,i "II lil'l < i •-htm rop (djmMll Utttomitg pitmg ..A-M.Q.(oS.^ 8817 Cornell University Library olin.anx The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031267242 BOWSER'S MATHEMATICS. ACADEMIC ALGEBBA, WITH NUMEROUS EXAMPLES. COLLEGE ALGEBBA, WITH NUMEROUS EXAMPLES. AN ELEMENTARY TBEATISE ON ANALYTIC GEOMETRY, EMBRACING PLANE GEOMETRY, AND AN INTRODUCTION TO GEOMETRY OF THREE DIMENSIONS. AN ELEMENTARY TREATISE ON THE DIFFERENTIAL AND INTEGRAL CALCULUS, WITH NUMEROUS EXAMPLES. AN ELEMENTARY TREATISE ON ANALYTIC MECHANICS, WITH NUMEROUS EXAMPLES. AN ELEMENTARY TREATISE ON HYDRO-MECHANICS, WITH NUMEROUS EXAMPLES. AST ELEMEITTAEY TEEATISE DIFFERENTIAL AND INTEGRAL CALCULUS. WITH NUMEROUS EXAMPLES. BT EDWAED A. BOWSER, LL.D., PBOl'ESaOB OF MATHEMATICS AND ENGIHEEBINS IN EUTGBE8 COLLEGE TENTH EDITION. NEW YORK: D. VAN NOSTRAND COMPANY, 33 Murray Street. 1889. /0\ «PYEIGHT, 1S80, by E. A. BOWSER PREFACE. fMHE present work on tlie Differential and Integral Calculus is designed as a text-book for colleges and scientific schools. The aim has been to exhibit the subject in as concise and simple a manner as was consistent with rigor of demonstration, to make it as attractive to the beginner as the nature of the Calculus would permit, and td arrange the successive portions of the subject in the order best suited for tlie student. I have adopted the method of infinitesimals, having learned from experience tliat the fundamental principles of the subject are made more intelligible to beginners by tlie method of infinitesimals than by that of limits, while iu the practical applications of the Calculus the investigations are carried on entirely by ihe method of infinitesimals. At the same time, a tliorough knowledge of the subject requires that the student should become acquainted with both methods ; and for this reason, Chapter III is devoted exclusively to the method of limits. In this chapter, all the fundamental rules for differentiating algebraic and transcendental functions are obtained by the method of limits, so that the student may compare the two methods. This chap- ter may be omitted without intecfering with the continuity of the work, but the omission of at least the first part of the chapter is not recommended. To familiarize the student with the principles of the subject, and to fix the principles in his mind, a large number of examples is given at the ends of the chapters. These examples have been carefully selected with the view of illustrating the most important points of the subject. The greater part of them will present no serious diffi- culty to the student, whUe a few may require some analytical skill. IV PRE FAG a. In preparing this book, I liave availed myself pretty freely of tie writings of the best American and English and French authors. Many volumes have been consulted vrhose titles are not mentioned, as credit could not be given in every case, and probably I am indebted to these volumes for more thaff I am aware of. The chief sources upon which I have drawn are indicated by the references in the body of the work, and need not be here repeated. For examples, I have drawn upon the treatises of Gregory, Price, Todhunter, Williamson, Young, Hall, Rice and Johnson, Ray, and Olney, while quite a num- ber has been taken from the works of De liorgan, Lacroix, Serret, Courtenay, Loomis, Church, Byerly, Dociiarty, Strong, Smyth, and the Mathematical Visitor; and I would hereby acknowledge my indebtedness to all the above-named works, both American and foreign, for many valuable hints, as well as for examples. A few examples have been prepared specially for this work. I have again to express my thanks to Mr. R. W. Prentiss, Fellovt in Mathematics at the Johns Hopkins University, for reading the MS and for valuable suggestions. E. A. B. Rutgers Collegb, New Brunswick, N. J., June, 1880. \ TABLE OF CONTENTS. PART I. DIFFERENTIAL CALCULUS. CHAPTER I. FIRST PRINCIPLES. AST. FAQS 1. Constants and Variables 1 2. Independent and Dependent Variables 1 3. Functions. Geometric Representation' 3 4. Algebraic and Transcendental Functions 4 5. Increasing and Decreasing Functions 5 6. Explicit and Implicit Functions 6 7. Continuous Functions 6 8. Infinites and Infinitesimals 7 9. Orders of Infinites and Infinitesimals 8 10. Geometric Illustration of Infinitesimals 10 11 . Axioms. 12 Examples 13 CHAPTER II. DIFFERENTIATIOir OF ALGEBRAIC AND TRANSCENDENTAL FUNCTIONS. 13. Increments and Differentials 15 13. Consecutive Values ". 16 14. Differentiation of Sum of a Number of Functions 16 15. To Differentiate y — ax±h 17 VI CONTENTS. ABT. PAOE 16. DiflEerentiation of a Product of Two Functions 18 17. Differentiation of a Product . . 20 18. Differentiation of a Fraction 30 19. DifEerentiation of any Power 31 Examples , 33 Illustrative Examples 36 30. Logarithmic and Exponential Functions 39 31 . Differentiation of an Exponential 31 32. Differentiation of an Exponential with Variable Base S3 Examples 33 S3. Logarithmic Differentiation. Examples 34 Illustrative Examples ^ 35 TRIGONOMETRIC FUNCTIONS. 34. To Differentiate y = sinx 37 35. To Differentiate y = cos x 38 33. To Differentiate y = tan x 38 37. To Differentiate y = cot x 39 33. To Differentiate y = sec x 39 39. To Diffei-entiate y = cosec x 40 30. To Differentiate y = versin x 40 31. To Differentiate y = covers x 40 33. Gfeometric Demonstration 41 Examples 43 Illustrative Examples 44 CIRCULAR FUNCTIONS. 33. To Differentiate y = sin-' x 4g 34. To Differentiate y = cos-' x 4g 35. To Differentiate y = tan-' x 47 36. To Differentiate y = cot-' x 47 37. To Differentiate y = see"' x 47 38. To Differentiate y = cosec-' x 47 39. To Differentiate y = vers-' x 43 40. To Differentiate y = covers-' x 48 Examples ,, » 4g Miscellaneous Examples , 49 CONTENTS. Vll CHAPTEE III. LIMITS — DBEIVED FUNCTIONS. ly>- lAQB 41. Limiting Values 59 43. Algebraic Illustration 59 43. Trigonometric Illustration 60 44. Derivatives 63 45. DifEerential and Differential CoefScient 63 46. Algebraic Sum of a Number of Functions 63 47. Product of Two Functions 64 48. Product of any Number of Functions 65 49. Differentiation of a Fraction 66 50. Any Power of a Single Variable 67 51. Differentiation of log x 68 53. Differentiation of a'^ 68 53. Differentiation of sin a; 68 54. Differentiation of cos x 69 CHAPTEE IV. - SUCCESSIVE DIFFERENTIALS AND DERIVATIVES. 55. Successive Differentials. Examples 71 56. Successive Derivatives 74 56a. Geometric Eepresentation of First Derivative. Examples . 76 CHAPTEE V. DEVELOPMENT OF FUNCTIONS. 57. Definition of Development of a Function 81 58. Maclaurin's Theorem 81 59 The Binomial Theorem 85 60. To Develop y = sin x and y = cos x 85 61. The Logarithmic Series 86 63. The Exponential Series 90 63. To Develop y = tan-i x 91 64. Failure of Maclaurin's Theorem 93 65. Taylor's Theorem. Lemma 93 66. To find Taylor's Theorem 05 67. The Binomial Theorem 97 yili CONTENTS. tar. icAGx 68. To Develop u' — sin (aj+y) 98 69. The Logarithmic Series 98 70. The Exponential Series 98 71. Failure of Taylor's Theorem 99 Examples , 100 CHAPTBE VI. EVALUATION OF INDETEEMINATE FORMS. 72. Indeterminate Forms 103 73. Common Factors. Examples 104 74. Method of the Differential Calculus 105 QO 75. To evaluate Functions of the form — 108 00 • 76. To evaluate Functions of the form Oxoo Ill 77. To evaluate Functions of the form oo — oo 113 78. To evaluate Functions of the forms 0", oo", and 1^"" 118 79. Compound Indeterminate Forms 116 Examples 116 CHAPTER VII. FTTNCTIOKS OF TWO OR MOKE VARIABLES. — CHANGE OP THE INDEPENDENT VARIABLE. 80. Partial Differentiation : 120 81. Differentiation of a Function of Two Variables 123 83. To find the Total Derivative of u -with respect to a; 125 83. Successive Partial Differentiation 130 84. Proof that Order of Differentiation is indifferent 131 85- Successive Differentials of a Function of Two Independent Variables 133 86. Implicit Functions 135 87. Differentiation of an Implicit Function 136 88. Second Derivative of an Implicit Function 13v 89. Change of the Independent Variable 140 90. General Values of ^, ^, ^,, etc 141 ax ax' aar 91. Transformation for Two Independent Variables 14C Examples 147 CONTENTS. IX CHAPTER VIII. MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. AKT. PAGE 93. Definition of a Maximum and a Minimum 151 93. Condition for a Maximum or Minimum 151 94. Geometric Illustration 153 95. Discrimination between Maxima and Minima 154 96. Condition given by Taylor's Theorem 154 97. Method of finding Maxima and Minima Values 155 98. Alternation of Maxima and Minima Values 156 99. Application of Axiomatic Principles 157 Examples 159 Geometric Problems 164 CHAPTBE IX. TANGENTS, NORMALS, AND ASYMPTOTES. 100. Equations of the Tangent and Normal 172 101. Length of Tangent, Normal, Subtangeut, etc 175 103. Polar Curves. Tangents, Normals, Subtangents, etc 178 103. Rectilinear Asymptotes 181 104. Asymptotes determined by Expansion 184 105. Asymptotes in Polar Co-ordinates. Examples 186 CHAPTER X. DIRECTION OF CURVATURE — SINGULAR POINTS — TRACING OF CURVES. .106. Concavity and Convexity 191 107. Polar Co-ordinates 193 108. Singular Points 194 109. Points of Inflexion 194 110. Multiple Points 196 111. Cusps 199 113. Conjugate Points 301 113. Shooting Points. Stop Points 303 114. Tracing Curves 205 Examples 806 115. Tracing Polar Curves. Examples 810 CONTENTS. CHAPTER XI. RADIUS OF CUKVATUKB, EVOLUTES AKD INVOLUTES — ENVELOPES. ART. PAGE 116. Curvature 316 117. Order of Contact of Curves 317 118. Dependence of Order of Contact on Arbitrary Constants 318 119. Radius of Curvature. Centre of Curvature 219 120. Second Method 330 131. Radius of Curvature in Polar Co-ordinates 333 133. Radius of Curvature at a Maximum or Minimum 383 133. Contact of Different Orders , 333 Examples 324 134. Evolutes and Involutes 338 135. Equation of the Evolute 338 136. Normal to an Involute is tangent to Evolute 330 137. Envelopes of Curves 381 138. Equation of the Envelope of a Series of Curves 333 Examples 333 PART II. INTEGRAL CALCULUS. CHAPTEE I. ELEMENTAKY FOEMS OF INTEGRATION. 139. Definitions 338 130. Elementary Rules for Integration 339 131. Fundamental Forms 343 133. Integration by Transformation into Fundamental Forms 843 133. Integrating Factor. Examples 847 134. Transposition of Variable Factors. Examples 849 135. Trigonometric Reduction, Examples 354 CONTENTS. xi CHAP TEE II. INTEGRATION OF RATIONAL FRACTIONS. ^■^'^- FAQS 136. Rational Fractions 256 137. Case 1. Decomposition of a Rational Fraction 356 188. Cases. " " " 259 139. Case 3. " " " 263 ■Examples 368 CHAPTER III. INTEGRATION OF IRRATIONAL FUNCTIONS BY RATIONALIZATION. 140. Rationalization 369 141. Monomial Surds 369 143. Binomial Surds of the First Degree 270 148. Functions of the Form =; 273 (a + WY 144. Functions containing only Trinomial Surds 373 145. Binomial Differentials 276 p_ 146. Conditions for Rationalization of x"" (a + ia;")' dx 377 Examples 280 CHAPTER IV. INTEGRATION BY SUCCESSIVE REDUCTIONS. 147. Formula of Reduction 385 148. Formula for Diminishing Exponent of s, etc 285 149. Formula for Increasing Exponent of a;, etc 287 150. Formula for Diminishing Exponent of Parenthesis 288 151. Formula for Increasing Exponent of Parenthesis 289 Examples. Applications of Formulae 289 Logarithmic Functions 295 153. Reduction of the Form / X(log xydx - 395 j^- 297 Xii CONTENTS. AKT. PAGE Exponential Forms ' 399 154. Reduction of the Form / a'^te'dx 399 — - 300 150. Trigonometric Functions 301 157. Formulse of Redaction for / sin"' 6 cos" 6 dd 303 158. Integration of sin-" 6 cos" d dB „ 805 159. Reduction of the Form I x" coa ax dx 307 160. Reduction of the Form fe" cos" x dx 308 161. Integration of f{x) sin-' x dx, f(x) tan-' x dx, etc 809 162. Integration of dy = r -^ 310 " a + 6 cos Examples 812 OHAPTEK V. II^-TEGKATION BY SEEIES — SUCCESSIVE INTEGRATION IN- TEGEATION OF FUNCTIONS OF TWO VARIABLES. DEFINITE INTEGRALS. 163. Integration by Series 319 164. Successive Integration 331 /n X dx" into a Series 166. Integrations of Functions of Two or More Variables 336 d^u 167. Integration of ^— ^ -.f{x,y) 326 168. Integration of Total Difterentials of the First Order 839 169. Definite Integrals. Examples .• 381 170. Change of Limits 334 Examples 337 Formulae of Integration 340 CHAPTER VI. LENGTHS OF CURVES. 171. Length of Plane Curves referred to Rectangular Axes 347 173. Rectification of Parabola 348 323 AllT. PAGE 173. Semi-cubical ParaboJa 349 174. The Circle ', 349 175. The Ellipse 850 176. The Cycloid 351 177. The Catenary 353 178. The Involute of a Circle 353 179. Recti6eation in Polar Co-ordinates 353 180. The Spiral of Archimedes 353 181. The Cardioide 353 18). Length o^ Curves in Space 354 183. Intersection of Two Cyliaders. Examples 355 CHAPTBK VII. AREAS OF PLANE CURVES. 184. Areas of Curves 360 185. Area between Two Curves 361 186. Area of the Circle 361 187. The Parabola 363 188. The Cycloid .363 189. The Ellipse : 363 190 Area between Parabola and Circle 363 191. Area iu Polar Co-ordinates 364 193. The Spiral of Archimedes 364 Examples 365 CHAPTER VIII. AREAS OE CURVED SURFACES, 193. Surfaces of Revolution 369 194. Quadrature of the Sphere 370 195. The Paraboloid of Revolution 371 196. The Prolate Spheroid 373 197. The Catenary 373 198. The Surface of Revolution generated by Cycloid 373 199. Surface of Revolution in Polar Co-ordinates 374 300. The Cardioide 374 301. Any Curved Surfaces. Double Integration 375 303. Surface of the Octant of a Sphere 376 Examples ,.,.,..., ,..,.,, dT' XIV CONTENTS. CHAPTBE IX. VOLUMES OF SOLIDS. AET. PASE 303. Solids of Revolution 381 304. The Sphere 381 205. Solid of Revolution of Cycloid 682 206. Solid of Revolution generated by Cissoid , 388 307. Volumes of Solids bounded by any Curved Surface 383 208. Mixed System of Co-ordinates 388 809. Cubature in Polar Co-ordinates c89 Bxamples . 390 PART I. DIFFERENTIAL CALCULUS, CHAPTER I. FIRST PRINCIPLES. 1. Constants and Variables. — In the Calculus, as in Analytic Geometry, there are two kinds of quantities used, condanis and variables. A constant quantity, or simply a constant, is one whose value does not cbange in the same discussion, and is repre- sented by one of the leading letters of the alphabet. A variable quantity, or simply a variable, is one which admits of an infinite number of values within certain limits that are debermined by the nature of the problem, and is represented by one of the final letters of the alphabet. For example, in the equation of the parabola, «/2 = 2px, X and y are variables, as they represent the co-ordinates of any point of the parabola, and so may have an indefinite number of different values. %p is a constant, as it represents the latns rectum of the parabola, and so has but one fixed value. Any given number is constant. 2. Independent and Dependent Variables. — Au independent variable is one to which any arbitrary value may 3 FUNCTION OF ONE OR MORE VARIABLES. be assigned at pleasure. A dependent variable is one whose value varies in, consequence of the variation of the inde- pendent variable or variables with which it is connected. Thus, in the equation of the circle y? -\- y^ =^ r\ if we assign to x any arbitrary value, and 6nd the correspond- ing value of y, we make x the independent variable, and y the dependent variable. If we were to assign to y any arbi- trary value, and find the corresponding value of x, we would make y the independent variable and x the dependent variable. Frequently, when we are considering two or more varia- bles, it is in our power to make whichever we please the independent variable. But, having once chosen the inde- pendent variable, we are not at liberty to change it through- out our operations, unless we make the cori'esponding trans- formations which such a change would require. 3. Functions. — One quantity is called a function of another, when it is so connected with it that no change can take place in the latter without producing a corresponding change in the former. For example, the sine, cosine, tangent, etc., of an angle are said to be functions of the angle, as they depend upon the angle for their value. Also, the area of a square is a function of its side ; the volume of a sphere is a function of its radius. In like manner, any algebraic expression in x, as a? — 2bx^ + hx + c, is a function of x. Also, we may have a function of two or more variables : a rectangle is a function of its two sides ; a parallelepiped is a function of its three edges ; the expres- sion tan {ax + by) is a function of two variables, x and y; a;3 4- y2 _|_ ^2 ig a, function of three variables, x, y, and z ; etc. When we wish to write that one quantity is a function of NOVATiON^GEOMETRIC REPRBSENfATlOlf. 3 one or more others, and wish, at the same time, to indicate several forms of functions in the same discussion, we use such symbols as the following : y^f{x); y = F{x); y = {x,y) = 0; f{x,y,z) = 0; which are read : " y equals the / function of x; y equals the large F function otx; y equals the function oix; y equals the /prime function of .t; y equals the /function of x and z ; the function of x and y equals zero ; the / function of X, y, and z equals zero;" or sometimes ^' y tz:^ f of x, y = Fof X," etc. If we do not care to state precisely the form of the function, we may read the above, "y = a func- tion of a; ; ^ = a function of x and z ; a function of x and p = 0; a function of x, y, and z = 0." For example, in the equation y =: aa? + hx + c, yis a function of x, and may be expressed, y =f(x). Also, the equation ax'^ + bxy + c?/^ = may be expressed, f(x, y) — 0. In like manner, the equations y = aoi? + lo^z + c^, and «/ = «a;2 + hxz + d'^, may be expressed, y — f{x, z) and y = 4>{x, z). Every function of a single variable may be represented geometri- cally by the ordinate of a curve of which the variable is the cor- responding abscissa. For if y be any function of x, and we assign any value to x and find the corresponding value of y, these two values may be regarded as the co-ordinates of a point which may be con- structed. In the same way, any number of values may be assigned to X, and the corresponding values of y found, and a series of points con- -1 ALOEBRAIC AND 'rRANSCBNDEXTAL FtTNCTIONS. structed. These points make up a curve of whicli the variable ordi- nate is y and the corresponding abscissa is. x. In lilie manner it may be shown that a function of two variables may be represented geometrically by the ordinate of a surface of which tlie variables are the corresponding abscissas. 4. Algebraic and Transcendental Functions.— An algebraic function is one in which the only operations indi- cated are addition, subtraction, multiplication, division, involution, and evolution; as, {a + lx^Y; {^-hxy)^; ^ -^^—-^--^.^ etc. Transcendental functions are those v?hich involve other operations, and are subdivided into trigonometric, circular, logarithmic, and exponential. A trigonometric function is one which involves sines, tan- gents, cosines, etc., as variables. For example, y ^= sm x; y ■= tan^ x; y = cos x sec x ; etc. A circular function is one in which the concept is a variable arc, as sin^^a;,* cos~i;c, sec~'^y, cot-^.T, etc., read, •'' the arc whose sine is x, the arc whose cosine is x," etc. It is the inverse of the trigonometric function ; thus, from the trigonometric function, y = sina', we obtain the circular function, x = sin^^ y. In the first function we think of the • right line, the sine, the arc being given to tell us which sine ; in the second we think of the arc, the sine being given to tell us tohich arc. The circular functions are often called inverse trigonometric functions. * This notation was suEgestecl by the use of the negative exponents in algebra. If wc have y = ax, v/e also have x = a~'y, vi'here yiss. function of a, and x is the corresponding inverse function of y. It may be worth while to caution the begin- ner against the error of supposing that sin-' y is equivalent to — . — ; while it is true that «-' is equivalent to - ■ INCnEASING AJS-D DECREASING FVNCTtONS. A logantlimic function is one which involves logarithms of the variables ; as, y = \ogx; «/ = log Vfl! — a;; f = Z\og^-^^^; etc. An exponential function is one in which the variable enters as an exponent ; as, y =. aP; y = a;* ; u ^ '3^ ; etc. 5. Increasing and Decreasing Functions. —An in- creasing function is one that increases when its variable increases, and decreases when its variable decreases. For example, in the equations y = ai?, y = log x, y =: Va^ + 3?, y =. OP, y is an increasing function of x. A decreasing function is one that decreases when its variable increases, and increases when its variable decreases. Thus, in the equations y = -, y ={a- xf, y = log -, x^ + y^ = r% 2/ is a decreasing function of x. In the expression, y = {a- xf, «/ is a decreasing function for 'all values of a; < «, but in- creasing for all values > a. In the expression y = sin x, y is an increasing function for all values of x between 0° and 90°, decreasing for all values of x between 90° and 270°, and increasing for all values of x between 270° and 360°. 6 CONi'tNUOUS FUXCTlONg. 6. Explicit and Implicit Functions. — An explicit function is one whose value is directly expressed in terms of the variable and constants. For example, in the equations y = {a — x)% y = Va* — x'^, ^ = "Zax^ — 3a;', y is an explicit function of x. An implicit function is one whose value is not directly expressed in terms of the variables and constants. For example, in the equations y^ — Zaxy + a:^ = 4, x^ — 3xy + 2y — 16, y is an implicit function of x, or x is an implicit function of y. If we solve either equation with respect to y, we shall have y as an explicit function of x ; also, if we solve for a; we shall have x as an explicit function of y. 7. Continuous Functions. — A function of x is said to be a continuous function of x, between the limits a and b, when, for every value of x between these limits, the cor- responding value of the function is finite, and when an infinitely small change in the value of x produces only an infinitely small change in the value of the function. If these conditions are not fulfilled, the function is discon- tinuous. For example, both conditions are fulfilled in the equations «/ = «a; + 6, «/ = sin x, in which, as x changes, the value of the function also changes, but changes gradually as x changes gradually, and there is no abrupt passage from one value to another; if x receives a very small change, the corresponding change in the function of x is also very small. The expression Vr^ — x^ is a continuous function of x for all values of x between + r and — r, while Vx^ — /^ is discontinuous between the same limits. tHPtmtM AND iNFINiTESiMALS. 7 8. Infinites and Infinitesimals. — An infinite quantity, or an infinite, is a quantity which is greater than any assign- able quantity. An infinitesimal is a quantity which is less than any assignable quantity. An infinite is not the largest possible quantity, nor is an infinitesi- mal the smallest ; there would, in this case, be but one infinite or infinitesimal. Influites may difEer from each other and from a quan- tity which transcends every assignable quantity, that is, from absolute infinity. So may infinitesimals differ from each other and from abso- lute zero. The terms infinite and infinitesimal are not applicable to quantities in themselves considered, but only in their relation to each other, or to a common standard. A magnitude which is infinitely great in com- parison with a finite magnitude is said to be infinitely great. Also, a magnitude which is infinitely small in comparison with a finite mag- nitude is said to be infinitely small. Thus, the diameter of the earth is very great in comparison with the length of one inch, but very small in comparison with the distance of the earth from the pole star ; and it would accordingly be represented by a very large or a very small number, according to which of these distances is assumed as the unit of comparison. The symbols oo and are used to represent an infinite and an infinitesimal respectively, the relation of which is 00 = ;r and = 00 The cipher is an abbreviation to denote an indefinitely small quantity, or an infinitesimal — that is, a quantity which is less than any assignable quantity — and does not mean absolute zero ; neither does 00 express absolute infinity. If a represents a finite quantity, and x an infinite, then - is an infinitesimal. If x is an infinitesimal and a is finite, X ^- is infinite; that is, the reciprocal of an infinite is infini- X tesimal, and the reciprocal of an infinitesimal is infinite. A number is infinitely great in comparison with another, 8 ORDBES OP INPtNltSS Ana INFtNtTESlMAlS. when no nwiiber can be found sufficiently large to express the ratio between them. Thus, x is infinitely great in relation to a, when no number can be found large enough to express the quotient -• Also, a is infinitely small in relation tc x when no number can be found small enough to express the quotient -; x and - represent an infinite and an infini- tesimal. One mUlion in comparison with one millionth is a very large num- ber, but not infinitely large, since the ratio of the first to the second can be expressed in figures : it is one trillion ; though a very large number, it is finite. So, also, one millionth in comparison with one million is a very small number, but not infinitely small, since a num- ber can be found small enough to express the ratio of the first to the second : it is one trillionth, and therefore finite. * 9. Orders of Infinites and Infinitesimals. — But even though - is greater than any quantity to which we can assign a value, we may suppose another quantity as large in relation to « as a; is in relation to a : for, whatever the mag- nitude of X, we may have the proportion .. .=^ t a ' ■ ' m which — is as large in relation to a; aa « is in relation to a a, tor - will contain z as many times as x will contain « ; hence, — may be regarded as an infinite of the second order, - being an infinite of \h& first order. • (^ Also, even though - is less than any quantity to which. we can assign a value, we may suppose another quantity as small in relation to a as a is in relation to x ; for we may have the proportion. OR Den s OF iNFiNifMs Ajsrb tNFimtSsiMALs. X : a :: a : - , X in which — is as small in relation to a as a is in relation to v, for - is contained as many times in a as a is contained X m X ; hence, - may be regarded as an infinitesimal of the second order, - being an infinitesimal oithB first order. We may, again, suppose quantities infinitely greater and infinitely less than these just named ; and so on indefinitely. Thus, in the series (Kx?, aa?, ax, a, -, ^, ^, etc., if we suppose a finite and x infinite, it is clear that any term is infinitely small with respect to the one that imme- diately precedes it, and infinitely large with respect to the one that immediately follows it; that is, mfi, ax*, ax are infinites of the third, second, and first orders, respectively ; _j _^ — sxe infinitesimals of the ^rsf, second, and third orders, respectively, while a is finite. If two quantities, as x and y, are infinitesimals of the first order, their product is an infinitesimal of the second order ; for we have the proportion, \ : X : : y : xy. Hence, if x is infinitely small in relation to 1, xy is infinitely small in relation to y; that is, it is an infinitesimal of the second order when x and y are infinitesimals of the first order. Likewise, the product of two infinites of the first order is an infinite of the second order. The product of an infinite and an infinitesimal of the same order is &, finite quantity. The product of an infinite 10 RAttoS 6P IkFINiiTESiMALS. and an infinitesimal of different orders is an infinite or an infinitesimal, according as the order of the infinite is higher or lower than that of the infinitesimal, and the order of the product is the sum of the orders of the factors. For example, in the expressions the first product is finite; the second is an infinite of the first order ; the third is an infinitesimal of the second order. Though two quantities are each infinitely small, they may have any ratio whatever. Thus, if a and b are finite and x is infinite, the two quantities - and - are infinitesimals; but their raiio is ;-> which is ^nte. In- XX *^ deed, two very small quantities may have a much larger ratio than two very large quantities, for the value of a ratio depends on the rela- tive, and not on the absolute magnitude of the terms of the ratio. The ratio of the fraction one-piillionth to one-ten-miUionth \Bten, while the ratio of one million to ten million is one-tenth. The latter numbers are respectively a million times a million, and ten million times ten mil- lion, times as great as the first, and yet the ratio of the last two is only one-hundredth as great as the ratio of the first two. Assume the series 10" ' (lOV ' .(io«) ' (icj ' \w) ' \w) ' ^^ in which the first fraction is ontt-millionth, the second one-millionth of the first, and so on. Now suppose the first fraction is one-millionth of an ineli in length, which may be regarded as a very small quantity of the first order; the second, being one-millionth of the first, must be regarded as a small quantity of the second order, and so on. Now, if we continue this series indefinitely, it is clear that we can make the terms bepome as small as we please without ever reaching hbsolute zero. It is also clear that, however small the terms of this series become, the ratio of any term to the one that immediately follows it is one million. 10. Geometric Illustration of Infinitesimals. — The following geometric results will help to illustrate the theory of infinitesimals. GEOMETRIC ILLUSTRATION OP INFINITESIMALS. 11 Let A and B be two points on the circumference of a circle ; draw tlie diameter AE, and draw EB produced to meet the tangent AD at D. Then, as the triangles EAB and ADB are similar, we have, and BE AB AE " AD AB BD AE ~ AD (1) (3) Now suppose the point B to approach the point A till it becomes infinitely near to, it, then BE becomes ultimately equal to AE; but, from (1), when BE = AE, we have AB = AD. AB . Also, -pj, becomes infinitely small, that is, AB becomes an infinitely small quantity in comparison ivith AE. Hence, from (2), BD becomes infinitely small in comparison with AD or AB ; that is, tolien AB is an infinitesimal of the first order, BD is an infinitesimal of the second order. Since DE — AE < BD, it follows that, when one side of a right-angled triangle is regarded as an infinitely small quantity of the first order, the difference letioeen the hypoth- enuse and the remaining side is an infinitely small quantity of the second order. Draw BN perpendicular to AD ; then, since AB > AN"," we have, AD - AB < AD - AN" < DN ; therefore, AD AB DN" AD BD ^ BD ^ DE' But AD is infinitely small in comparison with DE, there- fpf^ AD — AB is infinitely small in comparison with BD 5 13 AXIOMS. but BD is an infinitesimal of the second order (see above), hence AD — AB is a?i infinitesimal of the third order. In like manner it niay be shown that BD — BN is an infinitesimal of iln.Q fourth order, and so on. [The student who wishes further illustration is referred to Williamson's Dif. Cal., p. 35, from which this was taken.] 11. Axioms. — Prom the nature of an infinite quantity, a finite quantity can have no value when added to it, and must therefore be dropped. An infinitesimal can have no value when added to a finite quantity, and must therefore be dropped. If an infinite or an infinitesimal be multiplied or divided by a finite quantity, its order is not changed. If an expression involves the sum or difference of infinites of different orders, its value is equal to the infinite of the highest order, and all the others can have no value when added to it, and must be dropped. If an expression involves the sum or difference of infini- tesimals of different orders, its value is equal to the infinitesimal of the lowest order, and all the others can have no value when added to it, and must be dropped. These axioms are self-evident, and, therefore, axioms in the strict sense. For example, suppose we were to compare the mass of the sun with that of the earth : the latter weighs about six sextilliou tons, the • former weighs about 355000 times as much. If a weight of one grain were added to or subtracted from either, it would not affect the ratio appreciably ; and yet the grain, compared with either, is finite — ^it can be expressed in figures, though on the verge of an infinitesimal. If we divide this grain into a great many equal parts — a sextillion, for instance — and add one of these parts to the sun or the earth, the error of the ratio will be stUl less ; hence, when the subdivision is continued indefinitely, it is evident that we may obtain a fraction leas than any assignable quantity, Tumever small, which, when added to the sun or the earth, will affect the above ratio by a quantity less than any to which we can assign a value. By reason of the terms that may be omitted, in virtue of the prin- dples cojjtftined in th?s? axioms, the eijuation? formed i» tfee solutioi) EXAMPLES. 13 of a problem will be greatly simplified. It may be remarked that in the method of limits,* when exclusively adopted, it is usual to retain infinitely small quantities of higher orders until the end of the calcu- lation, and then to neglect them on proceeding to the limit ; while, in the infinitesimal method, such quantities are neglected from the be- ginning, from the knowledge that they cannot affect the final result, as they necessarily disappear in the limit. The advantage derived from neglecting these quantities will he evident when it is remem- bered how much the difficulty in the solution of a problem is increased when it is necessary to introduce into its equations the second, third, and in general the higher powers of the quantities to be considered. EXAMPLES. 3^ 4- ct 1. Find the value of the fraction - — —- r, if x is infinite, and a and b finite. Since a and h are finite, they have no value in comparison 3x with X, and must therefore be dropped, giving us -- = | as the required value of the fraction. 2. Find the value of the fraction ^ — -r—r , if « is infini- tesimal, and a and b finite. Since x is an infinitesimal, it has no value in comparison with a and b, and must therefore be dropped, giving us — ^ for the required value of the fraction. 8a^ _l_ 2x 3. Find the value of -^-5— — , when x is infinite ; also when X is infinitesimal. Ans. "Vyhen x is infinite, 4; when infinitesimal, 2. 4. Find the value of ■ — 5-^ 5-^; -— , when x is mx^ + nx^ +px + q infinite; and when infinitesimal. Ans. When a; is infinite, — ; when infinitesimal, -• m q * For a discussion of limits, see Chapter III. 14 EXAMPLES. ax'' 4- Zx^ 4- 3 5. Find the value of , . , — — r , when x is infinite; 5a;* — 4a; + 1 and when infinitesimal. Ans. When x is infinite, oo ; when infinitesimal, ?. ^ _. , , , , „ 4a;* + 3a;2 + '3a; — 1 , 6. Find the value of — tt-. r-5 s » when x is 3a;5 + 4kOi? + 2x infinite ; and when infinitesimal. Ans. When x is infinite, 0; when infinitesimal, oo. 7. Find the value of -7-= , when x is infinite ; and ,.„.,., 4a;3 _ fifx when infinitesimal. ^«s. When x is infinite, 0; when infinitesimal, 7m. 8. Find the value of -r -„ , when a; is infinite ; and V/jr fpA -^ when infinitesimal. ^jjs. When x is infinite, oo; when infinitesimal, 0. Ya; 2w 9. Find the value of j ^•- , when x and ?/ are infini- tesimals. ^ Ans. We do not know, since the relation between x and^ is unknown. c CHAPTER II DIFFERENTIATION OF ALGEBRAIC AND. TRANSCEN- DENTAL FUNCTIONS. 12. Increments and Differentials.— If any variable, as X, be supposed to receive auy change, such change is c^A&^Wi. increment ; this increment of x is usually denoted by the notation Aic, read "difference x," or "delta x," vi^here A is taken as an abbreviation of the word difference. If the variable is increasing, the increment is + ; but if it is decreasing, the increment is — . When the increment, or difference, is supposed infinitely small, or an infinitesimal, it is called a differential, and is represented by dx, read " differential x," where d is taken as an abbreviation of the word differential, or infinitely small difference. The symbols A and d, when prefixed to a varia- ble or function, have not the effect of multiplication ; that is, dx is not d times x, and Aa; is not A times x, but their power is that of an operation performed on the quantity to which they are prefixed. If M be a function of x, and x becomes x + ^x, the cor- responding value of u is represented by m + Am ; that is, the increment of u corresponding to a finite increment of x is denoted by Am, read "difference m." If 2; becomes x + dx, the corresponding value of n is rep- resented hy u + du; that is, the infinitely small increment of u caused by an infinitely small increment in x, on which u depends, .is denoted by du, read " differential u." Hence, dx is the infinitesimal increment of x, or the infinitesimal quantity by which x is increased; and du is the correspond- ing infinitesimal increment of u. 16 CONSECUTIVE POINTS — DIFFERENTIATION. The differential du or dx is + or — according as the variable is increasing or decreasing, i. e., the first value is always to be taken from the second. 