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Algebra for Schools, $1.20 Key to Algebra for Schools, .... 1.20 Plane Geometry and Trigonometry, with Tables, 1.40 The Essentials of Trigonometry, . . . . 1.25 II. COLLEGE COURSE. Algebra for Colleges, . . . . . . $1.60 Key to Algebra for OoUeges, .... 1.60 Elements of Geometry, 1.60 Plane and Spherical Trigonometry, with Tables, 2.00 Trigonometry (separate), 1.50 Tables {separate)^ . . . . . • . 1.40 Elements of Analytic Geometry, . • . 1.50 Calculus {in pi'eparation), Astronomy (Newcomb and Holden,) . . 2.50 The same, Briefer Course, .... 1.40 HE/\fRY HOLT d CO.. Publishers, New York. STBWCOifB'S MATnEMATIOAL COURSE ELEMENTS OP THB DIFFERENTIAL AND INTEGRAL CALCULUS BY SIMOI^ ISTEWOOMB Ptofem>r of Mathematics in the Johns Hopkins University NEW YORK HENRY HOLT AND COMPANY 1887 AS €•7 A COPYMOHT, 1887, BY HENRY HOLT & CO. use; iU 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 prin- ciples un'il he has become familiar with the mechanical pro- cess of differentiation, and with the application of the calcu- 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 a^ 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. nd best to and after- occasion to idvice and issor John )r. Fabian 3d sugges- utting the I of Wash- roblems as CONTENTS. PART I. THE DIFFERENTIAL 0/LGULU8. Chapter I. Of "V abiables and Functions ^1. Nature of Functions. 2. Their Classification. 3. anc- tional Notation. 4. Functions of Severa' Variables. 5. Func- tions of Functions. 6. Product of the First n Numbers. 7. Pi- nomial CoeflScients. 8. Graphic Representation of Functions. 9. Continuity and Discontiauity of Functions. 10. Many valued Functions. PAGE 3 •j^ Chapter II. Of Limits and Infinitesimals 17 § 11. Limits. 12. Infinites and Infinitesimals. 13. Properties. 14. Orders of Infinitesimals. 16. Orders of Infinites. Chapter III. Of Differentials and Derfvativrs 25 §16. Increments of Variables. 17. First Idea of Differentials and Derivatives. 18. Illustrations. 19. Illustration by Velocities. 20. Geom trical Illustration. Chapter IV. Dk::perentiation of Explicit Functions 31 § 21. The P "ocess of Dilierentiation in General. 23. 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. Logarithmic Functions. ?9. 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 of Several Variables and Impli- cit Functions 54 § 34. Partial Differentials and Derivatives. 35. Total Differen- tials. 36. Principles involved in Partial Differentiation. 37. Dif- ▼I CONTENTS. fcrentiation of Implicit Funciious. 88. Implicit Functions of Sev- eral Variables. 89. Case of Implicit Functions expressed by Simultaneous Equations. 40. Functions of Functions. 41. Func- tions of Variables, some of which are Functions of tlie Others. 42. Extension of the Principle. 48. Nomenclature of Partial Derivatives. 44. Dependence of the Derivative upon the Form of the Function. PASE Chapter VI. Derivatives op 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. 40. 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 op 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 op Functions op 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 Varir.bles connected by Equations of Condition. Chapter X. Indeterminate Forms 128 §70. Examples of Indeterminate Forms. 71. Evaluation of 73. Form co — oo. 00 75. Forms 0° and 00 °. the Form— . 72. Forms — and X oo, 00 CONTENTS. VU Vkom CnAPTER XI. Op Plane Curves 187 ^ 76. Forms of the Eiiuiitions c. Curves. 77. Indnitcsimal Ele- ments of Curves. 78. Properties of Intinitesimul Arcs and Chords. 70. Expressions for Elements of Curves. 80. Equa- tions of Certain Noteworthy Curves. The Cycloid. 81. The Lemniscate. 83. The Archimedean Spiral. 88. 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. Moditied Forms of the Equation. 88. Tangents and Normals to the Conic Sections. 80. Length of the Perpendicular from the Origin upon a Tangent or Normal. 00. Tangent and Normal in Polar Co-ordinates. 01. Perpendicidar from the Pole ujion the Tangent or Normal. 03. Equation of Tangent and Normal derived from Polar Equation of the Curve. Chapter XIII. Op Asymptotes, Singular Points and Curve-tracing 157 §03. Asymptotes. 04. Examples of Asympt'^tes. 05, Points of Inflection. 06. Singular Points of Curves. 07. Condition of Singular Points. 08. Examples of Double points. 00. Curve- tracing. Chapter XIV. Tiieory op Envelopes 169 §100. Envelope of a Family of Lines. 101. All Lines of a Family tangent to the Envelope. 103. Examples and Applications. Chapter XV. Op Curvature, Evolutes and Tnvolui'es 180 § 103. Position; Direction; Curvature. 104. Contacts of DiflFer- 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 ol Curvature when the Abscissa is not taken as the Independent Variable. 100. Ra- dius of Curvature of a Curve referred to Polar Co-ordinates. 110. Evolutes and Involutes. 111. Case of an Auxiliary Variable. 113. The Evolutc of the Parabola. 113. E volute of the Ellipse. 114. Evolute of the Cycloid. 115. Fundamental Properties of the Evolute. 110. Involutes. VIU CONTENTS. PART II. THE INTEGRAL CALCULUS. PAOB CiiAPTEB I. The Elementary Forms op Integration 201 ^117. Deflnition of Integrutiou. 118. Arbitrary Constant of InUjgratlon. 110. Integration of Entire Functions. 120. Tiie Logaritliinic Function. 121. Another Metliod of obtaining the Logarithmic Integral. 122. Exponential Functions. 128. The Elementary Forms of Integration. Chapter II. Inteqrals 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. lieduction to the Loga- rithmic Form. 127. Trigonometric Forms. 128. Integration of and ~ -::. 129. Integrals of the Form / - 131. a" 4" ** «* — «■■'' • o J a-\-bx-\-eji^^' 130. Inverse Sines and Cosines as Integrals. 131. Two Forms of Integrals expressed by Circular Functions. 132. Integration of 183. Integration of - ^ ^ ;;■ 134. Exponen- t/a'^ T x^ tial Forms. Chapter III. ^a-\-bx ± Gxi^ Integration by Rational Transformations. . 222 x^dx , xdx §185. Integration of ^-^^^-dir, and «« "''{a-\-bx)» a-\-bx±cx'*' 136. Reduction of Rational Fractions in general. 137. Integra- tion by Parts. Chapter IV. Integration op Irrational Algebraic Dip- perentials 233 §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 i^a -{- bx -{- cx^. dv 140. Integration of dO = — . 141. General Theory r yar"^ -\-hr - \ of Irrational Binomial Differentials. 142. Special Cases when m -f- 1 =: w, or m-\-\ -\- np = — n. 143. Forms of Reduction of Irrational Binomials. 144. Formulae A and B, in which m 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. PAOK Chapter V. Integration of Transcendent Functions 246 ^ 148. Integration of / «"•* cos nxdx and / «m* sin nxdx. 149. Integration of sin'" a; co8'» xdx. 150. Special Cases of / sin"* x (Ix cos^xdx. 151. Integration of ——T-;; , — j — ~. 152. Integra- * m' sm^ X -f- ii' cos^ x " dy tion of — I — i in' sin"'' X -f- n' cos" x' 153. Special Cases of the Last Two Forms. a + * cos //■ 154. Integration of sin mx cos nxdx. 155. Integration by Devel- opment in Series. Chapter VI. Of Definite Inteoralb 265 ^156. Successive Increments of a Variable. 157. Differential of an Area. 158. The Formation of a Definite Integial. 150. Two Conceptions of a Definite Integral. 100. Differentiation of a Definite Integral witli respect to its Limits. 161. Examples and Exerci.ses in finding Definite Integrals. 162. Failure of the Method when the Function becomes Infinite. 163. Clmnge of Variable in Definite Integrals. 164. Subdivision of a Definite In- tegral. 165. Definite Integrals through Integration by Parts. Chapter VII. Successivb 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. 173. Triple and Multiple Integrals. 173. Definite Double Inte- grals. 174. Definite Triple and Multiple Integrals. 175. Product of Chapter VIII. Rectification and Quadrature §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. 18G. The Hyperbola. 187. The Lemniscate. 188. The Cycloid. Chapter IX. The Cubature op VoiiUMES §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 Hyperboloid 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. 285 297 :•! PART I. THE DIFFERENTIAL CALCULUS. I ill USE OF 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 =, the latter may commonly be read is put for, or is defined as. Wnen the single letter follows the symbol, the latter may be read which let ns call. In each case the equality of the quantities on each side of E 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. I PART I. THE DIFFERENTIAL CALCULUS. s work indi- ed to repre- side of it. E, the latter s, 3 latter may sach side of is assumed tnption, any 3y the sign % CHAPTER I. OF VARIABLES AND FUNCTIONS. 1. In the higher mathematics we conceive ourselves to be dealing Avith pairs of quantities so related that the value of one depends upon that of the other. For each value which wo assign to one we conceive that there is a corresponding value of the other. For example, 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, r.nd 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 I hour, and if we ask hoAv long it will take the train to travel 15 miles, 30 miles, 60 miles, 900 miles, etc., we shall have for the corresponding times, or functions of the distances, half an hour, one hour, two hours, thirty liours, etc. THE DIFFERENTIAL CALCULUS. I In thinking thus we consider the distance to be travelled as the independent variable, and the time as the function of the distance. Example II. If between the quantities x and y wo have the equation y = 2ax^f we may suppose a; = - 1, 0, + 1, + 2, + 3, etc., and we shaU then have y = 2a, 0, 2a, Sa, 18a, etc. Here x is taken as the independent variable, and y as the function of x. 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 anal}i;ic 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 = x\ 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. Kemark. 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. be travelled as 1 notion of the ,nd y wo have and y as the liere is a corre- is is expressed tpressions, the er. jnts it stands Dlicit function n value of //, nsidering y as tion of X, be- ilue of y'y but le equation it and implicit the different Thus in the y = 2aa;', y — 2ax^ = 0, \y is the same function of x] but its form is explicit in the firs^-, [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 lor 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 Ifunttion 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- i lent. Indeed any algebraic expression containing a symbol is, |by definition, a function of the quantity represented by the jeymbol, because its value must depend upon that of the sym- Ibol. Every algebraic expression indicates that certain operations fare to be performed upon the quantities represented by the I symbols. These operations are: 1. Addition and subtraction, included algebraically in one lass. 2. Multiplication, including involution. 3. Division. 4. Evolution, or the extraction of roots. A function which involves only these four operations is ailed algebraic. Functions are classified according to the operations which ust be performed in order to obtain their values from the alues of the independent variables upon which they depend. A rational function is one in which the only operations ndicated upon or with the independent variable are those of ddition, multiplication, or division. 6 THE DIFFERENTIAL CALCULUS. \m 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-\- ex* -\- dx* is an entire function of x, as well as of a, h, c and d. The expression . m . c 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 Va -f bxy {a + mx* -\- 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 + «;)"», a^ are entire functions of x when n and y are integers. But they are exponential functions of y. Othor transcendental functions are; Trigonometric functions, the sine, cosine^ etc. IiOgarithmic functions, which require the finding of a logarithm. CirculsLT functions, which are the inverse of the trigo- nometric functions; for example, if y = a trigonometric function of x, sin x for instance, then :r is a circular function of y, namely, the arc of which y is the sine. VARIABLES AND FUN0TI0N8. h the variable integers. But ) of the trigo- 3. Functional Notation. For brevity and generality we lay represent any lunction of a 'variable by a single symbol laving a mark to indicate the variable attached to it, in any form we may elect. Such a symbol is called a functional lymbol or a symbol of operation. The most common functional symbols are F, f and 0; )ut any signs or mode of writing whatever may be used. [Then, such expressions as F{x), f(x), {bx), A^')» A^')> A^'h A^% 7. Let us put 0(m) = m(m — 1) (m — 3) (m — 3); thenco ind the values of K6), 0(5), 0(4), 0(3), 0(2), 0(1), 0(0), 0(- 1), 0(- 2). 8. Prove that if we put (f){x) = a*, we shall have 0(a: + y) - 0(^) X 0(y); 0(a:y) = [0(2:)]" = [0(2/)]'. 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. ExAMPTES. We may put 0(a:, y)^ax — ly, the comma being inserted between x and ;/ so that their )roduct shall not be understood. We shall then have 0(m, n) = am — 5w, 0(y, x)-ay — hx, jthe letters being simply interchanged; 0(« ■\-y,^-y) = ci{x-^y)- l{x - y) = (a - l)x + (a + h)y; (a, h) = a'- h') 2/) = 2^ + 3« — hy, p(z, y, x) = 2z + 3y - 6x; 0(w, m, —m) = 2m f 37?i -f- 5m = 10m; 0(3, 8, 6) = 3-3 -f 3-8 - 5*6 = 0. 10 THE DIFFERENTIAL CALCULUS. Let ns put \ 3. 0(3,4). 9. /(7, - 3). 12. /(^», rt, 2). 15. f(—a,—b,—ab). EXERCTGES. 0(a:, y) EE 32; - 4y; /(a;, y) E ax'\-by; A^f y» z) -, ax -f Jy Thence form the expressions: I. 0(y, x), 2. 0(a, 3). 4. 0(4, 3). s. 0(10, 1). 7. /(J, fl). 