/2c)6 ; so that j3 = (a + 2c)cos{ao)/(a + 2c)}, the polar equation of the pedal of an epicycloid, of the form r = b cos m6. ., u-r^- dp . aw . a# Also PY= — /- = asin — — T - = asm- ?r - ao) a + 2c 2c so that r 2 =0Y 2 +PY 2 = (a + 2c) 2 cos 2 ^r + a^sin 2 -^- K ' tc 2c 9 i a / , \ 9^0 4c(a + c) o or r 2 — or = 4c(a + c)cos 2 — = . v , ' ' p% 2c (a + 2e) 21 of the form r 2 — a 2 = (1 — m 2 )p 2 , m = a/(« + 2c), the relation connecting p and r in the epicycloid; proving incidentally- that the roulette of the centre of an epicycloid with respect to a straight line is an ellipse. If m = 0, then c = oo , and the curve becomes the in- volute of a circle, in which p 2 — r 2 — a 2 . 348 CURVES IN GENERAL. Again s^a+e)^—^, of the form s = I sin m\p-, with i//- for to, £ for 4c(a+c)/rt, and ?n for a/(a + 2c), as in § 104. And p = -r-= , I cos , , =^-=(l-m 2 )p. r da> a + 2c a + 2c dp l 165. The Teeth of Wheels. By cutting teeth on wheels we can make them engage and transmit power without slipping ; we thus secure the condition called perfect roughness by the theoretical mathematician ; perfect smoothness between two bodies on the other hand is sought practically by the inter- position of wheels. Suppose 0' to be the fixed centre of a wheel of radius 0'I=a', which is to be made to revolve in contact with the wheel of centre and radius a, without slipping. If the circle, centre G and radius c, rolls inside" this circle, centre 0' and radius a', and describes the hypo- cycloid A'P, then if the epi-- and hypo-cycloids AP and A'P start with the vertices A and A' in contact, the two curves will roll and slide on each other (fig. 50) so that the common normal at P passes through /, and therefore the constant velocity ratio of the wheels is maintained ; for this reason the wheels of teeth are shaped by epi- and hypo-cycloids. Only a small portion of each curve in the neighbour- hood of a cusp is made use of to form a tooth ; and the tooth is completed on the circle 0' by a portion of an epicycloid, and on the circle by a portion of a hypo- cycloid, each described by the ^ rolling of a circle of the same radius c'. CUR VES IN GENERAL. 349 For instance, if c' ' =\a, then the hypocycloid in the circle is given by x = 0, y = a sin 6 ; so that the hypocycloid degenerates into a straight line, a diameter; and the inside portion of the tooth is straight and radial. Any number of change wheels of a lathe may be made to work together indiscriminately, provided the teeth are shaped by rolling circles of the same radius c. In tig. 50 we have taken a/a' ' = ■£, and c = c ' = \a\ and four teeth on the circle engage with six teeth on 0'. When we make c and 0' infinite, the teeth are shaped by involutes of the circle and 0' ; involute teeth have the advantage of preserving the velocity ratio of the wheels unchanged for variable distances between the centres of the wheels, and are employed in rolling mills. When the radius a' is made infinite, the corresponding wheel becomes converted into a rack, and the teeth on the rack are shaped by cycloids. Sometimes to ensure greater regularity of working, helical teeth are employed, and now the tooth of one helix on the cylinder of radius a works with the tooth of the helix of equal pitch on the cylinder of radius a' ; so that when one helix rolls on another of equal pitch on a parallel axis, any point of the helix describes an epi- cycloid. (MacCord, Kinematics ; G. B. Grant, Odontitis). 166. The Double Generation ofEpi- and Hypo-cycloids. Produce PI both ways to meet the circles and 0' again in H and H', and draw EPE' parallel to 00' to meet OH and O'H' in E and E\ Then E and E' are the centres of circles, of radii a + c and a' — c, which touch each other at P and the circled and 0' at H and H' ; so that the epicycloid AP and the 350 CURVES IN GENERAL. hypocycloid A'P can be described by the rolling of these circles on the circles and 0'; we thus perceive that there is a double mode of generation of the epicycloid and hypocycloid. Relatively to the circle 0', any point on the circum- ference of the circle describes an epicycloid; and a hypocycloid if it is made to roll inside the circle 0'. The envelope of the diameter QGP is another epi- cycloid, described by a rolling circle of radius \c, as is readily perceived when we notice that the point of contact of the envelope is at the foot of the perpendicular drawn from I on PQ. The envelope of any other carried straight line, say parallel to PQ, will be a parallel curve to the epicycloid, the envelope of PQ, and will therefore have an epi- cycloidal evolute. 167. E'pi- and Hypo-Trochoids, or Bicircloids. A point fixed in GP at a distance he from G will describe a curve, given by x = (a + c)cos 6 + he cos(l + a/c)6, y = (a + c)sm 6 + he sin( 1 + a/c)6 ; these curves are called epi- or hypo-trochoids, and some- times bicircloids. The relative orbit of two planets is a bicircloid, if the planets describe circles round the Sun; figures of the relative orbits of the Earth and the different planets, drawn mechanically by Mr. Perigal, are given in Proctor's Geometry of Cycloids. For if a, /3 denote the (mean) distances from the Sun, and n, m the mean motions, the relative orbit is given by CURVES IN GENERAL. 351 x = a cos nt+/3 cos mt, y = a sin nt + /3 sin mi ; where n 2 a 3 = m 2 /3 3 , by Kepler's Third Law (§ 179); so that a + c — a, Icc = f3, and l+a/c = m/n=(al/3)* ; or a = a — /3%/a i , c = /3*/a*. When c = «, or m = 2n, the bicircloids are limagons (§ 156), as is otherwise geometrically obvious. Thus if we assume that the period of Mars is two years, the relative orbit of the Earth and Mars will be a limacon. Examples. (1) Prove that if an epicycloid rolls on a straight line, the centre will describe an ellipse. (2) Prove that, if a helix rolls on a straight line, any point on the helix will describe a cycloid. (3) The shadow of a helix on a plane perpendicular to its axis, thrown by parallel rays of light, is a trochoid. Find when the trochoid will be a cycloid. (4) The path of a steamer turning uniformly in a current will be a trochoid relatively to the land. (5) wShow that a variable velocity ratio of two wheels can be attained by cutting teeth in equiangular spirals of the same radial angle. (6) If an equiangular spiral rolls on a circle, the pole will describe an involute of a circle. (7) Show that the path of the Moon relatively to the Sun has no points of inflexion. 352 COR VES IN GENERAL. (8) The isoptic locus- of an epicycloid is an epitrochoid (Wolstenholme, Proc. London Math. Soc, vol. iv.). If the isoptic locus passes through the centre, its equation is of the form r = bsm{a6/(a + 2c)}. (9) Prove that the epicycloid has (i.) one cusp when c = a, 2a, 3a, 4«, ...; and then m = \, \, \, ... ; (ii.) two cusps when 2c = a, 3a, 5a, ...; and then m — hhh---'> ^ n( i draw them. (10) Prove that the equation of the Tricusp hypocycloid can be written (i.) p = %acos3w, (ii.) s = f a sin 3i/r, (iii.) \x 2 + y 2 ) 2 + %ax(x 2 - 3y 2 ) + 2a 2 (x 2 + y 2 ) - £a 4 = 0, (iv.) r i +%ar s cos36+2a 2 r 2 -$a i = 0. Prove that if the tangent at P meets the Tricusp again in Q and R, (i.) the length QR = ia, (ii.) the tangents at Q and R intersect at right aDgles in a point T on the inscribed circle of the Tricusp, (iii.) the normals at P, Q, R intersect in a point TV on the circumscribed circle, (iv.) NT passes through the centre. 168. Inversion and Inverse Curves. When the vector OP of a curve is produced to Q, so that OQ is inversely proportional to OP, or 0Q = c 2 /0P, then the locus of Q is called an inverse curve of the locus of P with respect to the origin 0, or with respect to the circle of the centre and radius c ; and c is then called the constant of inversion. Thus if u = f 8 is the polar equation of the locus of P, then r = cHd is the polar equation of the inverse curve, the locus of Q, with respect to the origin 0. CURVES IN GENERAL 358 The inverse of a circle (or sphere) is another circle (or sphere); except when the circle (or sphere) passes through the centre of inversion, when the inverse is a straight line (or plane). For if OPQ meets a circle (or sphere) in P and Q, then OP . OQ is constant and equal to OT 2 for an external origin 0, where OT is a tangent to the circle (or sphere) ; so that a circle (or sphere) is its own inverse with respect to any origin ; and varying the constant of inversion gives a similar curve (or surface) ; in this case another circle (or sphere). Maxwell calls the rectangle on the segments of a chord through made by the circle (or sphere) the power of the circle (or sphere) with respect to ; and now if the distances from to the centre of the circle or sphere and of its inverse are denoted by a and a', and the radii by b and b', then a 2 - b 2 , a' 2 - b' 2 are the powers of the circles , a' b' c 2 a 2 -b' 2 or spheres, and — = T = -= — rs = 5- — ■ r a b a^ — b* c A Any curve Pp and its inverse Qq with respect to will cut the vector OPQ at supplementary radial angles ; for since OP. OQ = c 2 = Op .0q,& circle can be drawn through PQqp, and therefore the angles pPQ, pqQ are supple- mentary, and Pp, Qq are ultimately the tangents at P and Q, when p and therefore q are brought up close to P and Q. Or, otherwise, denoting the radial angles by (j> and $', cot

; r' , r dp . dp' dr sin d> , dr'smd> « • , - + -=-£+?= — - T —^-+ — :r ^' = 2sin0; p p dr dr dr dr and the circle of curvature obviously inverts into the circle of curvature of the new curve. Lines of curvature (§ 130) will correspond on a surface and its inverse ; and if a circle of principal curvature is drawn, the normal plane of this circle and the sphere through the circle and origin of inversion will invert into the sphere and plane through the corresponding circle of principal curvature. "When the Cartesian equation of a plane curve is given in rectangular coordinates x and y, the equation of the inverse curve with respect to the origin is obtained by writing c 2 x/(x 2 + y 2 ) for x, and c 2 y/(x 2 + y 2 ) for y. Thus the inverse of the parabola y 2 = px with respect to the vertex is, writing a for c 2 /p, the cissoid x(x 2 +y 2 ) = ay 2 , or y 2 = x s /(a — x); or in polar coordinates, r=a(sec6 — cos 6), the locus of t in a circle, for an origin on the circumfer- ence (ex. 11, p. 42). CURVES IN GENERAL. 355 169. Mechanical Invertors and exact Parallel Motion. When P describes a given curve, the point Q can be made to describe an inverse curve by means of the mechanical invertors, invented by Peaucellier and Hart. Peaucellier's motion consists of a rhombus LP, PM, MQ, QL, formed of four links of equal length, jointed at JO, M, P, Q, and of two other equal links OL, OM, pivoted at a fixed point (fig. 51). Then, in whatever way the link motion is displaced by the motion of P, OP .0Q = OE 2 -EP 2 = 0L 2 -LP 2 =c 2 , a constant, so that P and Q describe inverse curves. For instance, if P is compelled to describe an arc of a circle by a link CP, pivoted at a fixed point C, and of length GP=OG, then Q will move in a straight line perpendicular to OC. A parallel motion is the name given in Mechanism to an arrangement of bars intended to guide a point Q, the head of a piston rod, in a straight line ; and this is effected with exactness by Peaucellier's arrangement, but only approximately by Watt's parallel motion. Hart's parallel motion accomplishes the same purpose with four bars, while Peaucellier's requires six. 356 CURVES IN GENERAL. A jointed parallelogram of rods, FGHK, is taken, and the longer rods are crossed ; any fixed point in one of the rods FG is taken, and OPQR is drawn parallel to FH or GK, to meet FK in P, GH in Q, and HK in R (fig. 51). , Then P, Q, R are also fixed points in the bars, such that when again opened into a parallelogram, OPRQ will form another parallelogram, the sides of which are parallel to the diagonals of FGHK, and the area of which bears a constant ratio to the area of FGHK ; so that OP . OQ is constant; and thus the parallelogram FGHK, when crossed, will act as an invertbr, P and Q describing in- verse curves when is fixed. - _ We can show Hart's parallel motion working in con- junction with Peaucellier's by drawing FG parallel to PM or LQ, GQH parallel to OL, and FPK parallel to OM ; and now 0, P, R, Q are the middle points of the bars of the crossed parallelogram FGHK. . We may join LR and MR by bars, and now the two rhombuses LPMQ, OLRM are said to make a complete Peaucellier cell. When P and Q are inside the cell, the cell is called positive (fig. 51), where it is shown acting a"s the parallel motion of a beam engine. . But when P and Q are outside, it is called a negative cell (fig. 52) ; and now it forms a compact mechanism for drawing not only straight lines, but circles of very large radius, as required in Architecture. (Kempe, How to Draw a Straight Line.) While Q describes a straight line and P a circular arc,, the point R will describe a curve whose polar equation is of the form r = a sec 6 ± b cos 6, the inverse of a conic section with respect to a vertex, ■ CURVES IN GENERAL. 