13. Consecutive Values. — Consecutive values of a function or variable are values which differ from each other by less than any assignable quantity. Consecutive points are points nearer to each other than any assignable distance. Thus, if two points were one-millionth of an inch apart, they might bj considered praotieaUyaa consecutive points ; and yet we might have a million points between them, the distance between any two of which would be a millionth of a millionth of an inch ; and so we might have a million points between any two of these last points, and so on ; that is, however close two points might be to each other, we could still suppose any number of points between them. A differential has been defined as an infinitely small in- crement, or an infinitesimal; it may also be defined as the difference between two consecutive values of a variable or function. The difference is always found by taking the first value from the second. In the Differential Calculus, we investigate the relations between the infinitesimal increments of variables from given relations between finite values of those variables. The operation of finding the differential of a function or a variable is called differentiation. 14. Differentiation of the Algebraic Sum of a Number of Functions. Let u ^= V -{- y — z, (1) in which u, v, y, z, are functions of x.* * We might also, in a similar manner. And the differential of a flinction of sev- eral variables ; but we prefer to reserve the inquiry into the differentials of ftinctlons of several variables for a later chapter, and confine ourselves at present to functions of a single variable. DIFFERENTIATION OF A PRODUCT. 17 Give to X the infinitesimal increment dx, and let du, dv, dy, dz, be the corresponding infinitesimal increments of u, V, y, z, due to the increment which x takes. Then (1) becomes ti + du = V + dv -^-y + dy — {z-\- dz). (2) Subtracting (1) from (3), we have du = dv + dy — dz, (3) which is tJie differential required. Therefore, the differential of the algebraie sum of any number of functions is found by talcing the alge- iraic sum of their differentials. 15. To Differentiate y = ax ±b. (1) Give to X the infinitesimal increment dx, and let dy be the corresponding infinitesimal increment of y due to the increment which x takes. Then (1) becomes y + dy = a{x-\- dx) ±i. (3) Subtracting (1) from (2), we get dy = adx, (3) which is the required differential. Hence, the differential of the product of a constant by a variable is equal to the constant multiplied by the differential of the variable ; also, if a constant be connected with a variable by the sign + or — , it dis- appears in differentiation. This may also be proved geometrically as follows : Let AB (Fig. 2) be the line whose equation is y = ax + l, and let {x, y) be any point P on this line. Give OM (= x) 18 GEOMETRIC ILL USTRA TION. Fig. 2, the infinitesimal increment MM' (= dx), then the cor- responding increment of MP (=y) will be CP' (= dy). Now in the tri- angle OPP' we have OP' = CPtanCPF;* or letting a = tan CPP', and suhsti- tuting for CP' and CP their values dy and dx, we have, dy = adx. It is evident that the constant 6 will disappear in differentiation, from the very nature of constants, which do not admit of increase, and therefore can take no increment. 16. Differentiation of the Product of two Func- tions. Let u = yz, (1) where y and z are both functions of x. Give x the infini- tesimal increment dx, and let du, dy, dz be the correspond- ing increments of u, y, and z, due to the increment which X takes. Then (1) becomes u + du = {y + dy) {z + dz) = yz + zdy + ydz + dz dy^ (3) Subtracting (1) from (3), and omitting dzdy, since it is an infinitesimal of the second order, and added to others of the first order (Art. 11), we have du = zdy + ydz, (3) which is the required differential Hence, the differential of the product of two func- tions is equal to the first into the differential of the second, plus the second into the differential of the first. * In the Calculus as in the Analytic Geometry, the radius Is always regarded as 1, unless otherwise mentioned. 6 c (f A C Fig. 3. ^^ ♦ GEOMETRIC ILLUSTRATION. 19 This may also be proved geometrically as 'follows : ' . Let z and y represent the lines AB and BC respectively ; then will u rep- resent the area of the rectangle ABCD. Give AB and BC the infinitesimal in- crements Ba (=(?«) and Cc (— dy) respectively. Then the rectangle ABCD will be increased by the rectangles BaGh, DCSc, and Ghcd, the values of which are ydz, zdy, and dzdy respectively; therefore du = ydz + zdy + dz dy. But dzdy being an infinitesimal of the second order and connected with others of the first order, must be dropped (Art. 11) ; if this were not done, infinitesimals Would not be what they are (Art. 8) ; the very fact of dropping the term dz dy implies that its value, as compared with that of ydz ^- zdy is infinitely small. The statement that ydz + zdy + dzdy is rigorously equal to ydz + zdy is not true, and yet by taking dz and dy suflBciently small, the error may be made as small as we please. Or, we may introduce the idea of motion, and consider that dz and dy represent the rate at which AB and BC are increasing at the instant they are equal to z and y respec- tively. The rate at which the rectangle ABCD is enlarging at this instant depends upon the length of BC and the rate at which it is moving to the right, + the length of DC. and the rate at which it is moving upward. If we let dz repre- sent the rate at which BC is moving to the right, and dy the rate at which DC is moving upward at the instant that AB = z and BO = y, we shall have du = zdy + ydz as the rate at which the rectangle ABCD is enlarging at this in- stant. (See Price's Calculus, vol. i, p. 41.) 20 DIFFERENTIATION OF A FRACTION. 17. Differentiation of the Product of any Num- ber of Functions. Let u = vyz, (1) Then giving to x the infinitesimal increment dx, and letting du, dv, dy, dz be the corresponding increments of u, v, y, z, (1) becomes u + du = (v + dv) (y + dy) {z, + dz), (2) Subtracting (1) from (2), and omitting infinitesimals of higher orders than the first, we have du = yzdv ■{■ vz dy + vy dz, (3) and so on for any number of functions. Hence, the differential of the product of any num- ber of functions is equal to the sum of the products of the differential of each into the product of all the others. CoE. — Dividing (3) by (1), we have du dv dy , dz ,,. — = — + -* H (4) % V y z ^ ' That is, if the differential of each function he di- vided by the function itself, the sum of the quotients mill be equal to the differential of the product of the functions divided by the product. 18. Differentiation of a Fraction. Let u = -, y then uy = X', (1) therefore, by Art. 16, we have udy + ydu = dx. Substituting for u its value, we have DIFFERENTIATION OF A POWER. ' 21 X - dy + ydu = dx. d Solving for du, we get ^„ ^ ydx-xdy \ . which is the required differential. Hence, the differential of a fraction is equal to the denominator into the differential of the numerator, minus the numerator into the differential of the de- nominator, divided by the square of the denominator. Cor. 1. — If the numerator be constant, the first term in the differential vanishes, and we have zdy yi Hence, the differential of a fraction with a constant numerator is equal to minus the numerator into the differential of the denominator divided by the square of the denominator. ^" = - ^^^ CoE. 2. — If the denominator be constant, the second term vanishes, and we have , dx du =: — , y which is the same result we would get by applying the rule of Art. 15. 19. Differentiation of any Power of a Single Va- riable. Let y = af. lat. W%en n is a positive integer. Regarding a?" as the product x, x, x, etc., of n equal fac- tors, each equal to x, and applying the rule for differentiating a product (Art. 17), we get 23 DIFFERENTIAL OF A POWER OF A VARIABLE. dy = x"-^ dx + a;""' dx + a;""' dx -f etc., to n terms. .•. dy = nT~^dx. (1) 2d. When n is a positive fraction. Let y ■= X" ; then y = a;". Differentiating this as just shown, we have, ny"-'^ dy = ?m"'~' &. Therefore, dy = — — .dx ^ n ^""' ' m x"'~^y - = z—^^ n y"- m — t^i^ ^^ (since y" = a;"*). m ™— 1 ■ ■ 3d. When n is a negative exponent, integral or fractional. Let y = x~" ; then V = — " X" Differentiating by Art. 18, Cor, 1, we have dy^ ^^ = - n^"'^ — 3) — {2a?— 3) d{ix + 3«) au _ ™ (4a; + x>y _ (4a; + g;^) jx dx ^ (2a;g — 3) (4 + 2a:) dx " (4a; + a;')" ' + 6a; + 12) dx ~ (4a; + Q^f 2x^ , 8ffl2a;s — 4a?, EXAMPLES. 25 1 + x , (l—'^x — x^)dx 16- y = ^r dy=--^,i 17. y = (aa? — s^f. dy = 4: {aa? — x^Y {2ax — Za?) dx. 18. y = {a + lx^)k dy = ^{a + ba^)i hx dx. i« a J &ax , 20. 2/ = Va^ — flS (Art. 19, Cor.). ^2^ = -^ '- 2^2^ — a' 2\/a;^ — a» (fl — x) dx 31. .y = 's/^ax — a"^. % = V'2aa; — ai^ " 1 , xdx 22. 7/ = — • -x''+ 2xWa^ -x^ — -7== dx L Vcr — x^ J , a* + a?x^ — ix^ , or, du = ; — dx. ^/a^ — x^ 1 + a?^ 2. u = 5- Passing to logarithms, we have J. ^^ w log u = log (1 + x^) — log (1 — a;2). du 2xdx 2xdx Axdx du = l+x^ ^ l—x^~ {l+x^){l—a?) 4xdx 1 — x^ 3. u = (a'' + 1)2. du = 2a'>'{a'' + 1) log a dx. 4 ,, - ^''-l _ 2a^ log a dx „ Vl + X , dx 0. ti = - , du = Vl — x (1 - «) Vl — a^ ILLUSTRATIVE EXAMPLES. 1. Which increases the more rapidly, a number or its logarithm ? How much more rapidly is the number 4238 increasing than its common logarithm, supposing the two to be increasing uniformly ? While the number increases by 1, how much will itg logarithm increase, supposing the 36 ILLUSTRATIVE EXAMPLES. latter to increase uniformly (which it does not) while the number increases uniformly. Let X = the number, and y its logarithm ; then we have y = log a;; .•. dy = — dx, which shows that if we give to the number (a;) the infinitely small increment (dx), the corresponding increment of y is — times as great; that is, the logarithm (y) is increasing it — times as fast as the number. Hence, the increase in the X common logarithm of a number is >, =, < the increase of the number, according as the number {x) <, =, > the modulus (m). When X = 4238, we have m , .43439448 ^ ^^ = 4238^* = -1338- ^^^5 4338 hence, dx = j^gj^gilg ^y = ^^^^^ ^^^^ ^^5 that is, the increment of the logarithm is '- — j^ot. — part of the increment of the number, and the number is increasing about 9758 times as fast as its logarithm. While the number increases by 1 , its logarithm will in- crease (supposing it to increase uniformly with the number) .434394 48 ^.^^^ ^ _ _oooi0347; that is, the logarithm of 4339 would be .00010247 larger than the logarithm of 4238, if it were increasing uniformly, while the number increased from 4238 to 4339. Besiabk. — While a number is increasing uniformly, its logarithm is increasing more and more sloidy ; this is evident from the equation iy = — dx, whicli shows that if the nuwher reoeiveg a very small in. TRIGONOMETRIC FUNCTIONS. 37 crement, its logaritliin receives a very small uicrement ; but on giving to the number a second very small increment equal to the first, the corresponding increment of the logarithm is a little less than the first, and so on ; and yet the supposition that the relative rate of change of a number audits logarithm is constant for comparatively small changes in the number is sufficiently accurate for practical purposes, and is the assumption made in using the tabular difference in the tables of loga- rithms. 2. The common logarithm of 337 is 2.514548. What is the logarithm of 327.12, supposing the relative rate of change of the number and its logarithm to continue uni- formly the same from 327 to 327.12 that it is at 327 ? Ans. 2.514707. 3. Find what should be the tabular difference in the table of logarithms for numbers between 4825 and 4826 ; in other words, find the increment of the logarithm while the num- ber increases from 4825 to 4826. Ans. .0000900. 4. Find what should be the tabular difference in the table of logarithms for numbers between 9651 and 9652. Ans. .0000450. 5. Find what should be the tabular difference in the table of logarithms for numbers between 7235 and 7236. Ans. .0000601. TRIGONOMETRIC FUNCTIONS. 24. To Differentiate y — sin x. (1) Give to X the infinitely small increment dx, and let dy represent the corresponding increment of y ; then we have y + dy = sin {x + dx) = sin X cos dx + cos x sin dx. (2) Because the arc dx is infinitely small, its sine is equal to the arc itself and its cosine equals 1 ; therefore (2) may be written y,-{- dy — siiv a; + cos a; dx. (3) 38 TRIGONOMETRIC FUNCTIONS. Subtracting (1) from (3), we have dy = cos X dx. (4) Hence, the differential of the sine of an arc is equal to the cosine of the arc into the differential of the arc, 25. To Differentiate y = cos x. Give to X the infinitely small increment dx, and we have y -\- dy =■ cos {x + dx) = cos X cos dx — sin x sin dx = cos a; — sin x dx (Art. 34). .'. dy =z — sin x dx. Otherwise thus: We have y =z cos x z= sin (90° — x). Differentiating by Art. 24, we have d^= cos (90° — x)d (90° — x) = sin X d (90° — x). .: dy — i — sin a; dx. Hence, the differencial of the cosine of an arc is negative and equal to the sine of the arp into the dif- ferential of the arc. (The negative sign shows that the cosine decreases as the arc increases.) 26. To Differentiate y = tan x. We have y = tan x = gbyA: dy — cos^ Differentiating by Arts. 18, 34, and 25, we have cos X dsinx — smx d cos x cos^ X cos^ X + sin' X ^ dx ■dx == cos' X cos' X s; sec' X dx. .: dy = see' z (?^, TRIGONOMETRIC FUNCTIONS. 39 Otherwise thus: Give to X the infinitesimal increment dx, and we have y + dy = tan {x + dx) .'. dy = tan {x + dx) — tan x tan X + tan cZa; tan X tan (^a; — tan X tan a; + & . , . , , , . = -1 7 , tan X (since tan dx = dx) 1 — tan X dx ^ ' dx + tan' X dx » , = : r— = sec^ X dx 1 — tan X dx (since tan x dx, being an infinitesimal, may be dropped from the denominator). .•. dy = sec* x dx. Hence, the differential of the tangent of an arc is"- equal to the square of the secant of the arc into the differential of the are. < 27. To Differentiate y = cot .-51?. We have «/ = cot « = tan (90° — x). .: dy = see' (90° — x) d (90° — x). .: dy = — cosec' x dx. The minus sign shows that the cotangent decreases as the arc increases. Hence, the differential of the cotangent of an arc is negative, and equal to the square of the cosecant of the arc into the differential of the arc. 28. To Differentiate y — sec x, 1 We have y = sec a; = cos X 40 TRIGONOMETRIC FUNCTIONS. d COS x smx dx .-. aw = 5 — = — = sec X tan x dx. ^ cos^ X cos^ X Hence, the differential of the secant of an arc is equal to the secant of the same arc, into the tangent of the arc, into the differential of the arc. 29. To Differentiate y = cosec x. "We have y = cosec x = sec (90° — x). .: dy = d sec (90° — x) = sec (90° — x) tan (90° —x)d (90° — x) = — cosec X cot X dx. Hence, the differential of the cosecant of an arc is negative, and equal to the cosecant of the arc, into the cotangent of the arc, into the differential of the arc. 30. To Differentiate ij — vers x. We have y = vers a; = 1 — cos x. .: dy = d{l — cos x) = sin x dx. Hence, the differential of the versed-sine of an arc is equal to the sine of the arc into the differential of the arc. 31. To Differentiate y = covers x. We have y = covers x — vers (90° — x), .: dy = d vers (90° —x) = sin (90° — x)d (90° — x) = — cos a; dx. Hence, the differential of the coversed-sine of an arc is negative, and equal to the cosine of the arc into the differential of the arc. GEOMETRIC DEMONSTRATION. 41 32. Geometric Demonstration at in the preceding Articles admit also of easy demonstra- tion by geometric construction. Let P and Q be two consec- ntive points* in the arc of a circle described with radius = 1. Let X = arc AP ; then dx = arc PQ. From the figure we have, PM = sin x; -The results arrived /T' Fig. 4. NQ = sin {x + dx) ; :. QE = d sin x. OM =: COS a; ; ON = cos {x + dx) ; .'. NM = — d cos X (minus because decreasing). AT = tan x; AT' = tan {x + dx) ; .: TT' = d tan x. OT = sec a;; OT' = sec {x + dx) ; .-. DT' = d sec X. Now, since EP and QP are perpendicular respectively to MP and OP, and since DT and TT' are also perpendicular to OT and OA respectively, the two infinitely small triangles PQE and DTT' are similar to MOP. Hence we have the following equations : * This notation, as already explained, means y = the arc whose sine is x. CIRCULAR FUNCTIONS. 47 35. To Differentiate y = tan-' x. We have z = tan y ; therefore, dx = sec^y dy — {1 + tan^ y) dy = (1 + a;«) dy. .'. dy = 5 = d (tan"* x). " 1 + a;^ ^ ^ 36. To Differentiate y = cot~^ a;. We have x = cot ^ ; therefore, dx=: — cosed^ ydy = — (1 + cot^ y) dy = - (1 + x^) dy. .: dy=— —^3 = d (cot-i x). 37. To Differentiate y = sec~' x. We have x = sec y; therefore, dx = sec y tan y dy = aec y Vsec' y — 1 dy dx ■, , 1 \ wy — x^x^- -1 ^ \^\J AlJ. 38. To Differentiate y = cosec" ^x. We have X = cosec y ; therefore, dx = — cosec y — cosec y coiy dy Vcosec^ y — 1 dy — x^Jx? - ^\dy. • » f dy = dx — // {nc\!ion~' ■^4 X,,/nA _ — U tUUoCv 48 EXAMPLES. 39. To Differentiate y = vers"' x. We have x = vers y ; therefore, dx = sin y dy ^ Vl — cos^ y i = Vl — (1 — vers yY dy = -v/^ vers y — vers^ y dy = VSa; — a:^ (?y. .-. (?w = — . = ^(vers~ia;). ^/'Hx-a? 40. To Differentiate y = covers"' x. We have x = covers y ; therefore, dx = — cos?^ dy = — Vl — mi?y dy = — Vl — (1 — covers y'f dy = — V^ covers y — covers* y 7. M = cosec ^ — dy = adx ^^/x^ - a^ 8. y = vers""i — ^ a dy = dx 'S/%ax — x^ 3.11 — dx 's/^ax — x^ 10, y = a cos~i -• adx dy = - a adx 1 x^ Va* - x^ \l^~a^ dx 11. y = a tan-1 — dy ■■ Of a c?dx ~ . X^~ V? -\-^ 1+ -2 OP dx , ,x -, a aHx 13. y = a cot-i -• % = - -—^, = - "Tip^- dx ,x, a a^dx 13. V = a sec"i -• dy = 7^= = ^ , „ - • a; /a;' MmcStLANS6m MXAMPLBS. 51 14. y =z a cosec~i — a dx dy= " «'^^ X Ix* cB-y/a-a — ^8 dx « — ie _i ^ 7 Oj adx 15. y — a vers 1 -• dy = — - =: — a / ^ ^2 V2ax — s? 16. y ^ a coTers"^ - • a dx V~''a- 0^ V^aX — 3? MISCELLANEOUS EXAMPLES. , a -\- X , 3a — X ^ 1. y = =• dy = ^dx. Va — X 2{a — x)^ 2. y — y X — Vi c? — a;^. , _ (a; + \/a*—l^dx ^ ~ 2Va^ — x>{x — V^"^^^*' „ a; , dx 4. 2/ + Vl - a;^ 2a; (1 — a;''') + Vl - ar* V(l — a;3)s -^ (1— a;2)t a* , — a«(a2-3a;2) + VI - a;']. rfa; dx dy X Xa^\ _ a4 17. 2/ = log(a;-«)--^^-^. dy = j^^j^dx. 18. y = ai"'. ^ c?^ = 3a'»' log a xdx. 19. y = e"' (1 — a^), J%ry — y* 46. y = x's/a^—7?+a^ sin-i -• dy = li.'ija^—ifidx. 4/r+ X , , , , dx 47. ^ = log y j^-^3^ + i tan-i a;. dy = r^—-^- , 3a; , dx 48. y = ymr^-^' dy = •v/9a; — a;* MtSdHLLAjYEdUS EXAMPLES. 61 60. y — af'°" '. dy = a;"""" 51. y = sec~* nx. h%. y = sin~i 'x log a; + (1 — x^)^ sin~^ a;" a: (1 — a;'*)^ , dx dy = dx. ^Vn^x^ — 1 adx Va^ + X 53. y = siu~i Vsin a;. :. y = tan-i y - 54. — cos X + COS a; a^ + x^ dy = ^(Vl + cosec x)dx. dy = \dx. 55. y = —p=z==. + log V 1 — a;^ ay = sin~^ X Vl —a;! 56. y z= {x + a) tan~i A / Vi 57. y = sec~i ^ 58. y = tan~i 59. y = sm~i ca/5 ^Va;^ + a; — 1 3a^a! — a:^ a* — 3aa;^ a;A/a — i (l-a;2) fla;. % = 'we make x — 1000000, and get _ 1 y — 1 + 1000000 which is less than ^^aoooo - If "^^ ^ish y to be less than one-trillionth, we make x = 1000000000000, aiid the re- quired result is obtained. We see that, however great x maj be taken, y can never become zero, though it may be 60 TRIGONOMETRIC ILLUSTRATION. made to differ from it by as small a quantity as we please. Hence, the limit of the function — — - is zero when x is infinite. We are accustomed to speak of such expressions thus : "When X is infinite, y equals zero." But both parts of this sentence are abbreviations: "When x is infinite" means, " When X is continually increased indefinitely," and not, " When X is absolute infinity ;" and " y equals zero " means strictly, "y can be made to differ from zero by as small a quantity as we please." Under these circupstances, we say, " the limit of y, when x increases indefinitely, is zero." 43. Trigonometric Illustration — An excellent exam- ple of a limit is found in Trigonometry. To find the values of 7 — 2 and — ^ , when diminishes indefinitely. Here we have 7 — ^ = eosO ; and when = 0, cos 6 = 1. tan 6 Hence, if 6 be diminished indefinitely, the fraction 7 — ^ will approach as near as we please to unity. In other words, the limit of 7 — 5, as d continually diminishes, is unity. We usually express this by saying, " The limit of sin , „ „ . ., „ "sine , , „ „„ ^^,whene = 0,isumty;' or, — - = 1, when = 0;" that is, we use the words " when 6 = 0" as an abbreviation for " when d is continually diminished toward zero." Since 7 — a = 1» vj^qvl 6 = 0, tan d we have also -; — „ = 1, when = 0, em TRIGONOMETRIC ILLUSTRATION. 61 It is evident, from geometric considerations, that if be the circular measure of an angle, we have tan (? > > sin0; or. tan e sin sin 6 >i; but in the limit, i. e., when 8 — 0, tan _ we have sine ' and therefore we have, at the same time, , , sin . -. — a = 1, and .-. —5— = 1, sin 6 6 ' which shows that, in a circle, the limit of the ratio of an arc to its chord is unity. a gin 8 In the expression, —^- = 1, when 8 = 0," it is evident u that —^ is never equal to 1 so long as 8 has a value dif- ferent from zero ; and if we actually make fl = 0, we render the expression — ^ meaningless.* That is, while —^ approaches as nearly as we please to the limit unity, it never actually attains that limit. If a variable quantity be supposed to diminish gradually, till it be less than anything finite which can be assigned, it is said in that state to be indefinitely small, or an infinitedmal ; the cipher is often used as an abbreviation to denote such a quantity, and does not mean abso- lute zero ; neither does express absolute infinity. 'Rem.. — The student may here read Art. 13, which is applicable to this method as well as to that of infinitesimals, which it is not neces- sary for us to insert again. , * See Todhunter's Dif. Cal., p. 6. 63 DERIVATIVES. 44. Derivatives. — The ratio of the increment of u to that of X, when the increments are finite, is denoted by — ; the ratio of the increment of u to that of x in the limit, i. e., when both are infinitely small, is denoted by y , and is called the derivative * of m with respect to x. Thus, let u =f{x) ;■ and let x take the increment h (= Ak), becoming x + h, while u takes the corresponding increment Ati ; then we have, 11 ■\- l^u = f(_x + h); therefore, by subtraction, we have Mt=f{x + }i)-f{x); and dividing by A (= Aa;), we get AM _f(x + h)—f {x) Ax- ~h ' ■ ^^^ It may seem superfluous to use both A and t^x to denote the" same thing, but in finding the limit of the second member, it will sometimes be necessary to perform several transformations, and therefore a sin- gle letter is more convenient. In the first member, we use Aa on account of symmetry. The, limiting value of the expression in (1), when h is infinitely small, is called the derivative of u or f{x) with respect to x, and is denoted by f (.x). Therefore, passing to the limit, by making A diminish indefinitely, the second member of (1) becomes /' (x), and the first member becomes, at the same time, -r- ; hence we , dx have ♦ Called »l80 the derined function and the differential coe^lcierit, DIFFERENTIAL COEFFICIENT. 63 45. Differential and Differential Coefficient. Let M = /(a;) ; then, as we have (Art. 44), we have du = df{x) = f (x) dx, where dx and du are regarded as being infinitely small, and are called respectively (Art. 12) the differential of x and the corresponding differential of u. f {x), which represents the ratio of the differential of the function to that of the variable, and called the derivative of f{x) (Art. 44), is also called the differential coefficient of f{x), because it is the coeflBcient of dx in the differential oifix). Some writers * consider tlie symbol — only as a wTiole, and do not assign a separate meaning to du and dx ; others.f who also consider tlie symbol -=- only as a whole, regard it simply as a convenient nota- tion to represent ^ , and claim that du and dx are each absolutely zero. 46. Differentiation of the Algebraic Sum of a Number of Functions. Let y = au + iv + ctv + z + etc., in which y, u, v, w, and z are functions of x. Suppose that when X takes the increment A (= Aa;), y, u, v, w, and z take the increments Ay, Am, C^v, Aw, A2. Then we have, y + C^y = a {u-\- ^u) + b (t; -f- At') + c (w + Aw) + (2 + Az) -f etc. .•. Ay = a Am + 5 Af + c Aw + Az + etc. Dividing by Ji or Ace, we have * See Todhunter's Dif. Cal., p. 17 ; also De Morgan's Calculus, p. 14, etc. t See Young's Dif. Cal., p. 4. 64 DIFFERENTIATION OF A PRODUCT. Aw Am , ,A!; , Mu . A^ , . — ^ =■ a {■ i he 1 h etc., ^x Aa; Ak AiB Aa; wliicli becomes in the limit, when h is infinitely small (Art. 12), dy du ^ dv dw , dz , , , « i ij\ 47. Differentiation of the Product of two Func- tions. Let y = uv, where u and v are both functions of x, and suppose Aj/, Alt, Ay to be the increments of y, u, v corre- sponding to the increment Aa; in x. Then we have y + ^y — {u + Am) (w + Av) :^ uv + uAv + vAu + Ml Av. .: Ay = uAv + V Am + AuAv; Am Az; Am Am . or, -^ = M — + i» \- -— Av. Ax Ax AX Ax Now suppose Ax to be infinitely small, and Ay Av Am Ax' Ax' Ax' become in the limit, dy dv du dx' dx' dx Also, since Ai' vanishes at the same time, the limit of the last term is zero, and hence in the limit we have du dv , du ,„ » ^ , „ X -^ = u-rr + v^- (See Art. 16.) dx dx dx ^ ' It can easily be seen that, although the last term vanishes, the remaining terras may have any finite value whatever, since they con- tain only the ratios of vanishing quantities (see Art. 9). For examplCj — = -- when a; = ; but by canceling x we get — = a. But the expression — x a:, which equals ;r x when a; = 0, becomes - x = X \j 1 when a = 0. DIFFERENTIATION OF A PROBUCT. 65 Otherwise thus: Let fix) ^ {x) denote the two functions of x, and let u =f{x) {x). Change x into x + h, and let m + A« denote the new product; then M + Am =f{x + h){x+h) Au _ f{x + h)(x + h) —fix) ^ (x) _ " Ax ~ h Subtract and add fix) (j) (a; + h), which will not change the value, and we have AM ^/(a' + ^^-/(g)^(^ + h) +/(^)i(i±^-il£). Now in the limit, when h is diminished indefinitely, t^i+J^-^ = ^' ix) (Art. 44); and ipix + h) = (a;) ; therefore, ^ = /' (a;) ^ ix) +fix) ' (oo), which agrees with the preceding result. 48. Differentiation of the Product of any Number of Functions. Let y = uvw, u, V, w being all functions of x. Assume z = vw, then y = uz, 66 DIFFERENTIATION OF A FRACTION, and by Art. 47 we have dy _ udz zdu dx ~dx dx Also, by the same Article, dz _ vdw wdv ^ dx ~ dx dx ' hence, by substitution, we have dy dw , dv , du ,„ . , ^„^ -f- ^= uv-^ — {■ uw -^ + VW-J-- (See Art. 17.) C(/X Cit& O/tlu uX The same process can be extended to any number of functions. 49. Differentiation of a Fraction. Let y = -• Then we shall have u + Au y 4- Ay = ; ti + Au u V -{- Av V _ vAu — uAv v^ + vAv Dividing by Ax and passing to the limit, du dv dy _ dx dx dx~ v^ (since vdv vanishes). (See Art. 18.) Cor. — If tt is a constant, we have udv dy _ dx dx ~ v> DIFFERENTIATION OF ANT POWER. 67 50. Differentiation of any Power of a Single Va- riable. 1st. WTven n is a positive integer. Let y = x"; then we have y + Ay = {x + h)" ; therefore, Ay — wa?-" h + V ' x"-^ h^ + etc h\ Dividing by h or Aa;, we get Aa; ^2 Passing to the limit, we have ^ = wa;"-i. (See Art. 19, 1st.) (1) dx 2d. When n is a positive fraction. Let y = u", where m is a function of x ; then y" = vT, and hence. by(i), d ny^~ • (r) dx dy dx = d (m") ; dx m M"^^ dv, ~ n iy"~' dx = — M" -T- (Art. w dx 19, 2d). (3) 3d, When n fractional/. is a negative exponent, integral or Let y = «-"; then 2^ = ^' 68 DIFFERENTIATION OF LOGARITHMS. and by Art. 49, Cor., we have dy nu''~^du „ i^u , , . ^f. qa\ /q\ i = --u^di = - ^'*"" Tx (^^*- 1^' ^•^)- (^) 51. Differentiation of log x. Let 2^ = log*; therefore, y + Ay = log (a; + h), and Ay = log (x + /*) — log x ^'"[x-^x^+z^-^^r' therefore, ^ = ^g _ -| + etc.); therefore, passing to the limit, we get dy m 1 -^ = — or - dx X X (according as the logarithms are not or are taken in the Naperian system. See Art. 20). 52. Differentiation of a''. Let y = a*. Proceeding exactly as in Art. 31, we get -f- = —]oga or a" log a (Art. 21). 53. Differentiation of sin x. Let 2/ = sin a; ; therefore y + Ay = sin (a; + A) ; hence. Ay = sin {x + A) — sin x. DIFFERENTIATION OF A COSINE. 69 But from Trigonometry, . . ■ T3 o A + B . A — B sm A — sm B = 3 cos — ^ — sm — - — .: Ay — sin {x + h) — sin x = 2 cos \x + ^j sm ^ ; • ^ hence, -^ = cosIcb + „) ^r— • 2 By Art. 43, when A is diminished indefinitely, the limit of . h sm ^ / h\ = 1 ; also, the limit of cos lx + ^)= cos x. 2 Therefore, -p = cos «. (See Art. 24) 54. Differentiation of cos x. Let y = cos a; ; therefore, y + Ay = cos (a; -|- h) ; hence. Ay = cos (x + h) — cos a; 2' A-B = — 2 sin (a; + ^1 sin because cos A — cos B = — 2 sin f — ^ — ) sm 7 . sm ■ Av . I li\ ■ Therefore, ^ = - sm (a; + ^j-y 70 COMPARISON OF THE TWO METHODS. Hence, in the limit, ^ = - sin X. (See Art. 35.) dx Of course this differentiation may be obtained directly from Art. 53, in the same manner as was done in the 3d method of Art. 35. Since tan x, cot x, sec x, and cosec x are all fractional forms, we may find the derivative of each of these functions by Arts. 18 or 49, from those of sin x and cos a;, as was done in Arts. 36, 37, 38, and 39 ; also, the derivatives of vers x and covers x, as well as those of the circular functions, may be found as in Arts. 30, 31, 33 to 40. From the brief discussion that we have given, the student will be able to compare the method of limits with the method of infinitesimals; he will see that the results obtained by the two methods are identically the same. In discussing by the former method, we restricted ourselves to the use of limiting ratios, which are the proper auxiliaries in this method. It will be observed that, in the former method, very small quantities of higher orders are retained till the end of the calculation, and then neglected in passing to the limit; while in the infinitesimal method such quantities are neglected from the start, from the knowledge that they necessarily disappear in the limit, and therefore cannot affect the final result. As a logical basis of the Calculus, the method of limits may have some advantages. In other respects, the superiority is immeasurably on the side of the method of infinitesimals. CHAPTER IV. .iUCCESSIVE DIFFERENTIALS AND DERIVATIVES. S5. Successive Differentials. — The difEerential ob- tu-ned immediately from the function is the first differential. Tne differential of the first differential is the second differ- ential, represented by d^y, cPu, etc., and read, "second differential of y," etc. The differential of the second dif- ferential is the third differential, represented by d^y, d^u, etc., and read, "third differential of «/," etc. In like man- n6r, we have the fourth, fifth, etc., differentials. Differen- tials thus obtained are called successive differentials. Thus, let AB be a right line whose equation is y z= ax + b; .: dy = adx. Now regard dx as constant, i. c, let x be equicres- cent;* and let MM', M'M", and M"M"' represent the successive equal increments of x, or the dn^s, and R'P', E"P", R"'P"' the corre- sponding increments of y, or the dy'B. We see from the figure that R'P' = R"P" = R"'t>"' ; therefore the dy's are all equal, and hence the difference between any two consecutive difs being 0, the differential of dy, i. e., d''y = 0. Also, from the equation dy = adx we have d^y = 0, since a and dx are both constants. Take the case of the parabola y^ = 2px (Pig. 7), from pdx Fig. 6. which we get dy = ' Regarding dx as a constant, we * When the variable increases by equal increments, i, e., when the dilTercntif^l j? (Qnslcfnt, the Ti^riable js called an e^uicrescejit imnable, n EXAMPLES. have MM', M'M", M"M"' as the successive equal increments of X, or the dz's ; while we see from Pig. 7 that R'P', R"P", R"T"', or the dy's, are no louger equal, but diminish as we move towards the right, and hence the difference be- tween any two consecutive dy-s is a negative quantity (remembering that the difference is always found by taking i\i& first value from the second. See Art. 12). Also, from the equa- tion dy = - dx we see that dy varies inversely as y. The student must be careful not to confound d^y with dy^ or d{y^): the first is "second differential of ^;" the second is "the square of dy\" the third is the differential of y'^, which equals Zydy. EXAMPLES. 1. Find the successive differentials ot y =.oiP. Differentiating, we have dy = 5a:* dx. Differentiating this, remembering that d of dy is d!^y and that dx is con- stant, we have d!^y = 20a:* da?. In the same way, differen- tiating again, we have d'^=60a;* da?. Again, d^y = 120.r da^. Once more, d^y — 12Ma?. If we differentiate again, we have d^y = 0, since dx is constant. 2. Find the successive differentials oi y =z i3fi—3a? + 2x. idy= (12a^ — 6x + 2)dx; Ans. I cFy = (24a; — 6)d^; ( d^y = 2ida?. 3. Find the first six successive differentials oiy =. sin x. {dy = cos X dx ; d?y = — sin a; dx'^ ; d^y = — cos X dx^ ; (Py = cos a;, da? ; d*y = sin x dx* ; d^y = — sin x dx^. EXAMPLES. 73 4. Find the first six successive diflferentials of y = cos x. {dy = ■— sin x dx d^y = sin x do? d^y = — sin a; da^ d^y = — cos X di? ; d^y ■= cos X dx^ ; d^y = — cos a; dx^. 6. Find the fourth differential of y= x". Ans. d^y — n{n — 1) (n — 2){n — 3) x^'^da^. 6. Find the first three successive differentials of y = a". !dy = a^ log a dx ; d^y = a" log^a da?; d^y = aP lo^ a da?. 7. Find the first four successive differentials of y == log x. Ans. \dy = d^y = dx x' d'y=^ do? x^ ' a? ' dhj = Ma^ a^ 8. Find the first four successive differentials of «/ = 2«'v/a^ adx _ -„ atZa;' '^x Zada? Ans. dy d?y = 2x^' ix'^ ,, 15adx* d'y= --— • 9. Find the first four successive differentials of y = log (1 + x) in the common system. mdx ,„ nidx' Ans. ^y =i + x' 2mda? d^y= - d'y^ - {l + xf 6mdx^ 10. Find the fourth differential of y = 6". Ans. d*y = ^da^. 74 SUCCESSIVE DEBIVATIVES. 56. Successive Derivatives. — A first derivative* is the ratio of the differential of a function to the differential of its variable. For example, let y = !ifi represent a function of a;. Differentiating and dividing by dx, we get I = 6.. (1) The fraction -^ is called the first derivative of y with respect to x, and represents the ratio of the differential' of the function to the differential of the variable, the value of which IS represented by the second member of the equation. Clearing (1) of fractions, we have dy = Qafidx; hence, -j- or &x^ is also called the first differential coefficient of y with respect to x, because it is the coefficient of dx. A second derivative is the ratio of the second differential of a function to the square of the differential of the variable. Thus, differentiating (1) and dividing by dx, we get (since dx is constant, Art. 55), g = 30.^ (2) either member of which is called the second derivative of y with respect to x. A third derivative is the ratio of the third differential of a function to the cube of the differential of the variable. Thus, differentiating (2) and dividing by dx, we get ♦ §?e Arts, 44 and 45t MPFmMNMAL COEPFtCtENVS. YS g = 130^, (3) either member of which is called the third derivative of y with respect to x. In the same way, either member of is called the foiirth derivative of y with respect to x, and so on. ^^°' M' jl' ^' ^' ^*°-' ^""^ '^^"^'^ respectively ^/^e ^/^/•s^, second, third, fourth, etc., differential coefficients of y with respect to x, because they are the coefficients of dx, dx^, da?, dod^, etc., if (1), (2), (3), (4), and so on, be cleared of fractions. In general, if y ■=f{x), we have % = ^-^ = -^'(^^ (Art. 45); .-. dy = f'{x)dx. S = ^^ = f" (^) 5 ••• ^y = f" (^) ^^; ^ = lfip =f"'{x); :.d?y=r(x)dx\ ^ = ^1^} =r{x); ,.d*y = f-{x)dx^. etc, ^ etc. = etc. .•. etc. = etc. rfa;" dx I \ I' a J \ I That is, the first, second, third, fourth, etc., derivatives are also represented by /' {x), f" {x), /'" {x), /" {x), etc. 16 GBOMSTMd RMPBSlSSNfAftdN. Strictly speaking, -^ or/' (a;) are symbols representing the ratio of an infinitesimal increment of the function to the corresponding infinitesimal increment of the variable, while the second member expresses its value. For example, in the equation y = aa:*, we obtain ~ or /' {x) is an drlitrary syinbol, representing the value of the ratio of the infinitesimal increment of the function {aoc^) to the corresponding infinitesimal increment of the variable {x), while 4aic* is the value itself. It is usual, however, to call either the derivative. 