8. /(y, x), ^^' f{qy -P)- "• /(^J. a;, y). 13. /(«. ^ c). 14. /(«', *', c')- Sometimes there is no need of any functional symbol except the parentheses. For example, the form {m, n) may be used to indicate any function of m and n, EXERCISES. Let us put (m, n) ~ -^. —} ^^, I ^ ' n{n — 1) (w — 2) ' then find the values of — I. (3, 3). 2. (4, 3). 3. (5, 3). 4. (6, 3). 5. (7, 3). 6. (8, 3). 7. (2, - 1). 8. (3, - 2). 9. (4, - 2). 6. Functions of Functions, By the definitions of the pre- ceding chapter, the expression /(0(^)) will mean the expression obtained by substituting (/){x) for x mf{x). We may here omit the larger parentheses and write f(p{x) instead of /[ \x) = 0(0(3:)). Continuing tho same system, we hare 0*(^) = 0(0'W) =0'(0(^)); 0*(a:) = 0(0*(a;))=0'(0(a:)); eto. etc. etc. Examples. 1. If (p{x) = ax*, then 0'(^) = «(«^')' = «V; 0*(a;) = a{a*xy = a V; etc. etc. etc. 3. If then /{x) ~ a — Xy P{x) =z a — {a — x) = X) f\x) = a — f'ix) — a — x; and, in general. 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 fpx or 10, X > 100, X > 100000, and so on without end, then x is called an infinite quantity. If of a quantity h we either suppose or prove h < 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 approajhes 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 qvantity. But the second has lately been used also as a Doun, and we shall therefore use the word infinite as a noun meaning in unite quantity. LIMITS AND INFINITESIMALS. 19 Bred in the can name; inite nor in- For example, let us have y X -\- a X — a 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, moans *' approaches the limit." The symbols ioo mean ^'increases without limit" or "becomes infinite." Hence the three last statements may be expressed symbolically, as follows: X -\- a When X = a. When ic i CO, etc. then then X — a X -{- a X — a etc. 00 = + 1; The same statements are more commonly expressed thus: a) lim. {x oo X — a lim. ^-±^(a;ico)= +1; X — a^ ' lim. ?-±-^ (a; i 0) = - 1. X — a^ ' 13. Properties of Infinite and Infinitesimal Quantities. Theorem I. The product of an infinitesimal ly any finite factor, however greats 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. we shall have n nh < a. (Axiom III.) um :•'! ■■ I! 20 TFE DIFFERENTIAL CALCULUS. That is, nh is less than a and not less than a, which is absurd. Hence nh becomes less than any quantity we can name, and is therefore infinitesimal, by definition. Theorem II. The quotient of an infinite qumitity ly any finite divisor, hotvever great, is infinite. Proof. Let X be the infinite quantity, and n the finite divisor. It X — n does not increase beyond every limit, let K be some quantity which it cannot exceed. Then \jy taking we shall have X> nK, X ^ (Ax. III.) that is, — greater than the quantity which it cannot exceed, which is absurd. Hence X—-n increases beyond every limit we can name when X does, and is therefore infinite when X is infinite. Theorem III. TJie product of any finite quantity, how- ever small, hy an infinite rnultiplier, is infinite. 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 IV. The quotient of any finite quantity, how- ever great, by an infinite divisor is i?ifinitesimaL 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 an infinitesimal is an in- finite, and vice versa. Let h be an infinitesimal. If j- is not infinite, there must be some quantity which we can name which j ^o^s not ex- LIMIT8 AND INFINITESIMALS. 21 3 can name, innot exceed, nal is an in- ceed. Let K be that quantity. Because h is infinitesimal, we may have A^; 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 Infinitesimals. Def. If the ratio of one infinitesimal to another approaches a finite limit, they are called infinitesimals of the ,samo order. If the ratio is itself infinitesimal, the lesser infinitesimal is said to be of higher order than the other. Theorem VI. If we have a series proceeding according to the powers of h, A+Bh-i- Ch' + Dh' + etc., in luhich the coefficients A, B, (7, are all finite, then, if h he- comes infinitesimal, each term after the first is an infinitesi- mal of higher order than the term p}receding. Proof. The ratio of two consecutive terms, the third and fourth for example, is Dh' _D Ch' ~ D . By hypothesis, Cand 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 or'ler than Ch''. Def. The orders of infinitesimals are numbered by taking some one infinitesimal as a base and calling it an infinitesi- mal of the first order. Then, an infinitesimal whose ratio to h ■''i '^ !' m I' I ! ^ i : P !3- 22 riC& DIFFERENTIAL CALCULUS. the wth power of the base approaches a finite limit is caUod an infinitesimal of the nth order. Example. If h be taken as the base, the term Bh is of cho first order • . * Bh : h — the finite quantity B\ Gie " " second" ','Gli^\W = " " C; EU'' " " ?ith " • . • Eh'' : 7i" = " " E. Cor. 1. Since when ?i = we have Bh"" = Bh° = B for all values of //, 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 VII. In a series of terms arranged according to the powers of x, A + Bx -\- Cx* -\- Dx^ + etc., if A, B, C, etc., are all finite, then, when x becomes infinite, each term, after the first is an infinite of higher order than the term jweceding. For, the ratio of two consecutive terms is of the form ~^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 ni\\ power of the base approaches a finite limit is called an infinite of the nth. 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 INFINITE8IMAL8. 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 fonn 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-Bi, tio corresponding increment of //. 5. Then, by Plane Trigonometry, the quotient --.-- will be the tangent of the angle PQX\ that is, the fangent of the angle Avhich the secant PP' makes with the axis of abscissas. Thus we have geometrical representations of the five fun- damental quantities under consideration. 17. Firf^t Idea of Dijferoifiah and Derivatives, Let us take, for illustration, the function y = nx^. (1) Giving to x the increment Ax, the new value of ;/ will be n{x + Ax)\ Hence y -\- Ay = n(x + Ax^ ~ n^"" -f %nxAx -f nAx", (2) niFFKHKNTIA LS AND DERI VA TI VE8. 97 Subtracting (1) from (2), wo have, for tho iiicrement of y, Jy = ?i{2x -f Jx)Jx, (8) Because, when ^x becomes inliuitosimal, lim. {2x + Jx) = 2x, we luivo, for tho ratio of the increments. -^ = 2nx -\- nAXf and, when Ax becomes infinitesimal, lim. -~ = 2wa;. nx (*) (») Dcf. Tho di£ferential of a quantity is its infinitesimal increment; tliat is, its increment considered in the act of ap- proaching zero as its limit, or of becoming smaller than any quantity we can name. Nofaiion, The differential of a quantity is indicated by < he symbol d written before the symbol of the quantity. For example, the expressions (Ix, du, d{x -\- y), mean any infimtesimal increments of x, ii, {x -f- ?/)> respect- ively. Thus the substitution of d for A in the notation of incre- ments indicates that the increment represented by A is sup- posed to be infinitesimal, and that we are to consider the limit toward which some quantity arising from the increment then approaches. Using this notation, the equation (5) may be written f- = 2«x. ax We also exDtess this value of the limiting ratio in the form dy — 2nxdx', meaning thereby that the ratio of the two members of this equation has unity as its limit. This is evident from Eq. (3). ■'■ t ihl tl >»J r4^ 28 TUPJ DIFFERENTIAL CALCULUS. dv Def. If 7/ is a function of x, the ratio ~- of the differential of y to that of x is called the derivati 76 of the function, or the derived function. 18. Illustrations. As the logic of infinitesimals offers great difficulties to the beginner, some illustrations of the subject may be of value to him. Consider the following proposition: The error introduced by neglecting all the powers of afi in- crement above the first may be made as small as we please by diminishing the increment. Let us suppose ?i = 2 in the equation (1). We then have the equations y = 2x'; Ay — 4:xAx + 3 Jrc'; Ay (a) Ax = 4.C -f 2Ax. I'li- 11 The ratio of the two terms of the second member is 2Ax Ax 4.x ' '^'' W ■^ ^ Let us now neglect this quantity and write the erroneous V "Iv*. equation V <^ ":s- Ax If, now, we suppose Ax < Ax < Ax < x 100' X J_ 200 1 10000' X the equation > {b) will still be {x -f- Ax). (4) We thus have (f)(x + Ax) — {x). We may put A(f){x) = 4>{x + Ax) — a;''f/** + kn^v\ 13. (^• + 5') (^ + 5'). 15. aa;' — - J_?/2;. 17. {a -i- x') {b - f). 19. .r(a -f •'*^) (^ ~ ^')' a ' X — uv 22. 24. a 26. rt 29. («7/' - ?A'6') (.-c - y). 30 (I IX^ 7/ bj\a "^ b 33. (rt + rry)'. 27. x\x''-^y{(i-x)\. 31. (rt+a;)'. 32. n{a -\-xy. 34. (rta; + Z»y)'. Th. IV.) (§ ^^3) v., Cor.) (§33) (Th. V.) v.. Cor.) J, suppos- sent con- DIFFERENTIATION OF KXPLIUIT FUNCTIONS. 'M 36 Differential of a Quotient of Two Variable.'^. Let the variables bo x and y, and let q be their quotient. Then X and qy = x. Differentiating, we have ydq -\- qdy = dx. Solving so as to find the value of dq. Hence: , dx — qdy ydx — xdq dq = i-^- = ^ '-, y y Theorem VI. The differential of a fraction is eqtial to the denominator into the differential of the nuvierator, minus the numerator into the differential of the denominator , divided by tlte square of the denominator. Remark. If the numerator is a constant, its differential vanishes, and we have the general formula d— = .dx. X X m EXERCISES. )V Form the differentials of the following expressions: a -^x X I. a + y a — X a -y a X !• a -\-bx a -{-by X + y x-y 2. (I + //' X' y 6. 8. lO. a {b + yf m -\- nx^ m — nx*' mx!^ -|- ny* mz' — ny** 'I I I 38 II. U- TUE DlFFEliENTlAL VALCUim. a . X -\- yz a 4- bx -\- cx^* m + 'xy VI — x^y^' a , b ^ y 17. a >^,»* xy 4- x'y Tn ^' + -V' 19. -5 j. x' - f . ^m V -j- a;z* 1 1 14. 5' X X 16. m n ? " f' 1 1 18. X y • a -yn x^ - f x' + y a* 37. Differentials of Irrational Expressions. Let it be re- quired to find the differential of the function m m and n being positive integers. Raising both members of the equation to the nih. power, we have ?<" = x^. in X n m — 1 Taking the differentials of both members, nu*^~hlti = mx"*~\lx, whence du _ 7n x^~^ _ m x^~^ 7ix~n u"-^ ~ n 7"^^" -~* a formula which corresponds to the corollary of Theorem V., where the exponent is entire. Next, let the fractional exponent be negative. Then mn — m X ^* m ?*- -1 = -X » n , {<') ' _1i 1 m ' Xn and, by Th. VI., / ■5\ --1 ; ^ dyx'') m.T" dx \ ^^'* — VH: ~ n 2m ~ X n dx, n and, for the derivative. dtc m -HL-i dx n DIFFERENTIATIUxV OF EXPLICIT Fl\NUTWjSti. 39 , it be re- jmbers of From this equation and from {a) wo conclude: Theorem VII. The formula d{x^) = nx^~^dx holds true whether the exponent n is entire or fractional, yosi- (ire or negative. Wo thus derive the following rule for forming the differen- tials of irrational expressions: Express the indicated roots by fractional exponents, positive or negative, and then form the differential hy the preceding methods. Examples. dx I. d Va-\-x=^ d{a -\- re)* = ^{a -f x) - Hx = - 2{a-{-xy 2. d^y^ = d [*(« + ^) - *] = hd{a + x) - * \hia -\-x)- idx — — - — r-T^dx. bx 3. d{a + bx*)^ = i{a + bx') - 4 Uxdx = I ,i! ^1 EXERCISES. eorem V., Form the differentials of the following expressions: I . Va -\- X. 4. Va — x'. a 2. Vb — x. 3. Va — bx. :hen 5. Va — bx"". b Q . 6. Vx -j- y. b '' Vx + y' °' Va + bx* • 9- )/a-bx'' 10. {a-\- x)\. II. {x — a)i. 12. {}>x*-a)\. 13. xVa-{-x. 14. X Va — X. 15. fVa-by" '^dx, (l7t Find the values of -7- in the following cases: m j6. u = mx -\ — . 17. tc = (mx' — w)*. I :] 40 77//i' DIFFEIIENTIAL CALCULUS. 1 8. u = V((x + bx\ 19. w a 20. u = X ^/a — X, n 4- X 22. u = — ■ — , a — X b -h ex'' 21. u = X i/x' -{- a, a — X a -f- a; 38. Logarilhmic Functions. It is required to difTerentiate the function ?t = log X, Wo h;ivo An = log (r + Z/.C) -- log x = log '^^- = log (l -f -^)- It is shown in Algebra that we have log (1 -I- h) = M{h - W + W - etc.), M being the modulus of the system of logarithms employed. Hence, puting — ^ for h, we find /^ iTA/, 1 J.r , 1 Ja:' , \ and, passing to the limit. du = ; X du _ M dx x' In the Napcrian system M = 1. In algebraic analysis, logarithms are always understood to be Naperian logarithms unless some other system is indicated. Hence we write J-loff X 1 , , dx -^~=-; d-\of!:x = — . dx X' ° X Example. , , d(axy) axdy + aydx dy , dx fZ-log axy = --^^ — — = ^-^ — ■■ — = -- H . ^ ^ axy axy y x Remark. We may often change the form of logarithmic 3* 5. 7. 9. II. 17. Diffe DlFFKlttJATlATW.N OF KXPUCIT FUNCTIONS. 41 rentiate fiiuctiouH, 80 as to obttiiii iliuir (lilTonJutials in vuriouH ways. Tlius, in tho last exaniplo, wo havo log {axy) = log a + log x -\- log y, from which wo obtain tho samo ditToroiitial found abovo. Tho studont should find tho following (.lilfurontials in two ways when practicable. EXERCISES. ^^)- ployed. analysis, Tarithms rite dx X garithmic Differentiate: I. log {a -f- x), Ans. 3. log {x' + b'). 5. log mx. 7. log {ax'' -f- b), 9. log {x + y), 1 1 . log xy. 13. \og{a-^b)\ ^ X -\-a 15. 10-.T — —7. 17. ylogx. dx a-{- X 2. log {x — a). 4. log (.0' - b). 6. log 7nx^. 8. log m''. 10. log {x - y). 12. log (a;' + ?/'). X 14. log -. y . , a — X 16. log T . b-y 18. log (r« — .r)*". 29. Exponential Functions, It is required to differentiate the function u = «'", a being a constant. Taking the logarithms of both members, log u =■ X log a. Diff.Qrentiating,, wo have, by the last article, fl?'log u = — = dx log «. ,11 I i! i i : '\ 42 THJ^ DIFFERENTIAL CALCULUS. Hence du = it log a dx = a" log a dx', which is the required derivative. If a is the Naperian base, whose value is e = 2.71828 we have log « = 1. Ilcnce d'e" s • • a dx = c*. Hence the derivative of c* possesses the remarkable prop- erty of being identical with the function itself. i i EXERCISES. 