357 Fig.52. If equal bars QB, BE are pivoted at Q, B, and a fixed point E in the straight line described by Q, then, as in fig. 8, any point D in the bar QB will describe an arc of an ellipse ; and the normal DI will pass through /, fixed in EB produced, so that EI=1EB. Any complicated system of linkwork in mechanism can always be analysed into three bar motion, such as OA , AP, PC in fig. 45, when a bar PC is pivoted about a fixed point at C; thus representing the beam, connecting rod, and crank in a beam engine ; the beam being practically of infinite length in the direct action engine. If OA, CP produced intersect in I, then / is the instantaneous centre of rotation of AP ; and for uniform velocity of the crank P, the velocity of A will be a maximum when OC and AP intersect in a point J, the foot of the perpendicular from / on AP. A point Q in AP will describe a figure of 8 curve ; Watt's parallel motion consists essentially of a three bar motion, such as AO, AP, PC, by means of which the head of the piston rod Q is guided for a short distance on each side of the point of inflexion on this figure of 8. 358 CURVES IN GENERAL. The curves described by any point Q on the connecting rod AP in three bar motion have been studied by Koberts, Cayley, and Clifford ; any such curve being capable of a triple generation by means of bars (Clifford, Kinematic). When a crossed parallelogram is employed, as in figs. 51, 52, and in fig. 49, the relative motion of the short bars FO and HK is imitated by rolling an ellipse with foci F, G, on an equal ellipse with foci H, K\ and the relative motion of the long bars OH, FK is imitated by rolling a hyperbola on an equal hyperbola ; and relatively to one bar any point carried by the opposite bar will describe the pedal of an ellipse or hyperbola. 170. Polar Reciprocals. The inverse of the pedal of a curve with respect to the same pole is called a polar reciprocal of the curve. For instance, the pedal of a circle with respect to any point is the limacon (§ 1 56) r=a+b cos 6; and therefore the polar reciprocal of a circle is the inverse of a limacon, and its equation is r=c 2 /(a+ b cos 6), the polar equation of a conic section, with a focus at the pole, excentricity e = b/a, and semi-latus-rectum l = c*/a. Again, the polar reciprocal of an epicycloid with respect to the centre is r cos m6 = b, a Cotes's spiral (§ 164). With the usual notation of 6, r, 0, p, p, ... for the original curve, and 0', r', ' = 7T, / = c 2 lp, p' = c 2 Jr ; and PP^ r d p -- r 'aW = C p^ CC0Se ^- CURVES IN GENERAL. 359 171. Orthogonal and Oblique Trajectories. A curve cutting at right angles a system of curves is called an orthogonal trajectory of the system : and a curve cutting the system at any constant angle y other than a right angle is called an oblique trajectory. In § 101, it was shown that the orthogonal trajectories of the cycloids described by all the points on the circum- ference of a wheel are equal cycloids. Familiar instances of orthogonal trajectories are seen with (i.) horizontal and vertical straight lines, (ii.) straight lines radiating from a centre, and the concentric circles, (iii.) confocal and coaxial parabolas, (iv.) rectangular hyperbolas, the asymptotes of one system being the axes of the other system, (v.) confocal ellipses and hyperbolas ; while the oblique trajectories of system (i.) are inclined straight lines, of (ii.) are equiangular spirals, of (iii.) are confocal but not coaxial parabolas, and of (iv.) are con- centric, but not coaxial rectangular hyperbolas. Since any two curves and the corresponding inverse curves cut at the same angle, it follows that the inverse of a system of orthogonal or oblique trajectories forms a new system of orthogonal or oblique trajectories. We have seen (§ 133) that the graphs of the conjugate functions u and v, given by u + iv = t(x+iy), when u or v is equated to a constant, form an orthogonal system. Also, since in an oblique trajectory cutting the curves u and v (§ 133) at constant angles y and \ir — y dSjCOS y + ds 2 sin y = 0, and dsjdu = ds 2 /dv = J - * ; therefore du cos y + dv sin y = 0, or u cos y + v sin y = a constant, is the general equation of an oblique trajectory. 360 CURVES IN OENERAt. Mr. Larmor has shown (Proc. London Math. Soc, vol. xv.) that in a field of force (§ 83), of which the potential is J, a particle will describe an oblique trajectory with velocity V, if started so that \ V 2 = J ; or that a ray of light will describe this oblique trajectory if the refractive index varies as J~*. Then, if R denotes the radius of curvature of the trajectory, resolving normally, F 2 2J dJ , d J . Ti /dJ ,dJ. \ 1 dJi , dJi . Any small contour in the (x, y) diagram will be transformed into a similar contour in the (u, v) diagram, since angles are unaltered in this transformation, J* denoting the scale of the transformation, or the ratio of the linear dimensions of corresponding elements ; this is the condition to be satisfied in maps and charts. Thus, for instance, if u + iv = (x + iy) n = r n cos n6 + ir u sin n6, by De Moivre's Theorem (§ 111), then, on putting u + iv = c n ~ \x' + iy'), u = c n - V = r"cos nd,v = c n ~ 1 y' = r n sin n6 ; and a new system of curves is obtained in which r is changed into i^/c" -1 and 6 into nd; so that we may put and now the radial angle

=rcosTC0=r n+1 /a' 1 , (tiw \ " +1 cos j " , pm _ Q.mgQg ^j^ w here m = «/(m + 1) ; so that the pedal curve AY oi the critical orbit AP is also a critical orbit ; and the equation of the polar recip- rocal of AP may be written r m cosm6=a m , another critical orbit. Consider the path of a ray of light in the Earth's atmosphere, on Simpson's assumption that the refraction index fx varies inversely as the n+l th power of the distance r from the centre of the Earth. Then (§ 73) p. sin or /xp is constant along the ray, or p varies as r™ +1 , so that the path of the ray is either the critical orbit r n = a n cosn9, or an oblique trajectory. Again, in these curves, the chord of curvature through = 2pdr/dp = 2rj(n + 1), so that (fig. 10) p = PgJ(n + 1). 173. Roulettes of Critical Orbits. When AP (fig. 48) is the roulette of the pole P of the critical orbit r n =a n cosn6 with respect to the straight line Ox, then denoting the angle IPM by i/r, and MP by y, \[r = n6 = ma>, and y m =p m = a m cosrnw = a m cos\ls, the relation connecting y and \]s in the roulette. CURVES IN GENERAL. 363 Differentiating logarithmically with respect to s, mdy , d\Is r- = — tan yb—r- '■> y as r as „ j . dy , ds andsmce ^=-sm^^ = p, therefore p = y sec \fr/m = PI fm, so that p the radius of curvature of the roulette is 1/m or 1 + ljn times the length of the normal PI, and there- fore m/(m+l) times the radius of curvature of the rolling pedal A'P; and the evolute of the roulette will possess a similar property. For instance, if n = l, the rolling curve is a circle, and the roulette is a cycloid ; and then p = 2PI (§ 101), and the evolute of AP is an equal cycloid. As applications of these roulettes AP, we may instance (i.) that AP is a catenary curve with Ox horizontal, when the line density is made proportional to the 2 + l/n th power of sec yfr, or of the tension ; reducing, for n = — \, to the ordinary catenary, which is therefore the roulette of the focus of a parabola ; (ii.) that, with m negative and = — p, AP can be described as a trajectory by a projectile in a resisting medium, in which the retardation due to the resistance is where v denotes the velocity at P, and v at A ; reducing, for p = J, to the ordinary parabolic trajectory, the roulette of the pole of r^cos£# = a*, the first negative pedal of the parabola r$ cos \Q = «* ; (iii.) that AP is also the path of a ray of light, when the refractive index varies as y* ; a catenary for p = 1. 364 CURVES IN GENERAL. 174. Vectors and Vectorial Equations. The quantity z = x + iy is now called the vector or step from to the point (x, y) ; being composed of a step of length x parallel to Ox and then a step of length y parallel to Oy, according to Argand's representation. Then if z' = x' + iy' represents another vector, z— z'=x — x'+i . y — y' represents the vector or step from (x', y') to {x, y). It is convenient in physical application to consider the logarithm of the vector, and now it' = log(0 — z') or u+iv = log(x — x'+i. y — y') = i, representing Cassinians, and the orthogonal rectangular hyperbolas, passing through A and A v When n equal negative electrodes are placed on the circle, at B v B 2 , ... B n , midway between the positive electrodes, the vector of B s being aexp{(2s — l)ir/n}, then their vector potential is log II \z — a exp(2s- l)ir/n} =log(z n + a n ) ; s = l so that for the system of positive and negative electrodes w = log(z" - a w )/(a" + a n ) _ r 2n — 2a n r n cos nd + a 2n — z °S r 2» + 2a n r n cos n8 + a 2n , r n smn6 ., , r"sinw6> + % tan " l — n — % tan - 1 — ^ -- r"cos nv — a n r"cos »0 + a" 2a n r n cosw0 . , , 2a"r n sin w0 = -tanh- 1 rfc + a „ ^an- 1 r2 „_ ffl2 „ , giving another orthogonal system of curves, r 2n + 2a"r"cos n6 coth u + a 2 " = 0, r 2 " + 2a"r»sin n6 cot t> - a 2n = 0. When « = 1, these curves are circles, the dipolar circles of the next article. CURVES IN GENERAL. 367 175. Dipolar Circles. When we put x + iy = c tan ^(u+iv), and write it . _ sin |(w + iv)cos \{u — iv) _ sin u + i sinh v cos^(u+iv)cos %(u— iv) cos it + coshi/ . , c siwii c sinh v then & = — , - -> y cosh v + cos u a cosh v + cosu thus dividing tan£(u+ iv) into a real and imaginary part. Also tan u = tan{ J(w + w) + £(w — w) } _ x+iy+x — iy _ 2cx ~ c 2 —x 2 —y 2 c 2 — x 2 — y 2 ' and t&niv = t&n{^(u + iv) — ^(u — iv)} _ cc + iy — x+iy _ 2icy ~ c 2 + x 2 -\-y 2 ~ c 2 + x 2 + y 2 ' so that x 2 +y 2 +2cx cotu — c 2 =0 (i.), x 2 + y 2 — 2cy coth v + c 2 = (ii.), the equations of a system of orthogonal dipolar circles (fig. 54.i.) ; called dipolar because the system (i.) represents circles passing through two poles N and S, while system (ii.) represents orthogonal circles having the same radical axis Ox ; thus representing the meridians and parallels in the stereographic projection of the terrestrial globe. Putting uor« = nt, we obtain circles described under excentric centres of force ; and writing them (aj+ccotu) 2 +2/ 2 =c 2 cosec 2 u (i.) x 2 + (y — c coth vf = c 2 cosech 2 t» (ii.), then (i.) represents a circle, centre ( — c cot u, 0), and radius c cosec u ; and (ii.) a circle, centre (0, c coth v) and radius c cosech v ; and with respect to 0, the power of (i.) is — c 2 , and of (ii.) is c 2 . At any point P, the angle NPS=tt — u, while the ratio 8PjNP=e\ or v=\og(SP/NP). 368 CURVES IN GENERAL. If X denotes the latitude of any parallel of latitude circle (ii.), then since on the stereographic projection the radius of this circle is c cot X, therefore (§ 34) cot X = cosech v, tan X = sinh v, sin X = tanh v, cos X = sech v, tan |X = tanh \v, or X = gd v, v = log(secX+tanX) = logtan(-j7T+JX). • while X = gd(u cot y) represents a loxodrome or rhumb line, cutting the meridians at a constant angle y. The inverse system of curves with respect to a pole N or S will be radiating straight lines and concentric circles, representing meridians and parallels in the neighbourhood of a pole, when projected stereographically from the other pole, as in Godfray's Great Circle Chart. . A great circle on this chart will be represented by a straight line ; and a rhumb line, that is an oblique trajectory of the meridians and parallels, will be an equi- angular spiral ; so that on the stereographic projection it will be an inverse of an equiangular spiral. The general operation of inversion with respect to any pole, combined with a change of origin, is represented by the linear substitution (Schwarz) z' = (az+b)/(Az + B), by which a polygon in the z plane, bounded by straight lines or circles, is transformed into a polygon bounded by circles, cutting at the same angles, in the z' plane. In Godfray's Great Circle Chart we transform to z 1 = x' + iy' = c exp(ia — /3) = ce " ^(cos a + i sin a) , so that a is the longitude, and the latitude X is given by r'=ce~P = c tan(^7r— JX), or /3 — log(sec X + tan X), X = gd /3. CURVES IN GENERAL. 