GIFT OF Ll -^ >£ Digitized by the Internet Archive in 2008 with funding from ■^ IVIicrosoft Corporation http://www.archive.org/details/elementsofdifferOOnewcrich NEWCOMB'S MATHEMATIGA'L COURSE ELEMENT'S DIFFERENTIAL AND INTEGRAL CALCULUS BY SIMOI^ NEWCOMB Professor of Mathematics in tJie Johns Hopkins University NEW YORK HENRY HOLT AND COMPANY 1887 • e « • .«t • ••• • . • • < cc* «• «*•* •••••• • Copyright, 1887, BY HENRY HOLT & CO. GIFT OF ENGINEERING LIBRARY lU PREFACE. The present work is intended to contain about as much of the Calcuhis as an undergraduate student, either in Arts or Science, can be expected to master during his regular course. He may find more exercises than he has time to work out; in this case it is suggested that he only work enough to show that he understands the principles they are designed to elucidate. The most difficult question which arises in treating the subject is how the first principles should be presented to the mind of the beginner. The author has deemed it best to be- gin by laying down the logical basis on which the whole superstructure must ultimately rest. It is now well under- stood that the method of limits forms the only rigorous basis for the infinitesimal calculus, and that infinitesimals can be used with logical rigor only when based on this method, that is, when considered as quantities approaching zero as their limit. When thus defined, no logical difficulty arises in their use; they flow naturally from the conception of limits, and they are therefore introduced at an early stage in the present work. The fundamental principles on which the use of infinitesi- mals is based are laid down in the second chapter. But it is not to be expected that a beginner will fully grasp these j^rin- ciples until he has become familiar with the mechanical pro- cess of differentiation, and with the application of the calcu- 8G5679 IV ^BEFACE. lus to special problems. It may therefore be found best to begin with a single careful reading of the chapter, and after- ward to use it for reference as the student finds occasion to apply the principles laid down in it. The author is indebted to several friends for advice and assistance in the final revision of the work. Professor John E. Clark of the Sheffield Scientific School and Dr. Fabian Franklin of the Johns Hopkins University supplied sugges- tions and criticisms which proved very helpful in putting the first three chapters into shape. Miss E. P. Brown of Wash- ington has read all the proofs, solving most of the problems as she went along in order to test their suitability. CONTENTS. PART I. THE DIFFERENTIAL CALCULUS. PAGE Chapter I. Op Variables and Functions 3 § 1. Nature of Functions. 2. Their Classification. 3. Func- tional Notation. 4. Functions of Several Variables. 5. Func- tions of Functions. 6. Product of the First n Numbers. 7. Bi- nomial Coefficients. 8. Graphic Representation of Functions. 9. Continuity and Discontinuity of Functions. 10. Many-valued Functions. Chapter II. Of Limits and Infinitesimals 17 §11. Limits. 12. Infinites and Infinitesimals. 13. Properties. 14. Orders of Infinitesimals. 15. Orders of Infinites. Chapter III. Op Differentials and Derivatives 25 § 16. Increments of Variables. 17. First Idea of Differentials and Derivatives. 18. Illustrations. 19. Illustration by Velocities. 20. Geometrical Illustration. Chapter IV. Differentiation of Explicit Functions 31 § 21. The Process of Differentiation in General. 22. Differen- tials of Sums. 23. Differential of a Multiple. 24. Differential of a Constant. 25. Differentials of Products and Powers. 26. Dif- ferential of a Quotient of Two Variables. 27. Differentials of Ir- rational Expressions. 28. Locjarithmic Functions. 29. Expo- nential Functions. 30. The Trigonometric Functions. 31. Cir- cular Functions. 32. Logarithmic Differentiation. 33. Velocity or Derivative with Respect to the Time. Chapter V. Functions op Several Variables and Impli- cit Functions 54 § 34. Partial Differentials and Derivatives. 35. Total Differen- tials. 36. Principles involved in Partial Differentiation. 37. Dif- IV PREFACE. lus to special problems. It may therefore be found best to begin with a single careful reading of the chapter, and after- ward to use it for reference as the student finds occasion to apply the principles laid down in it. The author is indebted to several friends for advice and assistance in the final revision of the work. Professor John E. Clark of the Shefiield Scientific School and Dr. Fabian Franklin of the Johns Hopkins University supplied sugges- tions and criticisms which proved very helpful in putting the first three chapters into shape. Miss E. P. Brown of Wash- ington has read all the proofs, solving most of the problems as she went along in order to test their suitability. CONTENTS. PART I. THE DIFFERENTIAL CALCULUS. PAGE Chapter I. Op Variables and Functions 3 § 1. Nature of Functions. 2. Their Classification. 3. Func- tional Notation. 4. Functions of Several Variables. 5. Func- tions of Functions. 6. Product of the First n Numbers. 7. Bi- nomial Coefficients. 8. Graphic Representation of Functions. 9. Continuity and Discontinuity of Functions. 10. Many-valued Functions. Chapter II. Of Limits and Infinitesimals 17 §11. Limits. 12. Infinites and Infinitesimals. 13. Properties. 14. Orders of Infinitesimals. 15. Orders of Infinites. Chapter III. Of Differentials and Derivatives 25 § 16. Increments of Variables. 17. First Idea of Differentials and Derivatives. 18. Illustrations. 19. Illustration by Velocities. 20. Geometrical Illustration. Chapter IV. Differentiation of Explicit Functions 31 § 21. The Process of Differentiation in General. 22. Differen- tials of Sums. 23. Differential of a Multiple. 24. Differential of a Constant. 25. Differentials of Products and Powers. 26. Dif- ferential of a Quotient of Two Variables. 27. Differentials of Ir- rational Expressions. 28. Loi^arithmic Functions. 29. Expo- nential Functions. 30. The Trigonometric Functions. 31. Cir- cular Functions. 32. Logarithmic Differentiation. 33. Velocity or Derivative with Respect to the Time. Chapter V. Functions op Several Variables and Impli- cit Functions 54 § 34. Partial Differentials and Derivatives. 35. Total Differen- tials. 36. Principles involved in Partial Differentiation. 37. Dif- VI CONTENTS. PAGE ferentiation of Implicit Functions. 38. Implicit Functions of Sev- eral Variables. 39. Case of Implicit Functions expressed by Simultaneous Equations. 40. Functions of Functions. 41. Func- tions of Variables, some of which are Functions of the Others. 42. Extension of the Principle. 43. Nomenclature of Partial Derivatives. 44. Dependence of the Derivative upon the Form of the Function. Chapter VI. Derivatives of Higher Orders 74 §45. Second Derivatives. 46. Derivatives of Any Order. 47. Special Forms of Derivatives of Circular and Exponential Functions. 48. Successive Derivatives of an Implicit Function. 49. Successive Derivatives of a Product. 50. Successive Deriva- tives with Respect to Several Equicrescent Variables. 51. Result of Successive Differentiations independent of the Order of the Differentiations. 52. Notation for Powers of a Differential or Derivative. Chapter VII. Special Cases of Successive Derivatives. . . 86 § 53. Successive Derivatives of a Power of a Derivative. 54. De- rivatives of Functions of Functions. 55. Change of the Equicres- cent Variable. 56. Two Variables connected by a Third. Chapter VIII. Developments in Series 95 -§ 57. Classification of Series. 58. Convergence and Divergence of Series. 59. Maclaurin's Theorem. 60. Ratio of the Circum- ference of a Circle to its Diameter. 61. Use of Symbolic Nota- tion for Derivatives. 62. Taylor's Theorem. 63. Identity of Taylor's and Maclaurin's Theorems. 64. Cases of Failure of Taylor's and Maclaurin's Theorems. 65. Extension of Taylor's Theorem to Functions of Several Variables. 66. Hyperbolic Functions. Chapter IX. Maxima and Minima of Functions of a Sin- gle Variable. 117 §67. Definition of Maximum Value and Minimum Value. 68. Method of finding Maximum and Minimum Values of a Func- tion. 69. Case when the Function which is to be a Maximum or Minimum is expressed as a Function of Two or More Variables connected by Equations of Condition. Chapter X. Indeterminate Forms 128 §70. Examples of Indeterminate Forms. 71. Evaluation of the Form— . 72. Forms - and X oo. 73. Form oo — cx>. CO 75. Forms 0° and oo ". CONTENTS, vii PAOB Chapter XI. Op Plane Curves 137 §76. Forms of the Equations of Curves. 77. Infinitesimal Ele- ments of Curves. 78. Properties of Irrfinitesimal Arcs and Chords. 79. Expressions for Elements of Curves. 80. Equa- tions of Certain Noteworthy Curves. The Cycloid. 81. The Lemniscate. 82. The Archimedean Spiral. 83. The Logarith- mic Spiral. Chapter XII. Tangents and Normals 147 § 84. Tangent and Normal compared with Subtangent and Subnormal. 85. General Equation for a Tangent. 86. Sub- tangent and Subnormal. 87. Modified Forms of the Equation. 88. Tangents and Normals to the Conic Sections. 89. Length of the Perpendicular from the Origin upon a Tangent or Normal. 90. Tangent and Normal in Polar Co-ordinates. 91. Perpendicular from the Pole upon the Tangent or Normal. 92. Equation of Tangent and Nonnal derived from Polar Equation of the Curve. Chapter XIII. Of Asymptotes, Singular Points and Curve-tracing 157 §93. Asymptotes. 94. Examples of Asymptotes. 95, Points of Inflection. 96. Singular Points of Curves. 97. Condition of Singular Points. 98. Examples of Double points. 99. Curve- tracing. Chapter XIV. Theory of Envelopes 169 §100. Envelope of a Family of Lines. 101. All Lines of a Family tangent to the Envelope. 102. Examples and Applications. Chapter XV. Of Curvature, Evolutes and Involutes 180 §103. Position; Direction; Curvature. 104. Contacts of Differ- ent Orders. 105. Intersection or Non-intersection of Curves ac- cording to the Order of Contact. 106. Radius of Curvature. 107. The Osculating Circle. 108. Radius of Curvature when the Abscissa is not taken as the Independent Variable. 109. Ra- dius of Curvature of a Curve referred to Polar Co-ordinates. 110. Evolutes and Involutes. 111. Case of an Auxiliary Variable. 112. The Evolute of the Parabola. 113. Evolute of the Ellipse. 114. Evolute of the Cycloid. 115. Fundamental Properties of the Evolute. 116. Involutes. VUl CONTENTS. PAET II. THE INTEGRAL CALCULUS. PAGE Chaptek I. The Elementary Forms op Integration 201 § 117. Definition of Integration. 118. Ai-bitrary Constant of Integration. 119. Integration of Entire Functions. 120. The Logarithmic Function. 121. Another Method of obtaining the Logarithmic Integral. 122. Exponential Functions. 123. The Elementary Forms of Integration. Chapter II. Integrals immediately reducible to the Elementary Forms 209 § 124. Integrals reducible to the Form / y^dy. 125. Appli- cation to the Case of a Falling Body. 126. Reduction to the Loga- rithmic Form. 127. Trigonometric Forms. 128. Integration of -^r-^ — ? and —^ ^. 129. Integrals of the Form / ——: — ; — ■, 130. Inverse Sines and Cosines as Integrals. 131. Two Forms of Integrals expressed by Circular Functions. 132. Integration of dx diX — . 133. Integration of - . 134. Exponen- ^cb^ ipx'^ ya-\-bx ± cx'^ tial Forms. Chapter III. Integration by Rational Transformations. . 222 § 135. Integration of ^^ — ' dx, - — t-t-t- and xdx a!~ ' {a-]-1)xY a -\-hx ± cx^' 136. Reduction of Rational Fractions in general. 137. Integra- tion by Parts. Chapter IY. Integration of Irrational Algebraic Dif- ferentials 283 §138. When Fractional Powers of the Independent Variable enter into the Expression. 139. Cases when the Given Differen- tial Contains an Irrational Quantity of the Form ^a -\-'bx -\- cx^. dv 140. Integration of (Z9 = — . 141. General Theory r \/ar'' + hr-l of Irrational Binomial Differentials. 142. Special Cases when m -\- 1 = n, or m -]- 1 -\- np = — n. 143. Forms of Reduction of Irrational Binomials. 144. Formulae A and B, in which 7n is increased or diminished by n. 145. Formulae C and D, in which p is increased or diminished by 1. 146. Effect of the Formulae. 147. Case of Failure in this Reduction. CONTENTS. ix PAGE Chapter V. Integration op Transcendent Functions 246 § 148. Integration of / e^"-^ cos nxdx and / Cmx sin nxdx. 149. Integration of sin"* a? cos" .cdc. 150. Special Cases of / sin*" a? dx co^^xdx. 151. Integration of ., . „ , — ^ t-- 1^2. Integra- ° W sm^ X + n^ COS'' x ^ dy tion of — ■ — f- . 153. Special Cases of the Last Two Forms. a -\- cos y 154. Integration of sin mx cos nxdx. 155. Integration by Devel- opment in Series. Chapter VI. Of Defenite Integrals 255 §156. Successive Increments of a Variable. 157. Differential of an Area. 158. The Formation of a Definite Integral. 159. Two Conceptions of a Definite Integral. 160. Differentiation of a Definite Integral with respect to its Limits. 161. Examples and Exercises in finding Definite Integrals. 162. Failure of the Method when the Function becomes Infinite. 163. Change of Variable in Definite Integrals. 164. Subdivision of a Definite In- tegral. 165. Definite Integrals through Integration by Parts. Chapter VII. Successive Integration 272 § 166. Differentiation under the Sign of Integration. 167. Ap- plication of the Principle to Definite Integrals. 168. Integration by means of Differentiating Known Integrals. 169. Application to a Special Case. 170. Double Integrals. 171. Value of a Func- tion of Two Variables obtained from its Second Derivative. 172. Triple and Multiple Integrals. 173. Definite Double Inte- grals. 174. Definite Triple and Multiple Integrals. 175. Product of /-foo e-x^dx. QO Chapter VIII. Rectification and Quadrature 285 § 177. The Rectification of Curves. 178. The Parabola. 179. The Ellipse. 180. The Cycloid. 181. The Archimedean Spiral. 182. The Logarithmic Spiral. 183. The Quadrature of Plane Figures. 184. The Parabola. 185. The Circle and the Ellipse. 186. The Hyperbola. 187. The Lemniscate. 188. The Cycloid. Chapter IX. The Cubature op Volumes 297 §189. General Formula?. 190. The Sphere. 191. The Pyra- mid. 192. The Ellipsoid. 193. Volume of any Solid of Revolu- tion. 194. The Paraboloid of Revolution. 195. The Volume gen- erated by the Revolution of a Cycloid around its Base. 196. The Hypcrboloid of Revolution of Two Nai>pes. 197. Ring-shaped Solids of Revolution. 198. Application to the Circular Ring. 199. Quadrature of Surfaces of Revolution. 200. Examples of Surfaces of Revolution. PART I. THE DIFFERENTIAL CALCULUS. USE OP THE SYMBOL = The symbol = of identity as employed in this work indi- cates that the single letter on one side of it is used to repre- sent the expression or thing defined on the other side of it. When the single letter precedes the symbol e, the latter may commonly be read is j^ut for, or is defined as, Wnen the single letter follows the symbol, the latter may be read loliicli let ns call. In each case the equality of the quantities on each side of = does not follow from anything that precedes, but is assumed at the moment. But having once made this assumption, any equations which may flow from it are expressed by the sign =, as usual. PART I. THE DIFFERENTIAL CALCULUS. CHAPTER I. OF VARIABLES AND FUNCTIONS. 1. In the higher matliematics we conceive ourselves to be dealing with pairs of quantities so related that the value of one depends upon that of the other. For each value which we assign to one we conceive that there is a corresponding value of the other. For exami)le, the time required to perform a journey is a function of the distance to be travelled, because, other things being equal, the time varies when the distance varies. We study the relation between two such quantities by as- signing values at pleasure to one, and ascertaining and com- paring the corresponding values of the other. The quantity to which we assign values at pleasure is called the independent variable. The quantity whose values depend upon those of the inde- pendent variable is called a function of that variable. Example I. If a train travels at the rate of 30 miles an hour, and if we ask how long it will take the train to travel 15 miles, 30 miles, GO miles, 900 miles, etc., we shall have for the corresponding times, or functions of the distances, half aa hour, one hour, two hours, thirty hours, etc. 4 THE DIFFERENTIAL CALCULUS. In thinking thus we consider the dista7ice to be travelled as the independent variable, and the tirne as the function of the distance. Example II. If between the quantities x and y we have the equation y = 2ax'*f we may suppose . . ; ; ..iC =-. - 1, 0, + 1, + 2, + 3, etc., ^hd we shall then have ; y : '; , .' ' : ; , y — 2c:, 0, 2a, Sa, ISa, etc. Here x is taken as the independent variable, and y as the function of a;. For each value we assign to x there is a corre- sponding value of y. When the relation between the two quantities is expressed by means of an equation between symbolic expressions, the one is called an analytic function of the other. An analytic function is said to be Explicit when the symbol which represents it stands alone on one side of the equation; Implicit when it does not so stand alone. Example. In the above equation y is an explicit function of X, But if we have the equation y' + ^y = ^^% then for each value of x there will be a certain value of y, which will be found by solving the equation, considering y as the unknown quantity. Here y is still a function of x, be- cause to each value of x corresponds a certain value of y; but because y does not stand alone on one side of the equation it is called an implicit function. Kemakk. The difference between explicit and implicit functions is merely one of form, arising from the different ways in which the relation may be expressed. Thus in the two forms VARIABLES AND FUNCTIONS. 5 y = 2aa;% y — ^ax" = 0, y is the same function of x; but its form is explicit in the first and implicit in the second. An implicit function may be reduced to an explicit one by solving the equation, regarding the function as the unknown quantity. But as the solution may be either impracticable or too complicated for convenient use, it may be impossible to express the function otherwise than in an implicit form. 3. Classification of Functions, When y is an explicit function of x it is, by definition, equal to a symbolic expression containing the symbol x. Hence we may call either y or the symbolic expression the function of x, the two being equiva- lent. Indeed any algebraic expression containing a symbol is, by definition, a function of the quantity represented by the symbol, because its value must depend upon that of the sym- bol. Every algebraic expression indicates that certain operations are to be performed upon the quantities represented by the symbols. These operations are: 1. Addition and subtraction, included algebraically in one class. 2. Multiplication, including involution. 3. Division. 4. Evolution, or the extraction of roots. A function which involves only these four operations is called algebraic. Functions arc classified according to the operations whicli must be performed in order to obtain their values from the values of the independent variables upon which they depend. A rational function is one in whicli tlie only operations indicated upon or with the independent variable are those of addition, multiplication, or division. 6 THE DIFFERENTIAL CALCULUS. An entire function is a rational one in which the only in- dicated operations are those of addition and multiplication. Examples. The expression a -{- hx -\- cx^ -\- dx^ is an entire function of x, as well as of «, h, c and d. The expression , m , c a-\- X x^ -^ nx is a rational function of x^ but not an entire function of x. An irrational function of a variable is one in which the extraction of some root of an expression containing that vari- able is indicated. Example, The expressions ^a + Ixy (a + wic^ + nx^) are irrational functions of x. Functions which cannot be represented by any finite com- bination of the algebraic operations above enumerated are called transcendental. An exponential function is one in which the variable enters into an exponent. Example. The expressions (a + xY^y Q?^ are entire functions of x when n and y are integers. But they are exponential functions of y. Other transcendental functions are: Trigonometric functions, the sine, cosine, etc. IjOgarithmic functions, which require the finding of a logarithm. Circular functions, which are the inverse of the trigo- nometric functions; for example, if y = a trigonometric function of x, sin x for instance, then a; is a circular function of y, namely, the arc of which y is the sine. VARIABLES AND FUNCTIONS. 7 3. Functional Notation. For brevity and generality we may represent any function of a variable by a single symbol having a mark to indicate the variable attached to it, in any form we may elect. Such a symbol is called a functional symbol or a symbol of operation. The most common functional symbols are F, f and 0; but any signs or mode of writing whatever may be used. Then, such expressions as F(x), fix), cp{x), each mean " some symbolic expression containing a;." The variable is enclosed in parentheses in order that the function may not be mistaken for the product of a quantity FyfoTcf) by X. Identical Fimctions. Functions which indicate identical operations upon two variables are considered as identical. Example. If we consider the expression a + iy as a certain function of y, then a-{'ix is that same function of x, and a + i(x + y) is that same function of 2; + ^• When the functional notation is applied, then: Identical functions are represented by the same functional symbols. Examples. If we put F(x) = a + bx, we shall have F{y) = a + by; F(f) = a + bf; F{x' -y') = a + b{x' - f). 8 THE DIFFERENTIAL CALCULUS, In general. If we define afunctional symbol as representmg a certain function of a variable, that sa^ne symbol a])plied to a second variable will represent the expression formed by sub- stituting the second variable for the first. In applying this rule any expression may be regarded as a variable to be substituted, as, in the last example, we used x^ — 7' as a variable to be substituted for x in the original expression. EXERCISES. 1. If we put (t>{x^ = ax^i it is required to form and reduce the functions 0(x + ay), (pix - ay), (f>{x'). VARIABLES AND FUNCTIONS. 9 6. Suppose /(a;) = x", and thence form the values of /(I), /(*'), f{^% f{^% /K), f{^% 7. Let us put 0(m) ~ m{m — 1) (m — 2) (7/i — 3); thence find the values of 0(6), 0(5), 0(4), 0(3), 0(2), 0(1), 0(0), 0(-l), 0(-2). 8. Prove that if we put ct){x) ^ a^, we shall have cp{x + y) = cf>{x) X {xy) = [cp{x)]y = [0(.v)]^ 4. Functions of Several Variables. An algebraic expres- sion containing several quantities may be represented by any symbol having the letters which represent the quantities at- tached. Examples. We may put 0(x, y) = ax- by, the comma being inserted between x and y so that their product shall not be understood. We shall then have 0(m, 7l) == a77i — hi, (p{y, x) = ay — bx, the letters being simply interchanged; ct){x + y,x-y) = a{x + y)-' b{x - y) = {a- b)x + (rt + ^)y; 0(a, b) = a'' - b'; 0(J, a) = ab — ba = 0; — 5c, we shall have (p{x, z, y)=2x + dz- 5y; y, x) = 2z + 3y - 5x; 0(w, 771, — 7n) = 27)1 + 3771 -f 5771 = 10/w; 0(3, 8, 6) = 2-3 + 3-8 - 5*6 = 0. 10 THE DIFFERENTIAL CALCULUS, EXERCISES. Let us put (p{x, y) =dx — 4:7/; f{x, y) = ax + by; f(x, y^ z) ^ ax -{- by — abz. Thence form the expressions: I. cp{y, x). 2. 0(a, b). 3. 0(3, 4). 4. 0(4, 3). 5- 0(10, 1). 6. f{a, b). 7. f{b, a). 8. f{y, x). 9. /(7, - 3). to- /to -j^)- II- /(^^ ^> y)* 12. /(z^, «, 2). 13- /(«^. ^. 0* 14- f{ci\ b\ &). 15. f^—ay-b.-ab). Sometimes there is no need of any functional symbol except the parentheses. For example, the form (7;?, n) may be used to Indicate any function of m and n. EXERCISES. T i. i. I \ ^^(^ — 1) (m — 2) Let us put (m, n) = — ) -^-^ ^r^, ^ \ ^ / ^(^^i _ 1) (^^ — 2) then find the values of — I. (3, 3). 2. (4, 3). 3. (5, 3). 4. (6, 3). 5. (^. 3). 6. (8, 3). 7. (2, - 1). 8. (3, - 2). 9. (4, - 2). 5. Functions of Functions, By the definitions of the pre- ceding chapter, the expression f[4>{x)) will mean the expression obtained by substituting 0(:c) for x mf{x). We may here omit the larger parentheses and write /0(^) instead of /( (p(x)\. For example, using the notation of exercises 1 and 3 of § 3, we shall have . , , . ax^ — a x^ — 1 f(p{x) = ax' + a x' + V VARIABLES AND FUNCTIONS. 11 For brevity we use the notation Continuing the same system, we have 0»E0(0'Cr)) =0^(0(0;)); 0» = 0(0'(^))-0^(0(:r)); etc. etc. etc. Examples. 1. If 0(.t) E ax^y then 0''(^) = a{ax^y = ^V; (P'{x) = a{a'xy = aV; etc. etc. etc. 2. If f{x) ~ a-x, then f^{'^) = a — (a — x) = x; f\x) = a- r{x) =a-x; and, in general, r-\x) =r{x). Remark. The functional nomenclature may be simplified to any extent. 1. The parentheses are quite unnecessary when there is no danger of mistaking the form for a product. 2. When it is once known what the variables are, we may write the functional symbol without them. Thus the symbol may be taken to mean (px or (p{x), 6. Product of the First n Numhers, The symbol nl, called factorial n, is used to express the product of the first n num- bers, 1-2-3-...W. Thus, 1!=1; 2! = l-2=:2; 3! = 1-2-3 = 6; 4! = l-2-3-4 = 24; etc. etc. 12 THE DIFFERENTIAL CALCULUS. It will be seen that 2! = 2-1!; 3! = 3-2!; and^ in general^ 7i\ — 7i ' {n — 1)\, whatever number 7i may represent. EXERCISES. Compute the values of — I. 5! 2. 6! 4. 7! 3! 4! 5- 8! 3! 5! 3. 8! 6. Prove the equation 2 • 4 • 6 • 8 • . . . 27^ = 2^n ! 7. Prove that^ when n is even, 11, _ n{n — 2) (^ — 4). . .4*2 — - . 2^ 7 . Binomial Coefficients. The binomial coefficient n{n ~1) {71 — 2) to 5 terms 1-2-3-...5 is expressed in the abbreviated form e). the parentheses being used to distinguish the expression from n the fraction -. s EXAMPLES. /7\ 7-6-5-4-3_ (n\ n /^\ _ 7i{n — 1) (7^ — 2) 1-2-3 VARIABLES AND FUNCTIONS. 13 EXERCISES. Prove the formulae: 5 3. Ur I) 5\ _ J. %l "2! 3] '' © = i^^- ^•e)- n\ ! (« - s)\ ( n + 1 _ U + 1/ s + i) + (!) = ( H 7i + 1 r)+(i)=m Ihu 8. Graphic Representation of Fnnctions, The methods of Analytic Geometry enable us to represent functions to the eye by means of curves. The common way of doing this is to represent the independent variable by the abscissa of a point, and the corresponding value of the function by its ordinate. Let x^, a\, x^, etc., be different values of the in- dependent variable, and !/i^ y'2> y,y etc., the cor- responding values of the function. We lay off upon the axis of abscis- sas the lengths OX^, T 1 ^1' 2/i P2 .Pa j!L\ X2 X3 -X Fig. 1. OX^, OJTj, etc., equal to a:,, x^y x^, etc., and terminating at the points X„ X,, X„ etc. At each of these points we erect a perpendicular to rep- resent the corresponding value of y. The ends, P„ F^, P,, of these perpendiculars will generally terminate on a curve line, the form of which shows the nature of the function. It must be clearly seen and remembered that it is not the curve itself which represents the values of the function, but the ordinates of the curve. 14 THE DIFFERENTIAL CALCULUS. ^:^ Fig. 2. 9. Continuity and DiscontinuUy of Functions, Let us consider the graphic representation of a function in the most general way. We measure off a series of values, OX^, OX^, 0X3, etc., of the independent variable, and at the points X^, X^, X^, etc., we erect ordinates. In order that the variable ordinate may actually be a function of x it is sufficient if, for every value of the abscissa, there is a corresponding value of the ordinate. ]^ow we might conceive of such a function that there should be no relation between the different val- ues of the ordinates, but that every separate point should have its own separate ordinate, as shown in Fig. 2. If this remained true how numerous soever we made the ordi- nates, then the ends of the latter would not terminate in any curve at all, but would be scattered over the plane. Such a function would be called discontinuous at every point. Such, however, is not the kind of functions commonly considered in mathematics. The functions with which we are now concerned are such that, however irregular they may appear when the values of x are widely separated, the ends of the ordinates will terminate in a curve when we bring those values close enough together. If a function is such that when the point representing the independent variable moves continuously from X^ to X^ (Fig. 1) the end of the ordinate describes an unbroken curve, then we call the function continuous between the values x^ and x^ of the independent variable. If the curve remains unbroken how far soever we suppose X to increase, positively or negatively, we call the function continuous for all vahies of the independent variable. VARIABLES AND FUNCTIONS. ir> But if there is a value a of x for which there is a break of any kind in the curve, we call the function discontmuous for the value a of the independent variable. Let us, for example, consider the function y = 6{a — x)' Let us measure off on the axis of abscissas the length OX = a . Then as we make our varying ordinate approach X from the left it will increase positively without limit, and the curve will extend upwards to infinity; if we approach Xfrom the right-hand side, the ordinate will be negative and the curve will go downwards to infinity. Thus the curve will not form a continuous branch from the one side to the other. Thus the above function is disco?ifi7mous for the value a of x. Fio. 3. 10. Many-valued Functions, In all that precedes, we have spoken as if to each value of the independent variable corresponded only one value of the function. But it may 16 THE DIFFERENTIAL CALCULUS, happen that there are several such values. For example, if y is an implicit function of x represented by the equation y^ + mxy"^ -f '^^^'^V + P^^ == ^> then we know^, by the theory of equations, that there will be three values of y for each value assigned to the variable x, Def. According as a function admits of one, two or n values, it is called one-valued, two-valued or 7i- valued. Infinitely -vahied Functioyis, It may happen that to each value of the variable there are an infinity of different values of the function. A case of this is the function sin ^" ^^ x, or the arc of which x is the sine. This arc may be either the smallest arc which has x for its sine, or this smallest arc in- creased by any entire number of circumferences. Take, for example, the arc whose sine shall be+i. The two smallest arcs will be 30° = \7t and 150° = \n. But if we take the function in its most gen- eral sense it may have any of the values (2+i)7r; (4 + i);r; (^^-^\)n, etc., or (2+|)7r; (4 + |)7r; (6 + |)7r, etc. When we represent an 7^-valued function graphically, there will be n values to each ordi- nate. Hence each ordinate will cut the curve in n points, real or imaginary. The figure in the margin represents the infi- nitely-valued function y a sm (-1)^ When — a 10, X > 100, X > 100000, and so on without end, then x is called an infinite quantity. If of a quantity li we either suppose or prove li < 0.1, h < 0.001, h < 0.00001, and so on without end, then h is an infinitesimal quantity. The preceding conceptions of limits, infinites and infinitesi- mals are applied in the following ways: Let us have an inde- pendent variable x, and a function of that variable which we call y. Now, in order to apply the method of limits, we may make three suppositions respecting the value of x, namely: 1. That X approaches some finite limit. 2. That X increases without limit (i.e., is infinite). 3. That X diminishes without limit (i.e., is infinitesimal).. In each of these cases the result may be that y approaches a finite limit, or is infinite, or is infinitesimal. * Strictly speaking, the words infinite and infinitesimal are both adjec- tives qualifying a quantity. But the second has lately been used also as a noun, and we shall therefore use the word infinite as a noun meaning infinite quantity. LIMITS AND INFINITESIMALS. 19 For example, let us have X -{■ a y = — — • Then— When X approaches the limit a, y becomes infinite. When X becomes infinite, y approaches the limit + 1. When X becomes infinitesimal, y approaches the limit — 1. The symbol ^^ followed by that of zero or a finite quantity, means ^^ approaches the limit/^ The symbols :^oo mean ^^ increases without limit ^^ or ''^ becomes infinite/' Hence the three last statements may be expressed symbolically, as follows: X -\- a , When X ^ a, then X — a When X ^ CO , then = + 1; X — a etc. etc. The same statements are more commonly expressed thus: x-\- a . lim. (x — a) = 00 ; lim. (X r= 00 ) = +1; lim. ^±^ (,; ^ 0) = - 1. X — a^ ^ 13. Properties of Infinite and Infinitesimal Qicantities. Theorem I. The product of an infinitesimal hy any finite factor, however great, is an infinitesimal. Proof. Let h be the infinitesimal, and n the finite factor by which it is multiplied. I say how great soever n may be, nh is also an infinitesimal. For, if nh does not become less than any quantity we can name, let or be a quantity less than which it does not become. Then if we take, as we may, h < -, (Axiom III.) n ^ ' we shall have nh < a. 20 THE DIFFERENTIAL GALGULU8. That is^ nil is less than a and not less than a, which is absurd. Hence nli becomes less than any quantity we can name^ and is therefore infinitesimal^ by definition. Theokem II. The quotient of an infinite quantity hy any finite divisor , hotoever great, is infi7iite. Proof Let X be the infinite quantity, and n the finite divisor. 11 X -^ n does not increase beyond every limit, let K be some quantity which it cannot exceed. Then by taking X>7iE, (Ax. III.) X we shall have — > K\ n that is, — greater than the quantity which it cannot exceed, which is absurd. Hence X -^ 71 increases beyond every limit we can name when X does, and is therefore infinite when X is infinite. Theorem III. The product of a^iy fi^iite qua7itity, how- ever S7nall, hy a7i iiifiriite 7nulti2Mer, is inji7iite. This follows at once from Axiom I., since by increasing the multiplier we may make the product greater than any quan- tity we can name. Theorem TV. The quotient of a7iy finite qua7itity, how- ever great, hy a7i i7ifinite divisor is i7ifinitesi7nal. This follows at once from Axiom II., since by increasing the divisor the quotient may be made less than any finite quantity. Theorem V. The reciprocal of a7i infinitesimal is an in- finite, a7id vice versa. Let h be an infinitesimal. If j- is not infinite, there must be some quantity which we can name which - does not ex- LIMITS AND INFINITESIMALS. 21 ceed. Let K be that quantity. Because h is infinitesimal, we may have which gives j- > K; that is, Y greater than a quantity it can never exceed, which is absurd. The converse theorem may be proved in the same way. 14. Orders of Lifinitesimals, Def, If the ratio of one infinitesimal to another approaches a finite limit, they are called infinitcsinials of the same order. If the ratio is itself infinitesimal, the lesser infinitesimal is said to be of higher order than the other. Theorem VI. If lue have a series proceeding according to the poivers of A, A-\-Bh+ Ch' + Dh' + etc., m which the coefficients A, B, C, arc all finite, then, if h he- comes infinitesimal y each term after the first is an infinitesi- rnal of higher order than the term preceding. Proof The ratio of two consecutive terms, the third and fourth for example, is Dh' _D By hypothesis, (7 and D are both finite; hence ^ is finite; hence when h approaches the limit zero, -^h becomes an in- finitesimal (§13, Th. I.). Thus, by definition, the term Dh' is an infinitesimal of higher order than CT'. Dcf, The orders of infinitesimals are numbered by taking gome one infinitesimal as a base and calling it an infinitesi- mal of the first order ^ Then, an infinitesimal whose ratio to 22 THE DIFFERENTIAL CALCULUS. the ni\i power of the base approaches a finite limit is called an wfinitesimal of the ntli order. Example. If li be taken as the base, the term Bli is of the first order ' r Bli\li — the finite quantity B\ C¥ '' " second ^^ ' r Ch' : h' = '' '' C; Eh"" '' '' wth '' • . • Eh'' : 7i~ = '' " E, Cor. 1. Since when n — ^ we have Bli'' — BW — B for all values of li, it follows that an infinitesimal of the order zero is the same as a finite quantity. Cor, 2. It may be shown in the same way that the product of any two infinitesimals of the first order is an infinitesimal of the second order. 15. Orders of Infinites, If the ratio of two infinite quantities approaches a finite limit, they are called infinites of the same order. If the ratio increases without limit, the greater term of the ratio is called an infinite of higher order than the other. Theorem VIL In a series of terms arranged according to the powers of x, A+ Bx+Cx" + Dx' + etc., if A, B, C, etc.y are all finite, then, when x hecomes infinite^ each term after the first is an infinite of higher order than the term jireceding. For, the ratio of two consecutive terms is of the form -^a:, x> which becomes infinite with x (Th. III.). Def Orders of infinity are numbered by taking some one infinite as a base, and calling it an infinite of the first order. Then, an infinite whose ratio to the nth. power of the base approaches a finite limit is called an infinite of the ni\\ order. Thus, taking x as the standard, when it becomes infinite we call Bx infinite of the first order, Cx'' of the second order, etc. LIMITS AND INFINITESIMALS, 23 NOTE ON THE PRECEDING CHAPTERS. In beginning the Calculus, conceptions arc presented to the student which seem beyond his grasp, and methods which seem to lack rigor. Really, however, the fundamental principle of these methods is as old as Euclid, and is met with in all works on elementary geometry which treat of the area of the circle. The simplest form in which the princi- ple appears is seen in the following case. Let us have to compare two quantities A and B, in order to determine whether they are equal. If they are not equal, then they must differ by some quantity. If, now, taking any arbitrary quantity h, we can prove that A-B (b) will still be { ^q^ part; true within part; 1000000' J t 2000000 etc., etc. So long as we assign any definite value to Ax, it is clear that there will be some error in neglecting Ax. But there is no error in the equations dy = 4:xdx and -j- — 4:Xy DIFFERENTIALS AND DERIVATIVES. 29 provided that we interpret them as expressing tlie limit which -p approaches as Ax approaches the limit zero, and interpret all our results accordingly. 