1 Di^erentiate: I. a^. A us. 2«"^ log rt r/.T. 2. «"*. 3. c" + "- 4. ^«--. 5- ^^ma: + ny ^ 6. /i"**-". 7. 7i-"*. 8. d'ay. 9. a**". TO. cf^b\ II. ah^'b-K 12. e"' + «. 13. t;^. 14. e«* + bv^ 30. T'/t'' Trigonometric Functions. The Sine. Putting h for the increment of x, we have, by Trigonometry, sin (:r + h) — sin .t = 2 cos (a; + i^O ^"^ i^** Now, let h approacli zoro as its limit. Then, sin {x -\- h) — sin x becomes d sin x] h becomes dx, because it is the increment of x; cos {x -\- ^h) approaches the limit cos x; sin ^h approaches the limit ^h or ^dx, because whe\i an angle approaches zero as its limit, its ratio to i':s sine approaches unity as its limit (Trigonometry). Hence, passing to the limit, d'sin X = cos xdx. DIFFEUENTIATIOK OF EXPLICIT FUNCTI0:NS. 43 The Cosine, By Trigonometry, cos {x-\-h) — cos X = — sill (x -f ^A) sin \1u Hence, as in the case of the sine, d cos X •=■ — sin x dx. Taking the derivatives, we have d sin X dx d'COB X dx = cos x; = — sm .r. M N Fig. 8. PB = A sin X, KP — A cos x. Geometrical Illustration. In the figure, let OX be the unit- o radius. Then, measuring lengths hi terms of this radius, we shall have NK = sin x; MB = sin (:r -|- h) ; 02^ = cos x; OM = cos {x i- Zt); Ab », supposing a straight line from A' to B, PK = - KP = KB sin PBK; PB = KB cos PBK. When B approaches K as its limit, the angle PBK ap- proaches XOK, or Xy 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 a« X increases, whence d'co^ x must have the nega- tive sign. The Tangent. Expressing the tangent in terms of the sine and cosine, we have tan X =■ sin X cos x' Differentiating this fractional expression, cos xd's'in X — sin xd'cos x sin' xdx + cos" xdx d tan X = cos X cos X — sec'' xdx, which is the rerjuired differential. i; m ' '1 '; 5 I' 44 THE DIFFEliENTIAL CALCULUS, We find, by a similar process. , , - cos X , - dx a cot X = d'—. — = — CSC xdx = c?'sec x=: d sm X 1 sm X d'Gos X sin xdx cos X cos* X = tan a; sec xdx; f/'cosec a; = — cot a; esc xdx. cos » Differentiate: I. cos {a -f ?y). 4. sin 1/ cos 0. 7. sin (i!:c. 10. sin {h 4" wr?/). EXERCISES. 2. sin {b — y). 5. tan 2c cos v. 8. cos ay. ir. COS (7i -1- my). 3. tan (c + z)' 6. sin « tan v. 9. tan 7/i2;. 12. sin (7i — ?w^). 13. cos' X ' [^Z'cos"^ 2; = 2 cos xd'coa a; = — sin 2xdx]. 14. 17. sm X. sin X 15. sm' y. 16. sin" 7iz. 18. sm a; 19. cos X COS ?/ cos y ' sm* ?/ 20. Show that J(sin' ?/ -[- cos' y) = 0, and show why this result ought to come out by § 24. 21. Differentiate the two members of the identities cos {ci -\- y) = cos a cos y — sin a sin y, sin (a -\- z) = cos a sin 2; -f- sin a cos 2;, and show thn^ the differentials of the two members of each equation arc identical. 22. Show that d'log sin x = cot x dx; d'log cos X = — tan x dx. 31. Circular Functions. A circular function is the in- verse of a trigonometric function, the independent variable being the sine, cosine, or other trigonometri(3 function, an(i the function the angle. The notation is as follows: If y =: sin z, we write z = sin ^~ '* y or arc-sin y; If u = tan X, we write x = tan ^~ ^^ u or arc-tan u; etc. etc, etc. OF' BTFFERENTIATTON OF EXPLICIT FUNCTIONS. 4^ Differentiation of Circular Functions. If we have to dif ferentiate z = sin (- *> y, we shall have y = sin z; dy = cos z dz = Vl ::~^£^ dz; dy dy . • . dz ~ Vl — sin' z \/\ ~— y^ (a) The Inverse Cosine. If z be the inverse cosine of y, we find, m the same way. dz =: -^ dy The Inverse Tangent. If we have z = tan ^- ') ^; then, y = tan .; r7f/ ... sec' ^ rf^ = (1 + tan' ^),?,; .'.dz=:^., r/^e /7^^;me Cotangent. We find, in a similar way, ^^cot<-ly= ^_ ^ 1 + y- ^Ae /wve.'se /Sfecaw^. If we have z = sec ^~ ^> ^; then, y = sec z-, dy = tan z sec zdz = y Vf~^^ dz; . '. dz __ dy yVf~\ The Inverse Cosecant. We find, in a similar way. d-QBG^-^^y y ^y' ~ 1 (^') (^) OO W ill r I ■li Ji 46 THE DIFFERENTIAL CALCULUS. EXERCISES. Differentiate with respect to a; or 2: 2. cos ^~ *) (x + a). I. sin<~*^ ax. 3. sin<~^> (tnx -f- a). 5. tan<-«(0-^). 7. tan <-"(-+-). \a xj 9. see(-») (^+-j. II. sin ^~ *> «a; cos ^" ') X a 4. cos^ *' -. X 6. tan<-« (2 + -). 8. tan<-»)(a;'). 10. sec<~^) Iz ]. 1 2. sec ^~ *> a;' tan <~ *> a;. Note. —The student will Hometimes find it convenient to invert the function before differentiation, as we have done in deducing the diflferen- tlal of sin (- »> x. 13, We have, by comparing the above differentials, ^Z(sin~ ^ y + cos" * ?/) = 0; ^7(tan~ \?/ + cot" ^ y) = 0; ^(sec~ ^ j/ + CSC" * y) = 0. Show how these results follow immediately from the defini- tion of complementary functions in trigonometry, combined with the theorem of § 24 that the differential of a constant quantity is zero. 33. Logarithmic Differentiation. In the case of products and exponential functions, it will often be found that the dif- ferential is most easily derived by differentiating the logarithm of the function. The process is then called logarithmic dif- fcrentiation. Example 1. Find -.^ when v = a;*"*. ax -^ We have log y = m.x log x\ ). )• 1) X, o invert the the dififeren- ils. the defini- , combined a constant 3f products lat the dif- 3 logarithm ithmic dif- DIFFERENTIATION OF EXPLICIT FUNCTIONS. 47 --^ = 7n loff X dx 4- mdxi dy , ^ ^;; = y/("i log ^ + w) = 7«.r«»'(l + log x). Example 2. y = sm"* X cos" X We have log y — m log sin a: — n log cos x\ dy _ m cos x n sin a; ydx sin n; fZ?/ _ sin *" ~ ^ a^ cos X dx cos " + — {m cos' x-\-n sin' a:). a; MISCELLANEOUS ilXERCTSES IN DIFFERENTIATION. Find the derivatives of the following functions with re- spect to a;: I. y = X log X. -4^''*?- '■£■ = 1 -h log X. 2. y = log tan x. 3' y = log cot X. X Ans. — = ■ i i I 48 THlil DIFFERENTIAL CALCULUS. II. ?/ = y 1+ Vi'^-x') 12. y = tan «*. 13- ^ = 2;" t4. y = sin (log a;). yl?i5. -4w." <«o ^^y _ ^(^ + a;) + ^(^ + a:) (a* 4- x^) tan~ * — . Ans, — = 2x tan ~^ — I- a. ^ ' a ax a = Am- = a; + log cos ^^ - = X sin ~ * X. tan X tan ~ * a;. . dy 1 (/« (1 - X) fl _ a;« \ A ^y 2 J dx 1 4- tan a;* A dy Ans. -f- dx dy __ sin ~ ^ ic -j — — X Ans. -y- — sec" x tan ~^a; -|- dx tan X 30. y = sin %a:(8in x^ ins. dy dx = n (sin a;)"~^sin {n -f- l)a;. 31- y _ (sin nxY ~ (cos mxy Ans. -r- dx dy _ *"^^ (sill ^Ja^) "*~^cos {mx — nx) (cos ?rta;) n + l 32. y = = e • a'x' cos ra; ^ws. 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, b) 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 Ans. —J 2b 7t dy dt sin" a' dt 2a7r 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 the end of t hours. (d '>:t| I :ii 64 TUE mFFEUEJSTlAL CALVULUIS CHAPTER V. FUNCTIONS OF SEVERAL VARIABLES AND IMPLICIT FUNCTIONS. 34. Def. A partial differential 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 with respect to a quantity is its derivative formed by supi)osing that quantity to change while all the others remain constant. Remark. The adjective /?rt?//rtZ may be omitted when the several variables are entirely independent. Example. Let us have the function u = x'{y + 2) + yz, (a) Differentiating it with respect to a; as if y and z wore con- stant, the result will be (lit ■=. 2x{y -}- z)dx, 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* -f- z)dy, (c) fdu^ (b) (du\ \dyl x' + 2, PA li TJA L DKlil VA TI VE8. 65 of sov- ) of the Lt. 3rcntiul osod to ct to a uantity len the which are tho partial difforcntial aiuJ durivativo with respect to y. For the partial diilfjreutial aud derivative with respect to z wo have du = («" -f y)dz\ (d) Notation of Partial Derivatives. 1. A partial derivative is sometimes enclosed in pa; jnthoses, as wo have done above, to distinguish it from a total derivative (to be hereafter de- fined). But in most oases no such distinctive notation is necessary. 2. In forming partial derivatives the student is recom- mended to use the form Dm instead of -r-, * dx * because of its simplicity. It is called ///« />j, o/*?/. The equa- tions following {b), {c) and {d) would then be written: D^u = ^x{y + z)'y ^^yU = X"^ + Z'y D^u = x' -i-y. m I {a) Ire con- io. EXERCISES. have (0) Find the derivatives of the following functions with respect to X, y and z'. DyV = —x-^2y; D^v = 0. 3. u = x^y'z*. 1. V = x^ — xy + y^' Ans. D^v = 2x — y; 2. w ^ x' -\- x^y -f- xz, 4. u = X log y + y log a;. 5- ^* = (^ + ^ + z)\ 6. u — \/{x -\- my). 7. u = {x + 2?/ -f- 3z)*. Note. In forms like the last three, begin by taking the total differential, thus: du = i{x -\-2y + dzy^d • {x + 2y + 3z) = U^ -\r2y-^3z)-^ {dx -f- 2dy + 3dz), II 56 I'HE DIFFERENTIAL CALCULUS. Then, supposing x alone to vary, D^u = supposing y alone to vary, DyU = supposing z alone to vary, D.ti = *• 2{x+2y+3z) 1 {x + 2y-{- dzf 3 ]i{x-{-2y-i-3z) i' 8. 7v = {x -f y -\- ^)^ lo. w =■ cos {inx ~j- ?/). 12. V — tan {x — ?/"). 14. V = cos* (rt.i' -j- Z'^;). 16. u = xe" -f- 2/^'"« 9. ?y = (.t" + y' + 2;')". II. w = sin {x + 2jy + 3z). 13. ?; = sec (wa; + nz). 15. y = 6'* •■''. 17. ?^ = ic" -|- y*. iS. ?i = sin {x-\-y) GOs{x—y). 19. u = x sin y — y sin ic. 35. Fundamental Theorem. T'Ac /o^a^ differential of a function of several var tables j all of tvhose derivatives are continuonti, is equal to the sum of its partial differentials. As an example of the meaning of this theoreni take the example of the preceding article; where w^e have found three separate differei als of it, namely, (b), (c) and (d). The theorem asserts .lat when x, y ani z all three very, the re- sulting differential of u will be the sum of these partial differ- entials, namely, du = 2x{y -f- z)dx -j- (x^ -f- z)dy -f (a;' + y)^^» To show the truth of the theorem, let us first consider any function of two variables, x and y, u z=z (p{x, y). (1) Let us now assign to x an increment /?.t, while // remains unchanged, and let us call u' the new value of u, and ^^.U' thQ resulting increment of u. We shall then have /t^u = (p(x -f Jx, y) - 0(.r, y). (3) ? TOTAL DIFFERENTIALS. 57 32) *• 32) i* f 32) h32). 0. 3m 2;. j^mZ of ms are (ds. ,ke the three The ne re- differ- er any (1) emains JO the (2) In the same way, if x retains its vahie while y receives the increment ^y, and if we call z/„w the corresponding incre- ment of w, we have Jy2i = ,t., presupposes (1) that we have the quantity u given, really or ideally, as an explicit function of .r, 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 functiou of x, we want rules for finding the values of Dg,Uy 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 u = y) = y -ax = 0. We have d d(p . di/ — ::^ fi' — ^^ ]_• '^ — d' dx * dy * dx * 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 tA\o differentials of each of the quantities x. y and (p, all that we are concerned with being the limiting ratios dy : dx\ d(p : dx, and d(p : dy, which limit- ing ratios are functions of x and y. We may, indeed, if we choose, suppose the two dr's equal and the two (fy's equal. But in this case the two rf0's must have opposite algebraic signs, because their sum, or the total differential of 0, is necessarily zero. Now, if we change the sign of either of the tZ^'s, wc shall get a correct result by a fractional reduction. 1 DIFFERENTIATION OF IMPLICIT FUNCTIONS. 61 ast be J as to lation. (1) uction irant to ng the clearing e. This the two :)ncerned oh limit- the two ilgebraic 'ily zero, a correct EXERCISES. Find the values of -j-, -j- or -j- from the following equa- tions: I. y — ax = 0. 2. y* — yx -^ x^ = 0. 3. x' + 4-.XZ 4- z" = 0. 4. u{a~x)-^tt'^{b -\-x) = 0. 6. log (x-^y) + log (x-y) = c. 8. sin ax — sin by = c. 10. X {1 — e cos z) = a. 5. log a: + log 2/ = c. 7. sin a; + sin y = c. 9. u -\- e sin ?* = a;. 38. Implicit Functions of Several Variables. The pre- ceding process may be extended to the case of an implicit function of any number of variables in a way which the following example will make clear. Let u be expressed as a function of x, y and z by the equation u' + xu^ + {x' 4- y^)ic + a;' + y* + z' = 0. Since this expression is constantly zero, its total differential is zero. Forming this total differential, we have (3w' + 'Hxu + a;' + y'')du + (w' + %ux + 3a;')^a; + {^uy + ^^)dy + ^z\lz = 0. By § 34 we obtain the derivative of n with respect to x by supposing all the other variables constant; that is, by putting cly = 0, dz = 0, and so with y and z. Hence du ^ u^ 4- 2ux 4- 3x^ ^^ - ^x^ - 3^^u _|_ 2ux + a;' + 2/" fZ?* ^ 2uf/ -{- 3?/' dy ~ " ~ 3^" + %ux + a;' -f- f' du^_ r. 3«^ dz " '^'' "" 3^" -h 3wa; + a;' -f iy'' m ■if M I ipi, 62 THE DIFFERENTIAL CALCULUS. EXERCISES, Find the derivatives of u, v or r with respect to x, y and z from the following equations: 1. xu^ + ifu^ -\~ z*u = x^yz» 2. a CC3 {x — u) A^h sin {x-\-u) ^=. y. 3. u'^-\-uy = ii\ 4. r* + »' + r*-" = r*. 5. V log X -\- z log V = y. 6. G^ cos a; + c* cos y = <;". 7. «' — 3wa; cos z -{- x' = a\ 8. v" 4- 3y.r cos 2 -f- a;' = Z^'. 39. Case of Implicit Functions expressed hy Simvlta- ncons Equations. If we have two equations between more than two variables, such as F^it, V, X, y, etc.) = 0, -^,(w, v, x, y, etc.) = 0, then, if values 01 all but two of these variables are given, we may, by algebraic methods, determine the values of the two which remain. We may therefore regard these two as func- tions of the others, the partial derivatives of which admit of being found. In general, suppose that we have n independent variables, a:,, x^ . . . .r„, and m other quantities, n^, u, . . . w„, connected with the former by m equations of the form FXu,, u, . • • '^^m) -^i) -^a • ' . ^«) = F,{u^, ti, . • • '^^m> "^if "^"a • . . a;„) = ^m\^\i ^a • • • ^m? *^i* X^ . , . X„f — {). J (") By solving these m equations (were we able to do so) we should obtain the m u'b in terms of the n x'8 in the form u. — 0,(^1, X^ , , . X„)', • • u, m — Y*m(''^j> X^ . . , Xf^j'f ^ (*) i DIFFEBESTxATION OF IMPLICIT FUNCTIONS. 