369 176. Mercator's Chart. Draw any contour on the (u, v) diagram (fig. 54, ii.) corresponding to a contour, the outline of a country, in the (x, y) stereographic diagram ; we thus obtain a new map, called Mercator's projection, in which the meridians and parallels are orthogonal systems of parallel straight lines ; and while a length u represents the longitude, a length v = log(secX+tanX) will represent the latitude X; also dv/dX = sec X, so that the minute of latitude in Mer- cator's chart increases as the secant of the latitude, being equal to the minute of longitude only at the equator. Twenty-four standard meridians at equal intervals of 15° in longitude from Greenwich mark the standard time at a place ; the standard time being the mean solar time at the nearest standard meridian. Taking the Godfray Chart as representing any system of parallels and meridians with respect to an axis through any zenith Z, say in longitude and latitude gd b, then the representation on Mercator's Chart will be given by t&n\{u+i . v — 6) = exp (ia — /3) = x'+iy' = z'; or, dropping b as representing a mere change of origin, i a — y3 = log tan £(u + iv) , sinw + isinh-y . , ^inhv ,. .coshv = log — ; — =itan -1 — : coth -1 — ; ° cosu + coshv smu cosu so that sinh v = tan a sin u, cosh v = coth /3 cos u ; the representation on Mercator's Chart of a system of parallel small circles and their meridians ; these lines are called Sumner lines (p. 203), being the lines on Mercator's Chart on which a celestial body in the zenith at Z will at any instant have the same altitude. J 2a 370 CURVES IN GENERAL. Fig. 55 177. Confocal Ellipses and Hyperbolas. As another practical illustration of conjugate functions, put x + iy = c cosh (u + iv) = c(cosh u cos v + i sinh u sin v) ; then x = c cosh u cos v, y = c sinh it sin w ; and alternately eliminating u and -y, yi c 2 cosh 2 w, c 2 sinh 2 % = 1. .(>.) ar y* c 2 cos 2 « c 2 sin 2 v .(ii.) representing a system of confocal ellipses and hyperbolas for constant values of u and v the polar reciprocals with respect to N or S of the circles v of § 175; and sech u will be the excentricity of the ellipse, sec v of the hyperbola (fig. 55); while (§ 35) v will be the excentric angle or excentric anomaly of a point P on the ellipse AP, and u will be hyperbolic excentric anomaly, of the point P on the hyperbola KP. CURVES IN GENERAL. 371 Put v=nt, then x + iy = c cosh (it -f- int) = c cos (w£ — iv) represents confocal ellipses described under a central attraction to 0, varying as the distance, the period being 2Tr/n ; while hyperbolas are described with u = nt, and x+iy = c cosh {nt + iv). In an oblique trajectory u cos y + v sin y = constant, we may put v = n{t — t) cos y, w = — n(t — t) sin y ; so that £c+i2/ = ccosn{(f — r)cosy+t(i — T')siny}; and the equation is obtained by eliminating t between x= ccos{n(t — r)cos y}cosh{n(t — T')siny}, i/ = — c sin{n(t — t)cos yjsinhf'n^ — T-')siny}. The equations of the tangent and normal of the ellipse at P, which are also the normal and tangent of the hyperbola at P, are cost; , sinv ..... x — i f- ty-^j — = c , (in-), coshw ^ sum it ^ ' coshw sinhti .. . x y—. = c (iv.). cos v u sin v Denoting by p and q the lengths of the perpendiculars from upon the tangents at P of the ellipse and hyperbola, p = c cosh u sinh u/(cosh 2 u — cos V)^, q = c sin 1 ?/ cos v /(cosh 2 u — cos 2 ^ ; The pole of (iii.) with respect to the hyperbola (ii.) will be (c cos^/cosh u, — c sm 3 v/smh u) , and this pole will therefore be Q, the centre of curvature of the ellipse at P (§ 98) ; and PQ = /> = e(cosh 2 ii — cos 2 w)*/(cosh u sinh u). Similarly (c cosh 3 ii/cos v, — c sinh%/sin v), the pole of (iv.) with respect to (i.), is the centre of curvature of the hyperbola at P ; and the radius of curvature is c(cosh 2 i6 — cos 2 'i;)^/(cos v sin v). 372 CURVES IN GENERAL. We may consider w = u + iv as representing a vector of curvilinear steps u and v; u along a hyperbola and v along a confocal ellipse, meeting at the point z = x + iy = c cosh(u + iv). If P' is any other point given by z' = x'+ iy' = c cosh(u' + iv'), then the vector P'P is given by z — z' = c{ cosh(w + iv) — cosh(u' + iv') } = 2csinh %(u + u'+i.v+v')smlni %(u — u'+i.v — v'); so that PP' 2 = 4e 2 sinh %(u + u'+i.v+ t/)sinh ^(u — u'+i.v — v') smh$(u+u' — i.v + v')smh ^(u — u' — i.v — v') = c 2 {cosh(u + u') — cos(t> + v') } {cosh(u — u') — cos^ — v')} . This expression is unaltered by an interchange of u, u', or v, v' ; so that if the points p, p', called corresponding points, are given by (u, v'), (u', v), then pp' = PP'. Further, if denotes the angle between PP' and pp', ,, . , , cosh(u+w) — coshfu' +£■?/) then 10 = log — =-7 — — r-4r . ; , , . ' r ° cosh(i6 + %v ) — cosh(u + %v) _, tanh -^(u — u') + i tan \(v — v') °tanh \{u — v!) — i tan \{v — v') = Si tan- 1 tan K t; - t . tanh£(u — w') Captain Weir's Azimuth Diagram consists of a series of confocals, the hyperbolas marking the longitude or hour angle v, while the ellipses are marked with the degrees in gd u, to represent latitude or declination. Now, if the hour angle is the complement of v, and if the latitude and declination are gd u and gd d, a straight line joining the points (u, v) and (d, 0) will give the azimuth or bearing of the object from the meridian Oy. (Godfray, Astronomy, § 222.) CURVES IN GENERAL. 373 Changing to the focus 8 or 8' as origin x — c + iy = 2c sinh 2 £(u + iv), x + c + iy = 2c cosh 2 ^ + iv), so that if r, r' denote the focal distances 8P, S'P, r = 2c sinh J(u + ii>)sinh \{u — iv) = c(coshu — cos v) = a( 1 — e cos v); r'= c(coshtt + cos , y) = a(l+ecosi;). Then r'+r = 2ccoshtt = J 4J.' (v.), r' — r = 2c cos v = KK' (vi.); and hence SP = AK, S'P = KA' ■ also OP = tf£. Denoting the angles ASP, AS'P by 9, 9', then in Astronomy 9 is called the true anomaly of P, reckoned from perihelion at J., the Sun being supposed to be at the focus S ; but 9' is the true anomaly from aphelion A, with the Sun now at the other focus 8'. Then n x — c cosh u cos v — 1 n , cosh u cos v + 1 cos ft = = 5 . cos 9 = i ; ; r cosh u — cos « coshu+cosv . . sinh u sin t; . ., sinh u sin v or sintf= — , . sm9 = cosh « — cos v cosh u + cos v ' tan £0 = coth |u tan \v, tan £#' = tanh \u tan £u ; and r = SA cos 2 £v sec 2 £0, r' = SA'cos^v sec 2 |0'. . . . l+cos0coshu e + cos0 Again, since cos v — , ■ -^ = =— - -k, ° coshw+costJ 1 + ecostf or (1 + e cos 9)(1 — e cos v) — 1 — e 2 ; therefore r — c sinh 2 u/(cosh u + cos 9), of the form r = 1/(1 + e cos 0), with I = c sinh%/cosh u, the polar equation of the ellipse, with origin at S. Similarly r' = c sinh 2 u/(cosh u — cos 9') is the polar equation of the ellipse, with origin at S' ; while for the hyperbola the corresponding polar equations c sin 2 ii . c sin 2 v are r= 2 , r = -& cos v — cos 9 cos v — cos v 374 CURVES IN GENERAL. 178. Confocal Limacons. By inversion of this system of conies with respect to either focus, say 8, taking 2c as the constant of inversion, x + iy = 2c cosech 2 |(« + iv), x + 2c + iy = 2c coth 2 J(w+w); and r sinh 2 ii = 4c(cosh u + cos 6) ( vn -)> r sin 2 ?; = 4c( cos v— cos 6) (viii.), are the inverses of the ellipse (i.) and hyperbola (ii.), and the pedals with respect to N or 8 of the circles v of § 175, a system of confocal limacons (§ 155). Denoting now by r' the distance from the other focus 8', the position of which is unaltered by inversion, then > a J.T. 1/ , • \ j.t 1/ • \ « COshtt + COSW r = 2c coth Uu + %v) coth *( u — %v) = 2c — = , 2V ' 2K ' coshw-cosv' while r = 2c cosech |(u + i^cosech |(u — -i-y) = — r ; cosn ii> ~~ cos i) ,, , , , „ 4c cosh u , ,. . so that r +2c= — = = rcoshtt (ix.), coshw — cosv , „ 4c cos v , . r— 2c = — , = r cos?; (x.), cosh u — cos?; ' equations of the form r' — It = constant, which are Cartesian ovals in the general case, the inverse of a conic with respect to any point on the transverse axis. The limacon in which v=$tt, l=\, is called the Trisectrix. For if A' OR is any angle to be trisected (fig. 47, ii.), and if A'R is joined cutting the curve in p; then, if A'Op = 6, Op = 0A'{2 cos - 1) ; and A'p = 204 'sin \Q = A'q, if Op meets the arc A'R in q. Therefore 6 = qpA'- OA'p =pqA'- OA'p = OA'q - OA'p = pA'q = iqOR; so that Op is a trisector of the angle A'OR. CURVES IN GENERAL. 375 179. Kepler's Laws of Planetary Motion. Kepler, by long continued observations and measure- ment of the Sun's diameter and motion in longitude, noticed that (i.) the variation of the Sun's apparent diameter d could be expressed by the formula D(l+e cos d), where D denotes the mean angular diameter (about 32') and 6 the Sun's longitude from peri- helion, e being a small constant, about 1/60 ; (ii.) that the Sun's daily motion in longitude was proportional to the apparent area or square of the diameter. Since the apparent diameter is inversely proportional to the distance, he deduced the laws called Kepler's Laws — (i.) that the relative orbit of the Earth (or any planet) and of the Sun is an ellipse, with a focus at the Sun, if the Sun is supposed fixed, given in polar coordinates by l/r = djD = 1 + e cos 9. (ii.) that r 2 d6/dt = h is constant, and that the Earth therefore sweeps out by its radius vector from the Sun equal areas in equal times. The third law of Kepler— (iii.) the squares of the periodic times of the planets round the Sun are proportional to the cubes of their mean distances — was easily inferred by arithmetical calculation, when once the distances of the planets from the Sun were measured, in terms of the Sun's distance from the Earth. Newton inferred from law (ii.) that the earth is attracted by the Sun ; and from law (i.) that the attraction must be inversely proportional to the square of the distance : while law (iii.) showed that the attraction of the Sun was of the same nature on all the planets (§ 84). 376 CURVES IN GENERAL. By Newton's Law of Universal Gravitation, employing C.G.s. units (§ 142), the attraction between two spherical bodies, for instance, the Sun and the Earth, weighing 8 and E grammes, when their centres are a centimetres apart, will be given by the expression GSEa' 2 (dynes); and C is called the constant of gravitation, being the attraction in dynes between two spheres, each weighing one gramme, when their centres are one centimetre apart. Then Kepler's Third Law, in a mathematical form, asserts that if T is the period in seconds of the Sun and planet, and if n = 2irlT denotes the mean motion, then by ex. 1, p. 175, n 2 a s = 4:7r 2 a a /T 2 = G(S+E). According to the Cavendish experiment, now being re- peated with improved apparatus by Mr. C. V. Boys, we may take G= 10" 8 x 6-48, 1/0= 10 7 X 1-54. (Everett, Units and Physical Constants.) Denoting by g the acceleration (in C.G.S. spouds) of the attraction of the Earth, then g = CE/R\ or CE=gR 2 ; so that we can determine E when C is known, and vice versa; and till this is done, Newton's Principia is merely Kinematics. With the above value of C, and # = 981, J? = 10 9 -h|7t, E=gR 2 /C= 1(F x 612 grammes ; giving a mean density p = E/(^ttR s ) = 5 - 67. Denoting by LT the Sun's parallax (8"76), and by T the number of seconds in one year, then the attraction between the Sun and the Earth is ^=4x 2 £flcosecn/r 2 = 10 27 x3-65 dynes, while S = Fa 2 /CE= 10 33 x V2 grammes, with the above numerical values. CURVES IN GENERAL. 377 180. Elliptic Planetary Motion. Considering an elliptic orbit AP (fig. 55) in which c cosh u = 0A = a, c sinh u = OB = b, c sinh 2 u/cosh u = I, the semi-latus rectum, and sech u = e ; then the sector ASP = sector A OP — triangle 08P = \ab{v — e sin v). If the ellipse is described in period T round the Sun at S, with mean angular velocity n=2ir/T, and if the time to P from perihelion at A is denoted by t, then by Kepler's Second Law, t _\ab(v — esinw) T~ ^ab ' or nt = v — e sin v ; and now nt is called the mean anomaly, being the true anomaly of a planet moving uniformly in a circular orbit in the same period T. Expressed in terms of the true anomaly 6 from peri- helion A, e + cos0 . ^/(l— e 2 )sin0 cosi> = =— ^, smi;=^Y- — 3 — , l + ecos# l+ecos0 , , . ix /(l-e z )sin0 ej(l - e 2 )sin 6 and Ti^sm" 1 ^-- n ^~~ ; a — ■ 1 + e cos 6 1 + e cos 6 Reckoned from aphelion A with the Sun at 8', nt' — v + esinv _ , in - jq - e> n y , e x/( 1 - e2 > in e ' l-ecos0' "*" l-ecos0' ' For completeness we may consider the hyperbolic branch KP, of excentricity e", described round the Sun at S, and now the mean anomaly n't = e's'mhu — u e'J(e'*-l)smd ^^-1^0 ~ 1 + e'cosfl l + e'cos0 ' with n'a'* = C(S+JS), 378 CURVES IN GENERAL. a' denoting the semi-transverse axis OK, and the true anomaly 9 being now K8P, reckoned from perihelion K. With the Sun at 8', the hyperbolic branch KP will be described under a repulsion from 8', and n't' = e'sinh u + u ey( (%p sin v = sin m + S-; — -^rrr -j — - (sin m)P +1 . (p + l)!dm^ v ' Similarly sin 2v, sin3i>, ..., sinpv, and cosv can be expanded in powers of e ; and r , dm - = 1 — e cos w = -T- ; ^_^._1 y eP d p , ■ .