56a. Geometric Representation of the First De- rivative. — Let AB be any plane curve whose equation is y-=zf{x). Let P and P' be consecutive points, and PM v e^ and P'M' consecutive ordinates. The part of the curve PP', called an ele- ment* of the curve, does not differ , from a right line. The line PP' pro- ^~~'^ rsr longed is tangent to the curve at the '^' ' point P (Anal. Geom., Art. 43), Draw PR parallel to XX', and we have MM' = PE = dx, and EP' = dy. Denote the angle CTX by a, and since CTX = P'PR, we have tan « = -^• dx And since the tangent has the same direction as the curve * tn this work, the word " element " will be used for brevity to denote an " in- finitesimal element." 4 at the tangent point P, a will also denote the inclination of the curve to the axis of x. Hence, the first derivative of the ordinate of a curve, at any point, is represented by the trigonometric tan- gent of the angle which the curve at that point, or its tangent, maJces with the axis of ac. In expressing the above differentials and deriTatives, we have assumed the independent variable x to be equicrescent (Art. 55), which we are always at liberty to do. This hypothesis greatly simplifies the expressions for the second and higher derivatives and difEerentials of functions of x, inasmuch as it is equivalent to making all differentials of "x above the first vanish.. Were we to find the second deriva- tive of y with respect to x, regarding dx as variable, we would have icPx ^y _ ^ i^y\ ^^ ^y ~ ^y ' d3? ~ dx \dx) do? ich less simple than i by supposing dx to be constant. which is much less simple than the expression -5^, obtained EXAMPLES. , 1. Given «/ = iK", to find the first four successive deriv- atives. dx ^ = n{n — l){n — 2)af~^ ; g = «(«-l)(«-3)(«-3)^H % EJiAMPLSk If w be a positive integer, we have g = ^(»i-i)(»»-3)....3.a.i. and all the higher derivatives vanish. If w be a negative integer, or a fraction,, none of the suc- cessive derivatives can vanish, 3. Given y = a? log x ; find -t£' -^ = Sa^log z + x^; T^ = Qx log a; + 3a; + 2a! = 6a; log a; + 5x; J = 61oga; + 6 + 5. ^ = -• It can be easily seen that in this case all the terms in tlie successive derivatives which do not contain log x will dis- appear in the final result ; thus, the third derivative of o^ is zero, and therefore that term might have been neglected ; and the same is true of 5a;, its second derivative being zero. „ 1+x , , , d^y 340 4. y = 6"^; prove that -j\ = «*e<"'- 5. ^ = tan X ; find the first four successive derivatives. dy , T^ = sec^a;; dx T-? = 3 sec* a; tan a;; da? -V? = 6 sec^ a; — 4 sec* x ; da? da^ 8 tan X sec* x (3 sec* x — 1). EXAMPLES. 19 6. y = log sin x ; prove that -5-^ = 3 cot a; cosec'' x. 7. ^'^ = 2pa;; find d^y da?' dy^_ dx .P. cP^ _ :1/^^ ^dx P^~ ■y ^ ^ dx _ ^ y d3? dx\ y^f y^ y^ y* 8. ^ = af ; prove that ^ _ eilf\_ _dx _ f_y _ Zf dx\ W ^ = ce" (1 + log xf + a?-i. 9. c?y^ + Vh? = ffl'S"; prove that ^ = — -|^. ^ doi? a^y' 10. ^ = i ;:; prove that -^ = l-x' ^^ ^"'' dx* - (1 - x)s 11. ^2 _ sec 2x ; prove that ^ + -^ = 3y^. d*v 12. y = ff-* cos a; ; prove that 4^ + -5^ = 0. (Z% 48 13. 2^ = a;* log {x") ; prove that ^ = — • 80 EX AMI' h JUS. 14. y z= a?; prove that d^y = 6 {dxy + 18xdxdJ>x + 3x^^, when X is not equicrescent, 15. y=f{x); prove that d^y = /'" (x) {dxf + 3/" {x) dxd^x +/' {x) 1.3.3 We may also use the following notation for the function and its successive derivatives : f(x), f (x), f" {x), f" (x), f"{x), etc., as given in Art. 56, and write the above theorem, y =/(«') = /(o) + /' (0) f + /" (0) ^^ + /'" (0) j-|— in which /(O), /' (0), /" (0), /'" (0), etc., represent the values which /(a?) and its successive derivatives assume 84 MAcLAtlBtN'S tSMOMSM. when X = 0. We shall use this notation instead of -^ , -~ , etc., for the sake of brevity. This theorem, which is usually called Maclaurin's Theorem, was previously given by Stirling in 1717 ; but appearing first in a work on Fluxions by Maclaurin in 1743, it has usually been attributed to him, and has gone by his name. Maclaurin, however, laid no claim to it, for after proving it in his book, he adds, " this theorem was given by Dr. Taylor." See Maclaurin's Fluxions, Vol. 3, Art. 751. To Develop y = {a + as)* Here f{x) = (a + xf; hence, /(O) = laf. f'{x) = Q{a-Vxf; f (0) = 6«5. f"{x) = 6-Q{a + xY; f" (0) = 5 • QaK fix) = 4.5.6(a + a;)S; /'" (0) = 4 • 5 . GflS. /"(z) = 3.4.5 •6(a + a;)2; /"(O) = 3.4.5. 6a2. /"{x) = 2.3.4.5.6(a + a;); /' (0) = 3 • 3 . 4 . 5 . 6a ; /"(a;) = 1.3.3.4.5.6; /"(o) = 1.3.3.4.5-6; Substituting in (7), we hare, « = (a + a;)6 = ff6 + Qa^x + 5 . 6a*-^ + 4 ■ 5 . GaS-^^r q 4. fi fl Q^a^ 2.3.4.5- 6aa:5 1.3.3.4.5. 6a;« + ^'*"^'^1.2.3.4'^1.3.3.4.5^1.2.3.4.5.6 = a» + Qa?x + 15aV + aOa^a^ + l5aW + 6aa^ +sf^, which is the same result we would obtain by the binomial theorem. THE BINOMIAL THEOREM. 59. To Develop j/ = (a + x)\ Here f(x) = (a + a;)" ; hence, /(O) = a". f'(x) = n(a + «)"-!; /' (0) = wa"-!. /" {x) = n{n — 1) {a + x)'^'>. /"(O) = w(w— !).«"-«. /'" (x) = n{n — l){n-2){a + a;)"-»; /'" (0) = m (m — 1) (ra — 2) a-'-l /"(a;) = w (w — 1) (w — 2) (re — 3) {a + a;)""*; /"(O) = n{n — l)(n — 2){n — 3) a"-*, etc. Substituting in (7), Art. 58, we haye, ff = (a + x)" = a" + nce'-h: -\ ^^ — - — '- ,0 1 • * n{n — l){n — 2) a-'-^a^s ■*■ 1-2^3 , w (w — 1) (w — 2) (w — 3) g"-V ■•" 1.2.3.4 ^ + ^*°- Thus the truth of the linomidl theorem is established, apphcable to all values of the exponent, whether positive or negative, integral or fractional, real or imaginary. 60. 1. To Develop y = sin x. Here f{x) = sin a; ; hence, /(O) = 0. " /' (x) = cos a; ; " /' (0) = 1. " f"(x) = —sin a: J " /"(O) = 0. 86 THS LOOARITHMIC SERIMS. Here /'" {x) = — cos a; ; hence, /'" (0) = - 1. " f"{x)=smx; " f"{0)=0. " f{x) = cosx; " /'(0) = 1. Etc., etc. Etc., etc. Hence, y = smx — x — :^ — ^ — - + 1-2-3 ' 1.2.3. 4. 5 + etc. 1.3-3.4. 5.6.7 2. To Develop y = cos x. Ans. y = cos ^ = 1 - ^ + iTaTsTl + 1 o b J K a S 5 — ^tC. 1.2. 3. 4. 5. 6^ 1.2. 3- 4. 5. 6-7. 8 The student will observe that hy taking the first derivative of the series in (1), we obtain the series in (3), which is clearly as it should be, since the first derivative of sin x is equal to cos x. Since sin {—x) = — sin x,. from Trigonometry we might have inferred at once that the development of sin x in terms of x could con- tain only odd powers of x. Similarly, as cos (— x) = cos a*, the development of cos x can contain only even powers. By means of the two f orlnulse in this Article we may compute the natural sine and cosine of any arc. For exam- ple, to compute the natural sine of 20°, we have x = are qt 20° = ^ = .3490652, which substituted in the formalsB, gives sin 20° = .342020 .and cos 20° = .939693. THE LOGARITHMIC SERIES. 61. To Develop y = log (1 + a;) in the system in which the modulus is m. Here /(x) = log (1 + x); hence, /(O) = 0. TBS LOGARITHMIC SERIES. 87 Here/" {x) = - ^^, ; hence, /" (0) = -m. "/"'(^) = (^|s; " /"'(o) = i.am. " -^"^^^ = - ^rT$5 " /"(O) = -..2.3m. Etc. Etc. Substituting in (7), Art. 58, we have, y = log (! + «) = m (a;— |cs3+ia;'— :Ja^+^a^— etc.), (1) which is called the logarithmic series. Since in the Naperian system m = 1 (see Art. 20, Cor.), we have, 2/ = log (1 + a;) = CB - I + I - - + - - etc. (2) which is called the Naperian logarithmic series. This formula might be used to compute Naperian loga- rithms, of very small fractions ; but in other cates it is useless, as the series in the second number is divergent for values of a; > 1. We therefore proceed to find a formula in which the series is convergent for all values of x ; i. e., in which the terms will grow smaller as we extend the series. Substituting — a; for a; in (2), we have, /inS /v3 QJ* oj5 log(l-ar) = -a;-^---^-g-- etc. (3) Subtracting (3) from (2), we have, ,' , X , ,, X „ 2a;8 2a^ , 2a;^ log (1 + 3!) - log (1 - 2;) = 2a; + -g- + -g- + ^ + etc. 88 CALCULATION OF LOGARITHMS. _ , 1 1 +a; 2 + 1 Let X = - , ^ ; .-. = 22 + 1 1 — x z Substituting in (4), we have, , 2 + 1 «r 1 1 + 4- 1 ^ 3 (3^ + l)s ^ 5 (20 + 1)« + 7T2^"Ip + ^*°-]' or log(.+ l) = log.+^3[2^+^2~3y3 + 5 (2. + 1)«+ 7127^-17' + "*4 (^) This series converges for all positive values of z, and more rapidly as z increases. By means of it the Naperian loga- rithm of any number may be computed when the logarithm of the preceding number is known. It is only necessary to compute the logarithms of privie numbers from the series, since the logarithm of any other number may be obtained by adding the logarithms of its factors. The logarithm of 1 is 0. Making 2 = 1, 2, 4, C, etc., successively in (5), we obtain the following Napekian ok Hyperbolic Logarithms. log2 = logl + 2(| + 3L +_!. +_!. + _!. +_J_, "•■ rs'-'P "•" 15T3»s + I7T3" + ^^^') ' or, since log 1 = 0, .33333333' .01234568 .00082305 log 2 = 2 ( .00006532 ) = 3 (0.34657359) = 0.6931471& .00000565 ' .00000051 .00000005 , CALCULATION OF LOGARITHMS. 89 log 3 = log 2 + 2g + 3I3 + ^, + A. + A. + etc.) = 1.09861228. log 4 = 2 log 2 = 1.38629436. log 5=log4 + 2g + 3l3 + A.+A-, + gL+etc.) = 1.60943790. log 6 = log 3 + log 2 = 1.79175946. log 7 = log6+2(l + 3i-3 + ^_ + _l^, + etc.) = 1.94590996. log 8 = 3 log 2 = 2.07944154. log 9 = 2 log 3 = 2.19722456. log 10 = log 5 + log 2 = 2.30258509. In this mannei', the Naperian logarithms of all numbers may be computed. "Wljere the numbers are large, their logarithms are computed more easily than in the case of small numbers. Thus, in computing the logarithm of 101, the first term of the series gives the result true to seven places of decimals. Cor. 1. — Prom (1) we see that, the logarithms of the same number in different systems are to each other as the moduli of those systems ; and also, that the logarithm^ of a number in any system is equal to the Jiaperian logarithm of the same number into the modulus of the given system. Cob. 2. — Dividing (1) by (2), we have Common log (1 + x) _ ^ . Naperian log (1 + a;) Hence, the modulus of the common system, is equal to the common logarithm of any number divided by the JVaperian logarithm^ of the same number. 90 EXPONENTIAL SERIES. Substituting in (6) the Naperian logarithm of 10 com- puted above, and the common logarithm of 10, which is 1, we have '^ = o Qr.oLKnQ = .4342944819032518276511289 . . . which is the modulus of the common system. (See Serret's Calcul Differentiel et Integral, p. 169.) Hence, the common logarithm of any number is equal to the iN'aperian logarithm of the sam,e number into the modulus of the common system, .434-^944^- Cor. 3. — Representing the Naperian base by e (Art. 21, Cor. 2), we have, from Cor. 1 of the present Article, com. log e : Fap. log e (= 1) :: .43429448 : 1; therefore, com. log e = .43429448 ; and hence, from the table of common logarithms, we have e = 2.718281+. EXPONENTIAL SERIES. 62. To Develop y = a*. Here f{x) = a^ ; hence, /(O) = 1. /' {x) = a== log a; " /' (0) = log a. " /" {x) = a» {log af; « /" (0) = (log af. " /'" {x) = a» (log aY ; « /'" (0) = (log a)K and the development is X X^ iK^ t/ = a=» = l + logaj + \og>aj-^ + hg^a j-^ + log^«j;^| + etc. (1) EXPONENTIAL SERIES. 91 OoR. — If a = e, the Naperian base, the development becomes 2, = «» = ! + _+_. + __ + ___ Putting a; = 1, we obtain the following series, which en- ables us to compute the value of the quantity e to any required degree of accuracy : 111 1 2 ' 2-3 ' 2.3-4 ' 2.3.4.5 + ----2:3^.-^ + ^*°- = 2.718281838 + . 63. To Develop y — tai\~^ x. In the applications of Maclaurin's Theorem, the labor in finding the successive derivatives is often very great. This labor may sometimes be avoided by developing the first derivative by some of the algebraic processes, as follows: Here f{x) = tan~ia;; hence, /(O) = 0. = (by division) 1 — x^ + x^ — ofi -\- a?; ' /'(0) = 1- /" {x) = — 2a; + 4x3 — 6a;5 + 8a;' — lOa;* +etc. ; /"(0) = 0. /'"(a;) = _2 + 3-4ai2-5-6a;^+7-8a;« — etc.; /"'(0) = -2. /" (x) = 2-3-4a; — 4-5-6a;3 + etc.; /"(O) = 0. 93 FAILURE OF MACLAUBIN'S THEOREM, f{x) = 2.3-4 — 3-4.5-6a;« + etc.; /'(0) = 2.3-4. /"(a;) = — 2-3-4-5-6a; + etc.; /"(0) = 0. /'"(a;) = — 3.3.4-5-6 + etc.; /•"(O) = — 2-3-4-5.6. Substituting in (7) of Art. 58, we get /TfO /*>5 fvl y = tan~i a; = a; — — + ^ ~kr + ^^• o 7 64. It sometimes happens in the application of Maclau- rin's Theorem tliat the function or some of its derivatives become infinite when a; = 0. Such functions cannot be developed by Maclaurin's Theorem, since, in such cases, some of the terms of the series would be infinite, while the function itself would h& finite. For example, take the function y = log x. Here we have f(x) = loga;; hence, /(O) = — co. /'(^)=^; " /'(0) = «. ^"(^) = -^; " /"(0) = -<». etc. etc. Substituting in Maclaurin's Theorem, we have y — \ogx = — 00 + 00- — 00^+ etc. Here we have the absurd result that log a; = oo for all values of x. Hence, y = log z cannot be developed by Maclaurin's Theorem. Similarly, ^ = cot a; gives, when substituted in Maclau- rin's Theorem, TAYLOR'S THEOREM. 93 y = cot .r = 00 — 00 - + etc. ; that is, cot a; = 00 for all values of x, which is an ahsurd result. Hence, cot x cannot be developed by Maclaurin's Theorem. Also, y = x^ becomes, by Maclaurin's Theorem, y = a;* — + ooa; + etc. ; that is, x^ = 00 for all values of x, which is an absurd result. Whether the failure of Maclaurin's Theorem to develop correctly is due to the fact that the particular function is incapable of any devel- opment, or whether it is simply because it will not develop in the particular form assumed in this formula, the limits of this book will not allow us to enquire. TAYLOR'S THEOREM. 65. Taylor's Theorem is a theorem for developing a function of the sum of two variables into a series arranged according to the ascending powers of one of the vainables, with coefficients that are functions of the other variable and of the constants. Lemma. — We have first to prove the following lemma: If we have a function of the sum of two variables x and y, the derivative will be the same, whether we suppose x to Tary and y to remain constant, or y to vary and x to remain constant. For example, let u = (x + yf. (1) Differentiating (1), supposing x to vary and y to remain constant, we have Au , , , — = n{x + yY-^ (2) 94 TAYLOR'S TSEOBEM. Differentiating (1), supposing y to vary and x to remain constant, we have p^ = n{x + yr-^; (3) from which we see that the derivative is the same in both (2) and (3). In general, suppose we have any function of a; + y, as u=f(x + y). (4) Let z = X + y; (5) •■• u=f(z). (6) Differentiating (5), supposing x variable and y constant, and also supposing y variable and x constant, we get dz ^ , dz ^ -7- = 1, and T- = 1. dx dy Differentiating (6), we have .•. du = f (z) dz. du ., , , dz jr, , s I • dz A And similarly, du ., , ^dz j.t I \ i ■ dz A du _ du dx ~ dy That is, the derivative of u imth respect to a;, y being constant, is equal to the derivative of u with respect to y, X being constant. MSTitoD Of Taylor's tHEORHM. 06 66. To prove Taylor's Theorem. Let vl z=L f (x -\- y) be the function to be developed, and assume the development of the form =.A^By^Gf^ Dy^ + Ef + etc., (1) in which A, B, C, etc., are independent of y, but are func- tions of X and of the constants. It is now required to find such values for A, B, G, etc., as will make the assumed development true for all values of x and y. Finding the derivative of «', regarding x as constant and y variable, we have dv' ^ = B + Wy + Wy^ + 4%" + etc. (2) Again, finding the derivative of u', regarding x as varia- ble and y constant, we have du' dA dB dC . dD , ^ . ,„, ^ = ^+^2/ + ^2^^ + ^2^' + etc. (3) By Art. 65, we have ^j- = -y- : therefore, dy dx B+2Cy+BDf+^Bf+ei.. =f + f ^ + §f + §f + etc. (4) Since (1) is true for every value of y, it is true when y = 0. Making y = in (1), and representing what u' becomes on this hypothesis by u, we have u=f{x)=zA. (5) Since (4) is true for every value of y, it follows from the principle of indeterminate coefficients (Algebra) that the coefficients of the like powers of y in the two members must be equal. Therefore, 96 ^AtLOR'S >tHBdRMM. 31) -^ . 7)_ ^ '^' dx' .. -- — J 2 3 ^^, dx' .: M — 1.2.3-4 " tP' Substituting these values of ^, 5, C, D, etc., in (1), we have , XI , \ , duy , ^u y^ , d^u «* Or, using the other notation (Art. 56), we have u' =f{x + y) =f{x) +f'(^x)l+r{x) -t^ +/"'(^) j^- + /"(a') 17^1 + etc., (7) which is Taylor's Theorem. It is so called from its discov- erer. Dr. Brook Taylor, and was first published by him in 1715, in his MetJiod of Increments. Hence, by Taylor's Theorem, we may develop a function of the sum of two variables; as u =/(a; + y), into a series of terms, the first of which is the value of the function when y = ; the second is the value of the first derivative of the function when y = 0, into y ; the third is the value of the second derivative when w = 0, into -^^ , etc. ^ l-S The development of /(cc — y) is obtained from (6) or (7), by changing + y into — y ; thus. BlNdMtAL THEORSM. ^^ J,. . _ duy ^^ d^u y^ d'hi y* . or, fix -y)= fix) -f {X) f +/" (,x) ^ -/'" [x) ^ Cor. — If we make a; = in (7), we have W =f{y) =/(0) +/'(o)f +/"(0)^ +/"'(0) j|-- which is Maclaurin's Theorem. See (7) of Art. 58. THE BINOMIAL THEOREM. 67. To Develop u' = {x + «/)". Making y = 0, and taking the successive derivatives, we have f{x) = ar", /' (x) = wa;"-', /"(a;) = «(» — !) a;"-2, /'" (x) —n{n- 1) (ra — 2) aj'-s, /" (a;) = « (w — 1) (re — 2) {n — 3) a;"-*, etc. etc. Substituting these values in (7), Art. 66, we have , / , N„ _ , nx?-~^y , w(re — l)a;"-2y m' = (a; + «/)»= af + — j-^ + -^ j-^ ^ n{n — 1) (w — 2)ar'~'^ + ' 172'. 3 + ^™-' which is the Binomial Theorem (see Art. 59). 5 98 APPMCAVION^ Off TAYLOR'S tSSORlUM. 68. To Develop u' = sin (a? + y). Here f{x) = sin sc, f (x) = cos x, f" (x) = — sin X, /'-' (x) = — cos x, etc. Hence, m' = sin {x + ^) + '°' ^ (f - rCs + 1:21:4:5 - 1:2:371-576-7 + '^) — sin a; cosy + cos a; sin?/. (See Art. 60.) THE LOGARITHMIC SERIES. 69. To Develop u' = log {pc + y). Here /(^) = log a;, /'" (a;) = |, /"(a;) = -l, etc. Hence, u' = log {x + «/) ==^°S^ + ^-2& + 3&-4l + «*''- CoE. — If x = l, this series becomes log(l + 2/)=f-|V|'-|* + etd., which is the same as Art. 61. EXPONENTIAL SERIES. 70. To Develop u' = «*+». Here f{x) = a'", /" (z) = «=« log^ a, f (x) = a" log a, f" (x) = a« logS a, etc. FAILURE OF TAYLOR'S THEOREM. 99 Hence, u' = aP'+y = a='(l+ loga-y + ^og^a^ + log^^]^ + etc.). Cob. — If x = 0, this series becomes ay = 1 + log a-y + log^a^^ + ^"^'"rts + ®*^"' which is the same as Art. 63. 71. Though Taylor's Theorem in general gives the cor- rect development of every function of the sum of two variables, yet it sometimes happens that, for particular values of one of the variables, the function or some of its derivatives become infl'hite ; for these particular values, the theorem fails to give a correct development. For example, take the function u' = Vet + x + y. Here, f{x) = Va + x, /"(^) = 2'\/a + X 1_ 4:{a + x)^' f"{x) = ?— ,-, etc. Substituting in (7) of Art. 66, we have u' — Va + X + y ^V'^in^ + ^_. ^ + _J^_etc. 2Va + x 8 {a + xY 16 («+a;)^ Now when a; has the particular value — a, this equation becomes ^' = '^y = 0-)-« — 00 + 00 — etc. ; 100 EXAMPLES. that is, when x=. — a, ^/y = oo . But y is independent of z, and may have any value whatever, irrespective of the value of X, and hence the conclusion that when x= — a, \/y = 00 , cannot be true. For every other value of z, however, all the terms in the series will be finite, and the development true. Similarly, u' = a + Va — x + y gives, when substituted in Taylor's Theorem, u' ^ a + \/a — X -\- y 3-v/a — X which, when x=.a, becomes vl ■=■ a ■\- '\/y = a — 00 4- etc. ; and hence the development fails for the particular value, a; = a. It will be seen that when Taylor's Theorem fails to give the true development of a function, the failure is -only for particular values of the variable, all other values of both variables giving a true development; butiwhen.Maclaurin's Theorem fails to develop a function for one value of the variable, it fails for every other value. Many other formulae, still more comprehensive than these, have been derived, for the development of functions; but a discussion of them would be out of place in this work. EXAMPLES. 1. Develop y = Vl + xK \ Put a?=zz, and develop ; then replace 2 by its value. Ans. y = Vl + a;* . , a? a^ sfi ba? EXAMPLES. 101 ^- y = T=r,- y = z = 1 + x-J-3fi + 3? + os* + etc. ^ 1 — X y = (a + x)-\ y = {a + x)-^ = a-s _ 3^-4^; + ga-^a^ _ I0a-«a;» f etc. y = e"°'. a;' a^ a;^ a:* y = e"° ' = 1 4- a; + 3 2-4 3-5 2. 4-5. 6 + etc. 5. 2^ = ^^' yz=xe''-=x + x^ + ^ + 2^3 + ^^' 6. y = '\/2x — 1. «^ = VSa; — 1 = V— 1 (l— a; — ^— -— etc.). 7. t/ = {a!> + x>)l y ■= {a? ■\- x^Y = a^ + ^a^x' + |a~5ra^ — ^ja~ia;« + etc. ^ V^a^ + a;* _ 1 _ 1 ^ ba? _ S-Oa;*" ^-^"+^^ -a 4fl5 + 4.8a9 4.8.12a« 5.9-13a;i6 + 4.8.12.16^7 ~^^' 9. y = (a^ + a*a; — a?)^. Put a% — a^ = 2, as in Ex. 1. X 4 a;2 4.9 0(? 5 S^a 1-2 "^5%2 1.2-3 4.9.14 a^ ^*«» 1.2.3.4 + etc. 103 EXAMPLES. 10. u = {x + y)K u = x^ + ixr^y — ix-^y^ + ^x-^y^ — etc. 11. u = cos {x + y). (See Art. 68.) u = cos (a; + ^) = cos a; cosy — sin* sin^. 12. y = tan x. a? 2a^ 2/ = tana; = a;+g+ — + etc. 13. y = sec a;, ^ = sec a; 14. y = log (1 + sin x). y = log (1 + sin a;) = a; — |- + - — — + etc. , , a?! 5x* 61a* 2, = seca. = l + -+^ + — + etc. CHAPTER VI. EVALUATION OF INDETERMINATE FORMS. 72. Indeterminate Forms. — When an algebraic ex- pression is in the form of a fraction, each of whose terms is variable, it sometimes happens that, for a particular value of the independent variable, the expression becomes inde- terminate; thus, if a certain value a when substituted for X makes both terms of the fraction 4-?-^ vanish, then it reduces to the form - , and its value is said to be indetermi- nate. Similarly, the fraction becomes indeterminate if its terms both become infinite for a particular value of x; also the forms 00 X and oo — oo , as well as certain others whose logarithms assume the form oo x 0, are indeterminate forms. It is the object of this chapter to show how the true value of such expressions is to be found. By its true value is meant the limiting value which the fraction assumes when X differs by an infinitesimal from the particular value which makes the expression indeterminate. It is evident (Arts. 9, 43) that though the terms of the fraction may be infinitesi- mal, the ratio of the terms may have any value whatever. In many cases, the true values of indeterminate forais can be best found by ordinary algebraic and trigonometric processes. For example, suppose we have to evaluate -^ — - when a; = 1. This fraction assumes the form - when x =: 1; but if we divide the numerator and denominator by a' — 1 104 EXAMPLES. a? -\- X 4- 1 before making x = 1, the fraction becomes — - — ; and now if we make x = l, the fraction becomes 1+1+1 _ 3 1 + 1 ~ 2' which is its true value when a; = 1. 73. Hence the first step towards the evaluation of such expressions is to detect, if possible, the factors common to both terms of the fraction, and to divide them out ; and then to evaluate the resulting fraction by giving to the variable the assigned value. EXAMPLES. 1. Evaluate ^-^J^— ^ , when x = l. This fraction may be written {x — l){x' + x + l) a? + x + l „ , , I iwT-^ 7-^ = "1 TT = 3, when X = 1. {x — l){x^—x + l) x^ — X + 1 X 2. The fraction . = -, when .r = 0. V fl + X— V a — X " To find its true value, multiply both terms of the fraction by the complementary surd, Va + x + -v/a — x, and it becomes X iVa + X + Va — x) Va + x + \/a — x -^ or ~ ; and now making a; = 0, the fraction becomes Va, which is its true value when a; = 0. 3. — when x = a. Ans. i. X — V2a? — V? ' MEftBOD OF EVALVA^tOH: 105 4. 5 -, when a; = 0. Ans. -r—- 5. :,- , when x = l. Ans. 5. X — 1 6. -v/a;* + ax — X, when a; = oo Ans. ^• There are many indeterminate forms in which it is either impossible to detect the factor common to both terms, or else the process is very laborious, and hence the necessity of some general method for evaluating indeterminate forms. Such a method is furnished us by the Differential Calculus, which we now proceed to explain. METHOD OF THE DIFFERENTIAL CAL- CULUS. 74. To evaluate Functions of the form ^■ Let f(x) and (/> (x) be two functions of x such that f(x) = and ^ (x) = 0, when a; = a. Then we shall have ^^-r = 7^- (t>{a) Let X take an increment li, becoming x -\- h; then the fraction becomes fix + A) {x + h)' Now develop f{x + h) and 0(a: + A) by Taylor's Theo- rem ; substituting h for y in (7) of Art. 66, we have /(. + A) ^ /(^)+.r(^)l+/"(^)2-+etc. _ •^ ("^ + '*) (^) + 0' (a;) ^ + r {X) ^ + etc. ' 106 METSOb OP BVALVATtOK. or wiien x ■= a, ■^ ("" + ^) (a) H- {x)' (1.) If /' (a) = and {a) (3.) If /' (a) = 0, and 0' (a) = 0, the new fraction ■ {, W is still of the indeterminate form -it- Dropping in this case the first tioo terms of the numerator and denomi- nator of (1), dividing both by ^, and making h = 0, we have. EXAMPLES. 107 /(«) _/"(«) 0(a) -f'(«)' fix) as the true value of the fraction -rr-x , when x ■= a. {x) . O' — i^ , 4. , when X = 0. — 1 = — 1. 5. X Ans. log •=-• (a; — a)' :, when a; = a. Here /'(^) ./.'(a;) according as s > or < 1 X — sin X we™ "1 = ~7 vTT = 00 or 0, s{x — a)"-' Jo ' 6. x^ , when a; = e»= — e^ — 2a; , 7. V , when a; = X — sm X ga> g-x, 8. — : , when a; = sm X e* — 3 sin a; — e~* X — sm X derivative. Ans. \. Ans. 2. Ans, 2. , when X = 0. Take the third. Ans. 4. 10. ~, when a; = a. Cancel the factor (a — x)i (a — a;)« ^ ^ 75. To evaluate Functions of the form §-• Let ^-^ = — , when x = a. '{x) ^{x)- _i_- f'{x) -fix) [) />) . ^-/'(a)' .A (a)' whence ^^ -(^- Hence the true value, of the indeterminate form — is 00 found in the same manner as that of the form -• In the above demonetration, in dividing the equation bv ^^, when X = ayyre assumed that —.--{ is neither nor oo , so that the proof 0(a) would fail in either of these cases. 110 EXAMPLES. It may, however, be completed as follows : Suppose the true value of 4--! to be ; then the value of £M±.M^ is n, where h may be any constant. But" as this latter fraction has a value which is neither nor «, its value by the above method is —,—, {a) or , I ■! + ^> ^i"i since the value of this fraction is li, the first term ^ =. : i. .., where 4^ = 0, £^ is also 0. ¥{a) {.a) ' f(asj fix) Similarly, if the true value of ^4 he oo when x = a, then 0(0!) —-, = ; and therefore we have .}, . = 0, by what has just been shown ; .•. . , / = (» . (a) Therefore, in every case the value of '^-r determines the value of 9 (a) ^~ for either of the indeterminate forms ;r or — . (See Williamson's (a) 00 Dif. Cal., p. 100.) EXAMPLES. 1. Evaluate , when a; = oo . Here (j) {x) (pi {x) nx""^ Ma?'Joo loff X 2. ETaluate — ^ — , when a; = 0. cot X Here 1 /' (a;) _ X _ _ sin^ x ~\ _ , 0' (x) ~ — cosec'a; x Jo~ ' /" (p") _ 3 sin a; cos x~ ^^ ~ 1 _ -^f-^ = 0, when a; = Q, cot a; = 0; EVALUATION. Ill 1 — log a; , „ T" » when a; = 0. Ans. 0. 7T , when a; = 0. to' 8 cot-^ log tan (%x) 1 „ . , 5. ° , — ^^ — -, when a; = 0. Jws. 1. log tan X 76. To evaluate Functions of the form x oo . Let f(x) X (a;) = X 00, when a; = a. The function in this case is easily reducible to the form - ; for if f(a) = 0, and {x) n ,nx TT 1 7T ^ ' — jr COSeC* -jr- 2. Evaluate ar* log a;, when x = 0. 1 , log a; X a;""! »'' log a; = — §— = — T = =0. f^ 2^ Jo —TO"'"-' nJo 113 EXAMPLES. 3. e"* log X, when a; = qo . Ans. 0. 4. sec a; (a; sin x — ^), when ck = ^- Ans. — 1. 77. To evaluate Functions of the form oo — oo . Let f{x) and

"{x) 2 _o TT 3. Evaluate sec a; — tan x, when a; = -• 1 — sin a; , tt sec X — tan x = • = - , when a; = -• cos a; 2 f (ps) __ — cos x~ fp' (x) — sin a; = 0. n 77 Hence, sec ^ and tan ^ are either absolutely equal, or differ by a quantity which must be neglected in their alge- braic sum.* 4. ^ when a; 1 a; — 1 log X 5. 1 X ^ T , when a; = 1. log X log X Ans. — 1. 78. To evaluate Functions of the forms 0°, co", and 1*°°. Let f{x) and (x) be two functions of x which, when x=: a, assume such values that [/(a;)]*^*^ is one of the above forms. Let 2/ = [/(a')]*^"^ ••• log y = (x) log/(a;). (1.) When f{x) = oo or 0, and ' (x) 1 x_ = 0; 00 * In general, the value of the indeterminate form 0° is 1. (See Note on Inde. terminate Exponential Forms, by F. Franklin, in Vol. I, No. 4, of American Journal of UathematicE.) EXAMPLES. 115 .'. log af =: 0, when a; = oo ; hence, af = 1, when aj = oo. 3. Evaluate (l + -| , when x = cc. i^^tf- Let a; = - , and denote the function by u. Then u = {1 + azy]o (since when a; = oo , « = 0) ; and , log (1 + az) , . log M = —^-^ , when z = 0. Taking dematives, we have a log Woe = a; 1 + a2jo 1 + - j = a, when a; = oo ; 1 + - 1 = e«, when a; = oo . If a = 1, we have that is, as x increases indefinitely, the limiting value (Art. 41) of the function (l + -I is the Naperian base. -I , when x = 0. Ans. 1. 5. af'°% when a; = 0. Ans. 1. 3 1 ", when x = a. Ans. e\ 116 COMPOUND INDETERMINATE FORMS. 79. Compound Indeterminate Forms. — If an inde- terminate form be the product of two or more expressions, each of which becomes indeterminate for the same value of X, its true value can be found by evaluating each factor separately ; also, when the value of any indeterminate form is known, that of any power of it can be determined. EXAMPLES. 1. Evaluate — -, when a; = oo. This fraction may be written ©" We first evaluate — , when x=.ck>. = 1=0. CO Here -Jj^ = ^ ^' {x) 1 ^ n Hence, -, = 0" = 0. giC 3. Evaluate a;" log" x, when a; = 0, and m and n are positive. Here (a;" log xY = f^^X- \x~n / We first evaluate — ^, when x = 0. We have X n 1 i>' (x) m -5^1 X n »» -1 = a;" =0. m J„ .*. or- log" a; = 0" = 0. EXAMPLES. 117 3. a;" ■m-l-n , a-m-l - , when a; = 1. 1— a?? This function can be written in the form X^ 1 _ ^n 1 + a-'i- 1 a:* We have to CTahiate only the latter function for a; = 1, since the former is determinate. Here ^' (x) —px''-'^ nx"~^ n ^_„ P x^ P _ 1 Y+xP ~ % ' £v.n ___ rvjm.-\-n m x-p 2p' when a; = 1. when a; = 1. when a; = 1. {x^ — a^) sm — 4. , when x = a. a? cos TTX 2a {a? — a^) sin ■nx 2a . TTX x^-a^ ™2^ TTX OP cos s- 2a cos Tra: 2a a;2 We have only to evaluate the first factor, and x^ — a2- a = 2a; 4fl2 rra; 2a _ TT . TTX -2a''''Ya_ TT . TTX~ Sin — 2a 1 118 > j — x^ y/W — y^ 6. « = «/"»'. Jw = y-'»Mog2/.cosxrfa; + ?i^if. TOTAL DBRlVATtVE. 1^5 , « 7 ydx — xdy 7. u = Yers~i— du = ■ , — -^ - y «/A/2a;2/ — x^ 8. M = log a^y. du z=-dx-\- log a; {x), we have dy = ~-dx; dz since z = 0i (a;)j we have dz ■= -^ dx. Substituting these values for dy and (?« in (1), we get ._^dydudz^ ~ dii fir ^ dz dr ^ ' Dividing by dx, and denoting the total derivative by ( ), we have (du\ _ dudy du dz dx) ~ dy dx dz dx ^ Cob. 1. — If z = x, the proposition becomes u =.f{x, y) dz and y =z{x); and since -j- = 1, (3) becomes (du\ _ du du dy dx) ~ dx dy dx CoK. 3. — If u =-f(x, y, z), and y = (x), and z = (l)i {x), we have , du T du , du ^ ,^. 1^6 ilXPLANATtOlf Off TSMMS. dv ■=. -^ dx, and dz ■= ^r dx. '^ dx dx Substituting the values of dy and dz in (1), and dividing by dx, we get idu\ _ du du dy du dz \dx} ~ dx dy dx dz dx CoE. 3.— If u =/(«/, z, v), and y = 2 (x), we have, , du ^ du -, . dv.- , ,,. ^'' = Ty^y-^Tz^^-^dv^'"- (1) , du -, , dz ; , dv ^ dii ::= -f- dx; dz ^= ^rdx dv = ^- dx. " dx dx dx Substituting the values of dy, dz, dv, in (1), and dividing by dx, we get ldu\ __ du dy du dz du dv \dxi ~ dy dx dz dx dv dx CoE. 4. — If u =f{y) and y =: (x), to find -j-- Since u=f{y), we have du : du , ^Ty'y- Since y = 1> (a;), we have dy : = fdx. dx erefore , dudu ^ , du =z -^^ dx, dydx and du dx dudy ~ dydx ScH. — The student must observe carefully the meanings of the terms in this Art. Thus, in the Proposition, u is indirectly a function of x through y and z. In Cor. 1, u is directly a function of x and indirectly a function of x tlu-ough y. In Cor. 2, u is directly a function of x and MXAMPLES. l^l* indirectly a function of x through y and z. In Cor. 3, u is indirectly a function of x through y, z, and v. In Cor. 4, M is indirectly a function of x through y. The equations in this Article may seem to be inconsistent with the principles of Algebra, and even absurd ; but a little reflection will remove the difficulty. The du'B must be carefully distinguished from each other. In Cor. 1, for example, the du in -r- is that part of the change in u which results directly from a change in x, while y remains constant ; and the du in j- is that part of the change in u which results indirectly from a change in x through y ; and -^1 is the entire change in u which results directly from a change in x, and indirectly from a change in X through y. 1. M = tan""i y EXAMPLES X and y = {r^ — x^)i, to find (^j- Here ^ = t du ^ _x_ ^^^ dy ^ _ a;. dx r'' dy r^' dx y have Substituting in (|) = g + J | (Art. 82, Cor. 1), we \dxl ~ r^^\ rV\ yl _ y^ + x^ _ 1 ~ r^y ~ V^?iZ^3' and this value is of course the same that we would obtain if we substituted in m = tan-i - for y its value in terms of X, and then differentiated with respect to x. 138 EXAMPtSS. 2. u — tan~i {xy) and y = ^, to find (-^)- Here -r- = ■ y du _ X dy _ dx~l + x^y^' dy~l + ^f dx ' .: (Art. 82, Cor. 1), ldu\ _ y + e^x _^ (1+a;) . \dx) ~ 1 + xY ~ 1 +a^' and this value is of course the same that we would obtain if we differentiated tan~i {xe"') with respect to x. 3. M = z^-\-y^-\-zy and 2 = sin a;, y =^ ^, to find (-t-)* Here ^ = 3^^ + ., ^ = ^^ + 2^» dz dy T- = cos a:, ~ ^= &^. dx dx .: (Art. 82), C^) = (^^' + «) e* + (22 + y) cos a; = (3e^ + sin x) (^ + (3 sin x + e»') cos x = Z^" + e* (gin a; _|_ cos x) + sin 2a;. (See Todhunter's Dif. CaL, p. 150.) Let the student confirm this result by substituting in u, for y and z, their values in terms of x, thus obtaining u =^ ^ -\- e'^ sm X ■\- sin^ x, and then differentiate with respect to x. 4. u — sm-i (y — 2), «/ = 3a;, 2 — da^. y) —f{^, y)- This expression being regarded as a function of y, let y become y + dy, a; remaining constant ; then ThX'^^^l'^y =/(^+^^' y+dy) -f(x, y + dy) ^ -[fi=c+dx,y)-fix,y)] — f{x+dx, y+dy) —f{x, y+dy) -f{x + dx,y)+f(x,y). In hke manner, ^dy =f{x, y+dy) —f{x, y). ■^\-2,^y)^^=f{^+'^^> y+^y) -f{x+dx, y) ^ -U{^,y+dy)-f{x,y)] = f{x + dx, y + dy) —f{x + dx, y) —f{^,y + dy)+f{x,y). These two results being identical, we have d (du , \ , d Idu -, \ -. TyW'')'^y = Tx\Ty^yr''' ,, ,. d^u , , d^u , , that IS, -^ — =- dit dx = -^r—^ dx dy. dy dx ^ dxdy " Dividing by dy dx, we get d^u d^u dy dx ~ dxdy In the same manner, it may be shown that , „ , dx^ dti = -^ — ^ -„ dy dx\ dx^dy •' dy da? ^ ' dH d^u or :!= da? dy dy d^' and so on to any extent. E-XAMPLES. 133 EXAMPLES. 1. Given u = sin {x + y), to find the successive partial derivatives with respect to x. ^ = cos {x + y), -^3 = - sin (^ + y)' ^ = - cos (x + y), -^ = sin {x + y), etc. 3. t« = log {x + «/), to find the successive partial de- rivatives with respect to x, and also with respect to y in the common system. du _ m d^u _ m d^u _ 3mi dx X + y' dx^ ~" {x + yY' da? ~ {x + yY' du _ m d^u _ m d^u _ 2m dy ~ X + y' dy^ ~ {x + yf dy^ ~~ {x ■{■ yY (See Art. 65, Lemma.) 3. If M = a; log y, verify that 4. If M = tan~i (-), verify that 5. If M = sin (aa" + iy°), verify that dH dydx d?u ~ dxdy dhi dyHx d^u ~ dxdy^' dHi d^u dxHy^ dy^d3? 85. Successive Differentials of a Function of Two Independent Variables. Let u =f (x, y). "We have already found the first differential (Art. 81), , du ^ du , ,, . 134 SUCCESSIVE DIFFERENTIALS. Differentiating this equation, and observing that -r, -j-, are, in general, functions of both x and y (Art. 83), and remembering that x and y are independent, and hence that dx and dy are constant, we have, d(^r dx] d[^r dx] d(-^ dy] \dx I -, \ax I , , \dy ^ I , d?u — f ■ dx H J dy -\ f dx dx dy "^ dx „ d^u ' , dhi , , , d^u -, J d^u J „ dH = -^,dx^ + a^Jyd^ ^my^'^^y + d^^^y (since -5-^ dydx = , , dxdy, Art. 84). DifEerentiating (3), remembering that each term is a function of x and y, and hence that the total differential of each term is equal to the sum of its partial differentials, we get, ^^ = £^ ^^ + ^ iky '^''y + ' mf '^'f+pf'(') and so on. It will be observed that the coefficients and exponents in the different terms of these differentials are the same as those in the corresponding powers of a bino- mial ; and hence any required differential may be written out. • The total differential of eacli of the terms (-p dx) and (— dy\ is equal to th« Enn of its partial differentials, IMPLICIT FUNCTIONS 135 EXAMPLES. 