63 (") (*) s mn values of the derivatives -r-*; -r-*; . . . and by differentiating these equations (b) we should find the — '• 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^ J dF, , , -J—- du. 4- -r~ du, + . . . + -7—' dUn = — T"^ dx, — etc. : du^ ' du^ ' du^ dx^ ^ ' dF^ , , dF^ ^ . , dF, J Fd, , , -f— ^ du, 4- -rr-- du. + . . . + -7— dti^ = — t^ dx. — etc. : du^ * du, • dUn dx^ ' dFm. J dFm J . , dF m du d^K dF^ dx. dx, — etc. du^ ' du. By solving these m equations for the m unknown quantities du^, du, . . . du^j we shall have results of the form du^ = M^dx^ -\- M,dx, + • • • + MJlx^', du, = N^dx^ + N,dx, + . . . + iV„c?a;„; etc. etc. etc. etc.; where M^, JV„ etc., represent the functions of «, . . , u x^ . . . Xn, which ai'e formed in solving the equations. We then have for the partial derivatives die m> dx, ^*^>' dx. ■ = M,; etc. Example. From the equations roos0 = x,) (^,j r 8m = y, ) it is required to find the derivatives of r and with respect to X and y. i ill fi '1! - I ) I ii 64 THE DIFFERENTIAL CALCULUS. By differentiation we obtain cos ^dr — r sin Odd = dx; sin 6dr -\- r cos 6d6 = dy. Multiplying the first equation by cos and the second by sin 0, and adding, we eliminate dO. Multiplying the first by — sin and the second by cos 0, and adding, we eliminate dr. The resulting equations are dr = cos 0-2i' -\- sin 0dy; rdd = *:'>,3 Air - sin 0dx. ft. Hence, as in the last sec. v>n, /^^ _ _ EL?. (^ _ \dxj ~ r ' V/v/ ~ cos r EXERCISES. 1, From the equations r sin 6^ = a; — y, r cos = X -\- y, find the derivatives of r and with respect to x and y» 2, From the equations US'" = r cos 0, ue~^= r sin ^, find the derivatives of u and v with respect to r and 0. Ans. £)=i(<3«8ine + o-«cos«); (^) = ^C^" cos ^-c- sine); = lr<^""^'"''+^" "»='')• ■ FUNCTIONS OF FUNCTIONS. 00 0); 3. From the equations w' -}- rw = a;' + y*» ru = xy, u find the derivatives of r and u with respect to x and y. 4. From the equations x'-\-y'-\-z' - 2xyz - 0, n ;i dZ ^ dZ find -7- and -7-. «a; ay 5. From ?*' — 2wz cos ^ -f 2;' = rt", w' + 221Z cos 6 -\- z^ = Z»', -, , r??< f/?f fZw (hv ^ dz' dO' dz' de' 40. Functions of Functions. Let us have an equa .or. of the form ?*=/(0. t, Qy etc.); {a) where 0, ?/', B, etc., are all functions of x, admitting ol being expressed in the form 0=/iG'^); ^/'=/,(^); ^=fz{^)\ etc. (*) If any definite value be assigned to .r, the values of 0, //', ^'^^ etc., will be determined by (i). By substituting these val- ues in {a), u will also be determined. Hence the equations («) and {}}) determine u as a function of x. By substituting in (a) for 0, ?/', B, etc., their algebraic expressions /,(:r), f^{x), etc., we shall have u 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 («) we have du = -n(l

) expresses the total change. It is in distinguisliing the two vahics of a derivative thus obtained that the terms pdrfial derivative and total derivative become necessary. If we have a function of the form u =f{x, y,to... z), in which any or all of the quantities x, ?/, m, etc., may be functions of z, then the partial derivative of u witli respect to z means the derivative when we take no account of the variations of x, y, tv, 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 ])y being enclosed in parentheses (§ 34). This is why the last equation is written du _ (du\ Uhi\ dx dz ~ \dzl \dxl dz' 42. Extension of the Principle. 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 = (p{u, V, w . . . X, y, z . . .), in which x, y, z, etc., are the independent variables, while iCf V, w, etc., are functions of these variables, we shall have Then, since u, v, zv, etc., arc functions of x, y, z, etc., wo have du = -y-dx + -j-dy + etc. ; dx dy dv = -j-dx 4- -j-dy + etc, dx ^ dy ^ fin FUNCTIONS OF FUNCTIONS. CO By subsiitutiug those values in the preceding equation we find* _\(ly) ''' [dul dy "^ \ ^/0\ do . ~1 , + TTenco, writing r for 0, its equivalent, r [dr\ , f dr\dic , (dr\dif , , dx etc. etc. etc. etc. EXERCISES. The independent variables r and ^ being connected with x and y by the equations .'c = r cos ^, y = r sin ^, it is required to find the derivatives of the following functions of X, y, r and 8 with respect to ?' and 0. We call each of the functions ti. I. 11 z= 0'^ -\- 2xy cos 36'. Here we have © = -= idu\ \do) f^ = 3Z/ cos 20; ax dii dy dx . -— = cos 0; dy dr -y, dy dO = 2a! cos 20; = sin ; = r cos = 2!. I I 'I * Here, wlien we use the symbol cj) instead of r, there is really no need of enclosing the partial derivatives in parentheses. We have done it only for the convenience of the student. ilf - J hi 70 TJT£? DIFFERENTIAL CALCULUS. rr ^^^ _ /^'^\ j^du dx du dy dr ~~ \dri ' ; V' I -j- w^ — 2p sin 6. \ • • • • («) It is required to find the total derivatives of the following functions with respect to p and 6 respectively: 8. u =^ v"^ -{- w' nh 10. n = vw (J, u = v^ — 2vw cos -\- id^, II. u-=^ {v -\- w) sin 6, 12. u = {v — w) cos ^, 13. 7^ = 'w^ — v'' -[- 2{w -f- ?')p cos (^. 1 J PARTIAL DEIilVATIVES. 71 by sub- of r and innot be thod. f id ^by . («) lowing From the pair of equations (a we find dv V dw w dp ~V dp ~2p' dv dO = ivc. ■~0; dw , dO''' ziiW cot 0; which values are to be substituted in the symbolic partial derivatives of u. 43. Remarks on the Nomendature of Partial Derivatives* There is much diversity among mathematicians in the no- menclature perta-ning to this subject. Thus, the term " par- tial derivative^' is sometimes eii:tended 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 " nd other German writers put the total deri- vatives in pareni.'.oses and omit the latter from the partial ones, thus reversing the above notation. If we have to express the derivative of 0(.r, y, z, etc.) with respect to z, the English writers commonly use the symbol -y in order to avoid writing a cumbrous fraction. have such forms as We thus d [x dx 7) f^' _L -^ 4- ^"V 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, y) (a) a function of these variables. AVithout making any change in u or .r, let us introduce, instead of y, another independent m i ! 72 variable making %, s y '■'i the TUE DIFFERENTIAL CALCULUS. upposed to be a function of x and y. Then, after substitution, we shall have a result of the form ?f = F{x, z). iP) Now, it is to be noted that although both u and x have the du same meaning in {b) as in {a), the value of -j-; will be differ- ent in the two cases. The reason is that in {a) y is supposed constant when we differentiate with respect to x, while in {b) it is z which is supposed constant. Analytic Illustration. Let us have u = ax^ -\- by^. du This gives dx = 2ax. (c) Let us now substitute for y another quantity, z, determined by the equation z = y -{- X or y = z — X. We then have ^i = ax-^ -\'b{z — xY', . —- = 'Zax -f- '^b{x — 7)\ which is different from {c). Our general conclusion is: The jjnrtial derivative of one variable luith respect to another depends not only upo}i 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 B be the polar co-ordinates of a point P, and x and ?/ its rectangular co-ordinates. Then Fig. 9. X = r cos B\ 7/ = r sin 8) r' = ic' + y\ (d) fr th th m( fn Tl: no] gre PARTIAL DERIVATIVES. 78 after >rm (*) ve the diller- 3pose] -.- i lis value may call . If we II ;m 11 DERIVATIVKS OF EIOnER ORDERS. 75 » f'J Then we may put Au^y' ~y = MQ, A'y = y''-y' = NR, as the two correspondirg increments of y. It is evident that these increments will not, in general, be equal; in fact, that they can be equal only when the thre*^ points of the curve are in the same straight line. If D is the point in which tlie line PQ meets the ordinate of Ry then DR will be the difference between the two values of Ay, so that we shall have DR — A'y — Ay — increment of Ay. Hence, again using the sign A to mark an increment, we shall have DR = A Ay = A'y, (h) in which the exponent does not indicate a square, but merely the repetition of the symbol A. Theorem I. When Ax becomes infinitesimal, A'^y becomes an infinitesimal of the second order. For, if D be the point in which PQ produced cuts the ordinate X^R, we shall have, in the triangle QRD, 7-. 71 ^ Tasini? 07) .. (^) sin QRD When Ax becomes an infinitesimal of the first order, so do both QD and the angle RQD, but the angle QRD will remain finite, because it will approach the angle QDN as its limit. Hence the expression will contain as a factor the product of two infinitesimals of the first order, and so will be an infini- tesimal of the second order. Since both the quantities QD nnd RQD depend upon Ax, we conclude that the ratio Ay Ax' may remf.',Hi UrAu'' when Ax l)ecomes infinitesimal. In fact. M 76 THE DIFFERENTIAL CALCULUS. f from the way we have formed these quantities, we have lim. 2^. = lim. -^ = ^^ = 0"(^). Hence — Theorem II. If we take tiuo equal consecutive infinitesimal increme7its, = dx, of the independent variable, then — 1. The difference between the corresponding infinitesimal increments of the function divided by dx^ will approach a certain limit. 3. This limit is the derivative of the derivative of the function. Def. The derivative of the derivative is called the second derivative. The derivative of the second derivative is called the third derivative, and so on indefinitely. Notation. The successive derivatives of y with respect to X are written dy d^u d^xi dx' dx'' dx'' ^^^'' I ? or I^xV; DxVr J^xVy etc. 46. Derivatives cf any Order. The results we have reached in the last article may be expressed thus: If we have an equation y = :r. 4*7. Special Forms of Derivatives of Circular and Ex- ponential Functions. Because cos X = sin (x 4- ^tt) and — sin x = cos {x -f- i^r), the derivativet: of sin x and cos .1* may be written in the form Djg sin ./; = sin [x -\- ^n) and /)x cos x = cos (.?; + ^tt). Hence, the sine and cosi^ie are such functions that their derivatives are formed by increasing their argument by ^tt. Differentiating by this rule 71 times in succession, we have ^" sin X Dj" sin x = DJ* cos X — fZ" COS X sin \x-\-^^7rj; n cos(:r + -;r); W. results which can be reduced to the forms found in Exercises 29 and 30 preceding. = COS X rivatives X what? X what? and Ex- he form id their \Q have ixercises f m DERIVATIVES OF UIGIIEli 0RDER8. 79 48. Successive Derivatives of an Implicit Function. If the relation between y, the function, and x, the independent variable, is given in the implicit form f(x, y) = 0, then, putting u for this expression, we have found the first derivative to be du dy _ dx . . dx ~ Sw' ^ ' Tiy The values of both the numerator and denominator of the second member of this equation will be func^tions of x and 2/, which we may call X^ and Y,. Wo therefore write dy dx Y (b) Differentiating this with respect to x, we shall have d'y _ ' dx "^ ' dx dx' y; (c) X^ and Yf being functions of both x and ?/, we have (§ 41) dX, ^ (d,X\ IdXXdji. dx \ dx I \ dy Idx^ dY, ^ (^A , I^^sYM dx \ dx 1 \ dii Idx' dy Substituting in these equations the values of ~ from (J), and then substituting the results in {c), we shall have the re- quired second derivative. The, process may then be repeated indefinitely, and thus the derivatives of any orders be found. Example. Find the successive derivatives of y with re- spect to X from the equation X* — xy -j- ^' E ?^ = 0, 11 '(| V : i I 80 TUB DIFFERENTIAL CALCULUS. Wehave ^| = 2:. - y; ^^ = - a: + gy; (1^1 ^ 2a; - y , rtfa; a: — 2y' which is a special case of {a) and (/>), and where X^ = 2.6' — y/ and Y^ = — x -\- 2fj. Differentiating tlie equation (a'), wo have d^y ^'' ^''^ IbT ^^'' ^^ dx («') nfa:" (.-2,)(2-|) + (2.-41 - Substituting the vahie of ~ from [a'), we have (Vj _ {x - 2//) ( - 3//) -t- 3r(2. r - //) dx'' ~ {x - tjiY - Cf-27/r' -(x--2.yf EXERCISES. Find by the above method the first two or three derivatives of V with respect to x, y or z, from the following equations: I. zv = a(v - z). An,. ^- = j^,. 2. V y + vy = a. 3. y' + z^.T + //' = b. 4. r(^< — .r)' + v''{b -{- x) = c. 5. log {v 4- z) + log (y - 2;) = c. 6. sin 7/;?; — sin y^?/ = h. 7. ?'(1 — (( cos z) = h. 8. If ?<^ — (3 sin ?^ = (J, show that ded(j (1 — c cos u)*' («') fivatives tions: ) DKltlVATIVKS OF HIOUKlt OltDKltS. 81 41). Leibmtz'h Thkohem. ToJUkI the sucvcHsice dcruut- tives of a product in terms of the successive derivatives of Us factors. Let /'^ N? iV \\ ^^ «' 6^ % A> '^ r^^" 23 WEST MAIN STREET WEBSTER, NY. 14S80 (716) 872-4503 ■!< i 82 THE DIFFERENTIAL CALCULUS. 50. Successive Derivatives with respect to Several Equi- crescent Variables. Studying the process of § 45, it will be seen that we supposed the successive increments of the inde- pendent variable to be equal to each other, and to remain equal as they became infinitesimal, while the increments of the functions were taken as variable. This supposition has been carried all through the subsequent articles. Def. A variable whose successive increments are supposed equal is called an oquicrescent variable. We are now to consider the case of a function of several equicrescent variables. If we have a function of two variables, the derivative of this function with respect to x will, in general, be a function of x and y. Let us write du dx 7 = (t>x{^, y)- Now, we may differentiate this equation with respect to y with a result of the form div dx ~dy = ^x,A-'^,y)' Using a noteUin similar to that already adopted, we rep- resent the first member of this equation in the form d'^u dxdy In the /^-notation this is written In either notation it is called ^Uhe second derivative of u with respect to x and y." As an example: If we differentiate the function i{> = y^ sin {pxx — ny) {a) DERIVATIVES OF HIGHER ORDERS. 