-. p . r~dm~ pi dm p ' while from tan|0 = coth £tt tan Jw, we deduce by logarithmic differentiation dO dv d6 sinh u 1— c 2 or sin sinv' cfa; cosh it, — cos 1; 1 — 2ccosv + c 2 ' with c= e~ u ; and, resolved into partial fractions, d6 = 1 _J dv~l-ce iv + l-ce~ iv = 1 + 22c^cos pv, c p 6 = v + 22— sin »y, which can now be expressed in terms of m. CURVES IN GENERAL. 381 We can rearrange these series so as to proceed accord- ing to cosines or sines of multiples of the mean anomaly m, the coefficients being functions of e, and this is more convenient for Kepler's Problem. A series of this nature is called a Fourier Series, and we proceed to show how the coefficients of such a series can be calculated, for any arbitrary single valued function fee, in a series proceeding by cosines or sines of multiples of x/l, when I is any arbitrary quantity. 183. Fourier's Series. Assume that, between the limits x = ± I, the function fa; can be expressed by a series of the form fx = | A + '2A p coa('p-7rx/r) + ~2B p sm(pTrx/l), where 2 denotes the sum of the series obtained by giving p all positive integral values from 1 to oo ; it is required to determine the A's and B's. Divide the function fx into its odd and even part (§ 46), and denote them by f x x and f 2 x respectively ; thus f 2 x = i{f(x) + f( — x)} = %A + J,ApCOB(pirx/l) (i.), i x x = i{i(x) — f( — x)} = 2,B p sia(pirxll) (ii). To determine A p , change x into v in (i.), multiply both sides by cos(p7rv/r), and integrate with respect to v be- tween the limits and I ; then since /cos ^p cos %j^dv =J J \ cos(p - q)^j-+icos(p+q)^- \dv _ [ ~Bin{(p-g)WE} + sin {(P + gV^} ~|' n . L 2(p-q)ir/l 2(p + q)7r/l J ~ ' while y^Efdv =f\\ + i cos 2 -Pf)dv = \l ; therefore /f 2 v cos(pirv/l)dv = \IA V ; /f 2 vdv — %IA . 382 CURVES IN GENERAL. y J V f \ 1 ' ' X (i) y /A y*. ''■-■'' V/ \ ,/ '• x (ii) Fig. 56 Similarly, changing x into v in (ii.), multiplying by sin (p-n-v/l), and integrating with respect to v between and I, we find /fj-u sm{pirvlV)dA}=^lB p . Therefore, between the limits x = ± I, 2. ix-- and this is called Fourier's Series. By supposing p to take all integral values, positive or negative, we may write Fourier's series, x = wS / iv cos ^j dv. CURVES IN GENERAL. 383 y y'\ \ X (i) ' y y y / s y y y y y / y y / x y / (ii) Fig. 57 At the limits x = ± I, the value of the series is f 2 l, so that in general there is discontinuity at the limits; and outside these limits Fourier's series represents periodic repetitions of the function fee between the limits. This is exhibited by drawing the graphs of (i.) ^A + ~ZA p cos(pttx/1) and (ii.) 1,B p sm(pTrx/l), as exhibited in fig. 56, (i.) and (ii.) When the limits between which fee is to be expressed by Fourier's Series are and I, then either series (i.) or (ii.) may be chosen at pleasure, to represent fee, but it is best to choose the series which introduces the least dis- continuity at the limits. For instance, suppose fx = x ; then between and I, 00 x = l -^f E ^if C0S ( 2 P- 1 ^T' or (ii.) x = I ~(-i)>- sin pwx M p . l but outside these limits the series (i.) and (ii.) represent the dotted lines in fig. 57, (i.) and (ii.). 384 CUR VES IN GENERAL. The student is recommended to draw the graphs of the first two or three terms in a Fourier series, to see how quickly the series approximates to the given function. For instance, draw the graphs of sin x — | sin 2cc, sinx — |sin 2x + % sin Bx, cos x + £ cos Bx, cos x + % cos Bx + ¥ X T cos hx, Fourier made great use of his series in problems on the Conduction of Heat, where a solution of the partial differential equation ^r = %rr2 is required, giving the temperature u at any time t ; for if the initial state of temperature U is expressed by the Fourier Series £/"= \A +*2,(A p cospx+B p sm.px), then at any subsequent time t the temperature u = \ A + 2( J^cos px + .Bpsin px)e ~ p2M . The geometrical meaning of the coefficients A p and B p is explained by Clifford (Proc. London Math. Soc, vol. v.) according to the method employed in Tidal Harmonic Analysis (G. H. Darwin, Brit. Ass. Report, 1883), where a pencil attached to a float registers fx, the rise and fall of the tide, on a cylinder turning uniformly by clockwork round a vertical axis. To analyse the tides of period P, the cylinder is made to revolve in a period P/p, when a closed curve will be traced on the cylinder after p revolutions. The plane on which the projection of this closed curve has a maximum area A, with attention to the sign of the area (§ 137), will cut the cylinder in an ellipse of area A/p. Now, if the pencil is made to follow this ellipse as the cylinder revolves in the period P/p, the pencil will have the component harmonic travel A p cos(pirx/l) + B p sm(pTrx/l). CORVES IN GENERAL. 385 184. Fourier's Series in Kepler's Problem. Given the relation connecting the mean and excentric anomalies, v = m + e sin v, , i civ 1 1 + e cos a , , „ . then - T - = - = -— ¥ —=- = l+2A p cospm, dm 1 — ecos-y 1 — e 2, r " L suppose, when expressed by a Fourier series ; and now %A P = — I -5 — cos pmdm=— /cosp(v — esmv)dv; and this definite integral is a function of pe, called Bessel's function of the order p, and denoted by J p (pe) ; and it can be verified, as in § 207, that v = J m {qr) satisfies Bessel's differential equation on p. 185 (§ 88). Now - = -t— = 1 + 22 J J »e)cos pm ; r dm and integrating with respect to m, v = m + 22 P ^P ' sinjom, P . giving the excentric anomaly v in terms of the mean anomaly m. Again dv dv n-r, T , , x • amV dm = de =2i:Jp{pe)Smpm ' and integrating with respect to m, cos v=G+ 22 A-P e ' ) C os mm, where 7rC= /cos i>cZm = /cos v(l — e cos w)cfo = — |«re, C= — |e; o j . v — m n-^Jpipe) . and sini> = = 22 ^ sinrom; e pe L whence x = a cos i>, i/ = b sin v, and r = a(l — e cos v) can be expressed by a Fourier series in terms of the mean anomaly m. A B 886 CURVES IN GENERAL. Examples on Fourier Series. (1) Prove that, in a series of cosines, between ± %l, Prove that, between and tt, sin a? sin Sx sin 5a; _sin(2» — l)x . 9 , > 18- + -3S- + — 5^ + "" or2 (2p-l)3 = ^ 7ra;(7r ~ a! ) ; and give the value of the series outside these limits. (2) In a series of cosines, cosh mx = 2tt sinh ml\ a „,. + 2— i~ — 4-2 C0S ?- h (8) In a series of sines, sinh mx = 2?r sinh mlX 010 „ o ff sin^— . 7)W+pV^ I Deduce the expansions of e mx and e'™*- (4) Expand cos mx in a series of cosines, and sin mx in a series of sines, of multiples of irxjl. (5) Show that the Fourier series which represents the density or temperature in an endless wire must in general contain both sines and cosines. Determine the Fourier series to represent the temperature of a circular wire, in which the temper- atures of the four quadrants in order are 1, 2, 3, 4. (6) Prove that the equation represents a staircase, of vertical and horizontal steps of length I. (7) Prove that the equation x 2 (_ \)P 24 = 2 — ^2- C0 4p0 + y)cos Jp(aj-y) represents circles of radius v; and draw them. CHAPTER VII. INTEGRATION IN GENERAL. 185. In the preceding chapters a sketch of Integration has been given, and then a number of applications, in- tended to show the practical use of the method ; and now it is proposed to resume the consideration of Integration from a more general and systematic standpoint, so that the student may more readily perceive and write down the required result. The process of Integration is necessarily of a tentative nature, depending on a previous knowledge of Differen- tiation ; and in general the most convenient order of the mental operations employed in the integration of a given function will be found to be : (i.) to guess the' function required for the integration; (ii.) to assign the argument of this function ; (iii.) to write down the proper constant or numerical factors of the integrated function. Of these three operations, the first is of the most fundamental importance, depending as it does on the principles of the Calculus, but it is the second operation which presents the greatest practical difficulty, while the third only requires verification by a mental differentiation. 387 388 INTEGRATION IN GENERAL. 186. General Integration of Algebraical Functions. The most general algebraical function of x which is capable of integration by means of the preceding stock of functions can be written 8+TJR u+ vjr' where 8, T, U, V are rational integral algebraical func- tions of x, and R is a linear or quadratic function of x, and therefore of the form ax 2 + 2bx + c. If R is of the third or fourth degree in x, elliptic functions are in general required for the integration ; and if R is of the fifth or higher degree, hyperelliptic functions are required. We first rationalize the denominator, when S+TJR _ (S+TJR)( U- VJR) M N 1 U+ VJR U 2 -V 2 R ~D + D JR' where D= U 2 - V 2 R, M = SU-TVR, N=(TU-SV)R; and D, M, iV are thus rational integral functions of x. To integrate the rational function MjD, this function is split up into its quotient and partial fractions, in the manner explained in treatises on Algebra ; and then the integration of each term is in general easily effected. To integrate the irrational part, NjD^/R, we may resolve the rational function N/D into its partial fractions, and integrate each term by appropriate substitutions. 187. Integration of Rational Algebraical Functions. To integrate any rational function MjD or fcc/Fcc, the numerator and denominator of which are rational integral algebraical functions of x (§ 15), and therefore of the form INTEGRATION IN GENERAL. 389- ix _ ax m +bx m - 1 + cx™- 2 + . .. ¥x~Ax n + Bx n - 2 + Cx n - 3 +... where m and n are positive integers, the function is first resolved into its partial fractions by the ordinary rules of Algebra (Smith, Algebra, chap, xxiii. ; Hall and Knight, Higher Algebra, chap, xxiii.) ; if the degree m of the numerator is equal to or greater than the degree n of the denominator, the quotient must be first obtained by division. Thus to integrate — ^ — ., 9 > we must suppose it re- solved into the form , „ , A B x s x + 6 + x-l + x-2 = x*-3x + 2' x + 3 being the quotient and A /(x — 1), B/(x — 2) the partial fractions of the remainder. To determine the numerator A, multiply both sides of the identity by its denominator x—1; then (x-l)(x + 3) + A + B°^±=^-; X — 'l x — z and now put x— 1 =0, in the identity : then A = — 1. Similarly to determine B, multiply both sides by its denominator x - 2, and then put this denominator x - 2 =0 ; this gives B = 8. Now /^Sr9=y&+ 3 - J -r+- yr- 3a3 + 2 _/ v x — 1 i dx x-2J = \x % + Sx - log(a - 1) + 81og(a; - 2), and the integration is effected.' Every rational integral algebraical function of x, such as Fx, can be resolved into real linear factors (factors of the first degree in x), or real quadratic factors (factors of the second degree) ; thus, for example, 390 INTEGRATION IN GENERAL. x 2 -l = (x-l)(x+l); x 2 + l is not decomposable into real linear factors ; x s -l = (x-l)(x* + x+l) ; x s +l = (x + l)(x 2 -x+l); ^-l = (a-l)(ai + l)(x 2 +l); x i + 1 = (x 2 - Jlx + l)(x 2 + J2x + 1) ; x 6 -l = (x-l)(x + l)(x 2 -x+l)(x 2 + x + l); x e +l = (x 2 + l)(x 2 - ls /3x + l)(x 2 + s /3x+l) ; x 8 -l = (a;-l)(a;+l)(a; 2 +l)(a; 2 -V-^+l)(« 2 + x / 2 a ; + 1 ); and so on. Corresponding to a quadratic factor in the denominator Yx we must assume a partial fraction with a numerator of the form Hx + K; thus, as resolved into partial ... 1 ^. Ifa+Z tractions, we must put — s — T = =-+ -?-r — r^r- r X s — 1 a: — l ar+cc + 1 The numerator J. is then determined as before by multiplying both sides by x — 1 and then putting 33-1 = 0; thus A = £ ; and now, by transposition, Hx + K 1 1 1 -g-2 33 2 ~+a:+l - x 3 -l 3x-r3(a! 2 +a! + l) (so that H=-i,K= -|). /"das /71 1 1 x + 2 \ N0W Jw=l = J U 5=1-8 fa 1. , .. 1 /"2a?+l , 1 /* dx =§iog(*-i)-ey^ i ^+ I ^-gy 5 : 2 + a3+l = ilog(c C -l)-ilog( a; 2 + a! + l)- ; ^tan-i^+i 1. O-l) 3 1 , _ 1 2cc + l (1) INTEGRATION IN GENERAL. 391 Examples. — Resolve into partial fractions and integrate 1/y* ,-yi2 /yiO "I rp /V«2 /yi3 1 /yt /yi" f .'-■, if. , >/0 .L , i.Oj iO j i ',' X , lOj i.O , • . • (2) (as-l)(a>-2)' (aj-oXaJ-6)' (a;-l)(a:-2)(a;-3)' X j I/O j iX", . . . J j lX' j iC J . . . (£C-o,)(x-6)(a;-c)' (cc 2 +a 2 )(x 2 +6 2 )' 1, cc, sc 2 , a; 3 , ... ce 2 — 1, fl3 2 + l, a; 3 — 1, £ 3 +l (3) -, ; — 7z (substitute mx + n = y), v ' (mx+n) r v a " xp , , ... , cc — a , (s-a)-( g -6)» C^stitute^-^^2/), -7 — m , .