1. « = {x^ + f)k du X du _ dy ~ y dx ■ ~ (^ + /)*' (x? + f)i dH _ y' . d^u dxdy ~ — xy dx> ■ (a^ + 2/2)1' {x> + f)^ cPu a;3 d^u dx^ ~ -Zxf df ~(a;2 + y^)«' {^ + f)^ dhi dxHy dhi dxdy^ ~ x{%f-xP) {7? + f)^ d^u _ -dyx^ dH = [— ^xyHoi? + 3«/ {%x^ — y^) dxHy + Zx (2/ — a;2) dxdy^ — Sj/x^^z/S] (a;2 + 2/2)4 3. M = e^M+sy). (Z^zf = [aHx^ + %abdxdy + S^^^j e»'+6» 86. Implicit Functions (see Art. 6). — Thus far in this Chapter, the methods which we have given, although often convenient, are not absolutely necessary, as in every case by making the proper substitutions we may obtain an explicit function of x, and differentiate it by the rules in Chapter 11. But the case of implicit functions which we are now to consider is one in which a new method is often indis- pensable. Let / [x, y) = be an implicit function of two varia- bles, in which Jt is required to find -p- If this equation 136 IMPLICIT FUNCTIONS. can be solved with respect to y, giving for example y = 4> (x), then the derivative of y with respect to x can be found by previous rules. But as it is often difficult and sometimes impossible to solve the given equation, it is necessary to iitvestigate a rule for finding -^ without solving the equation. 87. DifTerentiation of an Implicit Function. Let f(x, y) = 0, in which y is an implicit function of x, to find -p- Let f{x, y) — u. Then u ^^f{x,y) = 0. Hence by (Art. 82, Cor. 1), we have, ldu\ _ du du dy \dxl ~ dx dy dx But u is always — 0, and therefore its total differential = ; hence {-^\ = 0, and therefore, du du dy dx dy dx ' from which we get, du dy _ dx^ dx~ du' dy ScH.— It will be observed that while \-f\ = 0, neither du du . . , ^ -r, ^ nor -^ IS in general = 0. For example, x^ + y^ — r^ = is of the form f(x, y) = 0. "We see that if x changes while y remains constant, the function changes, and hence is no EXAMPLES. 137 longer = 0. Also, if y changes while x remains constant, the function does not remain = 0. But if when x changes y takes a corresponding change by virtue of its dependence on X, the function remains = 0. EXAMPLES. 1. y^ — %xy + fflS = 0, to find J. dx du „ 1 = .,,.„ du therefore, -r- = r- dx du _ -%y _ y '"■ly — %x y —X dy 3. ay + ¥x^ — aW = = 0, to find ^• dx f = Wx; dx ' % = ^-' du therefore, ^ = t— dx du - Wx ¥x dy J. (1) Since .y^- Va' — ^, from the given equation, we may solve this example directly by previous methods, and obtain dy hx dx a^a^ — x^' (2) which agrees with (1) by substituting in it the value of y in terms of x. In this example we can verify our new rule by comparing the result with that obtained by previous rules. In more complex examples, such as the following one, we can find -~ only by the new method. 138 SECOND DERIVATIVE OF AN IMPLICIT FUNCTION. 3. a? — ax^y + IxY — y^ = Q, to find -^^ -^ =z boi^ — da3?y + Zlxy'^; p = —a3^ + 2bx^y — 5y*; ,, „ dy 5x^ — 3ax^y + %bxifi therefore, -f- — -^—. — „, ^ 1- " " ax y — ax 6. y3_3y + ^ = o. 1 = 30^^- ~- 7. .3 + 3a., + ,3 = o. |=-^t2- 88. To Find the Second Derivative of an Im- plicit Function. Let u = f{x, y) = 0. du dy du dudy . ,„, -dx-^Tytx = ^' (^) it is required to find -r^- Differentiating (2), remembering that -^ , -^ , are func- tions of X and ?/, we get ^ £?% where the values of -y-_, ~, -j-, -^, can be found from (1). Whenever equations (1) can be solved for r and 6 sepa- rately, we can find by direct differentiation the values of dr dr dd dd -, ^ , ,,.,,■ -p, -J-, -1-, -T-, and hence by substitutmg m du dtidr du dd dx ~ dr dx dd dx ' , du du dr du dd , . , „„. ^^* Ty-drcTy + dery^^^-^^^' we can obtain the values of t- and t— dx dy When this process is not practicable, we can obtain their values by solving (2) and (3) directly, as follows : Multiply (2) by ^ and (3) by -^ and subtract ; then multiply (2) by ~ and- (3) by -^ and subtract. We shall tJi?n hS'Ye two equations, from which we obtain, MSAMPLlllS. li^ du dy du dy du _ dr do dO dr dx dx dy dy dx ' 'WrM~dr dB du dx du dx (*) , du _dd dr dr dd . , dy ~ dx dy dy dx ^ ' dr dO dr d.d The values of y-^, -i— j, etc., can be obtained from these, but the general formulae are too complicated to be of much practical use. (See Gregory's Examples, p. 35.) Cor. — If a; = »• cos 9 and y =^ r sm B, (4) and (5) become du „ du sin 6 du du cos 6 du . . du dx dr r da ' dy r dd dr EXAMPLES. 1. M = ?Jll, to find du (Art 81). _ a {xdy - ydx) 2. M = sin ax + sm by + tan~i -• du =: a cos axdx + 5 cos 5wi?v H ^ — ^ • o • -1 ^ 3. M = sm 1 -• y , _ ^& — xdy y 's/y^ — ^ 4. « = sin (a; + y). du = cos (a; + y) (dx + (?y). 5. __ jgy z^ — a^ _ X if —c?) (^ydx + xdy) — 'Hx^yzd z ~ (z^ — aFf vr _— V^ — yy y Vx^ — y* 7. u = cot a^ to find (^ (Art. 82, Cor. 1). (-=- ) = — 3? cosec^ a;?' ( - + log ;;; -^Y 8. M = sin (?/^ — z), and ?/ = log x, z=x% to find (-^') (Art. 83). ^"'^' ldu\ _ 2 (y — x^) cos (y^ — a ) 9. M = log tan -• y-'^tx if' sin - cos - ^ y y 10. u = log (x — a -\- '^^ — 2aa;). ld' increases till the latter reaches 90°, after which cosec (p increases as increases ; that is, cosec is a minimum when is 90°, since it is less than the immediately preceding and succeeding values. 93. Condition for a Maximum or Minimum. — If y be any function of x, and y be increasing as x increases, the differential of the function is positive (Art. 13), and hence the first derivative -~- will be fositive. If the func- tion be decreasing as x increases, the differential of the 153 GEOMETRIC ILLUStttAMON. function is negative, and hence the first derivative -^ ^^ be negative. Therefore, since at a maximum value the function changes from increasing to decreasing, the first derivative must change its sign from plus to minus ; as the variable increases. And since, at a minimum value, the functiou changes from decreasing to increasing, the first derivative must change its sign from minus to plus. But as a function which is continuous* can change its sign only by passing through or oo , it follows that the only values of the variable corresponding to a maximum or a minim/um, value of the function, are those which make the first derivative O or oc . 94. Gi-eometric Illustra- tion. — This result is also evident from geometric con- siderations ; for, let y = f(x) be the equation of the curve AB. At the points P, P', P", P", the tangents to the curve are parallel to the axis of x, and therefore at each of these points the first derivative /' {x) = 0, by Art. 56^ We see that as x is increasing and y approaching a maximum value, as PM, the tangent to the curve makes an acute angle with the axis of x ; hence, approaching P At P the tangent becomes parallel M M' M" M^'M" Fig. 9. from the left -,- is '+ dx dy to the axis of x ; hence, -.- = 0. dx Immediately after pass- ing P the tangent makes an obtuse angle with the axifc of x ; hence, -j- is — . * In this discasBioD the f anction ie to he regarded as conHntiout. CklTMRioN OF Maxima and Minima. 153 Also in approaching a minimum value, as P'M', from the left, we see that the tangent makes an obtuse angle with the axis of x, and hence -i- is — . At the point P, -f- = 0. CbX dx After passing P', the angle is acute and -=,- is +. C130 In passing P'", -^ changes sign by passing through qo , P"'M"' is a minimum ordinate. In approaching it from the left the tangent makes an obtuse angle with the axis of x, and hence -^ is — . At P'" the tangent is perpendicular to the axis of x, and -^ = qo . After passing P"'M"', the angle is acute and -7^ is +. While the first derivatiTe can change its sign from + to — or from — to + only by passing through or 00 , it does not follow that because it is or 00, it therefore necessarily changes its sign. The first derivative as the variable increases may be +, then 0, and then +, or it may be — , then 0, and then — . This is evident from Pig, 9, where, at the point D, the tangent is parallel to the axis of X, and ~- is 0, although just before and just after it is — . Hence the values of the variable which make -;- = or 00 , ax are simply critical* values, i. e., values to be examined. As a maximum value is merely a value greater than that which immediately precedes and follows it, a function may have several maximum values, and for a like reason it may have several minimium values. Also, a maximum value may be equal to or even less than a minimum value of the same function. For example, in Pig. 9, the minimum P'M' is greater than the maximum P'M''. * See Price's Cal., Vol. I, p. Wi, 154 CONDtTtON GlVElf Bt VAtLOk'S tsMRBM. 95. Method of Discriminating between Maxima and Minima. — Since the first deriTative at a maximum. state is 0, and at the immediately succeeding state is — , it follows that the second derivative, which is the difference between two consecutive first derivatives,* is — at a maxi- mum. Also, since the first derivative at a minimum state is 0, and at -the immediately succeeding state is +, it fol- lows that the second derivative is + at a minimum. There- fore, for critical values of the variable, a function is at a maximum or a minimum state according as its second derivative at that state is — or +. 96. Condition for a Maximum or Minimum given by Taylor's Theorem. — Let m =f{x) be any continuous function of one variable ; and let a be a value of x corre- sponding to a maximum or a minimum value of f{x). Then if a takes a small increment and a small decrement, each equal to h, in the case of a maximum we must have, for small values of h, /(a) > /(« + h) and /(a) > f(a ^ h) ; and for a minimum, /(«) < / (« + h) and /(«) < f{a - Ji). Therefore, in either case, f[a + h)-f{a) and f{a-h)-f{a) have both the same sign. By Taylor's Theorem, Art. 66, Eq. 7, and transposing, we have /(« + A) -/(«)= /'(«) /^ +/"(«) I' + etc.; (1) f{a - h) -/(«) = -/' (a) h +/" (a) J - etc. (2) * Remembering that the flrst value is always to be subtracted from the second. FINDING MAXIMA AND MINIMA VALUES. 155 Now if h be taken infinitely small, the first term in the second member of each of the equations (1) and (2) will be greater than the sum of all the rest, and the sign of the second member of each will be the same as that of its first term, and hence /(a + A) —/(a) and /{a — Ti) — f{a) cannot have the same sign unless the first term of (1) and (3) disappears, which, since h is not 0, requires that f{a) = Q. Hence, the values of x ivhich make f{oc) a maxi- mum or a minimum are in general roots of the equa- tion, f (pc) = 0. Also, when /' {a) = 0, the second members of (1) and (2), for small values of h, have the same sign as /" {a) ; that is, the first members of (1) and (2) are both positive when /" {a) is positive, and negative when /" (a) is nega- tive. Therefore, /(a) is a maximum or a m^inijnum/ according as f" (a) is negative or positive. If, however, /" (a) vanish along with /' (a), the signs of the second members of (1) and (3) will be the same as /'" (a), and since /'" (a) has opposite signs, it follows that in this case /"(«) is neither a maximum nor a mini- mum unless f" {a) also vanish. But if /'" (a) = 0, then f{a) is a maximum when /"' {a) is negative, and a minimum when /" {a) is positive, and so on. If the first derivative which does not vanish is of an odd order, f{a) is neither a maxijjnum nor a minimum ; if of an even order, f(a) is. a maximum or a minimum, according as the sign of the derivative which does not vanish is negative or posi- tive. 97. Method of Finding Maxima and Minima Values. — Hence, as the result of the preceding investiga- tion we have the following rule for finding the maximum or minimum values of a given function, f{x). Find its first derivative, f (a?) put it equal to 0, and solve the equation thus formed, f {x) = 0. Sub- 156 MAXIMA AND MINIMA VALUES ALTERNATE. stitute the values of x thus found for x in the second derivative, f" (x). Each value of x which m.akes the second derivative negative will, ivhen substituted in the function f {x) make it a maximum; and each value which niahes the second derivative positive will make the function a minimum. If either value of X reduces the second derivative to 0, substitute in the third, fourth, etc., until a derivaMve is found which does not reduce to 0. If this he of an odd order, the value of X will not make the function a maximum or minimum ; but if it be of an even order and nega- tive, the function will be a maximum ; if -positive, a Tninimum. Second Rule. — It is sometimes more convenient to ascertain whether a root a of /" (o:) = corresponds to a maximum or a minimum value of the function by substi- tuting for X, in /' (x), a — h and a + h, where h is infini- tesimal. // the first result is + and the second is — , a, corresponds to a maximum ; if the first result is — and the second is + , it corresponds to a minimum. If both results have the same sign, it corresponds to neither a maximum nor a minim^um. (See Arts. 93, 94) 98. Maxima and Minima Values occur alternately. — Suppose that f{x) is a maximum when x ^ a, and also when a; = &, where b y a; then, in passing from a to b, when .r = a + A (where h is very small), the function is decreasing, and when a;= 5 — h, it is increasing; but in passing from a decreasing to an increasing state, it must pass through a minimum value ; hence, between two maxi- ma one minimum at least must exist. In the same way, it may be shown that between two minima one maximum must exist. This is also evident from geometric considerations, for in Fig. 9 we see that tliere is a maximum value at P, a mini- mum at P', a maximum at P", a minimum at P'", and so on. A.PPLICATI0N8 OF AXIOMATIC PBINCIPLES. 157 99. The Investigation of Masima and Minima is often facilitated by the following Asdomatic Prin- ciples : 1. If M be a maximum or minimum for any value of x, and « be a positive constant, au is also a maximum or mini- mum for the same value of x. Hence, before applying the rule, a constant factor o?" divisor may he omitted. 3. If any value of x makes u a maximum or minimum, it will make any positive power of u a maximum or mini- mum, unless %i be negative, in which case an even power of a minimum is a maximum, and an even power of a maxi- mum is a minimum. Hence, the function may he raised to any power ; or, if under a radical, the radical may he om,itted. 3. Whenever m is a maximum or a minimum, log m is a maximum or minimum for the same value of x. Hence, to examine the logarithm of a function ive have only to examine the function itself. When the function con- sists of products or quotients of roots and powers, its exam- ination is often facilitated by passing to logarithms, as the differentiation is made easier. 4. When a function is a maximum or a minimum, its reciprocal is at the same time a minimum or a maximum j this principle is of frequent use in maxima and minima. 5. If M is a maximum or minimum, t( ± c is a maximum or minimum. Hence, a constant connected hy + or — may he om,itted. Other transformations are sometimes useful, but as they depend upon particular forms which but rarely occur, they may be left to the ingenuity of the student who wishes to simphfy the solution of the proposed problem. It is not admissible to assume a; = oo in searching for maxima and minima, for in that case x cannot have a suc- ceeding value. 158 EXAMPLES. EXAMPLES. 1. Find the values of. x -which will make the function uz=Qx -\- Zx^ — 4a;* a maximum or minimum, and the cor- responding values of the function u. Here ^' ^- 6 + 6a; - l%xK dx Now whatever values of x make u a maximum or mini- du mum, will make -i- = (Art. 97) ; therefore, 6 + 6a; — l%x^ = 0, or x^ — \x = \; .-. a; = i ± f = + 1 or - i. Hence, if u have maximum or minimum values, they must occur when a; = 1 or — ^. To ascertain whether these values are maxima or minima, we form the second derivative of u; thus, ^ - 6 - 24a; da? - ^ ^^■ When x = l, y-^ = — 18, which corresponds to a maxi- mum value of u. When a; = — ^, ;7-j = +18, which corresponds to a minimum value of u. Substituting these values of x in the given function, we have When x=-l, u =6 + 3 — 4 = 5, a maximum. When a;=— |, ?« = — 3 + f + |=— J, a minimum. 3. Find the maxima and minima vahies of u in u = a^—%x^ + 23a;' — 24a; + 12. ^ = 4a;s _ 34^2 + 44^; _ 24 = 0, dx EXAMPLES. 159 or, a? — 6x^ + 11a; — 6 = 0. By trial, a; = 1 is found to be a root of this equation ; therefore, by dividing the first member of this equation by X — 1, we find for tlie depressed equation, x^ — ox + 6 =0; .: a; = 2 or 3. Hence the critical values are a; = 1, a; = 2, and a; = 3. ^ = 12x» _ 48a; + 44 = + 8, when a; = 1. = — 4, when a; = 2. = + 8, when a; = 3. Therefore we have, when a; = 1, u = 3, a minimum ; when a; = 3, m = 4, a maximum ; when a; = 3, m = 3, a minimum. 3. Find the maxima and minima values of u in u — (x — ly {x + 2)». ^ = 4 (a; — l)3(a; + 2f + 3 {x + 2f(x — ly = {x-iy{x + if [4 {x% 2) + 3 \x — 1)], "'■ ^=(^-l)M^ + 3)^(7^ + 5) = 0; (1) .'. (a; — I) = 0, (a; + 2) = 0, (7a; + 5) = 0. .'. a; = 1, a; = — 2, and a; == — f, as the critical values of X. In this case, it will be easier to test the critical values by the second rule of Art. 97; that is, to see whether -j- changes sign or not in passing through a; = 1, — 2, and — 4" in succession. 160 EXAMPLES. If we substitute in the second member of (1), (1 — h) and (1 4 A) for x, where A is infinitesimal, we get ^!f = (l-A-l)i'(l-A + 2)«[7(l-A) + 5] = _ A3 (3 _ ]iY (13 _ 7/j) = _. and ^ = (1 + A - 1)3(1 + A + 3)2 [7 (1 + li) + 5] Therefore, as -j- changes sign from — to + at a; = 1, the function u at this point is a minimum. When a; = — 2, y- does not change sign ; /. u has no maximum or minimum at this point. When a; = — f, -^ changes sign from + to — ; .•. u, at this point, is a maximum. Hence, when rr = 1, m = 0, a minimum. 134.93 when x= — f , u = — ^;n— , a maximum. It is usually easy to see from inspection whether -^ changes sign in passing through a critical value of x, with- out actually making the substitution. 4. Examine u = b + {z — a)^ for maxima and minima. — = 3 (a; — a)2 = ; .•. x = a, and u = b. d u Since x = a makes -v-^ = 0, we must examine it by the dii second rule of Art. 97, and see whether t- changes sign at z = a. EXAMPLES. 161 -^ = 3 (a — A — a)^ =r 3^2 is the value of -r- immediately preceding a; = a. -^ = 3 (a + A — a)2 = SA^ is the value of -j- immediately succeeding x=za. Therefore, as -^ does not change sign at a; = a, m = 5 is neither a maximum nor a minimum. 5. Examine u = 1 + {x — a)^ for maxima and minima. T- = 4 (a; — a)^ = ; .•. x z= a and u = b. ax ^ ' d'U/ It is easy to see that -5- changes sign from — to + at x-=a', .•.»;■=« gives m = 5, a minimum, (x + ^Y 6. Examine m = 7 —~ for maxima and minima. {x - dy du (x + 2)2 (x — 13) .: X = — 2, 13, or 3. "We see that when a; = — 2, -j- does not change sign ; .•. no maximum or minimum ; when ar = 13, -^ changes sign from — to + ; .•. a minimum ; when a; = 3, -j- changes sign from + to — . .•. a maximum ; hence when a; = 13, u — J-p, a minimum value ; and when a; = 3, m = 00 , a maximum value. 163 EXAMPLES. 7. Examine m = 6 + (a; — «)^ for maxima and minima.- -^ = Ka; — fl)i = ; .'. x z= a and u = I. When x = a, -T- changes sign from — to +. X =. a gives u =■ i, a minimum. 8. Examine u=-'b — {a — x)^ for maxima and minima. -=- = !■(« — x)^ = : .•. a; = ffl and u = h. dx ^^ When a; = a, J- changes sign from + to — . .'. a; = a gives w = 5, a maximum. 9. Examine m = 5 + ^/a^ — 3a*a; + aa;^ for maxima and minima. If « is a maxima or minima, u — t will be so ; therefore we omit the constant 5 and the radical by Art. 99, and get ■u! = 0? — %a?x + ai& ; dv! -^-^ =z — 3a* + 2ax = 0; .•. a; = a and u = i. ax du' When a; = a, -5— changes sign from — to +. .*. x=:a gives u = b, a minimum. a^a; 10. Examine u = -. cm for maxima and minima. (a — x)^ Using the reciprocal, since it is more simple, and omitting the constant a* (Art. 99), we have u = ^ = 2a + x; X X ' du' a^ , ■, r. J ePiJ'' 2a2 ••• ^ = -^. + 1 = 0, and ^ = ^; EXAMPLES. 163 , cPu' 2 .'. X = ± a, and .-. -5--^ = ± — Hence, a; = + « makes u' a minimum, and a; = — a makes it a maximum ; therefore, since maxima and minima values of u' correspond respectively to the minima and maxima values of u (Art. 99, 4), we have, when X ^^ a, u = x, a maximum. " X = — a, u = —-, a minimum. 4 Find the values of x which give maximum and m/inimum values of the following functions : " '' 1. u = ^ — Zx^ — 2ix + 85. ■^-■ Ans. X = — 2, max, ; a; = 4, min. 2. u = 20? — 2\x> + 36a; — 20. X z= 1, max.; a; = 6, min. 3. M = a;s — 18a;2 + 96a; — 30. a; = 4, max. ; a; = 8, min. 4. u ■= —-• X = ±a, mm. a — 2x * K 1 + 3a; , , 5. M =: — — a; = — 1^, max. V 4 + 5x 6. u = 'X? — 3x» — 9a; + 5. a; = — 1, max. ; a; = 3, min. 7. u = a? — 3x> + 6x+'r. Neither a max. nor a min. 8. u = {x- 9)5 (x — Sy. X = S, max. ; x = 8^, mia 9. M = :; — ; 7 a; = COS a;, max. 1 + X tan a; 10. u ^= sin^ a; cos x. x = 60°, max. t-i sma; .„ 11. u = :z — — r X = 45 , max. 1 +-tan X 164 GEOMETRIC PROBLEMS. 13. u = sin a; + cos x. :c = 45°, max. ; x = 225°, min. 13. u = ^^- an* X = e", max. GEOMETRIC PROBLEMS. The only difficulty in the solution of problems in maxima and minima consists in obtaining a convenient algebraic expression for the function whose maximum or minimum value is required. No gen- eial rule can well be given by which this expression can be found. Much will depend upon the ingenuity of the student. A careful ex- amination of all the conditions of the problem, and tact in applying his knowledge of principles previously learned in Algebra, Geometry, and Trigonometry, with experience, will serve to guide him in form- ing the expression for the function. After reducing the expression to its simplest form by the axioms of Art. 99, he must proceed as in Art. 97. 1. Find the maximum cylinder which can be inscribed in a given right cone with a circular base. Suppose a cylinder^jiKScribed as in the figure. Let AO = i, DO = a, CO =:x, CB = y. Then, denoting the volume of the cylinder by v, we have Fig. 10. (1) From the similar triangles DOA and DOE, we have DO : AO :: DO : EC, or a I b :: a — X : y; which in (1) gives V = iT-.(a — xfx. (2) GEOMETRIC PROBLEMS. Dropping constant factors (Art. 99), we have (a — xfx = aH — lax^ + x^ ; a^ — 4aa; + Sa;^ =. 0, — \c? ; .-. a; = ff or ^a. 166 u = dx ~ or a;^ — 4aa; = T-: = — 4a + 6a; aa;2 = 3a, when a; = «, .•. minimum; = — 2a, when a; = ^a, .-. maximum. Hence the altitude of the maximum cylinder is one-third of the cone. The second value of x in (3) gives ,a ¥)-, Volume of cone = \T^aV. .: Volume of cylinder — ^ volume of cone. y z= -(a — \a) = |5 = radius of base of cylinder. 2. What is the altitude of the maximum rectangle that can be in- scribed in a given parabola ? Let AX = a, AH = x, DH == y, and A = area of rectangle. Then _ we have A = 2y{a — x). But from the equation of the parabola, we have y = V2px, which in (1) gives A = 2V^px (a — x). u' = v'a; (a — x) = aa;' — xk du' m -j- = i^ax-i — fa;^ = 0. .-. x = ^a. dx 166 GEOMETRIC PROBLEMS. Since this value of x makes -7- change sign from + to — , it makes the function A a maximum; therefore the altitude of the maximum rectangle is fa. 3. What is the maximum cone that can be inscribed in a given sphere ? Let ACB be the semicircle, and ^ ACD the triangle which, revolved about AB, generate the sphere and cone respectively. Let AO = r, AD and V = volume of cone. Then v = \-nyh>. (1) But 2/2 = AD X DB = {%r — x) x, which in (1) gives v z= \-n {2r — x) x% (2) or u = %rx^ — 0? ; .-. ^ = 4ra; - 3a;a = 0. .,)(/f . ax --' (, 1 and ^r. dx dtb The latter maikes -^ change sign from + to — ; .-. it makes v a maximum. Hence the altitude of the maximum cone is f of the diameter of the sphere. The second value of x in (2) gives v^\-n (2r - |r) {^f = ffTrr^ = ^ X %-r^r^. Volume of sphere = ^-nr^ ; •'. the cone = ^ of the sphere. 4. Find the maximum parabola which can be cut from a given right cone with a circular base, knowing that the area of a parabola is | the product of its base and altitude. GEOMETRIC PROBLEMS. 16? Let AB —a, AC = b, and BH = x; then AH = « — x. FE = 2EH = 2a/AH^"BH = 2'\/(« — x)x. Also, BA:AC :: BH : HD, or i :: a;: HD=-a;. a Calling the parabola A, we have A — fFE X HD ,= %-x\^{a — x) x. or aoif — x\ du dx :=.3aa? — 4r3 = 0; X ^ Q anJ X = fa. The second value makes -y- change sign from + to — , and .-. makes the function A a maximum. A = ■!■ -• faVCa — |«) |a = {abVS, which is the area of the maximum parabola. Rem. — In problems of maxima and minima, it is often more con- venient to express the function u in terms of two variables, x and y, wliicli are connected by some equation, so that either may be regarded as a function of the other. In this case, either variable of course may be eliminated, and u expressed in terms of the other, and treated by the usual process, as in Examples 1, 3, and 3. It is often simpler, however, to differentiate the function u, and the equation between x and 1/, with respect to either of the variables, x, regarding the other, V, as a function of it, and then eliminate the first derivative, ■^. The second method of the following example will illustrate the process. 168 GEOMETRIC PROBLEMS. 5. To find the maximum rectangle inscribed in a given ellipse. B Let CM = X, PM = y, and A = area of rectangle. Then we have _Aj A = A:xy, (1) and ay + S^a;' = aW. {%) 1st Method. — Prom (3) we get Fig. 14. « = - Va^ — x\ ^ a which in (1) gives ^ = 4 - a; -\/a' — x^, or u = a'x^ — .-c*. ^- = 2a^x — 4:3^ dx .x= ± a/2 X = -\ — makes -;- change sign from + to — ; .-.it makes A a maximum. Hence, the sides of the maximum rectangle are a a/S and i V2, and the area is 2ai. id Method. — Differentiate (1) and (3) with respect to x after dropping the factor 4 from (1), and get dA , dy . dx dx ^d^y^ + 25'« = 0; dy dx — l. X ■ dx ~ ¥x d>y : ^ = ^, or i^x^ = aV; d^y which in (3) gives GEOMETRIC PROBLEMS. 169 2a?y^ = aW, :. y = — = and x — -—=• 6. Find the cylinder of greatest oonyex surface that can be inscribed in a right circular cone, whose altitude is h and the radius of whose base is r. „ „ '^hr Surface = -^• 7- Determine the altitude of the maximum cylinder which can be inscribed in a sphere whose radius is r. Altitude = \r Vs. 8. Find the maximum isosceles triangle that can be inscribed in a circle. An equilateral triangle. 9. Find the area of the greatest rectangle that can be inscribed in a circle whose radius is r. The sides are each r V^. 10. Find the axis of the cone of maximum convex sur- face that can be inscribed in a sphere of radius r. The axis = |r. 11. Find the altitude of the maximum cone that can be inscribed in a paraboloid of revolution, whose axis is a, the vertex of the cone being at the middle point of the base of the paraboloid. Altitude = ^a. 12. Find the altitude of the cylinder of greatest convex surface that can be inscribed in a sphere of radius r. Altitude = r V2. 13. From a given surface s, a vessel with circular base and open top is to be made, so as to contain the greatest amount. Find its dimensions. (See Eemark under Ex. 4.) The altitude = radius of base =^ i/ ^r-- ' an 14. Find the maximum cone whose convex surface is constant. The altitude — V^ times the radius of base. 15. Find the maximum cylinder that can be inscribed in an oblate spheroid whose semi-axes are a and i. - 2 The radius of base = a Vf ; the altitude = b —-• V3 170 GEOMETRIC PROBLEMS. 16. Find the maximum difference between the sine and cosine of any angle. When the angle = 135°. 17. Find the number of equal parts into which a must be divided so that their continued product may be a maximum. Let X be the number of parts, and thus each part equals -, and therefore u = (-) , from which we get a; = - ; a therefore each part = e, and the product of all = (e)'. 18. Find a number x such that the xth root shall be a maximum. x = e = 2.71838 + . 19. Find the fraction that exceeds its m'* power by the greatest possible quantity. / 1 \si-T \m I 20. A person being in a boat 3 miles from the nearest point of the beach, wishes to reach in* the shortest time a place 5 miles from that point along the shore ; supposing he can walk 5 miles an hour, but row only at the rate of 4 miles an hour, required the place he must land. One mile from the place to be reached. 21. A privateer wishes to get to sea unmolested, but has to pass between two lights, A and B, on opposite head- lands, the distance between which is c. The intensity of the light A at a unit's distance is a, and the intensity of B at the same distance is 6 ; at what point between the lights must the privateer pass so as to be as little in the light as possible, assuming the principle of optics that the intensity of a light at any distance equals its intensity at the distance one divided by the square of the distance from the light. a^ + b^ 22. The flame of a candle is directly over the centre of a circle whose radius is r ; what ought to be its height above the plane of the cii'cle so as to illuminate the circumfer- ence as much as possible, supposing the intensity of the GMOMB'Mta PRoliLUMS. Vil light to vary directly as the sine of the angle under which it strikes the illuminated surface, and inyersely as the square of its distance from the same surface. Height above the plane of the circle = r Vi- 33. Find in the line joining the centres of two spheres, the point from which the greatest portion of spherical surface is visible. The function to be a maximum is the sum of the two zones whose altitudes are AD and ad; hence we must find an expression for the areas of these zones. Put CM = E and cm — r, Gc = a and CP = x. The area of the zone on the sphere which has R for its radius (from Geometry, or Art. 194) = SwRAD = '^■nW — 27rROD = 2:1- ( R^ ^l, and in the same way for the other zone, from which we readily obtain the solution. _ aRi R^ + r-* 24. Find the altitude of the cylinder inscribed in a sphere of radius r, so that its whole surface shall be a maximum. Altitude = r 2 {'-Vl CHAPTER IX. TANGENTS, NORMALS AND ASYMPTOTES. ±00. Equations of the Tangent and Normal — Lei P, {x\ y') be the point of tangency ; the equation of the tangent line at (x', y') will be of the form (Anal. Geom., Art. 35) y — y' = a{x — x'), (1) in which a is the tangent of the angle which the tangent line makes with the axis of x. It was shown in Ai'ticle 56ffl that the value of this tangent is equal to the derivative of the ordinate of the point of tangency, with respect to X, dy' Fig. 16. or a = dx' He?i:;e ^-^'^i^^^-'^')' (3) is the equation of the tangent to the curve at the point (x', y'), X and y being the current co-ordinates of the tangent. Since the normal is perpendicular to the tangent at the point of tangency, its equation is, from (3), y-y dx' dy -, {x — x'). (3) (Anai. Geom., Art. 37, Cor. 3.) EXAMPLES. 11^ Rem. — To apply (2) or (3) to any particular curve, we dy' dx' substitute for -pj or -^-7 , its value obtained from the equa- tion of the curve and expressed in terms of the co-ordinates of the point of tangency. EXAMPLES. 1. Find the equations of the tangent and normal to the ellipse ay + 6V = aW. ^ - , dy Wx dy' W and this value in (2) gives, which by reduction becomes, c?yy' + T^x'il = aW, which is the equation of the tangent ; and y-y' = w^''-''"^ is the equation of the normal. 2. Find the equations of the tangent and normal to the parabola y^ = 2px. We find f=P, .:% = ^„ dx y ax' y' and this value in (3) gives .V - y = ^ (a; - a;')» or yy' — y'^ = px—px'. But y'i = %px' ; IH EXAMPLES. •■• yy' = p(,3: + x'), which is the equation of the tangent ; and y-y' = --p{^-^') is the equation of the normal. 3. Find the equations of the tangent and normal to an hyperbola. Tangent, a^yy' — ly'xx' = — aW. Normal, y-y' = -^{x- x'). 4. Find the equation of the tangent to 3y* + a;^ — 5 = 0, at x = l. Here ^^ = — ^ = , „ .„.. = T -39 about, dx 6y ± 3.465 ■which in (3) gives 2^^1.155 = T. 29(^ — 1), or y — ^ -392; ± 1.44. Hence there are two tangents to this locus at a; = 1, their equations being y = — .29a; + 1.44 and y = + .29a; — 1.44. 5. Find the equation of the tangent to the parabola y'^ =z 9a;, at a; = 4. At (4, 6) the equation is y= fa; + 3. « (4, — 6) " " " y—— |x- — 3. 6. Find the equation of the normal to y^ = 2x^ — a?, at a; = l. At (1, + 1) the equation is y = — 2a; + 3. « (1, — 1) " " " y= 2x- 3. 7. Find the equation of the normal to y^ = 6x — 5, at y =z 5, and the angle which this normal makes with the axis of a;. y = — ^a; + -^ ; angle = tan~^ (— -f). LMNCt'tS OP tATfGSHf, NORMAL, ETC. m 101. Length of Tangent, Nonnal, Subtangent, Subnormal, and Perpendicular on the Tangent firom the Or^^. Let PT represent the tangent at the point P, PN the normal ; draw the ordinate PM ; then MT is called the subtangent, MN " " " subnormal. Let « = angle PTM; Fig. 17. then tan « 1st. 2d. 3d. dx' ^ (Art. 56a). TM = MP cot « = y'^,; dx' Subtangent = «/'-=-;. MN = MP tan MPN = j^'tan «; Subnormal = y' -^,- PT = a/PM* + MT2 4th. / ldx'\^ Tangent = y^ 1 + (^j • PN = -v/PM^ + MN2 5th. The equation of the tangent at P {x', y') is (Art. 100)^^ IHB MXAMPLBS. or xdy' — ydx' — x'dy' + y'dx' = ; which,, written in the normal form, is 'xdt-y^^g^^±idl ^ ^^ 'Vidx'f+WY hence, OD = f__ ^ = 0. ^/{d:df + {dy'f :. Perpendicular on the tangent from the origin , _ y'dx' — x'dy' ~ ^/(dx!Y + {dy'f ScH. — In these expressions for the subtangent and sub- normal it is to be observed that the subtangent is measured from M towards the left, and the subnormal from M towards dv' the right. If, in any curve, ^' t^ is a negative quantity, it denotes that N" lies to the left of M, and as in that case dx' y' -J-, is also negative, T lies to the right of M. EXAMPLES. 1. Find the values of the subtangent, subnormal, and perpendicular from the origin on the tangent, in the ellipse Here ^Jl - -^It. dx a?y Hence, the subtangent = «/'-=—, = — ^|^- , the subnormal = «' t^. = -x'', ^ dxs 0? ' SXAMPLSS. Ill the perpendicular from origin on tangent (ays ^ jva)^ 2. Find the snbtaugent and subnormal to the Cissoid yi = 2a — a; (See Anal. Geom., Art. 149.) Here Hence, the subtangent = dy' _ ,x^ (3a — x) (2a — a X (2fl — x) A^ (2a - xf 3a — X the subnormal = -7^ ^4 • (2a — x)^ 3. Find the value of the subtangent of y^^Sa? — 12, at a; = 4. Subtangent = 3. 4. Find the length of the tangent to y^ = 2x, at a; = 8. Tangent = 4a/17. 5. Find the values of the normal and subnormal to the cycloid (Anal. Geom., Art. 156). = r vers~* - — V^ry — y^; dx y_ _ '\/2ry—y'^ - " ^~~ ^/^^—f ~ '^r — y' dy _ 2r — y dx Fig. IS. '\/2ry — y^ .'. Subnormal = V^ry — ^ = MO. Normal = \/2ry = PO. It can be easily seen that PO is normal to the cycloid at P; for the motion of each point on the generating circle at m POLAR CtlttVES. the instant is one of rotation about the point of contact 0, ('. e., each point for an instant is describing an infinitely small circular arc whose centre is at ; and hence PO is normal to the curve, i.e., the normal passes through the foot of the vertical diameter of the generating circle. Also, since OPH is a right angle, the tangent at P passes through the upper extremity of the vertical diameter. 6. Find the length of the normal in the cycloid, the radius of whose generatrix is 2, at y = 1. Normal = 2. POLAR CURVES. A/ 102. Tangents, Nonnals, Subtangents, Subnor- mals, and Perpendicular on Tangents. Let P be any point of the curve APQ, the pole, OX the initial line. Denote XOP by 0, and the radius-vector, OP, by r. Give XOP the infinitesi- mal increment POQ = dO, then OQ = ?• + dr. Prom the pole 0, with the radius OP = r, de- scribe the small arc PR, sub- tending dd ; then, since dd — ab is the arc at the unit's distance from the pole 0, we have PE = rde and EQ = dr. (1) Let PQ, the element* of the arc of the curve, be repre- sented by ds. :. PQ' = PR^ + EQ^ or d? = T^Ti -f- Tr^. (3) Pass through the two points P and Q the right line QPT; * See Art. 56a, foot-note. \ POLAR CURVES. 179 then, as P and Q are consecutive points, the line QPT is a tangent to the curve at P (Art. 56a). Through P draw the normal PC, and through draw COT perpendicular to OP, and OD perpendicular to PT. The lengths PT and PC are respectively called the polar tangent and the polar normal. OC is called the polar subnormal; OT the polar subtangent ; and OD, the perpendicular from the pole oh the tangent, is usually symbolized by ^. The value of each of these lines is required. tan EQP = |^ = ^, from (1). (3) Since OPT = OQT + dd, the two angles OPT and OQT differ from each other by an infinitesimal, and therefore OPT = OQT, and hence, tan OPT = ^, from (3), (4) sin OPT = sin OQP = 5| = ?^, from (1). (5) Hence, i^d6 OT = polar subtangent = OP tan OPT = -^, from (4). (6) 00 = polar subnormal = OP tan OPC = OP cot OPT dr dd = J, from (4). (7) PT = polar tangent = VOP* + OT^ = r- V 1 + r^^, from (6). (8) PC = polar normal = Vo? + OC^ = yr^ + ^, from (7). (9) 180 EXAMPLES. OD=p = OF sin OPD = ^ from (5) = —-^ - from (a). (10) See Price's Calculus, Vol. I, p. 417. EXAMPLES. 1. The spiral of Archimedes, whose equation is r = aB. (Anal. Geom., Art. 160.) Here ^^ 1_ dr ~ a' Subt. = -,from(6). Subn. = a, from (7) Tangent ;ent = »• y 1 + -,, from (8), Normal = Vr^ + d?, from (9). P = Vr^ + a 2. The logarithmic spiral r Art. 163.) Here -j- ^ a^ log a = »• log a ; r . , from (10). = a^. (Anal. Geom., Subt log a = mr, (where m is the modulus of the system in which log a = 1). Subn. = -• P = Iformal = mr ^/wfi + 1 Fig. 20, ^2 ^ m^ , and what values of y will make x = ao . For example, in the equa- tion xy = m, a; = makes y = co, and y = makes a; = CO ; hence the two axes are asymptotes. Also in the equation xy — ay — bx = 0, which may be put in either of the two forms, bx ay y = or X — — •^ • X — a y — y = 00 when a; = ff, and jc = op when y = EXAMPLES. 