19. Illustration hy Velocities. Let us consider what is meant by the familiar idea of a train which may be contin- ually changing its speed passing a certain point with a certain speed. To fix the ideas, suppose the train has just started and is every moment accelerating its speed in such manner that the total number of feet it has advanced is equal to the square of the number of seconds since it started. Put d = the distance travelled expressed in feet; t ~ the time expressed in seconds. We shall then have (^ = f , and for the distances travelled: Number of seconds, 0; 1; 2; 3; 4; 5; etc.; Distance travelled, 0; 1; 4; 9; 16; 25; etc.; Distance in each second, 1; 3; 5; 7; 9; 11; etc. B ill 8 I 5 , 7 I 9 , 11 , Fig. 6, Let this line represent the space travelled the first five seconds from the starting time, and let us inquire with what velocity the train passed the point B at the end of 4\ Since distance travelled = velocity x time, the mean ve- locity is found by dividing the space by the time required to pass over that space. Now, the train had travelled 16 feet in the time 4 seconds, and (4 + At^ feet in (4 + At) seconds, or 16 + 8 J/5 + Af feet in (4 + At) seconds. Subtracting 16 feet and 4 seconds, we see that in the time At after the end of the 4 seconds the train went 8 J/ + ^^' = As feet. Hence its mean velocity from 4^ to 4" + At is 30 THE DIFFERENTIAL CALCULUS. As At (8 + ^0 ^^^^ P®^ second. ~^ow it is clear that, since the train was continually accel- erated how small soever we take At, the mean velocity during this interval will exceed that with which it passed B. But it is also clear that by supposing At to approach the limit zero, we shall approach the required velocity as our limit. Hence the velocity with which B was passed is rigorously (is dt = 8 feet per second. Fig. 7. 30. Geometrical Illustration, If, in the figure, we sup- pose the point P' to approach P as its limit, the increments Ax and Ay will approach the limit zero, and the secant P'P will approach the tangent at the point P as its limit. We have already shown that Ay -—- — tangent of angle made by secant with axis of abscissas. Passing to the limit, we have the rigorous proposition -^ = tangent of angle which the tangent at the point P makes with the axis of abscissas. DIFFERENTIATION OF EXPLICIT FUNCTIONS. 31 CHAPTER IV. DIFFERENTIATION OF EXPLICIT FUNCTIONS. 31. Def. The process of finding the differential and the derivative of a function is called dijBFerentiation, As exemplified in §§ 16, 17, it may be generalized as fol- lows: We have given (1) An independent variable ~ x, (2) A function of that variable = 0(a:). (3) We assign to x an increment = Ax', whereby (p{x) is changed into (p{x + Ax), (4) We thus have 0(a; + ^^) — 0(^) ^is the increment of (p{x). We may put A(f>{x) = (p{x + Ax) — (p{x). (5) We then form the ratio Ac/>{x) Ax (a) and seek its limit when Ax becomes infinitesimal. Using the notation of the last chapter, we have ; - = hm. — V~ ■ (^^ = 0), (Ix Ax ^ ' which is the derivative of 0(.r). In order to find the ratio (//), it is necessary to develop 0(^ + ^^) ill powers of Ax to at least the first power of Ax. Let this development be 0Gr + Ax) = A\ + A\Ax + X,Ax^ + . . . . (1) In the second member of this equation X^, X,, etc., will be functions of .r; and it is evident that X^ can be nothing but 32 THE DIFFERENTIAL CALCULUS. 0(.^•) itself, because it is the value of 0(a; + Ax) when Ax = 0. Thus we have A(P{x) = (p{x + Ax) - cp{x) = (X, + X,Ax) Ax + . . . ; Passing to the limit, d(p{x) = X/Jx; Thus, by comparing with (1), we have the following: Theorem I. The derivative of a function is the coefficient of the first pmuer of the increment of the independent variable when the function is developed in ptoioers of that increment. If we have to differentiate a function of several variable quantities, x, y^ z, etc., we assign an increment to each vari- able, and develop the function in powers and products of the increments. Subtracting the original function, the remainder will be its increment. The terms of highest order in this increment, considered as infinitesimals, are then called the differential of the function. The following are the special cases by combining which all derivatives of rational functions may be found. 32. Differentials of Su?ns, Let x, y, Zy ii, etc., be any variables or functions whatever. Their sum will be ^' + 2/ + ^ + ^^ + 6^c. Assigning to each an increment, x will become x -f Ax, y will become y + Ay, etc. Hence the sum will become X -^ Ax -\- y -^ Ay -\- z -^ Az -\- u -\- All -\- etc. Subtracting the original expression, we find the increment of the sum to be Ax 4- Ay -\- Az+ Au -\- etc. DIFFERENTIATION OF EXPLICIT FUNCTIONS. 33 Hence, when the increments become infinitesimal, ^K^ + y +• ^ + ^^ + 6tc.) = dx -\- dy -\- dz -\- du + etc., (3) or, in words: Theorem II. The differential of the stint of any number of variable.'^ is equal to the sum of their differentials. In this theorem the quantities x, y, z, u, etc., maybe either independent variables, or functions of one or more variables. 23. Differential of a Mnltiple. Let it be required to find tlie differential of ax, a being a constant. Giving X the increment /!x, the expression will become a{x + /Jx). Then, proceeding as before, we find d{ax) = adx. (4) 24. Theorem III. The differential of any constant is zero. For, by definition, a constant is a quantity which we sup- pose invariable, and to which we cannot, therefore, assign any increment. We therefore have, from Tlieorom I. when x is a variable and a is a constant, d{x + n) = dx -\~0~ dx, <^r, in words: Theorem IV. The differential of the sum of a constant and a variable is equal to the diff^erential of the variable alone. IIemark. It will be readily seen that the conclusions of §§ 22-24 are equally true whetlier we suppose the increments to be finite or infinitesimal. This is no longer the case when powers or products of some finite increments enter into the expression for other finite increments. 34 THE DIFFERENTIAL CALCULV8. EXERCISES. It is required^ by combining the preceding processes, to form the differentials of the following expressions, supposing a, 1) and c to be constants, and all the other literal symbols to be variables. I. U — V. 2, 2u — Vo 3. V -\-x-\- c. 4- ax + hy. 5- a^x + yy + c. 6. Sx + 4:ay + K 7» 4:ax -\- 61) X — y. 8. 6bx — abc. 9- dx -- a -{- ah. 10. ahx — aU. II. c{2x + a). 12. a{bx + etc). 13. ac{bu + ax). 14. bc{2ax - dby). 15. X — y — z. 16. — ax — by — cz. 17. — aiftx — cy). 18. - b{2ax - 3cv). 19. X a' 20. X -\- y — z b 21. (« + b-\-c){s-{-t + ^u- 4:y). 35. Differentials of Products and Poiuers, Take first the product of two variables, Avhich we shall call n and v. Then Product = nv. Assigning the increments An and Av, the product becomes [it -J- An) {v -\- Av) = uv + vAu -\- uAv -f AuAv, Subtracting the original function, nv, we find A{uv) = vAu + (?^ + An) Av, Supposing the increments to become infinitesimals, the co- efficient of Av in the second member will approach ^c as its limit. Hence, passing to the limit (§14), d{uv) — vdu + lidv. DIFFERENTIATION OF EXPLICIT FUNCTIONS, 35 To extend the result to any number of factors, let P be the l)ro(iuct of all the factors but one, and let the remaining fac- tor be X, so that we have Product = Px, By what precedes, we have d{Px) = xdP + Pilx. Supposing P to be a product of the two variables ^l and v, this result gives d{iivx) = xd{vu) + uvdx = vxdti + itxdv + uvdx, {a) If we add a fourth factor, y, we shall have d{uvxy) = yd{uvx) + tivxdy. If we substitute for d{uvx) its value (a), we see that we pass from the one case to the other by (1) multiplying all the terms of the first case by the common factor y, (2) adding the product of dy into all the other factors. We are thus led to the conclusion: Theorem V. The differential of the product of any nuin- her of variables is equal to the stem of the products formed hy replacing each variahle hy its differentiaL Corollary. If the n factors are all equal, their product will become the ni\\ power of the variable, and the n differentials will all become equal. Hence, when n is an integer, we have the general formula d^x"") = x''~^dx + x''~^dx + etc., to n terms, or ^Z(.r") = nx"-~^dx. By combining the preceding processes we may form the differential of any entire function of any number of variables. Examples. I. d{ax + hxy -\- cxyz) = d{ax) + d{hxy) + d{cxyz) (Th. II. 22) = adx + hd(xy) + cd{xyz) (Th. III. 23) = adx + h{ydx + ^dy) -\- c(yzdx + xzdy ~\- xydz) = {a + by + cyz)dx + {bx + cxz)dy + cxydz. 36 THE DIFFERENTIAL CALCULUS. 2. d{ax' + l) =. d{cix') (Th. IV.) ^ad(x^) (§23) == ?>ax\lx. (Th. Y., Cor.) 3. d{ax'ir) = ad{xY) (§ ^3) := a[2/'d{x') + .TV/(y«)] (Th. V.) = dajfx'^dx + naxhj''-^dy, (Th. V., Cor.) 4. J(« + xY = n{a-\- x'') ^ " '^d{a + .t') == 2n{a + r^-') ^ " ^xdx. EXERCISES. Form the differentials of the following expressions, suppos- ing the letters of the alphabet from a to n to represent con- stants: T. a-^-hx"^ -{- ex*. Ans, {2bx -{-4tcx^)dx, 2. B -\- Cy + Dy''' 3. ctxy. 4. hxyz, 5. a{x -\- yz). 6. a{x'^ + ^?^^). 7. ctxy -\- hiiv. 8. 7^(0;'^ + xy""). 9. fl^rr"'?/'. 10. hx'y''. II. dbx'y'^ -\-'ku'^v'^, 12. ;Ej(wia; + ^z-?/). 13. (r + ^)(^ + ^^). 14. ?i(a — i?;^). 15. <7.^•'^ — hyz, ,6. (^ + ,;)(^_^). x7. {a + x^){h-f), 18. (n^ — :?;) (a — .'t'*)^ 19. rt'(a + rf) (Z> — a:''). 20. (^. + ^:r + fe^) (y + z). 21. (^+/^^^+6^/^)(«^+^x) xy 22. — -. 23. (« + ^nt') {ex'' — 7l7f), x~ u v 25. {a — x){I) — x*){e — a;'). . X — nv, . . 26. (it 4- v) 27. xlx"" + y{a — a:)}. 28. I - + - \xy. - e'+i> 29. (mf - Z^:^'') (.'?; - y), ^ • U ^/W "^ ^j* 32. ^^^(«+a:)^ 33. {a + xy)\ 34. (^.0; + %)' DIFFERENTIATION OF EXPLICIT FUNCTIONS. 37 *ZG Differential of a Quotient of Two Variables, Let the vjiriables be x and y, and let q be their quotient. Then X and qy z=z X, Differentiating, we have ydq -{- qdy = dx, Solving so as to find the value of dq, _ dx — qdy _ ydx — xdy iiq — — ^ • y y Hence: Theorem VI. The differential of a fraction is equal to the denominator into the diff^erential of the numerator, minus the mcmerator into the diff^eremtial of the denominator, divided by the square of the denominator. Kemark. If the numerator is a constant, its differential vanishes, and we have the general formula ^a a ^ d— = MX. X X EXERCISES. Form the differentials of the following expressions: X a -\- X (^ + y a — X a -y a 5- -^ X a -f bx a + by ^ + y 4. -. 8. 10. « + / a (6 + yy m + wa;' m — nx*' mx^ + ny^ mx* — ny** 38 THE DIFFERENTIAL CALCULUS, a X + yz a -{- bx -{- cx^ m + xy '^' m - xy' a h '^'x+f a 17. xy + x^y^ y + 2:z' 1 1 14. — — — X X' 16. m n x^-f 1 1 t8. X y a x" + V ^ - y 19. , \ , 20. , , '\ . x' - y' x' + y' 37. Differentials of Irrational Expressions. Let it be re- quired to find the differential of the function m IC — X^, m and 71 being positive integers. Raising both members of the equation to the nth power, we have Taking the differentials of both members, 7m " ~ ^du = mx *" ~ V/a;, du n ?^"~* n ^m-l jji x"^"^ dx ' i mVn— 1 ^j mn — % U»; X »» - = -." ,(.) a formula which corresponds to the corollary of Theorem V ., w^here the exponent is entire. Next, let the fractional exponent be negative. Then -V!l 1 x^ and, by Th. YI., dyx"^) mx^ dx in - — -1, du = ^ = ^— = X ^ dx, X n 2: « and, for the derivative, du m -^-1 dx n DIFFERENTIATION OF EXPLICIT FUNCTIONS. 89 From this equation and from {a) we conclude: Theorem VII. The formula d{x'') = nx'^-'^dx holds true whether the exponent n is entire or fractional , posi- tire or negative. We thus derive the following rule for forming the differen- tials of irrational expressions: Express the indicated roots by fractional exponents, positive or nefjativey and then form the differential by the precediiig methods. Examples. 1. d Va-\-x = d{a -f .t)* = i(a + x)- klx = — — , — r-. 2. d-r-^ri = d \b{a + x) - *] =: bdia + x)-^ '=.^j,^a+x)-ldx^-^-^^^dx. 3. dia + bx^)i = Ma + bx') - i 2bxdx = - — ^^^,-^dx. "^ ' {a^bx^y EXERCISES. l\)rm the differentials of the following expressions: I . Va + X. 2, 5- 8. I r. 14. Vb-x. 3- 6. 9- 12. '5- Va - bz. 4. S/a — x\ a Va - bx\ b Vx + y. b 10. (a-\- xyi. 1 3. X Va + X, Va + bx' • {x-a)l^ X Va — X. Va - bz'' {hx' - a)i. y' Va - by'. Find the values of -- in the following cases: 16. n = mx -\ — \ 17. u = (mx* — ^i)♦. 40 THE DIFFERENTIAL CALCULUS. 1 8. 2^ = Vax + bx^, lo. u =^ ^, -\- ex 20. 20 = X Va ~ X. 21, 21 = X Vx^ -f- U. a -{- X a — X 22, 2C =^ . 2^. U =1 . a — X ^ a-\- X 38. Logarithmic Functions. It is required to differentiate the function 11 = log X. We have All ^ log (x -\- Ax) — log X = log -^ = log [1 -1 ]. X \ X J It is shown in Algebra that we have log (1 H- h) = M{h - W + W - etc.), M being the modulus of the system of logarithms employed. Hence, puting — ^ for h, we find An ~A. ^i Ml^ 1 Ax ^ Ax' , \ v = -x[^-2V + 3-x'-''^''h and, passing to the limit, _ 3Idx^ du _ M X ' dx x' In the Naperian system if— 1. In algebraic analysis, logarithms are always understood to be Naperian logarithms unless some other system is indicated. Hence we write ^•lo^ a; 1 , , dx Example. -, ^ d(axy) axd2j + aifdx dy , dx d'log axy = -^ — ^ = ^— ' — - — = -- H . ° ^ axy axy y x Eemaek. We may often change the form of logarithmic DIFFERENTIATION OF EXPLICIT FUNCTIONS. 41 functions, so as to obtain their differentials in various ways. Thus, in the last example, we have log {(ixy) = log a + log x + log y, from which we obtain the same differential found above. The student should lind the following differentials in two ways when practicable. EXERCISES. Differentiate: I. log {a + x). Ans. dx a-\- X 2. log {x - «). 3. log {x^ + b^). 5. log mx, 7. log {ax'' + b). 9. log {x + y). 1 1 . log xy. 4. log {x^ - b). 6. log mx"^, 8. log iif'. 10. log (x - y). 12. log {x' + y'). 13. log{a + b)y. 14. log-. x + a 17. ylogx. 16. Io2r , . 18. log {a — a:)"*. 29. Exponential Ftmctloiis. It is required to differentiate the function a being a constant. Taking the logarithms of both members, log u = X log a. Differentiating, we have, by the last article, iT'log u = — = dx log a. 42 THE DIFFERENTIAL GALGULU8. Hence du = u log a dx = cv^ log a dx\ ^^ = «^ log «' which is the required derivative. If a is the Naperian base, whose value is ^ = 2.71828 « . . , , we have log a = 1, Hence d-e ^ dx Hence the derivative of c^ possesses the remarkable prop- erty of being identical with the function itself. EXERCISKS. Differentiate: I. «^^. Ans, 2«'^ log a dx. 2. a"^. 3. c« + ^ 4. (f'^''. 5. J^mx + nv^ 6. /r^-J'. 7. 7^-"^. 8. a'^aK 9. d'hK lO. «2^Z^3^. II. ah'^h-'^y. 12. 6^+«. 13. eV^ 14. ^ax + &i/^ 30. The Trigonometric Ftinctions, The Sine, Putting h for the increment of x, we have, by Trigonometry, sin {x -\- h) — sin :?; = 2 cos {x -\- ^h) sin ^h. Now, let h approach zero as its limit. Then, sin {x -\- h) — sin x becomes ^ sin rr; h becomes dx, because it is the increment of x; cos {x 4- ^h) approaches the limit cos x-^ sin ^h approaches the limit \h or ^dx, because when an angle approaches zero as its limit, its ratio to its sine approaches unity as its limit (Trigonometry), Hence, passing to the limit, d'mi X = cos xdx. DIFFERENTIATION OF EXPLICIT FUNCTIONS, 43 The Cosine. By Trigonometry, cos (:c ~\- h) — COS a; = — ^sin {x + \h) sin -JA. Hence, as in the case of the sine, (I cos X = — sin X dx. Taking the derivatives, we have (I sin X dx d'cos X = cos x: dx =: — sm .r. M N Fig. 8. Pn = ^ sin :r. /vV^ = J COS X. Geometrical Illustrafion, In the figure, let OX bo the unit- o radius. Tlicn, measuring lengths in terms of this radius, we sliall have NK = sin ./;; MB = sin (:r + //) ; ON^ =z cos x; OM — cos {x + h)\ Ab' >, supposing a straight line from K to //, P/r = - KP = KB sin PBK; PB = KB cos PBK When B approaches K as its limit, the angle PBK ap- proaches XOK, or X, as its limit, and the line KB becomes dx. Hence, approaching the limit, we find the same equa- tions as before for d sin x and d cos x. It is evident that so long as the sine is positive, cos x di- minishes as X increases, whence ^Z'cos x must have the nega- tive sign. The Tangent, Expressing the tangent in terms of the sine and cosine, we have sin X tan X = . cos X Differentiating this fractional expression, , , cos xd'sin x — sin xd'C08 x sin' xdx + cos' xdx d tan x = r — =^ 4 • cos X cos X = sec* xdx. which is the required differential. 44 THE DIFFEBENTIAL CALCULUS. We find, by a similar process, , , , cos a: » -, dx d cot X = d'- — = — CSC xdx = ^-^- ; sm X sm x 1 fZ'cos X sin xdx a -sec X ~ d' • = ^ — = ^ — cos X cos X cos X = tan a: sec xdx; c?*cosec a; = — cot x esc a;^a;. EXERCISES. Differentiate: I. cos {a + .y). 2. sin (Z> — ?y). 3. tan {c + 2;). 4. sin y cos 2;. 5. tan ii cos ?;. 6. sin w tan t'. 7. sin «a;. 8. cos ay^ 9. tan ???2;. 10. sin {h + m?/). 11. cos {h -\- my), 12. sin {h — my). 13. cos"" X ' [f^'cos" X = % cos xd'GO^ i2; = — sin 2xdx]. 14. sin'' X. 15. sin' ^. 16. sin^ 7iz, sin :2; ^ sin* a; cos'* a; 17. . 18. . 19. -T-^-. cos y cos y sm y 20, Show that fZ(sin^ y + cos'* y) = 0, and show why this result ought to come out by § 24. 21, Differentiate the two members of the identities cos («-[-?/) = cos a cos y — sin a sin ?y, sin {a -\- z) = cos a sin z -\- sin « cos z, and show that the differentials of the two members of each equation are identical. 22, Show that d'log sin x = cot x dx; d'log cos X ■= — tan x dx, 31o Circular Functions, A circular function is the in- verse of a trigonometric function, the independent variable being the sine, cosine, or other trigonometric function, and the function the angle. The notation is as follows: If y — sin z, we write z — sin ^~ ^^ y or arc-sin y; If II =■ tan X, we write x = tan ^~ ^^ w or arc-tan u; etc. etc. etc. DIFFERENTIATION OF EXPLICIT FUNCTIONS. 45 Differentiation of Circular* Functions, If we have to dif- [ferentiate z = sin ^~ ^^ y. we shall have y = sin z; dy = cos z dz = Vl — sin* z dz; . dz = ^ ^_dy__ Vl - sin' z Vl- f The Inverse Cosine. If z be the inverse cosine of y, we find, in the same way. The Inverse Tangent. If we have z = tan^~ ^^ y; then, y = tan ;$;; r/// — sec^ z dz = {1 -\- tan' 21)^/2;; .•..fo = ^,. (.) 7^/ie Inverse Cotangent. We find, in a similar way, '^ cot <-?/ = - 5^. 00 3^Ae Inverse Secant. If we have 2; =r sec^~^^ i/; then, «/ = sec ;2; dy = tan 2; sec z dz = y Vy^ — 1 dz; .■.dz = --^L=. (e) yVy'-l The Inverse Cosecant. We find, in a similar way, d'csc^ ^^y = , yVf-1 4. X 6. \ z 1 8. i2^n^-'^ (x'). lO. ( 1, f 1 \ 12. sec^""^> x^ tan^-^> a;, 46 TRE DIFFERENTIAL CALCULUS. EXERCISES. Differentiate with respect to x or z: I. sin^~^^ rtx. 2. cos^~*^ (a; + a). 3. sin^"^^ {mx + ^). 5.tan-(.-l). 7. tan<-«p+^). \a xj 9. sec^~^^ f^;-! — j. II. sin^~^^^a;cos^"^^ — . a Note. — The student will sometimes find it convenient to invert tl function before differentiation, as we have done in deducing the differei tial of sin <- D x. 13. We have, by comparing the above differentials, eZ(sin ~ \v + c^s - ^ y) = 0; c?(tan~\^/ + cot~^ y) = 0; ^(sec~^ y + csc~ ^ y) = 0. Show how these results follow immediately from the defini tion of complementary functions in trigonometry, combine with the theorem of § 24 that the differential of a constan quantity is zero. 33. Logarithmic Differentiation. In the case of produci and exponential functions, it will often be found that the dii f erential is most easily derived by differentiating the logarithr of the function. The process is then called logarithmic dij ferentiation. Example 1. Find -,- when y — a;"*^ dx ^ We have log y = mx log x; DIFFERENTIATION OF EXPLICIT FUNCTIONS, Al -— = m log X dx -\- mdx'y -| = y{^r^ log X + m):=^ mx^'^{l + log x). Example 2. y = — -— . ^ cos"" X We have log y = m log sin x — n log cos a:; dy _ 7/1 cos X n sin a:^ y^/a; "~ sin x cos cc ^ Jy sin*"-^ r?:. , , . , . "7- = z—T— {m cos X + n sm a:). f?a; cos'^ + ^a;^ ' ^ MISCELLANEOUS EXERCISES IN DIFFERENTIATION. Find the derivatives of the following functions with re- sj^ect to a*: 1. y = X log X, Ans, -^ = 1 + log x, 2, y — log tan a*. Ans, -—■ = 3- y = log cot X, X 4* ^' - 4/(<«' - x^y 5. a." ■^ (1 + a:)"- A e' - e-" y- ^^e-^- 7. y = \og{e' + e-'^). 8. y = log tan (f + |). 9. a; lO. „^4/(l+^)+i^(l-^) y^^^. //-I 1 --\ //-I „\ dx sin 2a; . dii 2 A71S, -^ — dx sin 2a;' dt/ a^ A71S. -J— = dx {a' - x')r dy nx " ~ ^ Ans, -r- — A71S, dx ~ (l + a;)" + ^* (Iy__ 4__ dx ~ {e^+ e-'^y . dy c^ — e~ Ans. -T- = dx 6'* + ^ ~ * Ans, -r- = . dx cos X dy e'(l - a;) - 1 . dy _ 1 ^'''' d^~ X ^(1 - x'S 48 THE DIFFERENTIAL CALCULUS. 11. „ = j ^ I " , Ans, -f- — -^ 5 ^, . a ' aa: a cos x — b sm'' a: 22. li y =z —, prove the relation — '^ - -| - = 0. ^ 1/1 + 7/* Vl + a;* 23. y =e-^^^^. Ans. — = — 2a*xy. DIFFERENTIATIOK OF EXPLICIT FUNCTIONS, 49 1 1 ""^^ '^ ~ {a + x)^{b-\-xY' (ly Ans, dx m{h -\- x) -\- n{a -\- x) 25. y = (a* 4- x"^) tan~^ — . Ans, -j- = 2^tan~^ — |- a. (I (IX ci Ans, -r- = dx (1 -a:) /i _^^« • + tan x' 27. y = x + logco^l^-x\A7is. ^=j • 1 A dy . . ^ X 28. y ^= X sill" ^ X. A71S. -—- = sin~^a; -A . ^ dx l/T^^ 29. y = tan x tan "" ^ x, A 30. y = sin ?^a:(sin a;)", (sin ?ia;)' 4 dy 2 . _i , tana: .1 ^i.s. -—- = sec .T tan ^ a: + -— — .:. dx 1 + ^ A71S. -f- — n (sin a:)"~^sin {71 -\- \)x, dx ^ ^ ^ ' 31. y (cos ?>ia;)*** dii mn (sin ?Lr)*"~*cos (7^10: — nx) A7i^ -^ — ^= ^^ 32. y = e cos rx A ns. dx 'X dy^ ^ dx 33. y rr log. a + Z> tan - )'*A7lS, a — b tan - (cos ?/ia;) ** + ^ e - "''^^ (2rt''a: cos ra: -f- ^' sin ru). Jt/ _ ah 34. y = a;«. 35. y = sin '— ^. A71S, a cos^ o ~- " sm^ -r-. (ZV _ a:^(l — lo g x) dx A dy A71S, -T- = dx yi - 2a; - a;' Jy _ ^ 36. t/ = tan~ * (?i tan a:). ^7^5. -7^ = — = ; — « . . . * ^ ^ f/a; cos X + ^r sm a; 50 THE DIFFERENTIAL CALCULUS. .a , dy 37. y = sec-^— r-^ -^. Ans. -£ = \/{a'' - x'Y dx "' j^((f - x'Y 38. y^{x + a) tan- ^ (j/^ -'^{ax) . Ans. ^ = tan-V^. 39. ^ = sin- ^ |/(sin a;). Ans. -f~ — i Vi^ + cosec x). 40. y = tan- > j^^^. ^..s. ^ = 5^^,. \ - , Z> + a cos a; . dy — x/la^ — Z>^) 41. V = sm-^ — '—, . A71S. -f- = 7~ -. ^ a -{- b cos X dx a -\- cos x 42. V = cos- ^ on , T' ^^^5« -/- = on , V 43. 2/ = Bee- ^^--. Ans. -^- =. ~ -^rZT^y 44. 2/ - t^n-^-^^^^-^.Ans. % = ^^^^. 33. Derivatives luith Respect to the Time, — Velocities, If we have a quantity which varies with the time, so as to have a definite value at each moment, but to change its value con- tinuously from one moment to another, that quantity is, by definition, a function of the time. We now have the defini- tion: If we have a quantity 0, expressed as a function of the time = ty the derivative, -77-, is the velocity of increase, or rede of valuation of (p at any moment. This is properly a definition of the word velocity; but it may be assumed that the student has already so clear a con- ception of what a velocity is, that he needs only to study the identity of this conception with that of a derivative relatively to ty which he can do by the illustration of § 19. The student is recommended to draw a diagram to rep- resent the problem whenever he can do so. DIFFERENTIATION OF EXPLICIT FUNCTIONS. 51 EXERCISES. 1. It is found that if a body fall in a vacuum under the in- fluence of a constant force of gravity, the distances through which it falls in the first, second, third, fourth, etc., second of time are proportional to the numbers of the arithmetical progression 1, 3, 5, 7, etc., or, putting a for the fall during the first second, the total fall will be a + 3a -]- 6a -{- 7a + etc., continued to as many terms as there are seconds. It is now required to find, by summing t terms of this progression, how far the body will fall in^ seconds, and then to express its velocity in terms of t, and thus show that the velocity is proportional to the time. Ans. (in part). The total distance fallen in t seconds will be aP. The velocity at the end of t seconds will be 2at 2. The above motion being called nnifornily accelerated, prove this theorem: If a body fall from a state of rest with a uniformly accelerated velocity during any time r, and if the acceleration then ceases, and the body continue with the uni- form velocity tlien acquired, it will, during the next interval r, fall through double the distance it did during the first interval. Find (1) how far the body falls in r seconds; (2) its velocity at the end of that time; (3) how far, with that velocity, it would fall in another interval of r seconds; then show that (3) = 2 X (1). 3. The radius of a circle increases uniformly at the rate of m feet per second. At what rate per second will the area be increasing when the radius is equal to ;• feet ? PHnd (1) the expression for the value of the radius r at the end of t seconds, and (2) the area of the circle at that time. Differentiate this area, and then substitute for t its value in terms of r. Note that (t= — ). We shall thus have ^Ttvir for the velocity of increase of area. 52 THE DIFFERENTIAL CALCULUS. 4. A body moves along the straight line whose equation is with a uniform velocity of 7i feet per second. At what rate do its abscissa and ordinate respectively increase ? Ans, -—=. and -— =. Vb Vb 5. A man starts from a point h feet south of his door, and walks east at the rate of c feet per second. At what rate is he receding from his door at the end of t seconds? Ans, If we put ii = his distance from his door, we shall have du __ c^t 6. A stone is dropped from a point b feet distant in a hori- zontal line from the top of a flag-staff 9a feet high. At what rate is it receding from the top of the flag-staff (1) after it has dropped t seconds, and (2) when it reaches the ground, assuming the same law of falling as in Ex. 1 ? At the end of t seconds the square of the distance from the top of the flagstaff — u^=zb'^-\- a'^t*. On reaching the ground we should have du _ 6ia^ dt ~ V¥+'S1^''' 7. The sides of a rectangle grow uniformly, both starting from zero, and the one being continually double the other. Assuming one to grow at the rate of m feet and the other 2w feet per second, how fast will the area be growing at the end of 1, 2, 10 and t seconds? How fast, when one side is 4 and the other 8 feet ? 8. The sides of an equilateral triangle increase at the rate of 2 feet per second. At what rate is the area increasing when each side is 8 feet long ? Note that the area of the triangle whose sides = s is jf—- — . DIFFERENTIATION OF EXPLICIT FUNCTION. "6. 63 9. A man walks round a lamp, 20 feet from it, keejjing the distance with a uniform motion, making one circuit per minute. Find an expression for the rate at which his shadow travels on a wall distant 40 feet from the lamp. 10. The hypothenuse of a right triangle is of the constant length of 10 feet, but slides along the sides at 2)leasure. If, starting from a moment when the hypothenuse is lying on the base, the end at the right angle is gradually raised up at the uniform rate of 1 foot per second, find an expression for the rate at which the other end is sliding along the base at the end of t seconds, and explain the imaginary result when t> 10. 11. Two men start from the same point, the one going north at the rate of 3 miles an hour, the other north-east 5 miles an hour. Find the rate at which they recede from each other. 12. A body slides down a plane inclined at an angle of 30° to the horizon, at such a rate that it has slid 3^ feet at the end of t seconds. At what rates is it approaching the ground (1) at the end of t seconds, and (2) after having slid 75 feet ? 13. A line revolves around the point {a, h) in the plane of a system of rectangular co-ordinate axes, making one revolu- tion per second. Express the velocity with which its intersec- tion with each axis moves along that axis, in terms of a, the varying angle which the line makes with the axis of X. dx __ %h7t ^ cly __ 2a7r dt ~" sin' a' (it'" cos' a 14. A ship sailing east 6 miles an hour sights another ship 7 miles ahead sailing south 8 miles an hour. Find the rate at which the ships will be approaching or receding from each other at the end of 20, 30, 60 and 90 minutes, and at tliQ end of t hours. 54 THE DIFFERENTIAL CALCULUS CHAPTER V. FUNCTIONS OF SEVERAL VARIABLES AND IMPLICIT FUNCTIONS. 34. Def, A partial diflferential of a function of sev- eral variables is a differential formed by supposing one of the variables to change while all the others remain constant. The total differential of a function is its differential when all the variables which enter into it are supposed to change. A partial derivative of a function luith respect to a quantity is its derivative formed by supposing that quantity to change while all the others remain constant. Eemark. The adjective pai^tial may be omitted when the several variables are entirely independent. Example. Let us have the function u =: x\y + z) + yz. {a) Differentiating it with respect to x as if y and z were con- stant, the result will be du ■= '^x{y + z)dx, (b) which is the partial differential with respect to x. Also, is the partial derivative with respect to x. In the same way, supposing y alone to vary, we shall have du = (x' + z)dy, {c) (1)=^- I . , + z, \dyl PARTIAL DERIVATIVES, 65 which are the partial differential and derivative with respect to y. For the partial differential and derivative with respect to ;2; we have Notation of Partial Derivatives, 1. A partial derivative is sometimes enclosed in parentheses, as we have done above, to distinguish it from a total derivative (to be hereafter de- fined). But in most cases no such distinctive notation is necessary. 2. In forming partial derivatives the student is recom- mended to use the form Djii instead of -r-, dx because of its simplicity. It is called the D^ of u. The equa- tions following (h), {c) and {d) would then be written: D^u = 2x{ij -\-zy, Dyii = x' + z; D,u = x' + y. EXERCISES. Find the derivatives of the following functions with respect to X, y and z\ 1. V — x"^ ^ xy + ^'. Ans, D^v = 2x — y; DyV = ^ x -\- 2y; D^v = 0. 2. w = x^ -\- x'^y -\- xz. 3. ?^ = x*y^z\ 4. u =x log y + y log x. s. 7c = {x + y + z)'. 6. u = i/{x + my), 7. 21 = {x -}- 2y + 3z)K Note. In forms like the last three, begin by taking the total differential, thus: du = i{x + 2y + Sz)" * d • {x + 2y + 3z) - ii^ + 3// + 32)~* {dx + 2dy + ddz). 56 THE DIFFERENTIAL CALCULUS, Then, supposing x alone to vary, D^it = supposing y alone to vary. 8. w = {x -{- y -{■ %Y, 9. 10. w = cos [mx -\- y), 11, 12. V = iduTi{x — y), 13. 14. -y = cos^ {ax 4- ^^). 15. 16. i^ — ire^ + ^^'"' 1 7* 18. ?^ = sin(:r4-^)cos(.'r— ?/). 19. DyU supposing z alone to vary, D^tt ■ 2{x+2y+3zf 1 (^ + 2^+3^)** 3 2{x+2y+3z) i- w = {x' + .v' + zy, w — sin (x -\-2y -\- 3z), V =: sec (rnx -\- nz). V = c* + ^ -2^ = CC^ -]- t/"". w. = a; sin y — y sin ic. 35. Fuiq^DAMEi^TAL THEOREM. TliG total differential of a function of several variables, all of tohose derivatives are continuo7is, is cqnal to the sum of its palatial differentials. As an example of the meaning of this theorem, take the example of the preceding article, where we have found three separate differentials of v, namely, {h), {c) and {d). The theorem asserts that when x, y and z all three vary, the re- sulting differential of u will be the sum of these partial differ- entials, namely, du = 2x{y -\- z)dx + (^'^ + z)dy -\- {x^ + y)^z. To show the truth of the theorem, let us first consider any function of two variables, x and y, u = cp(x, y). (1) Let us now assign to x an increment ^x, while y remains unchanged, and let us call n' the new value of n, and ^^u the resulting increjnent of u. We shall then have n' = (p{x + /!x, y); J^ti = cp{x + Jx, y) - (p{x, y). m TOTAL DIFFERENTIALS. bl In the same way, if x retains its value while y receives the increment ^y, and if we call ^yU the corresponding incre- ment of u, we have ^yU = cp{x, y-\- Ay) -- (P{x, y). (3) When Ax and Ay become infinitesimal, these increments (2) and (3) become the partial differentials with respect to x and y. Now, to get the total increment of ^t, we must suppose both X and y to receive their increments. That is, instead of giv- ing ^ in (1) its increment Ay, we must assign this increment in (2). Then for the increment of u we shall have, instead of (3), the result Ayu' = cp(x + Ax, y + Ay) - 0(rc + Ax, y), (4) Note that (3) and (4) differ only in this: that (3) gives the value of Ayti before x has received its increment, while (4) gives AyU after x has received its increment, and is therefore the rigorous expression for the increment of u due to Ay, Now, what the theorem asserts is that, when the increments become infinitesimal, the ratio of Ayu' to AyU approaches unity as its limit, so that we may use (3) instead of (4). To show this, let us put Then, supposing Ay to become infinitesimal, and putting dyii for that part of the differential of u arising from dy, we shall have, from (3) and (4), dy2i = cp\x, y)dy; (3') dyu' = 0'(x + Ax, y)dy. (4') When Ax approaches zero as its limit, 0'(.r + Ax, y) must approach the limit 0'{./', y), unless there is a discontinuity in B8 TEE DIFFERENTIAL CALCULUS, the function (f>\ which case is excluded by hypothesis. Thus, using (3') for (4'), we have Total differential of ^ = du — (-- \dx + (p'{x, y)dy The same reasoning may be extended to the successive cases of 3, 4, . . . n variables. The following are examples of finding some differential al- ready considered in Chap. IV., by this more general process. 1. To differentiate ^i — xy. du dx ^y\ du -- = X. dy Total differential. du =: ydx -\-xdy. 2. X u ^^ — ^=- Xlf y •' ■ 1 du di-^y -1 . du dy xy - Hy\ - 1 7 - 2^ y^^ ~ ^^y du == y ^dx — xy ^dy = ^ — -. 3. u = ax -\- Jjxy + cxyz, ^^ = « + ^2/ + ^y^'y du , , dy du du = (fl5 + Z>y -f oyz)dx + (bx -\- cxz)dy + cxydz, as in § 25, Example 1. DIFFERENTIAriON OF IMPLICIT FUNCTIONS, 59 EXERCISES. Write the total differentials of the functions given in the exercises of § 34. 36. Principles Involved in Partial Differentiation. All the processes of tlie present chapter are aimed at the following object: Any derivative expression, such as du ^ presupposes (1) that wo have the quantity 7i given, really or ideally, as an explicit function of a:, and perhaps of other quantities; (2) that we are to get the result of differentiating this function according to the rules of Chap. IV., supposing all the quantities except x to be constant. Now, because it is often difficult or impossible to find u as an explicit function of x, we want rules for finding the values of D;eii, which we could get if we had u given as such a func- tion of X. For example, we might be able to find the equa- tion 21 = ' ^ ^ dy which is the required form in the case of an implicit function of one variable. Cor. If from an equation of the form x —f{y) we want to derive the value of D^y, we have (P{x, y)=^x -/{y) = 0; d(f) _ ^^0 _ dfiy) _ _ ^^ dx ~ ' dy ~^ dy ~' dy* Hence ~- = -?—. dx dx dy Example. To find B^y from the equation 0(^. y) = y - ax = 0. d(p d(p dy the same result which we should get by differentiating the equivalent equation y = ax. Remark. If we should reduce the middle member of (1) by clearing of fractions, the result would be the negative of the correct one. This illustrates the fact that there is no relation of equality between the two differentials of each of the quantities x, y and 0, all that we are concerned with being the limiting ratios dy : dx; d