„ (substitute ax m +c = lly), x(ax m +c) n ^ ,a " 1 1, #, x 2 (x+l)(x 2 -l)' (a;+l) 2 (cc 2 +l)' r 2 — a? 1 (4) a^ + aV + a*' x(l + 13a: 2 + 36a;*) 188. Generally, if x—p denotes a real linear factor of the denominator ~Fx, so that Yx = {x — p)x, , ix A R we put =-= \--—, hx x—p x and now A Jx-R{x-p) = fy cj>x tj>p on putting x — p = 0. But (x — p)(px = Fx, p = Fp ; and A = fp/Fp. Now the integral of the corresponding partial fraction A/(x-p) is A \og(x—p). 392 INTEGRATION IN GENERAL. If (x- : a) 2 +/3 2 denotes a quadratic factor of Fx, splitting up into the conjugate imaginary linear factors x — a — i(3 and x — a + 4/8, we take two corresponding partial fractions L+iM . L-iM x — a — i/3 x — a+ifi' coalescing into, the real partial fraction 2L(x-a)-2Mp (x-a?+P ' the integral of which is L log { (x - a) 2 + /3 2 } - 2M tan - * { (as - <,)/£}. We generally begin by assuming £b_ Hx+K 8 Fx (x-af+l3 2+ irx\ where Fx={(x-af+^ 2 }^x; and a+g^-^-'W ,fe Y*-a; - \jsx on putting (as — a) 2 + /3 2 = 0. This gives imaginary values of x ; but we may avoid the use of imaginaries by continually writing 2ax — a 2 — /3 2 for a; 2 , when finally ix and (Hx + K)-^rx are each reduced to the form 5» + and B'x + C; and thus 5 = £', C=C", whence H and .ST are determined. 189. When Fa has a factor (x — q) r , a linear factor x — q repeated r times, so that Fx=(x — q) r x x > we must assume r corresponding partial functions of the form Br fl-i A = fc y . (a;-g') r " , "(a!-g)'- li ""' , " t "a;-g Fa; x a;' and then to determine the -B's put x — q = y, ox x = q+y; so that multiplying by f, B r +B,.-iy+...+B 1 y>-i = {f(q+y)-Tyr}/ x (q+y); B r , B r .\, ... B 1 are thus the coefficients of y°, y 1 , ... y r ~ l INTEGRATION IN GENERAL. 393 in the algebraical expansion of f(g+2/)/x(2'+2/) m ascend- ing powers of y ; or, in other words, B r +B r .. 1 y + ...+B 1 y- 1 is the quotient and Ty r is the remainder, when i(q + y) is divided algebraically by xil + V) m ascending powers of y. Now the integral of a partial fraction B s (x — q)~" will be , ~ ' — , — - ; and of • — L will be BAogOx — q). (s-iyx-q)" 1 x-q 1 8V ^ When Fx has the repeated factor {(x — a) 2 + /3 2 } r , we can proceed in the same way, and assume r corre- sponding partial fractions of the form (H 8 x + K s )/{(x-a) 2 + P*}°; but in this case it is generally preferable to employ the conjugate imaginary factors ; or else to employ the sub- stitution x — a = /3 tan or /3 sinh u. , where m and n are positive integers, required in the vertical motion of a body when the resistance varies as the n th power of the velocity. If x — a is a factor of x n — 1, then a = l, or cos(2rTrjn)+i sin.{2rTr/n) = ex-p(2 im/m), where r has the values ±1, ± 2, . . . For then, by De Moivre's Theorem (§ 111), a™ = cos 2r7r+isin2r7r=l. ^.m-i a A Put x^l = x~=l + ^x~=a' here f% = x m - 1 ,~Fx=x n -l,~F'x=nx n - 1 ; so that ^= S =^-»-(-^=l) 2mr7r . . 2mr7r\ 1 2imr-7r , cos hi sm— ) = -exp , by De Moivre's Theorem ; also A = ljn. 71A 394 INTEGRATION IN GENERAL. ™ p fx m - l dx It/ in Therefore J -^^ = - log(a; - 1) + - 2 cos— — - + % sin )log ( a; - cos * sin ■ To express this result in a real form, the partial frac- tions with numerator A r and A. r , corresponding to conjugate imaginary values of a, must be combined into a real form as follows — A r A- r cc-cos(2r7r/n.)-isin(2r7r/ri.) x-cos(2r-!r/n)+i am(2rir/n) _ 1 2mrir 2a; — 2 cos(2r7r /to) to n x 2 — 2x cos{2rTrjn) + 1 _2 . 2mrir smftrir/n) . n to x 2 — 2x cos (2rirjri) + 1 ' and the corresponding integrals are 1 2mr 7 r 1 , » „ 2r7r , 1N -cos log(ar — 2a; cos f- 1) . ■n, to ° TO 2 . 2mr' ■ - sin- r, A ( 2r7r\ I . 2r7rl - tan " M ( a; — cos J / sm J-. TO TO When the denominator is x n -\-\, the result is of the same form, with 2r— 1 written for 2r; and from the above we infer that the typical quadratic factors of x n —l and x n +l are a; 2 -2a:cos(2r7r/TO) + l, and a; 2 -2a;cos{(2r-l)7r/TO}+l. If to is changed into to — m, as for instance by the substitution x=l/y, then cos(2mrTr/n) is unchanged, but sin(2mr7r/TO) changes in sign, so that (Euler) i -dx /■ ■ x m-l ±x n-m-l . X n ±l is expressed entirely, either by logarithms, or inverse circular functions. Degenerate cases occur for to = to, or 2m. INTEGRATION IN GENERAL. 395 When n is even, = 2p, and m is odd, = 2g + l, then, by a rearrangement of terms, fx m -Hx _ f x^dx J x n + l ~J «?p+1 2p^-v =1 2p x 2 + l 2^2/ sin §£ tan 5^1 • Taken between the limits and oo, the part depending on the hyperbolic functions vanishes ; and the result is f x^dx ttW = ? (2r-lW tt 2g+l / liTT=o"Z sm =75-cosec-^ 7T. J x ip +l 2p*-* r=1 p 2p 2p We may put x 2p = v n , and (2q+l)/2p = m/n, where tn and n are any integers ; and now, with the restriction that m < n, I „ , ., = — cosec — . J v n + l n n Thus, for instance, the area of the loop of the curve x n + -if 1 = a 2 x n ~ m - 1 y m ~ 1 on putting y = xv (§ 63), is given by V «**»=** J T+^- = ^ cosec ' n Putting v n = t, then with the restriction that m< 1, ° t m - 1 dy _ r I -i ,/ — 7r cosec m-w. /"It To determine y( tan 6)^d6, put tan0 = a^; then the integral becomes S"° pxP+v-Wx , p + q J i+i^=^ cosec V 7r - * (tanh u)vlPdu. 396 INTEGRATION IN GENERAL. Examples. — Resolve into partial fractions and integrate l,x x 2 , x s ... f 46 8 v ' x n -\ or x n +l , 9 . (1+ft 2 ) 2 (1+fli^ (1-cc 2 ) 2 (1-a; 2 ) 4 W (1-a; 2 ) 3 ' (1-x 2 ) 5 '"" (1 + rc 2 ) 3 ' (1+rc 2 ) 5 '"" (the integrations required when we integrate (sec 9) s , (sec 0) B , ... or (cosh uf, (cosh it) 4 , ... or cos 2 0, cos 4 0; ... or (sech u) 3 , (sech it) 5 , ... by means of the substitution tan \Q or tanh \u = x. 191 . Integration of an Irrational Algebraical Function. In integrating the irrational part N/B^/R we may suppose the rational function NjB resolved into its quotient and partial fractions ; and now we shall con- sider the integration of the simplest elements, J_ * d Hx+K JB' (x-p)JR' (Ax 2 +2Bx+G)JR' corresponding to the constant term in the quotient of N/D, and to the partial fractions corresponding with the linear and quadratic factors x—p and Ax 2 +2Bx + C of the denominator B ; and afterwards investigate formulas of reduction for x n 1 , Hx+K JR' (x-qYJR' an (Ax 2 +2Bx+GY s /R By formulas (i), (w), (x) (§ 39), ^ J jw-x*r sm m = cos A 1 -**)' (i i.) /" d f =sinh- 1 - = cosh-v(l+^); v ; J ls /(m 2 +x 2 ) m * \ mV (iii.) /' f 2 , = cosh-^ = sinh- Vfc-0 .= v ' _y „J(x 2 — m 2 ) m \in J INTEGRATION IN GENERAL. 397 in which it will be noticed that the analogy with the first integral breaks down when only the logarithm is employed to express the second and third integral. The quadratic R = ax 2 + 2bx + c can always be expressed as the sum or difference of two squares in one of the three forms m 2 — x 2 , m 2 + x 2 , or x 2 — m 2 ; and then the integ- ral fdxj/JR has the corresponding form (i.), (ii.), or (iii.). But substituting R = y 2 , we find ax + b = +/(ay 2 + b 2 — ac) ; so that / -m /'dx _ r JRJJ dy JR J J(ay 2 +b 2 -ac)' and now, as in ex. 13, p. 77, (i.) when a is negative, but b 2 — ac positive, fda__ 1 ■ n -i v / (-a)y y^V(-«) s/(P 2 -ac) 1 . ,J(-a)JR 1 , ax + b sin w „, x , =—77 — rcos * — ~V(-«) J(p 2 -ac) ,/("«) V(^ 2 -«c)' (ii.) when a, is positive, and b 2 — ac positive, (iii.) when a is positive, and b 2 — ac negative, f dx _ ! conh- 1 ^^^ - 1 cinh- 1 ffla; + 6 J */R~J(a) C ° x /(ac-6 2 )- /V /(«) x/(«c-& 2 )' We cannot have both a and b 2 — ac negative, because R would then be negative, and ,^/R imaginary, for all real values of x. When > we ma y substi- tute x — p = 1/y, when we obtain an integral of the same form as Jdx/^/R ; but it is more direct to substitute y = s /R/(x—p) ; and now 2 ;v = ^+A(a:-ff) 2 = (a+A>; 2 +2(&-j3A)3;+c+Ap 2 , (x — p) 2 (x — p) 2 a perfect square when A = — „ , „, ; so that r ^ O23 z +2op+c (a^+2&p + C ) 2/ 2 +6 2 - a c = {^±g±^±_ c } 2 ; and taking the negative sign with the radical dx ydy (x-pf~ *J{(ap 2 + 2bp+c)y 2 +b 2 +ac} ydx _ /* dx r dy (x-p)JR J(x-pfy J > /{(ap 2 + 2bp + c)y 2 + b 2 -ac} = 1 „;„-i s/(-aP 2 -2t>p-c)y s /(-ap 2 -2bp-cf l y/(b 2 -ac) 1 „ in -i N/(-^ 2 -2^-c)^/.R s /(-ap 2 -2bp-cf n Jitf-acXx-p) —W' or = * sillh - WW + 26p + p) W> J(ap 2 +2bp + c) C0&il ^/(ac-^^-y) -■< m ->' the real form to be chosen (ex. 18, p. 78). Here again, when a'p 2 + 26^ + = 0, so that x—p is a factor of R, the integral assumes an indeterminate form and the limiting value is the algebraical function JR ,JQ) 2 -ac)(x-p)' 400 INTEGRATION IN GENERAL. 193. We may consider the integral fdxj{x-p) s /R as the degenerate form of the elliptic integral Jdx/^/X, where X is a quartic function of x, as in ex. 39, p. 82, when X splits up into the factors (as — pf and R. Similarly the canonical elliptic integral fdsl,JS, where, as in ex. 39, p. 82, iSf = 4s 3 - g 2 s -g t = 4(s - e^s - e 2 )(s - e 3 ), when resolved into factors, becomes one of the degenerate forms of exs. 14, 15, p. 77, if two of the quantities e v e 2 , e s are equal. "We suppose e 1 >e i >e s ; and that the middle quantity e 2 is made equal either to the smallest e 3 , or the greatest e 1 ; and then with (i.) e 2 = e 3 , as in ex. 14, p. 77, /'ds _ f ds x ~^- 1 cin -i V( e i- e 3^-6i) . JB J2(x-e s ) /s /(x-e 1 ) 1 ain _i h_ .... fds (n) H=*vJ-j$ V2(e 1 -x)^(x-e 3 ) = ^(e 1 -ey inh ' 1 ie^ (x< ^' = 7(i^) cosh " 1 VS; (a;>ei) - Similarly, as in ex. 15, p. 77, f ds _ 1 . cos _! Mr^. Jl{e, x -x) f J{e z -x) s/(e x -e s ) *V e 1 -x' ^(e 3 -^ 1 -^) = 7(^ C0Sh " 1 V^^ < ^ = V(^) Sinh "" 1 V^ (a;>e 3>- INTEGRATION IN GENERAL. 401 In physical problems we require the integrals A=f— ^ , and C=f- /(a 2 + X) 2 (c 2 + X)* ■/(a (a 2 + A) 2 (c 2 + X) 3 / (a 2 + X)(c 2 + X)* \ + X /"» d\ 2 /c 2 N ° W J (« 2 + X) x /(c 2 +X") = s/Ja 2 ~c 2 ) C08_1 W+X A or =-77-5 57 cosh V(c 2 -ci 2 ) Va 2 + X' and, differentiating with respect to a 2 or c 2 , we deduce . V(c 2 + X) 1 x /c 2 + X .4 = ^^ — , cos 1 A / , (a 2 -c 2 )(a 2 +X) (a 8 -<**)* Va 2 +X V( c2 +X) | 1 cosh -! /£+*■ (c 2 -a)(a 2 +X 2 ) (c 2 -a 2 )* \a 2 + x' « 2 2 _ t /c 2 + X 0= ; COS 1 a/ ) (a'-^V^ + X) (a 2 -c 2 )* \a 2 + X 2 2 , , /c 2 + X or — cosn -l /_!_; (c 2 - a*)*/(c* + X) (c 2 - a 2 )* \ a 2 + X so that 2A + G= 7- a,^ 2 ,,,,^ . (a 2 + X) > /(c 2 +X) 194. To determine the integral Hx+K dx J A Ax 2 +2Bx + C J {ax 2 +2bx+ c)' where we may suppose A and therefore Ax 2 +2Bx+G positive for all real values of x, substitute y = (ax 2 + 2bx + c)/(Ax 2 +2Bx+C). Now, with the notation of ex. 8, p. 146, dy _ 2(Ab — aB)(x 1 — x)(x — x 2 ) d^~ (Ax 2 + 2Bx+Cf .. .. (Ayi-dKxi-x)* „ (a-Ay 2 )(x-x 2 ) 2 , yi *~ 4a* + 2.Ba;+C ' * ^ 2 .Ice 2 + 25a; +(7 ' where 2/ p t/ 2 denote the maximum and minimum of y, and x v x 2 the corresponding values of x. 2c r 402 INTEGRATION IN GENERAL. We may write L(x-x 2 )+M(x 1 -x) for Hx + K, and V=Ax 2 + 2Bx+C=P(x 1 -x) 2 +Q(x-x 2 f, R = ax 2 + 2bx +c = p(x 1 -x) 2 +q(x — x 2 ) 2 , as before, on p. 147 ; and now we find Kx-x^ + M^-x) ^ VJR dX __f L{x- x 2 ) + M(x x - x) V 2 dy ~J V % Jy ' 2(Ab-aB)(x x -x)(x-x 2 ) so that the expression for the integral is - LL' cos - V(2/M) + MM ' cosh - Wiv/Xtl or LL' sin - V(y/yO + -^^' sinh " V( - 2//2/ 2 )> according as i/ 2 is positive or negative ; that is, according as ac — b 2 is positive or negative ; that is, as R is always positive, or as R can vanish, for real values of x. When V = R, these forms become illusory ; and now fR-^dx = (ax + b)/(ac - b 2 ) JR, f(ax+b)R-?dx = R- i ; whence f(Hx+K)R-%dx can be determined, as an alge- braical function of x; and similarly f(Hx+K)R~%dx,.... Again, when Ab — aB = 0, the results are again illusory; but in this case we can choose a new variable, by changing x + b/a or x+B/A into x, so that we may make b and B vanish ; and now the integral 'Lx+M dx A Ax 2 +C J{ax 2 +c) consists of two parts, of which the first is a function of x 2 , of the form given in § 193 ; while, as in ex. 19, p. 79, INTEGRATION IN GENERAL. 403 r Mdx ... M I/O ax *+c \ J(Ax 2 + G)J{ax 2 + c) JCJ(Ac-aCy Q * y\cAx 2 + C) M _, , l/Cax 2 + c or rcosh" JCjtaC-Acr™' *V\cAx*+C M sinh -i If C ax*+c M . 