83 (a) with respect to x, and then differentiate the result with respect to y, we have BxU = — = my cos {tnx — ny) ; i>'x..?* ^'w — 2my cos (//ia; — wy) + mny^ sin (ma; — uy). 51. We now have the following fundamental theorem: d^u (I'll dxdy ~ dydx^ or, in words. The second derivative of a function luith reH2)ect to two equicrescent variables is the same whether we differentiate in one order or the other. Let u = M' (1) From those equations it is required to find the successive derivative of y with respect to x. The first derivative is given by the equation dy_ dy_ _du^_ DuV dx " dx " Dyfl du • From the manner in which the second member of this equa- tion is formed, it is an explicit function of u alone. Hence (§ 54) we obtain its derivative with respect to x by taking its derivative Tvith respect to u, and multiplying by -r-. Thus dx d*y dy d'x d^y __ du du^ da^ dx\' I dx ' \du du du* du ' dx dx d*y dy d*x __ du du* du du* [du I This, again, being a function of u, further derivatives with respect to x may be obtained by a repetition of the process. EXERCISES. Find the second derivative of x with respect to y, and also of y with respect to x, when the relation gt x aud y is given by the following equations: I. a; = a cos u; y = h sin u, z, X = a cos 2u; y = h sinu, 3, a; = rt cos 2u; y = J(cos u — sin w), " j 4, x = u — eBinu; y = u-\-6sinu. n I , 1 ■■ ; 'I . ■ 1 'hi; 1 14 5. x^^ u. y = ue^. ill 1 1, 94 THE DIFFERENTIAL CALCULUS. I i 6. Show that if « ,, d*u 3 sin u y = e" cos u, then -r-,- = -^. ; ;-. ay e*'*(cos u — sm u) '/. Show that the wth derivative of »" + «^"''* + ^a;""* is n\, n being a positive integer > 1. 8. Show that 9. Show that if v = u"*, then i» = nu^'-'D^'u -f- 3w(w - l)tc''-W^uDJ'u + ?^(7^ - 1) (?^ - 2)u''-\D^u)\ 10. If 7/ = a cos wia; + h sin wa;, show that DJu + w'w = 0. Then, by successively differentiating this result, show that, wiaatever the integer w, 11. If «^ = e* COS ic and v = e* sin a;, then D^'tc = — 2v and D^^v = 9,2^. Also, I>a,V 4- 4v = 0; 12. If w = e"* CDS ma; and v = e"* sin yna;, show that the successive derivatives of u and v may always be reduced to the form DJu = Afic — BiV; DJv = AiV -{• BtU, (a) where A and B are functions of v% and w. Also, find the values of ^,, A^, B^ and 5„ and sho\^ by differentiating (a) that At + i = A,At-B,Bt; B,^,^ B,A,-{- A.B^, f DEVELOPMENTS IN SElilES. 96 k»* n uy 4- Jaj^'Ms i ^u)\ IL^-\D^U)\ ;, show that. ay always be (II, (a) Iso, find the entiating (a) I CHAPTER VIM. DEVELOPMENTS IN SERIES. 5*7. 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 the 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 the function. We may designate a series in the most general way, in the form «*, + ^^ + W, + . . . + «n + W„ + i + . . . , the nth. terms being called w„. 58. Convergence and Divergence of Series. No universal criterion has been found for determining whether any given series is convergent or divergent. There 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 convergent unless, as n lecomes in- finite f the nth term approaches zero as its limit. For if, in such case, the limit of the terms is a finite quantity oc, then each new term which we add will always I 1 :" m 06 THE DIFFERENTIAL CALCULUS. change the sum of the series by at least or, and so that sum cannot approach a limit. As an example, the sum of the series 1 — 1 + 1 —1 + 1 — 1, etc., ad infinitum, will continually change from + 1 to 0, and so can approach no limit, and so is divergent, by definition. II. A series all of whose terms are positive is divergent unless nUn = when n^^ ao. < To prove this, we have first to show that the harmonic series i + | + i + -^+ etc., ad infinitum, is divergent. To do this we divide the terms of the series, after ihe first, into groups, the first group being the 2 terms ^ + :|-, the second group the following 4 terms, the third group the 8 terms next following, and, in general, the nih. group the 2" terms following the last preceding group. Wo shall then have an infinite number of groups, each greater than ^. Now, if, for all the terms of the series after the nth, we have nUn > a {a being any finite quantity), then Un > -, n 1 (I 1 and Un + Um+i +...>«- H rr + • o ' ^ \w w + 1 m + 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 limit, then the series is convergent* Let the series be w, — w, + «*, — w* + w. — Then, by hypothesis, w, > w, > w, > w, > . * • • • DEVELOPMENTS IN SEIilES. 97 the series, he 2 terms the third il, the 7ith. roup. We ich greater [le nihy we + ...). this eqiia- aj which e and nega^ IS a limit f Let us put Sn for the sum of the first n terms of the series, 71 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 S= Sn + l — {Un + 9 — Un + s) — (Wn + 4 " ^n + b) — • • • • Since all the differences in the parentheses are positive, by hypothesis it follows that, how many terms soever we take, the sum will always be greater than S^ and less than S^+i* The difference of these quantities ib tin ^ i, which, by hypothe- sis, approacliea zero as a limit. Since the two quantities /S'„ and Sn+i .ipproach 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 « = 6, O^ Se Ss Sia— S Sn S9 Sr I I t- 1 I I 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 3^ 5 of which the 7ith term is — y + --etc., (-1)« We shall hereafter see 2n - 1 that the limit of the sum of this series is Itt. If we divide the terms into pairs whose sums are negative, the series may be written 2 3'5 7-9 11 13 etc. i m H rf ' 3 Mi ;< !( n i I M ]^ I* t 98 2UIB DIFFERENTIAL CALCULUS. Pairing the terms so that the sum of each pair shall be posi- tive, the series becomes 2 2 3 2 3 +F7 + 901 + 1305 + ^^''' We may show by the preceding demonstration that these series approach the same limit. IV. If J after a ccrtai7i finite mcmler of terms , the ratio of tivo consecuiivG terms of a series is continually less than a cer- tain qnaittity a, luhich is itself less than unity j then the series is convergent. Let the nth term be that after which the ratio is less than a. We then have «*n+2 < ««*« + ! < «X; • • t • • • Taking the sum of the members of these inequalities, we have But a -{- a* ■{- a* -\- » , , is B>n infinite geometrical progres- sion whose limit when a < 1 is a a: , a finite quantity. Hence, putting *S^ for the limit of the sum of the given series, we have S x, and it approaches zero as a limit. Hence the series is convergent for all values of X. Corollary. A series proceeding according to the jjowers of a vai'iable, x, is conver- gent when X < 1, provided that the coefficients a^ do not in- crease indefinitely. Remakes. — (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 unity as a limit. But most of the series met with come into this class. (3) The sura 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 dx d'cfy = cf)\x) = ^, + ^A^x + ZA,x' + . . . ; dx ay dx' P= 0"(a:) = l-2^, + 2-3^3rc + 3-4^,a;' + = 0'"(a;) = l-2 dy dx"" = 0<'»)(a:) = 1-2 3^, + 2-3-4^,2;4- . . . ; 3 • 4 . , . 7iAn + etc. By hypothesis these equations are true for all values of x small enough to rei. ler the series convergent. Let us then put X = in all of them. We then have DEVELOPMENTS IN SERIES. 101 } a method 5S proce«,d- , the serios ; (1) ntirc func- ble depends s admit of 1 which the Dpment will nmonly this ertain mag- e that mag- levelopment ts A may be ^uation (1). "T • • • 5 0(0) = ^„; 0'(O) = A,) .*. A^ • * • ^j 0(0). 0'(O). 0"(O) = l-2^.; .•.^. = ji-^0"(O). 0'"(O) = 1 2-3^.; .-. ^, = p^0"'(O). 0(«)(O) = n\An\ 1^ .•.^„ = -^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 aerivatives are formed, suppose the variable to he zero in the original function and in each derivative. Divide the quantities thus formed, in order, l?yl', 1; 1*2; 1'2*3, etc., the divisor of the nth derivative leing n\ The quotients will he the coefficients of the powers of the variahls in the development, commencing with the zero power, 1 or absolute term, EXAMPLES AND EXERCISES. I. To develop (a + ^)" ^ ** ^^ powers of x. We have • • -A. ^^ a , u = {a ■-{- xY; lu dx dx .•. -4, = na^-K ^'^ ' f i\ / I \n_2 M n{n — 1) „ , m km '"nil t} i At 1,1 .1 t lvalues of x Let us then 'dx^ = n(n — l),,,{n — s-\-l)(a-\- a;)"-". 'I 4 ; ! 1 f 102 TITE DIFFERENTIAL CALCULUS. Thus the development is {a + xy = a" + m--'x + (|)a"- V + (|)«— V -j- . . . , which is the binomial theorem. 2. Develop (« — a:)" in the same way. 3. Develop log (1 + x). Here we shall have = (i + a^)-»; du _ 1 fl?aj "" 1 + a; I? = 1-3(1 + ^)-'; etc. etc. Noticing that log 1 = 0, we shall find log {l + x)=x-ix'-\-^x'-ix* + 4. Develop log (1 — x). 5. Develop cos x and sin x. The successive derivatives of sin x are cos x, — sin x, — cos x, sin «, etc. By putting x—0, these become 1, 0, — 1, 0, 1, 0, etc. Thus we find /p'i /}A /i^ 3?^ 2!^ 3j' 8in« = aj-3j+--- + .... 6. Develop e*, where e is the Naperian base. X' . X' Ans, 6»' = l+.T + ^ + ^^ + 7. Develop e"'". 8. Show that • • • • -111 . i^^^SC'V , (xlosa)' , fl'" = 1 + aj log ^, + ^ , ° ^ 4- ^ ,.°.^^ + 1-2 1-2-3 9. Deduce e«»°' = i4.a; + ^_^4. 2 4! DEVELOPMENrS IN 8ERIE8. 103 10. Develop sin (a + x) and cos (n -^x) and thonco, by com- paring with the results '^f Ex. 5, prove the formulaB for the sine and cosine of the sum of two arcs. Find first -a ^j + . .) 4-C0S« (^-gj 11. Develop (1 -f <'*)'* and show that the result may bo re- duced to the form n . n^ -\- n x* n^ -f 3n' x^ X' X sin (tt -f- x) — sin « (1 — n-r + • •) + cos a {x — ^+ . .). 2 r^2^^ 2' 1.2^ 2- 31 ^' 'I* 1 2. Develop e* sin x ana c* cos x and deduce the results x* ^x' . .'t' „ a:" .«sin^ = :. + 2^+23j-4:gj-8gj -... e" cos a; = 1 + a; -2^ -- 4?-r - 4^ + 2! '4! 51 13. Develop cos' x. Begin by expressing co8^ x in the form i cos Sa; 4- f cos x. 1 4. Develop tan ^"" %. This case affords us an example of how the process of de- velopment may often be greatly abbreviated. It has been shown that f?-tan(-^)a; 1 _ -— — - =z I — X* -{- X* — X* 4- etc. dx 1 + «• ' Now assume tan^~%' = A -\- A^x -\- A^x' -\- etc. This gives » ~ ft • 9. » 2 3-2 a' = =^- 6-3' • • • > 3-5" ' 5-6 • • • > [-etc. i7s Diameter, a method of he series (c) snce would be ybe obtained 1 is 45° = \^' a, «', «"> e*c- ,n have, by the a" = 1- -— ^' and ^" of the If y one of these 8 3-8 6-8' — jt = a -{■ a' -\- a". These series were used by Daso in computing n to 200 decimals. A combination yet more rapid in ordinary use is found by determining oc and a' by the conditions tan a = -' 5 ^a — a' = -r Tt, 4 We then have tan 2ar = tan 4ar = 120, 119' and because «' = 4^ — \7t = 4a' — 45°, we have , _ tan 4a' — 1 _ 1 ^^"""^ ~ tan 4« +1 ~ 239- Hence we may compute tt tl: an : a = -~ «' = 1 JL . J L. 4. 3-5''"^5-5' 7-5^"^* • •' "T" K.0QQ6 239 3-239" ' 5*239' 7f = 4.a — a'. Ten or eleven terms of the first series, with four of the [second, will give ;r to 15 places of decimals. ]'hA i ii. ■•I ;1 til *. 1, •; » 106 TUE DIFFERENTIAL CALCULUS 61. In developing functions by Maclaurin's theorem wo may often bo abb to express the derivatives of a certain order as functions of those of a lower order. Tlic process of find- ing the higlier derivatives may then be abbreviated by retain- ing the derivatives of lower orders in a symbolic form, so far as possible. EXAMPLES. I. Lot us develop n = log (1 -f- sin a:) E 0(x). We now have cos X 1 — sin X '"[x) = — sec X tan x (f>'{x) — sec x'{x) + sec' x(/)'{x) = - Hi 1; H I .1 lOS THE DIFFERENTIAL CALCULUS. i !; i! 11 ' :i ill I I When a: = 0, we have sin x — 0, cos x = 1, u = 1, and hence M= M' = . . , =0 in all the equations. Thus we find, for x = Qj B^n = M = 1; BJ^u = 6-1 = 5; D^'u = 75 - 15 4- 1 = 61; etc. etc. ; while the odd derivatives all vanish. Hence sec a; = 1 + - a;' + ^j x' -f -j- a;" + 63. Taylor's Theorem. Taylor's theorem differs from Maclaurin's only in the form of stating the problem and ex- pressing the solution. The problem is stated as follows: Having assigned to a variable x an increment h, it is re- quired to develop any function ofx-{-h in powers ofh. Solution, Let