in the form u^ = 0,(^1^ ^^ • • • ^n); Un,= (pJx^,x,. . . x„): (^) DIFFERENTIATION OF IMPLICIT FUNCTIONS. 63 and by differentiating these equations {b) we should find the mn values of the derivatives y-*; -y-*; . . . y-'; etc. Now, the problem is to find these same derivatives from {a) without solving (a). The method of doing this is to form the complete differen- tial of each of the given equations {a), and then to solve the equations thus obtained with respect to du^, du^, etc. The results of the differentiation may, by transposition, be written in the form dF^ , , dF^^ , , dF^ ^ dF^ - , ^du, +--^du,+...+^Ju^=--^dx, -etc.; dF, . , dF, . , , dF^ . Fd, . . d^du, + ^du^ + ... +p^du^ ^-^ip^dx,- etc. du^ * du^ ' dUm, dx^ ' By solving these rn equations for the m unknown quantities du^, dic^ . . . dum, we shall have results of the form du^ = M^dx^ + 3I^dx^ + . . . + M^dx^; du^ = N/lx^ + N/lx^ + . . . + J^ndXni etc. etc. etc. etc.; where M^, iV,, etc., represent the functions of ?i, . . . ic^, a;, . . . Xn, which are formed in solving the equations. We then have for the partial derivatives dx,-^^' dx,-^^' ^^''' Example. From the equations rcos6 = x, ) . ,v r sin 6^ = y, ) it is required to find the derivatives of r and 6 with respect to x and y. 64 THE DIFFERENTIAL CALCULUS. By differentiation we obtain cos 6dr — r sin Odd = dx; sin ddr + r cos 6d0 = dy. Multiplying the first equation by cos 6 and the second by sin 6, and adding, we eliminate dO. Multiplying the first by — sin 6 and the second by cos 0, and adding, we eliminate dr. The resulting equations are dr = cos 6dx -\- sin 6dy; rdd = cos Ody — sin 6dx. Hence, as in the last section, (I) = ^^^ ^' (J) = '^ ^^ EXERCISES. 1. From the equations r sin 6 = X — y, r cos (9 = 2: + ^^ find the derivatives of r and ^ with respect to x and y. 2. From the equations lie'" = r cos ^, ^te~'"= r sin ^, find the derivatives of u and v with respect to r and 6. Ans. (^)-i(e''sin^ + e-^cos^); (It) = £(«-"«- ^ + ''" cos ^). X ' + «/' + .= - 2xyz = 0, X ■ ■fy + « = a. id dz dx and dz dy- FUNCTIONS OF FUNCTIONS, 66 3. From the equations w* + rii = a;' + y% w^ — ru = a;y, find the derivatives of r and u with respect to x and y. 4. From the equations 5. From u^ — 2wz cos 6 -^ z'^ = a\ to^ + ^^^^ cos 6^ + 2;''= Z>% ^ , fZ?^ r/?^ tZw dw d^' 7w' ■^' r76r- 40. Fu7ictions of Fiuictions. Let us have an equation of the form ^* =/(0. ^/^ ^y etc.); (a) where 0, ?/', 6^, etc., are all functions of x, admitting of being expressed in the form 0=/.Oi-); i'=.a^); ^=/s(^); etc. {b) If any definite value be assigned to t, the values of 0, ^', ^, etc., will be determined by {b). By substituting these val- ues in {a), u will also be determined. Hence the equations {a) and (h) determine n as a function of x. By substituting in {a) for 0, tj^, 6, etc., their algebraic expressions fX^)y /aC^O? etc., we shall have ?/ as an explicit function of x, and can hence find its derivative with respect to X, But what we want to do is to find an expression for this derivative without making this substitution. By differentiating (a) we have du = -T-7^/0 + -TT ^0 f -rrrdO + etc. c/0 dtp ^ da By differentiating (Z>), dd> = -T-dxi dib = -r^c?a;; rW = -7-^/^; 6t;c. dx ^ dx dx m THE DIFFERENTIAL GALGULU8. By substituting these values in the last equation and divid- ing by clx, we have du __ du d(p die d'l^ die <^^ _, ^ /-, \ dx ~" d(p dx dip dx dO dx ' ^ The significance of this equation is this: a change in x changes it in as many ways as there are functions 0, ?/?, 6, etc. ■J- -j-dx is the change in it through 0; -yy -j~dx is the change in u through ^; etc. etc. The total differential is the sum of all these separate infinitesimal changes, and the derivative is the quotient of this total differential by dx, EXERCISES. 1. Find -T- from the equations dx u =. a sin (^mv -{- to) -\-h sin (mv — w)\ V = c -{- nx; w =^ c — nx. We find —- — am cos (mv -\- w) -\- bin cos (mv — w)\ -— =z w. dv V I / I ' dx da / . X 7 / \ ^^ -~ = a cos (mv -\- w) — o cos (mv — w): — ^ — — n: dw \ I / ^' dx whence, by the general formula, -— = ati{m — 1) cos {mv -{-w)-\- bn{m + 1) cos (mv — w). 2. Find -r^ from dx u = e'^ + e'f'; = ^^; ^ = ne-'', Aiis. e'^-^'^-ne'^-''. 3. Find --,- from dy ^' + ^0 + ^P"" = ^; (P = m{a + y); ?/; = 7iy, FUNCTIONS OF FUNCTIONS. 67 (It 4. Find -:- from dz r cos a; — r sin 2; = a — y; X = mz-\-h\ y = cos nz. 5. Find -Y from r' + a;r' + y'r + 0' = 0; a:^ -|- «2; = 0; y^ -\- az^ = 0\ = ti;?. 41. The foregoing theory applies equally to the case in which the function is one of two or more variables, some of which are functions of the others. For example, if It = c(>{x, z), (a) then, whatever be the relation between x and Zy we shall always have, for the complete differential of ii, '^" = (^)^^ + (Sh Suppose that x is itself a function of z. We then have dx = -z- dz. dz By substitution in the first equation we have du — (du\ dx . (du \ .,v = \di] dz + KTz)- ^^> The two values of -r- which enter into this equation are different quantities. A change in z produces a change in u in two ways: first, directly, through the change in z as it appears in (t/); second, indirectly, by changing the values of X in (a). The first change depends upon f-7-j in the second 68 THE DIFFERENTIAL CALCULUS. member of (Z>); the second uponf— j -y-; while the first mem- ber of {h) expresses the total change. It is in distinguishing the two values of a derivative thus obtained that the terms pmiial derivative and total derivative become necessary. If we have a function of the form ^ =/(^^ 2/^ ^^ . . • ^), in which any or all of the quantities x, y, lu, etc.^ may be functions of z, then the partial derivative of u with respect to z means the derivative when we take no account of the variations of x, y, w, etc.; and the total derivative, with respect to z, is the derivative when all these variations are taken into account. In such cases the partial derivative has to be distinguished by being enclosed in parentheses (§34). This is why the last equation is written du _ fdii\ fdu\ dx dz ~ \dzj \dx J dz' 4^2i, Extension of the Princii)le. The principle involved in the preceding discussion may be extended to the case of any number of independent variables and any number of functions. If we have r = cf){u, V, w . , , X, y, z . . .), in which x, y, z, etc., are the independent variables, while It, V, 2V, etc., are functions of these variables, we shall have *=(fV«+(f)* + ---+(s)'"+*- Then, since tc, v, to, etc., are functions of x, y, z, etc., we have d2i — -^r-dx 4- -r-dy + etc. ; dx dy '^ dv = —-dx -\- -^-dij -4- etc. dx ^ dy -^ ' FUNCTIONS OF FUNCTIONS. 69 By substituting these values in the preceding equation we find* +[(D+oi+(rr)i+---> + Hence, writing r for cp, its equivalent, dr _ fdr\ f dr\du fdr\dv dx - Vlx] + Wil'dTc + \dvUix + ^^''•' etc. etc. etc. etc. EXERCISES. The independent variables r and 6 being connected with x and y by the equations X — r cos d, y = r sin 6, it is required to find the derivatives of the following functions of X, y, r and 6 with respect to r and 6, We call each of the functions u. I. ?/ = 7-^ + 2.r3/ cos 26^. Here we have -- = 2y cos 20; - = 2x cos 29; da? dy dx ^ dy , . -— = cos 0; -f- = sm 0; dr dr dx , . dy . —^=—rsinO = — y; -y- = r cos = x. * Here, when we use tlie symbol instead of r, there is reiilly no need of enclosing the partial derivatives in parentheses. We have done it only for the convenience of the student. 70 THE DIFFERENTIAL CALCULUS, TT du _ ldu\ du dx du dy dr ~ \drf ~^ dx dr dy dr =z2r + 2y cos Q cos 29 + 2x sin cos 20 = 2r(l + cos 20 sin 20) = r(2 + sin 40); and, in the same way, ~ = 2r' cos 40, dQ We might have got the same result, and that more simply, by sub- stituting for X and y in the given equation their values in terms of r and 0. But in the case of implicit functions this substitution cannot be made; it is therefore necessary to be familiar with the above method. a^ x^ — if^ 2, '?^ = — -4 --^ COS 2^. r' a a' , ¥ 2ah 3- ''^ = -2 + -2 a-- X y r 4. ti=:r' - {x- y)\ 5- 1 ~ X mi 'ZO -\- y cos W 6. 1 1 X cos 26 y sin 26' 7. u = r^ -\- x"" — y\ Let V and w be given as implicit functions of p and 6 by the equations 70 = av; I v' + w' = 2p sin (9. ) ^^ It is required to find the total derivatives of the following functions with respect to p and 6 respectively: 8. u = v"" -{- w^ — p\ 9. 71 = v^ — 2vw cos 6 + tv^. 10. u = — . II. 7c = (v 4- 7a) sin 6. VW V ' / 12. 71 = {V — 7V) cos ^. 13. 21 = 70^ — t?^ -j- ^('^^ + ^OP ^^S ^« PARTIAL DEmVATIVES. 71 From the pair of equations (a) we find dv _ V dw _ w ^ dp ~ 2"/J' dft ~ 2p** -- = iv cot 0; ^-a^^ cot 0; wliich values are to be substituted in the symbolic partial derivatives of u. ^ 43. Remarks on the Nomenclature of Partial Derivatives^ There is much diversity among mathematicians in the no- menclature pertaining to this subject. Thus, the term '^ jiar- tial derivative^^ is sometimes extended to all cases of a deriva- tive of a function of several variables, with respect to any one of those variables, though there is then nothing to distinguish it from a total derivative. Again, Jacobi and other German writers put the total deri- vatives in parentheses and omit the latter from the partial ones, thus reversing the above notation. If we have to express the derivative of (p{x, ?/, z, etc.) with respect to z, the English writers commonly use the symbol -rr- in order to avoid writinor a cumbrous fraction. We thus dz ^ have such forms as D{t + ^y + t\ ^Aa'^ b' ^ c'l' each of which means the derivative of the expression in paren- theses with respect to x, and which the student can use at pleasure. 44. Dependence of the Derivative upon the Form of the Function, Let x and y be two variables entirely independent of each other, and u = 0(.r, ij) (a) a function of these variables. Without making any change in u or x, let us introduce, instead of y, another independent 72 THE DIFFERENTIAL CALCULUS. variable, z, supposed to be a function of x and y. Then, after making the substitution, we shall have a result of the form w = F{x, z), {h) Now, it is to be noted that although both ti and x have the du same meaning in {b) as in («), the value of -y-^ will be differ- ent in the two cases. The reason is that in {(t) y is supposed constant when we differentiate Avith respect to x, while in {b) it is z which is supposed constant. Analytic Illustratioii, Let us have II = ax"^ + by^. This gives du dx 2ax. (0) Let us now substitute for y another quantity, z, determined by the equation z = y -\- X or y — z — X. ^Ye then have w = ax^ -{- b{z — xY; dx ""^''^ "'^ ^^^"^ ~ ^^' which is different from {c). Our general conclusion is: Tlte partial derivative of one variable with respect to another depends not only %ipon the re- lation of those two variables, but upon their relations to the variables which we sup- pose constant in differen- tiating. Geometrical Illustra- tion, Let r and 6 be the polar co-ordinates of a point P, and x and y its rectangular co-ordinates. Then fiq. 9. X = r cos ^; y — r mi 8; r' = x' + y\ (d) PARTIAL DERIVATIVES. 73 Regarding r as a function of x and y, we have dr X ^ / V -7- = - = cos 6. ie) ax r ^ ' But we may equally express r as a function of a: and 6, thus: r = rr sec 6. (/) dv We then have -j- = sec ft (^) Referring to the figure, it will be seen that we derive (e) from (d) by supposing x to vary while ?/ remains constant; that is, by giving the point P an infinitesimal motion along the line PQ \\ to OX, In this case it is plain that the incre- ment of r (SQ) is less than that of x. But in deriving (g) from (/) we suppose x to vary while 6 remains constant. This carries the point P along the straight line OPR; and now it is evident that the resulting increment of r (PR) is greater than that of x. i4 THE DIFFERENTIAL CALCULUS. CHAPTER VK DERIVATIVES OF HIGHER ORDERS- 45. If we have given a function of x, we may, by differentiation, find a value of ~. This value dx will, in general, be another function of x, which we may call cj)\x). Thus we shall have dy dx -notation will he found most con- venient. Applying the rule for differentiating a square, the result is ^JduV du \dxl __ du dx __ du d'u dx '" dx dx ~ dx dx^^ or, in the /^-notation, D^{D^tcy = 2D^uD^*u. In the same way, we find d'{D^uY (duy-'d^u .^ ,n-in» SPECIAL CASES OF SUCCESSIVE DERIVATIVES 87 EXERCISES. Write the derivatives with respect to x of pressions, y being independent of x when equicrescent variable: lO 13 16, 19 \dx. y dx) ' die dti dx ~dy dz {dyV \dxl .dx) \dx') • yVfdyy "V \dx'J ' 14 17 \dx duy dy)' m- crnV dx') • duV dx) (hiV dx) [dx'J ' du dv dx dy du dy d^u dx ?)• du dv dy dx the following ex- it is written as an IduV '■ \d-y) • ( d'n V ^' Vdxdij)' '" (di) \di,) ' '5* \dfl \d/l' 54. Derivatives of Functions of Functions, Let us have, as in § 40, «=/-(^-), (1) where ^^ is a given function of x. It is required to find the successive derivatives of u with respect to x. We may evi- dently reach this result by substituting in (1) for t\) its ex- pression in terms of x, and then differentiating the result by methods already found. But what we now wish to do is to find expressions for the successive derivatives without making this substitution. To do this, assign to x the infinitesimal increment dx. The re- sulting infinitesimal increment in ip will be d.p = '^dx. 88 THE DIFFEBENTIAL CALCULUS. This, again, will give it the increment du = ~jj^i^9 or, by substituting for dip its value, and passing to the de- rivative, du __ du dip dx ~~ dip dx' ^ ' This is a particular case of the result already obtained in §40. The second member of (2) is a product of two factors. The first of these factors is formed by differentiating a func- tion of y^ with respect to ifr, and is therefore another (derived) function of rp\ while the second is, for the same reason, a function of x. Differentiating (2) with respect to x by the rule for a prod- uct, we have* d^u _ dip dip du d'^ip dx^ ~ dx dx dip dx^ ' ^ ' du Now, because -j— is a function of tp, its derivative with re- dip ^ spect to X is to be obtained in the same way as that of ti. If we put, for the moment, we have, as in (2), dti' _ du' dip _ d'^ii dip ^ dx dip dx dip^ dx ' Au dtp * The student should note that the expression — — cannot be put in the form , , , , because the latter form presupposes that ib and x are two in- d^dx dependent variables, which is here not the case. In fact, u does not con- tain X except in if). SPECIAL GASES OF SUCCESSIVE DERIVATIVES 89 and hence, by substitution in (3), dx' "~ di/Adx) ^ dtl^ dx'' ^' which is the required expression for the second derivative. From this we may form the third and higher derivatives by again applying the general rule embodied in (2), namely : If tf) is a function of x, we find the derivative of any func- tion, Uy of ip by differentiating u tuith respect to tp, and mul- tiplying the resulting derivative ly -j-. From the equation (4) we have ^ d^u d^ __ fdipy W , 2— ^ ^ dx* "~ \dx) dx dip* dx dx* J du d^tp dip du d*tp "^ Th'' ~d^' '^'dip'd^* By the rule just given, we have fd^n dtp"" _ d^\i dtp ^ dx ~ ~~ dtp^dx^ jdtc dtp __ d^u dip dx ~" dfp* dx ' Hence, by substitution and aggregation of like terms, cTtc __ d^u IdipV d^u d^ip dtp du d^tp . . d^* ■" 7fp*\d^) "^ cff' Tlx^ d^'^Thp W ^^ Repeating the process, we shall find d^n _ d^ufdfpy , (.(Pu^ ^TH^ItV dx' ~ dipAdx) + ^dip' di' \dx) . (Pu r.d^P dtp . Jd^yi .dud^tp "^ dtp* i^dx*' dx^\lx') A'^ dtp dx' • ^^ 90 THE DIFFERENTIAL GALGULU8. Example. Let us take the case of u = sin tp, ip being any function whatever of x. We may then form the successive derivatives as follows: du _ du dtp __ dip ^ dx ~" dtp dx ~~ dx ' d'^u ^(f ) + ^^^ 1'W'^ ^ - _ ftl^Jty^ 2 sin th^ ^ dx^ "" ^^\dx) ^dx dx"" . dtpd'tp , d'tp — sm t^—- -^-f + cos t^;--\- ^dx dx^ ^ dx^ Jdil^V ^ . ,dil^d'tp , , d'tp EXERCISES. Putting = a function of x, find the first three derivatives of the following functions of with respect to x: I, u = cos 0. 2. u = 0' 3. t^ = 0\ 4. u = 0\ 5, ?^ = log 0. 6. i^ = e'^ . y, u = sin 20. 8. «^ = cos 20, i '55. Change of the Equicrescent Variable, Let the relation between y and a; be expressed in the form ^ = 1. 8. Show that D^\u') = du'D^'u + ISuD^uDJ'u + 6(Z>^.?^)'. 9. Show that if v = u'^y then + ^z(^ - 1) (7z - 2)i^'»-3(Z)^?*)\ 10. If 10 = a cos mx + Z> sin ma;, show that BJ'u + m\i = 0. Then, by successively differentiating this result, show that, whatever the integer n, !>/ + % -m'Z>/«^ = 0. 11. li u = e"^ cos X and v = e^ sin x, then D^u = — 2v and Z>J^ = 2u, Also, D,^?; + 4?; == 0; i)^*^ + 4i^ = 0. 12. If to = e"^ cos mx and v = c"* sm mx, show that the successive derivatives of u and v may always be reduced to the form DJu = AfU — BiV; DJv = AiV -{■ Bi^i, (a) where A and B are functions of m and n. Also, find the values of A^, A^^ B^ and B^, and show by differentiating (a) that DEVELOPMENTS IN SERIES, 95 CHAPTER VIM. DEVELOPMENTS IN SERIES. 57. A series is a succession of terms all of whose values are determined by any one rule. A series is called Finite when the number of its terms is limited; Infinite when the number of its terms has no limit. The sum of a finite series is the sum of all its terms. The sum of an infinite series is the limit (if any) which tlie sum of its terms approaches as the number of terms added to- gether is increased without limit. When such a limit exists, the series is called convergent. When it does not exist, the series is called divergent. To develop a function means to find a series the limit of whose sum, if convergent, shall be equal to tlie function. We may designate a series in the most general way, in the form ^^ + ^^ + ^^3 + . . . + « the nth. terms being called w„. 58. Convergence and Diverge7ice of Series. No universal criterion has been found for determining whether any given series is convergent or divergent. Tliere are, however, a great number of criteria applicable to a wide range of cases. Of these we mention the simplest. I. A series cannot he cojivergent ufiless, as n becomes in- finite, tlie nth term approaches zero as its limit. For if, in such case, the limit of the terms is a finite quantity a, then each new term which we add will always 96 THE DIFFERENTIAL CALCULUS. change the sum of the series by at least a, and so that sum cannot approach a limit. As an example^ the sum of the series 1 — 1 + 1 — 1 + 1 — 1, etc., ad i7ifinitum, will continually change from + 1 to 0, and so can approach no limit, and so is divergent, by definition. II. A series all of ii^liose terms are positive is divergent unless nu,^ ~ when n= oo . To prove this, we have first to show that the harmonic series i + -|^ + i + -|^+ etc, ad infinittim, is divergent. To do this we divide the terms of the series, after the first, into groups, the first group being the 2 terms ^ + i, the second group the following 4 terms, the third group the 8 terms next following, and, in general, the 7ith. group the 2" terms following the last preceding group. We shall then have an infinite number of groups, each greater than i. Now, if, for all the terms of the series after the ?ith, we have nUn > a {a being any finite quantity), then Un > -, ' ^ ' \w'm + lm + 2 / Because the last factor of the second member of this equa- tion increases to infinity, so does its product by a, which proves the theorem. III. If the terms of a series are alternately positive and nega- tive, continually diminish, and approach zero as a limits then the series is convergent. Let the series be ^1 "^ '^3 + ^8 "* '^4 + ^^6 "" • • • • Then, by hypothesis, u,> u^> u,> u^> , , . . DEVELOPMENTS IN SERIES, 97 Let us put Sn for the sum of the first n terms of the series, n being any even integer, and S for the limit of the sum, if any there be. Then this limit may be expressed in either of the forms and Since all the differences in the parentheses are positive, by hjrpothesis it follows that, how many terms soever we take, the sum will always be greater than Sn and less than Sn+i. The difference of these quantities is ?< » + 1^ which, by hypothe- sis, approaches zero as a limit. Since the two quantities S^ and Sn+i approach indefinitely near each other from opposite directions, they must each approach a limit ;S^ contained be- tween them. Graphically the demonstration may be shown to the eye thus; Let the line OS^ represent the sum S^ when 71 = 6, O Ss S8 Sio— S Sn S-9 Sr Fio. 11. or any other even number; OS^ the sum S^, etc. Then every succeeding even sum is greater than that preceding, and every succeeding odd sum is less than that preceding, while the two approach each other indefinitely. Hence there must be some limit S which both approach. An example of such a series is of which the nth term is — ^^ -r. We shall hereafter see 271 — 1 that the limit of the sum of this series is ^tt. If we divide the terms into pairs whose sums are negative, the series may be written 2 2 2 3-5 7-9 1113 — etc. 98 THE DIFFERENTIAL CALCULUS, Pairing the terms so that the sum of each pair shall be posi- tive, the series becomes _j_i_j_ _i_f 3 +5^7 + 901 + 1305 + ^^''• We may show by the preceding demonstration that these series approach the same limit. IV. If, after a cert ai7i finite number of terms, the ratio of two consecutive terms of a series is continually less titan a cer- tain quantity a, ivMcli is itself less than unity, then the series is convergent. Let the nth. term be that after which the ratio is less than a. We then have Un-^2 < OCUn + i < Oc'Un; Un + 3 < ^^^n + 2 < «^X; Taking the sum of the members of these inequalities, we have ^^« + i + «*n+2 + «^n + 3+ ... <{a + a' + a' + . . . )Un. But «: + a' + ^' + . . . is an infinite geometrical progres- sion whose limit when a < 1 is , a finite quantity. Hence, putting S for the limit of the sum of the given series, we have The second member of this inequality being a finite quantity which S can never reach, S must have some limit less than that quantity. As an example, let us take the exponential series DEVELOPMENTS IN SERIES. 99 The ratio of the in + l)st to the nth term is -. This n ratio becomes less than unity when ii > x^ and it approaches zero as a limit. Hence the series is convergent for all values of X. Corollary. A series «o + ^1^ + ^a^' + ^3^' + • • • proceeding according to the powers of a variable, x, is conver- gent whe7i x <1, provided that the coefficients a^ do not in- crease indefinitely. Remarks. — (1) Note that, in applying the preceding rule, it does not suffice to show that the ratio of two consecutive terms is itself always less than unity. This is the case in the harmonic series, but the series is nevertheless divergent. The limit of the ratio must be less than unity. (2) If the limit of the ratio in question is greater than imity, the series is of course divergent. Hence the only case in which Rule IV. leaves a doubt is that in which the ratio, being less than unity, approaches amity as a limit. But most of the series met with come into this class. (3) The sum of a limited number of terms of a series gives no certain indication of its convergence or divergence. If we should compute the successive terms in the development of ^ — loo we should soon find our- selves dealing with numbers having thirty digits to the left of the deci- mal-point, and still increasing. But we know that if we should continue the computation far enough, say to 1000 terms, the positive and negative terms would so cancel each other that in writing the algebraic sum we should have 42 zeros to the right of the decimal-point. On the other hand, if the whole human race, since the beginning of his- tory, had occupied itself solely in computing the terms of the harmonic series, the sum it would have obtained up to the present time w^ould have been less than 44. For 1000 million of people writing 5000 terms a day for 2 million of days would have written only 10^^ terms. It is a theorem of the harmonic series, which we need not stop to demonstrate, that But Nap. ,0. 10.. = ^^^'=,3g_= 43.78. and yet the limit of the sum of the series is infinite. 100 THE DIFFERENTIAL CALCULUS. 1. 59. Maclauri7i's Theorem, This theorem gives a method of developing any function of a variable in a series proceed- ing according to the ascending powers of that variable. If X represents the variable, and the function, the series to be investigated may be written in the form ct>{x) = A, + A,x + A^x^ + A,x' + . . . ; (1) the series continuing to infinity unless is an entire f unc- 1 tion, in which case the two members are identical. >, Whether the development (1) is or is not possible depends '* upon the form of the function 0. Most functions admit of | being so developed; but special cases may arise in which the development is not possible. Moreover, the development will be illusory unless the series (1) is convergent. Commonly this series will be convergent for values of x below a certain mag- nitude, often unity y and divergent for values above that mag- j nitude. What we shall now do is to assume the development possible, and show how the values of the coefficients A may be found. Let us form the successive derivatives of the equation (1). We then have 0(a;) — A^-^ A^x^ A^x' + etc.; dx d'cp ^^ = cf>\x) = ^, + 2A,x + ^A^x' + . = 0"(a;) = 1-2^, + 2-3^30: + 3-4:jy + dx' ^0 = 0-(^) ^ 1.2. 3^3 + 2-3-4^,0: + —-= 0^^>(a:) =:l-2-3-4. . . ?iJn + etc. By hypothesis these equations are true for all values of x small enough to render the series convergent. Let us then put a; = in all of them. We then have DEVELOPMENTS IN ISEMES. 101 0(0) = ^.; .-.^. = 0(0). (O). By substituting these values in (1) we shall have the re- ;quired development. Noticing that the symbolic forms 0'(O), '0"(O), etc., mean the values which the successive derivatives ;take when we put x = after differentiation, we see that the coefficients are obtained by the following rule : Form the successive derivatives of the given function. After the derivatives are formed, suppose the variable to be zero in the original function and in each derivative. Divide the quantities thus formed, in order, by 1; 1; 1*2; '1*2*3, etc., the divisor of the nth derivative being n\ The quotients toill be the coefficients of the powers of the variable in the development, commencing with the zero power, or absolute term, EXAMPLES AND EXERCISES. I. To develop {a -\- xY^u in powers of x. We have u = [a-\- xY', . • . ^, = a\ -- = n{a + a;)**-*; .' . A^ = 7ia''~K '^i=n{n^l)(a + xY-'; r . A,= ^^.^ ^ a^-\ ^^ = n{^^-^)-'^n-s + l)(a + x)-- 102 :rHE DIFPEEENTIAL CALCULUS, Thus the development is {a + xY = a^ + na^'-^x + (|)^''~ V + (3^**"'^' + • • - which is the binomial theorem. I 2. Develop {a — xY in the same way. 3. Develop log (1 + ^)' ' Here we shall have ^! = 1-2(1 + .)-; etc. etc. Noticing that log 1 = 0, we shall find log {l-\-x)=x-ix'+ix' -lx'+ 4. Develop log (1 — x), 5. Develop cos x and sin x. The successive derivatives of sin x are cos x, — sin x, — cos a?, sin a?, etc. By putting oj = 0, these become 1, 0, — 1, 0, 1, 0, etc. Thus we find X^ , x^ ^ , 6. Develop e"", where e is the Naperian base. x^ x^ Ans, 6^ = 1 + a; + ^j + -J + . . . . 7. Develop e"*. 8. Show that 9. Deduce e^'^" = 1 + =*= + ^-^ + DEVELOPMENTS IN SERIES. 103 10. Develop sin {a + x) and cos (a -\-x) and thence, by com- paring with the results of Ex. 5, prove the formulae for the (sine and cosine of the sum of two arcs. Find first x"^ x^ sin (a + x) — sin ^ (1 — *-. + ..) -f cos a (x — ^-f- . .). 11. Develop (1 + <^'^)" ^^^ show that the result may be re- duced to the form "vA ri n^ -\-n x^ n^ ■\- 3?^^ a;' 1 2. Develop e^ sin x and (f cos x and deduce the results e" cos ic = 1 + ^ — ^ kr ""^;iT'~'^^+- •• /v ! 4! o ! 13. Develop cos' x. Begin by expressing cos^ x in the form i cos 3a; + f cos x. 14. Develop tan ^"^^o;. This case affords us an example of how the process of de- velopment may often be greatly abbreviated. It has been shown that -J = ::— i = 1-X* + X' -'X' + etc. (o) ax 1 + X ^ ^ Now assume tan^^^^x- = A + A^x + A^x* + etc. This gives ^^^^- = A, + 2A,z + 3A,z' + etc. (b) Comparing (a) and (b), we have J, = l; A^=-i; A^ = i; A, = - ^; etc. and A^ = A^ = A^ . . . = 0. The value of A^ is^ evidently zero. Hence tan<-*>a; = x - ix' + ^x' - ^x' + etc. (c) 104 TEE DIFFEBENTIAL CALCULUS. 15. Develop sin^~%. ^^''^^ — di = (^ "" ^ )" *> we may develop the derivative and proceed as in the last ex- ample. We shall thus find sm ^-1+^.3 +2.4*5 +2-4-67 + ^^''• 60. Ratio of the Circumference of a Circle to its Diameter. The preceding development of tan^~^^a; affords a method of computing the number n with great ease. The series {c) could be used for this purpose, but the convergence would be very slow. Series converging more rapidly may be obtained by the following device: Let a, a\ a", etc., be several arcs whose sum is 45° = \n. We then have tan (t»r + «' + a" + etc.) = 1. Let t, t' y V'y etc., be the tangents of the arcs a^ or', a", etc. If there are but two arcs, a and or', we then have, by the addition theorem for tangents. -^ = 1; or t-\-V = \-tV. lere are th V -\-t l-tf If there are three arcs, a, a', and a", we replace V by in the last expression, and thus get 1 - Vt" t + f + f' - tff' =zl^tf- ft" - tt''. We now have to find fractional values of ty V and V of the form — , m being an integer, which will satisfy one of these equations. Unity is chosen as the numerator because the powers of the fraction are then more easily computed. The simplest fractions which satisfy the last equation are ^~ %' ^ "5' ^ ~8- DEVELOPMENTS IN SERIES. 105 We then have, from the development of tan ^~*^ t, etc., oc =^ rr 1 1.1 2 3-2' ' 6-2' ,__l 1 , 1 ./I 1,1 3-8' ' b'^' ' - ;r = ^ + n'' + a". These series were used by Dase in computing n to 200 decimals. A combination yet more rapid in ordinary use is found by determining a and a' by the conditions tan a = —] 4a: — a =^ — Tt, 4 2^ 5 We then have tan 2a = 1 - f ~ 12' tan^a = ^^; and because a' = 4:a — ^tt = 4:a — 45°, we have , _ tan 4.^.|j + D^^Dyic |j \ + D^D^n ^ |J + Dy\v |J; rth. D^^J^^ + D^^-^DyU:^^^ \ + EXERCISES. I. Show that in the preceding development the terms of the ?*th order may be written in the form [ - j, ( -J, etc., denoting the binomial coefficients as in § 5. J. Extend the development to the case of three independent variables, and show that the terms to the second order in- clusive will be as follows : 114 If THE DIFFERENTIAL CALCULUS. ^=/(^^ y> ^)y then /• {x -\- h, y -\- k, z -\- 1) = lo + D^uB^u ' hi + DytiD^2i • ^;. A^f^"^^^ 66. Hyperbolic Fimctmis. The sine and cosine of an imaginary arc may be found as follows: In the developments for sin x and cos x, namely. sm X = X cos X : 3! + 5! •••' 2! + 4! • • • ' let us put yi for a:, (z = t^— 1). We thus have ^'^y' = \y + '^i + fi 3! ' 4 cos .v^• = 1 + I + I, + , (1) We conclude: The cosine of a purely imaginary arc is real and greater than unity y tvhile its sine is purely imaginary. We find from (1), cos yi + i sin yi :=: 1 — y -\- 2! y' etc. cos yi — i sin ^* = 1 + «/ + ^l + ^^c. — e^\ and, by addition and subtraction, cos yi r= |(e"^ + e^); ^ sin 1^/^ = \(e~'^ — e^); sin ^i = ii{e^ — e~^). The cosine of yi is called the hyperbolic cosine of y, and is written cosh y, the letter h meaning *^' hyperbolic/' (1) DEVELOPMENTS IN SERIES. 115 The real factor in the sine of yi is called the hyperbolic sine of y, and is written sinh y. Thus the hyperbolic sine and cosine of a real quantity are real functions defined by the equations sinh?/ = i{ev — e-^); ) cosh?/ = -J(c^ + c-^). ) By analogy, we introduce the additional function tanh y = — - — — . The differentiation of these expressions gives d sinh y , d cosh ?/ . , ,_. -^-•-=coshy; —^-^=smhy; (2) d tanh y = — ^ -. ^ cosh y They also give the relations cosh'' y — sinh^ y = ^- (3) Inverse Hyperbolic Fimctions, When we form the inverse function, we may put u = cosh y. Then, solving the equation ey -\-e-y = 2 cosh y = 2u, we find ey = u± ^u^ — 1. Hence y — log (xi ± ^iC" — 1) = cosh <~^^ w. (4) In the same way, if we put u = sinh y, we find y = log (u ± Vu^ + 1) = sinh^-*> u. (5) From the equations (2) and (3) we find, for the derivatives of the inverse functions: 116 THE DIFFERENTIAL CALCULUS. When y = cosh^~ ^^ ^i, or u = cosh y, then ^ = --L=-. (6) When y = sinh^~ ^^ t^, or to — sinh ^Z, then '^ = -J=. (7) du i^ic' + l Eemark. The above functions are called hyperbolic be- cause sinh y and cosh y may be represented by the co-ordinates of points on an equilateral hyperbola whose semi-axis is unity. The equation of such an hyperbola is x^-f = 1, which is of the same form as (3). EXERCISES. 1. By continuing the differentiation begun in (2) prove the following equations: VJ^ sinh X = sinh x; I) J cosh X — cosh x; Dx"^^^ sinh X = sinh x. etc. etc. 2. Develop sinh x, as defined in (1), in powers of x byMac- laurin^s theorem. A71S. smh x = y + --^-^^ + --^+,.,. 3. Develop sinh (x + h) and cosh {x + h) by Taylor^s theorem and deduce sinh (x+h) = sinh xil + ^^ + • • • )+ cosh xix + -^ + . . . j = sinh X cosh h -\- cosh x sinh Ir, cosh {x-\-h) — cosh a; cosh li + sinh 2; sinh h. i MAXIMA AND MINIMA. 117 CHAPTER IX. MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. 67. Def, A xnaxiinuiu value of a function is one which is greater than the yahies immediately preceding and follow- ing it. A xninimum value is one which is less than the values immediately preceding and following it. Remark. Since a maximum or minimum value does not mean the greatest or least possible value, a function may have several maxima or minima. 68. Problem. Having given a function y = 0(^). it is required to find those values of x for which y is a rnaxi- 771717)1 or a minimum. Let us assign to x the increments + h and — //, and develop in powers of h. We shall then have . , , 7 \ dy h ^ d'^y h^ , .y"=0(z+/O=y + ^j- + ^,p3 + ctc. In order that the value of y = 0(.r) may be a minimum, it must, however small we suppose //, be less than either ?/' or I/". That is, the expressions , dy h , d'^y ¥ , 118 THE DIFFERENTIAL CALCULUS. must both be positiye as h approaches zero. But if -j- is finite, li may always be made so small that the terms in h^ shall be less in absolute magnitude than those in h (§ 14), and the condition of a minimum cannot be satisfied. We must therefore have, as the first condition, | = 0'(-) = O. . (1) By solving this equation with respect to x will be found a value of X called a critical value. The same reasoning applies to the case of a maximum, so that the condition (1) is necessary to either a maximum or a minimum. Supposing it fulfilled, we have Since li^ is positive, the algebraic sign of these quantities, as h approaches zero, will be the same as that of -7-^. When this second derivative is positive for the critical value of X, y, being less than y' or ?/", will be a rtiinimtim. When 7iegative, y will be greater than either y' or y", and so will be a maximum. We therefore conclude: Conditions of minimum: -^ — 0; -7-^ positive, •^ dx ' dx^-^ Conditions of maximum: -J^ = 0; -— ^ negative. We have, therefore, the rule: Equate the first derivative of the fu7iction to zero. This equation will give one or more values of the independent vari- dbUy called critical values, and thence corresponding values of the function. MAXIMA AND MINIMA. 119 SuhstiMe the critical values in the expression for the second derivative. When the result is positive, the function is a ininimiim; wheii negative, a maximum. Exceptional Cases. It may happen that the second deriva- tive is zero for a critical value of x. We shall then have y -y= - .V" - // dx' 3!+^Z.T^4! ^^''•' ^i\ f^^\ etc. dx'31^ dx' 41 -here, and there can be neither a maximum nor a minimum unless d^y —^ = 0, If this condition is fulfilled, y will be a maximum when the fourth derivative is negative; a minimum when it is positive. Continuing the reasoning, we are led to the following ex- tension of the rule: Fi7id the first derivative in order which does not va7iish for a critical value of the independent variable. If this de- rivative is of an odd order, there is neither a 7naximum nor a minimtim; if of an even order, there is a minimum when the derivative is positive, a maxiimim when it is negative. The above reasoning may be illustrated by the graphic rep- resentation of the function. When the ordinate of the curve is a maximum or a minimum the tangent will be parallel to the axis of abscissas, and tlie angle which it makes with this axis will cliangc from positive to negative at a point hav- ing a maximum ordinate, and from negative to j^osi- tive at a point having a minimum ordinate. For example, in the fig- ure a minimum ordinate occurs at the point Q^ and maxi- mum ordinates at P and R. Fig. 12. 120 THE DIFIPERBNTIAL CALCULUS. EXAMPLES AND EXERCISES. I. Find the maximum and minimum values of the expres- sion y = 2x' + dx' - 36x + 15. By differentiation^ ^ :=: ex' + 6x- 36; ax Equating the first derivative to zero^, we have the quadratic equation x' + X — 6 = 0, of which the roots are x = 2 and x = — 3, d'x The values of -j-^ are + ^^ ^^^ — 30. Hence a; = 2 gives a minimum value of ^ = — 29; a; = •— 3 gives a maximum value of ^ = + ^^^ Find the maximum and minimum values of the following functions: 2. a:' + ^x" — 24:x + 9. 3. x' — 3x + 5. X x'^ — X -{-1 ' S.y = af. g, y =2 sin 2x — x, lo. y z=z {x + l){x — 2)\ II. y = {x — a'){x — Z>)'. _ {x + Sy _ (x-a){x-h) '^- ^-- {x+ 2y' '^' ^ ~ {x -p)(x - g)' 14. y =z cos 2.T. 15. y = cos 7^a;. 16. y = sin 3^. 17. ^ 1= sin ^io;. _ a; ^7^.9. A maximum when a: =+ cos a;. ^ ~ 1 -\- X tan a;* A minimum when x == — cosrK. MAXIMA AND MINIMA, 121 19. y = 31. y = sin X cos X. sin X y = sm X cos a;, cos a; y = 1 + tan a;' —' n -j^ _l_ ^^^^^ ^« — H 23. The sum of two adjacent sides of a rectangle is equal to a fixed line a. Into what parts must a be divided that the rectangle may be a maximum? Aris. Each part = ^a. Note that the expression for the area is x{a — x). 24. Into what parts must a number be divided in order that the product of one part by the square of the other may be a maximum? A 71s, Into parts whose ratio is 1 : 2. Kote that if a be the number, the parts may be called x and a — x, 25. Into what two parts must a number be divided in order that the product of the 7/^th power of one part into the 7ith power of the other may be a maximum? Ans. Into parts whose ratio is f/i : n, 26. Show that the quadratic function ax* -{-bx-^c can have but one critical value, and that it will depend upon the sign of the coefficient a whether that value is a maximum or a minimum. 27. A line is required to pass through a fixed point P, whose co-ordinates are a and b in the plane of a pair of rectangular axes OX and OV, What angle must the line make with the axis of X, that the area of the triangle XYO maybe a minimum? Show also that P must bisect the segment XY. Express the mtercepts which the line cuts off from the axes in terms of a, b and the variable angle a. The half product of these intercepts will be the area. We shall thus find b' \ \ a V i\ is \ c< \ \ Fio. 13. 2 Area = {a + b cot ct){b + a tan a) = 2ad-\-a^ tan a -f tan a 122 TEE DIFFERENTIAL CALCULUS. • \ Then, taking tan a = if as the independent variable, we readily find, for the critical values of t and cr, ■ ^=±-, or «sma=±5 cos a, a It is then to be shown that both values of i give minima values of the area ; that the one minimum area is 2a5, and the other zero ; that in the i first case the line YX is bisected at P, and in the other case passes I through 0. 28. Show by the preceding figure that whatever be the an- gle XOYy the area of the triangle will be a minimum when the line turning on F is bisected at P. / The student should do this by drawing through P a line making a \ \ ] small angle with XPY. The increment of the area XO Y will then be (, \ the difference of the two small triangles thus formed. Then let the small I angle become infinitesimal, and show that the increment of the area \ XOFcan become an infinitesimal of the second order only when PX== / PY, 29. A carpenter has boards enough for a fence 40 feet in length, which is to form three sides of an enclosure bounded on the fourth by a wall already built. What are the sides and area of the largest enclosure he can build out of his ma- terial? Ans, 10 X 20 feet = 200 square feet. 30. A square piece of tin is to have a square cut out from each corner, and the four projecting flaps are to be bent up so as to form a vessel. What must be the side of the part cut out that the contents of the vessel may be a maximum? Ans. One sixth the side of the square. 31. If, in this case, the tin is a rectangle whose sides are 2a and %l, show that the side of the flap is \(a-\-l - Va' - ah + ¥). 32. What is the form of the rectan- gle of greatest area which can be drawn in a semicircle? Note that if r be the radius of the circle, and X the altitude of the rectangle, \/r'^ — ^ will be half the base of the rectangle. Fig. 14. MAXIMA AND MINIMA. 123 69. Case tuhoi the function which is to be a maximum or minimum is exjyressed as aftuiction of two or more variables connected by equations of condition. The function which is to be a maximum or minimum may be expressed as a function of two variables, x and y, thus: u = 0(a;, y). (1) If X and y are independent of each other, the problem is different from that now treated. K between them there exists some relation A^, y) = 0, (2) we may, by solving this equation, express one in terms of the other, say y in terms of x. Then substituting this value of y in (1), u will be a function of x alone, which we may treat as before. It may be, however, that the solution of the equation (2) will be long or troublesome. We may then avoid it by the method of § 41. From (1) we have du __ ldu\ fdu\dy dx ~~ \dx J \dy Jdx ' and from (2) we have, by the method of § 37, dy_^ D,f dx Dyf Substituting this value in the preceding equation, we shall dii have the value of -z-y which is to be equated to zero. The equation thus formed, combined with (2), will give the critical values of both x and y, and hence the maximum or minimum value of w. 124 THE DIFFERENTIAL CALCULUS, EXAMPLES AND EXERCISES. I. To find the form of that cylinder which has the maxi- mum volume with a given extent of surface. The total extent of surface includes the two ends and the convex cylindrical surface. If r be the radius of the base, and h the altitude, we shall have : Area of base, nr'^. Area of convex surface, ^itrh. Hence total surface = 27t{r^ -f- rh) = const. E a, {a) Also, volume = 7tr%. {b) Putting u for the volume, we have, from (J), — - = 27trh 4- Ttr^—-, dr dr From (a) we find dh _ _ h + 2r dr ~ r ' Whence -r- — Ttrh — 27rr^. dr Equating this to zero, we find that the altitude of the cylinder must be equal to the diameter of its base. 2. Find the shape of the largest cylindrical tin mug which can be made with a given weight of tin. This problem differs from the preceding one in that the top is sup- posed to be open, so that the total surface is that of the base and con- vex portion. Ans, Altitude = radius of bottom. 