183 hence the two lines x = a and «/ = 5 are asymptotes to the cuiTe. In the logarithmic curve y = a', y = when x =z — oo , therefore the axis of x is an asymptote to the branch in the second angle. Also in the Cissoid ifi = , ^ 2a — x' y ■= oo when x =2 %a; hence a; = 3a is an asymptote. EXAMPLES, 1. Examine the hyperbola a^y2 _ ]jix^ = _ 05352^ foj. asymptotes. Here dy ¥x c^tfl a^ „ . -f- — -^ ; .: x^ = X — ^ = - — ^ lore X = ± ^ . ax a'y ' Wx x y^^y ^- = =Ofor«=± " ^ a^y y ^ -^ 00. Hence the hyperbola has two asymptotes passing through the origin. ., dy b^x h 1 .5 Also -f- = -^ = ± ; = ± - for a; = 00 . dx a'y a / rfl a Hence the asymptotes make with the axis of x an angle whose tangent is ± - ; that is, they are the produced diagonals of the rectangle of the axes. 2. Examine the parabola y^ = 'Hpx for asymptotes. 184 ASYMPTOTES DETERMINED BY EXPANSION. Here -^ z=?-; .: a;o = — J^ = — 00 when x or y = co, dx y' " %p " ^0 = ^ = 00 when y = ca or a; = oo . Hence the parabola has no asymptotes. The ellipse and circle haye no real asymptotes, since neither has an infinite branch. 3. Examine y^ = ax'^ + a^ for asymptotes. "When x=±cc,y— ±(x>; .: the curve has two infinite branches, one in the first and one in the third angle. dy _ 2ax + 3a;' _ dx ~ Sf^ ' — 3y^ _ ax^ _ a .: '^o — ^ — 2aa; + 3^3 - "" 2ax + 3x^ ~ ~ 3' when X = aa. 2ax^ + 3x^ _ 3{y^ — a^) —2a^ y^ = y 3^ 3^2 3yi ax^ a = 3 , when a; = 00 . 3 {ax^ + afi)^ 3 ' Hence the asymptote cuts the axis of a; at a distance — g , and that of y at a distance ^ from the origin, and as o o it is therefore inclined at an angle of 45° to the axis of x, its equation is (See Gregory's Examples, p. 153.) 104. Asymptotes Determined by Expansion. — A very convenient method of examining for asymptotes con- sists in expanding the equation into a series in descending EXAMPLES. 185 powers of x, by the Ibinomial theorem, or by Maclaurin's theorem, or by division or some other method. EXAMPLES. 1. Examine 'f = — ^ — for asymptotes. Then 2,= ±a:|/— - = ±«.(l + - + ^ + etc.) (1) When a; = 00 (1) becomes ^=±(a^+a). (2) We see that as x increases, the ordinate of (1) increases, and when x becomes infinitely great, the difference between the ordinate of (1) and that of (3) becomes infinitesimal ; that is, the curve (1) is approaching the line (2) and becomes tangent to it when a; = oo ; therefore, y=: ±{x+a) are the equations of two asymptotes to the curve (1) at right angles to each other. Another asymptote parallel to the axis of y is given by a; = a. 2. Examine a? — xy^ -{• ay^ = for asymptotes. Here y = ±sj ^^ Hence, ^ = ± I a; + - 1 are the equations of the two asymptotes. By inspection, we find that a; = a is a third asymptote. ^2 \ 3. Examine y^^^ -TT'^ f'°'' asymptotes. a; + 1 186 EXAMPLES, Here y = ± x\l — ^ + etc.) •*• y = ±^ are the two asymptotes, ji 105. As3anptotes in Polar Co-ordinates. — When the curve is referred to polar co-ordinates, there will be an asymptote whenever the subtangent is finite for r = oc . Its position also will be fixed, since it will be parallel to the r94i^s-vector. Hence, to examine for asymptotes, we find what finite values of 6 make r ^ oo ; if the corresponding polar subtangent, r^ -j- , which in this case becomes the perpendicular on the tangent from the pole, is finite or zero, there will be an asymptote parallel to the radius-vector. If for r = 00 the subtangent is oo , there is no corresponding asymptote. EXAMPLES. 1. Find the asymptotes of the hyperbola c?y^ — V^x^ = — (fl? by the polar method. The polar equation is a'sin^O — 5« 008^(9 = —^- (1) When r = 00 , (1) becomes, tan^ Q =. —^; .: 6 = tau- ■(±!)- Therefore the : asymptotes are inclined to the initial line at tan-i(± 3- from (1) we { , dB ?6t J- = ' dr aW r» (tf2 -1- 62) sin 6 cose' and .de dr a5 (52 cos' — fl« sin« 0)* ^ (a^ + ¥) sin cos ' (2) EXAMPLES. 187 which is equal to when 0=:tan~M±-); hence both asymptotes pass through tlie pole. 2. Find the asymptotes to the hyperbolic spiral rO = a. (See Anal. Geom., Art. 161.) Here »• = ^ , .: r = oo , when 6 — 0, u de a .odd ■J- = -„ , and r2 — = —a. dr r^ dr There is an asymptote therefore which passes at a distance a from the pole and is parallel to the initial line. 3. Find the asymptotes to the lituus rO^ = a. (Anal. Geom., Art. 162.) Here »• = — , .•. r = oo , when = 0. f=- ^, and r^f = - 2afl^ = 0, when 6 = 0. dr 1-^ dr Therefore the initial line is an asymptote to the lituus. 4. Find the asymptotes of the Conchoid of Nicomedes, r = pseod + m. (Anal. Geom., Art. 151.) Here r = oo when = zr ; and r^-=- = » when = -• 2 dr ^ 2 Therefore the asymptote cuts the initial line at right angles, and at a distance p from the pole. EXAMPLES. 1. Find the equation of the tangent to 3y^— 2x^—10 = 0, at a; = 4. Ans. y — ± .7127a; ± .8909. a? 2. Find the equation of the tangent to y^ = j , at 3 4 — X y = 2x — 2 and y = —2x + 2. 188 EXAMPLES. 3. Find the equation of the tangent to the Naperian logarithmic curve, Ans. y z=zy' {x — x' + 1). 4. At what point on y = 3? — ^x^ — 24a; + 85 is the tangent parallel to the axis of a; ? dti' {Here we must put -^ = 0. See Art. 56fl!.] At (4, 5) and (—2,113). 5. At what point on y^ = ^a? does the tangent make with the axis of x an angle whose tangent is 3, and where is it perpendicular ? At (3, 4) ; at infinity. 6. At what angle does the line y = ^x + 1 cut the curve y^ = 4rX? [Find the point of intersection and the tangent to the curve at this point; then find the angle between this tangent and the given line.] 10° 14' and 33° 4'. 7. At what angle does y^ = 10a; cut x^ + y^ := 144 ? 71° 0' 58". 8. Show that the equation of a perpendicular from the focus of the common parabola upon the tangent is «' 9. Show that the length of the perpendicular from the focus of an hyperbola to the asymptote is equal to the semi- conjugate axis. 10. Find the abscissa of the point on the curve ^ (a; — 1) (a; — 3) = a; — 3 at which a tangent is parallel to the axis of x. X =3 ± V^. 11. Find the abscissa of the point on the curve y^ = (x — fl)2 {x — c) at which a tangent is parallel to the axis of x. 2c + a X = — ■= — • EXAMPLES. 189 12. Find the subtangent of the curve y = Ans. V'^a — X X (3a — x) 3a — X 13. Find the subtangent of the curve y^—3axy+afi= 0. 2axy — x^ ay — x^ 14. Find the subtangent of the curve xy^ = a^{a — x). 2 {ax — a?) a 15. Find the subnormal of the curve y'^ = 2(j^ log x. t. x 16. Find the subnormal of the curve 3ay^ + a^ = 2a;*» a a? 17. Find the subtangent of the curve y^ = a — X 2x (a — x) 3a — 2x 18. Find the subtangent of the curve a?y^ = (a + a;)^ {&^ — x^). x{a + x)(¥ — x ^) 19. Find the subnormal, subtangent, normal, and tangent in the Catenary Subnormal = jle' — e «); normal = c Subtangent = — — — ; ; tangent = ^• 190 EXAMPLES. 20. Find the perpendicular from the pole on the tangent in the lituus rO^ = a. 2a^r 21. At what angle does y^ = 'Zax cut x^—3axy+y^ = ? cot~i A^i. 22. Examine y^^2x + 3a;' for asymptotes. y — Vs X + — - is an asymptote. V3 23. Examine y^ = 6x> + 3^ for asymptotes. y = x + 2 is an asymptote. 24. Find the asymptotes of j/^ (x — 2a)=:x^ — a\ x=2a; y= ±(x + a)/ 25 Find the asymptotes of « = ^Jzll^-±^. ^ ^ x^ — Zbx + W x = i; x = 2b; y = x — 3(a — b). CHAPTER X. DIRECTION OF CURVATURE, SINGLE POINTS, TRACING OF CURVES. 106. Concavity and Convexity. — The terms concav- ity and convexity are used in mathematics in their ordinary sense. A curve at a point is concave towards the axis of x when in passing the point it hes between the tangent and the axis. See Fig. HI. It is convex towards the axis of x when its tangent lies between it and the axis. See Fig. 32. If a curve is concave down- wards, as in Fig. 21, it is plain that as X increases, a decreases, and hence tan « decreases j that dy is as X increases, -r^ (Art. 56rt) decreases ; and therefore the de- rivative of ~ or -— is negative. dx dx' ° In the same way if the curve is convex downward, see Fig. 22, it is plain that as x increases, a in- creases, and therefore tan a in- creases ; that is, as x increases. Fig. 21, (Art. 12.) dy . increases, and therefore the de- dx rivative of dy d^y . -^- or -y^ IS positive. dx dx' Hence the curve is concave or convex downward according as ~ IS — or 4-, 192 POLAR CO-ORDINATES. This is also evident from Fig. 33, where MM' = M'M" = dx ; PP' is common to the two curves and the common tangent. PE = PR' = dx ; and P'R = P^R'. But P"R' > PjR' > PiR'. Now P'R and P"R' are consecutive values of dy in the upper curve, and P'R and P,R' are consecutive values of dy in the lower cuitc, and hence P"R' - P'R = 2; and therefore the curve is convex downward for all values of a; < 2, and concave downwards for all values of ic > 2. 2. Find the direction of curvature of y =: b + c {x -{- ay and y = a^ Vx — a. Ans. The first is concave upward, the second is concave towards the axis of x. 3. Find the direction of curvature of the lituus r = &i 194 SINGULAR POINTS. ^^^ dO "^ ~ 2d ~ ~ 2^^' ■'■ d(fi~ 4a*' which in (10) of Art. 102 gives, _ 2o?r . . dr_ _ (4a* + f^)^ ^ ~ (^ + 4a4)i' ■'■ dp ~ 2fl2(4«* — r*)' Therefore the curve is concave toward the pole for values of ?• < a V2, and convex f or r > a \/2. 4. Find the direction of curvature of the logarithmic spiral r = a^. By Art. 102, Ex. 3, mr dr Vm^ + 1 P Vm^ + 1 m which is always positive, and therefore the curve is always concave toward the pole. SINGULAR POINTS. 108. Singular Points of a curve are those points which have some property peculiar to the curve itself, and not depending on the position of the co-ordinate axes. Such points are : 1st, Points of maxima and minima ordi- nates ; 2d, Points of inflexion ; 3d, Multiple Points ; 4th, Cusps ; 5th, Conjugate points ; 6th, Stop points ; 7th, Shooting points. We shall not consider any examples of the first kind of points, as they have already been illus- trated in Chapter VIII, but will examine very briefly the others. 109. Points of Inflexion.— A point of inflexion is a point at which the curve is changing from convexity to concavity, or the reverse ; or it may be defined as the point at which the curve cuts the tangent at that point. When the curve is convex downwards, -=-| is + (Art. BXAitPtES. 195 106), and when concave downwards, -— \b — ; therefore, ax' ' at a point of inflexion ^ is changing from + to — , or from — to +, and hence it must be or oo. Hence to find a point of inflexion, we must equate -^-^ to or oo , and find the values of x ; then substitute for x a value a little greater, and one a little less than the critical value ; if -7^ changes sign, this is a point of inflexion. EXAM PLES. 1. Examine y = h + {x — a)* for points of inflexion. Here g = 6(a;-«) = 0; .•. x z= a and hence y ^=.h. This is a critical point, i. e., one to be examined ; for if there is a point of inflexion it is at a; = a. For « > a, -^ is + , and for x -Ca, -^, is — . Hence there is a point dx^ dx^ ^ of inflexion at (a, I). 2. Examine the witch of Agnesi, x^y = 4a' (3a — y), for points of inflexion. There are points of inflexion at I ± — -=, -^-l- 3. Examine y = i -^ {x — a)-^ for points of inflexion. There is a point of inflexion at {a, i). 4. Examine the lituus for points of inflexion. By Art. 107, Ex. 3, -~ is changing sign from + to — when r = a V^, indicating that the lituus changes at this 1% MMHOb dp PtNDINa M^LttPLS POtNTM. point from concavity to convexity, and hence there is a point of inflexion at r = a V^. 110. Multiple Points. — A multiple point is a point through which two or more branches of a curve pass. If two branches meet at the same point, it is called a double point ; if three, a triple point ; and so on. There are two kinds : 1st, a point where two or more branches intersect, their several tangents at that point being inclined to each other ; and 3d, a point where two or more branches are tangent to each other. The latter are sometimes called points of osculation. As each branch of the curve has its tangent, there will be at a multiple point as many tangents, and therefore as many values of -=^ as there are branches which meet in this point. If these branches are all tangent, the values of ~ will be equal. At a multiple point y will have but one value, while at points near it, it will have two or more values for each value of x. In functions of a simple form, such a point can generally be determined by inspection. After finding a value of x for which y has but one value, and on both sides of which it has two or more values, form ■-—• If this has unequal values, the branches of the curve intersect at this point, and the point is of the first kind. If ~- has but one value, the branches are tangent to each other at this point, and the point is of the second kind. When the critical points are not readily found by inspec- tion, we proceed as follows : Let /(x,y) = (1) be the equation of the locus freed from radicals. Then du dy dx ^ dx ~ du ' dy and as differentiation nerer introduces radicals vhen they do not exist in the expression differentiated, the value of -J- cannot contain radicals, and therefore cannot have sev- eral values, unless by taking the form ^• Hence we have -f-=.- or t- = 0, and -:=- = 0, from dx dx dy which to determine critical values of x and y. If these values of x and « found from -^ = and ^j- = are real " dx dy and satisfy (1), they may belong to a multiple point. If y has but one value for the corresponding value of x, and on both sides of it y has two or more real values, this point is a multiple point. "We then evaluate -j-^^-?., and if there are several real and unequal values of -j- , there will be as many intersecting branches of the curve passing through the point examined. (See Courtenay, p. 190.) EXAMPLES. 1. Determine whether the curve y = (a; — a)'^ + 5 a multiple point. Here y has two values for every positive value of a; > or < a. When a; =: or a, y has but one value, I ; hence there are two points to be ex- amined. When a; < 0, y is imagi- nary ; hence the branches do not pass through the point (0,5), and Fig. 25. 198 til^MpLSS. therefore it is not a multiple point. When a; > or < a, ^ has two real values, and therefore [a, i) is a double point. Therefore the point is of the first kind, and the tangents to the curve at the point make with the axis of x angles whose tangents are + Va and — Va. 2. Examine a^ + 2ax^i/ — ay^ = for multiple points. We proceed according to the second method, as all the critical points in this .example are not easily found by inspec- tion. ^ = 4x{a? + ay)=0; (1) ^ = a{2x^-3f)=0', (3) dy id? + iaxy dx~ Zay^ — 2aoi?' Solving (1) and (2) for x and y, we find e = oy ix = iflSA/A Ix = - ^a\/6\ = 0/' \y= -fa/' \^= -|« /• Only the first pair will satisfy the equa- tion of the curve, and therefore the ori- gin is the only point to be examined. Evaluating ^ in (3) for a; = and y = 0, and representing ~ by p, and -^ ^ ° dx •' ^' dx Fig. 26. by p', for shortness, we have (3) c^^pjS. 199 dx 43? + iaxy = p = 2ax^ 0' 3ay^ _ 12a;' + 4ay + iaxp Hayp — iax _ 24a: + 8ap + 4a^' ~ 6aj»' + 6ayp' — 4a _ 8a^ when = t; , when /a; = 0\ Vy = o/ /a. = 0\ W = o/ 6ap> — 4:a' .; p (6ap2 when /. = 0X \w = 0/ 4a) = 8ajB ; dy dx P = '^ = ^> + V2, or - ^2. Hence the origin is a triple point, the branches being in- clined to the axis of x at the angles 0, tan~i(v'2), and tan~i (— V^), respectively, as in the figure. (See Courte- nay's Calcnlus, p. 191 ; or Young's Calculus, p. 151.) 3. Examine «/' — a;' (1 — a;^) = for multiple points. Ans. There is a double point at the origin, the branches being inclined to the axis of a? at angles of 45° and 135" respectively. 4. Show that ay^ — oi?y — as? = has no multiple points. 111. Cusps. — A cusp is a point of a curve at which two branches meet a common tangent, and stop at that point. If the two branches lie on opposite sides of the common tan- gent, the cusp is said to be of the first species ; if on the same side, the cusp is said to be of the second species. Since a cusp is really a multiple point of the second kind, the only difference being" that the branches stop at the point, instead of running through it, we exam- ine for cusps as we do for multiple points; and to distin- Fig.27. sod cVSPS. guish a eusp from an ordinary multiple point, we trace the curve in the vicinity of the point and see if y is real on one side and imaginary on the other. To ascertain the kind of cusp, we compare the ordinates of the curve, near the point, with the corresponding ordinate of the tangent ; or ascertain the direction of curvature by means of the second derivative. In the particular case in which the common tangent to the two branches is perpendicular to the axis of x, it is best to consider y as the independent variable, and find the values of 3- , etc. dy' EXAMPLES. 1. Examine ^ = a? ± a;^ for cusps. We see that when x = 0, y has but one value, 0; when a; < 0, ^ is imaginary ; and when xyO,y has two real values; hence, (0, 0) is the point to be examined. ^ = 2a; ± ^a;^ = 0, when a; = ; hence the axis of x is a common tangent to both branches, and there is a cusp at the origin. j^ = 3±J^a!^ is positive when a; = 0; hence the cusp is of the second kind. The value of -j^ shows that the upper branch is always concave upward, while the lower branch has a point of inflexion, when x = -^ ; from the origin to the point of inflexion this branch is concave upward, after which it is concave downward. The value of -f- shows that the branch is horizontal dx when X = -Jf . Prom y z=.x^ — x^, we find that the lower branch cuts the axis of a; at a; = 1. The shape of the curve Is given in Fig. 28. Fig.29 com vo ATE pomts. ^01 2. Examine {y — if = {x — aY for cusps. Ans. Tlie point («, h) is a cusp of the first kind. 3. Examine cy^ = a? for cusps. The origin is a cusp of the first kind. 112. Conjugate Points. — A conjugate point is an iso- lated point whose co-ordinates satisfy the equation of thu curve, while the point itself is entirely detached from every other point of the curve. For example, in the equation «/ = (a + x)'\/x, if x is negative, y is, in general, imaginary but for the particular value a; = — a, y = 0. Hence, P is a point in the curve, and it is entirely detached from all others. When a; = 0, y = 0, which shows that the curve p_ passes through the origin. For positive values of x, there will he two real values of y, numerically equal, with opposite signs. Hence, the curve has two infinite branches on the right, which are symmetrical with respect to the axis of x. If the first derivative becomes imaginary for any real values of x and y, the corresponding point will be conjugate, as the curve will then have no direction. It does not fol- low, however, that at a conjugate point -^ will be imagi- nary; for, if the curve y =f{x) have a conjugate point at (x, y), from the definition of a conjugate point, we shall have f{x±h) = an imaginary quantity. But , dyh d^yh^ , d^y¥ , , therefore, if either one of the derivatives is imaginary, the first member is imaginary; hence, at a conjugate point some one or more of the derivatives is imaginary, 'Tice at a conjugate point some of the derivatives are -inary, let the n*^ derivative be the jirst that is imagi- 202 sxamples. nary. Suppose the equation of the curve to be freed from radicals, and denoted by u =f{x, y) = 0. Take the w'* derived equation (Art. 88, Sch.) ; we have dy dx" ' ' ' ' dx' ~ ' where the terms omitted contain derivatives of u with re- spect to X and y, and derivatives of y with respect to x, of lower orders than the n''K If, then, ^- be not 0, the value dy of j^ obtained from the deriyed equation will be real, which is contrary to the hypothesis; hence, y- = is a necessary condition for the existence of a conjugate point. But du du dy dx dy dx ' therefore, since -p == 0, we must have -=- = 0. Hence, at a conjugate point we must have -=- = 0, and -r- = 0. Rem. — Owing to the labor of finding the higher derivatives, it is usually better, if the first derivative does not become Imaginary, to substitute successively a+ h and a — h for x, in the equation of the curve, where a is the value of x to be tested, and Ji is very small. If both values oiy prove imaginary, the point is a conjugate poiut. EXAMPLES. 1. Examine ay^ — x^-\- iax^ — ba^x + 2aS = G for con- jugate points. ^ = - 3a;3 + %ax — bd? = 0. (1) g = 3«2/ = 0. (2) SMOOTING POINTS.— STOP POINTS. 203 Solving (1) and (3), we get (;="o) -^ czn Only the first pair of values satisfies the equation of the curve, and hence the point (a, 0) is to be examined. dy _ _ 3x> — Sax + 5a^ _ 6x — 8 a dx~ My ~ 2ap = , when ( "" „ ) ; therefore, ^ = — 1 ; .-. p — ± V— 1 — -^• This result being imaginary, the point {a, 0) is a conju- gate point. 3. Show that a;* — ax^y — axy^ + ay = has a conju- gate point at the origin. 3. Examine ((?y — 7?f = (a; — df {z — by for conjugate points, in which ay b. The point (5, -^j is a conjugate point. The first and second derivatives are real in this example ; hence the better method of solving it will be to proceed according to the Bemarb above given 113. Shooting Points are points at which two or more branches of a curve terminate, without having a common tangent. Stop Points are points in which a single branch of a curve suddenly stops. These two classes of singular points but rarely occur, and never in curves whose equations are of an algebraic form. 304 EXAMPLES. EXAMPLES. X 1. Examine y = ^ for shooting points. l + > Here dy dx 1 + 1.+ J.\2 (l+e') If a; is + and small, y is + ; if a; is — and small, y is —. When a; is + and approaches 0, y = 0, and ^ = : when x is — dx proaohes 0, y = 0, and and ap- dy dx 1. Fig, 30. Hence, at the origin there is a shooting point, one branch having the axis of x as its tangent, and the other inclined to the axis of x at an angle of 45°. (See Serret's Calcul Diffgrentiel et Integral, p. 267.) 2. Examine y = a; log x. "When a; is +, y has one real value ; when x = 0, y = ; when a; < 0, y is imaginary ; hence there is a stop point at the origin. 3. Examine « = a; tan~i -• ^ X c. If dy _ tan-i - X X dx x^ + 1 X = + or -0, y = 0; dy _ dx 7r 2 "^ o A Hence the origin is a shooting point, the tangent being inclined to the axis of x at angles tan~i (1.5708) and tan-i (— 1.5708). 4. Show that y =^e ^ has a stop point at the origin. TRAQilSQ CimVBS. 205 114. Tracing Curves.— We shall conclude this chap- ter by a brief statement of the mode of tracing curves by means of their equations. The usual method of tracing curves consists in assigning a series of different values to one of the variables, and cal- culating the corresponding series of values of the other, thus determining a definite number of points on the curve. By drawing a curve or curves through these points, we are enabled to form a tolerably accurate idea of the shape of the curve. (See Anal. Geometry, Art. 21.) In the present Article we shall indicate briefly the man- ner of finding the general form of the curve, especially at such points as present &nj 2)eculiarity, so that the mind can conceive the locus, or that it may be sketched without going through the details of substituting a series of values, as was referred to above. To trace a curve from its equation, the following steps will be found useful : (i.) If it be possible, solve it with respect to one of its variables, y for example, and observe whether the curve is symmetrical with respect to either axis. {2.) Find the points in which the curve cuts the axes, also the limits and infinite branches. (3. ) Find the positions of the asymptotes, if any, and at which, side of an asymptote tlie corresponding branches lie. (^.) Find the value of the first derivative, and thence deduce the maximum and minimum points of the curve, the angles at which the curve cuts the axes, and the multiple points, if any. {5.) Find the value of the second derivative, and thence the direction of the curvature of the different branches, and the points of inflexion, if any. (6.) Determine the existence and nature of the singular points by the usual rules. 206 EXAMPLES. EXAMPLES. 1. Trace the curve y ■■ When a; = 0, ^ = ; .•. the curye passes through the origin. For all positive values of x, y is positive ; and when a; = CO , y = 0. For negative values of x, y is negative, and when a; = — 00 , y =Q ; hence the curve has two infinite branches, one in the first angle and one in the third, and the axis of X is an asymptote to both branches. dy _ 1—x^ ^ _ 2x {3? — 3) ■ dx~ {i + x^f ' dc^~ {1+ c^' When a; = ± 1, -^ = ; .-. there is a maximum ordinate at a; =: + Ij and a minimum ordinate at a; = — 1, at which points y ^\ and — \ respectively. When a; = 0, -^ = 1 ; .-. the curve cuts the axis of x at an angle of 45°. Putting the second deriva- tire equal to 0, we get a; = or ± Vs. Therefore, there are points of inflexion at (0, 0) — and at a; = + \/3 and — -v/S, for which we have y — iVS, ■ — jVS. From a; = — VS to a; = 4- ■v/3, the curve is concave towards the axis of x, and beyond them it is convex. From this investigation the curve is readily constructed, and has the form given in the figure. 2. Trace the curve %f = 2ax^ — a?. y = x^ (2a — a;)' ; Fig. 31. EXAMPLES. 307 dx iax — 3x^ ~3f~' -8a2 9a;t (2a - x)i When a; = or 2a, y = ; .•. the curve cuts the axis of X at the origin and at a; = 2a. To find the equation of the asymptote, we have y /. 3aU /. 2a \ therefore, y=—x + ^a is the equation of the asymptote, and as the next term of the expression is positive, the curve lies above the asymptote. Evaluating the first derivative for a; = 0, y =■ 0, we have dy 4aa; — 3a;^ 4a — '6a; _ dx Zf 2a it is + ; hence the left branch is everywhere concave down- ward, and the right branch is concave downward from a; = to a; = 2a. At this last point it cuts the axis of x at right angles, and changes its curvature to concave upward; the two branches touch the asymptote at a; ^ + oo and — oo, respectively, i. e., they have a common asymptote. In the figure, OA = 2a, OB = ^a, 00 = fa. t^, J. ± a,\j\D ui Jt \yi.i L Y t< ''-^\x-a/ Let x = Q; : y = 0. '■ x<,a; • : y is positive. x = a; y = . / /V x> a< 2a; y is negative. y = 0. L // a; = 3a; / / r ' a; > 3a; y is positive. / o / a; = oo; y=. / When a; is —, y is always negative. Fig. 32.a To find the asympto te, we have 1 - -CM X ^ = ^('-|)(' + 5 + eto.) y = X ■ 1 - a X ■ "( 1 — a X 3a« , \ ^2 etc.) -a;- - a etc. X :. y = X — a is the equation of the asymptote. Hence, take OB = a =r OD, and the line BD produced is the asymptote; also take 00 = 2a. Then, since «/ = 0, both when a; = and x = 2a, the curve outs the axis of x EXAMPLES. 209 at and 0. Between and B, the curve is above the axis ; at B the ordinate is infinite ; from B to C, the curve is below ; from to infinity, it is above OX. Also, if x is negative, y is negative ; therefore the branch on the left of is entirely below the axis. . , dy a? — 2ax + 2d^ Also, -f- = ':- dx {x — a)2 Let x = a; .: -^ = as; and the infinite ordinate at the distance a to the right of the origin is an asymptote. li a; = 0, -^ = 3; if x =: 2a, -—- = 2 ; i.e., the curve cuts the axis of x at the origin and the distance 2a to the right, at the same angle, tan'^ (2). It x^ — 2ax + 2a^ or {x — a)' + a* = 0, a; is impossible ; hence there is no maximum or minimum ordinate. ^ . cPy 2(x — af—2Ux — a)^ + a^ Agam, -^. = — i '- — - — !-^-^- — i— ! — ^ " dx^ (x — a)* — 3a2 dx^ ~{x — af' is + if a; < a, and is — it x > a. But a; < a, y is + ; and a; > a < 2a, ?/ is — ; and a; > 3a, y is + ; therefore, from to B, and B to C, the curve is convex, and from C to infinity, it is concave to the axis of X. d^v 2(1? ^* ^ ^® ~' ^ = (STfflp ^^ +' ^"* 2/ Js - ; therefore the branch from the origin to the left is concave to the axis of X. (See Hall's Calculus, pp. 182, 183.) 4. Trace the curve y^ = a^a;'. The curve passes through the origin ; is symmetrical with respect to the axis of x ; has a cusp of the first kind at 310 TRACING POLAB CURVES. the origin ; both branches are tangent to the axis of x ; are conTex towards it ; are infinite in the direction of positive abscissas, and the curve has no asymptote or point of in- flexion. 115. On Tracing Polar Curves. — Write the equation, if possible, in the form r =f{0) ; give to 6 such values as to make r easily found, as for example, 0, ^n, n, frr, etc. Putting "52 = 0, we find the values of 6 for which r is a maximum or minimum, i. e., where the radius vector is perpendicular to the curve. Find the asymptotes and direction of curvature, and points of inflexion. After this there will generally be but little difficulty in finding the form of the curve. EXAMPLES. 1. Trace the lituus r = -r- When e = 0,r=z; when (9 = 1 (= 57°.3),* r= ±a; when = 2 (= 114°.6), r = ± .7a ; when — 3, r = ± .58a, etc. ; when 6 = co, r = 0. -jE = — K-ij and when ^- = 0, r = 0; hence, r and 6 at) aa' aa are decreasing functions of each other throughout all their values ; f and the curve starts from infinity, when fl = 0, and makes an infinite number of revolutions around the pole, cutting every radius-vector at an oblique angle, and reaching the pole only when 6 = . The subtangent r^-^=. = 0, when r = oo; hence the initial line is an asymptote (Art. 105). * The unit angle is that whose arc is equal to the radius, and is about 57° .29578. t If we consider alone the branch generated by the positive radius-vector. EXAMPLES. 211 6?r (4^4 + ^)1 a point of inflexion at ?• = aV^ ; from r = to r = aV^ the curve is concave toward the pole, and from r = a\/% to »• = 00 it is convex. 2. Trace the curve /■ = a sin 30. r=0, when 6=0, 60°, 120°, 180°, 240°, and 300°. When = 27r, or upwards, the same series of values recur. If = 30°, 90°, 150°, 210°, 270°, and 330°, r = a,—a, a, — a, a, and — a, successively. -^ — 3a cos 36, showing that r begins at when = 0, increases till it is a when d = 30°, diminishes to as passes from 30° to 60°, continues to diminish and becomes — a when 6 becomes 90°, and so on. dp ISaJ'r — 8r* i, • u i, . i, . -^ = ; , which shows that <^r (2a> - 8r2)* the curve is always concave towards the pole. There is no asymptote, as r is never oo . Hence the curve consists of three , _. „ r ig. 33. equal loops arranged symmetrically around the pole, each loop being traced twice in each revo- lution of r. A little consideration will show that the form of the curve is that given in the figure. (See Gregory's Examples, p. 185 ; also Price's Calculus, Vol. I, p. 427. ) 3. Trace the Chordel r = a cosec \2n)' If = 0, nn, 2nn, 3nn, 4«7r, bmr, etc., successively, r := 00 , a, 00 , — a, — qo , a, etc. dr a 6 B a ,0/ 0\ — - = — Tr- cosec s- cot s- = H- cosec^ jr- ( — cos ^-1 j dd 2n 2n 2n 2m 2n\ 2w/ which is negative from = to = mr, positive from 212 EXAMPLES. =nn to =z SriTT, negative from 6 = Swtt to 6 = Smr, etc. Hence we see that r begins at oo when 6 = 0; diminishes till it become* a when d = mr; increases as d passes from MTT to 2mr ; becomes oo when = 2nn; when 6 passes 2nn, r changes from + oo to — oo; when increases from 2mT to SwTT, r increases from — oo to — a; when increases from Stm to inn, r diminishes from — a to — oo ; when 6 passes inn, r changes from — oo to + oo . When in- creases beyond in, the same values of r recur, showing that the curve is complete. dr Fig. 34 e jfl = 5- cosec „ do 2n 2n (-'"'D gives = nn, 3nn, bnn, etc. ; i. e., the radius- vector is a minimum at = nn, Znn, 5nn, etc. d6 The subtangent = t^-j- = 2na W COSs- 2n = — 2na when = 0; and = + 2na when = 2nn ; EXAMPLES. 313 therefore the curve has two asymptotes parallel to the initial line, at the distances ± %na from the pole. r^ 2anr P = dp _ 2a?n (in^ — 1) .: the curve is always concave towards the pole. Thus it appears that while 6 is increasing from to 2nrr, the positive end of the radius-vector traces the branch drawn in Fig. 34 ; and wh"!le increases from 2mT to 4w:r, the negative end of the radius-vector traces a second branch (not drawn), the two branches being symmetrical with respect to the vertical line through the pole 0. EXAMPLES. 1. Find the direction of curvature of the "Witch of Agnesi x^y = ia^ (2a — y). The curve is concave downward for all values of y between 2a and |a, and convex for all values of y hetween |-a and 0. 2. Find the direction of curvature of y = h + {x — a)*. Convex towards the axis of x from a; > a to a; = oo ; and from x=:a — bi to x= — co ; concave towards the axis of X from a; < a to x=:a — b^. 3. Examine y ^ {a — x)^ -f- ax for points of inflexion. There is a point of inflexion at x = a. 4. Examine y = x + 36x^ — 2x^ — 0^ for points of in- flexion. Points of inflexion at x = 2, x= —3. 214 EXAMPLES. 5. Find the co-ordinates of the point of inflexion of the curve a? (a^ — a^) y = — ^5 — 6. Examine r = -^ z. lor points of inflexion. Here -^rs = dr^ _ 4r {r — a)^ _ etc. There are points of inflexion at r = fa and r = |«. 7. Examine y^ = {x — lyx for multiple points. There is a multiple point at x = 1. x^ ( ct^ x^) 8. Examine «/^ = — \ ^- for multiple points. There is a multiple point at the origin, and the curve is composed of two loops, one on the right and the other on the left of the origin, the tangents bisecting the angles be- tween the axes of co-ordinates. 9. Show that x^ + x^y^ — Gaa^y + a^y^ = has a multiple point of the second kind at the origin. 10. Show that y = a + x + bx'^ :tcx^ has a cusp of the second kind at the point (0, a), and that the equation of the tangent at the cusp is «/ = a; + a. 11. Show that y^ = ax^ + a? has a cusp of the first kind at the origin. 12. Show that ay^ — a? + h? = has a conjugate point 45 at the origin, and a point of inflexion at »; = —-• EXAMPLES. 215 ■ 13. Trace the curve ^' = a' — x^. The curve cuts the axes at {a, 0) and (0, a). It has an asymptote which passes through the origin. The points where the curve cuts the axes are points of inflexion. 14. Trace the curve y = ax^ ± 's/bx sin x. For every positive value of x there are two values of y, and therefore two points, except when sin x = 0, in which case the two points reduce to one. These points form a series of loops Uke pig. 35, the links of a chain, and have for a diametral curve the parabola y = ax^, from which, when x is positive, the loops recede and approach, meeting the parabola whenever a; = or tt, or any multiple of -n. But when X is negative, y is imaginary except when sin « = 0, in which case y = aa?, so that on the negative side there is an infinite number of conjugate points, each one on the parabola opposite a double point of the curve. (See De Morgan's Cal., p. 383 ; also, Price's Cal., Vol. I, p. 396.) CHAPTER XI. RADIUS OF CURVATURE, EVOLUTES AND INVO- LUTES, ENVELOPES. 116. Curvature.— OTe curvature of a curve is its rate of deviation from a tangent, and is measured by the external angle between the tangents at the extremities of an indefi- nitely small arc ; that is, by the angle between any infini- tesimal element and the prolongation of the preceding element. This angle is called the angle of contingence of the arc. Of two curves, that which departs most rapidly from its tangent has the greatest curvature. In the same or in equal circles, the curvature is the same at every point ; but in unequal circles, the greater the radius the less the curvature ; that is, in different circles the curvature varies inversely as their radii. Whatever be the curvature at any point of a plane curve, it is clear that a circle may be found which has the same curvature as the curve at the given point, and this circle can be placed tangent to the curve at that point, with its radius coinciding in direction with the normal to the curve at the same point. This circle is called the osculating circle, or the circle of curvature of that point of the curve. The radius of curvature is the radius of the osculating circle. The centre of curvature is the centre of the osculating circle. For example, let ABA'B' be an ellipse. If different circles be passed through B with their centres on BB', it is Fig. 36. ORDER OF CONTACT OP CURVES. 217 clear that they will coincide with the ellipse in very differ- ent degrees, some falling within and others without. Now, that one which coincides with the ellipse the most nearly of all of them, as in this case MN, is the osculating circle of the ellipse at B, and is entirely exterior to the ellipse. The osculating circle at A or A', is entirely within the ellipse ; while at any other point, as P, it cuts the ellipse, as will be shown hereafter. 117. Order of Contact of Curves. — .Let y=f{x) and y =