1 If C ax 2 +c\ or jcj(aC-Ac) sinh Vv~cI^qFUV ; x reducing, when aO— J.c = 0, to ^ /i a x 2 + c\ ' By differentiation of the integral Hx + K dx J A Ax 2 + 2Bx+G J(ax 2 + 2bx + c) with respect to A, B, or G, we can deduce the results of ix dx J(A {Ax 2 + 2Bx + C) n s J{ax 2 +2bx+cj The general linear substitution may be written , _ e(x 1 -x)+f(x-x i ) x ~E(x 1 -x)+F(x-x 2 )' and the form of the integral will then be unchanged ; in particular V and R are interchanged if we put E= J(P P ), F= JiQq), e = Ex v f= Fx 2 . Examples. (1) Integrate x—l x + 1 (X 2 + X+l) > /(x 2 -X+l) (x'-Xliy^ + JJ + l) L(x-l)+M(x-2) (Sxi-lQx+fyJibxL-lQx+U)' L\x-\)+M(x-2) (bx*-\6x+U) l J$x 2 -lQx+$) Hx+K (3a; 2 - 10a + 9)J{x 2 - 8x + 10)" 404 INTEGRATION IN GENERAL. (2) Prove that the maximum and minimum of (ax 2 + 2hxy + by 2 )/(Ax 2 + 2Hxy + By 2 ) are given by the common conjugate diameters of ax 2 -f- 2hxy + by 2 = c, Ax 2 + 2 Hxy + By 2 = G. (3) With the notation of ex. 8, p. 146. prove that ax + b = —p(x 1 —x) + q(x — x 2 ), bx + c= l^x 1 (x 1 — x)— qx 2 (x — x 2 ) ; Ax+B=-P(x 1 -x)+ Q(x-x 2 ), Bx + C= Px 1 (x 1 — x) — Qx 2 (x — x 2 ) ; and (B 2 -AC)R 2 + (Ac + aC-2Bb)RV+(b 2 -ac)V 2 = (Ab-aB) 2 (x 1 -x) 2 (x-x 2 ) 2 . If sb' = - (Bx + G)j(Ax + B), or - (bx + c)/(ax + b), B R' Ac+aC-2Bb V { V _ Ac+aC-2Bb V + V'~ AC-B 2 ' ° r R + R'~ , ac-b 2 - (4) Prove that if X denotes the reciprocal quartic ax 4, + ibx? + 6cx 2 + 4>bx + a, fx 2 -l dx 1 V(- a yi where aX = (ax 2 + 2bx + a) 2 + Dx 2 , B = 6ac- 46 2 - 2a 2 . fx 2 -\ dx 1 ' J(2a-Qc)JX (11 -V ~W+\ JX-J(2a-6c) sm J(-B)(x 2 +l)' 1 . , t J(Qc-2a)JX or V(6c-2«) V(-^)(^+i)' 1 1 ^/(6c-2a) x /Z , and (6c-2a)X = 4{te 2 -(a-3c)a;+6} 2 +i)(a; 2 +l) 2 . (5) Determine in a similar form /*a; 2 +l dec ,, f x 2 +\ dx J -~T 1JX' an V x 2 -l JX" where X' = ax i -4:bx 3 +6cx 2 +4 c bx+a. INTEGRATION IN GENERAL. 405 195. Integration by Rationalization. Any integral of the form f S+T(ax+byi r J U+V(ax+b)"' r ' where S, T, U, V are rational integral functions of x, can be rationalized, that is, can be made to depend on the integration of a rational algebraical fraction by the substitution ax + b = y r ; for then adx = ry r ~ 1 dy ; and the integral becomes the rational algebraical integral ' S+TyP ry*- 1 I U+Vyz a dy - Any integral of the form / 8 + TJR U+ VJR ax ' where R is the quadratic form ax 2 + 2bx + c, can be rationalized, (i.) when b 2 — ac is positive, and a negative, by the substitution J(-a) x /R _ 2y ax + b _ \-y 2 J{b 2 - ac) ~ 1 + y 2 ' tnen J{b 2 -ac) 1 + y 2 , — adx tydy and T^^ro+y? 5 (ii.) when b 2 — ac is positive, and a positive, by s/(a)JR _ 2y ax+b = l+y 2 J(b 2 -ac) 1-y 2 ' J(b 2 -ac) 1-y 2 ' adx tydy and J(P-ac)-(l=tfy' (iii.) when b 2 — ac is negative, and a positive, by J(a)JR _l+y 2 ax + b 2y J(ac -b 2 )~l-y 2 ' J(ac -b 2 ) 1 - y 2 adx _ 1+y 2 j and J(ac-b 2 )~ l TX^W V - i;2 406 INTEGRATION IN GENERAL. In (i.) we may suppose i/ = tan|0, and in (ii.) and (iii.) 2/ = tanh£w; and then the integration is changed to an integration of a rational function of cos 6 and sin 6, or cosh u and sinh u, with respect to 6 or u. Examples. — Integrate with respect to x x 3 x n „ „ , 1 — 77 — — *> — r, — 7—x' x n ,J(x — a), .-— — - — *-> 1 1 x xV(l+* 2 )' (*+l)V( a! + 2 )' (» + 2)^/(^ + 1)' _JL 1 1 s/(a?+x*) 1 ' 1 u+x ' x s /(x 2 + 3x + 2)' (x + l) s /(x 2 +AJ+1)' 1 1 bx + c (x 2 +l)$' (ax 2 + 2bx + cf (ax 2 +2bx+c)$' Hx+K 1 _ 2 , (ax 2 + 2bx + cf x s J '{ax 2 +c)' ( - 1 ~ 2x ^' J(x 2 + a 2 .x 2 + b 2 )' (aj*-l)*' ^ 1+e ^' (ffl^+J) -5 ?, (substitute a + bx~ n = y). 196. Integration of Circular and Hyperbolic Func- tions. To integrate powers and products of cos a; and sin a; the most general plan is to convert them into cosines and sines of multiples of x, which are immediately integrable (ex. 9, § 40). To integrate any odd power of cos a; or since, say (cos x) 2n+1 or (sin x) 2n+1 , we write them in the form (1 — sin 2 o;)"cos x, or (1 — cps 2 a;)"sin x, and expand by the Binomial Theorem ; then each term is immediately integrable, since by § 40 and (a), p. 84, INTEGRATION IN GENERAL. 407 {smx) m cosxdx= ( aiax T +1 ' m+1 (cos a:) m sin xdx = — — /c.. - y v m + l A similar method will serve to integrate (sina^cosa;)*, where either p or q is an odd integer : also to integrate any powers of vers x. The same processes apply when cos a; and sin a: are replaced by the hyperbolic functions cosh x and sinh x. Examples. — Integrate with respect to x, cos mx, sin(ma; + 'n.), sin 2x cos Sx, cos Sx cos 5x, sin 3x sin 5x, sin (mx + n) cos (px + q), sin x sin 2x sin 3a;, sin a cos a;, sin 2 a;cosa;, (sin a:) 3 cos x, (sin a;) m cos x, sina;cos 2 x, sina;cos 3 a;, sin x (cos x) m , sin 2 a!, sin 3 x, sin 4 a;, cos 2 a;, cos 3 a;, cos*a;, cos 2 ma; cos nx, cos s mx cos nx, ..., also the same functions with cosh x for cos x, and sinh x for sin x. 1 97. The integration of the remaining trigonometrical functions is a little more complicated ; thus, by (v). p. 85, cot xdx = 1 -. — dx = log sin a? = | log vers 2a;; ^/ sin cc t&nxdx = / dx = — log cos x = log sec x ; J COS X 6 & » wh ile _/coth a;cfcc = log sinh x, ytanh xdx = log cosh x. Again Jtax>?xdx=J( se*» a +2/ l-y = log(l + 2/) - log(l - 2/) = log (1 + 2/)/(l - y) , 1 + tanAa; , , . . /l+sina; = l0 gr3ta^ = l0gtan(i " + ia;) = l0g Vl^in^ = log(sec x + tan a;). Similarly /cosec a^a: = / ,y * 2 =/dy/y = log y = log tan Jo; = log ^"^ = log(cosec x — cot x). Similarly to integrate sech x or cosech x, we may rationalize by the substitution tanh \x = y ; and then / sech xdx = / -.', 2 = ^ tan ~ X V = % tan ~ x tanh \x, /cosech xdx = /dy/y =logi/ = log tanh %x. The substitution of 1/ for tanJQ will apply for any function of the form f (cos 6, sin 0)/F(cos 6, sin Q) ; the function being thereby reduced to the form U{y-b) I I sin £(0-/3) . Ii.(y-a) 0r TLsmi(e-a)' and this again by partial fractions (§ 187) to the sum of terms of the form A/(y — a), or A cos|0 cos|a cosec£(# — a) = A cos 2 |acot|(# — a) — \A sina- (Hermite, Proa London Math. Society, vol. IV.). INTEGRATION IN GENERAL. 409 1 98. By the substitution of z for sec x, cosec x, sech x, cosech x, the results can be expressed more directly ; thus with z = sec x, dx = — 77-5 — =--, z s/( z l ) and /sec xdx = / . — — =cosh _1 2, by (x.) (p. 85) = cosh - ^ec x = sinh ~ Han x = tanh ~ x sin x = 2 tanh " Han \x. Similarly, with z = cosec x, and as a corrected integral, /cosec xdx = / , 2 — yt = cosn " V(^ 2 -i)" = cosh - 1 cosec x = sinh - J cot x = tanh - 1 cos a*. Again, with z = sech x, or cosech x, /sech cedse = /— ttz sr = cos - l z = cos ~ 1 sech x = sin " ] tanh x = tan ~ 1 sinh x = 2 tan " Hanh \x. y cosech xdx = / .,, ", — sr = sinh - ^ / JQ-+Z 2 ) = sinh _ 1 cosech x = cosh ~ ^oth x = log coth £#. 199. To integrate 1/P, where P = a + b cosx + cs'mx, we can proceed in a similar manner, and put 1/P = y; and now — a = bcosx + csinx, V so that — 2 -^- = bsinx — ccosx=Jib 2 + c 2 — ( a) \ ; _. r dx f_ dj£ &n J a+b cos x+ c sinxZ/ +/{( — a 2 + b 2 + c 2 )y 2 + 2ay — 1}' which by the results of § 191 can be expressed by .. 1 ^ a + (-a 2 + b 2 + c 2 ) y W y/(a 2 -b 2 -c 2 ) COS J(b 2 +c 2 ) 1 , aP-a 2 + b 2 +c 2 -Jiat-V-c 2 ) 00 * J{b 2 + c 2 )P ' 410 INTEGRATION IN GENERAL. 1 ^P-gg+^+c 2 or ( u -) x / ( _ a 2 +6 2 +c 2 ) cosh ^(0+^ •• as in ex. 34, p. 81. Similarly to integrate 1/Q, where Q = a + b cosh aj + esinha;, we put 1/Q = 2/ ; and obtain the results of ex. 35, p. 81. Between the limits and tt, (i.) = Tr/, s /(ei 2 — 6 2 — c s ) but (ii.) is illusory, as the function to be integrated becomes infinite between the limits of integration, Again, between the limits and oo , Jdx/Q or Jdx/(a + b cosh x+c sinh a;) is given by means of ex. 35, p. 81, as 2 tan- ^ ( - Q2 + 62 - c2) J(-a?+b 2 -c 2 ) a+b+c or 2 tanh- 1 ( V(a»-6'+c') 1 J{a i -b i +c 2 ) coth-4 a+b + c )' 200. The integration of (a + 6 cos 6) ~ n is effected (i.) when b/a < 1, by the substitution a + frcos 6= r , or tanA0 = /V /( = JtanAv, a — bcosv 2 y\a — b/ 2 ' equivalent geometrically to a change from the true anomaly 6 to the excentric anomaly v (§ 177) in an ellipse of excentricity b/a ; and now (a + 6cos0)» (a 2 -b 2 ) n -iJ ' (ii.) when b/a > 1, by the substitution a + 6cos 6= j = — , or tan IQ=*{, jtanhAw, a — bcoshu ■* y\b — a/ i a change from the true anomaly 6 to the hyperbolic excentric anomaly u in a hyperbolic orbit of excentricity b/a ; and now INTEGRATION IN GENERAL. 411 = r /(b cosh u — a) n ~ l du. (a + b cos d) n (b 2 - a 2 ) n - *-/ When n = l, we obtain, as in ex. 31, p. 80, y r dd _ i> _ 1 _ 1 acos0+6 + 6^0 _ J(a* - b 2 ) ~ V(« 2 - & 2 ) C ° S " a + b cos ff u _ 1 , _ 1 acos8+fc ~ s/Q> 2 -a 2 ) ~ J{b 2 -a 2 ) ' a+b cos ff The integral for to = 2 is required, as in § 180, for the mean anomaly nt in an elliptic or hyperbolic orbit. Eeciprocally, since cos = (a cosh u — b)/(a—b cosh u), f- — — = — - — -. f{a+b cos ey-w, J (b cosh u - a) n (b 2 - a 2 )" " *-/ /^ — = , f(a + b cos e) n ~ Vsec 0)"d0 ; */(&-acoshu)» (b 2 -a 2 ) n ~iJ K ' including all possible cases required in the integration of (a+/3coshw)-". The results for n = 1 will be found in ex. 32, p. 80. To integrate (a + 6 sinh w.) - ™, substitute a + b sinh u = (a 2 + b 2 )/(a — b sinh v) ; and now f- — = j fla-b sinh v) n ~ l dv. J {a+b sinh u) n {a 2 + b 2 ) n " I J v If we put 6 = gdu, ^ = gdt;, and 6/a = tany, then sinhu = tan# = t-H^ =tan(0 + y); a — otanrf. ^ ' so that 6 = + y ; and now /^ — = — p- /"cos 0)»-*(sec 0)"d0 (or dd. ) y(a+6sinhu)» (o«+ &*)*"-' and the result for to = 1 is given in ex. 33, p. 8L 412 IN TEGRA T10N IN & EN ERA L. With the notation of the confocal conies of § 177,. tan 16 = coth \u tan §v, tan \& = tanh \u tan \v ; and taking the logarithmic differentials d6 _ — du _ dv dd' du _ dv sin 8 sinh u sin v sin 0' sinh u sin v ' while (cosh u — cos v)(cosh u + cos 0) = (cosh u + cos w)(cosh it — cos 0') = sinh 2 u, (cosh u — cos v) (cos -y — cos 6) = (cosh it + cos v) (cos 0' — cos v) = sin 2, y ; so that Asinhur-^, /• coshw _ cos y (cosh u + cost;)" ^/ v ' /Vsini0 2 ™ _1 ci!tt /" -, . 7 „, / , u , rr= /(cos 6- cos v) n dd; etc. ./ (cosh it+ cos v) y v As applications the student may evaluate the expres- sion for the area PpP'p' bounded by the elliptic arcs u, u' and the hyperbolic arcs v, v' of the confocals of § 177, or bounded by the circular arcs w, u' and v, v' of the system of dipolar circles of § 175, given by the integral II . ' " dudv = ^cyAcosh 2u - cos 2v)dudv, or //-, — , s ; and determine the eentroid. JJ (cosh v + cos uy Prove also that the C.G. is at N (fig. 54, i.) of (i.) a circular wire in which the line density varies as SP~ 2 ; (ii.) a circular area in which the surface density varies as SP _i ; (iii.) a spherical shell in which the surface density varies as SP' S ; INTEGRATION IN GENERAL. 413 (iv.) a solid sphere in which the volume density varies as $P~ 5 ; the boundary in each case being defined by u = a positive constant, so as to include N in the boundary. Determine also the C.G. of a quadrant of an elliptic wire, as in fig. 55, in which the line density varies as p, 1/p, xp, x/p, ...,f denoting the length of the perpendicular from the centre on the tangent. 201. The integrals (i.)/(#cos0 + K)d6/ (A cos^O +2B cos d +0), (ii.) f{R cosh u + K)du/(A cosh 2 w + 2B cosh u + C), (iii.) f{H sinh u + K)dvj(A sinh 2 u + 2B sinh u + G), are reduced to the form of the integral of § 194, by the substitution x = cos 0, cosh u or sinh v ; and now R is replaced by sin 0, sinh u, or cosh v ; while by writing sin 2 (9 = { (1 — cos a cos Of — (cos 6 — cos a) 2 } /sin 2 a, F=^cos 2 + 2J5cos0 + C = P(l — cos a cos 6) 2 + Q(cos Q — cos a) 2 , and 2/ = sin 2 asin 2 (9/F; 1 _ P y = (P + Q)/( cos e - cos a ) 2 / V, l + Qy = (P + Q)/(l - cos a cos 0) 2 / F ; then 1/P and- 1/Q are the maximum and minimum values of y; and the integration is expressed by inverse sines, circular and hyperbolic, of ^(Py) and ,J(Qy). Similarly for cosh u and sinh u, by writing sinh 2 u = { (cosh fi cosh u — 1 ) 2 — (cosh u — cosh /3) 2 } /sinh 2 /3, cosh 2 ii = {(sinh y sinh « + 1) 2 + (sinh it - sinh y) 2 }/cosh 2 y. As numerical examples, integrate {X(3 - cos 0) + if (1 - 3 cos 0) } /(5 cos 2 - 6 cos 0+5) {Z(cosh u- 2) + i/(2 cosh w- 1)}/(5 cosh 2 it-8 cosh u + 5), (£Tsinh i; + -fiT)/(sinh 2 v-sinh v + 1), (5'sinhw + A'')/(6sinh 2 w — 4 sinh w + 9). 414 INTEGRATION IN GENERAL. Examples. — Integrate with respect to *, (1) tan* sec 2 *, tan 2 * sec 2 *, tan 3 * sec 2 *, (tan x) m sec 2 a;, cot x cosec 2 *, cot 2 * cosec 2 *, (cot *) m cosec 2 *, tan x, cot a;, tan 2 *, cot 2 *, tan 3 *, cot 3 *, tan 4 *, cot 4 *, sec* tan*, sec 2 *tan*, (sec*) m tan*, (cosec*) m cot*, sec 2 *, sec 4 *, sec 6 *, cosec 6 *, sec*, sec 3 *, cosec 3 *, sec * cosec *, vers *, vers 2 *, 1/vers * ; also the cor- responding hyperbolic functions of *. 