3. Find the maximum rectangle which can be inscribed in a given ellipse. If the equation of the ellipse is h'^x'^ + <^V = <^^*^ the sides of the rectangle are 2x and 2y. Hence the function to be a maximum is ^xg, subject to the condition expressed by the equation of the ellipse. This condition gives dy _ hH dx ~~ a^y MAXIMA AND MINIMA. 125 We shall find the rectangle to be a maximum when its sides are proportional to the corresponding axes of the ellipse; each side is then equal to the corresponding axis divided by V§. 4. Find the maximum rectangle which can be inscribed in the segment of a parabola whose semi- parameter is p, cut off by a double ordinate whose distance, OX, from the vertex is a. Show also that the ratio of its area to that of the circum- scribed rectangle is con- stant and equal to 2 : |/27. By taking x and y as in the Fio. 15. figure, a — X will be the base of the rectangle, and we shall have 2y for its altitude. Hence its area will be 2y{a — x), while x and y will be connected by the equation of the parabola, y^ = 2px, 5. Find the cone of maximum volume which shall have a given extent of conical surface. A71S, Alt. = radius of base X V^. 6. Find the volume of the maximum cylinder which can be inscribed in a given right cone, and show that the ratio of its volume to that of the cone is 4 : 9. 7. Find the cylinder of maximum cylindrical surface which can be inscribed in a right cone. A71S, Alt. of cylinder = i alt. of cone. 8. Find the maximum cone which can be inscribed in a given sphere. If we make a central section of the sphere through the vertex of the cone, the base and slant height of the cone will be the base and equal 126 THE DIFFERENTIAL CALCULUS, sides of an isosceles triangle inscribed in the circular section. Thus the equation between the base and altitude of the cone can be obtained. Ans, Alt. = f radius of sphere. 9. Find the maximum cylinder which can be inscribed in an ellipsoid of revolution. 2 Ans, Alt. = — - of axis of revolution. Vd 10. Find the cone of maximum conical surface which can be inscribed in a given sphere. 11. Of all cones having the same slant height, which has the maximum volume ? 12. A boatman 3 miles from the shore wishes, by rowing to the shore and then walking, to reach in the shortest time a point on the beach 5 miles from the nearest point of the shore. If he can pull 4 miles an hour and walk 5 miles an hour, to what point of the beach should he direct his course? Ans. 4 miles from the nearest point of the shore. Express the whole time required in terms of the distance x of his point of landing from the nearest point of the shore. 13. Find the maximum cone which can be inscribed in a paraboloid of revolution, the vertex of the cone being at the centre of the base of the paraboloid. Ans, Alt.' = ^ alt. of paraboloid. 14. Find the maximum cylinder which can be described in a paraboloid of revolution. 15. Find the rectangle of maximum perimeter which can be inscribed in an ellipse. 16. On the axis of the parabola y^ = "Hpx a point is taken at distance a from the vertex. Find the aljscissa of the near- est point of the curve. Begin by expressing the square of the distance from the fixed point to the variable point {x, y) on the parabola. 17. Determine the cone of minimum volume which can be circumscribed around a given sphere. MAXIMA AND MINIMA. 127 1 8. Determine the cone of minimum conical surface which can be circumscribed around a given sphere. 19. Find that point on the line joining the centres of two circles from which the greatest length of the combined cir- cumferences will be visible. 20. Find that point on the line joining the centres of two spheres of radii a and h respectively from which the greatest extent of spherical surface will be visible. Ans, The point dividing the central line in the ratio «■ : h . 21. Show that of all circular sectors described with a given perimeter, that of maximum area has the arc equal to double the radius. 22. A ship steaming north 12 knots an hour sights an- other ship 10 miles ahead, steaming east 9 knots. What will be the least distance between the ships if each keeps on her course, and at what time will it occur? Ans. Time, 32 min.; distance, 6 miles. 23. What sector must be taken from a given circle that it may form the curved surface of a cone of maximum volume? Ans. V~i of the circle. 24. A Norman window, consisting of a rectangle sur- mounted by a semicircle, is to admit the maximum amount of light with a given perimeter. Show that the base of the rectangle must be double its altitude. 128 THE DIFFERENTIAL CALCULUS. CHAPTER X. INDETERMINATE FORMS. 70. Let us consider the fraction For any value we may assign to x there will be a definite value of (p{x) found by dividing the numerator of the frac- tion by the denominator. To this statement there is one exception, the case of a; = 3. Assigning this value to x, we have 0(3) = f ISTow, the quotient of two zeros is essentially indeterminate. For the quotient of any two quantities is that quantity which, multiplied by the divisor, will produce the dividend. But any quantity whetever when multiplied by will pro- duce 0. Hence, when divisor and dividend are both zero, any quantity whatever may be their quotient. But when we consider the terms of the fraction, not as ab- solute zeros, but as quantities approaching zero as a limit, then their quotient may approach a definite limit. We then regard this limit as the value of the fraction corresponding to zero values of its terms. As another example, consider the quantity We may compute the value of this expression for any value of X except 2, When x = 2 the terms will both become in- finite. Since if any quantity whatever be added to an infinite INDETERMINATE FORMS. 129 the sum will be infinite, it follows that any quantity what- ever may be the difference of two infinites. There are several other indeterminate forms. The follow- ing are the principal ones which take an algebraic form: ^; ^; Ox co; 00 - oo ; 0"; oo°; 1«. 71. Evahiation of the Form ^. In many cases the inde- terminate character of an expression may be removed by algebraic transformation. For example, dividing both terms of the fraction (1) by x — 3, it becomes .t -f- 3, a determinate quantity even for :t' = 3. Again, the expression (2) can be reduced to the form — — ^r, which becomes 1 when a: = 2. a;+ 2 The general method of dealing with the first form is as follows: Let the given fraction be and let it be supposed that both terms of this fraction vanish when X — a, ^o that we have (l>{a) = and ^^a) = 0. (3) Fut h =x — a, and develop the terms in powers of h by Taylor^s theorem. We shall then have cp{x) = \a) + ^) V.'(«) (4) 130 THE DlFFEREm'lAL CALCULUS. as a limits which is therefore the required lirait of the frac- tion when both its members approach the limit zero. It may happen that ct)'{a) and tp\a) both vanish. In this case the required limit of the fraction in (4) is seen to be 0"(«) ,p"(a)' In general: The required limit is the ratio of the first pair of derivatives of like order which do 7iot doth vajiish. If the first derivative which vanishes is not of the same order in the two terms, — for example, if, of the two quantities (p^{a) and fp^{a), one vanishes and the other does not, — then the limit of the fraction will be zero or infinity according as the vanishing derivative is that of the numerator or denominator. Eemaek. It often happens that the terms of the fraction can be developed in the form (4) without forming the succes- sive derivatives. It will then be simpler to use this develop- ment instead of forming the derivatives. EXAMPLES AND EXERCISES. x'-a' for X = a,* X — a (p(x) — x^ — a^\ 0'(^) =22:; .• . 0'(a) = 2a; ip{x) = X — a; tp\x) = 1; .•. ip^{a) = 1. x^ — a^ . • . lim. (x = a) = 2a* X — a^ ^ a result readily obtained by reducing the fraction to its lowesi terms. 2. — ^-r- for x = 1. Ans. 1. X —1 ff e~^ 3, for a; = 0. Ans. 2. X * Using strictly the notation of limits, we should define the quantity sought as the limit of the fraction when x approaches the limit a. Bui no confusion need arise from regarding the limit of the fraction as its value for a; = a, as is customary. INDETERMINATE FORMS. 131 2; — sin it - / ^v . , 4. i for (a; = 0). ^7i5. |. I Here the successive derivatives of the terms are: I i a — ' — ^^-^^ — ' — ' for ix = 0). Ans. 0. e* — 1 — a; ^ ^ 132 THE DIFFERENTIAL CALGULtTS. 00 72. Forms — and X oo . These forms may be reduced 00 to the preceding one by a simple transformation. Any frac- tion — - may be written in the form - — '- — ^-^. If N" and D both become infinite, \ -^ D and \ -^ N will both become infini- tesimal, and thus the indeterminate form of the fraction will be %. Again, if of two factors A and B, A becomes infinitesimal while B becomes infinite, we write the product in the form A -^ and then it is a fraction of the first form. \ -T- B But this transformation cannot always be successfully ap- plied unless the term which becomes infinite does so through . having a denominator which vanishes. For example, let it be required to find the limit of cc'^(log xy for X ^^, Here rr"* approaches zero, while log x, and there- fore (log xY^ becomes infinite for a; = 0. Hence the denomi- nator of the transformed fraction will be ^ (putting for brevity Z = log .t). The successive derivatives of this quantity with respect to x are -^ . ^M 1 ^ + ^V etc The successive derivatives of the numerator are mx'^~'^\ m(in — \)x'^~'^\ etc. The limiting values of the given quantity x'^V*- thus become • ^ L • ptp ■which remain indeterminate in form how far soever we may carry them. r INDETERMINATE FORMS. 133 [j In such cases the required limit of the fraction can be found only by some device for which no general rule can be laid down. In the example just given the device consists in replacing a; by a new variable y, determined by the equation log a; = - y. We then have a; = c ~ ^. Since for z ^0 y ^co ^ -wq now have to find the limit of for y = oo. By taking the successive derivatives of the two terms of the fraction -— we have the successive forms ^my ny''-\ n{n - l).y"~^ n{n - 1) {71 -2)y''-\ Whatever the value of n, we must ultimately reach an ex- ponent in the numerator which shall be zero or negative, and then the numerator will become n\ if n is a positive integer, and will vanish for ?/ = 00 , if n is not a positive integer. But the denominator will remain infinite. We therefore con- clude: lim. [:C{\og .t)"] {x ^ 0) = 0, whatever be m and n, so long as m is positive. From this the student should show, by putting z^x~'^ and m = 1, that the fraction (log^r becomes infinite with z, how great soever the exponent Uy and therefore that any infinite immber is an infinity of hiyher order than any power of its logarithm, 73. Form 00 — oo . In this case we have an expression of the form F{x) = ic — V, 134 THE DIFFERENTIAL CALCULUS. in which both u and v become infinite for some value of x. Placing it in the form we see that F{x) will become infinite with u unless the fraction V — approaches unity as its limit. When this is the case the expression takes the form oo x of the preceding article. 74. Form 1*. To investigate this form let us find the limit of the expression when n becomes infinite. Taking the logarithm, we have log u = hn log [l + -j Making n infinite, we have lim. log u = h; or, because the limit of log ii is the logarithm of lim. u, log lim. ti = h. Hence lim. \1 -\ — I (;i = oo ) = ,h In order that this result may be finite, h itself must not be infinite. We therefore reach the general conclusion: Theorem. In order that an expression of the form (1 + ay may have a finite limit luhen a becomes infinitesimal and x infinite, the j^Todnct ax must not lecome infinite. Cor. If the product ax approaches zero as a limit, the given expression will approach the limit unity. INDETERMINATE FORMS, 135 75. Forms 0"^ and oo\ Let an expression taking either of these forms as a limit be represented by w^^F, The problem is to find the limiting value of the expression when approaches zero and u either approaches zero or becomes infinite. From the identity u = c^^«" we derive F = iff* = c* ^"^^ \ We infer that the limit of F will depend upon that of log u. If lim. log uis + CO , then lim. F= oo. If lim. log 7^ is — 00 , then lim. F = 0. If lim. log u is 0, then lim. F = 1. If lim. log u is finite, then lim. F is finite. Hence the rule: To find the limit of w* when (p ^ and ti — or oo , put I = lim. log tc. Then lim. w^ = e'. EXAMPLES AND EXERCISES. 1. Find lim. of for a; =^ 0. Here of = ^^^^^ Since x log x has zero as its limit when x = 0, the required limit is e° or 1. 2. lim. a;***^ for x :^ 0, Ans. F = 1, 1 3. lim. a;* for a: £: 00 . A71S, F = 1. 4. a;J"^^ for a; = 1. Ans. —. n 5. a:i^-^ for a; = 1. ^W5. e~". 6. (1 — a;)* for a; = 0. ^7^5. e~\ ff ^— ^ 7. , jT— — T for a; = 0. Ans. 2, log (1 + x) ^ loer sin 2x . ^ . , . ^^ i 8. -r^ — ; for X = 0. Ans. fflEt. =" I log sm X ^ .- + log(l-:^)-l f^^ ^ ^ ^_ ^^^^ ^ a; — tan x 136 THE DIFFERENTIAL CALCULUS. 1 / log ^ y for a; = 00 . ^^^5. 1. 11. a; tan ^ — ^ sec x for a; = ^. ^^^5. — 1. 12, V sin — for y = co . Ans. a, if y ^ 13. x\cv^ ~^) ^^^ a; = 00. ^^5. log a. 14. f j for a; = 0. -47^5. 1. 15. ( J for a; = 0, Ans. e^, 16. (cos :r)x3 for x — 0. ^?«5. e 17. (1 - y) tan -y for ^ = 1. Ans, -. 1 18. ^^-?^^V for a; = 0. ^/^5. 1. 19. X — x'' log (1 + —) for a; = Qo. J^/^5. -J. 20. log (1 + for X ■■ ^^, 21. (a,^ + a: T for X =-^. Ans. a,a.. 23. Show that, how great soever the exponent n, X -p, r- — 00 when a; — Qo . (log xy - PLANE CURVES, 137 CHAPTER XI. OF PLANE CURVES. 76. Forms of the Equations of Curves, As we have here- tofore considered curve lines, they have been defined by an equation between the co-ordinates of each point of the curve, and therefore of one of the forms y=/(x); x=f{yy, (1) and F{x, y) — 0. The distinguishing feature of the equation is that when we assign a vahie at pleasure to one of the co-ordinates x or y, one or more cor respond mg values of the other co-ordinate are determined by the equation. But the relation between x and y may be equally well defined by expressing each of them as a function of an auxiliary variable, which is then the independent variable. Calling this auxiliary variable u, the equations of a curve will be of the form y = 0,(n). f ^^^ Assigning values at pleasure to n, we shall have correspond- ing values of x and y determining each point of the curve. An advantage of this method of representation is that for each value of u we have one definite point of the curve, or several definite j^oints when the equations give several values : of the co-ordinates for each value of u; and we thus have a relation between a point and the algebraic quantity u. It is also to be remarked that by eliminating u from the equations (2) we shall get a single equation between x and y which will be the equation of the curve in one of the forms 138 THE DIFFERENTIAL CALCULUS. ^ X 5^ py \ a i o X !x Fia. 16. Example 1. Let us put ay'b'^ the co-ordinates of any fixed point i? of a straight line; a E the angle which the line makes with the axis of x\ p = the distance of any point P of the line from the point {a, h). Then we readily see from the figure that the co-ordinates x and y of F are given by the equations x = a + pcosa;) y = h -\- p sin a; ) ^ * which are equations of the straight line in the independent form. Here p is the auxiliary variable, called ii in Eq. 2. By eliminating this quantity we shall have X sin oc — y cos a ^:^ a sin a — h cos a, which is the equation of the line in one of its usual forms. Example 2. The equation of a circle may be expressed in the form X =^ a -\- c cos II] \ y = h -\- c smw, (4) u being the independent variable. By writing (4) in the form X — a = c cos u, y — h = c sin Uy and eliminating ti by taking the sum of the squares of the two equa- tions, we have Fio. 17. PLANE CURVES. 139 {.c - ay + (// - by = c\ the equation of a circle of radius c, Notice ihe beautiful relation between (3) and (4). They are the same in form: if in (4) we write p for c and a for u, they will be the same equations. Then, by supposing p constant and a variable, we are carried round the point {a, h) at a constant distance p, that is, around a circle. By suppos- ing p variable and a constant we are carried through {a, h) in a constant direction, that is, along a straight line. 77. Infiriitesimal Elevients of Curves. Let P and P' be two points on a curve, P being supposed fixed, and P' variable. We may then sup- pose P' to approach P as its limit, and in- quire into the limits of any magnitudes associated with the curve. We may also measure the length of an arc of the curve from an initial point C to a terminal point P. Then, supposing C fixed and P variable, PP' may be taken as an increment of the arc. If we put s E arc GP, we shall have As = arc PP\ Axiom. The ratio of an infinitesimal element of a curve to the straight line joining its extremities approaches unity as its limit. We call this proposition an axiom because a really rigorous demonstration does not seem possible. Its truth will appear by considering that if the curve has no sharp turns, which we presuppose, then it can change its direction only by an in- finitesimal quantity in any infinitesimal portion of its length, Now, a line which has the same direction throughout its length is a straight line. 140 THE DIFFERENTIAL GALGULUS. 78. Theojiem I. If a straiglit line touch a curve at the point Fy a point P' on the curve at an infinitesimal distance will, in general, he distant from the tangent hy an infinitesimal of the second order. Let y = f (x) be the _o equation of the curve. fiq. 19. Let us transform the equation to a new system of co-ordi- nates^ x' and y% so taken that the axis of X' shall be parallel i to the tangent at F. This will make dx' : 0. Let x' and y' be the co-ordinates of F, and (a;' + h, ^z") the co-ordinates of a point P' near P. Developing by Taylor^s theorem, we have ^ ^ ~dx^^'~^ dx^' 1-2 + Since -r-. — 0. when h becomes infinitesimal dx' Now, f/" — y^ is the distance F^Q of the point P' from the tangent at P. d^y' h"^ the term of highest order in this distance is ~^ —-, a quan- ax X /v tity of the second order. Eemakk. In the special case when —-,2 = 0, the distance in question may be a quantity of the third or of some higher order, according to the order of the first differential coeffi- cient which does not vanish. Corollary. 7' he cosifie of an infinitesimal arc differs from tinity hy an infinitesimal of the second order. For if we draw a unit circle with its tangent at the initial point, the cosine of an arc will differ from unity by the dis- tance from the end of the arc to the tangent line. When the arc is infinitesimal, the corollary follows from the theorom, PLANE CURVES. 141 Theorem II. The area included between an mfinitesimal arc and its chord is not greater than an infinitesimal of the third order. From Th. I. we may readily see that the maximum distance between the chord and its arc is a quantity of the second order. The area is less than the product of this distance by the length of the chord, which product is an infinitesimal of at least the third order. 79. Expressio7is for Elements of Curves. Def An element of a geometric magnitude is an infinitesimal por- tion of that magnitude. The word implies that we conceive the magnitude to be made up of infinitesimal parts. Element of an Arc. Let us put s = the length of any arc of a curve; ds = an element of this arc. If P and P' be two points of a curve, we shall have (chord FPy = Ax' + Ay\ When PP' becomes infinitesimal, ^s the ratio of ds to PP' becomes unity (§ 77), and we have. y^ ^x Ay ds' = dx' + dl/; Fig. 20. ds = Vdx' + dif = yi + (vfj^^-^- Case of Polar Co-ordinates. To express the element of a curve referred to polar co-ordinates, differentiate the equa- tions X = r cos 6; y = r sin ^. Thus dx = cos Odr — r sin 6df^; dy = sin 6dr + r cos 6d0\ which gives d!<' = dr' -f r'dfP and ds = Yr' + (^^^' d6. 142 THE DIFFEEENTIAL CALCULUS. 80. Equations of certain Noteworthy Curves, The Cycloid. The cycloid is a curve described by a point on the circumfer- ence of a circle rolling on a straight line. A point on the circumference of a carriage-wheel, as the carriage moves, describes a series of cycloids, one for each revolution of the wheel. To find the equation of the cycloid, let P be the generating point. Let us take the line on which the circle rolls as the axis of X, and let us place the origin at the point where P is in contact with the line OX. Fio. ^. Also put a = the radius of the circle ; ^i = the angle through which the circle has rolled, expressed in terms of unit-radius. Then, when the circle has rolled through any distance OP, this distance will be equal to the length of the arc PP of the circle between P and the point of contact P, that is, to au. We thus have, for the co-ordinates of the centre, (7, of the circle, X = au; y = a; and for the co-ordinates of the point P on the cycloid, X = au — a Bin u = a{u — sin u); y = a —a cos u = a{l — cos u); which are the equations of the cycloid with u as an independ- ent variable. ^'1 (1) PLANE CURVES. 143 To eliminate Uy find its value from the second equation, ?^ = cos<-^>fl ~^). This gives sin 71 = Vl — cos' w = —. ci Then, by substituting in the first equation x = a cos<-» '^—^ - V2ay-y\ (2) CI • which is the equation of the cycloid in the usual form. ''^ 81, The Lemniscate is the locus of a point, the product of whose distances from two fixed points (called foci) is equal to the square of half the distance between the foci. Let us take the line joining the foci as the axis of X, and the middle point of the segment between the foci as the origin. Let us also put c = half the distance between the foci. Fio. 22. Then the distances of any point {x, y) of the curve from the foci are V{x - cY + y^ and V{x + cY + y\ Equating the product of these distances to c% squaring and reducing, we find {x^ + yy = 2c\x^-y-), (3) which is the equation of the lemniscate. 144 THE DIFFERENTIAL CALCULUS. \ Transforming to polar co-ordinates by the substitutions x = r cos 6, y = r Bin 6, we find, for the polar equation of the lemniscate, r^ = 2c' cos 20. (4) Putting ^ = 0, we find, for the point in which the curve cuts the line joining the foci. The line a is the semi-axis of the lemniscate. Substitut- ing it instead of c, the rectangular and polar equations of the curve will become r' = a' cos 26. ) ^^ 83. The Archimedean Spiral, This curve is generated by the uniform motion of a point along a line revolving uni- formly about a fixed point. To find its polar equation, let us take the fixed point as the pole, and the position of the revolving line when the generat- ing point leaves the pole as the axis of reference. Let us also put a = the distance by which the generating point moves along the radius vector while the latter is turning through the unit radius. Then, when the ra- dius vector has turned through the angle ^, the point will have moved from the pole through the distance ad. Fig. 23. Hence we shall have as the polar equation of the Archimedean spiral. PLANE CURVES. 145 If we increase by an entire revolution (27r), the corre- sponding increment of r will be 27ra, a constant. Hence: The Archimedean spiral cuts any fixed position of the ra- dius vector in an indefinite series of equidistant points. 83. The Logarithmic Spiral, This is a spiral in which the logarithm of the radius vector is proportional to the angle through which the radius vector has moved from an initial position. Hence, if we put 0^ for the initial angle, we have log r:=l{e- 0^ I being a constant. Hence 19 - 100 r = e = e w. Putting, for brevity, a — e % the equation of the logarith- mic spiral becomes fio. 24. r = ae^^y a and I being constants. EXERCISES. 1. Show (1) that the maximum ordinate of the lemniscate is |c, and (2) that the circle whose diameter is the line join- ing the foci cuts the lemniscate at the points whose ordinates are a maximum. 2. Find the following expression for the square of the dis- tance of a point of a cycloid from the starting point (0, Fig. ai): r^ = 2ay -\- 2uax — a^u*, 3. A wheel makes one revolution a second around a fixed axis, and an insect on one of the spokes crawls from the cen- tre toward the circumference at the rate of one inch a second. Find the equation of the spiral along which he is carried. 146 THE DIFFERENTIAL CALCULUS. 4. If, in that logarithmic spiral for which a = 1 and I = 1, r = e^ the radius vector turns through an arc equal to log 2, its length will be doubled. 5. 11, in any logarithmic spiral, one radius vector bisects the angle between two others, show that it is a mean propor- tional between them. 6. Show that the pair of equations X = au^y represent a parabola whose parameter is — . 7. If, in the equation of the Archimedean spiral, 6 and therefore r take all negative values, show that we shall have another Archimedean spiral intersecting the spiral given by positive values of 6^ in a series of points lying on a line at right angles to the initial position of the revolving line. This should be done in two ways. Firstly, by drawing the continua- tion of the spiral when, by a negative rotation of the revolving line, the generating point passes through the pole. It will then be seen that the combination of the two spirals is symmetrical with respect to the vertical axis. Secondly, by expressing the rectangular co-ordinates of a point of the spiral in terms of we have x = aO cos 0, y = aB sin 9. Changing the sign of G in this equation will change the sign of x and leave y unchanged. 8. Show that if we draw two lines through the centre of a lemniscate making angles of 45° with the axes, no point of the curve will be contained between these lines and the axis of Y. TANGENTS AND NORMALS. 147 CHAPTER XII. TANGENTS AND NORMALS. 84. A tangent to a curve is a straight line through two coincident points of the curve. Fia. 25. A normal is a straight line through a point of the curve perpendicular to the tangent at that point. The subtangent is the projection, TQ, upon the axis of Xy of that segment TP of the tangent contained between the point of contact and the axis of X, The subnormal is the corresponding projection, QNy of the segment PN of the normal. Notice that a tangent and a normal are lines of indefinite length, while the subtangent and subnormal are segments of the axis of abscissas. Hence the former are determined by their equations, which will be of the first degree in x and y, while the latter are determined by algebraic expressions for their length. But the segments TP and PN are sometimes taken as lengths of the tangent and normal respectively, when we con- sider these lines as segments. 148 THE DIFFERENTIAL CALCULUS. 85. General Equation for a Tangent. The general prob- lem of tangents to a curve may be stated thus: To find the condition which the parameters of a straight line must satisfy in order that the line may he tangent to a given curve. But it is commonly considered in the more restricted form: To find the equation of a tangent to a curve at a given point on the curve. Let {x^y y^ be the given point on the curve. By Analytic Geometry the equation of any straight line through this point may be expressed in the form y^y^^rnix- X,); (5) m being the tangent of the angle which the line makes with the axis of X, But we have shown (§ 20) that "^ ~ dx,' this differential coefficient being formed by differentiating the equation of the curve. Hence y-y. = |;(^-^.) (6) is the equation of the tangent to any curve at a point {x^, yj on the curve. Equatio7i of the Normal, The normal at the point {x^, y^ passes through this point, and is perpendicular to the tangent. If m' be its slope, the condition that it shall be perpendicular to the tangent is (An. Geom.) m' = - - = - — m dyl dx^ Hence the equation of the normal at the point (x,, y^) is ^^{y-y,) = x,-x. (7) TANGENTS AND NORMALS. 149 In these equations of the tangent and normal it is necessary to distinguish between the cases in which the symbols x and y represent the co-ordinates of points on the tangent or nor- mal line, and those where they represent the given point of the curve. Where both enter into the same equation, one set, that pertaining to the curve, must be marked by suffixes or accents. 86, Suhtangent and Subnormal. To find the length of the subtangent and subnormal, we have to find the abscissa x^ of the point T in which the tangent cuts the axis of abscis- sas. We then have, by definition. FiQ. 26. Subtangent = x^ — x^ The value of x^ is found by putting y = and a: = a;„ in the equation of the tangent. Thus, (6) gives Hence, for the length of the subtangent TQ, Subtangent =..-.. = |.. dx^ We find in the same way from (7), for QN, Subnormal = — y dyj dx^ (8) (9) 150 THE DIFFERENTIAL CALCULUS, 8*7. Modified Forms of the Equation, In the preceding discussion it is assumed that the equation of the curve is given in the form y =/(^)- ,, firstly, it may be given in the form F{^, y) = 0. shall then have (§37) dF dx^ ' dF' Substituting this value in the equations (6) and (7), we find (10) „ . dF, , dF, ■ Tangent: ^(y-y,)=^(r«,- a;); dF, dx. Normal: —{y-yj = —-{x- a;.). dF, dy. (11) Secondly, if the curve is defined by two equations of the form y = 0,(^), ) dy^ , dy^ du wo have —^ = -7-, dx^ dx die in which there is no need of suffixes to x and y in the second member, because this member is a function of u, which does not contain x or y. By substitution in (6) and (7), we find Eq. of tangent: {y ~ y^)-£- = (a; - xj£-. Eq. of normal: (y - y^)^ = {x, - x)^. (12) TANGENTS AND NORMALS. 151 (tX fill By substituting in these equations for a:,y,, -j- and -~ their values in terms of w, the parameters of the lines will be functions of u. Then, for each value we assign to u, (11) will give the co-ordinates of a point on the curve, and (12) will determine the tangent and normal at that point. r 88. Tangents and Normals to the Conic Sections, Writing the equation of the ellipse in the form aY + Vx' = a^h\ (a) we readily find, by differentiation, dy __ Vx dx "" a*y* Applying the suffix to x and y, to show that they represent co-ordinates of points on the ellipse, substituting in (6) and (7), and noting that x^ and y^ satisfy (a), we readily find: For the tangent: ^ + 1^ = 1. For the normal: -x y z= a* — b*. Taking the equation of the hyperbola, - aY + b'x* = aV, we find, in the same way. For the tangent: 5l? _ M = 1. a a* I* For the normal: -x A — y = a* + b*. Taking the equation of the parabola, y' = 2px, we find, by a similar process. For the tangent: y^y = p{x + x^). For the normal: y — yi = — (^i ~~ ^)» 152 THE DIFFERENTIAL CALCULUS. 89. Problem. To find the le^igth of the perpendicular dropped from the origi7i upo7i a tangent or normal. It is shown in Analytic Geometry that if the equation of a straight line be reduced to the form the perpendicular upon the line from the origin is G p=z VA' + JB' It must be noted that in the above form the symbol O rep- resents the sum of all the terms of the equation of the line which do not contain either x or y. If we have the equation of the line in the form we write it mx — y — mx^ -j- y^ = 0, and then we have A = m'y 0=^y^- mx^. Thus, the expression for the perpendicular is y^ — mx, Vm' + l Substituting for m the values already found for the tan- gent and normal respectively, we find. For the perpendicular o?i the tangent : /> + m For the perpendicular on the normal: dy^ ^ ^1 + y^dx^ ^ x,dx^ + y,dy . (2) TANGENTS AND NORMALS. 153 90. Tangejit and Normal m Polar Co-ordinates. Pkoblem. To find the angle which the tangent at any point makes tuith the radiiis vector of that point. Let PP' be a small arc of a curve referred to polar co-ordinates; KP, a small part of the radius vector of the point P (the pole being too far to the left to be shown in fiq. 27. the figure); K'P'y the same for the point P\ KSR, a parallel to the axis of reference. Drop PQ\_K' P\ Let SPThe the tangent at P. We also put y E angle KPS which the tangent makes with the radius vector. Then let P' approach P as its limit. Then QP' ^ dr; PQ = rdd; PQ . rdO tan y ^ QP' - dr • (1) We also have cos y — dr 4/(1 + tan' y) /l-'+lS)!'"" sin y = cos y tan y = /l'-+(S)T (2) Cor. The angle FSP which the tangent makes with the ^xis of reference is y -{■ 6^ 154 THE DIFFERENTIAL CALCULUS. 91. Perpendicular frovi the Pole iipon the Tangent and Normal, When y is the angle between the tangent and the radius vector, we readily find, by geometrical construction, that the perpendicular from the pole upon the tangent and normal are, respectively, p := r sin y and p ^= r cos y. Substituting for sin y and cos y the values already found, we have. For the perpendicular on tayigent : r' p For the perpendicular on normal : r V /{'■■H^n dr d~6' (3) 93. Pkoblem. To find the equation of the tangent and •normal at a given point of a curve whose equatiori is expressed in polar co-ordinates. It is shown in Analytic Geometry that if we put p = the perpendicular dropped from the origin upon a line; a ~ the angle which this perpendicular makes with the axis of Xy the equation of the line may be written X cos oc -\- y sin a — j) =^ 0, (1) Now, as just shown, the tangent makes the angle y -\- with the axis of X, and the perpendicular dropped upon it makes an angle 90° less than this. Hence we have a = y^6- 90°; cos a — sin i^y -^ 0) — sin / cos /9 -j- cos y sin 6\ sin or = -- cos {y -\- B) ~ — cos ;^ cos ^ -f- sin y sin 0, TANGENTS AND NORMALS. 155 By substitution in (1), the equation of the tangent becomes a:(sin y cos 6 -f- cos y sin 6) — ?/(cos y cos — sin y sin 0) — p = 0. Substituting for cos y, sin y and p the values already found, this equation of the tangent reduces to ir cos ^ + 7^ sin 8) x -\- Ir sin ^ — -jn cos 6^1 y — r* = 0, (2) ?• and 6^ being the co-ordinates of the point of tangency. In the case of the normal the perpendicular upon it is parallel to the tangent. Therefore, to find the equation of the normal, we must put in (1) a = y -{- 6. Substituting this value of a, and proceeding as in the case of the tangent, we find, for the normal, ( -^ cos 6 — 7' sin Ojx+ir cos ^ + jn siii ^) ?/ — ^'7^ = ^« (3) Generally these equations will be more convenient in use if we divide them throughout by r. Thus we have: Equation of the tangent : ^cos 6 + --^^8m0jx+ ^sin ^ - - /^ cos 6jy-^r = 0. (4) Equation of the normal : (i|co8^-sin^)..+ (^|sin. + cos^),-|j = 0.(5) In using these equations it must be noticed that the co- efficients of X and y are functions of r and 6, the polar co- dv ordinates of the point of tangency. When 7% 8 and -y^ are given, this point and the tangent through it are completely determined. 156 THE DIFFERENTIAL CALCULUS. I EXERCISES. I. Show that in the case of the Archimedean spiral the general expressions for the perpendiculars from the pole upon the tangent and normal, respectiyely, are ad^ , ad v{i + o') -^ ^ va + n Thence define at what point of the spiral the radius vector makes angles of 45° with the tangent and normal. Find also what limit the perpendicular upon the normal approaches as the folds of the spiral are continued out to infinity. Show also from § 92 that the tangent is perpendicular to the line of reference at every point for which r sin — a cos = 0, and hence that, as the folds of the spiral are traced out to infinity, the ordinates of the points of contact of such a tan- gent approach ± a as their limit. 2. Show by Eq. 12 that in the case of the logarithmic spiral the angle which the radius vector makes with the tan- gent is a constant, given by the equation tan y = J-. 3. Show from Eq. 12 that if a curve passes through the pole, the tangent at that point coincides with the radius vector, unless -^ = at this point. Thence show that in the lemniscate the tangents at the origin each cut the axes at angles of 45°. 4. Show that the double area of the triangle formed by a tangent to an ellipse and its axes is . Then show that the area is a maximum when — = ± t^. a Show also that the area of the triangle formed by a nor- mal and the axes is a maximum for the same point. ASYMPTOTES AND SINGULAR POINTS, 157 CHAPTER XIII. OF ASYMPTOTES, SINGULAR POINTS AND CURVE-TRACING. 93. Asymptotes. An asymptote of a curve is the limit which the tangent approaches when the point of contact re- cedes to infinity. In order that a curve may have a real asymptote, it must extend to infinity, and the perpendicular from the origin upon the tangent must then approach a finite limit. For the first condition it suffices to show that to an infi- nite value of one co-ordinate corresponds a real value, finite or infinite, of the other. For the second condition it suffices to show that the expres- sion for the perpendicular upon the tangent (§§ 89, 91) ap- proaches a finite limit when one co-ordinate of the point of contact becomes infinite. If, as will generally be most con- venient, the equation of the curve is written in the form F{x, y) = 0, (1) the value (1) of the perpendicular, omitting suffixes, may be reduced to dF , dF \ishm\' dy If this expression approaches a real finite limit for an infinite value of x or y, the curve has an asymptote. If the curve is referred to polar co-ordinates, we use the expression (3), § 91, for ;;, If this approaches a real finite limit for an infinite value of r, the curve has an asymptote. 158 THE DIt'FERENTlAL CALOXfLUB. The existence of the asymptote being thus established^ its equation may generally be found from the form (10), § 87, which we may write thus: dF dF _ dF dF dx. (3) by supposing x^ or y^ to become infinite. fl F dF Commonly the coefficients -j^ and -j- will themselves be- come infinite with the co-ordinates. We must then divide the whole equation by such powers of x^ and y^ that none of the terms shall become infinite. 94. Examples of Asymptotes. 1. F{x) = x'-^ y'- daxy = 0.{a) The curve represented by this equation is called the Folmm of Descartes, The equation (3) gives in this case, applying suffixes, = ^' + y' - ^«^i!/x = cix^y,' To make the coefficients of x and y finite for x^ = co , divide by x^y^, comes n> ^y FiQ. 28. Then the equation be- (?-) Let us now find from (a) the limit of y^ for ic^ = oo . We have 1 + h = 3a?^ The second member of this equation will approach zero as a limit, unless y^ is an infinite of as high an order as x^*, which is impossible, because then the first member of the equation containing y^^ would be an infinite of higher order ASYMPTOTES AND SINGULAR POINTS 159 than the second member, which is absurd. Hence, passing to the limit, lim. (|j(x, = <») = - 1. Then, by substitution in {h), we find, for the asymptote, x + y -\-a = 0. 