(x-\-7i). I Assume n' = X, + Xfi + X,/i' + A\7i' + etc where X„, X„ etc., are functions of x to be determined. Then, by successive differentiation, we have ^- = uY, -f 2XJt + dXJi' -f 4:XJi' 4- etc.; (1) (2) d'u' d/t d'u' J- = 2X, + 2-3X3/^ + 3-4Xyi' + etc.; (3) = l-2-3X3-f 3'3-4X,A + etc. etc. etc. etc. We now modify these equations by the following lemma: If we have a function of the sum only of several quantities, the derivatives of that function with respect to those quantities will he equal to each other. DEVELOPMENTS IN SERIES. 109 u = 1, and B. Thus we difEers from ablem and ex- i follows: nt h, it is re- ers of h. sloped, and let (1) • • • • (3) termined. etc. ;0« « :1 (3) wing lemma: feral quantities, those quantities For if in f{x -\- h) we assign an increment Jh to x and to h separately, the results will hef{x -\-h-\- Ah) and /(a; + JA -f- h), which are equal. It follows that we have du' _ di^' dh ~ dx' Now these equal derivatives, like u' itself, are functions of X + h alone, so the lemma may be applied to as many suc- cessive derivatives as we please, giving dW d\t' dh' dhi' dx"' d'u' dh' dx' ' etc. etc. Now let the derivatives with respect to x bo substituted for those with respect to h in equations (3), and let us suppose h to become zero in equations (:^) and (3). Then ii' and its de- rivatives will reduce to u and its derivatives, and we shall get du x = dx' ^r- 1 d'u ~ I'^dx'' • 1 d'u ~ l-'Z'^dx'' • • _ 1 d^'u " n\ dx'' ' Then, by substitution in (^), we shall have, for the required development, , du h , d'u h"* , d*u W , , U' — u -\- -J- - -}- -j-T, r—- -f -y-3 r— t-t; + etc. dx 1 ' dx' 1-2 ' dx'1'2-3 This formula is callecj. Taylor's Theorem, after Brook Taylor, who first discovered it. :m i ' ■ I I I VH : i\ 110 THE DIFFERENTIAL CALCULUS. I " I (i ;' I' II! EXAMPLES AND EXERCISBS. I. Develop (x + A)". We proceed as follows : du dx = nx^~^\ d u , -.vm. — " — = w(?i-l)ic» -; etc. etc. By substitution in the general formula we find (2;-h/0" = a:~ + p'i;»-^A + ^ '^*(^*-l)..n-s 2 h ~i r^2'^"3 -f • • • • 2. Devolop the exponential function «*"*"'' in powers of U, Am. (f\\ -f log a^ + (log w)'^^ + . . • V 3. sin {x -\- h). 5. sin {x — //). 7. log {x + 70. , x-\-h II. cos* {x -f- Zf). 13. tan(-»>(.c + //). 4. cos (:c + ^0* 6. cos (:c — /i). 8. log (.c — A). 10. log cos X. 12. sin' {x — Zi). 14. sin^~^^ {x — 7i). 15. Deduce the general formula X n^) -r;i ~-^' — etc. (1 + a:)' 1-2 16. Prove, by differentiation and applying the algebraic theorem that in two equal series the coefficients of like powers of the variables must be equal, that if we have log («o + «i^ + «a^' + •••) = ^0 + ^1^ +K^'' + ' ' ' , DEVELOPMENTS IN SERIES. Ill then the coefficients a and b are connected by the relations K - log a,; ctoK = «.; ^aj)^ + a J), = 2a,; 3a,b, + 2a,b, + a,&, = 3a,; etc. etc. etc. 1 1 7. Hence show that ^ is the logarithm of the sum of 1 — X ° an infinite series whose first terms are 63. Identity of Taylor's and Maclaurin's I'heorems. These two theorems, though different in form, are identical in principle. To see how Taylor's theorem flows from Maclaurin's, notice that h in the former corresponds to x in the latter. The de- rivatives with respect to x in Taylor's theorem are the same as the derivatives with respect to 7^, and if we suppose A = after differentiation Taylor's form of development can be de- rived at once from Maclaurin's. Conversely, Maclaurin's theorem may be regarded as a special case of Taylor's theorem, in which we take zero as the original value of the variable, and thus make the increment equal to the variable. That is, if we put/(u) in the form /(O + ^'). and then, using x for //, develop in powers of x by Taylor's theorem, wo shall have Maclaurin's theorem. 64. Cases of Failure of Taylor's and Maclaurin's Theorems. In order that a development in powers of a vari- ble may have a determinate value it is necessary that none of the coefficients in the development shall become infinite and that the developed series shall be convergent. For example, cosec x cannot be developed in powers of x, because when x = the cosecant and all its derivatives be> come infinite. :'f ■■■ I > 1 1 i U i li B I.' ' r' 1 f 112 THE DIFFERENTIAL CALCULUS. 65. Extension of Taylor's Theorem to Functions of Several Variables, Let us have the function n=f{x,y). (1) It is required to develop this function when x and y both re- ceive increments. Let us first assign to x the increment hy and suppose y to rema'n constant. We then have, by Taylor^s theorem, /(^ + 7,,t,) = » + ^^+^,^-, + ^.3, + ...,{3) in which «, y-, etc., are all functions of y. Next, assign to y the increment k. The first member of (2) will become /'(a; -{- h, y -^ k). Developing the coefficients in the second member in powers of k, the result will be: i* will be c]ianged into du -T- E i^a;?^ will be changed into ^ , d-Dji k , d\0^uk^ , d^u dy 1 6/?/' T-j E jC^x'^* will be changed into ^, , d-^,'uk , d'D^uk' , etc. dy 1 etc. dy"" /*; etc. Substituting these changed values of the coefficie-its in (2) it will become dii k . d^u ¥ . d^u k^ f{x + h,y -^ k) = u + -^- j-\- - I ,7^,3 Q I I • • • ly 1 ' dy'-M ' dy'dl du h d^u h k d'u h k' dxdyl 1 ' dxdy'l 2!~^' * * d'u lek . dSi h'k' ^ dx' 21 ^ da-'dy'^l 1 ^'dx'dy' 2! 2! d'u h' '^ dx'3r W ''^- DEVELOPMENTS IN SERIES. 113 Thus the function is developed in powers and products of the increments h and Tc. The hiw of the series will be seen most clearly by using the /)-notation. For each pair of positive and integral values of m and n we shall have the term " m\ n\ If we collect in one line the terms of the development which are of the same order in li and ky we shall have; Order of Terms. h h Ist. D^U^ + DyUzr, 2d. 2>>^j + A^„?*^--j + i>»'w^j. ''ill \ I I' fficieiits in (2) 7i** h^~^ h rth. z>j-w-j + ^*''~'^w^«77:_-iyi x + • • • EXERCISES. I. Show that in the preceding development the terms of the rth order may be written in the form =-j, (^ j, etc., denoting the binomial coefficients as in § 5. 2. Extend the development to the case of three independent I variables, and show that the terms to the second order in- iclusive will be as follows ; --S ill hi i III : i i I V i? 114 If THE DIFFERENTIAL CALCULUS. u=f{x,y, z), then /• (x -{- h, y -{- k, z + I) — u + D^uD^u ' hi 4- DyiiD^u ' hi, 66. Hyperbolic Functions. The sine and cosine of an imaginary arc may be found as follows: In the developments for sin x and cos x, namely, 3"! + 5! sin 2; =:. a; - I, + 1^ - , . 1 x" ^ X* cos re = 1 - -J + -J - ... , let us put yi for a:, (i e 4/— 1). We thus have 8m?/i = i'^;?/ + ^j + ^j+ . . . cos y t = 1 + |j 4- |j 4- . ... V (1) We conclude: The coaine of a purely imaginary arc is real and greater than U7iity, while its sine is jmrely imaginary. We find from (1), cos yi 4- * sin yi = 1 — y -\- |- — etc. = e~^; ?/» cos yi — i sin yi = 1 -\- y -\- -j- 4- etc. = e"; z I and, by addition and subtraction, cos yi = ^{e~'" -\- e^)'j i sin yi =. ^[g~^ — e"); sin yi — \i{e^ — e~^). The cosine of yi is called the hyperbolic cosine of y, | and is written cosh y, the letter h meaning ^Miyperbolic' jy DEVELOPMENTS IN SERIES. 115 ^D^DyU-hh cosine of an developments (1) .al and greater = 6-"; = 6"; c cosine of y, hyperbolic." The real factor in the sine of yi is called the hyperbolic sine of y, and \l written sinh y. Thus the liyperbolic sine and cosine of a real quantity are real functions defined by the equations sinh y = ^a" — c~^); cnshy = ^{e" -{-e-") By analogy, we introduce tlie additional function -'h) '-").) (1) (,V (,-V tanh y — — — -. The differentiation of these expressions gives d sinh y , d cosh y . , i — ■- = cosh y: 7- — = smh y: dy -^^ dy ^ (2) d tanh y = — ~ •^ cosh y They also give the relations cosh' y — sinh' y = 1. (3) fn verse Hyperbolic Functions, When we form the inverse function, we may put u E cosh y. Then, solving the equation e^ + e-" = 3 cosh y = 2w, we find 6" = 2^ ± Vu"^ — 1. Hence y = log {n ± Vu"" — 1) = cosh <-^> In the same way, if we put u ~ sinh y, we find u. (4) y — log {u ± Vii" -f 1) = sinh<-*) u. (5) From the equations (2) and (3) we find, for the derivatives of the inverse functions: ft ^ ill i '11 ^ ; 1 1 P ' m i-\i *M m I -n' lit 116 TUF. DIFFEUENTIAL GALGUL US. When y = cosh^"" ^> Uf or u = cosh y, ii then ^iy _. 1 du i^tt^ _ 1 , When y = sinh^~ ^^ w, or u = sinh y. then dy _ 1 du Vu* 4- 1 frii .1L (6) (7) Kemark. 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. I. By continuing the differentiation begun in (3) prove the following equations: Bx sinh X = sinh x; D^ cosh X = cosh x\ D^^~^' sinh x — sinh x. etc. etc. 2. Develoj) sinh .r, as defined in (1), in powers of x by Mac- laurin^s theorem. Ans. sinh ^' = 2/ + of + «"?+••• • 3. Develop sinh {x -\- h) and cosh {x -f- //-) by Taylor^s theorem and deduce sinh {x-\-h) = sinh xll -\- -^ -{- . . .1+ cosh xlx + -^ + . . . J = sinh X cosh h -\- cosh x sinh //; cosh (x-\-h) — cosh x cosh h -j- sinh x sinh h. I MAXIMA AND MINIMA, 117 (6) yperboUc be- 3 co-ordinates -axis is unity. in (3) prove •s of X by Mac- ) by Taylor's CHAPTER IX. MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. 67. Def. A maximum value of a function is one which is greater than the values immediately preceding and follow- ing it. A minimum 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 = )', (a: — a){x — h) 13. y = 7 r) (. 15. ?/ = cos ?i.T. 17. y ~ sin ?ia;. ^ ?w. A maximum when x = -f-cos x, 1 -\- X tan a;' A minimum when x ~ —cos x. 8. y 10. y:^{x + l){x-2)\ _ {x + 3)' '^* •^^~ (.'?;+ 2)'* 14. y = cos 2.^*. 16. 7/ - sin 3x. '8. y = h MAXIMA AND MIA' I MA. 121 19. ^ = sm X cos X, sin X 2i» y = z — r~i • ^ 1 + tun X 20. y = sin' X cos x, cos 2; 32. y = 1 -f- tan x' 23. The sura 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 bo a maximum? A718. Each part = ia. Note that the expression for the area i« 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? A71S. Into jiarts wliose ratio is 1 : 3. Note 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 miXi power of one i)art into the nth power of the other may be a maximum? Ans. Into parts whose ratio is m : n, 26. Show that the quadratic function ax^ -\-J)x-\-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 h in the plane of a pair of rectangular axes OX and OY. "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 intercepts 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 FlO. 13. «'i ,\ 'I ; n I 2 Area = {a-\-b cot cc)ip -f- a tan a) = 2ad -{- a^ tan a + tan a a f ' 1 T" 1 I 5 1 1. ■ ;!:' 1! h I' L iili; i 1 i ^ 1 r i 'in, v:' f l> I Jli I ■'. 122 7!ffiS^ DIFFERENTIAL CALCULUS. Then, taking tan a: — < as the independent variable, we readily find, for the critical values of t and or. «= ± a' or a sin a = ± 6 cos a. It is then to be shown that both values of t give minima values of the area ; that the one minimum area is 2ad, and the other zero ; that in the first case the line YX is bisected at P, and in the other case passes through 0. 28. Show by the preceding figure that whatever be the an- gle XO Y, the area of the triangle will be a minimum when the line turning on P is bisected at P. The student should do this by drawing through P a line making a small angle with XPT. The increment of the area XOY will then be the difference of the two small triangles thus formed. Then let the small angle become infinitesimal, and show that the increment of the area XOFcan become an infinitesimal of the second order only when PX=z 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 tlio 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 2h, show that the side of the flap is 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. Fio. 14. MAXIMA AND MINIMA, 123 lily find, for values of the ; that in the case passes be the an- muin when ine making a " will then he 1 let the small t of the area y when PX = e 40 feet in ire bounded re the sides t of his ma- juare feet, ut out from DO bent up so the part cut imum? he square, ose sides are -^ 1 X '9^B / \ ''^H \ '-^^H y \ ' -'-'i^^H f \ ^^^H / \ l^M. / \ '^^B f \ ' '^^B / o \ 'Wk 69. Case when the function which is to he a maximum or minimum is expressed as a function of two or more variables connected hy 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 = y) = 0, (3) 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), to 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 dii _ ldu\ fdu\dy dx ~~ \dx I \dy Idx ' and from (2) we have, by the method of § 37, dy ^ BJ^ dx Byf Substituting this value in the preceding equation, we shall have the value of -7-, 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. m \u: ;'!'' io. U. 124 THE DIFFERENTIAL CALCULUS. I EXAMPLES AND EXERCISES. 1. To find the form of that cylinder which has the maxi- mum volume with a given extent of surface. The total eirtent of surface includes the two ends and the convex cylindrical surface. If ?• be the radius of the base, and h the altitude, we shall have : Area of base, nr"^. Area of convex surface, ^icrh. Hence total surface = 27r(r' + ^^) = const. = a. (a) Also, volume = itr^h. (6) Putting u for the volume, we have, from (&), du f. , . jdh ^ dr dr From (a) we find "Whence 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 JV -j- aV = »'^*. the sides of the rectangle are 2x and 2y. Hence the function to be a maximum is 4ry, subject to the condition expressed by the equation of the ellipse. This condition gives dy h^x dh n + 2r dr~ • r ' du dr~ nrh - 2nr\ dx a?y MAXIMA AND MINIMA. 126 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:i/27. By taking x and y as in the pi) ■ f i: 11 1 1 ill : ! ! 1 t '; fi )i.:: ' i '■I ii si, i! s ii i lIlN : 130 THE DIFFERENTIAL CALCULUS, as a limit, which is therefore the required limit of the frac- tion when both its members approach the limit zero. It may happen that 0'(«) and ip'(a) both vanish. In this case the required limit of the fraction in (4) is seen to be In general: The required limit is the ratio of the first pair of derivatives of like order tohich do 7iot both vanish. If the first derivative which vanishes is not of the same order in the two terms, — for example, if, of the two quantities 0'(«) and ^'(«), 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. Remark. 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. I. X — a EXAMPLES AND EXERCISES, for X = «.* 0(a;) = a;' — «'; (t>'(x) =2x; . • . 0'(a) = 2a; ip{x) = X — a; ^'(x) = 1; .•, ^'(«) = 1. lim. x — a (x = a) — 2«, a result readily obtained by reducing the fraction to its lowest terms. 2. — ^-rr for x=l. x-1 , — m X for x = 0. Ans. 1. Ans. 2. * Using strictly the notation of limits, we should define the quantity sought as the limit of the fraction when x approaches ihe limit a. But no confusion need arise from regarding the limit of the fraction as its value for a; = a, as is customary. INDETERMINATE FORMS. a; — sm a , , . 4. j for {x = 0). -4m«. |. 