1 1 11 ^ ' cos*+sin*' 6cos* + csin*' = r-s- — T-s-, sec*sec2*,tan*tan2*. sira-sm^ 1 — sin^a sin'n (3) Prove that (i.) sin * sec(* - a)sec(* — b) = cos a cos b cosec(a-6){sec a sec(*-a)-sec 6 sec(*-6)} ; .... sin* _„ sinacot(* — a) sin(* — a)sin(* — 6)sin(* — c) sin(a — 6)sin(a — c)' ..... sin 2 * _ sin 2 acosec(* — a) (111 \ == > i £. • sin(* — a)sin(* — 6)sin(* — c) sin(a — 6)sin(a — c)' .. . sinm* 1 . . (wherea^k-jr/n, (iv.) — = s -2(-l)*smmacotl(*-a) v , J I v sinm* zn ' ' and m 6 - ,6 . . /**"■ c&» _ 1 -1_ COS" a or —ma. 2\ cosn " - ■ ' ? =0-23180. + 3cos0 <(t ' =0-27465. $ + 5cos0 ( vii -) /* £■ a** 8 m2 = 0-03494. >/ (5+3 cos 0) 2 ( viii / *£■ -^ m2 = 0-05267. ./ (3 + o cos 0) , R . .. . /•»• g cos 6+K j._ Hsm^a + iTcos|a {b) (1 ' ) ycos 2 0-2sinacos0+r y_ iS /(2coB 8 a) ""' if cos a is taken positive. ("■) /-r v ,y ere i?cos0 + .A: •dfl : cos 2 # — 2ab cos a cos + 6 2 _ j^( a 2 +6 2 -c 2 )+ir v /(a 2 +6 2 + c 2 ) , and Jxe x dx = xe x —Je x dx = xe x — e x . (ii.) to integrate log a-, take u = logx, dv = dx, v = x; then _/log xdx = x log x —Jxdxjx = x log x — x. (iii.) sin ^Jx, put s /x = z, then Jsin n/xdx =ysin z . 2zdz INTEGRA T10N IN GENERAL. 417 = — 2z cos z+ 2/cos zdz = — 2z cos z+ 2 sin 2; = — 2^/a; cos ^x + 2 sin „/a;. (iv.) /sin _1 a;dx = a3sin" 1 a;— / .,, — =- = x sin ~ *a; + ^/(l — a? 2 ). (v.) to integrate cos mx cosh nx ; here cos mx or cosh ma; may be taken indiscriminately for u, and we must integrate by parts twice. „ . .„ , cfo sinhwa: ±or instance 11 cosmx = it, cosh nx=- T -, v= , ax n /cosmxcosh.nxdx = -cosmrcsinh nx-\ /smmxsmh.nxdx J n nj =-cosma;sinh7iaH — i smmxcQshnx — ~- i /cosmxcosh nxdx; n n* ni/ and therefore, by transposition and division, y, , msinmxcoshrae+w.cosmajsinlma; ,. ^ cos mx cosh nxdx = 5— — 5 (1. ) m i +n i In a similar manner . . 7 m sin mx sinhna; + ?i cos mx cosh nx ... . cosmxsmhnxdx= 5— — 5 (11.) m 2 +n 2 7isinma;sinh?ia! — m cos ma; cosh ma; m?+n 2 smmaicosh nxdx = ^ , ^ 2 — — (111.; , wsmma;cosh«#— mcosmajsinhna;,. . sin mx smhnxax = «-. — 5 (iv.; By addition of (i.) and (ii.), (iii.) and (iv.), or independ- ently by integration by parts, f t «,cosma; + msinma; , s e nx cosmxdx = e nx m 2 +n 2 v v Tismma;— mcosma; , . > e»sinw;(ii;=« M m '+m» (V1-) 2d 418 INTEGRA TION IN GENERA L. , With a subsidiary angle a given by tana — 7n/n, /e^cos mxdx = e nx cos(mx — a)cos ajn, ie nx &in mxdx — e™* sin(ma; — a)cos ajn. Again, by successive integration by parts, e^udx-ae^Hu — a-p +a 2 -r-2—a s -T-^ + ...j. y n d n v, _ d n -h) du d"- 2 v U dx n dx"- 1 dxdx n -* + -" the integrated terms of which can again be integrated by parts. We can establish Taylor's Theorem by successive inte- gration by parts ; for i(a+h) — ia=/ i'xdx ; a a+h r*a+h = - {{a+h-x)i'x} + /{a+h-x)Fxdx a =kta-l 2l (fl+h- xfi"x\ + - j /(o + h - xff'xdx a = hi' a + T ,f "a + . . • + Jyf n a +~/( a +h- xfl^xdx ; a so that, as in § 114, f(a+ft)-fa-Afe*-&"a- - --Fa • 2! n\ . = R=-j fi L a+h-x) n i n + 1 xdx. INTEGRATION IN GENERAL. 419 It is assumed here that fas, i'x, i"x,... i n x, i n+1 x are all finite between the limits a and a+h of x; and now if P denotes a certain average value of f n+1 as between the limits a and a + h, which we may denote by i n+1 (a+6h), a Examples. — Integrate by parts, asV, asV~, x i e a * +i , ...; e Vx t c ^» exp^/a; ; a; m loga;™ (logo;) 2 , (log a;) 3 , (logo;) 4 , (logo;) 5 , ... ; as 2 cos x, a: 3 sin x, ...; cos 4/*, sin^/as, . . . ; e aa!+b cos(mx + n), e^ +h sm{m,x + n), ...; cos _1 a;, tan -1 a;, coWas, sec -1 *, cosec _1 aj, vers -1 *; cosh" 1 ^, sinh _1 a;, tanh -1 *, ... ; (sin* 1 *) 2 , (sinh _1 a;) 2 , (cosh -1 *) 2 , (vers -1 *) 2 ; tan-^as, tanh _1 4/a3, ... ; J(a 2 -x 2 ), J(x 2 -a 2 ), s /(a 2 + x 2 ), (a 2 - x 2 f, x 2 J{a 2 -x 2 ), ...; a;sin _1 a;, ^cos" 1 *, a3 3 cosh -1 a; ... ; log x . sin -1 *, log x . cosh" 1 * ; (sec as) 3 ,(sec as) 5 ,(cosec x) 5 , ... sec astan 2 as,seca:tan 4 a;, . . . 203. The integration of J {a 2 - x 2 ), J(x 2 - a 2 ), J (a? + x 2 ) is required in the quadrature of the circle, ellipse, and hyperbola (§§ 50-54); thus for example, integrating by parts for the quadrature of the circle, J {a 2 - x 2 )dx = x J (a? - x 2 ) +J j {p? _ x 2- ) = xj(a 2 - xS ) +a y u a ^ x i) -JJW ~ x *) dx > (and, by transposition and division) = % s /{a 2 -x 2 ) + ^a 2 am-\x/a) (§ 50). 4,20 INTEGRATION IN GENERAL. Similarly fsjifl- -x.x— ft)dx = |(*-^)V(a-z.z-/3)H(«-/3)W^; so that taken between the limits a and /3, at which ^/(a — x.x — fi) vanishes, and between which it is real, the integral is ^(a — /3) 2 . More generally, denoting ax 2 + 2bx + c by R, and integrating by parts y r /r>7 ,ax + b /r> ,b 2 — ac f dx JR dx = i — r> /R- ir—J -^ , and the result has one of the three forms of ex. 28, p. 80. When a is negative and b 2 — ac positive, the value of x is restricted to lie between the roots, a and /3 suppose, of the equation R = 0, for R to be positive and ^/R real ; and now, taken between the limits a and /3, /^ /D , ac-b 2 7T a Suppose for instance that the area of the ellipse is required, when it is given by the general equation of the second degree (§ 13) ax 2 + 2 hxy + by 2 + 2 gx + 2fy + c = 0. Solving this equation as a quadratic in y, hx + by +f= J{{h 2 -ab)x 2 + 2(fh-bg)x+P-bc} ; so that (§ 49) %i " 2/ 2 ) = 2 J{ (h 2 - ab)x 2 + 2(fh - bg)x +f 2 -bc}; and the area of the ellipse isjty^ — y^dx, taken between the limits for which the quantity under the radical is positive; and therefore the area is 7rA/(a& — A. 2 )*, where A denotes the discriminant (ex. 9, p. 139). This result must be multiplied by sin w, if the co- ordinate axes are inclined at an angle w. INTEGRATION IN GENERAL. 421 204. Formulas of Reduction. A formula of reduction in the Integral Calculus is a formula, obtained in general by integration by parts, by which one integral, say u n , is made to depend upon a simpler integral, say ia„_i, or tt„_ 2 ; then by successive substitution in the formula of reduction, we finally arrive at an integration which can be effected. Suppose, for instance, u n =_/(sm Q) n dQ ; integrating by parts, with u= (sin 0) m-1 , dv = sin 6d9, v= —cos 6, u n =J (sin 8) n dd=f (sin dY^sm BdB = -(sin flf-icos + (w-l)/(sin 0)»- 2 (cos QfdB = +(ti-1)u,„_2 — (n— l)u n ; or nu„ = +(?i-l)u„_ 2 , 1 n-1 u n = — (sin0Y ,, ~ 1 cos(H ttn-2, ■nr n a formula of reduction. Similarly /"cos 0) n d0 = -sin 0(cos fl)»-i+Ztjli/'cos 0)"- 2 d0, another formula of reduction. Taken between the limits and ^7r, y"iT 17—1 /~i ,r (sin 0)»cM = ^-y(sin 0)»-W, o o (cos 6) n de=^-/(cos dy- ^j^$jn-w^ +M ^ + ' N ^ ,,^ /I n«7 (tana;)' 1 " 1 (13) u n =/(ta,nx) n dx = ± '- « n _ 2 . /* (coto;)™ -1 (14) u n = /(cotx) n dx=—- -^r M„-2- . „, /! ._ 7 (sec a;) n_2 tan a; , n — 2 (15) U n =/(seCX) n dx= K - ^-^ + — y^-2. /7 . 7 (cosecoi)' ,i ' -2 cotrc ,« — 2 (16) u„ =/(cosec x)»cZa! = - v ^ + i ^ Z jU»_ s . "^ ~,2j.^2 u n-i- 424 INTEGRATION IN GENERAL. (17) u n = /(versx) n dx = * 1 w„_i. /io\ /"„ xn ■ .nx"- 1 n(n-l) (18) w,„=/aj' l cosma3 = — sin«+ — s-cosmx — - — ^m,,.* v J m m 2 m 2 (19) u n =/eP*(cosx) n dx = -+ ni +p > ■»i(tc— 1) (20) « (Wi B) = /(sin a;) m (cos a:)"da: _ (sin a;)™ +1 (cos a;)" -1 tc — 1 ~ rn + w " + 7^+^°"' " - 2) ' (sin ccY™ - 1 (cos x) n+1 in — 1 Or — ^ ^ f '- ; U(m-2,»), m+n m+n (ro-1 )(sin a;) m+1 (cos a;)"- 1 -^-! )(sin jb^Xcob a;)" +1 (m + nX m + n ~ 2) (m-l)(w-l)u(^ 2|W . 2 > (m+n)(m+n— 2) ' (21) /(sinx) m (cosx)* n +idx = , ,,„,'", ,,,.. , Y (m+l)(m+3)...(m+2n+l) (22) /^Jn^to^) 2 ^- 1 - 3 - 5 "^ 171 - 1 ^ 1 - 35 "^ 271 - 1 ^ (U)J(Bmz) (cosz) ^- 246 ..2 m (2m+-2)...(2m+2«) 2" /2!n=(»-i)i /"■CO since Tl = /e ~ x dx = 1, also TO = oo . Changing x into x 2 gives Ym = 2/e-° ? x im - 1 dx; o so that we may express the product of Tm and Tn by TmTn = 4 le - x "x^ n - 1 dx/e ~ Af n ~ Hy o 426 INTEGRATION IN GENERAL'. = 4 / /e-*- y i x im - V" " 1 dxdy = 4 /e-'V 2m + to - 1 dr/(sin e^-^cos ff^-WQ (on changing to polars, with o? = r cos 6, y = r sin 0) = 2r(m+ n)/(sin fl^-^cos fl) 2 ^" 1 ^. o The definite integral /x m ~ 1 (l-x) n - 1 dx = 2/(sin e) 2m -Kcos df^MO, o o on putting a3 = (sin0) 2 ; and is called the First Evlerian Integral, and denoted by B (m, n) ; so that TmTn = T(m+n)B(m, n). v Also, on substituting a — x = (a — /3)sin 2 #, x — f3 = (a—ft)cos 2 6, J (l-x) m -\x- p) n -Hx = (a-^)" 14 "- 1 ^™,, w). This integral has been evaluated for positive integral values of m and tc (p. 424) ; and for fractional values it depends on the tabulated, values of the Gamma Function, by means of the above relation B(m, n) = TmTn/T(m + n). When m + n=l, then (§ 189) Tm T(l - ra) = 5(m, 1 - m) = 2 /(tan 0) 2m - 1 cZ0= TrcosecmTr. o Putting m = £, gives I^a/ir, and T$ = \T% = %Jir; and now if Tm is calculated from m = 05 to m = 1, then Tm is known from m = 05 to m = ; and thence Pn. or log Tn can be tabulated from n =■ 1 to n = 2 (Bertrand, Calcul Integral, p. 285), and the graph of Tx can be drawn for all positive values of x. INTEGRATION IN GENERAL. 427 The substitution of x = z/(l+z) or 1/(1+0) shows that As an application, let us determine the area A of the positive quadrant of the curve x n +y n = a n ; then A =j(a n - x n )ndx = ^a?f(In 0)1 - *(cos 0)^+W (on substituting- ce" = a"sin 2 0) =^VI,I +1 ) = ^r(I+i)/if+i) = 2 -( ri ) 2 A-- 2n\ nJ I n For instance, if n = 2, we obtain the area of the quad- rant of the circle \ir(^. Also -y=|«(r|)y(rir|). By integrating, as in § 134, throughout the space in which x, y, z are positive, but (x/a) m +(y/b) n +(z/c)P-l is negative, we can show in a similar manner, that the volume V of this space within the surface is given by abc \m / \n J \p // \m n p 1 while 4^4r(I +1 )r(I+i)r(i+i)/rP+l+i+i),.. a?bc 2 \m / \n J \p // \m n p I agreeing with § 134 when we put vi = n=p = 2. Again, the length of a branch of the curve r n = a ,l cos nQ (§ 172) can be expressed by By g— , ^ 428 INTEGRATION IN GENERAL. 206. Bernoulli's and Eider's Numbers as Definite Integrals. Making use of the relation, with mu = x, pas /- /e-^uP-Wu = m- p /e- x x p ~ 1 dx = m- p Tp, o o then, with the notation of § 112, T p Tp=/(e- u +e- 3u +e- 5u + ...)u p ~ l du o 2 P T 2 2,_1 Z* 00 so that S p = ^j — ^r = j^ — ,-p /cosech u . u p - l du ; and Jg„ = (22m _ 1 . in J cosech u. u^-Hv, ; ^ ' o so that, on putting u=\-kv, y-as cosech £7™ . v 2 " " ^ = 2 2 »(2 2 » l - l)B„/2n, o the coefficient of a3 2n_1 /(2'n, — 1)! in the expansion of tan x, and this is always an integer ; namely 1, 2, 16, 272, .... Similarly (§§ 112, 113) UpVp = f(e- u -e- 3u + e- bw ...ya p - 1 du = \ /sechu.u p - 1 du; o E n = 2(2 n)\ U 2 n+i/(Wf n+1 =/sech ^ . v 2n dv ; o which, since E„ is an integer, shows that this definite integral is always an integer, namely 1, 5, 61, 1385, .... Put sech \irv = cos , \-kv = log(sec (p + tan ), and we find (§ 1 13) \alf{\ INTEGRATION IN GENERAL. 429 /***■ J {log (sec + tan J {log (sec + tan )} 2n - 1 d = 2(2n-l)\U'2 n . o Thus, for example, the area between a branch of the curve m6 = gd (r/a), or r/a = log (sec m.0 + tan md), and its asymptote is {log (sec m6 + tan m$)} 2 <2# o = $a\ fafEJm = ^a'/m. 207. Differentiation and Integration of a Definite Integral, and Applications. Denoting the indefinite integral with respect to x of a function i(x, c), involving a parameter c (§ 105), by i^x, c), then the definite integral 1 = li [x, c)dx = i x {a, e) — f-j(b, c) ; b and now, if a, b, c are functions of some variable t, dl_d£ da ol db dl dc dt da dt 36 dt dc dt b of which a geometrical interpretation can easily be constructed. We can also integrate I according to a similar rule. By means of this method of differentiation and inte- gration we can show the connexion between different classes of integrals, as already done in §§ 193, 194, and also verify the solution of certain differential equations, when the solution is given as a definite integral, as in § 184. 430 INTEGRATION IN GENERAL. . Thus, if a and /3 are positive, and 1= , ^ tan' 1 ^/-, /•fr d6 1 _ 1 / acoB»e-j8Bin'fl \* ir . Jacos i 0+psin 2 6~2 > /(apf OS \a cos 2 6 + /3 sinW f"° du _ 1 _ 1 /acosh 2 w,-/3sinh 2 «\ J aCOsh 2 w+/3sinh 2 tt _ 2 x /(a/3) VaCOsh 2 u+/3sinh a u./ 1 =1; and /^ d0 - (-^Y 3 | 9 Vj y(acos 2 0+^sin 2 a)' 1 + 1_ n! \3a 3/3/ ' ./(« cZtt _ (-l)V 3 d\ n T («cosh 2 u. + /3sinh 2 w)' l + 1 n\ Xda dfiJ We can verify that the definite integral cos(2W — x sin v)dv, fi which we have denoted by irJ p (x) in § 184, and called the Bessel function of x of the £>th order, is a solution of Bessel's differential equation given on p. 185. For dividing the definite integral into two parts y=/ cos pv cos (x sin v)dv, and z = / sin pv sin(ai sin v)dv ; o o and using accents to denote differentiation, x % y" + xy' + (x 2 —p 2 )y = / { — aj 2 sin 2 u cosyv cos(a: sin v) — x sin v cospv sin(cc sin v) o + (x 2 —p 2 )cospvcos(xsmv)}dv = {x cos v cos pv sin(a; sin v)—p sin py cos(a; sin v)} g = p sin. p-n-; while x 2 z"+xz'+(x 2 —p 2 )z = { — x cos v sin pv cos (a: sin -y) +p cos pu sin(a: sin v)} g = —xsinp-Tr; and therefore 2/ and ~z each satisfy Bessel's equation if p is an integer ; and y or z vanishes as p is odd or even. INTEGRATION IN GENERAL. 