2. Take next the equation F{x, y) = x"" — 2x^y — ax^ — a^y = 0. (a) With this equation (3) becomes (3a;/ - 4:X^y^ - 2ax^)x - (2a:/ + a')y = Sx^' - 6x;y, - 2ax: - a'y^. {b) FiQ. 29. We notice that the terms of highest order in the second member are tliree times those of highest order in (a). From (a) we have ^:' - ^-^I'y = ^^/ + «Vi- Subotituting in tlie second member of (b), and dividing by x/, (/>) becomes (•^-*i;-l>-K?V=«+?'- <*') Solving (a) for y, we find an expression which approaches the limit i when 2:^:^00. Thus, passing to the limit, {b') gives, for the equation of the asymptote, X — 2y= a. 160 THE DIFFERENTIAL CALCULUS. 3. The Witch of Agnesi. This curve is named after the Italian lady who first investigated its properties. Its equation is ic'y + a'^y — «^' = 0. (a) The equation of the tangent is 2x^y^x + {x^' + a')y = ^x^y, + a'y, = da' - 2«Y- (^) By solving {a) for x and y respectively we see that x^ may become infinite, but that y^ is always positive and less than a. Hence, to make the coefficient of y in {h) finite for 0^^ = 00, we must divide by x^, which reduces the equation of the asymptote to y = 0. Hence the axis of x is itself an asymptote. 95. Points of Inflection, A point of inflection is a point where the tangent inter- sects the curve at the point of tangency. It is evident from the figure that in passing along the curve, and con- sidering the slope of the ^^^- ^^• tangent at each point, the point of inflection is one at which this slope is a maximum or a minimum. Because we have slope = g, the conditions that the slope shall be a maximum or minimum are dx' and —^ different from zero. If the first condition is fulfilled, cVy , but if -tS is also zero, we must proceed, as in problems of maxi- ASYMPTOTES AND SINGULAR POINTS. 161 ma and minima, to find the fir^fc derivative in order which does not vanish. If the order of this derivative is even, there is no point of inflection for -j-^ = 0; if odd, there is one. As an example, let it be required to find the points of in- flection of the curve xy^ = a^{a — x). Keducing the equation to the form f = a X -a\ dy dx a" we find The condition that this expression shall vanish is 4:xy* = rt', which, compared with the equation of the curve, gives, for the co-ordinates of the point of inflection, 3 . a X = -a\ y = ± ~~. EXERCISES. Find the points of inflection of the following curves : X \ X = ae^. ins. [ I. xy = a' log — . A),., « (y = lae~^ ix = a{l — cos w); (y = a{7iu + sin ti), ' _ (MJ>. X — , Ans, < ^=4»'-"(-^)+^^) 162 TBE DIFFBBENTIAL CALCULUS. <^.y Fig. 32. 96. Singular Points of Curves, If we conceive an infini- tesimal circle to be drawn round any point of a curve as a centre, then, in general, the curve will cut the circle in two opposite points only, which will be 180° apart. But special points may sometimes be found on a curve where the infinitesimal circle will be cut in some other way than this: perhaps in more or less than two points; perhaps in points not 180° apart. These are called singular points. The principal singular points are the following: Double-Joints ; at which a . curve intersects itself. Here the curve cuts the infinitesimal circle in four points (Fig. 33). Cusps; where two branches of a curve terminate by touching each other (Fig. 34). Here the infinitesimal circle is cut in two coincident points. Stopping Points; where a curve suddenly ends. Here the infinitesimal circle is cut in only a single point. Isolated Points; from which no curve proceeds, so I _y that the infinitesimal circle is not cut at all. fig. 36. Salient Points; from which proceed two branches making v/ith each other an angle which is neither zero nor 180"". Here the infinitesimal circle is cut in two points which are neither apposite nor coincident. There may also be multiple-points y through which the curve passes any number of times. A double-point is a special kind of multiple-point. A multiple-point through which the curve passes three times is called a triple-point. Fig. 33. Fig. 34. Fig. 35. ASYMPTOTES AND SINGULAR POINTS. 163 :i: X / "^\^ I y^^\ Xo 1 ^--- = i^"i a:?; a.T^^/ dy Noting that when p = then x = x^, we see that the de- velopment by Maclaurin's theorem will be F{x, y) = F{x,, ^J + p(cos ^g + sin 6^^ + etc. = 0. Here -r- means the value of -7- when x^ is put for x, etc. dx^ dx ° ^ Because {x^y y^) is by hypothesis a point on the curve, w( have F{x^, y^ = 0, and the only terms of the second membei are those in p, p', etc. Thus the polar equation (2) of the curve may be written F/P + F/'p^ + FrP' + etc. = 0, ) or FJ + FJ'p + i^/' V + etc. = 0. f ^^ To find the points in which the curve cuts a circle of radiuf Py we have to determine ^ as a function of p from this equa- tion. When p is an infinitesimal, all the terms after the firsi will be infinitesimals. Hence, at the limit, where p becomes infinitesimal must satisfy the equation dF dx which gives tan 6 •= — — ^. dF ^0 This is the known equation for the slope of the tangent a1 (^0* y^y ^^d gives only the evident result that in general the ASYMPTOTES AND SINGULAR POINTS 165 irve cuts the infinitesimal circle along the line tangent to e curve at Q. But, if possible, let the point {x^y^ be so taken that ilF (IF Then we shall have FJ — 0, and the equation (3) of the irvo will reduce to FJ'p + F^p' + etc. = 0, ^o" + ^o"'P + etc. = 0. Again, letting p become infinitesimal, we shall have at the mit ," = cos^ .||; + , sin .cos .^- + sin^ ^11 . 0. (5) Dividing throughout by cos' 6, we shall have a quadratic }uation in tan 6, which will have two roots. Since each lue of tan 6 gives a pair of opposite points in which the iirve may cut the infinitesimal circle, and since (5) depends Q (4), we conclude: The necessary condition of a douUe-jwint is that the three quations F(x,y)-^, —-^—^0, ____0, hall be satisfied by a single pair of valnes of x and y. If the two values of tan derived from FJ' — are equal, VG shall have either a cusp, or a poii^.t in which two branches f the curve touch each other. If the roots are imaginary, he singular point will be an isolated point. 98. Examjjles of Double-points, A curve whose equation ;ontains no terms of less tlian the second degree in x and y las a singular point at tiic origin. For example, if the equa- ion be of the form F{x, y) = Px' + Qxy + Rf = 0, :hen this expression and its derivatives with respect to x and y will vanish for a; = and y = 0. 166 THE DIFFERENTIAL CALCULUS. = — 6(rt'a; -\- ax') = — 6ax{a -f x); My' - «') = ■iyiy + «) (y - «)• (1) (3) Let us now investigate the double-points of the curve {y" - ciy - 3aV - 2az' = 0. We have dF dx dF dy The first of these derivatives vanishes for x = ot — a\ The second of these derivatives vanishes for y —0, — a or -\- a. Of these values the original equation is satisfied by the fol- lowing pairs: ^. — y. which are therefore the co-ordinates of singular points, r„= 0; 0; -a-,) (3) Differentiating again^ we have d-'F .... d'F -zr-- = — 6a — V^ax\ -^ — =- : dx dxdy Forming the equation i^" = 0, it gives ^' 1?=!^^'-*'^'' ASYMPTOTES AND SINGULAR POINTS. 167 (12y' ~ 4a') tan' 6* = Ga' + nax. Substituting the pairs of co-ordinates (3), we find: At the point (0, — a), tan 6 = ± ^ V^-, At the point (0, + a), tan = ± ^ Vd\ At the point (— a, 0), tan 6^ = ± Vl. The vahies of tan 6 being all real and unequal, all of these points are double-points. The curve is shown in the figure. Remark. In the preceding theory of singular points it is assumed that the expression (2), § 97, can be developed in powers of p. If the function F is such that this development is impossible for certain values of x^ and y^, this impossibility may indicate a singular point at (:c„, y^). 99, Curve-tracing, We have given rough figures of va- rious curves in the preceding theory, and it is desirable that the student should know how to trace curves when their equations are given. The most elementary method is that of solving the equation for one co-ordinate, and then substitut- ing various assumed values of the other co-ordinate in the solution, thus fixing various points of the curve. But un- less the solution can be found by an equation of the first or second degree, this method will be tedious or impracticable. It may, however, commonly be simplified. 1. If the equation has no constant term, we may sometimes find the intersections of the curve with a number of lines through the origin. To do this we put y = mx in the equation, and then solve for x. The resulting valuea of :r as a function of m. are the abscissas of the points in which the curve cuts the line y — mx = 0. Then, by putting m = ± 1, ±2, etc.; vi = ± -J, ± J, etc., we find as many points of intersection as we please. 168 THE DIFFERENTIAL CALCULUS. To make this method practicable, the equations which we have to solve should not be of a degree higher than the second. If the curve has a double-point, it may be convenient to take this point as the origin. 2. If the equation is symmetrical in x and y or x and — t/, the curve will be symmetrical with respect to one of the lines X — y = and x -{- y = 0. The equation may then be simplified by referring it to new axes making an angle of 45° with the original ones. The equations for transforming to such axes are x={x' + y') sin 45°; y=z{x' - y') sin 45°. Application to the Folium of Descartes. If, in the equa- tion of this curve, x' + y' = 3axy, we put y = mx, we shall find 3am Bam'' 1 ^ — 11 ...3; y — l + m'' ^ 1+m'' We also find, from the equation of the curve and the pre- ceding expressions for x and y in terms of m, dy __x^ — ay __ 2m — m* dx ax • -y'~ 1 - 2to'* 1, 3 x==^-a; y = 3 dy _ dx ~ - 1. 2 x = ~a; y^ 4 3«; dy dx ~ 4 5* 3 2' 36 x==-^a; ?/ = 54 35"' dy dx 33 92 2, 6 x=:-a; y = - 13 dy dx ~ 20 i7' etc. etc ;. etc. Then, for m = m = m = etc. Thus we have, not only the points of the curve, but the tangents of the angle of direction of the curve at each point, which will assist us in tracing it. THEORY OF ENVELOPSS. 169 CHAPTER XIV. THEORY OF ENVELOPES. lOO, The equation of a curve generally contains one or more constants, sometimes called parameters. For example, the equation of a circle, {x - a)' +(y- by = r\ contains three parameters, a, b and r. As another example, we know that the equation of a straight line contains two independent parameters. Conceive now that the equation of any line, straight or curve, (which we shall call ^^the line^' simply,) to be written in the implicit form 0(^, y. ^) = 0, (1) a being a parameter. By assigning to a the several values a, a', a", etc., we shall have an equal number of lines whose equations will be {^, y> oi) = 0; (f>{Xy y, a') = 0; 0(rr, y, or") = 0; etc. The collection of lines that can thus be formed by assign- ing all values to a parameter is called a family of lines. Any two lines of the family, e.g., those wliich have a and a' as parameters, will in general have one or more points of intersection, determined by solving the corresponding equa- tions for X and y. The co-ordinates, x and y, of the point of intersection will then come out as functions of a and a\ Suppose the two parameters to approach inlinitesimally near each other. The point of intersection will then approach a certain limit, which we investigate as follows: 170 THE DIFFERENTIAL CALCULUS. Let us put a' =z a -{- Aa. The equations of the lines will then be cf)(xy y, a) ::^0 and (p{xy y, a -{- Aa) ==0. If we develop the left-hand member of the second equation in powers of ^ a: by Tajlor^s theorem, it will become cpix, y, a) + £Aa + -^ ^-^ + etc. = 0. Subtracting the first equation, dividing the remainder by A ay and passing to the limit, we find d(p(Xy y, a) da = 0. Hence the limit toward which the point {x, y) of intersec- tion of two lines of a family approaches as the difference of the parameters becomes infinitesimal is found by determining X and y from the equations 0(.,,,«) = O and M^|^)=0. (2) The values of x and y thus determined will, in general, be functions of a; that is, we shall have a; =/>(«); y =/,(«); (3) which will give the values of the co-ordinates x and y of the limiting point of intersection for each value of a, Now, suppose a to vary. Then x and y in (3) will also vary, and will determine a curve as the locus of x and y. Such a curve is called the envelope of the family of lines, ct){x, y, a) = 0. In (3) the equations of the curve are in the form of (2), § 76, a being the auxiliary variable. By eliminating a either from (2) or (3), we have an equation between x and y which will be the equation of the curve in the usual form. THEORY OF ENVELOPES. 171 101 • Theorem. The envelope and all the lines of the family which generate it are tangent to each other. Geometrically the truth of this will be seen by drawing a series of lines varying their position according to any con- tinuous law, as in the first example of the following sec- tion. Taking three consecutive lines and numbering them (1), (2) and (3), it will be seen that as (1) and (3) approach (2) their points of intersection with (2) approach infinitely near each other. Since these infinitely near points of inter- section also belong to the envelope, the line (2) passes through two infinitely near points of the envelope and is therefore a tangent to the envelope. Analytic Proof, The equation of the envelope is found by eliminating a from the equations (2), and we may conceive this elimination to be effected by finding the value of a from the second of these equations (2), and substituting it in the first equation. That is, the equation ct>(x, y,a) = (4) represents any line of the original family when we regard a as a constant; and it represents the envelope when we regard a as a function of x and y, satisfying the equation M^l^) ^ 0. (5) Let the value of a derived from this last equation be a = F{x, y). (6) Now, to find tlie slope of the tangent to the original line of the family at the point {x, y), we differentiate (4), regarding a as a constant. Thus we have d0 d^dy_^^ or -^ = - ^ (7) dx dy dx dx JJyCp' If the original line is a straight one, this equation will give its slope. To find the slope of the tangent to the envelope at the same 172 THE DIFFERENTIAL CALCULUS. point, we differentiate this same equation, regarding a as hav- ing the value (6). Thus we have dx dy dx da\dx dy dx I ' ^ ^ But, because -~ = 0, this equation will also give the value (7) for the slope; whence the curves have the same tangent at the point {xy y), and so are tangent to each other at this point. 103. We shall now illustrate this theory by some examples. 1. To find the enveloije of a straight line which moves so that the area of the triangle lohich it forms with the axes of co- ordinates is a constant. Fig. 39. Since the area of the triangle is half the product of the intercepts of the axes cut off by the line, this product is also constant. Calling a and h the intercepts, the equation of the line may be written in the form 0(aj, y, a) ^+1-1 = 0. a (1) THEORY OF ENVELOPES. 173 Here we have two varying parameters, a and h, while, to have an envelope, the change of the parameters must depend on a single varying quantity. But the condition that the product of the intercepts shall be constant enables us to elimi- nate one of the parameters, say Z>. We have, by this condition, h = -, (2) , db c whence -?- = a« da a Now differentiating the equation (1) with respect to a, re- garding Z> as a function of a, we have d^^ _^_ _y (]^_ cy - ^'^ _ ^ _ ?_ _ n\ da ~ a' b' da ~ a'b' ~ c a' ^^ We have now to eliminate a from the equations (1) and (3), using (2) to eliminate b from (1). The easiest way to effect this elimination is as follows: From (3) we have a'y = cx; a = a/ ^-. (4) Multiplying (1) by a, and substituting for b its value from (2), we have x-\ ^ = a. c Substituting from (4), this equation becomes and thus the equation of the envelope becomes xy = \c, which is that of an hyperbola referred to its asymptotes. This result coincides with one already found in Analytic Geometry, that tangents to an hyperbola cut off from the asymptotes intercepts whose product is a constant. 174 THE DIFFERENTIAL CALCULUS. 2. To find the envelope of the line for which the sum of the intercepts cut off from the co-ordinate axes is a constant. Fig. 40. Let c be the constant sum of the intercepts. Then, if a be the one intercept, the other will he c — a. Thus the equa- tion of the line is X a -^-1, c — a in which a is the varying parameter. Clearing of fractions, we may write the equation ct){x, y, a) = cx + a{y — X- c) -{- a"" ~ 0, whence ~=:y — x — c-\-2a=^0. da ^ ' From the last equation we have « = *(^ - !/ + ^); this value of a being substituted in the other gives cx-\{x- y ~\- cY = 0, or {x - yY - 2c{x + i/) + c' = 0. THEORY OF ENVELOPES. 175 This equation, being of the second degree in the co-ordi- nates, is a co7iic section. The terms of the second degree forming a perfect square, it is a parabola. The equation of the axis of the parabola is To find the two points in which the parabola cuts the axis of X we put y = 0, and find the corresponding values of x. The resulting equation is x' — 2cx + c^ = 0. This is an equation with two equal roots, x = c, showing that the parabola touches the axis of X at the point {c, 0). It is shown in the same way that the axis of Y is tangent to the parabola. It may also be shown that the directrix and axis of the parabola each pass through the origin, and that the parame- ter is V~2c, 3. If the difference of the intercepts cut off by a line from the axes is constant, it may be shown by a similar process that the envelope is still a parabola. This is left as an exer- cise for the student, who should be able to demonstrate the following results : (a) When the sum of the intercepts is a positive constant, the parabola is in the first quadrant ; when a negative con- stant, the parabola is in the third quadrant. (/?) When the difference, a — by of. the intercepts is a posi- tive constant, the parabola is in the fourth quadrant; when a negative constant, in the second. {y) The co-ordinate axes touch the parabola at the ends of the parameter. In each case the parabola touches each co-ordinate axis at a point determined by the value of the corresponding inter- cept when the other intercept vanishes, and each directrix intersects the origin at an angle of 45° with the axis. 176 THE DIFFERENTIAL CALCULUS. 4. Next take the case in which the sum of one intercept and a certain fraction or multiple of the other is a constant. Let m be the fraction or multiplier. We then have h -\- ma = c = u, constant. The equation of the line then becomes a c — ma Proceeding as before, we find the equation of the envelope to be {mx — yY — 2c(ma; + y) + c' = 0, which is still the equation of a parabola. 5. To find the envelope of a line which cuts off intercepts subject to the conditio7i ^+-^,=h {a) m and n being constants. We may simplify the work by substituting for the varying intercepts a and b the single variable parameter a determined by either of the equations m n sm or E - ; cos « = 7-. a The equation of the varying line will then oecome OJ 4/ d)(x. y) = — Bin a -4- ~ cos a = 1. (1) By differentiating with respect to a, we have T- = — cos «? — — sm or = 0. (2) da m n ^ ^ We may now eliminate a by simply taking the sum of the squares of these equations, which gives 7)1 n the equation of an ellipse whose semi-axes are m and 7u THEORY OF ENVELOPES, 177 6. To find the envelope of a circle of constant radius whose ^centre moves on a fixed circle* For convenience let us take the centre of the fixed circle as the origin, and put: a, by = the co-ordinates of the centre of the moving circle; c E its radius; d E the radius of the fixed circle. The equation of the moving circle now becomes (x - ay + {y- by - c» = 0. (1) By differentiation with respect to a, The condition that {a, b) lies on the fixed circle gives a' + b' = d\ (2) , db a whence -—=: — --. da b Then, by substituting this value, ay — bx = 0. (3) We have now to eliminate a and b from (1), (2) and (3). Firstly, from (1) and (2), we find a:* + 3/* - 2aa; - 2% = c* - d\ (1') From (2) and (3) we find the following expressions for a and b: xd , yd a = . : b = Vx^ + f' Vx^ + y^' By substitution in (1'), and putting for brevity r' = x' + y\ we find r' ± 2rd + d* = c\ Hence r* = x* + y^ = (c ± d )% the equations of two circles around the origin as a centre, with radii c -\- d and c — d. 178 THE DIFFERENTIAL CALCULUS. 7. Find the envelope of a family of ellipses referred to their centre and axes, the product of whose semi-axes is equal to a certain constant, 6'\ Ans, The equilateral hyperbola xy = ic^, 8. To find the e^ivelope of a family of straight li7ieSy such that the product of their distances from two fixed points is a constant. Let {a, 0) and {—a, 0) be taken as the two fixed points, and let c' be the constant. Also, let X cos a -\- y ^m a — p = (1) be the equation of any one of the lines in the normal form, p and a being the varying parameters. The distances of the line from the points {a, 0) and {—a, 0) are respectively — p -{- a GO^ a and — p — a qob a. Hence we have the condition j»' - a^ cos' a = c\ (2) Differentiating (1), regarding jt? as a function of a, we have — XBm a -\- y cos a f- = 0. ^ da From (2) we obtain dp __ a^ sin a cos a da ^ p ' We thus have the three equations X cos a '\- y mi a =z p^ («) a^ sin a cos a (*) ;?' = c' + a' cos' a — c^ -\- a^ — d^ sin' o'. (<') from which to eliminate p and ^. TnRORT OF ENVELOPES. 179 I To effect the elimination of a and p we find the values of Z and y from (a) and (b) by taking (a) X cos a + {h) X sin a and (a) x sin a — {h) x cos a. We thus find, by the aid of (c), px = ^' cos a -\- a* sin' a cos a; cos O' a: = (c' + a')- ,8in a y = c' • i^ ^ a; cos a Hence -=— - — i = ; c + a p y _ sin a c* '~ p ' If we multiply the first of these equations by x and the second by y and add, then we have X* y* __x COB a -\- y sin a _^ ?~+'a' "^ ? "" p ~ Hence the equation of the envelope is c' + a' ' c' This represents an ellipse whose foci are the two fixed points. This interpretation, however, presupposes that the product c* of the distances of the line from the two points is positive; that is, that the points are on the same side of the enveloping line. If the product is negative, the equation of the envelope will be a — c c which is the equation of an hyperbola. These results give the theorem of Analytic Geometry that the product of the distances of a tangent from the foci of a conic is constant. 180 THE DIFFEEENTIAL CALGTJLXIB. 1.^ . J , , CHAPTER XV. I OF CURVATURE; EVOLUTES AND INVOLUTES. t 103, Position; Directio7i; Curvature. The positmi of any point P on a curve is fixed by the values of the co-ordi- nates, X and y^ of F» This is shown in Analytic Geometry. If we have given^ not only x and y, but the value of ■— for the point P, then such value of the derivative indicates the direction of the curve at the point P, this direction being the same as that of the tangent at P. The curve may also have a greater or less degree of curva- ture Sit P. The curvature is indicated by a change in the di- rection of the tangent, that is, in the value of -3^, when we ax pass to an adjacent point P'. But such change in the value di/ of -^ when we vary x is expressed by the value of the second d^V derivative -7^. If this quantity is positive, the angle which ax the tangent makes with the axis of X is increasing with x at the point P, and the curve, viewed from below, is convex. If y4 is negative, the tangent is diminishing, and the curve, seen from below, is concave. To sum up: If we take a value of the abscissa a:, then the corresponding value of y gives the position of a point P of the curve; -p- gives the direction of the curve at P; d^V -t4 depends upon the curvature of the curve at P. ax CURVATURE; EVOLUTES AND INVOLUTES. 181 104. Contacts of Different Order i>. Let two different sjirves be given by their respective equations: y =/(^) an^ y = 0(^)- If for a certain value of x, which value call x^, the two values of y are equal, the two curves have the corresponding point in common; that is, they meet at the point {x^, y). If the values of -^ are also equal at this point, it shows tliat the curves have the same direction at the point of meet- ^ ing. They are then said to touch each other. It the values of -y4 are also equal at this point, the two ax j curves have also the same curvature at this point. i To show the result of these several equalities, let us give j th(3 abscissa x^ (which we still take the same for both curves), I uii increment A, and develop the two values of y in powers of cli/ h by Taylor^s theorem. To distinguish the values of y, -f-y etc., which belong to the two curves, we assign to one the suffix 0, and to the other the suffix 1. Then, for the one curve, '='.+(i).^+p/.+--+(g)l+-. and, for the other, '•-.+@),"+(S'),iV---+(g)|+- The difference between the values of y' and y is the inter- cept, between the two curves, of the ordinate at the point whose abscissa is x^ -f- h. Its expression is 2/i - ^0 + [(i),-(t]"4(£o,-(m]o+- Now, consider the case in which the curves meet at the point r, whose abscissa is x^. Then 182 THE DIFFERENTIAL CALCULUS. and the intercept of the ordinate will be which, when h becomes infinitesimal, is an infinitesimal of the first order. If we also have ldy\ __ (dy\ \dxir \dxi: the ordinates will differ only by a quantity containing h^ as a factor, and so of the second order. Hence: Wlien two curves are tangent to each other, they are sepa- rated only ly quantities of at least the second order at an in- finitesimal distance from the point of tangency. In the same way it is shown that if the second differential coefficient also vanishes, the separation will be of the third order, and so on. Def When two curves are tangent to each other, if the first n differential coefficients for the two curves are equal at the point of tangency, the curves are said to have contact of the nth order. Hence a case of simple tangency is a contact of the first order. If the second derivatives are also equal, the contact is of the second order, and so forth. 105. Theorem. In contacts of an even order the two curves intersect at the point of C07itact ; in those of an odd order they do not. For, in contact of the nth. order, the first term of y' — y (§ 104) which does not vanish contains h"^"^^ as a factor. If 71 is odd, 7i + 1 is even, and y' — y has the same alge- braic sign whether we take h positively or negatively. Hence the curves do not intersect. If n is even, n -\- Ih odd, and the values of y' — y have opposite signs on the two sides of the point of contact, thus showing that the curves intersect. CURVATURE; EVOLUTEt^ AND INVOLUTES. 183 ^ 106. Radius of Curvature, The curvature at any point is measured thus: We pass from the point F to a point P' in- jfinitesimally near it. The teurvature is then measured by the ratio of the cliange in the direction of the tan- gent (or normal) to the distance PP\ Let us put aEthe angle which the tangent at F makes fiq. 41. with the axis of X. a + da = the same angle for the tangent at F\ ds E the infinitesimal distance FF\ Then, by definition. Curvature = -^. ds Now, because tan a we have, by differentiation. dy d^' sec' a da = -~{dx. Also, sec' « = 1 -4- tan' a = 1 -f and ds = \/{l-\.'^^dx. From these equations we readily derive cry dz' dl dx^ Curvature = -7- = ds Kg rtf.2/'\i Now, draw normals to the curve at the points P and P', and let C be their j)oint of intersection. Because they are perpendicular to the tangents, the angle FGF' between them will be da, and if we put 184 THE DIFFERENTIAL CALCULUS. we shall have Hence p = -j— = PF' = ds = pda. fi + ^T 1 r ^ dx^i da curvature d'^y d^^ The length p is called the radius of curvature at the point F, and C is called the centre of curvature. Corollary. The centre of curvature for any point of a curve is the intersection of consecutive normals cut- ting the curve infinitely near that point. p^ 10*7. The Osculating Circle, li, on the normal PC to any curve at the point P, we take any point ^^^ ^2 as the centre of a circle through P, that circle will be tangent to the curve at P; that is, it will, in general, have contact of the first order at P. But there is one such circle which has contact of a higher order, namely, that whose centre is at the centre of curvature. Since this circle will have the same curvature at P as the curve itself has, it will have contact of at least the second order at P. This proposition is rigorously demonstrated by finding that circle which shall have contact of the second order with the curve at the point P. Let us put Xy ?/, the co-ordinates of P; dij P = -;f- for the curve at the point P; q^-~ior the curve at the point P, CURVATURE; EV0LUTE8 AND INVOLUTES. 185 These last two quantities are found by differentiating the equation of the curve. dij d'^u Now, -j^ and -7-^ must have these same VpJues at the point {xy y) in the case of the circle having contact of the second order (§ 104). Let the equation of this circle be {X - ay + (2/ - by = r\ (a) By differentiation, we have {z - a)dx + {y - h)dy = 0, whence -/- = ^ ~ ^ = p. (b) dx b — y ^ ^ ' Differentiating again, J> ^ 1 , (^ - a ) dy_ _ (y - by + (r - a)' dx' b-y'^ {b-yY dx ~ (y - d)' r 3 = q- {c) "^ (y - ^) From {b) combined with (a) we find {X - ay _ 1+^' = 1 + (?/ - *)' ~{y- W (1 + ff r Dividing this by (c) gives the equivalent of the expression already found for the radius of curvature. Hence if we determine a circle by the condition that it shall have contact of the second order with the curve at the point P, its radius will be equal to the radius of curvature. This circle is called the osculating circle for the point P. Each point of a curve has its osculating circle, determined by the position, direction and curvature at that point. 186 TUB] DIFFERENTIAL CALCULUS. Cor. The osculating circle will^ in general, intersect the curve at the point of contact, for it has contact of the second order. This may also be seen by reflecting that the curvature of a curve is, in general, a continuously varying quantity as we pass along the curve, and that, at the point of contact, it is equal to the curvature of the circle. Hence, on one side of the point of contact, the curvature of the curve is less than that of the circle, and so the curve passes without the circle; and on the other side the curvature of the curve is greater, and thus the curve passes within the circle. If, however, the curvature should be a maximum or a minimum at the point of contact, it will either increase on both sides of this point or diminish on both sides, whence the circle will not intersect the curve. Xl08. Radius of Curvature tuhen the Abscissa is not taken as the Independent Variable. Suppose that, instead of x, some other variable, to, is regarded as the independent vari- able. We then have Now, it has been shown that, in this case, we have (§56) d'^y dx d'^x dy d^y __ du^ du du'' du d^' ~ ^ Y ' (2) Also, we have 1 I (^y V— 1 I ^^^^^ ^ _ \duJ_^ \du I . . \du / \du I These expressions being substituted in the expression for the radius of curvature, it becomes CURVATURE ; E VOLUTES AND INVOLUTES. 187 \\du} + \d^l \ (Py dx d^x dy dtc^ dii du^ du (4) 109. Radms of Curvature of a Curve referred to Polar Co-ordinates, Let the equation of the curve be given in the form r = 0(6'). The preceding expression (4) may be employed in this case by taking the angle as the independent variable. By differ- entiating the expressions X = r cos 6, y = r sin 0, regarding r as a function of 0, we find, when we put, for brevity, r' = - ' r" - — - dO' ~ dO'' -^ = — r sin 8 -f r' cos 6; du '^ = (7-" - r) cos - 2r' sin 6; -il= r cos + r' sin 0; du '^ = (r" - r) sin 6 + 2r' cos 0, By substituting these derivatives with respect to for those with respect to u in (4) and performing easy reductions, we find i'-+(l)T r' - rr" + 2r" err ^^ (dry which is the required expression for the radius of curvature. 188 THE DIFFERENTIAL CALCULUS. EXAMPLES AND EXERCISES. 1. The Parabola, To find the radius of curvature of a curve at any point, we have to form the value of p from the equation of the curve. The equation of the parabola is whence we find dx y ^ (ly _ _f dx' ~ y'' Then, by substituting in the expression for p, we find ^_{ y' + f? the negative sign being omitted, because we have no occasion to apply any sign to p. At the vertex ?/ = 0, whence Hence, at the vertex, the radius of curvature is equal to the semi-parameter, and the centre of curvature is therefore twice as far from the vertex as the focus is. 2. Show that the radius of curvature at any point {x, y) of an ellipse is ^ ~ a'V and show that at the extremities of the axes it is a third pro- portional to the semi-axes. 3. Show that the algebraic expression for p is the same in the case of the hyperbola as in that of the ellipse. 4. What must be the eccentricity of an ellipse that the cen- tre of curvature for a point at one end of the minor axis may lie on the other end of that axis? Ans. e = 4/^, CURVATURE ; EV0LUTE8 AND INVOLUTES, 189 5. Show that in the case supposed in the last problem the radius of curvature at an end of the major axis will be one fourth that axis. 6, The Cycloid. By differentiating the equations (1), § 80, of the cycloid, we find dx du d^x __ dy ' du d^y -r=— =z a — a cos u = y: du ^ 1-, = -7^ = asm u; du du , » = a cos 21, du (3) Then, by substituting in (4) and rec^ucing, we find, for the radius of curvature, p — 2^a Vl — cos 21 = 4rt sin ^u . We see that at the cusp, 0, of the cycloid, where u = 0, the radius of curvature also becomes zero. 7. The Archimedean Spiral. Show from (5) that the ra- dius of curvature of this spiral (r = a^) is 8. The Logarithnic Spiral, The equation of the loga- rithmic spiral being 19 r = ae , show that the radius of curvature is p = rVr+f. Hence show that the line drawn from the centre of curva- ture of any point P of the spiral to the pole is perpendicular to the radius vector of the point P. 9. Show that the radius of curvature of the lemniscate in terms of polar co-ordinates is _ ^ _ ^' 190 THE BIFFBBENTIAL CALCULUS. no, Evolutes and Involutes, For every point of a curve there is a centre of curvature, found by the preceding for- mulae. The locus of all such centres is called the evolute of the curve. To find the evolute of a curve, let {x^y^ be the co-ordi- nates of any point P of the curve ; PC, the radius of cur- vature for this point; and a, the angle which the tangent at P makes with the axis of X, Then, for the co-ordinates of (7, we have FiQ. 43. p em a\ y = y^ + P^OB a. Substituting for p its value (§ 106), and for sin a and cos a their values from the equation _dy, dx/ tan a we find 1 + X:=i X, y~y,-\- ^1 dx^ dx^ dx, (1) If in the second members of these equations we substitute the values of the derivatives obtained from the equation of the curve, we shall have two equations between the four vari- ables X, y, x^ and y^. By eliminating x^ and y from these equations and that of the given curve, we shall have a single equation between x and y, which will be that of the evolute. CURVATURE; EYOLUTES AND INVOLUTES. 191 111. Cane of an Auxiliary Variable, If the equation of the curve is expressed by an auxiliary variable, we have to make in (1) the same substitution of the values of —, ^-, etc., as in § 108. Thus we find, instead of (1), du ' du'^ X,— y = y^ + ^ dy^ \du J ' \dic J du d'^y^ dx^ d'x^ dy^ ' du^ du du^ du IdjcX Idyy dx, \du I '^ \du I (2) du d^y^ dx^ d'x^ dy ^ ' du'' du du^ du which are the equations of the e volute in the same form. EXAMPLES OF EVOLUTES. r 113. Tlie Evolute of the Parabola. If we substitute in (1) for the derivatives of y^ with respect to x^ the values already found for the parabola, these equations (1) become if p^ We now have to eliminate y^ from these two equations, x^ having already been eliminated by the equation of the curve. They give ?//' = ip{^ - p); y/ = - pY Equating the cube of the first equation to the square of the second, we find, for the equation of the evolute of the parabola, _ 8 (X - pY ^ ~27 p ' 192 TH^ DIFFmmTIAL CALCULUS. 113. Evolute of the Ellipse. From the equation of the ellipse, we find dy^ _ Z>X. ^/ ^ dx^ ~ d'y/ dx/ ~~ «V/* By substituting in (1) and reducing, we find Remarking that the equation of the ellipse gives „47,2 -.4-, 2 -,SZa^2 a — a y^ = a o x , and putting e'^ = a'^ — Vy the preceding equation becomes c^x' X = {a) In the same way we get In this case the easiest way to effect the elimination of x^ and y^ is to obtain the values of these quantities from (a) and (^), and then substitute them in the equation of the ellipse. From (a) and {b)y we find which values are to be substituted in the equa- J r tion ^ -i *r -^>.,^^^ -J. ( ¥-^^ y V "^^B F ) We thus find, for the \ N / ' / equation of the evolute of \. \ / J the ellipse. ^\ ^\ /^ ^..^ Fia. f D 44. form of the curve. The following properties should be de- duced by the student. CURVATURE; EVOLUTES AND INVOLUTES, 193 (a) The evolute lies wholly within the ellipse, or cuts it (as in the figure), according as e' < ^ or c* > i. {b) The ratio CD : AB (which lines we may call axes of the evolute) is the inverse of the ratio of the corresponding axes of the ellipse. 114. Evolute of the Cycloid. Here we have to apply the formula) (2) for the case of a separate independent variable. Substituting in (2) the values of the derivatives already given for the cycloid, we shall find d^y dx dic"^ du — a'^{l — cos ii)'y d^x dy du* du X z=z x^ -\- 2a sin u = a(u -{- sin u); y = y^ — 2a{l — cos ti) = — a{l — cos 2^). These last two equations are those of the evolute. Let us investigate its form. For ^^ = we have x = and y = 0, whence the origin is a point of the curve. For u = 7t we have X = uTt; y = - 2flj; giving a point C, below the middle of the base of the cycloid, at the dis- tance 2a. Let us take this point as a new origin, and call the co-ordinates referred to it a;' and y\ We then have x' = X — art = a{0 — tt + sin d)\ y' = y + 2a = a(l + cos /9). If we now put V V \ / V^X V Fig. 45. e'^e-n, these equations become 194 THE DIFFERENTIAL CALCULUS, x' = a{0' - sin 6'); / :=: ^(1 _ COS 6>'); which are the equations of another cycloid, equal to the original one, and similarly situated. The cycloid therefore j posesses the remarkable property of being identical in form with its own evolute. 