131 Here the successive derivatives of the terms are: <})'(x) = 1 — cos x) . If N and D both become infinite, 1-4-7) and 1 ~r N will both become infini- tesimal, and thus the indeterminate form of the fraction will bet. Again, if of two factors A and B, A becomes infinitesimal while B becomes infinite, we write the product in the form ; ^, and then it is a fraction of the first form. I -i- 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 a;'"(log xY for X ^0. Here x^ approaches zero, while log x, and there- fore (log xY, becomes infinite for x-=0. Hence the denomi- nator of the transformed fraction will be ^ (putting for brevity I = log x). The successive derivatives of this quantity with respect to x are xl — • ^M I ^ + ^V etc The successive derivatives of the numerator are wa;"*"^; m^ni — l)x^~^', etc. The limiting values of the given quantity x'^V*' thus become ^r^min + i m{m — l)x^ „m n n 1 , ri + iy n + 1 1 l^ +» J etc.. which remain indeterminate in form how far soever we may carry them. 'i! INDETERMINATE F0RM8. 133 ihus become ever we may In such cases the required limit of the fraction can be found only by some device for which no general rule can bo 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 x = c~'". Since for x i y i oo , we now have to find the limit of f or y i 00 . By taking the successive derivatives of the two terms of the fraction —-, we have the successive forms ny""' \ n{7i — l)y "-^^ n{n -- 1) {71 — %)y n — 3 etc. 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 jiositive integer, and will vanish for y = co , if 71 is not a positive ^.iteger. But the denominator will remain infinite. Wo therefore con- clude: lim. [a:'"(log x^] (x ^ 0) = 0, whatever be m and 71, so long as 7n i positive. From this the student should show, by putting z~x~''- and m = 1, that the fraction z becomes infinite with z, how great soever the exponent 71, and therefore that any iTiJinite munber is an infinity of higher order than any power of its logarithm, 73. Foi-m 00 — 00 . In this case we have an expression of the form F{x) = u — V, .' ''i > ''P 1(1 I ..' ^V)\ J ( 134 THE DIFFEUIUNTIAL CALCULUS. I. I in which both u and v become infinite for some value of x. Placing it in the form F{x) = «(i - g, wo see that F{j^ will become infinite with u unless the fraction v — approaches unity as its limit. When this is the case the expression takes the form qo x of the preceding article. 74. Form 1". To investigate this form lot us find the limit of the expression 'i+ir (- nl u when n becomes infinite. Taking the logarithm, we have log ?« = hn log ^1 + -j = hn \ — -. + j7—, — ...!• Making n infinite, we have lim. log u = U; or, because the limit of log u is the logarithm of lim. u, log lim, 11 = h. lim. (l + ^)''V:^oo) = e\ Hence In order that this result may be finite, h itself must not be infinite. Wo therefore reach the general conclusion: Theorem. In order that an expression of the form (1 + ccY may have a finite limit when a becomes infinitesimal and x infinite, the product ax must not become infinite. Cor, If the product ax approaches zero as a limit, the given expression will approach the limit unity. INDETERMINATE FORMS. 136 75. Forms 0* and oo*. Lot an expression taking either of these forr s as a limit be represented by u^^F. The problem is to find the limiting value of the expression when approaches zero and i« either approaches zero or becomes infinite. From the identity w = c '«« " we derive F=ti'^ = e^ •"« ". We infer that the limit of F will depend upon that of

l 143 THE DIFFERENTIAL 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 lin*^. A point on the circumference of a carriage-wheel, as the carriage moves, desciibes 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. ! ' O Q K B ^ Fia.si. Also put a E the radius of the circle ; u E the angle through which the circle has rolled, expressed in terms of unit-radius. Then, when the circle has rolled through any distance OR, this distance will be equal to the length of the arc PR of the circle between P and the point of contact R, that is, to au. We thus have, for the co-ordinates of the centre, G, of the circle, z = au'y y^a\ and for the co-ordinates of the point P on the cycloid. x = aw — fl sin w = a(\i — sin w); y = rt — fl cos w = rt(l — cos u) ;1 (1) which an the equations of the cycloid with w as an independ- ent variable. PLANE CURVES. 143 To eliminate u, find its value from the second equation^ w = cos<-«(l-^V This gives sin u = ^1 — cos' u = — ^. a Then, by substituting in the first equation X = a cos<~** — V2ay — y% which is the equation of the cycloid in the usual form. (2) 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 cEhalf 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'(a:' - y% (3) which is the equation of the lemniscate. 'I ( 'I '■ --'h I M * H t 144 J' i! ill 11 fr T5:fi? DIFFERENTIAL CALCULUS. (5) Transforming to polar co-ordinates by the substitutions X = r cos 6, y = r sin ^, we find, for the polar equation of the lemniscate, r' = 2c» cos 2^. (4) Putting ?/ = 0, we find, for the point in which the curve cuts the line joining the foci, x= ± i^2c = a. 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 {X' + fy = a\x^ - f); r^ = a^ cos 2/9. 83. The Arcliimedean 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 E the distance by which the generating point moves along the radius vector \/hile the latter is turning thr'^^igh the unit radius. Then, when the ra- dius vector has turned through the angle 6, the point will have moved from tlie pole through the distance aO, r = aO Fig. 23. Hence we shall have as the polar equation of the Archimedean spiral. PLANE GUnVES. 145 If we increase 6 by an entire revolution (27r), the corre- sponding increment of r will be llTia, a constant. Hence: The Archimedean spiral cuts any fixed jmsitioii of the ra- dius vector in an indefinite series of eqiiidistant 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 6^ for the initial angle, we have log r = l{e- 0X I being a constant. Hence 19 - 19, - Wo 10 r = e * = e "e , Putting, for brevity, a=ze , the equation of the logarith- mic spiral becomes fio. 24. r = ae^^, a and I being constants. EXERCISES. 1. Show (1) that the maximum ordinate of the lemniscate is |c, and (3) that the circle whose diameter is the line join- ing the foci cuts the lemniscate at the points whose ordinatea 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. 21): r = 2ay -{- 'Huax — a'w*. 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. ^1 ' 1 t I ' t '^1 146 THE DIFFERENTIAL CALCULUS. 4. If, in that logarithmic spiral for which ^ = 1 and 1=1, r = 6*, the radius Toctor turns through an arc equal to log 2, its length will be doubled. 5. li, 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 , represent a parabola whose parameter is a' 7. If, in the equation of the Archimedean spiral, and therefore r take all negative values, show that we shall have another Archimedean spiral intersecting the spiral given by positive values of 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 « = aG cos 0, y = aO sin 0. Changing the sign of 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 V. n of 0! and TANGENTS AND NORMALS. 147 CHAPTER XII. TANGENTS AND NORMALS. 84. A tangent to a curve is a straight liii^ 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, TQy upon the axis of X, of that segment TP of the tangent contained between the point of contact and the axis of X, The subnormal is the corresponding projection, QN, of the segment PN of the normal. Notice that a tangent and a normal are linos 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. M i t 1 4jl I. 'I t II 3 i 1 ' i'S ! } 148 y//^ DIFFERENTIAL CALCULUS. 85. General Eqiiat ion for n Tdwient. The general prob- lem of tangents to a curve may bo stated th«s: 7b find the conditvm which the parameters of a straight line must satisfy in order that the line may be tangent to a given cnrve. 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. Ijct {x^f ?/,) 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,-m{x- xy, (5) m being the tangent of the angle which the line makes with the axis of X, But we have shown (§ 30) that *" - dx: this differential coefficient being formed by differentiating the equation of the curve. Uenco (6) is the equation of the tangent to any curve at a point (a;,, y,) on the curve. Equation of the Normal. The normal at the point (a;,, 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. ) — = _ JL m ~ dyl dx. m' = = - Hence the equation of the normal at the point (a:,, y,) is dx iy - ?/,) = a;, x. m TANGENTS AND N0UMAL8, 149 In these equations of the tangent and normal it is necoBsary 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 pertainiag to the curve^ must be marked by suflixes or accents. 86. 8%ibtangent and Subnormal. To find the length of the subtangent and subnormal, we have to find the abscissa a;, of the point T in which the tangent cuts the axis of abscis- sas. We then have, by definition, Fio. 26. Subtangent = x^ — x^ The value of x^ is found by putting y = and x the equation of the tangent. Thus, (6) gives - ^' = ^;(^« - ''■>• Hence, for the length of the subtanorent TQ, Subtangent = a;, — a;„ = J^, We find in the same way from (7), for QN, Subnormal = — t/.^'« dx^ x^m (8) (9) i ' P ii» 1 jjlH!' « ; fault * ii H ! : \ ' •}\ I 160 TEE DIFFERENTIAL CALCULUS. :i :■ :i 5' 87. Modified Forms of the Equation. In the preceding discussion it is assumed that the equation of the curve is given in the form But, firstly, it may be given in the form F{x, y) = 0. We shall then have (§ 37) dF dy^ dx^ Substituting this value in the equations (6) and (7), we find ^ . dF, . dF. . "i Tangent: -^(y - yj = ^{x, - x); • Normal: -y~ (i/ — V,) = ^— (« — «,). dz, - ^*' dy^^ " (10) Sec adly;, if the curve is defined by two equations of the form (11) we have dy^ __ du dx^ ~ dx ' du 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,)^ = {x - a:,)^. ' Eq. of normal: (y - y,)'^ = (a;, - x)^. du y (13) TANGENTS AND NORMALS. 151 By substituting in these equations for a;,y„ -r- and ^ ay du their values in terms of u, the parameters of the lines will be functions of w. 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. 88. Tangents and Normals to the Conic Sections. Writing the equation of the ellipse in the form oY 4- *•«• = a^h\ (a) we readily find, by differentiation, dy _ b*x 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: ^ + ^tt = 1. a* h* For the normal: -x y — a* — h*. Taking the equation of the hyperbola, we find, in the same way. For the tangent: ^ — ^ = 1. a" h* For the normal: -xA — y = a' 4- 5". Tr-king the equation of the parabola, we find, by a similar process. For the tangent: y,y = p(x -f a;,). For the normal: y ^ y^:= ^ -{^i — ^)» I: ^1;' <-,: : i iih i t ■A ■ i s -i SI ','"' 5 162 THE DIFFERENTIAL CALCULUS. 89. Problem. To find the length of the perpendicular dropped from the origin upon a tangent or normal. It is shown in Analytic Geometry that if the equation of a straight line be reduced to the form Ax-\-By-\-G=0, the perpendicular upon the line from the origin is C VA' + B' It must be noted that in the above form the symbol C 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 y-y, = m{x- x^), ^ we write it mx — y — mx^ -f y^ = 0, . ' and then we have -4 = m; 0=y,- mx^. Thus, the expression for the perpendicular is y, — mx, p = ^' ' - Vw' + 1 Substituting for m the values already found for the tan- gent and normal respectively, we find. For the perpendicular on the tangent : P / ^ + 4)' ds (1) For the per2)endicular on the normal : x4-V^' p = ^^+(S:)' ds (2) TANGENTS AND NORMALS. 163 Fig. 27. 90. Tangent and Normal in Polar Co-ordinates, Problem. To find the angle which the tangent at any point makes with the radius 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 the figure); K'P', the same for the point P\ KSR, a parallel to the axis of reference. Drop PQlK'P'. Let SPT be the tangent at P, We also put y = angle KPS which the tangent makes with the radius vector. Then let P' approach P as its limit. Then QP' = dr; PQ = rdd; PQ . rdS ^^^y^-QP^-Tr- We also have 1 1 dr (1) cos y 4/(1 + tan" y) /k+en* sin y = cos y tan y = i/|-^(l)T \ (3) Cor, The angle RSP which the tangent makes with the ^xis of reference is ^ -j- 6*. ■ I X-. l[ ill t m » l!^ ! I'" m 1 :i n->ii th ' III 164 THE DIFFERENTIAL CALCULUS. 91. Perpendimdar from the Pole upon the Tangent and Normal. When y is the angle between the tangent and the radius vector, we readily find, by geometrical construction, that the perpenuicular from the pole upon the tangent and normal are, respectively, j3 = r sin ^^ and p — r cos y. Substituting for sin y and cos y the values already found, we have. For the perpendicular on tangent : p = /l^'+ST For the jwrpendicular on normal : r P = dr v^i^'^m"' (3) 92, Problem. 7b find the equation of the tangent and normal at a given poi7it of a curve -whose :'quatio7i is expressed in polar co-ordinates. It is shown in Analytic Geome(.ry that if we put p = the perpendicular dropped from the origin upon a line; a E the angle which this perpendicular makes with the axis of X; the equation of the line may be written X cos fx -}- y sin a — p = 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 (y -{- 0) = sin y cos d -f- cos y sin 6; sin « = — cos {y -{- 6) =: — cos y cos ^ + sin ;^ sin d. TANGENTS AND NORMALS. 155 By substitution in (1), the equation of the tangent becomes a:(8in y cos + cos y sin 6) — y(coB y cos — Bin y sin 6) — p = 0. Substituting f o- cos y, sin y and ji the values already found, this equation of the tangent reduces to Ir cos & -\- -,n sin 0jx-{-lr sin — -jh cos 6) y — r* = 0, (2) r and being the co-ordinates of the point of tangency. In the ease 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 -\- 0, Substituting this value of a, and proceeding as in the case of the tangent, we find, for the normal, -j^ cos — r sin 0jc -\-ir cos ^ + ;t^ sin ^J y — r-r-n = O* (3) Generally these equations will be more convenient in use if we divide them throughout by r. Thus we have: Equaiion of the tangent : cos ^ + - ~ sin 0jx-\- [sm ^ - - ^^ cos 0\y - r = 0. (4) Equation of the normal : 1 dr rdS cos 0-Bm0jx^ [^ ^~ sin + cos 0jy - ~ = 0. (5) I In using these equations it must be noticed that the co- efficients of X and y are functions of r and 0, the polar co- ar ordinates of the point of tangency. When r, and -^^ are given, this point and the tangent through it are completely determined. i W il 166 a: ii ? ? I f i'i ill Hi i i THE DIFFEBEyTlAL CALCULUS. EXERCISES. 