'{x) h + cj)" (x) - 7? 3 Now if, when x = a = OM, we have f{a) =0 (a), the two curves intersect at P, i. e., have one point in common. If in addition we have /' (a) = 0' (a), the curves have a common tangent at P, i. e., have two consecutive points in common ; in this case "they are said to have a contact of the first order. If also we have, not only/ (a) = (a) and/' {a) = 0' {a), but/" {a) ■=. 0" (a) ; i. e., in passing along one of the curves to the next consecutive point, — {i. e., the curva- ture), remains the same in both curves, and the new point 10 218 CONTACT OF THE SECOND ORDER. is also a point of the second curve ; i. e., the curves have three consecutive points in common ; in this case the curves are said to have a contact of the second order. If f{a) = ^ (a), f («) = ■ {a), /" {a) = f (a), /'" (a) = '" {«), the contact is of the third order, and so on. It is plain that the higher the order of contact, the more nearly do the curves agree ; if every term in (1) is equal to the cor- responding term in (3), then y^ = y^, and the two curves become coincident. 118. The Order of Contact depends on the num- ber of Arbitrary Constants. — In order that a curve may have contact of the n^^^ order with a given curve, it follows from Art. 117 that n + 1 equations must be satisfied. Hence, if the equation to a species of curve contains n + 1 constants, we may by giving suitable values to those con- stants, find the particular curve of the species that has contact of the w^ order with a given curve at a given point. For example, the general equation of the right line has two constants, and hence two conditions can be formed, / {x) = (j) {x) and/' (a;) = ' {x), from which the values of the constants may be determined so as to find the particular right line which has contact of the^rs^ order with a given curve at a given point. I?i general, the right line cannot have contact of a higher order than the first. Contact of the second order requires three conditions, f{x) =

" (x), and hence in order that a curve may have contact of the second order with a given curve, its equation must contain three constants, and so on. The general equation of the circle has three constants; hence, at any point of- a curve a circle may be found which has contact of the second order with the curve at that point ; this circle is called the osculating circle or circle of curvature of that point ; in general, the circle cannot have contact of a higher order than the second. The parabola can have contact of the SADttrs OP cu&vaWrm. )il9 third order, and the ellipse and hyperbola of the fourth. In this discussion we have assumed that the given curve is of such nature as to allow of any order of contact. Of course the order of contact is limited as much by one of the curves as by the other. For example, if the given curve were a right line and the other a circle, the contact could not in general be above the first order, although the circle may have a contact of the second order with curves whose equations have at least three constants. Also, we have used the phrase in general, since exceptions occur at particular points, some of which will be noticed hereafter. 119. To find the radius of curvature of a given curve at a given point, and the co-ordinates of the centre of curvature. Let the equation of the given curve be y^fix), (1) and that of the required circle be {x' — mf + (y' — nf = r^ ; (2) it is required to determine the values of m, n, and r. Since (3) has three arbitrary constants, we may impose three conditions, and determine the values of these con- stants that fulfil them, and the contact will be of the second order (Art. 118). From (3), by differentiating twice, we have, x'-m+(y'-n)^, = Q', (3) If (3) is the circle of curvature at the point {x, y) of (1), we must have, x' = x, y' = y; dy' _ dy cPy' _ i^y dx' ~ dx' dx'^ ~ dy? S^O ttAibtUS OP CVRp-AftRS. Substituting these values in (2), (3), and (4), we have, {x — mf + {y — nY - r^ ; (5) cc-m + {y-n)'^ = (6) Therefore, y — n— ^— (8) dx^ L ,df\dy \ '^dxVdx ,„, ^-'^ = ^ ^^^ By (5), (8), and (9), we have From (9) and (8) we have dx^ L ,df\dy V^dxVdx ,,,, "' = * ^ ^^^) dx^ l + ^-l dx^ 120. Second Method. — Let ds denote an infinitely small element of a curve at a point, and the angle which the tangent at this point makes with the axis of x. Imagine two normals to be drawn at the extremities of this elemen- tary arc, i. e., at two consecutive points of the curve ; these SAbWs OF ctriiVAfrtm^/^l'''''^ ^^1 normals will generally meet at a finite distance. Let r be the distance from the curve to the point of intersection of these consecutive normals. Then the angle included be- tween these consecutive normals is equal to the correspond- ing angle of contingence (Art. 116), i, e., equal to dcjt^ Since d^ is the arc between the two normals at the unit's distance of the point of intersection, we have ds = rd^, or r = ~ (1) Now this value of r evidently represents the radius of the circle, which has the same curvature as that of the given curve at the given point, and hence is the radius of curva- ture for the given point, while the centre of curvature may be defined as the point of intersection of two consecutive normals. To find the value of r, we have (Art. 56a), tan = -^ ; .•. = tan~i ~- ; dx ^ dx' dec ^__^^_^^_^_ and hence dl dx\dzV da?\ "^ da;>/ _ dx- '-^-a -0; ^dllM cPy _ dx\dxV . . d^ ~ ~~7TW ^ dx^ MFFEttENT ORbERS OF CONTACT. ^23 Differentiating (8) of Art. 119, we haye d^y _ dxXdxy , , dx^ ~ -. dy^ Hence the third derivative at a point of maximum or minimum curvature is the same as it is in the circle of curvature, and therefore the contact at this point is of the third order (Art. 117). Cor. — The contact of the osculating circles at the vertices of the conic sections is closer than at other points. 123. Contact of Different Orders. — Let y = f{x) and y ^{x) represent two curves, and let x^ be the ab- scissa of a point of their intersection ; then we have f{x,) =

ix,±h) =

" (x,) i^^' + 0"'(^,)%|-^' + etc. (2) Subtracting (3) from (1), we get, for the difference of their ordinates, corresponding to X: ± h, y,-y,^ [f'{xO-'(X:)] {±h) + [/"(^.) - 0"(^.)] ^^- + [/'"(a^O - r{x:)] ^l^' + etc. (3) Now if these curves have contact of the first order, the first term of (3) reduces to zero (Art. 117). If they have contact of the second order, the first two terms reduce to zero. If they have contact of the thu-d order, the first three terms reduce to zero, and so on. Hence, when the order of contact is odd, the first term of (3) that does not reduce to zero must contain an even power of ± h, and y^ — y^ does not change sign with h, and therefore the curves do not intersect, the one lying entirely above the other ; but when the order of contact is even, the first term of (3) that does not reduce to zero must contain an odd power of ± ^j and Vi — Vi changes sign with Ji, and therefore the curves inter- sect, the one lying alternately above and below the other, CoK. 1. — At a point of inflexion of a curve, the second derivative equals ; also, the second derivative of any point of a right line equals 0. Hence, at a point of inflexion^ a rectilinear tangent to a curve Kas contact of the second order, and therefore intersects the curve. Cob. 2. — Since the circle of curvature has a contact of the second order with a curve, it follows that the circle oj curvature, in general, cuts the curve as well as touches it. CoE. 3. — At the points of maximum and minimum curva- ture, as for example at any of the four vertices of an ellipse, the osculating circle does not cut the curve at its point of contact. EXAMPLES. 1. Find the radius of curvature of an ellipse, a;2 a 2 + Here ^ - - ^- • \ ^^V^ - «V + ^'^ ^^^^ dx- c?y' •• ^ + 5^8-"~^2 llXAMFL10/i \ ^^3 ^ - _ dx _ _ 53(fly + aV) _ _ J\ .-. (Alt 120), T= ^ ^' „/^.. (ay + ¥x^)i , , ,. ,, , , = ^4^,4 — (neglecting the sign). At the extremity of the major axis, ' ^ a At the extremity of the minor axis, a* x = 0, y = b, .: »' = j-- 2. Find the radius of curvature of the common parabola. y^ = 2px. Here ^=^ ^=-t. dx y' dy? y^' {y^ ^p')^ (normal)*' ^8 pi At the vertex, y z^ 0; .•. r ^ p. 3. Find the radius of curvature of the cycloid X =. r vers~i - — ^/^ry — y^. Here ^"^ = ^ - • • l + ^' = ?!!• <^«/ V^r?/ — /' ■ ■ dx"^ y ' which equals twice the normal (Art. 101, Ex. 5). 2^6 HVOtUTES AND tNV6LUTE8. 4. Find the radius of curvature of the parabola whose latus-fectum is 9, at x=. 3, and the co-ordinates of the centre of curvature. r = 16.04; in — 13^, n= — 6.91. 5. Find the radius of curvature of the ellipse whose axes are 8 and 4, at a; = 2, and the co-ordinates of the centre of curvature. r = 5.86 ; m = .38, n=— 3.9. 6. Find the radius of curvature of the logarithmic spiral r = a". dr de «* log a ; (See Ex. 2, Art. 102.) 7. Find the radius of curvature of the spiral of Archi- medes, r = ad. E = (gg + r^)t 2a2 + r^ 8. Find the radius of curvature of the hyperbolic spiral, 124. Evolutes and Involutes.— The curve which is the locus of the centres of all the osculating circles of a given curve, is called the evoluie of that curve ; the latter curve is called the involute of the former. Let P], Pj, Pa, etc., represent a series of consecutive points on the curve MN, and C„ Cs, C3, etc., the corresponding centres of curvature ; then the curve C,, Cj, C3, etc., is the evolute'ot MN", and MN is the involute of Ci, C2, C3, etc. Also, since the lines OiPi, C^Pj, etc., are normals to the involute at the consecutive points, the points Ci, O2, C3, etc., may be regarded as Fig^ 38. EQUATION OF THE EVOLUTE. 227 consecutive points of the evolute ; and since each of the normals PjC,, P2C2, etc., passes through two consecutiTe points on the evolute, they are tangents to it. Let r„ r = !' (^) in which we have only the variable parameter a. •'• -a3 + i- = <'5 .-. «^ = -; (6) which in (5) gives c + c^' •■• '^y = i''- 4. Find the envelope of the right lines whose general equation is y z= mx + («%' + ¥)i, (1) where m is the variable parameter. b X We find m = « a/«* — a;"' 3^ V^ which in (1) gives — ^ + ^ = 1 for the required envelope. Hence the envelope of (1) is an ellipse, as we might have inferred, since (1) is a tangent to an ellipse. (See Anal. Geom., Art. 74.) EXAMPLES. 1. Find the radius of curvature of the logarithmic curve X = log y. __ (^2 + ya)4 ~ my 3. Find the radius of curvature of the cubical parabola ys = a>x. _ (9^4 4. a4)| 3. Find the radius of curvature of the curve y = afl — x' + 1 236 EXAMPLES. where it cuts the axis of y, and also at the point of mini- mum ordinate. At the first pointy r = — i; at the second, r = J. 4. Find the radius of curvature of the curve y^ = 6a;' + afi. ~ — 8x^y 6. Find the radius of curvature of the rectangular hyper- bola xy = m\ _ (a;a + f)^ 6. Find the radius of curvature of the Lemniscate of Bernouilli r^ = o? cos %Q. „ a^ ^ = ¥r 7. Find the equation of the evolute of the ellipse diyi ^ j2^2 — ^,852. (flm)T + {in)^ = (a? — S^)!, 8. Find the equation of the evolute of the hyperbola a^yi — ]yix^ = — cfib^. (am)* - (5w)t = (a^ + 52)!. 9. Prove that, in Fig. 39, OM = 40A — 4p, and MP' = 2^a/2. 10. Find the length of the evolute AP' in Fig. 39. Ans. (3* — l)p. 11. Find the length of the evolute of the ellipse. (See Art. 123, Ex. 1, and Art. 124.) , ,a^ — b^ Ans. 4 5 ab 12. Find the length of the cycloidal arc OO'X, Fig. 40. Ans. 8r. 13. Find the envelope of the series of parabolas whose equation is y^ =zin(x — m), m being the variable parameter. , ^ EXAMPLES. 237 14. Find i,he envelope of the series of parabolas expressed 1 + a^ by tlie equation y ^ ax — a;', where a is the variable pai-ameter. The result is a parabola whose equation is 0^ = 2^,(1 -2/). This is the equation of the curve touched by the parabolas de- scribed by projectiles discharged from a given point with a constant velocity, but at different inclinations to the horizon. The problem was the first of the kind proposed, and was solved by John Bernouilli, but not by any general method. 15. Find the envelope of the hypothenuse of a right- angled triangle of constant area c. c ""y -% 16. One angle of a triangle is fixed in position, find the envelope of the opposite side when the area is constant = c. c 17. Find the envelope of a; cos a + y sin a = p, in which cc is the variable parameter. x^ + y^ z= p^. 18. Find the envelope of the consecutive normals to the parabola y^ = 2px. Q Ans. y^ = ^fT- (x — pY, which is the same as was found for the evolute in Ex. 1, Art. 135, as it clearly should be. (See Art. 124.) 19. Find the envelope of the consecutive normals to the ellipse ay + li^z^ = aW. Ans. {aw)^ + (by)^ = {a^ — h^)^, which is the same as was found in (7) for the evolute of the ellipse. PART II. INTEGRAL CALCULUS. _ »• ■♦ »< CHAPTER I. ELEMENTARY FORMS OF INTEGRATION. 129. Definitions. — The Integral Calculus is the inverse of the Differential Calculus, its object being to find the relations between finite values of variables from given relations between the infinitesimal elements of those vari- ables ; or, it may be defined as the process of finding the function from which any given differential may have been obtained. The function which being differentiated pro- duces the given differential, is called the integral of the differential. The process by which we obtain the integi-al function from its differential is called integration. The primary problem of the Integral Calculus is to effect the summation of a certain infinite series of infinitesimals, and hence the letter S was placed before the differential to show that its sum was to be taken. This was elongated into the symbol / (a long 8), which is the sign of integra- tion, and when placed before a differential, denotes that its integral is to be taken. Thus, C^xHx, which is read, « the integral of ^7?S,x," denotes that the integral of ZxMx is to be taken. The signs of integration and differentiation are ELEMENTARY RULES FOR INTEGRATION. 239 inverse operations, and when placed before a quantity, neutralize each other. Thus, / d {ax) = ax, and d I axdx = axdx. 130. Elementary Rules for Integration. — In the ele- mentary forms of integration, the rules and methods are obtained by reversing the corresponding rules lor differ- entiation. When a differential is given for integration, if we cannot see hy inspection what function, being differ- entiated, produces it, or if it cannot be integrated hy known rules, we proceed to transform the differential into an equivalent expression of known form, whose integral we can see iy inspection, or can obtain by known rules. In every case, a sufficient reason that one function is the integral of another is that tJie former, being differentiated, gives the latter.* (i.) Since d{v + y — z) = dv + dy — dz; (Art. 14.) / (dv + dy — dz) ^ d{v + y~z)^v + y — z = / dv + dy — I dz. Hence, the integral of the algebraic sum of any number of differentials is equal to the algebraic sum of their integrals. (£) Since d {ax ± 5) = adx ; (Art. 15.) * While there is no quantity whose differential cannot he found, there ie a large claps of differentials whose integrals cannot be obtained ; either because there is no quantity which, being differentiated, will give them, or because fhe inetbQ4 fpj Qjexr integrfttioj} has pot yet heeii fotind, 240 ELEMENTARY RULES FOR INTEGRATION. / adx = d {ax + h) ^ ax ±b = a I dx ±, b. Hence, a constant factor can be moved from one side of the integral sign to the other without affect- ing the value of the integral. Also, since constant terms, connected by the sign ±, disappear in differentia- tion, therefore in returning from the differential to the integral, an arbitrary constant, as 0, must he added, whose value must be determined afterwards by the data of the problem, as will be explained hereafter. (3.) Since d - [/(a:)]" = a [/(a;)]"-i df{x) ; ' (Arts. 15 and 19.) 11/ .: fa U{^)Y-' d/(x) = fd^ [f{x)y = |[/(^)]" + c. Hence, whenever a differential is the product of three factors, viz, a constant factor, a variable factor with any exponent except — 1, and a, differential factor which is the differential of the variable factor without its exponent, its integral is the product of the constant factor by the variable factor zvith its exponent increased by 1, divided by the new ex- ponent.* It will be seen that the rule fails when w = — 1, since if we divide by J — 1 = 0, the result will be qo . (4.) Since d (a log x) = ; (Art. 20, Cor.) •'■ J-^^ J ^{alogx) =0,\ogx. * The arbitrary constant is not mentioned since its addition is always under- stood, and in the following integrals it will be omitted, as it can always be supplied when necessary. EXAMPLES. 241 Hence, whenever a differential is a fraction whose numerator is the product of a constant by the differ- ential of the denominator, its integral is the product of the constant by the Jfaperian logarithm of the denominator. EXAMPLES. 1. To integi'ate dy = ax^dx. y = J axHx =: J a- a? ■ dx.= -^- [by (5)]. 2. To integrate dy = {a + b3?YxHx^ The differential of the quantity within the parenthesis being IbxHx, we write y = /t^ (« + ^^Y ^^^'dx = ^^. [by {S)l This example might also be integrated by expanding the quantity within the parenthesis, and integrating each term separately by (1), but the process would be more lengthy than the one employed. 3. To integrate dy ^ a {ax' + ix^)^ 2xdx + %hx^ {ax^ + boi?)^ dx. y = r[a {ax' + bx^)^ 2xdx + 3bx^ {ax^ + bx^)i dx] = r{ax^ + bx^)i {2ax + Sba?) dx = | {aa? + bifi)i [by {S)]. 4. To integrate dy = j-— Since the numerator must be bdx to be the differential of the denominator, we must multiply it by b, taking care to divide by b also ; hence, f adx a C bdx « , , , , x ri, / m 11 242 DIFFERENTIAL FORMS. 131. Fundamental Forms.— On referring to the forms of differentials established in Chap. II, we may write down at once the following integrals from inspection^ the ariitrary constant being always understood. 1. 2. 3. y = J ax^dx _ Padx y ~ J Off" _ radx y ~ J X ~ n+1 a ~ (w-^a;"-! = a log X.* 4. y = J a'^loga dx = ax. 5. y = J (^dx = (F. 6. y ■= 1 cos X dx = sin X. 7. «/ = / — smx dx = cos X. 8. y = J sec^a; dx = tan X. 9. y = 1 — cosec' x dx = cot X. 10. y z= / see a; tan x dx = sec X. 11. y =z 1 — cosec x cotan x dx = cosec X. 12. y = J sinxdx = vers X. 13. y = J —cosx dx = covers x. * Since the constant c to be added is arbitrary, log c is arbitrary, and we may write the integral in the form / a — = o log a; + log c = log c}f. INTEGRAttON S¥ TMAmPORMATtON. 243 dx ^/1 — x^ dx = sin~i X. = cos~i X. 14. y = f. 15. y-f- 17. y = f-^^ = cot-la;. 18. y — I — , = sec-i a;. a;-v/a;' — 1 dx 19. 2/ = / 21. 2, = / a;-\/2?^ — 1 V3a; — a;8 (?a; V^^ic — ^ = cosec-i a;. = Ters""^ X. = coTers"* X. These integrals are called the fundamental or elementary forms, to which all other forms, that admit of integration in a finite number of terms, can be ultimately reduced. It is in this algebraic reduction that the chief difiBculty of the Integral Calculus is found ; and the processes of the whole subject are little else than a succession of transformations and artifices by which this reduction may be effected. The student must commit these fundamental forms to memory; ihey are as essential in integration as the multiplication table is in arithmetic. 132. Integration of other Circular and Trigono- metric Functions by Transformation into the Ftm- damental Forms. 1. To integrate dy = — - • V a' — V^x^ We see that this has the general form of the differential 344 tntEGRAtlOn BY TRAN^PdiiMATlO^. of an arc in terms of its sine (see foi'm 14 of Art. 131) ; hence we transform our expression into this form, as follows : ^ r & ^ r dx r a To make this quantity the differential of an arc in terms of its sine, the numerator must be the differential of the square root of the second term in the denominator, which is -dx. Therefore we need to multiply the numerator by h, which can be done by multiplying also by the reciprocal of b, or putting the reciprocal of h outside the sign of integra- tion. Hence, y = r j^ _ = /"—l^ = 1 r '^^"^ '^ 's/a^ — Wx* '^ I V^x^ bJ / j2^ V-? \^ a. To integrate dy = ^^ ■\/«3 — b^x^ Here y ^ I ^ - = / 1 . ,bx b a dx AS -dx . , a 1 , bx V'-? y COS"' J2a;3 b a f. dx — Jl f. -1 ^^ a!> + b^x^ ~ ab a" 4. «= /'__^_-lcot-i^. ^ H J a3 + yix^ - ab ^°* « PROcms 6f IMTlEGttATlOtf. M5 _ r dx \ .Ix 5. w = / — , = - sec~i — •^ x\/¥x^ — a^ « <* „ n dx 1 ,hx «^ xV¥x^ — a^ « « „ C dx \ .tx •^ 'V^abx — V>x^ 5 « o /* dx 1 , Ja; 8. V = / , = V coTers"^ — •^ V^oSa; — js^a;' * « P, , /"gin a; VI — a;* '^ a;^(Za; 47. (Zy = V3 — 4a;3 Here y = J ■ x^dx a/3 • Vl — 3a)3 V3 ■ |a;'c?a; ^3 • f '^ ^3 • Vl — 3a;3 _ /" — -!- ' • y — %x Va^ + «*. i 66. (?^ = tan^ x sec' a; + 1 dx 1 + X + x^ r dx I + (a; + i)' vr V3- 1 + [<-«^J a/3 ' ^Vs ^^- ^^ = 2-tx + x^ - 2' = ta^-M«=-l). 72.^^2^-M±M^. ^ a^ ^ a;2 m , \,x , n y = - tan-i - + ^ log («' + «^)- J854 THIGONOMETRIV REDUVTIOX. 135. Trigonometric Reduction. — A very slight ac- quaintance with Trigonometry will enable the student to solve the following examples easily. After a simple trigonometric reduction, the integrals are written out by inspection. 1. dy = tan* xdx. Here ^ = / tan^ xdx = / (sec' x — V) tan xdx = / [sec' X tan xdx — tan xdx^ = \ tan« X — log sec x. [See (9) of Art. 132.] 3. dy = tan* xdx. y — \ tanS ar — tan a; + a;. [(15) of Art. 133.] 3. dy = tan^ xdx. y = i tan* x — \ tan' a; + log sec x. 4. ^y = cot^ xdx. y — —\ cot' a; — log sin x. 5. dy = cot* xdx. y = — ^ cot^ a; + cot a; + a;. 6. dy = cot^ xdx. y = — |- cot* X + ^ oof a; + log sin x. 7. or = n, (1) may, by common division, be reduced to the sum of an integral algebraic expression, and & fraction whose denominator will be the same as that of (1) and whose numerator will be at least one degree lower than the denominator. For example, — 3? + X- ofi — x^ ■\- X ■\-\ a^ — x^ + X + 1 The former part can be integrated by the method of the preceding chapter; the fractional part may be integrated by decomposing it into a series of partial fractions, each of which can be integrated separately. There are three cases, which will be examined separately. 137. Case I. — TVhen the denominator can he re- solved into n real and unequal factors of the first degree. f (oc\ dx For brevity, let ^ \ . denote the rational fraction {x) whose integral is required, and let (x—a) {x—l) . . . (x—l) be the n unequal factors of the denominator. Assume DECOMPOSITION OF FRACTIONS. 357 1^1 = ^- + -!- + ^- ..-li- (1) (/) (a;) X — a . X. — b x — c ' ' ' x — V ^ ' where A, B, C, etc., are constants whose values are to be determined. Clearing (1) of fractions, by multiplying each numerator by all the denominators except its own, we have f{x) = A (x~I}){x- c). . . {x—l)+B{x—a)(x—c).. . (x—l) + etc. + L (x—a) (x—i) . . . (x—k), (2) which is an identical equation of the (w— 1)'* degree. To find A, B, C, etc., we may perform the operations indicated in (2), equate the coeflS.cients of the like powers of x by the principle of indeterminate coefficients in Algebra, and solve the n resulting equations. The values of ^, B, C, etc., thus determined, being substituted in (1) and the factor dx intro- duced, each term may be easily integrated by known methods. In practice, however, in this first case, there is a simpler method of finding the values of ^, B, etc., depending upon the fact that (3) is true for every value of x. If in (3) we make x^=a, all the terms in the second member will .re- duce to 0, except the first, and we shall have f{a) = A{a — i)(a —~c) . . . (a — T), . Aa) _ f{a)_ " {a-b)(a-c) ...{a-l)~ '{a)' In the same way, making x=.h, all the terms of (3) disappear except the second, giving us /(5) = ^(5-«)(6-c)...(5-0, B- /w _ m_ {I)'-a)ib-c) ...{b-l)~ '{b) 258 CASE I. Or, in genet-al, the value of L is determined in any one of the terms, =, by substituting for a; the corresponding fix) f(T\ root I of d> (x) in the expression zrrr^; i. e., L = zttjx- EXAMPLES. ^ ^ . , , (a;^ + 1) dx 1. Integrate dy = ^^^e^^n^ + e ' In this example, the roots * of the denominator are found by Algebra to be — 1, — 3, — 3. .-. iB» + 62e2 + lla; + 6 = (a; + 1) (re + 2) (a; + 3). Assume a;S+6a^ + lla; + 6 a; + l^a; + 3 x + 3 .: x^ + 1 = Aipo + %) {x + Z) + B [x -^ 1) [x + 3) + G{x + \){x + %). Making x = —1, we have 2 = %A, :. ^ = 1. « x= —%, " " 5 = —B, :. B = —b. " x= —3, « « 10 = 2C, .-. = 5. Substituting these values of A, B, C, in (1), and multi- plying by dx, we have y ~ J a^ + 6x^+nx + 6 /dx r_^_ I K C ^^ ^-+1 - V ^Ta "^ V a; + 3' .-. y = log (a; + 1) - 5 log (a; + 2) + 5 log(a; + 3) - io^ (a: + 1) {X + 3)^ ~ ^^ (a; + 2)« • If the factors ot the denominator are not easily seen, put it eqnal to 0, an^ solve the equation for x ; the first root may be found by trial, x minus each of the Beveral roots In turn will be the factors. (See A.Igebra.) 2. Integi'ate dy = DECOMPOSITION OF FRACTIONS. 259 adx X' — w , , X — a ., Ix — a 3. Integrate dy = ^f^^- ATI i 7 (5X'+ l)dX 4. Integrate dy = ^-^■ y = log {x — ly {x + 2)'. 5. Integrate dy = ^^-,. 2/ = ^ log (|^). 138. Case II. — When the denominator can be re- solved into n real and equal factors of the first degree. f(x) Let the denominator of the rational fraction '^-)-4 con- tain n factors, each equal to a; — a. v* (. ; Assume m = _A_ 4. :?__ , ^_ (x) {x — a)" "'" {x — «)"-> "^ {x — af-"^ Clearing (1) of fractions by multiplying each term by the least common multiple of the denominators, we have f(x) = A + B{x-a) + C(x- of + . . . i {x-aY-\ (2) which is an identical equation of the (n — 1)'* degree. To find the values of A, B, C, etc., we equate the coefficients of the like powers of x, as in the preceding Article, and solve the n resulting equations. The values of A, B, C, 260 CASE II. etc., thus determined, being substituted in (1), and the factor dx introduced, each term may be easily integrated- by known methods. In this case we cannot find the values of A, B, C, etc., by the second method used in Case I, but have to employ the first. When both equal and unequal factors, however, occur in the denominator, both methods may be combined to advantage. BXAMPLES. 1. Integrate dy = ^-(^-Yy— 2 32;3 ^ BO .-. % — Za? = A + Bx + 2B+Cx^+4:Gx + 4:G. .: A + %B + 4.C =%. (2) B + W — 0. (3) C = - 3. (4) Solving (3), (3), and (4), we get A = —10, B = 12, G = —3. Substituting these values of A, B, and G in (1), and multiplying by dx, we have (2 — 3x^ dx _ _ ■10(?ig 12& _ Zdx {x + 2)3 ~ {x+ 2)8 "•" (a; + 2)2 x + % "■ y~J (a; + 2)8 K 19 8. Integrate c?^^ = ^^+^^. DECOMPOSITION OF FRACTIONS, %Q1 Assame (x — 2f{x — l) (x — 'iif^{x-^)^x-l ^^' .: x^ + x = A{x — 1) + B(x—2) (x — l) + G{x- 2)2. (2) Here we may use the second method of Case I, as follows : Making x = 2, we find A := 6. X = 1, " " G = 2. Substituting in (3) for A and. G their values, and making r = 0, we find = —6 + 2B + 8; .: B = —1. Substituting in (1), and multiplying by dx, we have _ P {x^ + x) dx y ~ J (^^^2)2 {x — 1) _ P 6dx P dx p 2dx ~ J (x — 2Y ~ J x — 2^ J x — 1 6 {x-2f — log {x — 2) + 3 log (x ^ 1). x — 2 6 , . {x-lf • • n - x — 2'^{x-2) - _ , , , (3x — l)dx 3. Integrate dy = , _ ',^ — y=--^ + 3\ogix-S). A T J. ^ J x^ — Ax + 3, L Integrate dy = ^—^-^-^-dx. y = log [x {X - 3f]i. 263 CASE HI. 139. Case III. — When some of the simple factors of the denominator are imaginary. The methods given in Arts. 137 and 138 apply to the case of imaginary, as well as to real factors ; but as the cor- responding partial fractions appear in this case under an imaginaiy form, it is desirable to give an investigation in which the coefficients are all real. Since the denominator is real, if it contains imaginary factors, they must enter in pairs ; that is, for every factor of the form x ±,a + bV— 1, there must be another factor of the form x ±a — bV— 1,* otherwise the product of the factors would not be real. Every pair of conjugate imaginary factors of this form gives a real quadratic factor of the form (x ± a)^ + b'. Let the denominator contain re real and equal quadratic factors. Assume f{x) _ Ax + B Cx + D

i resulting equa- tions, we find the values of A, B, G, etc. Substituting these values in (1), and introducing the factor dx, we have a series of partial fractions, the general form of each being {Ax + B) dx [{x±aY + V^Y' in which n is an integer. To integrate this expression, put x±a = z; .-. x-= z^a, * CaWei conjugate Imaginary factors. EXAMPLES. 263 dx = dz, (x ± aY — z\ Substituting these values, we have, J [0 {Ax + B) dx _ r {A% ^: Aa + B) dz — f ^^^ ^ 4. r{B_^Aa)dz ~ 2 (W - 1) (22 + &2)»-l "^ t/ (2« + b^' (wheu ^' = J5 T Ja) ; so that the proposed integral is found to depend on the integral of this last expression ; and it will be shown in Art. 151 that this integral may be made to depend finally upon y ^TjTp' S^^^g J ^^'^"^ V '^^'^^ ^''^^' ^'^ EXAMPLES. 1. Integrate dy = „ _ .. • The factors of the denominator are {x — 1) and [x^ + X + 1). therefore assume, a; A Bx+ O "f" ^2 _L ^ _L 1 ' \^) a? — 1 X — 1 x^ + X + 1 .: x=z Ax^ + Ax + A + Bx^ + Cx — Bx - C. .: A + B = 0; A- B + C =1; A — C = 0. .: A = \; B=-i; = i- _ r xdx _ r^ dx r^ {x — l) dx •■• y - J gfiZTl- J ^^-l J^x^ + X + 1 364 EXAMPLES. = ^log(._l)-i/J^)^^; (2) (by changing the form of the denominator,) Put a; + ^ = 2, then a; — 1 = z — |, and dx = dz, and the second term of (2) becomes = - i log {z^ + i) + -^ tan-1 ^ (Art. 133, 3.) V3 v3 = _^log(.. + .4-l)H--^tan-^\ (by restoring the Talue of z). Substituting in (3), we have, y = i [log {x-l)-i log {xi + x + 1) + V3tan-»^±i1. 3. Integrate dy = V3 xMx X* + x^ — 2 To find the factors of the denominator, put it = and solve with respect to ay' ; thug, X* + x^ — 2 = 0, or ic* + a^a = 3. .-. a^ = — J ± f = 1 or — 3. .-. x* + x^ — 2 = {ay'—l) (a;^ + 3), A '^^ A , B Cx + D ,:,. Assume ^j_j-^,_^ = -^^ + -_-^ + _-i_-. (1) Hence a:^ = ^ (a; - 1) {x^ + 3) + 5 (a; + 1) {x^ + 2) + {Gx ^D){x- 1) ix + 1). (2) EXAMPLES. 265 We may equate the coefficients of the hke powers of x, to find the values of A, B, C, D, or proceed as follows : Making « = — 1, we find J. = — ^. " x = l, " " B — i. Substituting these values of A and B in (2) and equating the coefficients of a? and a^, we have 6C=0 and 6Z> = 4; .-. C = and i> = |. dx •■■* = -*/ifi + */i^i + »/5 + 1 ' '^t/x — l ' * + 2x — 2 , (« — 1)^ la; V = log -5^ '-r — tan 1 — — • (a;« + 2)^ 3\/2 V^ 18. Jy= (x^ + x)dx_ , , _, . (x - 2)t ^ = I tan 1 a; + log ■ - (a^s + 1)1^ 19. % = ?-! + ^>^ - ^ a;3 + a;2 + a; + 1 , (.r + l)t _ - , 2/ = log - \^ /, - i tan-i as. (a;3 + 1)T „. , 9a;» + 9a; — 128 , 80. dy = -^ -^ — — -r dx. , (a; -3)" , 5 - 2a;<^?a; 21. dy = {a? + 1) (a^ + 3) , (x^ + l\i y = ^°^ ix^ • 268 SJCAMPLSS. oo J (*' — 1) ^^ 23. dy = 3/ = I + log [(a; + 2)1 (a; - 2)1]. xdx [x + l){x + 2) {a? + !)■ , r(a; + 3)i (a^ + 1)^b~I CHAPTER III. INTEGRATION OF IRRATIONAL FUNCTIONS BY RATIONALIZATION, 140. Rationalization. — When an irrational function, which does not belong to one of the known elementary forms, is to be integrated, we endeavor to rationalize it ; that is, to transform it into an equivalent rational function of another variable, by suitable substitutions, and integrate the resulting functions by known methods. 141. Function containing only Monomial Surds. — When the function contains only monomial surds, it can be rationalized by substituting a new variable with an exponent equal to the least common multiple of all the denominators of the fractional exponents in the given function. : For example, let the expression be of the form, f }dx. m' Put X = fS^MX.^ m' nf c' e' dx = mnce!^'''^-^dz. Hence dy = ^,^^^ ^ ^,^^^, mnce^^-^dz ; which is evidently rational. 270 FtJNCTlOm OOlf'tAiNtN& BtNdMtAL SUBM. 1. Integrate dy = j dx. (1) 1 — a;« Put a; = 2*; then a;^ = z*j a;^ = z^, and ; whence. aZ^ + P ^ - z^+1' /ct-^a _|_. /^ \ and Va + bx-x^ -[^^_^^ ^Jz _ (l3-ec)z z^ + 1 274 s^amPlJss. The values of x, dx, Va + bx — x^, being expressed in rational terms of z, the transformed function will be rational. 1. Integrate dy ^/a + bx + a? Assume Va + bx + x^ ^ z — z; then, as in Case I, we have a + bx — z^ — 2zx: .-. x — ^ ~ ^ .. _ 2 {z^ + bz + a) dz ax _ ^-p^^p / — ;— r — ; — 5 z^ ^bz ■\- a . fh, - 2 (jg^ + &g + a) t^g X (2g + 5) •• "2/- (J + 22)2 j< (2» + 5« + a) 2c?2 rfz 6 + 2« 5 2 + ^ = log If J = 0, we have dx ^ + a; + Vfl + te + a^ . /-^ = log(a;+V^T^); and if a = 1, we have dx y yf^p = iog(«' + vrT^). EXAMPLES. 275 Had we integrated the expression - — ^ without dividing both terms of it by 3, we would have found for the intfegral the following : S' = log(6 + 2z) = log[6 + 3a; + 2VaTte + rc^], which differs from the above integral only by the tei-m, log 2, which is a constant. (See Note to Art. 135.) 2. Integrate dy = ' V « -\- bx — x^ Let a and (i be the roots oi x^ —ix — a^O; then, as in Case II, we have Va + bx —x' = 's/Jx — a) (f3 — x) — (x — a) z. ««» + (3 (i8 — a;) = {x — a)7?; 2« + 1 7 _ 2 (« - S) zdz {z^ + 1) _ 2dz {z^ + ly (13 - a) z ~ 1 + z^' . y = / ^^ ^ _ 2 /__^_ "^ Va + bx-x^ '^ l+z^ = —2 tan-i 2 = — 3 tan-i \ /^-. V a; — « 3. Integrate dy = - — a;Vl + a; + a;2 Assume Vl + aj + a;^ = z—x, and we have, as in Case I, ^-1 + 2^' ''''- (l + 2ip 1 +%z . rh, _ a (^ + g + 1) t?^ (1 + 2g ) (1 + 2^) 2f7z (1 + 2zf (f + « + 1) (^^^^ - 22 -1 ,-^-[-~l(Art.m). 276 BINOMIAL DIFFERENTIALS. /dx _ /■ ^^ /• dz , Z — 1 , X — 1 + V^ + X + x> = log -T = log r 6^ + 1 ^a: + l + Vl + a;+^ 3x = log; 3 + a; + 2^1 + a; + a;« 145. Binomial Differentials.— Expressions of the form dy = X'" {a + bx^y dx, in which m, n, f denote any numbers, positive, negative, or fractional, are called binomial differentials. This expression can always be reduced to another, in which m and n are integers and n positive. 1st. For if m and n are fractional, and the binomial of the form ar*(a + hx^)" dx, we may substitute for x another variable whose exponent is equal to the least common multiple of the denominators of the exponents of x, as in Art. 141. We shall then have an expression in which the exponents are whole numbers. Thus, if we put x = ifi, we have ar^(a + Ix^J" dx — &z-^[a + Wydz, in which the exponents of z are whole numbers, and the exponent of z within the parenthesis is positive. 3d. If n be negative, or the binomial of the form a;"* (a + lyr'^y dx, we may put a; = - , and obtain x'"- {a + hTT^'Y dx = — 2r»'-2 {a + Iz^ydz, in which the exponents of z are whole numbers, and the one within the parenthesis is positive, CONDITIONS FOR RATIONALIZATION. 277 3d. If X be in both terms, or the binomial is of the form a;"* (ax* + baf'Y dx, we may take «* out of the parenthesis, and we shall have ^m+pt ^g ^ Ix^-'Y dx, in which only one of the terms within the parenthesis con- tains the variable. 146. The Conditions under which the General Form p dy = ar{a + baf'Ydx, can be rationalized, any or all of the exponents being frac- tional. (i.) Assume a + 5a;" = zf. p Then (a + Ix'^Y = zK (1) Also X = {^-rJ^) , and »"• = (^-^y- (a). Multiplying (1), (3), and (3) together, we have m+l di/ =zcii^(a + bx'')^dx = -^ ^p+s-' \^^^) " dz, (4) an expression which is rational when is an integer, or 0. (2.) Assume a + boif — ziaf'. -•■ Then a;" = a (2» — J)-i. (1) 378 CONDITIONS OF INTEGBABILITY. .: X — a^ {!fl — b)--K (3) VI m .: or = a" {ifl — h)~^, (3) dx = —^- ah {!fl — l)-^-^ zf-^ dz. (4) Multiplying (1) by b, adding a, and taking ^ power, we have ^ p p _]^ {a + Ix^Y = a^{z^ — b) ''^. (5) Multiplying (3), (4), and (5) together, we have «"(« + byffdx = — ^a^" « "'^(«' — b)^" « ^z'+^-^dz, 7)t -I- 1 7) an expression which is rational when — — h - is an in- teger, or 0.* ^ Therefore there are two cases in which the general bino- mial differential can be rationalized : 1st. When the exponent of the variahle without the -pa,renthesis increased by unity, is exactly divisible by the exponent of the variable ivithin the parenthesis, 2d. mien the fraction thus formed, increased by the exponent of the parenthesis, is an integer. Rem. — These two cases are called the conditions of integrahUity of binomial differentials.f and when either of them is fulfilled, the inte- gration tnay be effected. If, in the former case, 1 is a positive integer or 0, or in the latter case, h - + 1 is a negative integer * The student will observe that Art. 143 is a particular case of this Article, ve- snlting from malting m an odd positive integer, and re = 2. t These are the only cases of .the general form which, in the present state of analysis, can be made rational. When neither of these conditions is satisfied, the expression, if - be a Pactional index, is, in general, incapable of integration in a finite nqm^er of terms, EXAMPLES. 279 or 0, the binomial (z' — a) or (z? — 5) will have a positive integral ex- ponent, and hence can be expanded by the Binomial Theorem, and each term integrated separately. But if, in the former case, 1 IS a negative integer, or in the latter, — '- — I- - + 1 is a positive inte- ger, the exponent of the binomial (z' — 6) will be negative, and the form will be reduced to a rational fraction whose denominator is a binomial, and hence the integration may be performed by means of Chapter II. But as the integration by this method usually gives com- plicated results, it is expedient generally not to rationalize in such cases, but to integrate by the reduction formulm given in the next Chapter. 1. Integrate dy =: x^{a + t?)^ dx. Here 1 = 3, a positive integer, and therefore it can be integrated by the first method. Let {a -f V) = z\ Then (a + x^)i = z. (1) afi =: (2^ — a)% a^dx = li^ — ayz^dz. (3) Multiplying (1) and (3) together, we have dy := aP (a + rc')^ dx = f {z^ — d^^dz. 3. Integrate dy = 1 = ar-«(l + ^Y^dx. a;* (1 + x^y Here^i-f2-fl = ^i-|-fl = -l,aneg- n q 3 3 ative integer, and hence it can be integrated by the second method. 