115. Fundame7ital Pi'operties of the Evolute, ^ - Theorem I. The involute of a c%irve is the envelope of its \ normals, \ As we move along a curve, the normal will be a straight line moving according to a certain law depending upon the form of the curve. This line will, in general, have an en- velope, which envelope will be, by definition, the locus of the point of intersection of consecutive normals. But this point has been shown to be the centre of curvature, whose locus is, by definition, the evolute. Hence follows the theorem. Corollary. Ilie nor- mals to a curve are tan- gents to its evolute. For this has been shown to be true of a moving line and its envelope. Theorem II. If the os- culating circle move around the curve, the motion of its centre is along the line join- ing that centre to the point of contact. This theorem will be made evident by a study of the figure. If the line PgG, be one of the nor- mals from the point of contact P^ to the centre, then, since Fio. 46. I CURVATURE; E VOLUTES AND INVOLUTES. 195 ills normal is tangent to the locus of the centre, it will be tlie _ ine along which the centre is moving at the instant. TuEOREM III. The arc of the evolute contained between any (too points is equal to the difference of the radii of the oscillating circles whose centres are at these points. For, if we suppose the points C„ C^, etc., to approach in- finitesimally near each other, then, since the infinitesimal arcs C^C^, ^a^3> ^tc, are coincident with those successive radii of the osculating circle which are normal to the curve, these radii are continually diminished by these same infini- tesimal amounts. The analytic proof of Theorems II. and III. is as follows: Let the equation of the osculating circle be {X - ay + (y - by = p\ where a and b are the co-ordinates of the centre of curvature, and therefore of a point of the evolute. The complete differential of this equation gives {x — a) {dx ~ da) + (/y — b) {dy — db) = pdp. (a) If, in this equation, we suppose x and y to be the co-ordi- nates of the 2ioint of contact of the circle with the curve, then dx and dy will have the same value at this point whether we conceive them to belong to the circle, supposed for the mo- ment to be fixed, or to the curve. But in the fixed circle we have {x - a)dx + (?/ - b)dy = 0. (h) Subtracting this equation from {a) and dividing by p, we find da + db = — dp, (c) P ' p "^ ' ^ which is a relation between the differential of the co-ordi- nates of the centre and the differential of the radius. Now, if we put ft for the angle which the normal radius makes with the axis of X, we have X — a ^ y —W . a / 7\ = cos p; = sm p, (d ) 196 THE diffehential calculus. But this same normal radius is a tangent to the evolute. If we call a the arc of the evolute, we find by a simple con-' struction da = cos ft da] db = sin /3da, Multiplying these equations by cos /3 and sin-;^ respectively, and adding, we find do- =z cos ftda + sin /3db, Comparing (c) and (d), we find do- = — dp, or d{(7 -}- p) = 0, Now, a quantity whose differential is zero is a constant. Hence we always have cr -\- p =z constant, or o" = constant — p. If we represent by or the intercepted arc equal to the difference of the radii, as was to be proved. It must be remarked, however, that whenever we pass a cusp on the evolute, we must regard the arc as negative on one side and positive on the other. In the case of the ellipse, for example, those radii will be equal which terminate at equal distances on the two sides of any cusp, sls A, B, C or D, and the intercepted arc must then be taken as zero. 116. Involutes, The involute of a curve C is that CLtye which has G as its evolute. The fundamental property of the involute is this: The involute may be formed from the evolute by rolling a tangent CURVATURUE ; E VOLUTES AJSI) INVOLUTES. 197 ine upon the latter. A point F on the rolling tangent will then describe the involute. This will be seen by reference to Fig. 46. The rolling line, ling tangent to the evolute, coincides with the radius P, C\y ,nd as it rolls along the evolute into successive positions, C,, P^C^, etc., the motion of the point F is continually lormal to its direction. It will also be seen that the radius of curvature of the in- olute at each point is equal to the distance FC from F to ;he point of contact with the evolute. The conception may be made clearer by conceiving the rolling line to be represented by a string which is wrapped around the evolute. The involute is then formed by the mo- tion of a point on the string. The general method of determining the involutes of given curves involves the integral calculus. f/h<^' ^ . \-€ toU" .V PART II. THE INTEGRAL CALCULUS. PART II. THE INTEGRAL CALCULUS. CHAPTER I. THE ELEMENTARY FORMS OF INTEGRATION. 117. Definition of Integration. Whenever we have given \ function of a variable x^ say u = F(x), ^e may, by differentiation, obtain another function of x, ^ = ^ (^)' rhich we call the derived fn7iction. In the integral calculus we consider the reverse process. Ve have given a derived function F'{x), nd the problem is: What function or functions, tolieji differ- ntiatedy will have F\x) as their derivative? Every such function is called an integral of F'{x). The process of finding the integral is called integration. The operation of integration is indicated by the sign / , ailed ^^ integral of,'^ written before the product of the given unction by the differential of the variable. Thus the ex- ression fF\x)dx leans: that function whose differential with respect to x is ^'{x)dx. 202 THE INTEGRAL CALCULUS. Calling u the required function, then if we have we must also have As examples: Because d{x'') — 2xdxy we have / 2xdx = x^. Because d{ax^ -^ hx -\- c) = {2ax + ^)dxy we have / {2ax ~\- h)dx = ax' -\- hx + c. And, in general, if, by differentiation, we have dF{x) = F'(x)dx, we shall have I F'{x)dx — F{x). 118. ArMtrary Constant of Integration. The folio win] principle is a fundamental one of the integral calculus: If F{x) is the integral of any derived function of the va riaUe x, then ^ery function of the form Fix) + A, h heing any quantity whatever independent of x, will also h an integral. This follows immediately from the fact that h will dia appear in differentiation, so that the two functions F{x) and F{x) + h have the same derivative (cf. §24). The same principle may be seen from another point o view : Since the problem of differentiation is to find a f unc tion which, being differentiated, will give a certain result and since any quantity independent of the variable whicl may be added to the original function will have disappeare( by differentiation, it follows that we must, to have the moa i TUE ELEMENTARY FORMS OF INTEGRATION. 203 general expression for the integral, add this possible but un- known quantity to the integral. The quantity thus added is called an arbitrary constant. But it must be well understood that the word constant merely means independent of the variable with reference to which e integration is performed. It follows from all this that the integral can never be com- pletely found from the differential equation alone, but that Bome other datum is needed to determine the arbitrary con- stant and thus to complete the solution. Such a datum is the value of the integral for some one value of the variable. Let F{x) + h be the integral, and let it be given that when X = a, then the integral = K. We must have, by this datum, F{a) + h = Ky which gives h = K — F{a), and thus determines h. Remark. Any symbol may be taken to represent the ar- bitrary constant. The letters c and h are those most gener- ally used. We may affix to it either the porftive or the nega- tive sign, and may represent it by any function of arbitrary but constant quantities which we find it convenient to intro- duce. It is often advantageous to write it as a quantity of the same kind as the variable which is integrated. 119, Integration of Entire Functions. Theorem I. The integral of any power of a variable is the power higher by unity y divided by the increased exponent. In symbolic language, we have x^'dx = — -— -f ?i 71+1 X n + 1 For, by differentiating the expression — — :: -f h, we have 204 THE INTEGRAL CALCULUS. Theorem II. Aiiy constant factor of the given differen- tial may be written before the sign of integration. In symbolic language, faF'{x)dx = ajF\x)dx. This is the converse of the Theorem of § 23. By that theorem we have d(aF(x)) = adF(x)y from which the above converse theorem at once follows. In the special case « — — 1 we have J- F\x)dx = J*F'{x)d{- x)=^'- jF\x)dx. Hence the corollary: If the integral is preceded by the nega- tive sign toe may place that sign before either the derived function or the differential. Theorem III. If the derived function is a sum of several terms, the integral is the sum of the separate integrals of the terms. In symbolic language, f{X+ Y+ Z-\- . . ,)dx =fxdx+fYdx+fzdx+ . . This, again, is the converse of Theorem II of § 22. The foregoing theorems will enable us to find the integral of any entire function of a variable. To take the function in its most general form, let it be required to find the integral u— j (ax"^ + bx"" -\- cx^ -\- . . .)dx. By Theorem III., u^= I ax'^dx + / bx'^dx + / cx^dx + « o • • / THE ELEMENTARY FORMS OF INTEGRATION. 205 8y Theorem II., / ax'^dx = a I x'^dx; etc. etc. ; ad by Theorem I., x'^dx^ r-^ + h,; etc. etc. By successive substitution we then have m -{-1 n -\-l p -\-l sphere h^, h^, h^, etc., are the arbitrary constants added to the separate integrals. Since the sum of the products of any number of constants ly constant factors is itself a constant, we may represent the Bum ah^ -f hh^ -f ^^-^s by the single symbol li. Thus we have fiax"^ + hx"" + ca;P + . . .)dx ___ ax^^_ hx""-^^ cx^+^ , ~" nTfi + ^Tfl "^ ^Tfl + • • • + '^^ • EXERCISES. Form the integrals of the following expressions, multiplied by dx: I. x\ 2. x\ 3. x-\ 4- a;-». 5. ax*. 6. hx\ 7. ax-\ 8. ^o;-'. 9. aa; + ^• 10. ax^ — c. II. ax^ + ^^' 12. rta:' — cx^ 13. a:*. 14. a:i. 15. x-K 16. ax-h 17. ax^—bx-K 18. x^ 19. a h x' x'' 20. »+i. r 130. The Logarithmic Function. An exceptional case of Theorem I. occurs when n ^= — 1^ because then n -\-l = 0, and the function becomes infinite in form. But since 6?- log X = — = x~^dx. 206 THE INTEGRAL CALCULUS. it follows that we have for this special case / x~^dx = / — = log x-\-h. {a) Let c be the number of which h is the logarithm. We then have log X -^h — log X -\- log c = log cx^ We may equally suppose h= — log c = log -• X Then log ic + ^ = log -. c Hence we may write either rdx , J-=\ogcx, /dx , X — =log~; X ^ c ^ c being an arbitrary constant. We thus have the principle: The arbitrary constant added to a logarithm may be introduced by multiplying or dividing by an arbitrary constant the number whose logarithm is ex- pressed. 13 !• We may derive the integral (a) directly from Theo- rem I., thus: In the general form x-dx = ^-— + h 71 + 1 let us determine the constant h by the condition that the in- tegral shall vanish when x has some determinate value a. This gives + A = 0; .-. h= - (^) n+1 ' -, . . '^ ^^y Thus the integral will become / x'^dx = , «/ 71 -{-1 THE ELEMENTARY FORMS OF INTEGRATION. 207 in which a takes the place of the arbitrary constant. This expression becomes indeterminate for n = — 1, But in this [Case its limit is found by § 71, Ex. 5, to bo log x — log «. Thus we have / x'^dx = log X — log a = log — , as before, log a being now the arbitrary constant. 133. Exponential Functions, Since we have d{a^) = log a . a^dxy it follows that we have / log a . a'^dx = a* + h, or, applying Th. II., § 119, to the first member and then di- viding by log a, a'^ + h / a'^dx = log a' which we may write in the form / log rt ' because = is itself a constant which we may represent by h. 133. The Elementary Forms of Integration. There is no general method for finding the integral of a given differen- tial. What we have to do, when possible, is to reduce the differential to some form in wliich we can recognize it as the differential of a known function. For this purpose the fol- lowing elementary forms, derived by differentiation, should be well memorized by the student. We first write the prin- cipal known differentials, and to the left give the integral, found by reversing the process. For perspicuity we repeat the forms already found, and we omit the constants of in- tegration. 208 THE INTEOEAL CALCULUS, y d'logy d'sin y == cos ydy, d'cos y = — sin ydy, d'tsm. y — sec'' ydy, ^ sm y tan y ., rt'sec y = -d?/y ^ cos y *^ ^Z'sin^""^^ y = dy Vl-./ dy VI- f' d'tan^"^^ y d'ci^ . • d'sm]i^~^hj dy : a^ log ady, dy Jynay =£11, (1) + ■/f ='o,>. m . / cos ydy = sin ?/. (3) . / sin ydy = — cos y, (4) . / — ~ =-- tan ?/. (5) t/ cos y ./ \ / .fJl- =-coty.(6) t/ sm y J \ / /tan Vf7y -—^ = sec ^. (7) cos y J \ / ./-^^=sin<-»y.(8) "^^ VI— y •/iTT/-' =tan<-V.(10) .Jaydy = log a (11) Vf+: •.• d-cos h'- %= — --- — ■ V;?/"- 1 '.•d'ta.nh'^'^^y- Vf-1 _ dy : COS h<-"y = log (y + Vf- 1). (13) y' -'J^y^ =tanh->, = |logi± 1 + y (14) INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 209 >^ CHAPTER II. INTEGRALS IMMEDIATELY REDUCIBLE TO THE ELEMENTARY FORMS. 124. Integrals ReduciUe to the Form I ifdy. The fol- lowing are examples of how, by suitable transformations, we may reduce integrals to the form (1). Let it be required to find I {a-[- xYdx. We might develop {n -\- xY by the binomial thorem, and then integrate each term separately by applying Theorem III., § 119. But the following is a simpler way. Since we have dx = d{a + x)y we may write the integral thus: Ha + xYd{a + x). It is now in the form (1), y being replaced hy a -{- x. Hence /(« + :r)».Zx=.(^i±^--+A. (1) In the same way, f{a - xYdx = - f{a - xYd{a - x) = h - ^^ ~f^^ —' To take another step, let us have to find Ha + hxydx. We have dx = jd{hx) = jd{a -\- bx). Hence, by applying Th. 11. , f{a+bxrdx=lf{a+hxrd{a+bx)= i^+Mlll + k, (2) 210 THE INTEOHAL CALCULUS, We might also introduce a new symbol, y = a '\- hx, and then we should have to integrate y'^dy with the result in § 123. Substituting for y its value in terms of Xy we should then have the result (2).* These transformations apply equally whether n, a and h are entire or fractional, positive or negative. EXERCISES. Find: i. i (a -{- xydx, 2. f d{a — xydx, 3. I {(I — 2xydx, 4. / (a + x)'^^dx, 5. / (« — x)~^dx, I {a-\-mx)~^dx, 1- I {ci — mxydx. 8. / {a — mx)^^dx. r dx r dx r dx J {a + xY ^""'J {a - xy ''V {a - ^.xf (a-\- xydx, 13. / («^ + nxydx. 14. I {a -\- x^yxdx, Ai + 1-' + -xh' ^^•/(^^«• J [-(jr^xy + {a - xy + (^^^j ^''' J \{a — mxy (a — mxy {a ~ mxy) I {a-{-lx-\- cx''){h + %cx)dx, J (a -\'lxAr cxY{b + 2cx)dx. r {h-\- 2cx)dx J (a + Z>a; + cx'Y * The question whether to introduce a new symbol for a function whose differential is to be used must be decided by the student in each case. He is advised, as a rule, to first use the function, because he then gets a clearer view of the nature of the transformation. He can then replace the function by a new symbol whenever the labor of repeatedly writing the function will thereby be saved. 6 9 12 15 17 18 19 20, INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 211 135. A2)plication to the Case of a Falling Body, We have shown (§33) that if, at a time t, a body is at a distance z from a point, the velocity of motion of the body is equal to dz the derivative —-, Now, when a body falls from a height at under the influence of a uniform force g of gravity, unmodi- fied by any resistance, the law in question asserts that equal velocities are added in equal times. That is, if z be the height of the body above the surface of the earth, and if we count the time t from the moment at which the body began to fall, the law asserts that dz . , . the negative sign indicating that the force g acts so as to diminish the height z. By integrating this expression, we have z=h- igt\ {h ) Here the constant Ji represents the height z of the body at the moment when ^ = 0, or when the body began to fall. From the definition of h and z, it follows that h — z is the distance through which the body has fallen. The equation (b) gives h-z = igt\ (c) Hence: The distance through which the body has fallen is proportional to the square of the time. At the end of the time t the velocity of the body, meas- ured downwards, is, by (a), equal to gt. If at this moment the velocity became constant, the body would, in another equal interval t, move through the space gt X t = gt^. Hence, by comparing with (c) we reach by another method a result of §33, namely: In any period of time a body falls from a state of rest^ through half the distance through lohich it would move in the same period with its acquired velocity at the end of the period. 212 THE INTEOBAL CALCULUS, 136. Reduction to the Logarithmic Form. Let us have to find /mdx ax -\- 1) u Since dx = —d{ax) = —d{ax + ^), we may write this expression in the form /m d{ax + h) a ax -\-b ^ and the integral becomes m pdiax -\-l) m , ax-^-h %c— - -^ ~7-^ = - log -', aJax-\-b a° c c being an arbitrary constant. EXERCISES. Integrate the following expressions multiplied by dx: I. X -\ . 2. — . X X 1 1__ "^^ x + r ^* 2x - r 7. h^* 8" y^a: * 2aa; + b' X* -{- kx a-\-b 3. 6. m ex — Z>' 9- a' 2bx + a'' 2. m ~ 71 mx — 71 ^ 4 + ^ ' ' ax -[- b' \ T^, ^ p xdx p xdx Note that a; t?a; = id{x'') = id{l + x'). /xdx ^ /* X* dx pDs^s; x dx dx Note that log a; -— = log xd . log x. \og{l-\-y) p xdx n xdx i8. / ^ , -dy. 19. / -— =:. 20. / - + a;' */ (1 - a;')' r INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 213 127. Trigonometric Forms. The following are examples of the reduction of certain trigonometric forms: /cos mz dx = — I cos mx dhnx) = — sin 7nx + h. mj ^ ^ m /sin 7nx dx = — I sin 7nx d(vix) = h cos mx. mJ ^ ' 7)1 I cos (a + mx)dx = — / cos {a + 7nx)d{a + 'tnx) __ sin (a + mx) m mixdx rd'co^x + h. tan xdx = / = — / t/ cos a; ^ cos a; = h — log cos a; = log c sec x, i where 7^ = log c. In the same way, / cot xdx = log c sin a;. /Jrc /> 1 Ja: /^fZ'tan a; , -; = / 7 r = / -T = log c tan a:. sm x cos a; t/ tan x cos a; j tan x /dx 1 r dx 1x1 = o / -• — i r- = log c tan -a;. sm a: 2 1/ sm ix cos -Ja; ° 2 /dx __ n dx __ , fn x\ cos x'~ J sin (|;r — a;) """ ^ \4 2/ EXERCISES. Integrate : I. (1 + cos y)dy. 2. {1 — e sin u)dn. 3. cos 2y f?y. Atis. i I cos 2yd{2i/) = ^ sin 2y. 4. sin 2y dy. 5. cos 7iy dy. 6. sin y cos y dy. Atis. J / sin 2yd(2y) = -- i cos 2//. 7. tan 2a: c?a:. 8. cot 2a; dx. 9. 2 cos* X dx. A71S, / (1 + cos 2a;)^/a; = a; -f- ^ + i sin 2a:. 214 THE INTEGRAL CALCULUS. lo. 2 sin' xdx, 1 1. tan 2y dy, cos y dy ^ , A/(l + sin i/) , /-, , • x 1 + sm t/ «/ 1 + sm 1/ to V « ^^z sin V dy sin v ^V 1-2. —, 14. —, 1 + COS y 1 — cos y cos y dy ^ sin %y dy 15. :^ -V-^. 16. ^— ^-. 1 — sm y cos «/ cos' .-r — sin' X ^ ^ sin 2a; , 17. ; — ^ ax. 18. — r-^— dx, sm 2a; cos x — sm a; ^o; dx dx iQ. . 20. cos ma; sm wo; sm 7nx cos wo; 22. sin {mx + a) Jo;. 23. cos {a — 7ix)dx. 24. tan nx dx, 25. tan (2a; — a)dx. dx dx ^ dx 26. -. — ^ r. 27. jj r-. 28. sin {a — x)' ' cos (b -— nx)' * sin {a — ^ix)' cos w^6?v sin 7^v<^V sec' xdx 29. ; r^-^^. 30. -•?-^Z_. 3 J, ^ ^ a-j- sm 7^2/ « — cos 7iy a — m tan a; 138, hitegration of -5-— — ^ «^c? • a' + a;' «' — a;' We see at once that the first differential may be reduced to that of an inverse tangent ; thus, dx 1 dx . "© ic' + a' ^' ^ I T ^^£'11' Hence "We find in the same way J a -\- X ^^11 ^ ^ 9 ' -*- (1) /'_,^ = ltanh<->^ + A = llogcl±^, (2) ,/ a' — a;' a a 2a ^^ (? — a; ^ 6' being an arbitrary constant factor. r INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 215 139. Integrals of the form I — -- — 5. fj (I -y~ Ox ~\~ 6»C The reduction of integrals of this form depends upon the character of the roots of the quadratic equation ex" -\'hz-\-a = 0. (1) I. If these roots are imaginary, the integral is the inverse of a trigonometric tangent. II. If the roots are real and unequal, tlie integral is the inverse of an hyperbolic tangent. III. If the roots are real and equal, that is, if the above ex- -^* pression is a perfect square, the integral is an algebraic frac- tion. Dividing the denominator of the fraction by tlie coefficient of x", the given integral may be written dx cj , ^ . bx a ^ H h - c c («) Writing 2;j for - and q for -, the expression to be inte- grated may be reduced to one of the forms of § 128, thus: dx __ dx __ d(x -f- p) (^0 X' + 2px + q {x + j^)' + q-p' {^+py + q -y The three cases now depend on the sign oi q — p"", I. If q — jy' is positive, the roots of (1) are imaginary and the form is the first of the last article with x-\- p m the place of X, and q — p"^ in the place of a\ Hence we have dx __ /» d{x -\- p) X- + 2px +^ "" J i ■\-2px + q J {x + pY + q-p' = — =L= tan <- ^> - ^-P- + h. (1) Vq - p' ^^q-p Comparing this result with (a), we see that this integral may be reduced to its primitive form by changing p into 216 THE INTEGRAL CALCULUS. — — and q into — . Substituting and reducing, we have A/ C C dx 1 p dx /dx —^r a -\- bx -\- cz"^ ~ cj 2 . ^ I ^ ^ tan<->-J?=±i-- + /.. (2) II, If q — p^ is negative, that is, if 4ac — Z>' is negative in (2), the expression (2) will contain two imaginary quantities. But these two quantities cancel each other, so that the ex- pression is always real. When q — p^ is negative, we write {b) in the form _ d{x+p) The integral is now in the form (2) of § 128, and we have dx _ p d{x -\- p) /dx _ p x"" + 2px + q~ J f x" + 2px -\- q J f — q — (x-^- pY = A -A tanh<-i> ^J^P ^V^ — q Vp'^ — q = J, _ —±= log /7^ + ^+P ^ (3) ^ Vp"" — q yp^— Q—{^-\-p) Making the same substitutions in these equations that we made in Case I., we find — — T — \ a = A ; tanh^-^> — — a + l)x + ex"" |/^' _ 4,ac Vb'' - Aac 1 , Vb'-4:nc + 2cx + b ,,^ — h ; r:^ lOg C— -^-(4) Vb'-4.ac Vb'-4:ac-{2cx+b) III. If p' — q = 0, the expression to be integrated becomes dx We have already integrated this form and found i^+py {^+P? ~ " ~ x+p /dx ^ 1 I. INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 217 EXERCISES. Integrate the following expressions: dx clx dz z' -2x- r ^' (a;-a)(a;~70' ^' a + 2bx - x'' dx dx dx / "*' z' + ix + 2' ^' x{x - a)' '(jr^x){x~^y 130. Inverse Sines and Cosines as Integrals. From what las already been shown (§ 123, (8) and (9)), it will be seen that ^e have the two following integral forms: f - -j^^ = cos <-»:. + h' = u'; {b) rhere we have added h and h' as arbitrary constants of in- egration. Comparing the first members of these equations, we see hat each is the negative of the other. The question may herefore be asked why we should not write the second iquation in the form u' = - f—^= = h"- sin<-^>a:, (c) B well as in the form {h). The answer is that no error ould arise in doing so, because the forms (b) and {c) are quivalent. From {h) we derive X = cos {u' — A') = cos (^' — w'); (d) nd from (c), X = sin (A" — u'), (e) Now, we always have sin (a + 90°) = cos a. Hence (d) nd (e) become identical by putting A" = /^' + 90°, hich we may always do, because the value of A" is quite rbitrary. 218 THE INTEGRAL CALCULUS, 131. The preceding reasoning illustrates the fact that integrals expressed by circular functions may be expressed either in the direct or inverse form. That is, if the relation between the differentials of u and x is expressed in the form ^ dx du = — Vl - x'' we may express the relation between u and x themselves either in the form u = sin ^~ ^^ a; + ^* or in the form x = sin {21 — h). So, also, in the form (1) of § 128 we may express the rela- tion between x and u either as it is there written or in the reverse form, x:=^ a tan a{u — li), dx 132, Integration of - Va" ^ x" We have />A dx rh a , X . , ... / \/-. i = J -7~^ = sm< ''- + h. (1) «/ Va — x^ ^ ./ x' ^ a In the same way — dx f- ^^ z=o,os(-^)--f ^. or 7^-sin^-^>-. (2) Va' - X We also have d'- dx = log-(.T+4/^^+^).(3): ^ INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 219 d'- dx = log^(a; + Va;'_a'). (4) EXERCISES. Integrate the differentials: dx dv I. , 2. ^— Vc - x' 71 dy mdz 5. - =z. 6. dx 7. — — . o, i^4c' + a;' dll __ ^x 13. ^ 14. ^^ __ — cos xdx 15. If rm — ^ r= ^, then sin re == r? cos (?^ + ^)» 10. Sfia dx Vif -{X- dz ay Via ' - m\' vidx V^ + ni'x' dx V^ + wi'(a; dx -ay V{^ - ay - 4c' tix" -'dx Va' sin z e'dx dx 16. — — 17. Vl -e^ ^Vl- (log a:)' — sin a*r/:r cos xdx lo. -T --— . 10. —z :— ;— . a -\- cos X a -\- sin' x (x — a)dx (x 4- a)dx 20. ^ 21. , ^ :r:^=:. Vl- (x-ay Vl + (x + aY 220 THE INTEGRAL CALCULUS. 133. Integration of — — - . Every differential of Va-\-l)x ± cx^ this form can be reduced to one of the three forms of the preceding article by a process similar to that of § 129. The mode of reduction will depend upon the sign of the term cx^. Case I. The term ex" is negative. Putting, as before, 11 __a^ we have ^-2P Va + hx- ex'' = V7 Vq + 2px-x'= V^ Vp' + q- {x-p)\ Then, comparing with (1) of § 132, we find dx _ 1 /* d{^ — P) r dx - _1 /* t/ 4//y _L h/l - sin <9 + cos' /9^ sin cos ^tZ^ ^ a sin ^^(9 V4 - cos 26* - cos' 26> Va" '^ b\l - cos 6)' 134. Exponential Forms, Using the form (11) of § 123, we may reduce and integrate the simplest exponential dif- ferentials as follows: a^dx = ± / a^d(mx) =~ ^ h. (1) ra^+'^dx = fa'+'^dix + b) = ^- + h. (2) 7;iJ "^ ' ^ 7Wl0grt "^ ^ /-I 7. >y — ma; a-^dx = ~ - /a-^^d(- 7nx) = -^-— . (4) 7?2t/ ^ ^ m log a "^ ^ EXERCISES. Integrate: I. (fdx. 2. Z>''^r/. 3. a^'^dy. 4. (a + Z>)e^Jar. 5. a^-^'dy. 6. a-^'dx. 7. (a* + a-*)t?a;. 8. (a* - rt-*)d7a:. 9. (a + e^)rZa;. 10. («'^--a-*=^)f7a:. II. ^-j^. 12. j-qT^^. 15. (a*^ + a-"»^)'Ja:. 16. Ce^^xdx. 17. Ce^^'^x-dx, 18. A-«(^'-^)a:^a:. 222 THE INTEGRAL CALCULUS, CHAPTER Mi. INTEGRATION BY RATIONAL TRANSFORMATIONS; 135. We have now to consider certain forms which cannot be reduced so simply and directly as those treated in the last chapter. Before passing to general methods we shall consider some simple cases. ((I _i_ x\^ I. Integration of ^ :j^dx. Any form of this kind, when m is entire, may be integrated by developing the numerator by the binomial theorem. We then have and each term can be integrated separately. If n < m + 2, and entire, one of the terms of the integral will contain log x. x^dx II. Integration of -. — [ -. We may reduce this form to the preceding, by introducing a new variable, z, defined by the equation z = a -{- bx. This gives x = — 7 — ; dx = -j-. Substituting these values of x and dx in the expression to be integrated, it becomes {z - aY dz which may be integrated by the method of the last article. III. Integration of -—j-v — 3 . We reduce the denomi- a ~j~ ox jt ex INTEGRATION BY RATIONAL TRANSFORMATIONS. 223 iiator to the form ± {v'' — q) ± {x -\- pY as in § 129. Then, putting, for brevity, z = x+p, which gives dx = dz, the integration will have to be performed on an expression of the form {z — p)dx _^ zdz pdz V ± z^ - 1F±1'' " ¥~±~z'' Each of these terms may be integrated by methods already given (§§ 126, 128). The process is exactly the same if we have to find (a -\- hx)dx J V ± {x-pr EXERCISES. Integrate: (i -l]'dx (x — a) dx \a x ) I. ^^ T- • 2. . x^ X x*dx x^dx dx {x + a)d x x^dx TTZIX' xdx zdz \a xl X x^dx 8. 9- a' + {b - xf '''• {a + zy + (a - zY (y — b)di/ {z — c)dz ''• {y-'^r + (!/+W '"• a^-az + z^' (x - a)d x (y + a )dy ^' x{x^b)' '^- a^-{y-+br z'^dz ^ z\lz (1 + zY (1 - zY 224 THE tNTBGBAL GALClTLm, 136. Reduction of Rational Fractions in general, A ra- tional fraction is a fraction whose numerator and denominator are entire functions of the variable. The general form is ^0 + Qi^ + Q^^' + . . . + qnx'' ~ i> * If the degree m of the numerator exceeds the degree n of the denominator, we may divide the numerator by the de- nominator until we have a remainder of less degree than 7i. Then, if we put Q for the entire part of the quotient, and R for the remainder, the fraction will be reduced to If we have to integrate this expression, then, since Q is an entire function of x, the differential Qdx can be integrated by § 119, leaving only the proper fraction y-. Now, such a fraction always admits of being divided into the sum of a series of partial fractions with constant numerators, provided that we can find the roots of the equation D = 0, The theory of this process belongs to Algebra, but we shall show by ex- amples how to execute it in the three principal cases which may arise. Case I. The roots of the equation D = all real and un- equal. Let these roots be a, /?,;/... ^. Then, as shown in Algebra, we shall have I) = {x- a){x - /3){x -y) . . . (x-O). We then assume I)'~'x--a^x-/3^x^r^'"' A, B, Cy etc., being undetermined coefficients. To deter- mine them we reduce the fractions in the second member to the common denominator i), equate the sum of the numera- tors of the new fractions to R, and then equate the co- efficients of like powers of x. INTEGRATION BY RATIONAL TRANSFORMATIONS. 225 As an example, let us take the fraction X — X We readily find, by solving the equation a:' — cc = 0, X* -x = x{x- l){x + 1). Assume ^_+3 _A B C a:' — a:~" x ir — la; + l __ {A + B+C)x'-\-{B- C)x-A ~~ x^ — X Equating the coefficients of powers of x, we have ^ + ^ + 6^ = 0; B- C=l; ^ = -3; whence B = 2 and C = 1, Hence x + 3 __ _ 3 , 2 1 . ic'-ic"" x'^ x-1^ x + V and then, by § 120, O X — X Ox ^ X —1 ^ X -\-l = - 3 log a; + 2 log (.-r - 1) + log (x + l)+log G EXERCISES. Integrate: (x — l)dx xdx '• a;' -a;- 6* ^' ¥'^V xdx {x + a:')f7a: ^' 1 -a:*' "^^ (a;-l)(a;+l)(a;-2)(a;4-2)* (a;* + 2a : ^)^a; (a;^ + a:')rZa: ^' a:' + 2a;''- 8a;* a;(a; - l){x -f- l)(a; - 2)* a;*6?a; 67a; ^* a;' - (a -fb)x + ^^//' ^^' a;" - (re + Z>)a:' + a^a;* 226 THE INTEGRAL CALCULUS, Case II. Some of the roots equal to each other. Let the factor X — a appear in D to the nih. power. Then, if we followed the process of Case I., we should find ourselves with more equations than unknown quantities, because the n fractions would coalesce into one. To avoid this we write the assumed series of fractions in the form A B F H and then we proceed to reduce to a common denominator as before. The coefficients A, B, etc., are now equal in num- ber to the terms of the equation Z> = 0, so that we shall have exactly conditions enough to determine them. As an example, let it be required to integrate X — X — X -\- 1 We have x' - x"" - x -\~1 =^ {x - ly {x -^ 1). We then assume x'-h A . -^ , C' {x-iy{:c~\-l)~{x-iy^ x-1 ' x + 1 - (^ + Oy + {A - 2C)x + A-B+C {x-iy{x + i) We find, by equating and solving, A = -2; B = +2; 0= -1. Hence {x - iy{x + 1)" {x-iy~^x-i x + r IJSTEORAriON BY MATIONAL TUANSFORMATIONS, 227 The required integral is = ^ + 2 log (.T - 1) - log (x + 1) + log C = ^3^ + log ~^:p^. EXERCISES. Integrate: f/.r dx x(x-\-iy' x^x- If x^dx dx ^* {x - \)\x + 2)^• ^* (a: - afi^x - ^)'' (rt + x)dx (« — rr)^/a; ^' :r''(a; - af x\x + «p(;^"^- i( Case III. Imaginary roots. Were the preceding methods applied without change to the case when the equation D = has imaginary roots, we should have a result in an imaginary form, though actually the integral is real. We therefore modify the process as follows: It is shown in Algebra that imaginary roots enter an equa- tion in pairs, so that if x :=^ a -\- fii (where i = V — 1) is a root, then x = a — /3i will be another root. To these roots correspond the product {x - a - /3i)(x - a + /3i) = (x - a)' + /S". By thus combining the imaginary factors the function I) will be divided into factors all of which are real, but some of which, in the case of imaginary roots, will be of the second degree. The assumed fraction corresponding to a pair of imaginary roots we place iu the form A + Bx 228 THE INTEGRAL CALCULUS. and then proceed to determine A and B as before by equa- tions of condition. We then divide the numerator A -\- Bx into the two parts A + Ba and B{x — a), the sum of which is ^ + ^^« Thus we have to integrate The first term of (a) is, by methods already developed, A -\- Ba , , ,.x — a ___ tan<-«-^-, and the second is iBlog{{x-ay + /3^). We therefore have, for the complete integral, r_A±Bx_ __ A + Ba .,.,,x-a + iBlog{{x-ay + /3^}+h. EXERCISES. rx + 3x\ r dx I. / — 7-! — —dx. 2. / —^ -. ^ X — 1 ^ X —1 The real factors of the denominator in Ex. 1 are (aj^ + l){x + !)(« — 1). We resolve the given fraction in the form A+^x , C D and find it equal to -yzh^ + ^a.i + ^ _ i • Then the integral is found a;' + l ^ x-{-l^ x-V a^ + l^aj + l^aj-l to be i log {x^ + 1) + log (aj5 - 1). The factors of the denominator in Ex. 2 are « — 1 and x'^-\-x-\-\ — (x + \f + h r dx r {x" + i)dx 3- e/ ^' + 1- 4. y ^3 _ 2^ _)_ 4- Note that a; + 2 is a factor of the denominator in (4). y INTEGRATION BY RATIONAL TRANSFORMATIONS. 229 /l37. Integration hy Parts, Let u and i; be any two functions of x. We have d(uv) __ dv du dx ^ dx dx By transposing and integrating we have fu'^^dx^uv-fvf^dx + h, (1) which is a general formula of the widest application, and should be thoroughly memorized by the student. It shows us that whenever the differential function to be integrated can be divided into two factors, one of which [-^-^A can be integrated by itself, the problem can be reduced to the inte- gration of some new expression [v—dxY The formula may be written and memorized in the simpler form / udv = uv — vdu, (2) it being understood that the expressions dv and du mean dif- ferentials with respect to the independent variable, whatever that may be. It does not follow that tlie new expression will be any easier to integrate than the original one; and when it is not, the method of integrating by parts will not lead us to the integral. The cases in which it is applicable can only be found by trial. The general rule embodied in the formulae (1) and (2) is this : Express the given differential as the product of one function into the differential of a second function, Tlien its integral will be the product of these tivo functions, minus the iiitegral of the second fimction into the differential of the first. 230 THE INTEGRAL CALCULUS. EXAMPLES AND EXERCISES IN INTEGRATION BY PARTS. 1. To integrate x cos xdx. We have cos xdx = d'sin x. Therefore in (2) we have u = x; v = sin x; and the formula becomes / X cos xdx = / xd'sin x = x sin ^ — / sin xdx = X sin X -\- cos X -j- h, which is the required expression, as we may readily prove by differentiation. Show in the same way that — 2. / X sin xdx = — x cos x + sin ^ + /^. 3. f X sec' xdx = X tan x — (what ?). 4. / X sin a; cos a;c?a: = — \x cos 2a: + :|^ sin '^x -{- lu 5. / log xdx = X log ^ — / ^t?-log a; = 2: log a; —■ .t + ^^• 6. The process in question may be applied any number of times in succession. For example, / x^ cos xdx = I x^d'^m x = x^ sin x ~ 2 I x sin xdx. Then, by integrating the last term by parts, which we have already done, / x"^ cos xdx = X* sin x -{- 2x cos ic — 2 sin a; + >^^« 7. In the same way, / x^ cos xdx = / x^d'sin x = x* sin x— 3 / x^ sin xdx; I x^ sin xdx = — / a:'^* cos x = — x"* cos x -\-2 / x cos 0:^0;. INTEGRATION BY RATIONAL TRANSFORMATIONS. 231 Then, by substitution, / X* cos xdx = {x" — Qx) sin x + (3a;' — 6) cos x + h, 8. In general, / x^ cos xdx = / 2:"f/'sin x=^ x'^ sin x — n j x"^^^ sin a:fZx; — / z**"^ sin icc^a: = / a;"~^(Z*cos a: = a:**~^ cos X — {n — \) x"^^^ cos aJcZo:; — / a:**"* cos a:c?a; =— a;**~*sin x +(;i — 2) / a:**~^ sin a:Ja;; / a; ** ~ ' sin a;rfa; = — a; " ~ ^ cos x -\-{n — d) x"^'*" cos a:r7a;. etc. etc. etc. Then, by successive substitution, we find, for the required integral, |a;"-n(?i-l)a;«-2+?z(?i-l)(?i-2)(?i-3).T"-*- . . .| sin a: + {wa:'*~^ — n{n — 1) {n — 2)x''-^ + . . . ( cos .r. 9. In the same way, show that / x^ sin xdx = |-x"+w(?i-l).r»»-2-7?(M-l)(7j-2)(?i-3)a:'*-*+. ..1 cosrr + {?ia;"~^ — 7/(/i — 1) (n — 2)a;**~^ +. . . ! sin x, .T" log xdx = j^^^J log a:fl?- {x ~ + *) = ^^-— log a; 1 px""^^ , a;" + ^ , x""^^ / dx = — - lOP^ X — 7 r-^TT^. 7l-\-lJ X 71 + 1 ° {?l + 1)' II. Cxe-'^dx-^ r\xd'(e-^') = - ?^% 1 Ce-'^dx. J J a ^ a aj Now, we have / e~"^''dx = . J a Hence I xc'^dx =? — 232 THE INTEGRAL CALCULUS, 12. To integrate x'^e'^dx when w is a positive integer, we proceed in the same way, and repeat the process until we re- duce the exponent of x to unity. Thus, x^e-'^dx = h - / x'^-h-'^dx. a aj Treating this last integral in the same way, and repeating the process, the integral becomes ^m^-ox ^^m-ig-ox 7n{m — l)x"'-^e- — etc. a a a' a 13. From the result of Ex. 5 show that y (log xydx = x{r -21 + 2) +h, where we put, for brevity, I = log x. 14. Show that, in general, if we put u^ =J{\og xy dx, then Un = xV^ — nu^_i'y and therefore, by successive substitution, u^ = x^V - ?^/'»-^ + n{n — l)/"-^ - . . . ± ^!) + A. 15. Deduce (m + 1) C Px'^dx = Pa;*" + ^ - A^ + WP. 16. Show that \lJPdx = Q, then / Px'^dx = Qx"" — n I Qx'^'-Hx. Also, if we have / Qdx - E; I Rdx = S, etc., then PPx^'dx = Qx"" — nRx^-^ + n{7i — l)^'*-^ - etc. INTEGRATION OF IRRATIONAL DIFFERENTIALS, 233 CHAPTER IV. INTEGRATION OF IRRATIONAL ALGEBRAIC DIFFERENTIALS. 138. When Fractional Powers of the Independent Vari- able enter mto the Exjjvessmi. In this case we may render tlie expression rational by reducing the exponents to their least common denominator, and equating the variable to a new variable raised to the power represented by this denominator. Example. If we have to integrate idXy then, the L. C. D. of the denominators of the exponents being 6, we substitute for x the new variable z determined by the equation X = z% which gives dx = 6z\lz, The differential expression now reduces to z* -\-l By division this reduces to />/« 4.S.1 ^\7i ^zdz , 6dz 6(2« -z' + z' + z'-z- l)dz + ^-^^ + ^-^y Integrating and replacing z by its equivalent, x', wc find / ^ dx = -x^ — -x^ + -rx^ + -x^ — -a;* — 6x* ^ ^ 4 o /« -f 3 log {x^+ 1) + C tm'-'^ x^ + h. l+a^r 7" ~5" ■^4'^ ^3*^ 2 234 THE INTEGRAL CALCULUS. If the fractional exponent belongs to a function of x of the first degree, that is, of the form ax + i, we apply the same method by substituting the new variable for the proper root of this function. Example. To integrate {a + hxfdx 1 + (« + Ix)' We put {a + Ix)^ = z; a -\- bx = z^; , 2zdz ax = —7—. The expression to be integrated now becomes dz \ z'+ir 2z'dz _hfi hil + z') ~^ b\ z'-\-: of which the integral is ~{z- tan(-^>;3 + ^) = ?. j (« + Z>a:)*-tan<-i>(« + &a;)*+7i \ , Integrate: EXERCISES. x^dx x^dx 1 — a;' , 2. r. 3. r dx. ■ X '1 + ^' * 1+a;*' 1 + a;* {a — x)^dx {a — x)^dx 1 + a- ^^^^^ 4- 1 _|_ a^ x' ^* 1 _ (^ _ a:f • {a - x)^ (x + c)* o (^ — ^Y 7 (^^ — «)*^^ 7. -^ ■ — ^-dx. 8. ^. {-rdx. 9. -^^ — -:. {x + cf (^ -oy 1 + (2:^; - a^ [o. — ^—-^ ^-rdz. II. 5^ — ! — ^-dx, i + {z - cy 1 + (^ + ay V X ^ x' -, 12. —dx, 13. -aa;. (a; — a)* — {x - -< (x - a)* + {z - -«)* 14. z fi — ^^^' INTEGRATION OF IRRATIONAL DIFFERENTIALS. 