431 The student may verify in a similar manner that if PJju.) (p. 314) satisfies the differential equation of ex. 6, p. 276, |{( l -^f} +1l(% + l)P = °' wPnifi.) =/{fx - J(n 2 - 1 )cos 0}»cty =J{1 + -vV - l)cos V4 — ^- the one integral being transformed into the other by (§200) M -V(M 2 -l)cos = /x + V(/x2 1 _ 1)cos ^ = ^.. Putting yu = cos u,- or cosh v, according as /u. is < or > 1, S+ «*«g+«(n + l)P=0, P + co th^-^ + l)P = 0; so that the solution may be written pie 7rP„(cos u) = / (cos it — i sin w cos /(/u 2 - l)cosh \[r} n d\ls, by means of the substitution (§ 200) U+ V(m 2 - l)cosh 0} {/, - ^/( M 2 - l)cosh V-} = 1 ; and Qni/J-) = °° when /x = 0. Example. — Prove — — * = V {In + l)P„(cos u)Q n (cosh v). cosh v — cos u -*- 1 432 INTEGRATION IN GENERAL. 208. Mr. W. M. Hicks has shown (Phil. Trans., 1881) that similar functions are required in the solution of y 2 F=0 (§ 146), in connexion with the anchor rings or toroidal surfaces, generated by the revolution of the dipolar circles of § 175. With the columnar coordinates y, 6, x (§ 132) round Ox as axis, we obtain the transformation V dx 2 ydy\ y dy) + y 2 d6 2 ' or, on putting V— \/ri/ _i , and, on changing from x, y to new variables u, v, con- jugate functions given by u+iv = f(x + iy) (§ 133) With the dipolars of § 175, x + iy = c tan \(u + iv), Then, if \]s varies as cos nu cos md, _y2F=i/-^sinh 2 f|--i'-7i 2 i/r-(m 2 -^)^ cosech 2 v } . Further, if we put \jr = ^/(c sinh v)P, — y 2 V= (cos u + cosh vf/VP d 2 P , ., dP , , . 3 2 P , , _\ INTEGRATION IN GENERAL. 4.33 If we suppose P = P™cos nu cos m9 is a solution of y 2 F=0, then P™, as a function of v only,, satisfies the differential equation d 2 P dP -j-p + coth v-g O 2 - \)P — w 2 P cosech 2 v = ; and this equation, in the particular case when m = 0, is the same as that satisfied by the ponal harmonic (p. 276) with n — I for n; so that writing -G for cosh?; or /m., and S for sinhv or ^(^-1), the solution is AP n +BQ n , where xP t ^; + / c ^^^ ^ ^ + ^ co ^^ ))i+4 . For, as in ex. 6, v., p. 276, • dC J . (G + 8 cos 0)»+* = (%+ t)/kl^±lcos|) %/ (O+^cos0)"+» ^ = (^ + J)(P„ +1 -CP, 1 )'; /"•7T while taking the form irP n = I (C—8 cos (j>) n -hl4>, 5f2 w =( * l " J)(CP "- p ' i - l); and thence ^(&^f) = ( %2 ~ i) P » 5 and similarly with Q m . 209. In the general case, when m is any integer, we can verify that the solution of dcVdCJ {n i)r 8* U is p=ap';;+bq:, 2e 434 INTEGRATION IN GENERAL. where p,„ _ /"t cos md „ m _ /*°° cosh md n ~J {C+S cos ) n +i' ^ n y(C+8cosh) n +i' For now, taking Q™, we prove that (fl 1 ^- niG)Q:+ (» - m - J)^ 1 • - . f £sinh(m+l)0 + C'sinhm = (-l)»(»-i)(m-f)...(m-m-l)g:. Similarly (^+mC)Q:+C/i+m-^e- 1 _ f £sinh(m- 1)0 + Csinh m0 \ co _ ~ I Tr(0+5coah^)»+* Jo ; and thence ^fi^) = _mQ:-mC^<2:-(«+m-i)(g(2:-HS^Q:- 1 ) thus verifying the differential equation; and the procedure is the same with P™. The substitution tan id = <* , , C ? S , ^ will give cosh \u sinh <£ ° /(C-cos0)* /(C+Scosh^-^ w ^' and generally if we denote / ° s n a , by 7ri?„, then ^/ \\j ~~ COS v ) INTEGRATION IN GENERAL. 435 s#-(« + »<*, tl -™.>-{ 0si ^_-^ +i)e y r o, n being an integer ; so that R n satisfies the same equa- tions as Q n ; and since R = t J2Q , therefore R n = s/^Qn- By differentiation with respect to C, 135/ i\ f T S m cosn6d6 t ._._...^ TO _ 2 y ((7 _ cos0) m+i rl m 7? f]? n O = V2(n-iX»-t) - ^- m - ^rfcX% " When the dipolar system of § 175 is revolved round Oy, we shall find in the same manner, on putting V=\p-x~^, and \Js = ^/(c sin u)P, -*"---*{(£+!*> w "!£ + t*} (cosu+coshvf [??P 3P 3 2 P . 3 2 P . D \ and now if P = P™cos nv cos m6 is a solution of v 2 V— 0, then P^, as a function of u only, satisfies the equation ^ + cot w^ - (w 2 + i)P - m 2 P cosec 2 w, = 0, (XU 2 clw of which the solution is obtained from the preceding results by replacing cosh v and sinh v by cos u and sin u. In the particular case of m = 0, the order of the corre- sponding zonal harmonics would be - \ + in, and therefore imaginary ; these are called Mehler functions. 436 INTEGRATION IN GENERAL. 210. Lagrange's Theorem. We can establish Lagrange's Theorem (§ 150) by the rule for the differentiation of a definite integral ; for, if y = a + X(py, —,^-f (xtbz + a - zYi '%d% n\ ?>aj v ^ a a • ■ so that differentiating again n — 1 times with respect to a, and denoting nJ^y^ X ' pZ+a ~ Z) " i ' zdz by Rn ' a we find that while R = iy — ia; so that which is Lagrange's Theorem. 211. Integration considered as Summation. Hitherto we have considered Integration as an opera- tion to be performed by the reversion of Differentiation ; but now, by the aid of Taylor's Theorem, we can exhibit it in its fundamental conception as the Summation of a number of infinitesimal elements, and show that the result is the same as before. For let any function ix, between the fixed values a and b of x, be considered for the consecutive infinitesim- ally graduated sequence of values a, x v x 2 , ... x r , ... x n , b, of the variable x. INTEGRATION IN GENERAL. 437 Then by Taylor's Theorem (§§ 114, 118), using the symbol Ax r to denote the difference x r+ i — x r , ix x -la -Ay 1 + 2y 2 +iy s +... + 4 y m +3 ; and now the area of the three strips r%i fydx = Sl(a + %bl + fcZ 2 + f el 3 ) Jy' = ¥(y m + tym+i + 3y m+s + 2/ m+3 ). It is convenient to take an odd number of ordinates and an even number of strips, and a general formula for approximate quadrature, with 2n + l ordinates and 2n strips, has been given by Newton. With 7 ordinates and 6 strips, the expression for the area is, very approximately, A=^l{6u m +u m _ 1 +u m+ i + 5(w m _ 2 +u v , +2 )+u m _3 + u m+s }, called Weddle's Rule. In these expressions for the area, the y's may represent breadths as well as ordinates ; or they may represent the areas of parallel equidistant sections of a solid body ; for instance, horizontal and vertical sections of a ship. Simpson's Rule may also be used for finding the centroid and moment of inertia of the area, by putting Ax=fxydx = %l(x y + 403^+ 2x 2 y 2 +...), Ay=f\fdx = \l{ y*+ 4^ + 22//+...); Ak x *=f\y 3 dx = \l( y*+ 4y»+ 2y ■* + ...), Ak y 2 =fx 2 ydx = $l(x 2 y + 4^x 1 z y 1 + 2x*y 2 +...). As a numerical application, calculate these quantities for the horizontal section of a ship, given the half ordinates from the median line as "5, 6, 10, 12, 12'4, 12 - 5, 12-5, 12-5, 12-4, 12-3, 11, 8, and *5 feet; the ordinates being spaced at 12 feet. 440 INTEGRATION IN GENERAL. 213. Differential Equations. In Chapters III. and V. the principles of Differentia- tion have been illustrated in the formation of certain important Differential Equations from their primitive relations ; but now it is requisite to reverse this process, and to discover the primitive relation or solution where possible, when the differential equation is given. The complete subject of Differential Equations would soon lead us beyond the scope of the present work, and the reader is referred to the treatises of Forsyth and Woolsey Johnson ; at the same time, however, it is possible to indicate the procedure in a few simple cases, often required in Dynamics and Electricity. The simplest case of frequent occurrence is the linear differential equation with constant coefficients, of the „ d n y . d" -1 -?/ . dy . TT form dS +A ^ + --- +A ^c& +A «y= v > where the A's are constant, and Fis a given function of x. Using D to denote the operation d/dx, we may write the equation F(D)y = V, (1). where F(D) is a rational integral algebraical function of D ; and now the solution will consist of two parts, one of them being any particular solution of this equation (1), while the other, called the complementary function, is the general solution of the equation F(i% = (2). We have shown in § 68 that F(D)e hx =e bx Fb, so that y = e bx will be a solution of (2) if F6 = 0; and if the n roots b v b 2 , b z , ... of this equation Fb = are all real and distinct, the complementary function will" be y = B x e^ x + B 2 e b ^ + ... + B n e h «*, where B v B v ..., B n are n arbitrary constants. INTEGRATION IN GENERAL. 44 1 If m roots are equal to b, the corresponding terms in the complementary function are, as in ex. 3, p. 143, (G + G x x + . . . + C m _ 1 x m - 1 )e ix . If a pair of roots are conjugate imaginary, a±i/3, the corresponding terms, by De Moivre's theorem, are written e^A cos (3x + B sin fix) ; and if these imaginary roots are repeated m times, the corresponding terms are s=m- 1 e<^^x s (AsCos fix + i? 8 sin fix). 8=0 In determining the particular solution of (1), we write the equation symbolically where (FD)' 1 can be resolved into a series of partial fractions of the form A(D- b)'\ or B(D-b)~ r . Consider the separate terms of V, of the form x n , e ax , e te , sinrnx, (i.) (¥D)- 1 x n is found by expanding (FD) -1 in ascend- ing powers of D as far as B n , and performing the differentiations ; terms beyond D' 1 may be neglected, since D n+1 x n = 0, .... (ii.) (FD) -H"* = e ax (Fa) - \ and (FD)- 1 e ax fx = e ax {F(D + a)}-Hx, (§ 70). (iii.) Thus, if F6 = 0, (D-b)- r e bx = e b *D-r. 1 = e ?*(c o + C x x + ... + O r _ 1 x n - 1 + -^j, since D~ r represents the operation of integration, re- peated r times. (iv.) (D-b)- r cos'mx = (l) + by(I) i -b 2 )- r cosmx = (D + b) r ( - m 2 - 6 2 )- r cos mx, (§ 68); and so on for (D 2 + m 2 )~ r cos mx, ..., by employing the exponential values of cos ma;, ... (§ 111). 442 INTEGRATION IN GENERAL. Consider for instance the differential equation cftx dec --T72 + 2% cos /3~77 + n 2 x = E cos pt , for the forced vibrations, due to E cos pt, of a system of which the natural motion is given by the complementary function x =ue ' nt C0B cos (nt sin /3 + e) (ex. 6, p. 139), representing s.h.m. (§ 103) of period 2-7r/(n sin j3), with a modulus of decay n cos /3. The particular solution is _ E cos pt X ~ W+2n cos ~$.D + n 2 D 2 -2ncosp.D + n 2 p ~ (B 2 + n 2 ) 2 - 4n 2 cos 2 /3 . D 2 C0S P _ (n 2 — p 2 )E cos pt + 2np cos /3 . E sin p£ (% 2 — p 2 ) 2 + 4n 2 p 2 cos 2 /3 _2?cos[jrf — tan- 1 {2'n/p cos f3/(n 2 — p 2 )}] Jin* + 2n 2 p 2 cos 2/3 +p*) ' representing a S.H.M. of amplitude E/Jin* + 2n 2 p 2 cos 2/3 + p% and change of phase tan ~ 1 { 2np cos /3/(w 2 — p 2 ) } . Examples. — Solve the differential equations (i.) y" — 5y' + 6y = x 2 + e ix + sin 4a: + xe 2 * + x 2 e 3x , (ii.) y'' + iy = sin x + sin 2x, (iii.) i/" — 6y' +9y = cosh 2x + x cos 2a;, (iv.) j/" + 2y' + 5y = e^sin 2a; + cosh x cos 2a;, (v.) y'" + y" — y'—y =xe x + cosh x, (vi.) 1/'" - 8y" + 16y' = sinh 2a; + cosh 4a;, (vii.) y"' + y' -\-10y = cosh 2a; + sinh x sin 2a;, (viii.) y'" — 26y — 60y = sinh 6x + (A + B cosh 3a;)cos x, (ix.) y"" + 2i/" + x = sin 2a; + sin x, (x.) i/"" - 2y'"+ 3y"-2y' + y = x + e-i*(A+B cosy3a;), (xi.) y"" + S2y' + 4<8y = x sinh 2a- + cosh 2a; cos (2^/20;). INTEGRATION IN GENERAL. 443 214. With variable coefficients P, Q, R, ..., given functions of x, the general linear differential equation of the first order is written y'+Py=Q. To solve this, we can proceed as in § 88, and put y = uv, when u'v + u(v' + Pv) = Q ; and now if v is chosen so that v' + Pv = 0, or v = exp( -fPdx), then u'=Q expjPdx, and u=jQ(ex-pJPdx)dx+C, y — uv — exp( —J~Pdx) { fQ, (exipfPdx)dx -J- C } • Thus, for instance, the solution of the equation i, +iJi = 5 , cos»i, connecting ii the resistance in ohms, i the current in amperes, E cos pt the variable electromotive force of a dynamo in volts, and Ldijdt the counter electromotive force of induction, so that L is the inductance in secohms (Fleming, The Alternate Current Transformer) is i = e-X'IL(f E cos pt . e Bt ' L dt+C) _RE cos pt+pLE sin pt.p _ Rt/L W+pU? + Ecosjpt-t&n^ijoL/R)} a Rt/L ^(Rt+pW) But considered as a differential equation with constant coefficients, Ce~ ml1 is the complementary function, while the particular solution is, as before, . Ecospt R — LD j? nn „. ^m+R^RT^M^™^ _ RE cos pt + pLE sin pt E cos {pt— tan ~ *( pL/R) } R 2 +p 2 L 2 JiRt+fl? ' 444 INTEGRATION IN GENERAL. 215. The general linear differential equation of the second order (§ 88) y" + 2Py' + Qy = R, on putting y = uv, and choosing v so that v'+Pv = 0, becomes u"+Iw = S, where I = Q-P 2 -P', S = R/v; and now, if we can determine the complementary function Au^+Bu^, then, as in § 88, u^Uz - U1U2 = C ; and the particular solution is Gu = u t J~Su 2 dx — u 2 J~8u 1 dx. Thus, as on pp. 183-185, Bessel functions are required in the solution of Biccati's equation, in which I=kx m ; and the student can easily verify that ^(ma^sin 0)(cos 6) n dd +B/"^(maf , mnh 0)(cosh 0)""