380' EXAMPLES. Let (1 + x^) = z^x\ ■ Then a^ = {f — l)-\ x= (22-l)-i; X-* = {z^ - If. (1) {]. + x^) = l + (z^-l)-\ (1 + x^)-^ = a-i (2^ - l)i (3) dx = — (;^2-l)-*2^«. (3) Multiplying (1), (2), (3) together, we have dy = a;-* (1 + x^)-^ dx = — {f — 1) dz. .-. 2/ = — y (z3 _ 1) 6?2 = 2 _ ^23 _ (1 + a;S)^ ^(1 + a;3)t _ (1 + a:^)*, 32;s ■ (3a;8 - 1). EXAMPLES. (2a;i — 3a;^) + 2 (1 + a:2)t o^dx (1 + a;8)t" 2^= - 3 (1 + a;")' ■v/l + a?' 282 EXAMPLES. 12. dy = ^^ (Art. 144.) ■s/l + X + x> y = log (l + 2a; + 2\/l + x + a?). (See Art. 144, Ex. 1.) 13. dy = Vx^ — x—1 y = log (2a; — 1 + 2'\/a;2 — x — 1). 14. e?tf = ,— - =^- y = —2 tan-U /— ^ 15. dy = — X 2' Vl +X — '. 2, = -2 tan- V^^^-^^^ 16. dy = x-i- i-v/5 dx (Assume \/^ + x' = z — x, etc. I y = V log (bx + VoM^lVi). xVa^ + W ^ ~ a °^ \ ix ^) 18. dy=.^^^- , 4 tf = log (a; + 1 + V2a; + x^) , • a &\ -r -r -r / a; + V2a; + a;« 17. „ ^^ mS (^ + 1) ./ ^ ^ ^ _^M^+1 /'a,»(^ + 5a;»)'^a;. Transposing the last term to the first member and redu- cing, we have SFBCIAL FORMVLyE OF REDUCTION. 387 ,np + m. + 1\ f _ ^"'-"+'(« + i^ ^ri m. — n + 1 nb {p + 1) Therefore we have / x'^-" {a + bxfy dx. ij = I x'^{a + bx^y dx a;"'-"+'(a + 6a;")^*-' — (wi— ?i+ l)ft Cx'^-^ia + bx^ydx b {np + m + 1) luMch is the formula required. ;(^) 149. To find a formula for increasing the exponent of X vrithout the parenthesis by the exponent of x within, in the general binomial form y — jxr'*^ (a + hdC^'Y dx. Clearing (^) of fractions, transposing the first member to the second, and the last term of the second to the first, and dividing by {in — n -\-\) a, we have /'a;'"-" (« + br-y dx ^m-«+i (a ^ 5a;«)?-+i _ 5 (^^ + „ J + 1 ) /V' [a + b^ydx a {in — M + 1) (1) Writing — vi for m — n, and therefore — m + w for m, (1) becomes y = I «"'" {a + bx^'Y dx a;-"'+'(« + fa")P+' + 5(??i — np — n—1) /"a;-"'+"(a + bogydx = _______ ;(^) ?ir/?/c/« «*' the formula required. 388 SPECIAL FORMULA OF REDUCTION. 150. To find a formula for diminishing the expo- nent of the parenthesis by 1, in the general bino- mial form y — I ge^{a + bx^)v dx. fx'"- {a + bai^)p dx = J Making m = m, a ^ a% 5 = — 1, ra = 3, ^ = — ^, we have from formula (A), y = I X'"- (a^ — x^)~i dx = - [2 (- i) + m + 11 m (m — 1) a^ Cr^-^ {a^ — x^)~^ dx + - ^ (1) in ^ ' When m = 2, (1) becomes P x^dx X . . ,. 1 , a^ . _, a; When m = 3, (1) becomes ^ VO? — a;^ When ??? = 6, (1) becomes, by applying {A) twice in succession, XIXAMPLSS. 2S1 = /: ^ ^ sill" 4 • 4 (which the student may show.) 2. Integrate dy = Va> + x^ Here «/ = / a;"' (a^ + a^)-s t?a;. Making wi = m, a ^ c?, 5 = 1, ^^ = 2, ^ = — \, we have foom (^), y = I x"' (ff3 + a;2)-s dx a:™-' (a3 + a;2)4 (w - 1 ) a^ ^ = ^^ -^^ — — / 2! " M a^ + x^) * «a;. (1) By continued applications of this formula, the integral will finally depend on / —p = log {x + Va^ + «'), when m is eyen, "^ V «2 + x^ /ccdx 1 — — (fl^ + a;^)^', when m is odd. V a^ + x^ 3. Integrate «Zy = j- a;™ (fl!^ — a;^)' Here y = / a;-"" (a^ _ a;3)-i flfa;^ from which we see that by applying (B) we may increase m by 2, and by continued applications of (B), we may reduce m to or 1, according as it is even or odd, making the integral finally depend on a known form. 202 APPLtCATiOM OF PORMVLM Making m = m, a = a% b — — 1, n = 2, p = — '^, [B) gives us y = J a;-'" {a^ — x^)~^ dx a,-mfi (^2 _ a;2ji _ (to _(. 1 _ 2 _ 1) I'x-^+Xa^—x'y'^ doc ~ — a^ (vi — 1) _ _ {a> - a:^)^ (wz - 2) /' ^ When m = 2, (1) becomes r dx (a2 _ a;2)i . (since the last term disappears.) When m = 3, (1) becomes ^ f d^ ^ _ (C^_-Xj^ 4- A /* '^^ ^ ^ ^3 ./„2 _ o-a 2ffiV + 'la^J ; x^'s/a^ — x^ 2ffiV 2fl2./ 2;Va- — a^ _ Vft^ — ic' 1 , a — 'Ja^ -~ +57-3 log 2a2a;2 '^ ^a^ ^ a; (Ex. ] 7 of Art. 146.) 4. Integrate (?«/ = («^ — a=^)^ ^2;, when n is odd. Here we see that by applying (C) we may diminish n ^ by 1, and by continued applications of (C) we can reduce ^ to — \, making the integral depend finally upon a known form. Making m = 0,a = a^,h= -\,n = %,p = \, (0) gives us y = J {a^ — «2)^ dx X (a^ — a?f + na^ f\a^ - x^)^"' dx = ^U (1) APPLtcAtidm OP PoMMitlJS. 5J93 Wlien w = 1, (1) becomes /"/ 2 2\4 J ^ («^ — ^^)* , «^ ■ 1 a^ dx 5. Integrate cZe/ = :^, when w is odd. Here ^ = I (a? — x^)-^ dx, from which we see that by applying {D) we may increase the exponent ^ by 1, and by continued applications of iD) A/ we can reduce - to — ^, making the integral depend finally on a known form. Makii gives us 71 Making m = 0, a ^= a% i = — 1, n = 2, p := -, {D) y ■= J (o? — x^) ' dx ■ («» — a;2)~^"^' -{3 — n)J' {a^ - x'^)'^*' dx 2a2 (i-O + !^^1_ r ^^_. a) " (» - 2) «2 (a« - x^)i-' (w — 2)a^J (^2 _ ^^- When w = 3, (1) becomes _ r dx _ a; '^ ~ J (a2 _ a;2)f ~ fl2 (a2 - a;^)*' 6. Integrate dy = —7= \/3aa; — a? Here y = J x"' {2ax — x^)~i dx = J x"'-i(2a—x)~idx, which may be reduced by (^) to a known form. 204 APPLICATIONS OF •fORMUhJBl. Making m = ?w — |j a = 2a, 5 = — 1, w = 1, ^ =• —\, (A) gives us V'2aa; — x^ x'^h {2a — x)i — 2a (m — i) f'x'^-i {2a — x)-i dx — m = 'V2ax — x^ + ^ :— - / — (1) m m «/ ^2ax - x^ When m = 2, (1) becomes /x'dx X + 3a /s » — ;== = s — V2aa; — x^ V2ax — x» 2 I O (1 f ^^ |«^/. A/3aa; — x^ X + 3a /^ :, „ , , a; — s — V2ax — x^ + ia' Ters~i -• 2 ' ^ a x^dx 7. Integrate dy Vl — x> /a;5 1 ■ 5 „ 1 ■ 3 • 5 \ ,- 1-3.5. , 8. Integrate dy = - — — a;* V « + ^a?* 9. Integrate t/y = (1 — a;^)^ dx. y z= ^x{l — x^)^ + fa; (1 — x^)i + | sin"* x. dx 10. Integrate dy = (1 + x^f y = r(TT^^ + !• (TT^J + I tan- a;. LOGARtTJSMiC FUlfO&ONS. ^95 11. Integrate dy = — • V^ax - x^ V = - d' + 4| « + i • f«') V2^^^ + I • ffl8Ters-i|. dec 13. Integrate dy = • 1 • 3 \ /:; -„ 1 • 3 , 1 + Vn^^ / 1 , 1 • 3 \ /- -„ 1 • 3 , X These integrals migM be determined by one or other of the methods of Chapter III, but the process of integration by reduction leads to a result more convenient and better suited in most cases for finding the definite integrals.* LOGARITHMIC FUNCTIONS. 152. Reduction of the Form / X (log x)"' dx, in which X is an Algebraic Function of x. Put Xdx = dv and log" a; = m, .•. V =^ / Xdx and du =^ n log""* x — Substituting in / udv =. uv — I vdu, (Art. 147) we have y =: J X log" xdx = log" X J' Xdx - J^n lqg"-i x —f{Xdx) ; or by making / [Xdx) = X, , we have y ■= J X log" xdx — log"-' xdx ; * For a cl'scnSBion of definite integrals, see Chap, Y. 206 sxAMPzes. which diminishes the exponent of log x by 1, whereyer it is possible to integrate the form / Xdx. By continued applications of this formula, when w is a positive intege?, we can reduce w to so that the integral will finally depend on /^dx. •^ X ScH. — A useful case of this general form is that in which X = x^, the form then being y =^ J 3i^ log" xdx I and the formula of reduction becomes a;"" log" xdx = — — -y log" x -T J ^ log"-' xdx. m + by means of which the final integral, when m is a positive integer, becomes. r , a?"+i / x^'-dx = -• ^ m + 1 EXAMPLES. 1. Integrate dy = x/^ log^ xdx. Making m = 4, and n — 2, we have y = / X* log^ xdx = ^^i^- -if^ log ^^^' (1) Making m — i and n = 1, we have fx^ log xdx = "^^ - ^ fx^dx (= I"), LOGAttlTSMIC FVNCTtom. S97 which substituted in (1) gives us = I (log' a; -I log a; + ^). „ T i J. -7 a; log a; «?a; z. Integrate aw = — - ° Put ^ (Zw and log x =. u; then w = Vfl^ + a;* and du = — /xlogx dx , , , „. 1 , P'\/cfi+^ , - — ^ = (ffl2 + 2;2)4 log a; — / ■ — dx , „ oxi 1 /^ ffl'f^a; P xdx = {a^^x^y\ogx-J-j===-J-—== *^ ^ya + x^ "^ ya' + a? = {a^+x^)i loga; + a log / « + VaHa;^ _ ^^qi^j (See Ex. 17, Art. 146.) „ T , , 7 log X dx 3. Integrate ) ^dx ; either of which may be reduced by the above formulae. This process will always effect the integration when in and n are either positive or negative integers, and often when they are fractions. The method is exhibited by the following examples. EXAM PLES. 1. • dy = sin« Odd. Put sin = «, then dd = {l-x^)-idx. .-. y = y sin6 Odd = J yfi (1 _ a^)-i dx = - le + 476 + 2:176 J (^ - ^ )' + 3 .-476 «^^~' ^ (by Ex. 7, Art. 151); cos , . . „ , 5 . , - 5-3 . „, 5-3 „ = --g- (sm^e + jsm30 +_-sme) + ^-^0. a. % sin^ Put sin 6 = a;, then do — {I— x^)'^ dx. INTEGRATIOJSr. 305 (by Ex. 13, Art. 151); cos / 1 3 7_1 3 \ 1^ tan iO. 4 Vsiu* ^ 3 sin2 6/ "^ 3-4 ^ ^ / ■ , 1 + cos , sin 1 X , /, N (since — log ■ - = log — = log tan 4ft) ^ ° sm fl ° 1 + cos 6 ° ^ ' 3. dxj = sin* edd. + f sin e) + f ( (See Ex. 10, Art. 135.) COS ^ 2^ = J- (sinS 6( + f sin e) + f 0, 4. dy = cos* 0c?i9. I CO T sin cos^ ^ , , • /I o , ,fl ^ = 2 h I sm cos + |0. (See Ex. 9, Art. 135.) 158. Integration of sin"* cos" fZ0 in terms of the sines and cosines of the multiple arcs, "nrhen m and n are positive integers. The aboTe integrations have been effected in terms of the powers of the trigonometric functions. When m and n are positire integers, the integration may be effected with- out introducing any powers of the trigonometric functions by converting the powers of sines, cosines, etc., into the sines and cosines of multiple arcs, before the integration is performed. The numerical results obtained by this pro- cess are more easily calculated than from the powers. Three transformations can always be made by the use of the three trigonometric formulae. 306 EXAMPLES. {1.) sin a sin 5 = 1^ cos {a — h) — ^ cos {a + h). {2.) sin a cos S = ^ sin (a + J) + | sin {a — I). (3.) cos a cos 5 = ^ cos {a + b) + ^ cos (« — h). EXAMPLES- 1. dy = sin* 6 cos^ Odd. Here sin* cos^ = sin 5 (sin cos 0)" = sin e {^ sin 26)^ [by (2)] = I- sin (sin2 20) = i-^P^°^0 [by(i)] = ^ sin — -J sin cos 40 = -^ sin — ^ (^ sin 50 — -^ sin 30) [by (2)] = -J sin — -^ sin 50 + ^ij- sin 30 .-. y = Ain^ cos« 0(Z0 = / (i sin ede — -^ sin hOdd + -j^Sg- sin 30«?0) = — -J cos + ^ cos 50 — :jij cos 30. 2. dy = sin3 cos^ ddO. y = — -^ cos 20 + y^ cos 60. 3. dy = sin' Odd. y — ^ cos 30 — f cos 0. 4. dy = cos* 0«?0. 2^ = ^ij sin 30 + I sin 0. FORMULA OF REDUCTION. 3 159 . Reduction of the Form J oc" COS ax dx. Put u = x", and dv = COS ax dx ; then du = nx"~^ dx, and V 1 . — - sm ax. a ••■y = / a^cosaxdx = a ' sm.ax / a;""' sin ax dx. Agai n, put u — J, , and dv : = sin ax dx ; then du = (re — 1) a!"-2 ^2,^ and V 1 = cos ax. a 307 / a;''"^ sin axdx = x'^^ cos ax t/ a ■\ / a;""' cos ax dx, a 'J .'. y = f X" cos axdx ^^ - x" sin ax «^ a — 7:\—:i^ ^°s "^ H • / ^ cos ax dx] a \ a a fJ / af^* (ax sin aa; + w cos ax) n (n — 1) P „ = -^ ^—^^ — -J x"^^ cos axdx. The formula of reduction for /a!" sin axdx can be obtained in like manner. EXAMPLE. 1. di/ = ar' cos x dx. 1/ = a^sinx + 3x^ cos a; — 6a! sin a; — 6 cos x. 308 FORMULJE OF REDUCTION. 160. Reduction of the Form / e"* cos" X dx. Put U = COS" X, and dv — (T dx; then du = —n cos"~' x sin x dx. and V _ e . ~ a .'. y = /«- cos" xdx e" cos" X a ^If- COS"-' X sin X dx. (1) Again , put u = COS""' X sin X, and dv = e^'dx; then du = -{n- 1) COS "-' X sin* a; «?a; + COS" a; dx. and V _ ^ ~ a I e" COS"-' X sin a;rfa; 1 1 /* = - e" COS"-' a; sm a; / er\— (re — 1) cos"-' x sin* a; + COS" a;] b; (a* - b^)i (^2 _ j2)i cos 8 313 EXAMPLES. n a tan ^ + 5 — (S« — a^)i and = f log , when a < 5. There are other forms which can be integrated by the application of the formula for integration by parts (Art. 147). Those which we have given are among the most important, and which occur the most frequently in the practical applications of the Calculus. The student who has studied the preceding formulae carefully should find no difficulty in applying the methods to the solution of any expression that he may meet with, that is not too compli- cated. The most- suitable method of integration in every case can be arrived at only after considerable practice and famil- iarity with the processes of integration. EXAMPLES. vl — a;' 2. dy = a^dx Vl — x^ /afi , 1-3 \ /- -. , 1-3 . , „ , x'^dx 3. dy = — 7 ,_ /^ I 1-6^ I l-4-6^, . l-2-4.6 \ ,j-^ ^- \7 +5-7 +3.5.7^+1.3.5.7/^ . dy = — - Va + W 1 /„ . 4aa;2 , U\ , — ,-5-5 EXAMPLES. 313 dx \6a + 1-5 6a;6 ' 4:.6a;* ' 2-4:-6x^ + 1-3-5 iJA/r 1-3-5 , 1 + Vl • 4-6 log 6. dy = dx y = - a;Vl + X log {^^±^^\ a; • " \v^i + a; + (See Ex. 1, Art. 143.) dy = (a^ — x^^ dx. y — ^x {a^ — a?f + -^ a^a; (a^ — a;^)^ + /4-^«*a;(«'-a;^)* + a; 6-4:-3 6-4-3 a 8. (?t/ = a;^ (1 + a;^)^ c?a;. y = 5a;3 — 2 3 5 (1 + a«)i 10. dy = £?«/ = (1 — a;^)t dx. y = ix{l— a;2)t + |a; (1 — a;^)* + | sirr^x. dx (a + bx^)i y ^ Ifl + tea "^ a/ 11, dy = y = + (?a; BaVa + ix^ {a + Ix^Y 1 + + (a + fe«)2 ^ 3« (a+fe8) ^ 3a2 SaVa+Sa^' 314 EXAMPLES. 13. dy = ^dx (1 - ^f y = x^ — 3x 1 sin-la?. ^Vl — a;S 13 dy = x^dx V^ax — x^ 2/ = (x^ , tx^ , + 7-5.3 4-3-3 a 14. f?^ 15. dy 16. ? + 2a; + 2). 26. dy = a'^dx ^ "^ ~ £(1 + * log") + -^- (l°g =^ + l°g ''■'^ + ilog8«.|- + etc.)- 27- ^^2/ = -^^irn^- 3/ = log («- + e-). 28. dy = e^'e^dx. y = e^. ^xdx ^ ^^- ^^ = (iT^)V y = i^x „„ , (1 + a;2) e'^dx (x — 1\ 30. dy = ^ ^ / ,, ■ y = ef (—-—)■ ^ (1 + xf ^ Vl + a;/ [Put (1 + a;) = z ; then x^z — 1, dx-= dz, etc.] sm^d dd ,.,,.„ ^ 31. dy — ^-- (Art. 156.) " cos^ ^ ' y =. sec + 2 QoaO —\ cos^ 0. 33. dy = sin^ (9 cos^ Odd. «/ = | sin* (9 — f sin^ 6. 33. dy^^:^^^. 2^ = fcosi0-2cosie. cos^0 34. dy = '^^^-- y = ismie-^smie. 316 MxAMPLm. 35. sin^ e de y- .l«(«i"^^ + 4 36. dy^^ ^^ y sin* 6 cos* d 1 4 cos 8 cos d cos d sin.^ 9 3 sin^ 6 3 sin 6 37. % = - ■ '^^ , • 2/ = 2 tan* ^ (1 + ^ tan^ 0). sin* cos^ , sin^e d?9 o J. 4 fl 38. (^2/ = F— • «/ = I tan^ ^• cos'O 39. dy = . ,f^ ^^ ' y = — 8 cot 20 — | cot^ 26. ^ sin* cos* 6 " ^ 40. dy = sin* 6 cos* ' derivative p = x dx"^ '&) = ^^-' 323 SUCCESStVB tNtM&BANdK hence, by integi-ating we have Again, we get from this last equation, and by integrating, 1^^ = X, + ax + C,. Also from this we obtain d (y-s:|) = ^idx + CiX dx ^- C^sda;, and integi-ating, And continuing the process we get, after n integrations, y d^'y = J Xd^ - ^'' ■•" ^' 1.2.3 . . . (m - 1) + ^' 1-2.3 ... (w - 2) + . . . . C„_.a; + C„. (1), The symbol J Xdaf" is called the n^ integral of Xdx", and denotes that n successive integrations are required. The first term X„ of the second member is the »<* integral of Xdx", without the . arbitrary constants ; the remaining part of the series is the result of introducing at each integration, an arbitrary constant. DEVELOPMENT OF INTEGRALS. 323 165. To Develop the n*^ Integral / Xdx"' into a Series. — By Maclaurin's theorem, we have + (/H 1.2.3. M.- -1) + (-^) iT^I:::^ \rfa;/l.i + .2... (w + 1) in which the brackets (^fxdxA, (y"~X^af-') .... (^/Xdxj, are the arbitrary constants for that is what these expressions become respectively, when a; = 0. By Maclaurin's theorem, we have ^ ,^, , (dX\x , /(^2X\^ , /d^X\ x^ , , ... which may be converted into (1) by substituting for aP, x^, x^, a?, etc., in (2), the quantities , etc., 1.2...W' 2.3... (w + 1)' 3-4... (re + 2) Since — / Xax" = I Xdz"-'- 324 EXAMPLES. and prefixing the terms containing the arbitrary constants as above shown, viz.. X ^ X- C„, C„-ij, C'»-!i.3, • • • • ^>i.2.3...(ra-l)' (See Lacroix, Oalcul Integral, Vol. II, pp. 154 and 155.) EXAMPLES. 1. Develop / — •^ a/1 - a;2 Here X = (1 - x^yi , , 1-3 ^ 1-3-5 „ = 1 + ia;2 + — ja^ + ^-^-^ a;" + etc. Substituting in this series for a;", x*, a;*, a;*, etc., the quantities a^ x^ a;8 a;W etc.; 1-3-3-4' 3- 4-5 •6' 5- 6-7.8' 7-8-9-10' and prefixing c,, (73 -.3 ^^2 a;2 1-2' ^' 7? 1-2-3' we get /* ^^ - a, .+ a^ ^ ^' 4- a? ^^1.2-3 •^ Vl - a^ + 17 a^ 2-3 :4+2- a;6 3-4-5-6 l-3a;f 1 ' 2-4. 5-6. 7-8 + 7r- 1 ■3.5a;i'' --K + etc, 2. Integrate 6?^?/ = ^adx^. Dividing by dx^ we have ^ = 6ac?a;, or tZ(^^) = Qadx, dx^ \dxV EXAMPLES. 335 or -t4 = &CIX + C,. Multiplying by dx and integrating again, we haye ^ = Zax^ + C,x + Cs. Multiplying again by dx and integrating, we have 3. Integrate (?'«/ = sin x cos^ a; £?a^. Put sin a; = 2 ; .-. (?« = cos xdx, I and (?s^ = cos^ x dx^ ; .*. dJ^y = zdz^; from which we get y = ^-+ C,z+ C; sin^ a; _ . _, .-. y = — g h t/, sm X + Cb. 4. Integrate d^y = ax'dx^ 5. Integrate (^3/ = 2a;~8 ^ ' ■'^' It is evident that -^ must be a function such that if we dz differentiate it with respect to y, regarding x as constant, the result will be f{x, y). Therefore we may write ^^=J'f[x,y)dy. INTEGRATION OF PARTIAL DIFFERENTIALS. 337 Here, also, it is evident that u must be such a function that if we differentiate it with respect to x, regarding y as constant, the result will be the function Hence, ^ = V J f{^> y)dy ^^• Therefore, we first integrate with respect to y, regarding X as constant,* and then integrate the result with respect to X, regarding y as constant,* which is exactly reversing the process of differentiation. (Art. 83.) The above expression for u may be abbreviated into fff («. y) % ^« or fff i^^ y) d^ dy- We shall use the latter form ; f that is, when we perform the «/-integration before the a;-integration, we shall write dy to the right of dx. It is immaterial whether we first integrate with respect to y and then with respect to x, or first with respect to x and tlien with respect to y. (See Art. 84) In integrating with respect to y, care must be taken to add an arbitrary function of x, and in integrating with respect to x to add an arbitrary function of y. In a similar manner, it may be shown that to find the value of u in the equation d?u ., . = /('(^j y, z), dx dy dz we may write it u = I I J f{x, y, z) dx dy dz, * Calleel the ^-integration and a-integration, respectively, t On this point of notation writers are not quite nnifoCB. SeeTodhnnter'sOs' ■) p. 78 ; also Price's Gal., Vol. 11, p. 281. 338 EXAMPLES. which means that we first integrate with respect to z, regard- ing X and y as constant; then this result with respect to y, regarding x and z as constant ; then this last result with respect to x, regarding y and z as constant, adding with the 2-integratiou arbitrary functions of x and y, with the ^-integration arbitrary functions of x and z, and with the a;-integration arbitrary functions of y and z. (See Lacroix, Calcul Integral, Vol. II, p. 206.) EXAMPLES. 1. Integrate dJ^u = ix^ydx^ Here ^(t~) ~ bx^ydx. •'■ ^ ~ fi^V^ = \bci?y +f{y). du = \b7?ydx .+ / {y) dx. .: u = ^I>xH/+f{y)x + {y). 2. Integrate ^u = 2x^y dx dy. Here ^\^) ~ ^^^ydy. •'• & ^ S'^x^ydy = xy + {x). du = x^dx + (p (x) dx. ••• w = Wy' + fi> («) dx +f{y). 3. Integrate d^u = Sxy^dxdy. u = ixY + f (a;) dx +f(y). 4. Integrate dJ^u = a3?y^dxdy. u = ^^y^ + f(x)dx+f{y). INTEGRATION OF TOTAL DIFFERENTIALS. 339 168. Integration of Total Differentials of the First Order. If u — f{x, y), we have (Art. 81), , du , du , ft fit nil in which -j- dx and -j- dy are the partial differentials of u; cicc ciy also, we have (Art. 84), d?u d^u dx dy dy dx' d ldu\ _ d ldu\ ,^. dy \dx/ ~ dx \dy/ ^ ' Therefore, if an expression of the form du = Pdx + Qdy (3) be a total differential of u, we must have du _ ^ du _ ^ d^- ' dy~ ^'' and hence, from (1), we must have the condition dP _dQ , , dy ~ dx' ^ > which is called Euler's Criterion of Integr ability. When this is satisfied, (2) is the differential of a function of x and y, and we shall obtain the function itself by integrating either term ; thus, u = fPdx+f{y), (4) in which f{y) must be determined so as to satisfy the con- dition du _ dy-^- 330 EXAMPLES. Remabe. — Since the differential with respect to x of every term of u which involves x must contain dx, therefore the integral of Pito will give all the terms of « which involve x. The differential with respect to y of those terms of u which involve y and not x, will be found only in the expression Qdy. Hence, if we integrate those terms of Qdy which do not involve x, we shall have the terms of u which involve y only. This will be the value of /(y), which added with an arbitrary constant to fPdx will give the entire integral. Of course, if every term of the given differential contain x or dx, f(y) will be constant. (See Church's Calculus, p. 274.) EXAMPLES. 1. du = ^^fdx -I- '?,yhiHy. Here P = ^Y, Q = da^K .: ^ = nxY and ^ = na?y\ ay " dx '^ Therefore (3) is satisfied, and since each term contains x or dx, we have from (4), u = fiafly^dx = x*y^ + C. 2. du = f+{2y-^~;jdy. (3) is satisfied, therefore from (4) we have ^dx Since the term %ydy does not contain x, we must have, from the above Remark, f{y) = fiydy = y% which must be added to - , giving for the entire integral, M = - + «»+ C. 3. du = ydx + xdy. u = xy + C. 4. du = {&xy — y^) dx + (3a? — 2xy) dy. u = 3x^y — y^+ C. DEFINITE INTEGRALS. 331 0. du = i^axy — 363;^^) dx + («a^ — Ix^) dy. u = ax^y — byx^ + G. The limits of this work preclude us from going further in this most interesting branch of the Calculus. The student who wishes to pursue the subject further is referred to Gregory's Examples ; Price's Calculus, Vol. II ; Lacroix's Calcul Integral, Vol. II ; and Boole's Differential Equations, where the subject is specially investigated. 169. Definite Integrals. — It was shown in Art. 130 that, to complete each integral, an arbitrary constant C must be added. While the value of this constant remains unknown, the integral expression is called an indefinite in- tegral; such are all the integrals that have been found by the methods hitherto explained. When two different values of the variable have been sub- stituted in the indefinite integral, and the difference between the two results is taken, the integral is said to be taken between limits. In the application of the Calculus to the solution of real problems, the nature of the question will always require that the integral be taken between given limits. When an integral is taken between limits, it is called a definite integral.* The symbol for a definite integral is f{x) dx, /: which means that the expression f{x) dx is first to be inte- grated ; then in this result b and a are to be substituted successively for x, and the latter result is to be subtracted from the former ; b and a are called the limits of integra- tion, the former being the superior, and the latter the inferior limit. Whatever may be the value of the integral * In the Integral Calcnlns, it is often the most difficult part of the work to pass firom the indefinite to the definite integral. 332 DEFINITE INTEGRALS. at the inferior limit, that value is included in the value of the integral up to the superior limit. Hence, to find the integral between the limits, take the difference between the values of the integral at the limits. In the preceding we assume that the function is continu- ous between the limits a and b, i. e., that it does not become imaginary or infinite for any value of x between a and i. Suppose M to be a function of x represented by the equa- tion u =f{x); then du = f [x) dx. Now if we wish the integral between the limits a and I, we have u^ f'f'(x)dx=f{b)-f{a). If there is anything in the nature of the problem under consideration from which we can know the value of the integral for a particular value of the variable, the constant C can be found by substituting this value in the indefinite integral. Thus, if we have du =■ (abx — bx^)^ (ab — 25a;) dx, and know that the integral must reduce to m when x = a, we can find the definite integral as follows: Integrating by known rules, we have u = i {abx — fe2)t + C, which is the indefinite integral ; and since u = m when, x=^ a, we have m = 0+ C; .: G = m, which substituted in the value of u gives M = I [abx — bx^)^ + m. EXAMPLES. 333 EXAMPLES. 1. Find the definite integral of du = (1 + ^x)^dx, on the hypothesis that u = Q when a; = 0. The indefinite integral is Since when a; = 0, m = 0, we have « = m + ^' .-. 8 27ffl' which substituted in the indefinite integral, gives for the definite integral required. 3. Integrate du = Qoi?dz between the limits 3 and 0. Here f" Qx^dx = '«?2; = •°° «?a; /q a** + ^^ 4-a;3 3. u = / x"dx = «/o |_w + 1 4. u z= I e~' 5. u = 6. M = / - 2a;3 1 3 * 54. ra + 1 = -(0-l) = l. tan-i - L «_ ~tan-i -" a 7T * This notation signifies that the integral is to be taken between the limits 3 and 0. 334 7. CHANG Ji OF LIMITS. dx + X^ tan" 1 IT = - rtan~i 00 — tan"' (— oo )] = -• a'- ^ '■' a P" dx sin" 2' Eemaek. — It should be observed here tliat the value of the Infini- tesimal element corresponding to the superior limit is exelvded, while that corresponding to the inferior limit is included in the definite in- tegral : for, were this not the case, as — becomes equal to oo when x = a, the integral of Ex. 8 between the limits a and would not be correct ; but as the limit a, being the superior limit in Ex. 8, and that which -renders infinite the infinitesimal element, is not included, the definite integral is correct. (See Price's Calculus, Vol. 11, p. 89.) 9. « = fid" — x^)idx. (See Ex. i, Art. 151. J - {a^ — a;«)i + ^ sin" a'n 10. = i ^' x^dz Vl — a;' _ l-3-5- 7r ■" 2.4.6.a" (See Ex. 7, Art. 151.) 11. sin'' X cos* X dx = 4^ 3-5-7-11 170. Change of Limits. — It is not necessary that the increment dx should be regarded as positive, for we may consider x as decreasing by infinitesimal elements, as well as increasing. Therefore, we have CHANGE OF LIMITS. 335 f\' {x) dx^{a)-^ {b) = _ [0 (&) _ ,^ (a)] = — / 0' (2;) dx. - 7. )i dx. '^ —a ^0 (See Art. 170.) w = 3-5 r^ x*dx 15. u = I '^0 V, =z V2ax — a;2 7-5.3 4-3-3 a'rr. 16. . = ^'.3(l_,)f^,=_^ 13 17. M " / ' -^— (1 - e^xni (See Ex. 3.) _ TT 1 jTT 1-3 ^TT 1.3^-5e«7r - "-2~4^ 2~33.42^ 2 22.43.68 3" ^ ' •»!"• pxxdxdy 18. M a;^ + t/^' (Art. 167.) We first perform the z^-integration, regarding x as con- stant, and then the integration. '0 \ X X =y„ 4^^=16- /lo /^a; />8/ 19. ?< = / / / a;y2; [x) J X — a IX — 5 ' ' J I — x^ ' rf{x)dx ^^- J -Jlx)' _ P Adx n Bdx f Ldx -J {x-a)"^J (,r-a)»-i"^ ■■J (x-a)' ^^^^^ 18 rfj^dx^ n {Ax + B ) dx 'J 4) (x) J \{x ± a 4.4. C {Gx + D)dx , r {Kx + L) dx ■^■^ J '\ix + a)2 + 521"-' ^ "J (a; + fl)3 , 52- ^^^«J (p{x) ~ J [(a; + af + ¥Y + D)dx n {Kx -f [(a; ± a)8 + 52]"-' "^ ■ ■ V (a; ± flpT5''' ^ W^W ~ Mb ^^ \a — bxl' P adx . Ix — a 20- y.Tz:T^ = i°gV-"+^' 342 fOBMULJB OP tNTEGRATtON. CHAPTER III. IbHational Punctioks. (Page 269.) [where 2 = (a + lx)^\ (142) ' ^ {a + bx>)i ~ ^ *"+' ' [where e — (a + ix^)i]. (143) 23. /- ^^-^== = log(| + a:+ V^b^+^\ (144) 24. /'-=^= = - 2 tan-i , /^^ "^ Va + bx — x^ y x — a' [where a + bx — a? =: (x — «) — x). (—-1) 25. fx"'{a + baf')idx= 3^f!^+i-'->i^^::^ " dz,{U&) (where 2* = a + 5.?;") ; or =;— ^a(""^«^«)y(2'— 5)(^"^^'^') «(?+?-') (?2, (where a;V = a + 5a;"). 26. r^== -?= log (:^^«±§=lA^) . Ex. 6, (146) ^ a;A/a + 5a; V« V ^ / 27. f--M:^-^==\ log (5a; + Va'+SV). Ex. 16. 28. f ^— ^llog N^^-\ Ex.17 = _llog(^^?). fORMVLJB Of INTEGBATTON. 343 39. f--^^= = a log (a; + fl + V^ax+a?). Ex. 20. '^ V^ax+a? 30. f{a^^a?)^dx = %(a^+ x^)^ + ^ log \x + («2 + a;3)4]. Ex. 35. OHAPTEK IV. Successive Eeductiok. (Page 285.) 31. fudv — uv — fvdu. (147) 32. J'x'^ {a + Ixy dx (148) ar^+^{a + ia^y+^—{m.—n + l) a Cx'^-''{a-\-Wydx t^ (j\\ b {np + m + 1) 33. y r-"" (« + hx^'Y dx (149) ar-^+^a + ba^y+'^ + 5(m — »p — w — 1) y a;-"+"(« + Ixtydx -_ _a(m-l) •^^"> 34. /'a;'" (a + fe")^ fZa; (150) »;"'+' (a + Sa;")? + aw» I x'^ {a + iu^Y'^ dx ^ "l — ((7) np + m- + I 35. faf (a + te")-^ dx (151) af +Va + Ja!")~»'+' —(m + n + l—np) fx'"(a + I)3f)-^+^dx 36. jx^ {a!> — 3?Y^ ^^ - ^^ (fl8-a;2)i + i^inl) a^ fy^^ /^a _ ^ay^ ^j.. 344 fOnHtVLJS op tNTSGtlATtON. r 37. JtT (fl' + a;2)-J dx m ^ m J \ I / 38. /: dx a;" (a* — a?)^ _ — (a^ — a?)i m — 2 P d^ + t — 2 r (m — l)fl2af»-' "^ (»i — 1) fl2 1/ 2..B-2/^a _ ^%\\' 39. f(a>-a?fdx 40. /: n + 1 dx ia^ — x*)i n—3 P dx _x w— 3 /* {n-%)a\a^-x^yi-^ in—2)a'^ (a2-a;2)|-' - .^ /• afdx = y2ax — x^ + ^ — / - m mo. 42. / x^'dx Vffl -\- hx ■\- ex* V^ax—pe^ r »•- = a;"-> Va + 6a; + ca?* n — 1 a P . -»^-?& ^^c « c*^ Vfl + Saj + cajS 2n—l b r af-^dx * :/ 2w ct/ y'a^jaj.H-' .* See Price's Calonlus, Vol. 11, p. 63. 43. / S'ORMvrL^ OP iNTMGItATtOif. §45 V« + ix + cx^ dx , Ix U\ , IW a\ r + bx-\-cx^ xdx 44. f-^ ^ ya + bx + c^ _ Va + bx+cx^ I f _ ^^ i'. •#/ 2c '^ \/a + 6a; + ca;^ 4:5. / a:^ log" a;(?a; 7 log" X / x" log"-i a;Ja;. (152) m + 46. A^^ J log" a; .+ (w — 1) log"~"^ X n — \ 54. /• ^__ ^ a + h cos = _^ 1 r V^+Vg^tan j-| VS^-a^ ^I'^b + a—^b—atz.np /n T '7*2 'r"~^ Xrfa;" = C„ + C„_. ?+ CL, j^+. .C,j •2.3.(ra-l) ■•■ ^■^^l-%-i...n^ Wa;Jl-2-3... {n + 1) CHAPTER VI, LENGTHS OF CURVES, 171. Length of Plane Curves referred to Rectan- gular Axes. — Let P and Q be two consecutive points on the curve AB, and let {x, y) be the point P ; let s denote the length of the curve AP measured from a fixed point A up to P. Then PQ = ds, PR = dx, EQ = dy. Therefore, from the right-angled triangle PEQ we have m n Fig. 43. dx. ds = ^dx^ + dy'^; hence, s = fy/dx^ + df = J' {l + ^j To apply this formula to any particular curve, we find the value of -~ in terms of x from the equation of the dx curve, and then by integration between proper limits s becomes known. The process of finding the length of an arc of a curve is called the rectification of the curve. It is evident that if y be considered the independent variable, we shall have '=/(>-£)** dyh The curves whose lengths can be obtained in finite terms are very limited in number. "We proceed to consider some of the simplest applications ; 348 RECTIFICATION OF TBE PARABOLA. 172. The Parabola. — The equation of the parabola is y^ = %'px ; dtl p hence, -f- =^— • ; .- ax y or '"'s = - f{f + y^)^dy, (which, by Ex. 35, Art. 146) = r/^^_+/ + 1 log {^y + Vi7« + f) + a (1) If we estimate the arc from the vertex, then s = 0, y = 0, and we have 0=|logiJ+^;^ .-. C=-|logi>, which in (1) gives s 3- + 3 log ^^ ^ j , (2) which is the length of the "curve from the vertex to the point which has any ordinate y. If, for example, we wish to find the length of the curve between the vertex and one extremity of the latus-rectum, «/ = p, we substitute p for y in (2), and get s = W2+|log(l + A/2) for the required length. We have here found the value of the constant C by the second method given in Art. 169. We might have found the definite integral at once by integrating between the limits and p, as explained in the first method of Art. 169, and as illustrated in the examples of that Article. Hence, RECTIFICATION OF THE CIRCLE. 349 we need not take any notice of the constant G, but write our result * KJ ; ' s = - f^{f + y^)^ dy, (see Art. 169) '" and integrate between these limits. 173. Semi-Cubical Parabola.* — The equation of this curve is of the form 'f = ao?. (See Fig. 39.) A^A - '^^ TT % 3 /— ^ %' » ■ nn " ^^ Hence, -^ z= ^yax and -^^ = \ax. * ' - s = J (1 + ^ax)i dx =-- ^^Jl + iax)i + a If we wish to find the length of the curve from A to P, we must integrate between the limits and 3p (see Art. 128, Ex. 9) ; hence, s = f^{l + lax)Ux = "~ (1 + lax)^ = 4(H-¥«^)t^4 = 2^ [(1 + ^P)^ ^ 1] = ^ (3t - 1), g by substituting ^— for a. (See Art. 125, Ex. 1. Compare Ex. 10, Art. 128.) 174. The Circle. — From af + y^ = r% we have dx y * This was the first curve which was rectified. The anthor was William Neil, who was led to the discovery, about 1660, by a remark of Wallis, in his Arithmetica Infinitomm. See Gregory's Examples, p. 420, 350 RECTIFICATION OF THE ELLIPSE. Heuce, for the length of a quadrant, we have (since the limits are and r), J a \ fl Jo ^r« — ai2 = psin-i^ I =rir7r, which involves a circular arc, the very quantity we wish to determine. The circle is therefore not a rectifiable curve ; but the above integral may be developed into a series, and an approximate result obtained. By Ex. 1, Art. 170, we have _r (X _^ l-3.-r5 l-3-5 a;7 \n ^ ^ - \j\r + 2.3r3 + 2.4.5r5 + 2.4.6.7?-' + ' ]q [. 1 1-3 3-5 ^ \ = 'V + %r^+ 37475 +^2^76:7 + ^^•) ' 1 1-3 3-5 •"• i'^ = 1 + 2:3 + 2:4:5+ 3:4:6:7 +'*'■ By taking a sufficient number of terms, reducing each to a decimal, and adding, we have 7r= 3. 141592653589793 + . For the approximation usually employed in practice, w is taken as 3.1416, and for still ruder approximations as 3^. 175. The Ellipse.— From y^ = (1— e^) {a^—x% we have ^ = - (1 - e«) - = - ?-^^^2. dx 'y ^/a^ — x^ To find the length of a quadrant, we must integrate be- tween the limits and a ; hence. RECTIFICATION OF THE CYCLOID. 351 This integration cannot be effected in finite terms, but may be obtained by series. Put - = z; then dx = adz. When x — a, z = l, and when X = 0, 2 = 0; therefore the above integral becomes t/o ' Vl — 2* (by Ex. 17, Art. 170), which is the length of a quadrant of the ellipse whose semi-major axis is a and eccentricity e. 176. The Cycloid.— Prom x — j'vers~i- — V^ry—y^ we have dx _ y Jy 's/'Hry — y^ = v3r / {%r — y)-^dy 'Jo = 1—3 {%r)^ (ar - y)* a- = 4?-, which is 1^ tbe cycloidal arc; * hence the whole arc of the cy- cloid is Sr or 4 times the diam- eter of the generating circle. If we integrate the above ex- Fig. 44. pression between y and 2r, we get s = V^ f (3r — y)-^ dy = % (2r)* {%r — y)* = 2-v/3r {2r — y) = arc BP. But BD = VBA X BC = V2r{2r-y)', .: arc BP = 2 times chord BD.* * This rectification was discovered by Wren. See Gregory's Examples, p. 421. 352 INVOLUTE OF A CIRCLE. 177. The Catenary.— A catenary is the curve assumed by a perfectly flexible string, when its ends are fastened at two points, A and B, nearer together than the length of the string. Its equation is lif+e-^). y Hence, dy 1/1 -i\ If s be measured from the lowest point V, to any point P {x, y), we have ' = I X" (""" + ''') ^"^ = U'°- '~')- 178. The Involute of a Circle.— (See Art. 124.) Let C be the centre of the circle, whose radius is r ; APE is a portion of the involute, T and T' are two consecutive points of the circle, P and Q two consecutive points of the in- volute, and (j) the angle ACT. Then TCT' = PTQ = d^, and PT = AT = r(p. .-. ds = PQ = rcpdtj) ; '"'£• *S' .-. s = rj'(l>d(j} = iri)^ + G. If the curve be estimated from A, (7 = 0, and we have s = \rf. For one circumference, = 2Tr ; .■. s = ^r {^Ttf = grw*. For n circumferences, = "Unn ; .: s = \r (2w7r)' = %rn^-n\ TBE CARDIOIDE. 353 179. Rectification in Polar Co-ordinates.— If the cuiTe be referred to polar co-ordinates, we have (Art. 103), n I drW hence we get ^ — J \f + ^) ^^' « = y (1 + -^ j ^r- 180. The Spiral of Archimedes. — From r = ad, we have dd _1 dr a 1 /"■ 1 + 2 °^\ a /' 2a (see Art. 172), from which it follows that the length of any arc of the Spiral of Archimedes, measured from the pole, is equal to that of a parabola measured from its vertex, r and a having the same numerical values as y and p. 181. The Cardioide. — The equation of this curve is ?' = a (1 + cos d). Here -^ = .— a sin d, aa and hence s = J'[_a^ (1 + cos df + a^ sin^ 0]^ dB = aJ^{2 + 2cose)ide cos ^ dd = 4a sin 5 + C 354 LENGTHS OF CURVES IN SPACE. If we estimate the arc s from the point A, for which = 0, we have s = 0; C=0. Making =ztt for the superior limit, we have s = 4a sin ;t = 4a, Fig. 47. which is the length of the arc ABO; hence the whole perimeter is 8a. 182. Lengths of Curves in Space. — The length of an infinitesimal element of a curve in space, whether plane or of double curvature, from the principles of Solid Geom- etry (see Anal. Geom., Art. 169) is easily seen to be Hence, if s denote the length of the curve, measured from some fixed point up to any point P (x, y, z), we have /. =/D+(i)'H-(ir dx. If the equations of the curve are given in the form y = f{x) and z = ^ {x), dtj dz we may find the values of -^ and -r- in terms of x, and then by integration s is known in terms of a;. * The E^tudent who wants further demonstration of this, is referred to Price's Cal., Vol. I, Art. 341, and Vol. II, Art. 164; De Morgan's Dif. and Integral Cal., p. 444 ; and Homersham Cox's Integral Cal., p. 95. EXAMPLES. 355 183. The Intersection of Cycloidal and Parabolic Cylinders. — To find the length of the curve formed by the intersection of two right cyhnders, of which one has its generating lines parallel to the axis of z and stands on a parabola in the plane of xy, and the other has its generating lines parallel to the axis of y and stands on a cycloid in the plane of xz, the equations of the curve of intersection being y^ = 4:px, z =z a vers~i - + V'2«a; — x^. ^-y = JP and'f^^v/?^^^; \ X ax \. X Here , ax = (^+l + 'i-^fci^ = iP + 2a)i''' Vx Estimating the curve from the origin to any point P, we have _dx x^ /"" . /7'r , ,- = {p+ 2a)i —^^2{2J + 2a)i V^. «/o x^ EXAMPLES. 1. Rectify the hypocycloid whose equatiow is »;' + «/' = a^. Ans. The whole length of the curve is 6a. z 2, Rectify the logarithmic curve y z= ie. Ans. s = a log f + V^ + p + C. « + V «■■' + y^ • 3. Rectify the curve e"" = ~ r- between the limits x=.\ and a; = 3. s = log (e + e~^). 356 EXAMPLES. 4. Eectify the evolute of the ellipse, its equation being ©'+©* = '• Put a; = « cos' d, y = P sin' 0; then dx = — 3a cos' 6 sin d dO, dy = '6!ism^d cos Ode ; .: s = d f "(«« cos' d + !^ sin' (9)* sin cos e dO "~ ««-i3'' (»S flS therefore the whole length is 4 - " «' — /32 If /J = a, this result becomes 6a, which agrees with that given in Ex. 1. (See Price's Calculus, Vol. II, p. 203.) 5. Find the length of the arc of the parabola x^ + y^ =■ a^ between the co-ordinate axes. Put X = a cos* B, y = a sin* ; .: s = 4:a I (cos* d + sin* 0)^ sin 6 cos B dB = — -^ ^"(1 + cos«3(9)i, y = {2b — a) cos (p — (b — a) cos' 5, we have J ydx. (1) In like manner, if the area were included between the curve, the axis of y, and two abscissas at a finite distance apart, we would have ■A- = j xdy, where c and d are the y-limits. (2) QUADBATURE OF THE CIRCLE. 