235 lo9. Cases when the Given Differential contains an Irrational Quantity of the Form Va -\- bx -{- cx^. It is a fundamental theorem of the Integral Calculus that if we put E = any quadratic function of x, then every ex- pression of the form F{x, VR)dx, {F{x, V R) being a rational function of x and \/R), admits of integration in terms of algebraic, logarithmic, trigonometric or circular functions. But if R contains terms of the third or any higher order in x, then the integral can, in general, be expressed only in terms of certain higher transcendent func- tions know as elliptic and Abelian functions. We have three cases of a quadratic function of x. First Case : c positive. If c is positive, we may render the expression rational by substituting for x the variable z, de- termined by the equation Va + bx + cx^ = Vc(x + z)) . • . a '\' Ix -\' cx^ = cx^ -f- 2cxz + cz'. This gives X = ^^£; (a) , ^ a — hz + cz^ , ,-. — 2cz ^ ' By substituting the values given by (a), (b) and (6') for the radical, x, and dx, the expression to be integrated will become rational. Second Case : a positive and o negative. If the term in a;* is negative while a is positive, we put Va'\- bx — cx^ = Va + xz, Wq thus derivQ x = — ^-— ; (a) z ~j~ c 236 THE INTEOBAL CALCULUS, 2(Vaz' -Vac- Iz) ^ (z + c) ^ __ Vaz"^ — Vac — Iz , Va + hx — cx^ = —■ . (c z + c ^ The substitution of these expressions will render the equa tion to be integrated rational. Third Case : a and c loth negative. If the extreme term of the trinomial are both negative, we find the roots of th^ quadratic equation — a -\- hx — cx^ = (), which roots we call a and ^. We then have — a -{- Ix — cx"^ =: c{a -— x) {x — /3), and we introduce the new variable z by the condition V— a -{-bx — cx^ == Vc{a — x) {x — fi) ~ Vc{x — a)z, I.' I. ' az' + /3 . which gives X = ^ ' .. -; (a '^^- (2' + l)' ' ^^ z -\-l substitutions which will render the equation rational. {0 140. We have already integrated one expression of th^ dx form just considered, namely, - without ration Va -\-bx -{• cx^ alization. There is yet another expression which admits o being integrated by a very simple transformation, namely, d0. '^' r Var^ -\-hr — 1 This is the polar equation of the orbit of a planet arounc the sun. To integrate it directly, we put INTEGRATION OF IRRATIONAL DIFFERENTIALS. 237 1 , dx X = —: dr = -.. r x^ ffe thus reduce the expression to — dx /- Va -\-hx — x* Proceeding as in §133, Case I., we find the value of the ntegral to be dr , ,. 2x — h , ,, 2 — br = COS^"^^ — = COS^"*^^ r Var' + br - 1 V4:a + b' r Via + b' Thus, 6-7r = cos(-^> - 4^-^ , r V4:a + b' re being an arbitrary constant. Hence 2-br rVia + b^ cos {6 — tt). Solving with respect to r, we have, for the polar equation of the required curve, 2 ^~ b + |/(4a + b') cos {0 - Tt)' ^^^ This can be readily shown to represent an ellipse. The polar equation of the ellipse is, when the major axis is taken as the base-line and the focus as the pole, _ a{l - e ') 2 ^ ~ 1 + 6 cos 6* "" 2 , 2e -Pi n + -Pi n cos e a(\ — e) a{l — e^) Comparing with (a), we have 2 a{l — e"^) = J- = parameter of ellipse =p; or e = -^-^ — j——^ = eccentricity of ellipse. 238 THE INTEGRAL CALCXTLXI8. Irrational Binomial Forms. 141. General Theory, An irrational binomial differen- tial is one in the form {a + Ix'^Yx'^dx, (1) in which m and n are integers positive or negative, while p is fractional. To find how and when such a form may be reduced to a rational one, let the fraction p^ reduced to its lowest terms, b€ T — ; and let us put 1 y^{a + hx^)\ (2) This will give, when raised to the rth power and multi- plied by x'^dx, {a + Ix'^Yx'^dx — x'^y^'dz. (3] We readily find, from (2), hx"^ = y" ^ a; {a] dx — -I — --^; x'^y^dx = --a;'^~'* + ^y''+*~*J2/; or, substituting for x its value from (a). This last differential will be rational if -^- is an in- n teger, which will be the case if — ^i— is an integer. We shall call this Case I. To find another case when the integral may be rationalized, let us transform the given differential (1) by dividing the bi- nomial by x"" and multiplying the factor outside of it by 2;"^, which will leave its value unchanged. It will then be INTEORATION OF IRRATIONAL DIFFERENTIALS. 239 {h-\-ax-''Yx'^ + ''^dx, (1') which is another differential of the same form in which n is changed into — n and m into 7n + np. Hence, by Case I., this form can be made rational whenever — ^ "^ is an n integer; that is, when h /^ is such. We have, therefore, two cases of integrability, namely: Case I. : when — '^^^— = an inteerer. n ° Case II. : when — — 1- jt? = an integer. Remark. It will be seen that all differentials of the form r {a + ix^Yx^dx must belong to one of these classes, because — -^ — IS an integer when m is odd, and — -^ h ^ is sucn when 771 is even. In this statement we assume r to be odd, because if it is even the original expression is rational. 143. If, in Case I., the integer is + 1, that is, if m + 1 = n, then the expression can be integrated immediately. For (4) then becomes the integral of which, after replacing y by its value in (2), be- comes /(. + ..«)^.--^.= (|+i^l;i' + . (5) Again, if the integer in Case II. is — 1, we have TTi -{- 1 -{- np =z -^ riy or m + wjt? = — n — 1. The expression (1) reduced to the form (!') will then be {h + ax-''Yx-''~Hx =-(/> + ax-'^Y —d{b + ax-""), 240 THE mTEQEAL CALCVLU8. which is immediately integrable, and gives by simple reduc- tions /(«+ *.T--— '^- = ^ - ;^'ii^W (6) 143. Forms of Reduction of Irrational Binomials, Al- though the integrable forms can be integrated by the substi- tution (2), it will, in most cases, be more convenient to ap- ply a system of transformations by which the integrals can be reduced to one of the forms just considered. The objects of these transformations are: I. To replace m by m + ?^ or m — n\ II. To replace ;:> by ;? + 1 or ^ — 1. 144. Firstly, to replace m by m + n. Let us write, for brevity, Xe a + Ix"", which will give dX = Ijnx '^~'^dXy and the given differential will be X^x'^dXy which, again, is equal to rf>m — n + l ^m — n + l X^dX=f-r——rd{X^-^''). bn bn{p+ 1) Integrating by parts, we have J ^'^^ '^^ = MP + 1) ~ Mi+T)J ^'''--^-- («) Since X^ + ^ = X^{a + W) = aX^ + hX^x^, the last integral in the above equation is the same as a r X^x '^-''dx-\-h C X^x'^dx, of which the second integral is the same as the original one. Making this substitution in (a), and then solving the equa- tNTEORATION OF IRRATIONAL DIFFERENTIALS. 241 tion so as to obtain the value of / X^x^^dx, we find I X^x'^dx = 77 ■ — T - j} — ■ — -YTx / X^x'^-^dx. {A) Thus the given integral is made to depend upon another in which the exponent of x is changed from m to m. — u. By reversing the equation we make the given integral depend on one in which the exponent is increased by 7i, To do this we change 7)i into m -{- 7i all through the equation (.4), thus getting / X'^x'^^'^dx^-. , — ^-^-T7 — \ V , nx / X^x'^dx, Solving with respect to the last integral, we find / X^x'^dx = — —-r ^^-^-7^ — T TV-- / X^x'^^^'dx. (B) J a{m + 1) a\in + 1) ^z ^ ' The repeated application of i^A) and (/>) enables us to make the value of the given integral depend upon other in- tegrals of the same form, in which m is replaced by m + n\ m + 2;^; etc.; or by m — n\ m — 2n; etc. 145. Next, to obtain forms in which /; is increased or diminished by unity, we express the given differential in the form / />.m + 1 \ XPx'^dx = X^ \m + 1/ Integrating by parts and substituting for dX its value bnx''-^dx, we have p X^x '^ + ^ / X^x^dx = , , J 7;i + 1 l^m + \lx. (i) Now, we have x^( X-a) Xx"^ b ~ b ax"^ b ' 242 THE INTEGRAL CALCULUS, and therefore, by multiplying by X^^'^dx, Substituting this value in (J), and solving as before with respect to / X^x'^dx, we shall find Cx^x-^dx = ^''^"^' , + ^^-T Cx^-^x-^dx, (C) in which j9 is diminished by unity. If we write jt? + 1 ioT p in this equation, the last integral will become the given one. Doing this, and then solving with respect to the last integral, we find X^x'^dx = -. — rr + ^^i -,-^ I X^+^x'^dxJD) By the repeated application of the formula ((7) or (D) we change p into j9 — 1, j9 — 2, j9 — 3, etc., or 'p into jt? + 1, ^ + 2, ;? + 3, etc. ^ --U ^^•'— 1 ri46^ To see the effect of these transformations, let us / put, in the criteria of Cases I. and II., § 141: I. ' — = t, an mteger. II. — — hi> = ^'> an integer. Then when we apply formula i^A) or {E), since we replace m by 7w — ?i or m + /I, we have, for the new integers: n II. — - — \-p = i' If 1. It is also clear that by ((7) and (p) we change II. by unity. Thus, every time we apply formulae (^), (^), ((7) or {U) we change one or both of these integers by unity, so that we may bring them to the values unity treated in § 142. INTEGRATION OF IRRATIONAL DIFFERENTIALS, 243 147. Case of Failure in this Reductioji. If, in an integral of Case II., i' is positive, we cannot change it from zero to — 1 by the formula {A) or ((7), because, when — — \- p =z 0, ft we have m -{- 1 + np = 0, and the denominators in {A) and {C) then vanish. In this case we have to apply the substitution of § 141, without try- ing to reduce the integral farther. EXAMPLES AND EXERCISES. 1. To integrate {a' ± x')^dx. We see that if we diminish the exponent i by unity, we shall reduce the integral to a known elementary form of § 132. So we apply {0), putting m = 0; 71 = 2; p = i; a = a^; Z> = ± 1. Then (C) becomes We therefore have, from § 132, f{a' + x'Ydx = I I x{a^ + a;')» + «' log ^{x+{a'+ x')^) } ; /"(a* - x')\lx = ^ j x{a' - a;')* + «' sin*-" ^ + A } • Deduce the following equations: 2. J {c' - x') xdx =7i - i{c' - x'). ,.fic'+xr.ax =, + (^)p r dx ^ j^ __ ( c'^x^)* ^" •' xV + x')* <-"a; 244 THE INTEGBAL CALCULUS. Here apply formula ((7); in the following {A). 8. Hi - x'fx'dx ^h-{^ + ^^{1 - x'f. 9. To reduce and integrate (1 + x'^)^x^dx. Here t/i — 3; 71 = 2; i? = i; m + 1 = 4 = 2w<. We can therefore reduce the form to Case I. hy a transformation of m into m — n, for which we may use either {a) or {A) of § 144. Using (a), we have f{l + x'f x^dx = (1+|!)!£' _ |y*(i + aj^)i xdx. The last integral can be immediately found, and gives for the required integral i(l + «2) V - t\(1 + x')^. {a) Using (J.), we should find y (1 + oj^)* x^dx = (1 + ^f (J' - ^5), © a form to which (a) can be immediately reduced. The student will remark that the form {a) is reduced to (A) because in the former the exponent of X is increased by 1, which often makes the integration inconvenient. But when this increase of p does not in- terfere with the integration, we may use {a) more easily than (A), 10. To reduce and integrate (1 + x^)^x^dx. Applying (A), we find Al + x'f a^dx = ^^ + ^'^^ _ ^y*(i + a.2)i ^^^ A second application repeats the form (b) above, thus giving 11. Reduce and integrate (1 + x'')^x'^dx, where m is any positive odd integer, and show that INTEGRATION OF IRRATIONAL FUNCTIONS. 245 /'(I + x')^x'^ dx =(i+^')'(; ^__ (m-l)a: ^-^ (7?^-l)(m-3)a;^ ' ,m+2 (m + 2)7?t "^ (m+2)7w(m-2) Remark. Where the student is writing a series of transformations he will find it convenient to put single symbols for the integral expressions which repeat themselves. Thus: rx^x'"dx={l); Cx^x'^-''dx^{^)\ etc. Thus the equations of reduction in the present example may be written ^^)- m + 2 ^TTS^^^' m m etc. etc. 12. Deduce the result - § I «(a* + «')* + a' log t7 (a; + l/^M^) [ . 246 THE INTEGRAL CALCULUS. r CHAPTER V. INTEGRATION OF TRANSCENDENT FUNCTIONS. When the given differential contains trigonometric or other transcendent functions of the variable more complex than the simple forms treated in Chapter II., no general method of reduction can be applied. Each case must therefore be studied for itself. 148. To find the integrals / 6""* cos nxdx and / e"^ sin nxdx. (1) Since we have m ^ ' \ml the integration by parts of these two expressions gives ^mx ^Qg ^xdx = h — / ^"""^ sin nxdx\ m mj /«.^ • 7 ^"*^ sin nx n n ^^ , ^mx gjjj fixdx = / e^'^ cos nxdx, m mJ Solving these equations with respect to the two integrals which they contain, we find / -.^ , e"*^(m cos nx -\- n sm nx\ e^"" cos nxdx = — ^^ „ . ' ^; m + 71 /^^ . , e"^(m sin nx — n cos nx) e*^^ sm nxdx = — ^^ 5— — r -: m" + n^ (2) which are the required values. Kemark. These integrals can also be obtained by substi- tuting for the sine and cosine their expressions in terms of imaginary exponentials, namely, INTEGRATION OF TRANSCENDENT FUNCTIONS. 247 — ^turt 1^ ^ — nxi 2 cos iix =: e"^ -{- e 2 sin 7ix = -ie"^ — e"**^), and then integrating according to the method of § 134. The student should thus deduce the form (2) as an exercise. 149. Integration of sin*^ x cos** xdx. This form is readily reducible to that of a binomial, and that in two ways. Since we have cos xdx = d'sin x, cos X = {1 — sin' x)\ we see that the integral may be written in the fonn /< /( (1 — sin' x) 2 sin"* xd'sin x; or, putting y e sin x, {l-f)^y-^dy. (3) By putting z = cos x we should have, in the same way, -y*(l - z') ^z'^dz, (4) which is still of the same form, and is always integrable by the methods already developed in Chapter IV. If either 7)1 or 7i is a positive odd integer, then by develop- ing the binomial in (3) or (4) by the binomial theorem we shall reduce the expression to a series containing only posi- tive or negative powers of x, which is easily integrable. We can also, in any case, transform the integral so as to in- crease or diminish either of the exponents m and ?i by steps of two units at a time, as follows: sin"* x CDS'* xdx ■■ = cos"~'a:a sm m-f 1^ 7/i-f 1 Then, integrating by parts, we have 248 THE INI EQUAL CALCULUS, + ^\ r Ain^^ + ^-g cos**"^ a;^a;. (5) m -\rlj ^ ' / sin"" X cos*" xdx _ cos*"-^ X sin"' + ^2; Because sin "* + ^ a; = sin"* x{l ~ cos^ rz;), the last term is equivalent to — - / sin^ X cos**~^ X / sin"^ x cos** xdx, m-\-lJ m -\- IJ The last of these factors is the original integral. Trans- posing the term containing it, we find {m + n) / sin"" x cos^ xdx = sin "^ + ^ a; cos '^~^x -\- {n — 1) I sin*^ X cos''^^xdXy (6) in which the exponent of cos x is diminished by 2. We may in a similar way place the given differential in the form • «. 1 7C0S** + ^a; — sm'^""^ xd -— -, n -\-l and then, proceeding as before, we shall find (m + ^^) / sin"* X cos** xdx = — sin *" ~ ^ a: cos ** + ^ a; + (m — 1) y sin ^-^ X cos" xdx, (7) thus diminishing the exponent of sin x by 2. By reversing these two equations we get forms in which the exponents are increased by 2. Writing n -{- 2 for n in the first, and m + 2 for m in the second, we find {n + 1) / sin"* X cos** xdx = — sin *" + ^ cc cos ** + ^ a; + {ni + n + 2) / sin"* x 008** + ^ xdx; (8) (m + 1) / sin"* X cos" xdx = sin "* + ^ :?; cos " "♦" ^ a; + {7n + n + 2) y sin "* + ^ a; cos" xdx, (9) INTEGRATION OF TIUNSCENDENT FUNCTIONS. 249 150. Special cases of I sin"* x cos" xdx. If m is zero and u is positive, we derive, from (6), /* n 7 sin a; cos"-* a; , ?i — 1 /» „ . , / cos" a;rfa; = \- / cos"-^ xdx\ J n n J y„ ., , sin a; cos"-^ a; . 7i — 3 /» , , cos"~^ xdx = ^ / cos"-* xdx: 71 — 2 n — 2j etc. etc. etc. The integral to be found will thus become that of cos xdx when 71 is odd, and that of dx, or x itself, when 7i is even. The given integral is then found by successive substitution. We find in the same way, from (7), >m /. ^ , cosirsm'"-*^: , 7)i — 1 p . ^ ^ , sm"* xdx = / sm*"-^a:a:c: 771 771 J 7n — 3 (11) /• «. o 7 cosa:sin'"-^a: , 7n — 6 p . ^, , sm'^-^xdx — X I sin*"-*a;c?a;: 771 — 2 m — 2J etc. etc. etc. From (8) and (9) we derive similar forms applicable to the case of negative exponents. EXERCISES. 1. / sin^ X cos' xdx. A71S. \ cos' x — \ cos^ x. 2. / sin' X cos' xdx. Ans, \ sin' x — \ sin^a;. /cos' xdx . 3 sin' x — 1 sm* x 3 sm' x 4. y sin' X tan' xdx. 5. / 7 — ^—dx. 6. A'*' sin dydy. 7. / e'^^'' cos (x + b) dx. 8. / e'*' sin y cos ydy. 9. / e'^ cos' (y + a) ^y. 10. Derive the formulae of reduction /tan *** ^ ^ ar /* tan"* xdx = -— — — / tan "* "^ ^ xdx: 771 + 1 J 250 THE INTEGRAL GALCULU8. and hence / tan** xdx = / tan"""^ xdx. These equatious may be obtained independently by putting tan" x = tan " - 2 ^(sec2 a; — 1); or they may be derived from (5). Hence derive the integrals: 11./ tan' xdx = i tan'' x — log c sec x. (CI § 127) 12. / tan* xdx = ^ tan* x — tan x -{- x -{-Ji. 13. For all odd positive integral values of n, A «' 7 tan**-^:?; tan'*-^^; , . , / tan** xdx — — \- , . , ±\ogcsQQ, x. J n — 1 n — 3 14. When 71 is positive, integral and an even number, /», „ , tan~~^a; tsin''~^x . / tan** xdx = h • • • ± tan x ± x, J n — 1 71 — 3 15. When the exponent is integral, odd and negative, /• „ , cot'*-^^: , cot**-^a; , , / tan ~ ** xdx = . . . ± loer c sm x. J 71 — 1 71 — 3 ° 16. When the exponent is integral, even and negative, /„ , cot"~^:r , cot**~^a; tan~"'xdx — — . . . ± cot xl^ x, 71 — 1 71 — 3 /> . 5 , cos .t/ . , ,4.3 , 4-2\ 1 7. / sm xdx = —- 1 sm x ~\- ~ sm x + 57^ )• 18. I mn^ xdx 1 sm ''^ + T sm x -\- — sm x\ 19. / sin** x cos" xdx = — - / sin** cos 22; sin** ~^ 22: . 71-1 /> . ^ „ ^ , = T^m \- .^n • / sm**" ^ %xdx. cosxf . , . 5 . , , 5*3 . \ , 5-82; INTEGRATION OF TRANSCENDENT FUNCTIONS. 251 151. 7h inteqrate ., . .. ; — ^ j—^du. "^ m sin X + n cos x Dividing both terms of the fraction by cos' x, noticing that = J- tan X and writing t E tan x, we find cos X (It -J^iftT^n^' . (1^) The integral is known to be (§ 128) 7)1)1 n so that we have dx 1 , , ,^m = f , ■ , 1 , . = — taii<-'> - tan X + h, (13) J 711 sm X -j- n cos x mil n < > \ , ')t or tan x = — tan vi)i(tc — 70. 153. I)itcrjratio)i of ^ a -\- b cos ?/* We reduce this form to the preceding one by the following trigonometric substitution: a = a {cos'' iy + sin' ^i/); b cos y — Z>(cos' \y — sin' ^y); by which the expression reduces to the form O r ^^(^.^/) MAX J (a - b) sin' iy + (rt + ^) cos' iy' ^ ^ which is that just integrated, when we put m = l^a — ^; We therefore have r _,f^ = -^=1= tan<-' /^ tan ^y + A. (15) 252 TEE INTEGRAL CALCULUS. 153. Ifj, in the form of §151, m^ and u"^ have opposite signs, or if in § 152 we have d > a, imaginary quantities will enter into the integrals, although the latter are real. If, in the first form, the denominator is m'^ sin"" x — ii" cos^ x, we shall have, instead of (12), the integral r dt _ _1 /* (U 1 /» dt ^ J in'f — n^ 2inJ mt — n %nj mt -\- n^^ ' = — ^ log -— — ^ \- h. Hence, corresponding to (13), we have the result /dx 1 , m tan x + n , ^ ,.^. 7^?'^ sin'^ X — if cos'' X 27nn m tan x — n If, now, in § 152, 1) > «, we write (14) in the form d'^y V( {b — a) sin' t}?/ — (a -{- h) cos' ^y' and instead of (15) we have the result / '^y -,7^_|_ ^ _ log Vb-ai^n^y-\-Vb-\- a^ ^.^^^ a+b cos y Vb'-a' l^^-a tan iy- Vb + a 154. Integration of sin mx cos 7ixdx, Every form of this kind is readily integrated by substitut- ing for the products of sines and cosines their expressions in sines and cosines of the sums and differences of the angles. We have, by Trigonometry, sin mx cos nx = ^ sin {m -\- n)x -{- i sin {m — n)x. Hence , cos (m-\- n)x cos im — n)x , , sm mx cos nxdx = -r^^ — ; — ^ -r ^- — h ti 2(vi -\- 71) 2{m — 71) We find in the same way sin (fn + n)x , sin (m — 7i)x , , cos mx cos 7ixdx = —^ — -^ — ^ — ^77^^ ^ — [- h 2{m + 71) 2(7n — 71) , sin (ni 4- n)x , sin hn — 7i)x , , sm mx sm Tixdx = ttt — . — r - H kt r — r ^« 2(771 + ??) 2{7n — 71) f I I INTEO RATION OF TRANSCENDENT FUNCTIONS. 253 155. Integration hif Development in Series. When the given derived function can be developed in a convergent series, we may find its integral by integrating each term of the series. Of course the integral will then be in the form of a series. The development of many known functions may thus be obtained. EXAMPLKS AND EXERCISES. I. We may find / sin xdx as follows: We know that x^ x^ x'' sin a: = a;- -, + -,--, + ...; ■/ sin xdx = h + ^ ~ |] ^~ ll ~ ^^^'' which we recognize as the development of — cos x with an arbitrary constant 7i + 1 added to it. Of course we may find / cos xdx in the same way. dx 2. To integrate 1+x {1 + x)-' = 1 -^ X + x' -- x' + ■A dx x' x' x' l+-x = ^' + '^-2+3--I + -"' (^) /dx — - — =z log (1 + x). Hence (a) is the development of log (l-f- x), when we put h = log 1 = 0. The series (a) is divergent when a: > 1. In this case we may form the development by the binomial theorem in de- scending powers of x, thus: {x + l)-' = x-'-x-^ + x-^-x-^+ . . . . Hence we derive, when xy 1, 254 THE INTEGRAL CALCULtTS. The arbitrary constant is zero because, when x is infinite, log {x-{-l) — log X is infinitesimal. ^. To find / — '_^- -^ — sin ^~ ^^ x in a series. J Vl-x' Hence r dx . , ,, , 1 a;^ , 1-3 :?:' , I'S'S x' , y7r^^ = ^^^^^'^^^+2-3 +2^4-5 +2^-7 +•••• The arbitrary constant is zero by the condition sin^~^^ = 0. This series could be used for computing zr by putting x = \, because \ = sin 30° = sin — . But its convergence would be much slower than that of some other series which give the value of 7t. dx Vl + x'' rive the expansion 4. From the equation / — = log {x-\- Vl~{- x"") de- 1 x' , 1-3 x' 1-3-5 x' log{x+Vl+x^)=x^-,^-+-.~^^:^^.- + .... dx 5. By expanding , = ^-tan^"^^ x, derive J. ~|~ X tan<-» x = x-ix' + ix'-^x'+ . ... Derive: r r ^^ -7. , 1 «' , 1-3 a;' 1-3-5 x" DEFINITE INTEGRALS. 255 CHAPTER VI. OF DEFINITE INTEGRALS. 156, In the Differential Calculus the increment of a variable has been defined as the difference between two values of that variable. Let us then suppose w to represent any variable quantity whatever, and let us suppose n to pass through the series of values Then we shall have ^21, = u^ -^o; ^u, = u^ -^^; ^u, = ^K ^-^K; Taking the sum of all these equations, we have Jw, + ^21, + An^ + . . . + z/?^n-i = «^n - n,'y That is, the diffei^ence hettveen the two extreme values of a variable is equal to the sum of all the successive increjuents hy which it passes from oue of these values to the other. The same proposition may be shown graphically by sup- posing the variable to represent the distance* from the left- hand end of a line to any point upon the line. The differ- ence between the lengths Au^ and A^t^ is evidently Aii^ + jdu^ + . . , + Au^. I I Auq I AU| I AKa I Ana I ^^4 | _ , ^ X Uq ti, lia «3 t*4 lift Since the proposition is true how small soever the incre- ments, it remains true when they are infinitesimal. 256 THE INTEGRAL CALCULUS. Fig. 47. 157. Differential of an Area» Let P^PP' beany curve whatever, and let us investigate the differential of the area swept over by the ordinate XP. Let us supjK)se the foot of the ordinate to start from the position X^, and move to the position X. During this motion XP sweeps over the area X^P^PX, the mamitude of which will depend upon the distance OX, and will therefore be a function of x, which represents this distance. Let us put u = the area swept over; y = the ordinate XP. Then, if we assign to x the increment XX\ the corre- sponding increment of the area will be XPP' X, Let us call ?/' the new ordinate X'P\ It it evident that we may always take the increment XX' ~ Ax so small that the area XPP' X' shall be greater than yAx and less^^an y' Ax or vice versa. That is, if y' > y, as in the figure, we shall have yAx < Alt < y'Ax, - Now, when Ax approaches the limit zero, y' will approach y as its limit, so that the two extremes of this inequality yAx and y'Ax will approach equality. Hence, at the limit, du = ydx, (1) That is, theoHa w is such a function of x that its differen- tial is ydx, a7id its derivative loitli respect to x is y. From this it folloiirs by integration that :: / ydx -\- h (2) is a general expression for the value of the area from any initial ordinate, as X^P^ to the terminal ordinate XP, DEFINITE INTEGRALS, 257 158. The Conception of a Definit^ Integral, Suppose the area X^ P^ FX=u to be divided up into elementary areas, as in the figure. This area will then be made up of the sum of the areas of all the elementary rect- angles, plus that of the o^ triangles at the top of the several rectangles. That is, using the notation of § 156, we have T being the sum of the areas of the triangles; or, using the notation of sums, u = 2 y,Ax, + T, Xo Fia. 48. Now, let each of the inoretiients /dxi become infinitesimal. Then each of the small triangles which make up T will be- come an infinitesimal dPthe second order, and their sum T will become an infinitesimal of the first order. We may therefore write, for the area u, Xr=OX « = OX ti = lim. 2 yAx — 2 ydx. x=OXo x=OXo That is, 2c is the limit of the sum of all the infinitesimal products ydx, as the foot of the ordinate XF moves from X^ to Xby infinitesimal steps each equal to dx. Such a sum of an infinite number of infinitesimal products is called a definite integral. The extreme values of the independent variable x, namely, OX^^x^ and OX^ x^, are called the limits of integration. The infinitesimal increments ydx, whose sum makes up the definite integral, are called its elements. 17 258 THE INTEGBAL CALCULUS. 159. Fundamental Theokem. The definite integral of a conti7iuous function is equal to the difference letween the values of the indefinite integral corresponding to the limits of integration. To show this let us write (/){x) for y, and let us pufc^ for the indefinite integral, ^(p{x)dx = F{x) + c. /' Now, as already shown, this is a general expression for the area swept over by the ordinate y = (p{x), when counted from any arbitrary point determined by the constant c. If we count the area from X^Pq, the area will be zero when x = x^; that is, we must have F{x,) + c = 0, which gives c = — F(x^). If we call x^ the value of x at X, we shall have u = Area X,P,PX = F{x,) + c = F{x^) ~ F{xX (3) which was to be proved. We therefore have a double conception of a definite in- tegral, namely: (1) As a sum of infinitesimal products; (2) As the difference between two values of an indefinite integral; and it will be noticed that the identity of these two concep- tions rests on the theorem just enunciated. Notation. The definite integral is expressed in the same form as the indefinite integral, except that the limits of inte- gration are inserted after the sign / above and below the line; thus, (p{x)dx means the integral of (p{x)dx taken between the limits x^ and iCj, the first being the initial and the second the terminal limit. r DEFINITE INTEGRALS. 259 Example of the Identity of the Two Coiiceptions of a Defi- ite IntegraL The double conception of a definite integral jiist reached is of fundamental importance, and may be i'lirther illustrated analytically. To take the simplest possible ise, consider the definite integral „ adx, a being a constant. By definition this means the sum of all the products adx + adx -\- adx + . . . , as X increases from x^ to x^. The sum of all the dx^a must 1)0 equal to x^ — x^ (§ 15G). Hence (i{dx -\- dx -^ dx -\- dx -\- , , ,) =1 a{x^ — x^). But we have for the indefinite integral /« adx = ax; and the definite integral is therefore, by the theorem, ax^ — ax^ or a{x^ — x^), as before. 160. Differentiatio7i of a Definite Integral with respect to its Limits. — Because the definite integral / ydx = ti means the sum of all the products ydx as x increases by infinitesimal increments from the lower limit x^ to the upper limit x^, or u = y/lx + y'dx + y'^dx + • • • + ^"V/a;, therefore, assigning an increment dx^ to the terminal limit x^ will add the infinitesimal increment y^dx^ to u (see Fig. 48). That is, we shall have du = y.dx^y or ~=y^ = (P{x^). (4) In the same way, increasing the initial limit x^ by dx^ will take away from the sum the infinitesimal product y^dx^, so 260 THE INTEGRAL CALCULUS. that we shall have ^^ = -2/. = -0K)- (5) The equations (4) and (5) give us the derivatives of the definite integral u = I (p{x) . dx with respect to its limits x^ and x^. 161, Exa7nples a7id Exercises in fijiding Definite Inte- grals. The fundamental theorem gives the following rule for form- ing definite integrals: 1. Form the indefinite integral. 2. Substitute for the variable with respect to which we inte- grate, firstly, the upper U7nit of integration; secondly, the lower limit. 3. S^ibtract the second result from the first. The difference tuill be the 7^equired definite integral. 1. f \^dx =z ix,' - ix;. 2. I xdx — ^{V — a'). 3. / xdx = -J. 4. r sin xdx = — cos TT + cos = 2. 5. / COS xdx — sin \7i, 6. / azdz — \a(a^ — Z>'). 7. / sin 2xdx. 8. / cos 2xdx. 9. / sin* a;6?ir. 10. / cos* xdx. II. / x^xTixdx. 12. / 2; cos 2;^^;. 13. / ^'^ sin 2;J;2;. 14. I z^ cos ^j^Z^ DEFINITE INTEGRALS. 261 r z" cos "Zzdz. 1 6. / z" «/o z ' --. l8. / 7i;2V;2. / cos a;^a;. 22. /^ sin xdx. p+ ^ dz /.+ 1 \- b ^l - X 25. / (:c — a)dx. 26. / ?/^v. t/ a - b J a - X ^'^' fi + x ^'^~ ^^'^^' ^^' S"^ '^^ -a)(x- c)dx. 31. y ^ siu a^'ef:z:. 32. A cos (« + a:)c?a;. 2,2i' Deduce / cos (.?; + 7j)dx = sin 2y. J- y 34. Show that j^/{x)dx = - ry{j^dx. 35. Deduce / c~^df/ = l. 36. Deduce / c~''^dy:=^—. Jo a 37. Deduce / e^dy = 1. e/— 00 38. Deduce / ^ - Vydy = h 39. Deduce f -— — , = tt. t/_OD 1+2; T^ , p^ dz n 40. Deduce / — = — . 41. Deduce / - = n. J- a Va' - z' 262 THE INTEGRAL CALCULUS. 163. Failure of the Method when the Function becomes Infinite. It is to be noted that the equiyalence of the two conceptions of a definite integral does not necessarily hold true unless the function y or ct)(x) is continuous and finite between the limits of integration. As an example of the failure of this condition, consider the function 1 y {x - af the curve representing which is shown in the margin. The indefinite integral is = / ydx o Fig. 49. To find the value of this integral between two such limits as and k, k being any quantity OM less than a, we put X = and x = k, and take the difference as usual. Thus 1 1^_ k_ a{a — u = (5) k a a{a — k)' Now, if we suppose k to approach a as its limit, so that a — k shall become infinitesimal, then the area ?/ will increase without limit, as we readily see from the figure as well as by the formula. But suppose k > a; for example, k = 2a, Then the theorem would give __1_1___2 /o " a a a' a negative finite quantity; whereas, in reality, the area is an infinite quantity. The theorem fails because, when x = a, y becomes infinite, so that ydx is not then necessarily an infinitesimal, as is pre- supposed in the demonstration. i DEFINITE INTEGRALS. 263 163, Change of Variable i7i Definite Integrals. When, in order to integrate an expression, we introduce a new vari- able, we must assign to the limits of integration the values of the new variable which correspond to the limiting values of the old one. Some examples will make this clear. Ex. 1. Let the definite integral be rdx a -\' X Proceeding in the usual way, we find the indefinite integral to be log {a + 2;), whence we conclude Ji = log 2a — log a = log 2. a ~\~ X But suppose that we transformed the integral by putting y^a-{-x; dy = dx. Since, at the lower limit, x = 0, we must then have y = a for this limit, and when, at the upper limit, x = a, we have y = 2a. Hence the transformed integral is rdy which we find to have the same value, log 2. Ex. 2. u = / ^ sin x{l — cos x)dx. We may write the indefinite integral in the form / sin xdx + / cos xd{cos x). In the first term x is still the independent variable. But, as the second is written, cos x is the independent variable. Now, for a; = 0, cos x = 1; and for a; = — , cos x = 0, Hence, writing y for cos x, the value of u is u = / ^ sin xdx -f- / ydy = 1 — ^ = i. 264 THE INTEGRAL CALCULUS. Eemark. The variable with respect to which the integra- tion is performed always disappears from the definite integral, which is a function of the limits of integration, and of any quantities which may enter into the differential expression. Hence we may change the symbol of the variable at pleasure without changing the integral. Thus whatever be the form of the function 0, or the original meaning of the symbols x and yy we shall always have r (p{x)dx = f (l>{y)dy = f (p{y + a)dy, etc. t/a t/a t/0 164. Subdivision of a Definite IntegraL The following definitions come into use here: 1. An even function of :r is a function whose value remains unchanged when x changes its sign. 2. An odd function of x is one which retains the same absolute value with the opposite sign when x changes its sign. As examples: cos x is an even, sin x an odd, function. Any function of x^ is even; the product of any even func- tion into x is odd. It is evident, from the nature and formation of a definite integral, that if we have a sum of such integrals, Jr (t){x)dx + r (f}{x)dx + / (f)(x)dx +... + / {x)dx-{- r'' (p{x)dx = 2 f''cf){x)dx. J- a J- a c/o t/o Theorem II. If(p{^) is an odd function of x, tlicn^ 'what- ever he a, (p(x)dx = 0. L For in this case each element 0(— x)dx will be the negative of the element (p{x)dx, and thus the positive and negative elements will cancel each other. EXERCISES. Show that r e-'^x'^dx = C flog -j dz. Substitute x = \og -. 2. Show that whatever be the function 0, we have / 0(sin z)dz = I 0(cos xdx). As an example of this theorem. i'^a + b cos** x^ pi^'a + b sin** x b sin** x p^^a + COS" x^ _ n^^a -\- t/o a — b cos** X Jo a — dx. The truth of this theorem may be seen by showing that to each ele- ment of the one integral corresponds an equal element of the other. Draw two quadrants; draw a sine in one and an equal cosine in the other. Any function of the sine is equal to the corresponding function of the cosine. We may fill one quadrant up with sines and the other with cosines equal to those sines, and then the two integrals will be made up of equal elements. 266 THE INTEGRAL CALCULUS. To express this proof analytically, we replace ic by a new variable y = ^TT — w, which gives sin x = cos y; dx = ~ dy; and then we invert the limits of the transformed integral, and change y into x in accordance with the remark of the last article. IT ^ 3. Show that I /(sin x)dx = 2 y ^/(sin x)dx. 4. Show that / 0(siii x) cos xdx = 0. 5. Show that if be an odd function, then / 0(cos x)dx — 0. 6. Show that the product of two like functions, odd or even, is an even function, and that the product of an even and an odd function is an odd function. 7. Show that when is an odd function, 0(0) = 0. 165. Definite Integrals tlirougli Integration dy Parts. — In the formula for integration by parts, namely, / udv = uv — I vdu, let us apply the rule for finding the definite integral. To ex- press the result, let us put {nv)^ and {uv)^, the values of uv for the upper and lower limits of integration, respectively^ udv and / vdu, the values of the two indefinite in- tegrals for the upper limit, x^i I udv and / vdu, the values of the integrals for the lower limit, x^. We then have, by the rule of § 161, / udv = I udv — / udv = {uv)^ — / vdu — (uv)^ + / i = {uv)^ — {uv)^ — / vdu. flxQ vdu DEFINITE INTEQEALS, 267 In order to assimilate the form of this expression to that of a definite integral, it is common to write EXAMPLES AND EXERCISES. I. We have found the indefinite integral / log xdx = X log X — I dx. If we take this integral between the limits x = and x = 1, the term x log x will vanish at both limits, so that {x log x)^ - {x log x)^ = 0. Hence f log xdx = — yja;=— 1+0= — 1. 2. To find the definite integral, r sin'^ xdx. In the equation (11), § 150, the first term of the second member vanishes at both the limits x = and x= tt. Hence /»"■ 7U — 1 /*" I sm'^xdx=z / sm'^-^xdx. Jo m e/o Writing m — 2 for m, and repeating the process, we have / sin"*"^a:t7a: = ^/ ^\n'^~^xdxi Jo ni — 2 Jo J/*"^ 111 — 5 p^ f sin"*""*.Trfx = 7 / sin'^'^xdx: m — 4: Jo etc. etc. If VI is even, we shall at length reach the form r dx = 7t — = Tt, Then, by succossive substitution^ we shall havo 268 THE INTEOBAL CALCULUS, r sin- xdx = 0»-l)(»^-3)(m-5). 1 ^ Jq mym — 2){m — 4) ... 2 If m is odd, the last integral will be f sin xdx = + 2, and we shall have Jo m(m — 2)(m — 4 (m-2)(m-4:) . . . 3' 3. From the equation (6) of § 149 we have, by forming the definite integral and dividing by m + "^h r'" . ,„ „ -, /sin"* + ^a;cos**~^2;V / sm''* X cos"* xdx = I Jo \ m + n Jo + -^^^ — f sin*" X cos'*-^ xdx. m + nJo Since sin tt = sin = 0, the first term of the second member vanishes between the limits, and we have /»"" 71 — 1 />"■ / sm*" X cos"" xdx = ; — / sin"" x cos"*"^ xdx. Jo m + nJo Writing 71 — 2, and then n — 4, etc., in place of 71, this formula becomes / sin"* a; cos"*"^ xdx = ; .r / sin"" x cos""'* xdx: t/o m + 71 — 2 Jo / sin"^ a; COS**"* xdx = — ; I sin*^ a: cos''"^ xdxi Jo m + 7i — 4: Jo etc. etc. If 71 is odd, the successive applications of this substitution will at length lead us to the form / sin*^ X cos xdx = (sin*^ + ^ tt — sin'^ + ^ 0) = 0: t/o m + 1 "^ ^ ' and thus, by successive substitution, we shall find all the in- tegrals to b^ zero. DEFINITE INTEGRALS. 269 If 11 is even, we shall be led to the form I sin"* xdXf which we have just integrated. Then, by successive substi- tution, we find I sin"* X cos"* X _ (n- l){n - 3) . dx 1 r*-" m + 2) Jo sin"* xdx. Jo R 4. To find , T^:^^„. We transform the differential thus: /dx _ 1 Mx^ -\- a^ — x"^) dx {^+^-~a''J (x' + d'Y _ 1 /> dx ^ r ^^^^ I \ ~TrJ (2;'^+^r)""^ ~~ d'J (x^ + cef ^^^ Integrating the last term by parts, we have r x^dx _\_ r _j^xdx__ -\. r ^i(^!±^!) J (x' + a'Y "" 2 e/ ^'{a" + x'Y ~ %J ^ (.t^ + a^f -\s- xd. 1 r dx Substituting this value of the last term in (r/), we have r dx __ 1 X t/ (x^ + a^Y " %a\n - 1) {x^ -\- a^y-^ ^ a\ 2(n - l)/e/ (o;^ + a^y-^' Passing now to the limits, we see that the first term of the second member vanishes both for a; = and for a: = oo . We also have 1 2;i - 3 1 2(w - 1) 2(;i - 1)* 270 THE INTEGRAL CALCULUS. Hence we have the formula of reduction /^* dx _ 2n — 3 />^ dx . Jo {x' + ay "^ 2{n - l)7i'Jo {a' + x')^-''' ^^^ We can thus diminish the exponent by successive steps until it reaches 2. The formula (b) will then give dx 1 /»=° dx 7t Jn"^ ax __ 1 A" {x' + ay ~ 2^e/o a^ + a;'' -' 4^=*' Then, by successive substitution in the form {h), we shall have dx _ {271 - d) {271 - 6) . . .1 n j: k {x^ + ay {2n - 2)(2^ - 4) . . . 2*2a2"-^' ^^^ If in {c) we suppose « = 1, and write the second member in reverse order, we have jdx l-3'5 . . . (2/^-3) n /o (1 + iy 2-4-6. . . (2^- 2)*2" / t/o ' To find r - ^^— ^y^ Vl-x' Let us apply to the indefinite integral the formula (A), 1 144. We have in this case a = 1; Z> = — 1; n = 2; p =z — i. The formula then becomes r x^dx _ __ x"^-^ Vl — x" m — 1 n x'^-Hx In the same way vr- -x' >^m- -'dx vr -x" p x'^-Hx _ __ x^^Vl-_x2^ m- 3 r J 4/1 _ x^ ~ ^^~ ^ 771 — 2J Continuing the process, we shall reduce the exponent of x to 1 if 7n is odd, or to if m is even. Then we shall have Taking the several integrals between the limits and 1, we DEFINITE INTEGRALS. 271 note that in {a) tlie first term of the second member vanishes at both limits, while (b) gives J/>^ xdx _ /»^ dx __ 1 ~vf^^~ ' J vr^^'"^^' We thus have, by successive substitution. y2n + l=jQ ^ x^-^^dx __ 27i{2n - 2) {27i - 4) 2 ^ VfZ:^^ ~ {27i+l)(2n-l){2n-3) ... 3' _ p x^dx _ {2n-l){2n-'d){2n-b) . . . 1 ^ ^^" -Jo |/iZr^' "" 2?i(2;i - 2) (27i - 4) 2*2 * {c) Let us now consider the limit toward which the ratio of two values of y^ approaches as m increases to infinity. We find, from {a), a ratio of which unity is the limit. Next we find, by taking the quotient of the equations (c*), ^___ {2-4- 6 . . . {2n — 2)'2)iY y^^ 2 - ts^tt. . (2/i-i)r (2/1 + 1) >*,.+,• Since, when 7i becomes infinite, the ratio ^/gn • .Ven + i ap- proaches unity as its limit, we conclude that ^Tt may be ex- pressed in the form of an infinite product, thus: Tt 4 4' 6' 8' 10' 2 3'3-5*5-7'7-9*911 ad infinitum. This is a celebrated expression for tt, known as Wallis's formula. It cannot practically be used for computing tt, owing to the great number of factors which would have to be included. 