361 185. Area between Tvro Curves. — If the area were included between the two curves AB and db, whose equa- tions are respectively y ^ f {x) and y «/ = (/> (x), and two ordinates OD and EH, where OT) — h and OH — a, we should find by a similar course of reasoning, pa ^= / UiP^) —'t>{x)]dx. Fig. 50, The determination of the area of a curve is called its Quadrature. 186. The Circle. — The equation of the circle referred to its centre as origin, is y^ ^ a^ — x^ ; therefore the area of a quadrant is represented by A — r\a^ — a;2)i dx = r^i^^ + f gin-i -X (See Ex. 4, Art. 151.) L 2 a ajQ _ aV "~ T' therefore the area of the circle = -na^. Also, if OM = X, the area of OBDM becomes A = r(a^ - x^)i ^0 dx =c x (a^ — x^)i , a^ ■ -i^V — i — ;- — — + :r sm~i -• 2 ^3 a Fig. 51. This result is also evident from geometric considerations 16 3C3 QUADBATUJRB OF THE PARABOLA. for the area of the triangle OMD = "- {a^ — x^)^, and the area of the sector ODB = ^r sin"' -• a a Bemabk. — The student will perceive that in integrating between the limits a; = and x = a, we take in every elementary slice PQRN in the quadrant ADBO ; also integrating between the limits a! = and X ■= X = OM, we take in every elementary slice between OB and MD." 187. The Parabola. — From y^ — %px, we have y = 's/^px. Hence, for the area of the part 0PM, we have ^ A = V^p / x^dx — f V^p xi ; i. e., ^xy. Therefore the area of the segment POP', cut off by a chord perpendicular to the axis, is f of the rectangle PHH'P'. 188. The Cycloid. — Prom the equation X ^= r vers"' - — 's/%ry — y^, H O M / H \ / P Fig. 52. we hare dx = ydy ^/^ry f •|7rK yMy •s/^ry — y^ (See Ex. 6, Art. 151) = ^ the area of the cycloid. Since integrating between the limits includes half the area of the figure. * The student should pay close attention in every case to the limits of the integration, AkEA BETWEEN PAHAMoLA AHb CIRCLE. 363 Therefore the whole area = Znr^, or three times the area of the generating circle.* 189. The Ellipse.— The equation of the ellipse referred to its centre as origin, is ay + Trx^ = therefore the area of a quadrant is represented by h a< A=- (a^-xf-dx 7> 2 = - ~ (See Art. 186)' = JaSTr. Therefore the area of the entire ellipse is -nal. 190. The Area between the Parabola y"^ = ax and the Circle y^ = 2ax — xK — These curves pass through the origin, and also intersect at ^ ^^ the points A and B, whose common abscissa is a. Hence, to find the area included between the two curves on the positive side of the axis of x, we must integrate between the limits a; = and a; = a. Therefore, by Art. 185, we have A = f\(2ax — x^)i — (axyi] dx Fig. 53. ■na'' \a^ ; (See Ex. 6, Art. 151.) which is the area of OPAP'. * This quadratnre was first discovered by Eoberval, one of the most distin- gnished geometers of his day. Galileo, having failed in obtaining the quadrature by geometric methods, attempted to solve the problem by weighing the area of the curve against that of the generating circle, and arrived at the conclusion that the former area was nearly, but not exactly, three times the latter. About 1628, Koberval attacked it, but failed to solve it. After studying the ancient Geometry for six years, he renewed the attack and effected a solution in 1634 (See Salmon's Higher Plane Curves, p. 266.) Ui The spiral df' AkcHtMEDEs. 191. Area in Polar Co-ordinates. — Let the curve be referred to polar co-ordinates, being the pole, and let OP and OQ be consecutive radii-vectores, and PE an arc of a circle described with as centre ; let {r, 0) be the point P. Then the area of the infinitesimal element OPQ = OPE + PEQ ; but PRQ is an infinitesimal of the second order in comparison with OPE, when P and Q are infinitely near points ; conse- area OPE = — — quently the elementary area OPQ Hence if A represents the area included between the curve, the radius-vector OP, and the radius-vector OB drawn to some fixed point B, we have A = ij'rm. If 13 and « are the values of corresponding to the points B and C respectively, we have =^rrm. 192. The Spiral of Archimedes. — Let ?• = ;r- be its equation. .Then A = TT J.r'^dr = \-nr^ + C. If we estimate the area from the pole, we have .4 = when r — 0, and .•. C = ; hence, A =inr%. which is the value of the area passed over by the radius- vector in its revolution from its starting at c to any value, as r. EXAMPtm. 365 If we made B = Zn, we have r = 1 ; therefore A = l-rr, which is the area described by one revolution of the radius- vector. Hence the area of the first spire is equal to one- third the area of the measm-ing* circle. If we make d = 2 (sJtt), r == 2 ; therefore A = fTT, which is the whole area described by the radius-vector during two revolutions, and evidently includes twice the first spire -|- the second. Hence the area of the first two spires = f w — -J-t = ^n, and so on. EXAMPLES. 1. Find the area of y = a; — a:^ between the curve and the axis of x. Ans. \. The limits will be found to be a; = 0, k = + 1 ; also ic = 0, a; = - l.f 2. Find the area oi y :=z a? — V^x between the curve and the axis of x. Ans. ^¥. 3. Find the area of ^ = a;^ — ax^ between the curve and the axis of x. Ans. -^aK 4. Find the whole area of the two loops of a^y^ = s^ (a* — x^). Ans. ^a\ 5. Find the area of x^ = a^ between the limits y ■=!) ftnd y = c. A » t^ — c " Ans. 2a?— ^ — iic 6. Find the whole area of the two loops of a'^y^ = aWx^ — *'a^. Ans. ^ab. * See Anal. Geom., Art. 158. + The stndenl should draw the figure in every case, and determine the limits of the integrations. 366 MXAMPLM. 7. Find the whole area of a^ = a? {^a — x). (See Arts. 150 and 188.) Ans. to'. 8. Find the whole area between the Cissoid y^ = and its asymptote. (See Art. 103.) . „ ~~ 9. The equation of the hyperbola is a^y^ — W^ = — a^jsj find the area included between the curve, the axis of x, and an ordinate. xy ah , Ix + 's/x^ — a^ ,og(^+^ 10. The equation of the Witch of Agnesi is a?y = 4a« {2a — y) ; find the area included between the curve and its asymptote. Ans. AaH. 11. Find the area of the catenary VPMO, Fig. 45. Ans. n (e« — e"" j = a {y^ — a*)i. 13. Find the area of the oval of the parabola of the third degree whose equation is cy^ = (x — a) (x — b)^. (See Art. 142.) . 8 ,- ^B ' Ans. TT— p (o — ap. 13. Find the area of one loop of the curve ay'i = x^ {a" — x^)i. Ans. -fal 14. Find the whole area between the curve xY + aW = ay and its asymptotes. Ans. %-nal}. 15. Find the whole area of the curve ©'- (I)' = - Ans. JrraJ, MXAMPLSA Bet 16. Find the area included between the parabola y^ = 2px and the right line y = ax. These two loci intersect at the origin and at the point whose ab- scissa is -^ ; hence the a;-limits are and ~ ; therefore. Art. 185, A = J ( Y^2pa; — ax)dx = ^^, Ana. 17. Find (i) the area included between the parabola y^ = "ipx, the right line passing through the focus and inclined at 45° to the axis of a;, and the left-hand double ordinate of inter- section. (See Art. 185.) Also find {2) the whole area between the line and parabola. (1.) Here the CB-Iimits are found to be 1(^/2 + 1)'^ and | (^8—1)" | hence we have •^ — / [V^^ "^ dx — (x — ^p) dxj ^^(Vz-iY = iV^P <«' — W + iP"" KVa+i)" ^{V2-if = Vi'" - 3^/2p' + ■\/2p^ = p^ (V- - 3\/3). Ans. {S) Ans. ip^V^" 18. Find the whole area included between the four infi- nite branches of the tractrix. Ans. ira'. 19. Find the area of the Naperian logarithmic spiral. Ans. \r^. 20. Find the whole area of the Lemniscate r* = a^ cos 20. . Ans, a\ 368 H^AMPtES. 31. Find the whole area of the curve r — a (cos W + sin %d). Ans. TtaK 22. Find the area of the Cardioide. (See Art. 181.) Am. |7ral 33. Find the area of a loop of the curve »• = « cos nd. Ans. -r—' in 34. Find the area of a loop of the curve r =^ a COB nO + i sin nO. . a^ + i^ n Ans. — r^ < 4 n 35.. Find the area of the three loops of the curve »• = a sin 3d. (See Fig. 33.) Ans. -7— 4 36. Find the area included between the involute and the evolute in Fig. 46, when the string has made one revolution, Ans. 1?%*. CHAPTER VIII AREAS OF CURVED SURFACES. 193. Surfaces of Revolution. — If any plane be sup- posed to revolve around a fixed line in it, every point in the plane will describe a circle, and any curve lying in the plane will generate a surface. Such a surface is called a surface of revolution ; and the fixed line, round which the revolu- tion takes place, is called the axis of revolution. Let P and Q be two consecutive points on the curve AB ; let {x, y) be the point P, and s the length of the curve AP measured from a fixed point A to any point P. Then MP = «/, NQ = .y + dy, and PQ = ds. Denote by S the area of the surface generated by the revolution of AP around the axis OX; then the surface generated by the revolution of PQ around the axis of x is an infinitesimal element of the whole surface, and is the convex surface of the frustum of a cone, the circumferences of whose bases are 2-ny and 'In {y + dy), and whose slant height is PQ = ds ; therefore we have ds = ^-y + ^-^^y+ M PQ = ^^ygs, since the last term, being an infinitesimal of the second order, must be dropped. Therefore, for the whole surface, we have 8 = ^nj'yds = %7Tjy\/dx^ + dy% Fig. 55. 3'70 QVAbnAWRt! OP thM spMerM. the integral being taken between proper limits. If for example, we want the surface generated by the curve be- tween the two ordinates whose abscissas are a and t, where a > 5, we hare In like manner it may be shown that to find the surface generated by revolving the curve round the axis of y, we have S = 2tt J xds. 194. The Sphere. — Prom the equation of the gener- ating curve, x^ + y^ = r^, we have « = (f' — a;^)^ and ~ ■= ; n ^ ' Ax y^ .; S = %n J y{l + -S dx = 2tt J rdx = 2nrx + G. Hence, the surface of the zone included between two planes corresponding to the abscissas a and 5 is S =2Tr frdx = 27:r (a — b); that is, the area of the zone is the product of the circum- ference of a great circle by the height of the zone. To find the surface of the whole sphere, we integrate between + r and — r for the .^-limits ; hence we have S =2nr J dx = 2nr [r — (— r)] = inr^; that is, the whole surface of the sphere is four times the area of a great circle. Remark. — If a cylinder be circumscribed about a sphere, its convex surface is equal to 2ir?- x2r — iwr^, which is the same as the surface of the sphere. If we add Sirr'' to this, which is the sum of the areas of the two bases, we ishall have for the whole surface of the cylinder QUADBATUHE OF PARABOLOID OF REVOLUTION. 371 Qnr'. Hence the whole surface of the cylinder is to the surface of the sphere as 3 is to 2. This relation between the surfaces of these two bodies, and also the same relation between the volumes, was discovered by Archimedes, who thought so much of the discovery that he ex- pressed a wish to have for the device on his tombstone, a sphere inscribed in a cylinder. Archimedes was killed by the soldiers of Marcellus, B. c. 813, though contraiy to the orders of that general. The great geometer was buried with honors by Marcellus, and the device of the sphere and cylinder was executed upon the tomb. 140 years afterward, when (Jicero was questor in Sicily, he found the monument of Archimedes, in the shape of a small pillar, and showed it to the Syracusans, who did not know it was in being , he says it was marked with the figure of a sphere inscribed in a cylinder. The sepulchre was almost overrun with thorns and briars. See article '■ Marcellus," in Plutarch's Lives, Vol. HI, p. 120. 195. The Paraboloid of Revolution. — From the equation of the generating cuiYe if = "ipx, we have y = VW^, and | = i ^/l' ••• S = 2^^/ V3^ (l + Y^^dx = 2WpX{p + 2x)i dx = [i^ Vp(p + ^^)^]= i^ ^p Up + 3^)^ - /] . (1) which is the surface generated by the revolution of the part of the parabola between its vertex and the point h> y)- We might have found the surface in terms of y instead of X, as follows : dx _y dy~p' 372 TME PROLATE SI'BERaiD. = ^i{f + y^Y^ - p^l which result agrees with (1), as the student can easily Terify. 196. The Prolate Spheroid (See Anal. Geom., Art. 191). — From the equation of the generating curTC yi = (1— e2) {at — z% we have 27r yds = 2n Vl — e' Va^ — ^ ds — 2tt Vl — e^ Va^ — e# dx (Art. 175.) = ^TT - e ( - — x^y dx, therefore for half the surface of the ellipsoid, since the tc-limits are a and 0, we have a «/o \e^ I \e^ I d' 2Trbe ex + jj-s sm' -1 : 2 ' 2e« a_ (See Ex. 4, Art. 151.) = .4(i_..)i + .«y^n = TTtr -\ sm 1 e e 197. The Catenary. — From the equation of the gen- erating curve, a / ; , -;\ SURFACE GENERATED BY THE CYCLOID. 373 we have for the surface of revolution around the axis of x between the Umits x and 0, S = 2n I yds = na J le» + e "j ds = i-TO r(e» + e""«) dx (by Art. 177) Tia f^l — — ^\ '0 2x = Tr— le" — e -j + ax] =^T(\(^s + ax), (where s = VP, Fig. 45.) 198. The Smface generated by the Cycloid when it revolves around its axis. — From its ecfuation ' y = r vers~i - + V^rx — x\ (1) we have dx~\ X ' (3) * = (i + g)'^ = \/l'^- (3) .-. yS = 277 fyds = 2Tr fy y ^ dx. W Put u = y, dv — \ — dx; y 2/ .'. du = dy, and v = 2 VSra; ; therefore (by Art. 147) we have 374 POLAR CO-ORDIXATES. = 2y 's/Wx — 2 V^rJ'VJr — x dx [by (2)] = 2 ^/2rx \r vers-i - + ^/%rx — ^j + f V2r (2r — a;)t [by (1) and integrating.] which in (4) gives S = SttV — -^TTz-s = 87rr2 (tt — ^). 199. Surfaces of Revolution in Polar Co-ordi- nates. — If the surface is generated by a curve referred to polar co-ordinates, its area may be determined as follows : Let the axis of revolution be the initial line OX, see Fig. 54, and from P (r, S) draw PM perpendicular to OX. Then PM. = r sin Q, and the infinitesimal element PQ = ds will, in its revolution round OX, generate an infini- tesimal element of the whole surface, whose breadth = ds and whose circumference = 27rr sin B. Hence, 8 = J^nr sin d ds* = '^-^ fr sin 6 (r» + ^^dO, (Art. 179) the integral being taken between proper limits. 200. The Cardioide.— From Art. 181, we have ds = a{2 + 2 cos 6)i dO = 2a cos ^ d6. • This expression might have been obtained at once by substituting in Art. 193, for y, Its value r sin 9, DOUBLE INTEGRATION. 375. For the surface of reyolution of the whole curve about the initial line, we have tt and for the limits of e, there- fore we have S = / 2nr sin d ds = lira^ / (1 + cos 6) cos -r sin 6 P' d = IQna' / cos* ^ sin jr dO ^fl 201. Any Curved Surfaces. — Double Integration.— Let (x, y, z) and (x + dx, y -\- dy, z + dz) be two consecu- tive points p and q on the sur- face. Through p let planes be drawn parallel to the two planes xz and yz ; also through q let two other planes be drawn par- allel respectively to the first. These planes will intercept an infinitesimal element pq of the curved surface, and the projec- tion of this element on the plane of xy wiU be the infini- tesimal rectangle PQ, which = dx dy. Let S represent the required area of the whole surface, and dS the area of the infinitesimal element pq, and denote by a, (i, y, the direction angles* of the normal at p (x, y, z). Then, since the projection of d8 on the plane of xy is the rectangle PQ = dx dy, we have by Anal. Geom., Art. 168, dx dy = dS cos y. (1) Fig. 56. * See Anal. Geom., Art. 170. .376 SURFACE OF A SPHERE. Similarly, if d8 is projected on the planes yz and zx, we have dy dz = dS cos a ; (3) dz dx = dS cos j3. (3) Squaring (1), (3) and (3), and adding, and extracting the square root, we have dS = {dx>dy^ + dfdz^ + dz''dx^)i (since cos' a + cos' /S + cos' 7 = 1, Anal. Geom,, Art. 170). .-. S = f fidx^dy^ + dyHz^ + dzHx^)k r ri, dz^ dz^ w , , = J Jv + ^+df)'^'''^^' the limits of the integration depending upon the portion of the surface considered. 202. The Surface of the Eighth Part of a Sphere.— Let the surface represented in Fig. 56 be that of the octant of a sphere ; then being its centre, its equation is x^ + y^ + z'^ = aK dz __ X dz _ y dx~ z' dy ~ z - ^^Z/^ + p + S*"^* /> z' adxdy J J v'ff' — o& — y^ Now since pq is the element of the surface, the effect of a ^/-integration, x being constant, will be to sum up all the elements similar to 'pq from H to ?; that is, from «/ = to 2/ = L^ = «/, = Va' — ^ ; and the aggre- Hence, EXAMPLES. 377 gate of these elements is the strip 'Kpl. The effect of a subsequent a;-integration will be to sum all these elemental strips that are comprised in the surface of which OAB is the projection, and the limits of this latter integra- tion must be a; = and x = OA = a. Therefore, we have Va^ — x^ — y^ adxdy ^yi _ yi = / \adx sin~' - '• pa EXAMPLES. 1. Find the convex surface of a right circular cone, whose generating line is ay — ix ^ 0. A ns. nb Va^ + ^. Remark. — It is evident that the projection of the convex sur- face of a right circular cone on the plane of its base, is equal to the base; hence it follows (Alia,!. Geom., Art. 168) that the convex surface of a right circular cone is equal to the area of its base multiplied by the secant of the angle between the slant ieight and the base. Thus, calling this angle a, we have in the above example, «■ = Trjs sec a = ttJ'-^!-^— = ttS y^TP, which agrees with the answer. 3. Find the area of the surface generated by the reTolu- tion of a logarithmic curve, y — eF, about the axis of x, between the ^-limits and y. Ans. TT 12/ (1 + /)* + log [2^ + (1 + ff^ \. 378 EXAMPLES. 3. Find the area of the surface generated by the revolu- tion of the cycloid (1) about its base, and (£) about the tangent at the highest point. Ans. (1) ^na^; {2):_3^naK 4. Find the area of the surface generated by the revolu- tion of the catenary about the axis of y, between the a;-limits and x. Ans. 2it [zs — a{t/ ^ a)]. .•. - 5 = Stt / xdsi = Sir aa — / sdx , from which we soon obtain the answer. 5. Find the area of the surface of a spherical sector, the vertical angle being 2« and the radius of the sphere = r. Ans. 4:Trr^ (^^° 1). 6. Find the area of the surface generated by the revolu- tion of a loop of the lemniscate about its axis, the equation being r« = a'- cos 20. j^ns. na^ {% — 2*). Here find rd» = a''dd ; .•. etc. 7. Find the area of the surface generated by the revolu- tion of a loop of the lemniscate about its axis, the equation being r^ = a^ sin 20. Ans. 2na\ 8. A sphere is cut by a right circular cylinder, the radius of whose base is half that of the sphere, and one of whose ^ edges passes through the centre of the sphere. Find the area of the surface of the sphere intercepted by the cylinder. Let the cylinder be perpendicular to the plane of xy ; then the equations of the cylinder and the sphere are respectively y^-=ax — x* and «* + y* -|- 2« = d^. It is easily seen that the ^-limits are and y/ax — a?-= y^, and the a;-limits are and a. Therefore, Art. 201, we have EXAMPLES. 37.) ~ Jo Jo y/a^ — x^ — y/o? — x^ — y^ f"^ . , (ax — x°)i , = a I sin"i ^^ ^ ax Jo (cfi — x^y a (Art. 147) Therefore, the whole surface = 3fl^ (tt — 2). (In Price's Calculus, Vol. II, p. 336, the answer to this example is c? {-n — 3), which is evidently only half of what it should be.) ' 9. In the last example, find the area of the surface of the cylinder intercepted by the sphere. Eliminating y, we have z = Va^ — ax for the equation of the projection on the plane xz of the intersection of the sphere and the cylinder. Therefore the 2-limits are and z, = '/W— ax, and the a;-limits are and a ; hence, Art. 201, we have \_dx^ df + df dz> + dz^ dx^f = 1 + (J)' + (g)' dxdz CL d^ dz - for an element of the surface of the cylinder. 2'v/aS--^ _fl pa p^^ dxdz _a^ P'^dx _ j,_ i« 2 i/fl t/o ^^/ax — 3? '^'^0 x^ therefore the whole area of the intercepted surface of the cylinder is 4a'. (See Gregory's Examples, p. 436.) 10. The axes of two equal right circular cylinders inter- sect at right angles ; find the area of the one which is inter- cepted by the other. Ans. 8a*. 380 EXAMPLES. Let the axes of tlie two cylinders be taken as the axes of y and z, and let a — the radius of each cylinder. Then the equations are y? + z'2 = a?, d' + y'^ = a-. 11. A sphere is pierced perpendicularly to the plane of one of its great circles by two right cylinders, of which the diameters are equal to the radius of the sphere and the axes pass through the middle points of two radii that compose a diameter of this great circle. Find the surface of that por- tion of the sphere not included within the cylinders. Ans. Twice the square of the diameter of the sphere. 12. Find the area of the surface generated by the revolu- tion of the tractrix round the axis of x. Ans. ^■naK 13. If a right circular cone stand on an ellipse, show that the convex surface of the cone is ^ (0 A -h A') (0 A ■ A')* sin «, At where is the vertex of the cone, A and A' the extremities of the major axis of the ellipse, and a is the semi-angle of the cone at the vertex. (See Eemark to Ex. 1.) CHAPTER IX. VOLUMES OF SOLIDS. 203. Solids of Revolution. — Let the curve AB, Fig. 55, revolve round the axis of x, and let V denote the volume of the solid bounded by the surface generated by the curve and by two planes perpendicular to the axis of x, one through A and the other through P ; then as MP and NQ are consecutive ordinates, the volume generated by the revo- lution of MPQN round the axis of x is an infinitesimal element of the whole volume, and is the frustum of a cone, the circumferences of whose bases are Stt^ and 2n [y -\- dy) , and whose altitude is MN = dx; therefore we have ^^^ .f + .iy + dyy^-.y(y + dy)^^ ^ ^^,^^^ by omitting infinitesimals of the second order. Hence, for the whole volume generated by the area between the two ordinates whose abscissas are a and b, where ay h, we have ■ny^dx. In like manner, it may be shown that to find the volume generated by revolving the arc round the axis of y, we have V ^ -rr I xMy. 204. The Sphere. — Taking the origin at the centre of the sphere, we have y' = a^ — a;^ ; therefore we have F = TT ^{a^ — x^) dx 'J —a 77 {o?X — ^X^) a —a for the whole volume of the sphere. 383 VOLUME GENERATED BT CYCLOID. OoK. 1. — To find the volume of a spherical segment be- tween two parallel planes, let b and c represent the distances of these planes from the centre ; then we have V = TT f''{a^ - x^)dx = 7T [a^ (6 - c) — i(63 — cS)]. Cor. 2. — To find the volume of a spherical segment with one base, let h be the altitude of the segment; then 5 = a and c = a — h, and we have r=n f {a^~ x^) dx = n¥ (a — 5)- Cob. 3. ^-na^ = f of na^ x 2fl = | of the circumscribed cylinder. (See Art. 194, Remark.) 205. The Volume generated by the Revolution of the Cycloid about its Base. Here dx = ^^^ (Art. 176) ; V'iry — «/2 and integrating between the limits y = Q and y = 2r, we find for the whole volume V=2n (by Ex. 6, Art. 151) = 2n^rf^ V^ry — y" 2'- yMy V^ry — y^ = -yirn (!?•%) (by Ex. 6, Art. 151) We have SttV = ^n {2rY x 2TTr. Hence, the volume generated by the revolution of the cycloid about its base is equal to five-eighths the circumscribing cylinder. SOLIDS BOUNDED BY AN'Y CURVED SURFACE. 383 206. The Cissoid when it revolves round its Asymptote. — Here OM = x, MP = y, OA = 2a, MA = 2a — x, KD = dy; hence an infinitesimal element of the whole volume is genei-ated by the revo- lution of PQDH about AT, and is 'A y^esented by n {2a — x)~dy. The equation of the Cissoid is Fig. 57. r 2a . , _ (3«-a;) (a az-^)^ •• '^y - (2a-xf ''^' hence, between the limits x = and x = 2a, we have V =2n / (2a — xfdy = 2- {3a— x) {2ax—x^)i dx '2a ^a^x — 5ax^ + x^ 2n J V2ax — x^ ■dx 2-n^a^ (by Ex. 6, Art. 151). 207. Volume, of Solids bounded by any Curved Surface.— Let {x, y, z) and {x+dx, y + dy, z+dz) be two consecutive points E and F within the space whose volume ig to be found. Through E pass three planes parallel to the co-ordinate planes xy, yz, and zx; also through F pass three planes parallel to the first. The solid included by these six planes is an infinitesi- mal rectangular parallelopipe- don, of which E and F are two opposite angles, and the volume is dxdy dz ; the aggregate of all these solids between Fig. 58. 384 TRIFLE INTEGRATION. the limits assigned by the problem is the required volume. Hence, if V denote the required volume, we have F = / / I dx dy dz, the integral being taken between proper limits. In considering the effects of these successive integrations, let us suppose that we want the volume in Fig. 58 contained within the three co-ordinate planes. The effect of the 2-integration, x and y remaining con- stant, is the determination of the volume of an infinitesimal prismatic column, whose base is dxdy, and whose altitude is given by the equations of the bounding surfaces ; thus, in Fig. 58, if the equation of the surface is z^f{x,y), the limits of the ^-integration are /(a;, y) and 0, and the volume of the prismatic column whose height is P^ is f{x, y) dx dy ; hence the integral expressing the volume is now a double integral and of the form V^ J J f(x, y) dx dy. If we nowintegrate with respect to y, x remaining con- stant, we sum up the prismatic columns which form the elemental slice Uplmq, contained between two planes per- pendicular to the axis of x, and at an infinitesimal distance (dx) apart. The limits of y are hi and 0, LZ being the y to the trace of the surface on the plane of xy, and which may therefore be found in, terms of x by putting 2 = in the equation of the surface ; or, if the volume is included be- tween two planes parallel to that of xz, and at distances «/o and yi from it, «/o and y^ being constants, they are in that case the limits of y ; in the same way we find the limits if the bounding surface is a cylinder whose generating lines are parallel to the axis of s. In each of these cases the result of the z/-integration is the volume of a slice included between two planes at an infinitesimal distance apart, the length of which, measured parallel to the axis of y, is a BXAilPLES. 385 function of its distance from the plane of yz ; thus the limits of the ^-integration may be functions of x, and we shall have V = J J f{x, y) dx dy = J F{x) dx, where F{x) dx is the infinitesimal slice perpendicular to the axis of a; at a distance x from the origin, and the sum of all such infinitesimal slices taken between the assigned limits is the volume. Thus, if the volume in Pig. 58 be- tween the three co-ordinate planes is required, and OA = a, then the a-limits are a and 0. If the volume contained between two planes at distances x^ and «, from the plane of ■ yz is required, then the a-limits are x^ and x^. EXAMPLES. 1. The ellipsoid whose equation is x^ y"^ .^ _ -, a2 + j2 + ^ - 1- 1 j — 12) and 0, which call z^ and 0; the limits of y are \A = 1 — —J ^^^ ^> ^^ic^ ^^^^ yi ^^^ ; the a;-limits are a and 0. First integrate with respect to z, and we obtain the infini- tesimal prismatic column whose base is PQ, Fig. 58, and ■whose height is Pp. Then we integrate with respect to y, and obtain the sum of all the columns which form the elemental slice Uplmq.. Then integrating with respect to x, we obtain the sum of all the slides included in the solid OABC. .-. V = 8 f" f"^ f'dx dy dz •'0 «^o «^0 17 ^' 7 ax 386 EXAMPLES. = -bJ, J„ W-y')'p'ds _8c r^y?^. -~bJo '^'i = — J- / (ffl^ — a;') b , „ cb-T /'«, „ , = 8—;- I dz = 2abcn. (See Price's Calculus, Vol.11, p. 356.) 3. The volume of the solid cut from the cylinder x^ + y^ = a^ by the planes 2 = and 2 = a; tan «. Here the z-limits are x tan a and 0, or «, and 0; the ^/-limits are (rf^ — .t^)^ and — {a^ — x^)i, or y, and — y, ; the a;-liraits are a and 0. HJ^AMPiES. 38'}' A ^= f f^^ f^dx dy dz pa ny^ = / / {x tan a) dx dy — 2 tan a J X {a^ — x^)^dx — % a^ tan «. 4. The volume of the solid common to the ellipsoid ^3 + I + J = 1 and the cylinder x^ + y^ = b^. Here the limits of the ^-integration are ell ^) and 0, or «, and ; the limits of the a-integration are (53 _ ^2)1 and 0, or Xi and 0; the ^/-limits are and b* J J dy dx dz dy dx _8c r aJn .2„2 xi «'-w2/' ("-T-^') (b'-y')i dy 'I' In this ezample, this order of integration is simpler than it fvould he to tal^e It with respect to y and then x. 388 EXAMPLES. (See Mathematical Visitor, 1878, p. 36.) 208. Mixed System of Co-ordinates. — Instead of dividing a solid into columns standing on rectangular bases, so that z dx dy is the Tolume of the infinitesimal column, it is sometimes more convenient to divide it into infinitesimal columns standing on the polar element of area ahcd = r dr dd, in which case the corresponding parallelopipedon is represented by zr dr dd, and the expression for V becomes Fig. 59 V = J fzrdrde, taken between proper limits. From the equation of the surface, z must be expressed as a function of r and 0. EXAMPLES. 1. Find the volume included between the plane z = Q, and the surfaces a;^ + ^/^ = 4a3 and y'^ = 2cx — x\ Here z = a;3 + . ""la" -r- ; hence the 2-limits are ^- and 0. ia w The equation of the circle y^ = Sea; — o^, in polar co-or- dinates, is r ■= 2c cos 6 ; hence the r-limits are and 2c cos Q, or and r, ; and the ^-limits are and 5- ~ '^ Jo Jo 4:a Jo ' a Sttc* 8 a cos* 6 de dddr ^77. (Ex. 4, Art. 157.) HXAMPLES. 88d 2. The axis of a right circular cylinder of radius i, passes through the centre of a sphere of radius a, when a > J ; find the Tolume of the solid common to both surfaces.* Take the centre of the sphere as origin, and the axis of the cylinder as the axis of z; then the equations oi the surfaces are a? -\- y^ -\- z^ =: a^ and a;^ + y^ — j2. q^^ in terms of polar co-ordinates, the equation of the cylinder is r = J. Hence for the volume in the first octant, the z-limits are Va" — x^ — y^ or Va' — r^ and ; the r-limits are i and ; the 6-limits are ^ and 0. '/7 />5ir rtb \ J T{a b zr dr dd r !)* de dr de (See Gregory's Examples, p. 438.) 209. The polar element of plane area is r dr dO (Art. 308). Let this element revolve round the initial line through the angle Stt, it will generate a solid ring whose volume is Srrr sin Or dr dO, since %-nr sin is the circumfer- ence of the circle described by the point {r, 6). Let ^ denote the angle which the plane of the element in any position makes with the initial position of the plane ; then d is the angle which the plane in any position makes • This example, as well as the preceding one, might be integrated directly in terms of a; and y by the method of Art. 207, but the operation woald be more com- plex than the one adopted. 390 EXAMPLBM. with its consecutive position. The part of the solid ring which is intercepted between the revolving plane in these two consecutive positions, is to the whole ring in the same proportion as d is to 2^. Hence the volume of this intercepted part is r^ sin d d(j) dO dr, which is therefore an expression in polar co-ordinates for an infinitesimal element of any solid. Hence, for the volume of the whole solid we have V = fffr^ sin d dd dr, in which the limits of the integration must be so taken as to include all the elements of the proposed solid. In this formula r denotes the distance of any point from the origin, 8 denotes the angle which this distance makes with some fixed right line through the origin (the initial line), and denotes the angle which the plane passing through this distance and the initial line makes with some fixed plane passing through the initial line. (See Lacroix Cal- cul Int6gral, Vol. II, p. 309.) The order in which the integrations are to be effected is theoretically arbitrary, but in most eases the form of the equations of surfaces makes it most convenient to integrate first with respect to r ; but the order in which the 0- and ^-integrations are effected is arbitraiy. EXAMPLES. 1. The volume of the octant of a sphere. Let a = the radius of the sphere ; then the limits of r are and a ; hence, V = yy J' sin e dtji de. In thus integrating with respect to r, we collect all the elements like r^ sin d d^ dQ dr which compose a pyramidal EXAMPLES. 391 solid, having its vertex at the centre of the sphere, and for its base the curvilinear element of spherical surface which is denoted by a^ sin 6 d^ dd. Integrating next with respect to between the limits and - , we have ■F^/f[(-cos.)]% = /j^#. In thus integrating with respect to 0, we coUect all the pyramids similar to ^ sin 6 d dd, which form a wedge- shaped slice of the solid contained between two consecutive planes through the initial line. Lastly, integrating with respect to

dd dr «/o '-'0 'Jq = / ~ Bind d^dd t/O t'O " = / - (i — COS a)d^ '0 = fiftS (1 _ cos a). 393 EXAMPLES. EXAMPLES. 1. Find the volume of a paraboloid of reyolution whose altitude = a and the radius of whose base = b. Ans. -^ai^. 2. Find the volume of the prolate spheroid. * Also of the oblate spheroid. Ans. The prolate spheroid — ^naV: The oblate spheroid = ^nc^l. 3. Find the volume of the solid generated by the revolu- tion of y =: cf about the axis of x, between the limits x and -00, where «>1. Ans. I a^ {log a)-^ 4. Find the volume of the solid generated by the revolu- tion of y = a log X about the axis of x, between the limits X and 0. Ans. ■"C?x (log^ x — 3 log x + 3). 5. Find the volume of the solid generated by the revolution of the tractrix round the axis of x. Ans. \nc^. 6. Find the volume of the solid generated by the revolution of the catenary round the axis of x. Ans. ^ a {ys -|- ax). (Compare with Art. 197.) 7. Find the volume generated by the revolution of a parabola about its base 2b, the height being h.* (See Art. 306.) Ans. \^nbh\ 8. The equation of the Witch of Agnesi being y find the volume of the solid generated by its revolution round the asymptote. Ans. i^n^aK * This solid is called a parabolic spindle. EXAMPLES. 393 9. Find the volume of a rectangular parallelopipedon, three of whose edges meeting at a point are a, h, c. (See Art. 207.) j^ns. abc. 10. Find the yolume contained within the surface of an elliptic paraboloid * whose equation is ^ + - = 2a;, a b and a plane parallel to that of yz, and at a distance c from it. Ans. n(^{ab)i. 11. The axes of two equal right circular cylinders inter- sect at right angles, their equations being x^ + z^ = a^ and a? -\- y^ = a^; find the volume of the solid common to both. Ans. ^aK 13. A paraboloid of revolution is pierced by a right cir- cular cylinder, the axis of which passes through the focus and cuts the axis of the paraboloid at right angles, the radius of the cylinder being one-fourth the latus-rectum of the generating parabola ; find the volume of the solid com- mon to the two surfaces. , „/2 ■n\ . Ans.p''[^ + -^. Here the equations of the surfaces are y^ + e^ = 2px and x^+y^ =px. 13. Find the volume of the solid cut from the cylinder 3? + y^= 2az by the planes z = x tan « and z=^x tan /?. Ans. 2 (tan /J — tan «) -^ • 14. Find the volume of the solid common to both sur- faces in Ex. 8 of Art. 202. (See Art. 208.) Ans. 1(377 — 4) a8. 15. Find the volume of the part of the hemisphere in the last example, which is not comprised in the cylinder. Ans. fa'. * Called elliptic paraboloid becanse the sections made by planes parallel to the planes of xy and xz are parabolas, while those narallel to the plane of yz are ellipseB. (Salmon's Anal. Geom. of Three Dimensions, p. S8.) 394 EXAMPLES. 16. Find the volume of the solid intercepted between the concave surface of the sphere and the conveX'^.rface of the cylinder in Art. 208, Ex. 2. _^ns. ^tt (a^ _ ja)!. 17. Find the volume of the solid comprised between the SiTirface z^ae "°~ and the plane of a;«/. Ans. ■na(?. Here the r-limits are and oo ; and the fl-limits are and 3t. 18. Find the volume of the solid generated by the revo- lution of the cardioide r = a (1 + cos S) about the initial line. Here V ■= I I I r^ sm 6 dO d(f) dr ■= etc. (See Art. 209.) , 8-rra^ Ans. — 5— • 19. Find the volume of the solid generated by the revo- lution of the Spiral of Archimedes, r = aO, about the initial line between the limits 6 =.tt and = 0. Ans. fTT^ftS (ttS _ 6). 20. A right circular cone whose vertical angle = 2a, has its vertex on the surface of a sphere of radius a, and its axis coincident with the diameter of the sphere ; find the volume common to the cone and the sphere. . Arra^ ,^ , . Ans. —5- (1 — cos* a). 21. Find the volume of a chip cut at an angle of 45° to the centre of a round log with radius r. (Mathematical Visitor, 1880, p. 100.) Ans. ir\ 22. Find the volume bounded by the surface and the positive sides of the three co-ordinate planes. . ahc Ans. ^. SXAMPLES. 395 23. Find the volume of the solid bounded by the three surfaces x^ + y^ ^ cz, x' + y^ = ax, and z = 0. 3TTa* ^"^- 33c- 24. A paraboloid of revolution and a right circular cone have the same base, axis, and vertex, and a sphere is described upon this axis as diameter. Show that the volume intercepted between the paraboloid and cone bears the same ratio to the volume of the sphere that the latus-rectum of the parabola bears to the diameter of the sphere. 25. Find the volume included between a right circular cone whose vertical angle is 60° and a sphere of radius r touching it along a circle, by the formula F = / / I dx dy dz. Ans. -g- 26. In the right circular cone given in Ex. 13 of Art 202, prove that its volume is represented by ■ /..:■::.. ;.XIiiil