272 TEE INTEGBAL GALGULU8, CHAPTER VII. SUCCESSIVE INTEGRATION. 166. Differentiatio7i under the Sigii of Integration. Let us have an indefinite integral of the form u ■=^ j 4>(oc^ x)dx = F{a, x), (1) a being any quantity whatever independent of x. It is evi- dent that n will in general be a function of a. We have now to find the differential of ii with respect to a. The differentiation of (1) gives d^ii _ d(t>{a, x) dadx da -p. d'^u -^du -r^du ^ 1 . T Because -^ — 7- = Dorr- = J^xi—y we have, when we consider dadx dx da -z— as a function of x (cf. § 51), \da I dxda da Then, by integrating with respect to x^ du ^ Ma, X) da J da in which the second member is the same as (1), except that ^{a, x) is replaced by its derivative with respect to a. Hence we have the theorem: The derivative of aii integral with respect to any quantity which enters into it is expressed ly dijferentiating with re- spect to that quantity under the sign of integration. SUCCESSIVE INTEGRATION. 273 167. This theorem being proved for an indefinite inte- gral, we have to inquire whether it can be applied to a definite integral. If we take the integral (1) between the limits x^ and x^, and put u^ and u^ for the corresponding values of u, we have, for the definite integral, f '(p{a, x)dx = F{a, x^) - P{a, x^) = u, - u, = u,\ Then, by differentiation, du,' ^ dF{a, x^) _ dFja, x^) da da da ^ ' Comparing (1) and (2), we have r dcf>{a,x) ^^^^ dF{a,x) ^ tJ da da ' whence, if x^ and x^ are 7iol functions of a, />^i d{ci, x)dx. That is, tliG symbols of differeiitiation and integration with respect to two independent quantities may he iriterchanged in a definite integral, provided that the limits of integration are not functions of the quantity with respect to which we differ- entiate. If the limits x^ and x^ are functions of a, we have, for the total derivative of ti^' with respect to a (§ 41),- dii^ __ fdu^\ dul^ dx^ dul dx^ la \ da I dx^ da dx^ da da By § 160 we have dx 18 274 THE INTEGUAL CALCULUS. Thus from (3) and (4) we have This formula is subject to the same restriction as the theorem for the value of a definite integral; that is^ 0(^, x) and its dei'ivative with resjject to a must le finite and con- timious for all values of x lettueen the limits of integration. If this condition is not fulfilled, (5) may fail. EXERCISES. Differentiate: /dx ,,^ , , . n dx — ; — with respect to a. Ans, — / 7 — ; Tq. X + a ^ J {0^+ ex) 2. / {x -^ ocfdx with respect to a, Ans. n I (x-\-aY~^dx. 3. /^(a:'-f-^^)''^^ with respect toy. Ans. 2 l{x^-\-7!^y)dx. 4. / x^dx with respect to a. Ans. a^. t/o 5. / o^dx with respect to a. Ans. 8<^^. 3;*"^.?; with respect to or. Ans. =«^(2a'^ + ^--l). And show that we have the same results in the first three cases whether we integrate the differential with respect to a or ijy or differentiate the integral. 168. The preceding method enables us to find many integrals, indefinite and definite, by differentiating known integrals with respect to constants which enter into them. Thus, by differentiating with respect to a the integral a SUCCESSIVE INTEGRATION. 275 we find, after adding the constants of integration, etc. etc. which leads to the same results as integration by parts, and is shorter. 169. The following is an instructive application of this and other principles. We shall hereafter show that f e"""^ dx= Vrt, fj— CO From this it is required to find the value of / e~^^^^ dy. U — 00 If we put x^ay, (1% whence dv — — , the corresponding indefinite integral will be Now, when y = ± oo , we have also a: = ± oo . Hence / e ^ dy —- I e dx = . By differentiating with respect to a, and simple reductions, we find and from this, etc. etc. 276 THE INTEQBAL CALGULU8. EXERCISES. 1. By differentiating the integrals /cos axdx = — sin ax, a /sin axdx = cos ax, a twice with respect to a, prove the formulaB I X cos axdx =.[ -A ^in ax A — -^ cos ax\ J \a a J a /> , . ^ (2 x'X 2x , I X sm axdx =[-. cos a^ H — ^ sm ax. J \a a J a Thence show that we have Jy"" cos ydy = (y' - 2) sin y + 2y cos y; J 2/ sin ydy = {2 - y') cos y + 2y sin y. 2. Prove the formulae: /o 1 /^^ 1 e^^dx =-; {h) xe'^^'dx = - -,; 00 ^ t/ — CO a /o 2 />0 7iT ^'V^^^ = -3; {d) / c^^V^^Zo; = (~ 1)"-^. CO ^ t/ — 00 a 3. Show that the preceding formulaB are true only when a is positive, and find the following corresponding forms when a has the negative sign: J/»^ 1 />°° 1 i e^'^'^dx = — ; / xe'^^^dx = — ,5 c^ Jo a i x^e-'^'^dx = -3: / x^e-'^dx = -^ ; etc. a'' Jq a' ' 4. By differentiating the form of § 132, namely, /dx _ • (_ 1) ^ J^Zl^y - ^^^ a' with respect to a, show that /dx _ x {a" - x^)l ~ a\d' - ^i' SUCCESSIVE INTEORATION. 217 170. Double Integrals. The preceding results may be summed up and proved thus: Let us have an integral of the form w = y 0(^, y)(lx, (1) and let us consider the integral fndy or flf^i^^ y)(^^Ul/y which, for brevity, is written without brackets, thus: JJ^i^y y)dxdy. This expression is called a doiihle integral. Theorem. Tlie value of an indefinite double i^itegral re- mains unchanged token we change the order of the integra- tions, provided that we assign suitable values to the arbitrary constants of integration. Let us put u retaining the value (1). The theorem asserts that / %uly = / vdx. Call these two quantities U and F, respectively. We then have, by differentiation, dU d'U du ^, , dV d'V dv ^, , Therefore, because of the interchangeability of differentiations, d—d~ ' dx _ ' dx dy ~ dy Then, by integration with respect to y, dU_(lV Ox "dx '^''' 278 THE INTEGRAL CALCULUS, and, by integration with respect to x, U^ V+cx + c\ Putting c = and c' = 0, we have 11= V, as was to be proved. 1 7 1 . By the process of successive integration thus indi- cated we obtain the vahie of a function of two variables when its second derivative is given. The problem is, having an equation of the form dxdy where (p{x, y) is supposed to be given, to find u, as a func- tion of X and y. This we do by integrating first with respect to one of the variables, say x, which will give us the value of -r-9 because the first member of (2) is D^^-r-, Then we in- dy ^ ^ dy ' tegrate with respect to y, and thus get w. As an example, let us take the equation d'^u „ -. du „ , xy , or d.-j— = xy dx. dxdy '^ ' ' dy Integrating with respect to x, we have %-rY+K (3) li being a quantity independent of x, which we have common- ly called an arbitrary constant. But, in accordance with a principle already laid down (§118), this so-called constant may be any quantity independent of x, and therefore any function we please to take of y, Next, integrating (3) v/ith respect to ij, and putting Y ^Jhdy, we find u = \xY + F + X, in which X is any quantity independent of y, and so may be an arbitrary function of x. Moreover, since h is an entirely arbitrary function of y, so is Y itself. SUCCESSIVE INTEGRATION. 279 The student should now prove this equation by differenti- ating with respect to x and y in succession. 173. Triple and Multiple Integrals. The principles just developed may be extended to the case of integrals involving three or more independent variables. The expression /// cp{x, y, z)dxdyclz means the result obtained by integrating 0(.^, y, z) with re- spect to X, then that result with respect to y, and finally that result with respect to z. The final result is called a triple integral. If we call F{x, y, z) the final integral to be obtained, we have . d^F{x, y, z) .. . dxdydz -^(^-y-^)> and the problem is to find F{x, y, z) from this equation when 0(:r, y, z) is given. Now, I say that to any integral obtained from this equation we may add, as arbitrary constants, three quantities: the one an arbitrary function of y and z] the second an arbitrary function of z and x] the third an arbitrary function of x and ?/. For, let us represent any three such functions by the symbols [y, 2;], [z, x], {x, ?/], and let us find the third derivative of F{x, y, z) + [y, z] + [z, x] + [x, y]=u with respect to x, y and z. Differentiating with respect to X, y and z in succession, we obtain die ___ dF(x, y, z) d[z, x] d[x, ?/] ^ dx """ dx dx dx ' tfw _ d'^F{Xy y, z) d'lx, y~\^ dxdy ~~ dxdy dxdy ' d\i _ d*F{x, y,z) ^ dxdydz dxdydz * an equation from which the three arbitrary functions have 280 THE INTEGRAL CALCULUS. It is to be remarked that one or both of the variables may disappear from any of these arbitrary functions without chang- ing their character. The arbitrary function of y and z, being any quantity whatever that does not contain x, may or may not contain y or z, and so with the others. As an example, let it be required to find ^l — I / {x — a){y — b){z — c)dxdydz. Integrating with respect to z, and omitting the arbitrary function, we have J fW -«)(«/- ^)(^ - ofdxdy. Then integrating with respect to y, which gives, by integrating with respect to x, and adding the arbitrary functions, u = i{x - ay{y - by{z - cy + [y, z'] + [z, x^ + [x, y]. The same principle may be extended to integrals with re- spect to any number of variables, or to multiple integrals. The method may also be applied to the determination of a function of a single variable when the derivative of the func- tion of any order is given. EXERCISES. I. / I -^dxdy. 2. / I {x — a){y—hydxdy, 3. / / jxy^z^dxdydz, 4. / / i-^dxdydz, 5. I I I {x — ay{y — b){z — cydxdydz. 6. ff{x - aydx\ 7. fff{^ + ^^)'dz\ Ans. (6). y\(.T -■ ay -\- Cx -\- 0\ (7 and C being arbitrary constants. ISUCCESSIVE INTEGIUTION. 281 173. Definite Double Integrals. Let U be any function of X and y. By integration with respect to or, supposing y constant, we may form a definite integral udx E ur From what has been shown in § 163, Kem., U' will be a function of y, x^ and x^. AVe may therefore form a second definite integral by integrating U'dy between two limits y^ and y^. Thus we find an expression f^" V'dy = f^' r"' Udxdy, which is a definite double integral. The limits x^ and x^ of the first integration may be con- stants, or they may be functions of ?/. If they are constants, the two integrations will be inter- changeable, as shown for indefinite double integrals. If they are functions of y they are not interchangeable, un- less we make suitable changes in the limits. 1*74. Definite Triple and Multiple Integrals. A definite integral of any order may be formed on the plan just described. For example, in the definite triple integral /»^i />y> r*^\ III ^(^^ y^ z)dxdydz *Izo t/?/o t/xo the limits x^ and x^ of the first integration may be functions of y and z; while y^ and y^ maybe functions of z. But z^ and z^ will be constants. So, in any multiple integral, the limits of the first integra- tion may be constants, or they may be functions of any or all the other variables. And each succeeding pair of limits may be functions of the variable which still remain, but cannot be functions of those with respect to which we have already integrated. 283 THE INTEGRAL CALCULUS. EXAMPLES AND EXERCISES. I. Find the values of / / xifdxdy and / / xy^dxdy. It will be seen that in the first form the limits of x are condtants^ and in the second, functions of y. First integrating with respect to Xy we have for the indefi- nite integral Jxfdx = ia;>% and for the two definite integrals / xy^dx = \o^y^9 I xy'^dx = ^y\ t/y Then, integrating these two functions with respect to y, we have ££\y\lxdy = l£y'dy = ^b\ Let us now see the effect of reversing the order of the in- tegrations. First integrating with respect to y, we have / xy^dy — ixb^ = U. Then integrating with respect to x, we have f Udx = r TxyWydx = \a^l)\ the same result as when we integrated in the reverse order between the same constant limits. 2. Deduce / ^ / cos {x -\- y)dxdy = — 2. SUCCEi<^I Vh l^ TliUUATION. 2b3 3. Deduce / / cos (x — y)dxdy = -|- 4. /*b /*a 4. Deduce / / (u; — a){y — h)dxdy = lit'b''. 5. Deduce jT^j^'^u; - a){y - h)dxdy = i{2ab-a')(2ab-b'). 6. Deduce / / {.v—a){y—b)dxdy = a^b — iab""— ^a\ 175. Product of Ttvo Definite Integrals. Theorem. The product of the two definite integrals I Xdx and I Ydy is equal to the double integral / ' / 'XVdxdy, provided that neither integral contains the variable of the other. For, by hypothesis, the integi^al / Xdx^ f/ does not con- tain y, Tlierefore U / Ydy = I UYdy = / / XYdxdy, as was to be proved. / + » _ a e dx. This integral, 00 which we have already mentioned, is a fundamental one in the method of least squares, and may be obtained by the ap- plication of the preceding theorem. Let us put ;=:/ e-Ulx = 2 e-^dx^'^f e"^Jy.(§164) /.•=: Then, by the theorem, t'o t/0 * Jq t/o Let us now substitute for y a new variable t, determined by the condition y = tx. 284 THE MTEGRAL CALCULUS. Since^ in integrating with respect to y, we suppose x con- stant^ we must now put dy — xdt. Also, since t is infinite when y is infinite, and zero when y is zero, the limits of integration for t are also zero and infinity. Thus we have Jo Jo Since the limits are constants, the order of integration is indifferent. Let us then first integrate with respect to x. I Since I 1 I xdx = id'x' = ^.^ ,. <:?• (1 + f)x% j the integral with respect to x is Then, integrating with respect to ^, Hence f e~^^dx = Vtt^ J— CO RECTIFICATION OF CURVES, 285 CHAPTER VIM. RECTIFICATION AND QUADRATURE. 177. The Rectification of Curves, In the older geometry to rectify a curve meant to find a straight line equal to it in length. In modern geometry it means to find an algebraic expression for any part of its length. Let us put s for the length of the curve from an arbitrary fixed point (7 to a vari- able point P. If P' be another position of the variable point, we shall then have fig. 5o. As = FF\ If PP' becomes infinitesimal, it has already been shown (§ 79) that we have, in rectangular co-ordinates. ds = Vdx^ + df = 4/1 + i^£ldx = Ml + (^)^7y, (1) and, in polar co-ordinates, d. = l/r' + (^JrZft (2) If both co-ordinates, x and ijy are expressed in terms of a third variable Uy we have The length of any part of the curve is then expressed by 286 THE INTEOEAL CALCULUS. the integral of any of these expressions taken between the proper limits. Thus we have or -/i(rj+(i)T"«- (3) In order to effect the integration it is necessary that the second members of (3) shall be so reduced as to contain no other variable than that whose differential is written; that is, we must have ds=f{x)dx; f{y)dy; f{e)d6; or f{u)dit. Then we take for the limits of integration the values of X, y, 6 or u, which correspond to the ends of the curve. 178. Rectification of the Parabola, From the equation of the parabola y'^ = 2px we derive ydy = pdx. We shall have the simplest integration by taking y as the independent variable. We then have The formula (C) of § 145 gives The method of § 132 gives («) .n»' {f + yl dy {f + f) = A-log((;/ + 2/T-?/) RECTIFICATION OF CURVES. 287 Thus, j)utting A' E i;? (// — log j)), the indefinite integral of {a) is s = // + i^{f + f)' + ^p log ({/ + f)' + y). The arbitrary constant 7/ must be so taken thcit .9 shall vanish at the initial point of the parabolic arc. If we take the vertex as this point, we must have 5 = for y — 0. Then 7/ = -^p log p. We therefore have, for the length of a parabolic arc from the vertex to the point whose ordinate is y, s=)j^u^^+ff-viP^o^^-^-y^. (4) \ 179. Rectification of the Ellipse. The formulae for rec- tifying the ellipse take the simplest form when we express the co-ordinates in terms of the eccentric angle u\ then (Analyt. Geom. ) cr = a cos ?/; y = h sin u. We then have dx =. — a sin itdti] dy = h cos udu. Then if e is the eccentricity, so that a^e* = a* — ^% ds = {a^ sin'' u -{-V cos' u)^du = n{l — e' cos' u)^dUy s = a / {1 — e^ cos' u)^du. This expression can be reduced to an elliptic integral: a kind of function which belongs to a more advanced stage of the calculus than that on which we are now engaged. It may, however, be approximately integrated by develop- ment in series. We have, by the binomial theorem, 1 . , 11 ^008 U--. (1 — e* cos' uy =zl -^ -e* cos' u — r-;^ ^* cos* u 11-3 « , ,-—7—7 e cos u ~ etc. 2*4G 288 THE INTEGRAL CALCULUS. The terms in the second member may be separately in- tegrated by the formulEe ((>), § 149, by putting m == and ^ rz 2, 4, 6, etc. We thus find 2 / cos^ tidu = sin ^c cos n -\- u; 4 / cos* ndu = sin -2^ (cos' u + I cos u) + fw> etc. etc. etc. Since at one end of the major axis we have u = and at the other end u =:i tt, we find the length of one half of the ellipse by integrating between the limits and tt. Since sin u vanishes at both limits, we have cos' 7idtc = —tt; r ^.3 COS* udii — 77— .tt; 2*4 1-3-5 ''' ^ = 2^^- We thus find by substitution that the semi-circumference of the ellipse may be developed in powers of the eccentricity with the result = fiTti , 1 . 3 , 3^-5 2' 2' -4' 2' -4^ -6' 180. The Cycloid, The co-ordinates x and y of the cy- cloid are expressed in terms of the angle u through which the generating circle has moved by the equations (§ 80) X = a{u — sin w); y == a(l -— cos ti). Hence els' = dx'' + dy"" = rtM(i _ cos w)' + sin' u}du^ = 20" (1 — cos u)du'' = id' sin' ^tcdu^. By extracting the root and integrating, s ■= h — 4:a cos ^u. UECTIFICATION OF CURVES. 289. If we measure tlie arc generated from tlie point where it meets the axis of abscissas, that is, where n = 0, we must have s = for n = 0. This gives h = 4« and s = 4a(l — cos ^u) = 8^ sin' ^71. This gives, for the entire length of the arc generated by one revolution of the generating circle, that is, four times the diameter of the generating circle. 181. The Archimedean Spiral, From the polar equation of this spiral (§ 82) we find dr = add. Hence ds = a{l + O^fdO, Then the indefinite integral is (§ 147, Ex. 1) 5 = 1 1 0(1 + e^)' + log c{e +(1 + e^))' [ . If we measure from the origin we must determine the value of C by the condition that when 6 = 0, then 5 = 0. This gives log C' = ; .-,0=1. If instead of 6 we express the length in terms of r, the radius vector of tlie terminal point of the arc, we shall have / ' = 2 7S'' + ' ) + 2 ^^^^ a • 183. The Logarithmic Spiral, The equation of this spiral (§ 83) gives dr 1 le ■, ^-^ = ale =lr. Hence ds = (1 + r)Wft To integrate this differential with respect to 6 we should first substitute for r its value in terms of 6. But it will be 290 THE INTEGRAL CALCULUS. better to adopt the inverse course, and express dd in terms of dr. We thus have (1 + ly ds = J dr; whence s = — —r + 5^, s^ being the value of s for the pole. If we put y for the constant angle between the radius vector and the tangent, then (§§ 90-92) ^=:cot y, and we have s = r sec y -{- s^. Between any two points of the curve whose radii- vectors are r^ and r^ we have 5 = (r^ — r J sec y. Hence the length of an arc of the logarithmic spiral is pro- portional to the difference between the radii-vectors of the ex- tremities of the arc, EXERCISE. 1. Show that the differential of the arc of the lemniscate is , add ds = Vl-2 sin'' 6 (This expression can be integrated only by elliptic func- tions.) 183. The Quadrature of Plane Figures, In geometrical construction, to square a figure means to find a square equal to it in area. The operation of squaring is called quadrature. In analysis, quadrature means the formation of an algebraic expression for the area of a surface. In order to determine an area algebraically, the equation of the curve which bounds it must be given. Moreover, in order that the area may be completely determined by the bounding line, the latter must be a closed curve. Then whatever the form of this curve, every straight line QUADRATURE OF PLANE FIGURES. 291 X / T / 1 o / V Q /| Xo x» Fig. 51. must intersect it an even number of times. The simplest case is that in which a line paral- lel to the axis of V cuts the bound- ary in two points. Then for every value of x the equation of the curve will give two values of y corresponding to ordinates termi- nating at F and Q. Let these values be y^ and ?/,. Then, the infinitesimal area in- cluded between two ordinates infinitely near each other will be {y^ - y.)dx = d(T. The area given by integrating this expression will be ^ = f \y.-yy^> in which the limits of integration are the extreme values of x corresponding to the points X^ and X^, outside of which the ordinate ceases to cut the curve. The same principle may be applied by taking {x^ — x^dy as the element of the area. We then have 0- rr n\x^ - x^)dy. If the curve is referred to polar co-ordinates, let S and T be two neighboring points of the curve, and let us put Z6 = angle SOT. If we draw a chord from S to 7", the area included between this chord and the curve will be of the third order (§ 78). The area of the triangle formed by this chord Fio. 52. 292 THE INTEGRAL CALCULUS. and the radii vectors will be irr' sin A 6, ]S"ow let AS be- come infinitesimal. 08 will then approach r as its limit; the ratio of sin ^6^ to AS itself will approach nnity^ and the area of the triangle will approach that of the sector. Thus we shall have, for the differential of area, dor rr ^r\ie. If the pole is within the area enclosed b}'^ the curve, the total area will be found by integrating this expression be- tween the limits 0° and 360°. Thus we have, for the total area. /"•■«• 1 84. The Parabola, As the parabola is not itself a closed curve, it bounds no area. But we may find the area of any segment cut off by a double ordinate MN, The equation of the curve gives, for- the two values of y, y, = + V2px; yo= - ^^* Hence o da ~ VSp.x^dx. The indefinite integral is pj^ ^3 For the area from the vertex to MJVwe put x^ = OX, and take the integral between the lim.its and x^. Calling this area cr^, we have Because 2y^ = MX, it follows that the ^ltq^xABMN ='2x^y^. Hence: Theore^i. The area of a paraholic segrneJit is two thirds that of its circumscribed reciavgJe. qUADUATURE OF PLANE FIGURES, 293 185. The Circle and the Ellipse, Referring the circle of radius a to the centre as the origin, the values of y will bo Hence J\y.- y.)dx ^^.fia' - xydx X = xicv - x^ + a' sin c-^)- + h. ^ ' a This expression, taken between appropriate limits, will give the area of any portion of the circle contained between two ordinates. Taking the integral between the limits — a and + ^ gives, for the area of the circle, G = a' sin<-^> (+ 1) - a' sin(-^> (- 1) = 7ra\ The Ellipse. From the equation of the ellipse referred to its centre and axes, namely. we find V = ± — ^n' — x'. ^ a The entire area will be The last integration is performed exactly as in the case of the circle. 186. The Hyperbola, Since the hyperbola is not a closed curve, it does not by itself enclose any area. But we may consider any area enclosed by an hyperbola and straight lines. Let us first consider the area A PM contained between the curve, the ordinate MP, and the segment AM oi the major 294 THE INTEGRAL CALCULUS, axis. The equation of the hyperbola referred to its centre and axes gives, for the value of y in terms of Xy y ^-Vx" a\ If we put x^ for the value of the abscissa OM, then^ since OA = a, the area AMP will be equal to the integral a Fig. 54. - r\x' - a'')^dx', /(.' - a^)^d. = l^i.' - a')* - 1 log g + g - 1 )*]; and for the definite integral between the limits a and x, 1 ab ^ + S-)']- Now, ixy is the area of the triangle OFM; we therefore conclude that the second term of the expression is the area included between OA, OF and the hyperbolic arc AP, Much simpler is the area included between the curve, one asymptote, and two parallels to the other asymptote. The equation of the hyperbola re- ferred to its asymptotes as axes of co-ordinates (which axes are oblique unless the hyperbola is equilateral) may be reduced to the form fio. 55. xy db 2 sin a' qUADUATUUE OF PLANK FIGURES, 295 a being the angle between the axes. We readily see that the differential of the area is ydx X sin a instead of ydx simply. Hence for the area we have / y sin adz — I -clx = -- log ex. If we take the area between the limits OM=x^ and OM ~ x^, the result will be ab , ad ^ X, -dx = -^-log-\ X, 2a; 2 ^a;, We note that this area becomes infinite when x^ becomes zero or when x^ becomes infinite, showing that the entire area is infinite. 187. The Lomniscate. The equation of this curve in polar co-ordinates is (§ 81) r' = a" cos 26. It will be noted that r becomes imaginary when 8 is con- tained between 45° and 135°, or between 225° and 315°. The integral expression for the area is \Jr''de = ia'fcos 2ede = ia' sin 20. To find the area of the right-hand loop of the curve we must take this integral between the limits 8 = — 45° and 6 = -f 45°, for which sin 2/9 = — 1 and + 1. Hence Half area = ia^; Total area = a'. Hence the area of each loop of the lemniscate is half the square on the semi-axis. 188. The Cycloid. By differentiating the expression for the abscissa of a point of the cycloid we have ^x = a{l — cos u)du. Hence 296 THE INTEGRAL CALCULUS, j ydx —a" / (1— cos uYdu — a' / (| — 2 cos u-{-i cos 2ii)du. The indefinite integral is lu — 2 sin u -\- i sin 2v, To find the whole area we take the definite integral between the limits and 27r, Thus we find Area of cycloid = dTta", or three times the area of the generating circle. EXERCISES. I. Show that the theorem of § 184 is true only of the pa- rabola. To do this we must find what the equation of a curve must be in order that the theorem may be true. The theorem is / ydx = ^xy. Differentiating both members, we have ydx = ixdy + ^ydx ; . ^0 _^ ' ' y x' Then, integrating both members, log y'^ — log cx\ . • . 2/2 _ ^^^ c being an arbitrary constant. This is the equation of a parabola whose parameter is \c. 2. Show that the equation of a curve the ratio of whose area to that of the circumscribed rectangle is m : n must be of the form CUBATUBE OF VOLUMES, 397 CHAPTER IX. THE CUBATURE OF VOLUMES. 189. General FormulcB for Cuhaticre, In the ancient Geometry to mide a solid meant to find the edge of a cube whose volume should be equal to that of the solid. In Ana- lytic Geometry it means to find an expression for the volume of a solid. Let us have a solid the bounding surface of which is de- fined by an equation between rectangular co-ordinates. Let the solid be cut by a plane PL parallel to the pline of YZ, and let w be the area of the plane section thus formed. If we now cut the solid by a second plane, parallel to PL and infinitely near it, that portion of the solid contained between the planes will be a slice of area ?^ and thickness dx, dx being the infinitesimal distance between the planes. If, then, we put v for the volume of that part of the solid contained between any two planes parallel to YZ, we have Fig. 50. and dv = udx, V = I vdx, (1) x^ and :r, being the distances of the cutting planes from the prigin 0, 298 THE INTEGRAL CALCULUS. If we take for x^ and x^ the extreme values of x for any part of the solid, the above expression will give the total vol- ume of the solid. In order to integrate (1), we must express w as a function of X, That is, we must find a general expression in terms of X for the area of any section of the solid by a plane parallel to that of XY. This is to be done by the equation of the bounding surface of the solid. Of course we may form the infinitesimal slices by planes perpendicular to the axis of Y oic oi Z as well as of X. 190. The Sphere. The equation of a sphere referred to its centre as the origin is x^' + y^ + z' - a\ If we cut the sphere by a plane PMQ parallel to the plane of YZy and having the abscissa OM = Xy the equation of the circle of intersection will be 2/' -}- ;2^ = a' — X^*, Fig. 57. that is, the radius MP of the circle will be Va" — x% and its area will be 7t{a^ — x^). Hence the differential of the vol- ume of the sphere will be dv = 7r(a* — x^)dxy and the indefinite integral will be V = 7t{a^x ~ \x^) -\-C. The extreme limits of x for the sphere are x^^ — a and x^ = + a. Taking the integral between these limits, we have Volume of sphere = |;ra'. CUBATURE OF VOLUMES. 290 191. Volume of Pyramid. Let the pyramid be placed ^th its vertex at the ori- gin, and its base parallel to the plane of XY. Let us also put h = OZ its alti- tude; a, the area of its base. Let it be cut by a plane EFGH parallel to its base. It is shown in Geometry that the section EFGH is similar to the base, and that the ratio of any two homologous sides, as EFaniAS, is the same as the ratio OL : OZ, Because the areas of polygons are proportional to the squares of their homologous sides, .-.Area EFGH : Area ABCD = OL' : 0Z\ Putting Area ABCD = a, OL^z and OZ = h, FiQ. 58. Area. EFG 11 = az The volume of the pyramid is therefore •'^ nz^'dz 1 , That is, one third the altitude into the base. The same formulae apply to the cone. 193. The Ellipsoid. The equation of the ellipsoid re- ferred to its centre and axes is 1, a, h and c being the principal semi-axes. If we cut the ellipsoid by the plane whose equation is X = x', the equation of the section will be = 1 _ 300 THE INTBanAL GALOVLtfS. This is the equation of an ellipse whose semi-axes are a Hence its area is x'^ and 7tlc{a' — a ') Fig. 59. Then, by integration between the limits —a and -\-af we find Volume of ellipsoid = ^-Ttabc, From the known expression for the area of an ellipse {rtah) it is readily found that the volume of an elliptic cylinder cir- cumscribing any ellipsoid is ^nabc. Hence we conclude: The volume of an ellipsoid is two thirds that of a7iy right elliptic cylinder circumscribed about it. 193. Volume of any Solid of Revolution. In order that a solid of revolution may have a well-defined volume it must be generated by the revo- lution of a curve or un- broken series of straight or curve lines terminating at two points, Q and R, of the axis of revolution. As an element of the volume we take two planes infinitely near each other and perpendicular to the axis of revolution. Every such plane cuts the solid in a circle. If we place the origin at 0, take the axis of revolution as that of X, and let OM = a; be the abscissa of any point P of the curve, and MP = y its ordinate, then the section of the solid through M will be a circle of ra- dius y, whose area will therefore be ny^. Hence the volume contained between two planes at distance dx will be ny'^dx, and the volume between two sections whose abscissas are x^ and x^ will be V= r'jnfdx. (1) CUBATURE OF VOLUMES. 301 If the two co-ordinates are expressed in terms of a third variable ^i by the equations we have dx = (p'{ii)du. Putting u^ and u^ for the values of u corresponding to x^ and x^y the expression (1) for the volume will become V=7r r\ip{u)Ycp\n)d^t. Juq (2) The equations (1) and (2) give the volume AA'B'B gen- erated by the revolution of any arc AB oi the given curve, and of the ordinates MA and NB of the extremities of the arc. The limits of in- tegration for x are OM A'., ^--b = x^ and ON ~ x^. To fig. go. find the entire volume generated we must extend these limits to the points (if any) at which the curve intersects the axis of revolution. 194. The Paraboloid of Revolution, The equation of the parabola being y* = 2px, we readily find from (1) a result leading to the following theorem, which the student should prove for himself: Theorem. The volurtie of a para- boloid of revohUio7i is one half that of the circumscribed cylinder. 195. The Volume Generated by the Revolution of a Cycloid aroujid its Base. From the equations of the cycloid in terms of Fio. 61. 302 TBE INTEGRAL CALCULUS. the angle through which the generating circle has moved, we find the element of the volume to be dV= 7ta\l — cos uydu. Hence V = Tta^ / (1 — 3 cos 1^ + 3 cos^ u — cos' ii)du. By the method of §§ 149^ 150^ with simple reductions, we find / cos'^ udii — \\i -\-\Avl%u\ I cos' udxi = / (1 — sin'^ u)ditiTL ^ == sin w — ^ sin' u = I sin t^ 4- iV sin 3w. We thus find, for the indefinite integral, y — 7ta^(^u — ^- sin u -\- i sin 2w — y^ sin ^u). . The total volume formed by the revolution of one arc of the cycloid is found by taking the integral between the limits u = and u = %7t. The volume thus becomes F=5;rV, from which follows the theorem: The volume generated hy the revolittion of a cycloid around its base is five eighths that of the circumscribed cylinder, 196. The Hyperboloid of Revolution of Two Najjpes, This figure is formed by the revolution of an hyperbola about its transverse axis. The general expression for the volume is found to be h being the arbitrary constant of integration. If we consider that part of the infinite solid cut off by a plane perpendicular to the transverse axis, we must determine h by the condition CUBATURE OF VOLUMES. 303 that V shall vanish when x = a, because then the plane will be a tangent at the vertex of the hyperboloid, and the volume will become zero. This condition gives h =: da' - a' = 2a\ Thus we have By the same revolution whereby the hyperbola describes an hyperboloid of revolution the asymptotes witt" describe a cone. Let us compare the volume just found for the hyperboloid with that of the asymptotic cone, cut off by the same plane which cuts off the hyperboloid. The equation of the generat- ing asymptote being ay = iXy we find for the volume of the cone The difference between (1) and (2) will be the volume of the cup-shaped solid formed by cutting the hyperboloid out of the cone. Calling this volume F", we find F" = 7rb\x - ^a). (3) This is equal to the volume of a circular cylinder of which the diameter is the conjugate axis of the hyperbola, and the altitude x — fa. This result is intimately associated with the following theorem, the proof of which is quite easy: If a plane peiyendicular to the axis of revolution cut an hyperbola of two 7iappes and its asymptotic cone, the area of the plane contained between the circular sections is constant and equal to the area of the circle whose diameter is the con- jugate axis. IT P J r O M X Fig. 62. THE INTEGRAL CALCULUS, 197. Ring-sliapecl Solids of Revolution, If any com* pletely bounded plane figure J P^^ revolve around an axis OX lying in its own plane^ but wholly outside of it, it will describe a ring-shaped solid. To investigate such a solid, let the ordinate MP cut the figure in the points Q and P, and let us put The points P and Q will describe two circles which will contain between them the sectional area Ay." - y"y Taking two ordinates at the infinitesimal distance dx, the corresponding infinitesimal element of volume will be dV=7t{y:-y:)dx. (1) The integral V=7t r\y; - y^yix = n r\y^ + y^) {y^ - y;)dx will express the volume of that part of the solid contained be- tween the two planes whose respective aDscissas are x^ and x^. By taking for x^ and x^ the abscissas of the extreme points A and B, V will express the total volume of the solid. 198. Application to the Circular Ring, Let the figure ^^ be a circle of radius c, whose centre is at the distance h from the axis of revolution. Let us also put a = the abscissa of the centre. We then have y^ = h + Vc'- {x - aY; y. + y^ = ^^y y. - 2/i = ^ Vc'- {x-aY; CUBATURE OF VOLUMES. 306 V =4.711 r'[c' -(x- aVfclz. The limits of integration for the whole volume are x^z= a — c and x^ = a -{- c. If we put z^x — a, the total volume will become V = 4.7th f ^{c' -z'fdz. By substituting the known value of the definite integral, we have The area of the generating circle is ;rc% and the circumfer- ence of the circle described by its centre is ^nh. The product of these two quantities is ^it^lc^. Hence: The volume of a circular ring is equal to the product of the area of its cross-section into the circuinfereiice of its ceyitral circle. EXAMPLES AND EXERCISES. 1. Compare the cycloid with the semi-ellipse having the same axes as the cycloid, and show the following relations be- tween them: a. The maximum radius of curvature of the ellipse (at the point B) is greater than that of the cycloid in the ratio ;r' : 8, or 5 : 4, nearly. /?. The area of the semi-ellipse is greater than that of the cycloid in the ratio tc : 3. y. The volume of the ellipsoid of revolution around the axis OX is greater than that generated by the revolution of the cycloid in the ratio IG : 15. 20 306 THE INTEGBAL CALCULUS. o Fig. 63. R 199. Quadrature of Surfaces of Revohition. Let us put Js = a small arc PQ of a, curve re- volving round an axis OX; y = the distance of F from the " axis OX; y' = the distance of Q from the axis OX. Considering As as a straight line, the surface generated by it will be the curved surface of the frustum of a cone. If we put Aa = the area of this curved surface, we have, by Geometry, Aa = 7t{y + y')As» Now let As become infinitesimal. Then y' will approach y as its limit, and we shall have, for the diiferential of the sur- face, d(T = 27tyds = 27ry\ 1 -f [■jfj dx. This expression, when integrated between the limits x^ and iCj, will give the area of that portion of the surface for which the co-ordinates x are contained between x^ and x^. The modifications and transformations of this formula so as to apply it to cases when another axis than that of Y is the axis of revolution, or when the equation of the curve is not in the form y = (p{x), can be made by the student himself. 300. JExamples of Surfaces of Revolution, The process of applying the general formula for da to special cases is so like that already followed in quadrature and cubature that the briefest indications will suffice to guide the student. Surface of the S2)liere, Supposing the equation of the gen- erating circle to be written in the form SURFACES OF REVOLUTION. 307 we shall find the diflEerential of the surface to be da = %7tadx. From this we may easily prove the following : Theorem I. If a sphere he cut by any numher of jyarallel and equidistant 2)laneSy the curved surfaces of the spherical zones contained between the planes will all be equal to each other, Tpieorem II, The total surface of a sphere is equal to the product of its diameter aiid circumference. Surf ace generated by the Revolution of a Cycloid. We shall find the differential of the surface to be da = Sna^ sin^ ^udu. By a formula found in Trigonometry, we have 8 sin' ^ = 6 sin ^; — 2 sin 3v. Hence, putting v = ^u, da = 4;t«' (3 sin v — sin 3v)dv, The whole surface is obtained by integrating between the limits tc = and u = 27t; that is, v = and v = 7t, We thus find, for the total surface, a = -^7Ta\ Hence the theorem: The total surface generated by the revolution of a cycloid about its base is four thirds the surface of the greatest in- scribed sphere. The Paraboloid of Revohdion, Taking the integral be- tween the limits zero and x, we have for the curved surface me END. a 4 i i- ^^ -M 4-^ 5 -L c< ®( ^ ^ 1- — > v~:7 II xL -1- H \ i H UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. JAN ^9 19**i AUG 2 y 1957 • ^^^^^'^'' ^ ll^p'5;l:H REC Ulan'52LU DEC 1 5 1961 ..^^?'5TCR| '"L 1 1 2001 • / / / LD 21-100m-9,'47(A5702sl6)476 86S679 ^03 THE UNIVERSITY OF CALIFORNIA UBRARY