1. Show that in the case of the Archimedean spiral the general expressions for the perpendiculars from the pole upon the tangent and normal, respectively, are 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 6^ = 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 dr 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 — ' = ± p, 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 ''ondition it sufiices to show that to an infi- nite value cf 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 P = dF dF ^ dy dx m + \dv / ) (2) 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 j). If this approaches a real finite limit for an infinite value of r, the curve has an asymptote. U !ll Til H m :'t' I 168 TEE DIFFERENTIAL CALCULUS. % I is Lis m % li m I 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 (8) by supposing ar, or y, to become infinite. fl F (IF CommoD^ ^^)o coefficients -7- and -j— will them selves bo- dx^ dy, come infinite .p^itli b^ co-ordinates. We must then divide the whole equation by such powei s of x^ and y^ that none of the terms shall become infinite. 94. Examples of Asymptotes, 1. F(x) = x'-\- y*— daxy = O.(rt) The curve represented by this equation is called the Folium of Descartes. The equation (3) gives in this case, applying suffixes, (x* - ay,)x + (y/ - ax;)y = «.' + y/ - 2aa;,y, = ax^y,. To make the coefficients of x and y finite for a;^ = 00 , divide by a;,y,. Then the equation bo- comes Fio. 28. fx. a\ , fy, a \ (-' ]x-{- •-* ]y \y, xj ^W, y,r a = 0. W Let us now find from (a) the limit of y^ for a;, = 00 . Wo have X X 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 169 than the second member^ which is absurd. Hence, passing to the limit, Urn. (|')(rt,i=o) = -l. Then, by substitution in (b), we find, for the asymptote, X -\- y ■\- a =■ 0. 2. Take next the equation F{x, y) = x^ — 2x*tf — ax^ — «'y = 0. (a) With this equation (3) becomes = Zx* — 6a;/y, — 3a; - a'j/,. (*) Fig. 29. "We notice that the terms of highest order in the second member are three times those of highest order in (a). From {a) we have X* - 2x^'y ~ ax* + a'y,. Substituting in tlie second member of (b), and dividing by a:,*, (/>) becomes Solving (a) for y, we find h - ^.' - ^^i an expression which approaches the limit ^ when a;, = oo , Thus, passing to the limit, (b') gives, for the equation of the asymptote, 35 — 2y = a. |!» .il5 !ii!il 160 THE DIFFERENTIAL CALCULUS. I I; I I ll i - ■ 3. The Witch of Agnesi. This curve is named after the Italian lady who first investigated its properties. Its equation is x'y + a'y — a* = 0. {a) The equation of the tangent is 2x^y^x + (:c; + a')y = Sx^y, + a\ = 3a' - 2a\. (b) By solving (a) for x and y respectively we see that a;, may become infinite, but that y, is always positive and less than a. Hence, to make the coefficient of y in (b) finite for :r, = oo , we must divide by a:,', 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 = |., the conditions that the slope shall be a maximum or minimum are dx* ^ d'y and -— different from zero. If the first condition is fulfilled, but if -T^, is also zero, we must proceed, as in problems of maxi- I s! ASYMPTOTES AND SINGULAR POINTS. 161 ma and minima, to find tlie first derivative in order which does not vanish. If the order of this derivative is even, there is no point of inflection for -j^ = ^J 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). Eeducing the equation to the form we find f ~ X -a\ dy dx = — a* 2x'y' cry dx'~~ a* 2x*v ii^^y + ^,dy> dx J "" 2x*v The condition that this expression shall vanish is 4:xy^ = «*, which, compared with the equation of the curve, gives, for the co-ordinates of the point of inflection. x^-^a; y V'6 EXERCISES. Find the points of inflection of the following curves : X I. xy = rt' log -. Ans. X = ac^. a ix = a{l — cos u); y |«e *. 2. 1'^ = ''^ \y = a{nu + sin u). Ans. < X n ,=4o.-..(-L)+i^). , I. .^i! (i : ' HI 1- i ! ! il II 162 THE DIFFERENTIAL CALCULUS. Fio. 82. 96. Singular Points of Curves, If we conceiye 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 bo 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-points; 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 , . , . , Fio. 36. only a smgle point. Isolated Points; from which no curve proceeds, so l._^ that the infinitesimal circle is not cut at all. fio. 36. Salient Points; from which proceed two branches making with 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, 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. Fio. 83. Fio. 84. I infini- ) cut in lan two 3 called to. 84. 35. o Fio. 36. making r 180". ich are le curve al kind 3 three ASYMPTOTES AND SINGULAR POINTS. 163 97. Condition of Singular Points. Let {x^, y,) be any point on a curvo> and let it be required to invutstigate the question whether this point is a singular one. Wo Urst trans- form the equation of the curve to one in polar co- ordinates having the point (^0* .*/o) ^ ^^® polo. To do this we put, in the equation of the curve, x = x^-i- pcoBd;) jjj y = y,+ pemO.) The resulting equation between p and 6 will be the fio. 87. equation of the curve referred to {x^, y^) as the pole. More- over, if we assign to /o a fixed value, the corresponding value of 6 derived from the equation will be the angle 6 showing the direction QP from Q to the point P, where the circle of radius p cuts the curve. The limit which 6 approaches as p becomes infinitesimal will determine the points of intersection of the infinitesimal circle with the curve. If, now, the given equation of the curve is F(x, y) = 0, then, by the substitution (1), the polar equation will be F{x, + P cos e,y, + p sin 6) = 0. (2) Now, let us develop this expression in powers of p by Mac- laurin . theorem. Since p enters into (2) only through x and y in (1), we have dF dF dx , dF dy JF , . JF_ „, -7-=-^--T- + -j--^=cos 6-^ + sin 0-j~ = F', dp dx dp dy dp dx dy (because -^- = cos 6 and -:- = sin 6], \ dp dp i Then il M :,"f I ,i 164 THE DIFFERENTIAL CALCULUS. dp^ ~ dp ™ Kdx" dp dxdy dpi 4_ sin e{~ ^4.t.^ ^\ [dxdy dp dy^ dp) = cos' e^- + 3 sin 19 cos 6^^- + sin' ^^f = F". dx dxdy 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^, y,) + p(cos e'-^ + sin ^^ j . 1 ,/ , J'F . „ . 1 . d'F , . , J'F -\- -p \ cos ff^—-, + 2 Sin 6/ cos ff-^ — -. h sm 6-j—r 2"^ \ dx/ dx^dy, dy/ -\- etc. = 0. dW dF Here -j- means the value of y- when x^ is put for x, etc. Because (a-^, yj is by hypothesis a point on the curve, we have F{x^y ?/„) = 0, and the only terms of the second member are those in p, p', etc. Thus the polar equation (2) of the curve may be written FJP + i^o'V + Frp' + etc. = 0, ) 5. = 0. f (3) li ! OP F: + FJ'p + F/'^p" + etc. To find the points in which the curve cuts a circle of radius p, we have to determine ^ as a function of p from this equa- tion. When p is an infinitesimal, all the terms after the first will be infinitesimals. Hence, at tlie limit, ivhere p becomes infinitesimal 6 must satisfy the equation f: = 0, dF fix which ffives tan 6 = — -~. dF Wo This is the known equation for the slope of the tangrnt at {x^f yj, and gives only the evident result that ia general the M ASYMPTOTES AND SINGULAR POINTS. 165 curve cuts the infinitesimal circle along the line tangent to the curve at Q. But, if possible, lot the point (.^o^o) be so taken that (i) Then we shall have F/ ~ 0, and the equation (3) of the curve will reduce to F/'p + F/"p' + etc. = 0, or F/' + F/''p + etc. = 0. Again^ letting p become infinitesimal, we shall have at the limit 5T // (VF (PF cop' (^~j—i -\- 2 sin cos 6 fPF , , -fsin'^,--=0. (5) Dividing throughout by cos' 0, we shall have a quadratic equation in tan 0, which will have two roots. 8inco each value of tan gives a pair of opj)osite points in wliich the curve may cut the infinitesimal circle, and since (5) depends on (4), we conclude: The necessary condition of a doiihle-jwi] it is that the three equations 0, j^,,y) = 0, ^^^^=0, ^3^'^) dx, ' dij shall he satisfied hy a sinyJe pair of values of x and y. If the two values of tan 6 derived from FJ' = are equal, we shall have either a cusp, or a point in wliich two branches of the curve touch each other. If the roots are imaginary, the singular point will be an isolated point. 1)8. Examples of Douhh'-poials. A curve whose equation contains no terms of less tlum the second degree in x and y has a singular point at tlio origin. For example, if the equa- tion be of the form F{x, y) ^- Px' + Qxy + Bf = 0, then this expression and its derivatives with respect to x and y will vanish for x = and y — 0. 'r in ;H Hi' I m »• f ■ 1 i i m ?yk i) 166 THE DIFFERENTIAL CALCULUS. Let us now investigate the double-points of the curve (y' - a'Y - 3aV - 2ax' = 0. We have (1) dF dx dF dy Q{a^x -j- ax*) = — 6ax{a -f x); My* - «') = -^yiy + «) (2/ - «)• (3) The first of these derivatives vanishes for x = or — a; The second of these derivatives vanishes for y =0, — aor-{-a. Of these values the original equation is satisfied by the fol- lowing pairs: ^0 = 0; 0; -a;) a; -{-a; 0; ) ^0 = - «; which are therefore the co-ordinates of singular points. (3) Fio. 38. Differentiating again, we have d'F d'F ax ^^"^^ ^=^' s^ - = 12?/' - 4a'. Forming the equation F" = 0, it gives ASYMPTOTES AND SINGULAR POINTS. 167 (12^' - 4a') tan' 6 = 6a' -\- 12ax. Substituting the pairs of co-ordinates (3), we find: At the point (0, — or), tan 6^ = ± ^ Vd; At the point (0, + a), tan <9 = ± ^ V3; At the point ( — a, 0), tan — ± Vf. The values of tan 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 (x^, 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, howevor, 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 values of x 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.; m = ± i, ± |, etc., we find as many points of intersection as we please. it^ f I' 9 i 168 THE DIFFERENTIAL GALCULU To makt this method practicable, the equations which tto have to solve should not be of a degree higher than the secoTjd. 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 — y, 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=(^'- y') «in 45°. Ajmlication to the Folium of Descartes, If, in the equa- tion of this curve, x' + y' = 3«a;y, we put y = mx, we shall find oam Sain^ X = y We also find, from the equation of the curve and the pre- ceding expressions for x and y in terms of w, dy _ x^ — a',' 2m — m* dx ax ■~ y l-2m='* Then, for 771 = 1, 3 x =-- ^a; y 3 dy dx = — •1. m — o 2 x = ~a; y- 4 = 3^; dy dx = 4 5' m — 3 2' 3G ^ = 35"> y 54 dy dx = 33 92* m= — 2, 6 X = -^a; y 12 dy dx =:: — 20 17- etc. etc. etc. etc. Thus we have, not only the points of the curve, but the rt:.i»<.;er.ts of the angle of direction of the curve at each point, which will assist us in tracing it. Bli '^ f r '':Wlr^ THEORY OF JENVELOPFS. 169 seconu. lent to id -y, lie lines g it to 3S. le equa- tho pre- 3 2* 7* but the I I i CHAPTER XIV. THEORY OF ENVELOPES. 100. The equation of a curve generally contains one or more constants, sometimes called parameters. For example, the equation of a circle, {X - ay ^{y- by = r\ contains three parameters, a, h 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(a;, y, a) = 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 ^{x, y, a) = 0; cf)(x, y, a') = 0; (f){x, y, a") = 0; etc. The collection of lines that cr.n 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 Of' 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 ex'. Su^^p{Xy t/, a) = and (p{x, y, a -{■ /ia) =0. If we develop the left-hand member of the second equation in powers of J a by Taylor's theorem, it will become 0(^. y. «•) + a;i^« + ji-. 1-3 + etc. = 0. Subtracting the first equation, dividing the remainder by J a, and passing to the limit, we find d(x,y,c) = and M^|^) = 0. (2) The vail es of x and y thus determined will, in general, be functions of a; that is, we shall have ^ =/,(«); 2/ =/,(«); (3) which will give the values of the co-ordinates x and y of the iiLiitin;.{ p(.int 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, Fro. 40. Let c be the constant sum of the intercepts. Then, if a be the one intercept, the other will be a — a. Thus the equa- tion of the line is -+- a c y a = 1, in which a is the varying parameter. Clearing of fractions, we may write the equation cf){x, ijy a) = cx-\-a{ij — X — r) + a' = 0, d(f) whence da = y — X — c -\- 2it = 0. From the last equation we have a=:^x-y-{■ c); this value of a being substituted in the other gives ex — 'l{x — y -\- cY — 0, or {x - yY - M^ -\- y) + c' = 0. 5 sum of the istant. 'hen, if a be s the equa- 0, THEORY OF ENVELOPES. 176 This equation, being of the second degree in the co-ordi- nates, is a conic section. The terms of the second degree forming a perfect square, it is a parabola. The equation of the axis of the parabola is X — y = 0. 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. {/3) When the difference, a — b, of the intercepts is a posi- tive constant, the parabola is in the fourth quadrant; when a negative constant, in tlie 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. ' i •! :r; : f ii 'm I • * IMAGE EVALUATION TEST TARGET (MT-S) /> ^ J%s .>V%^ K° 1.0 I.I £ Ui 112.0 1.8 1-25 1.4 1.6 •• 6" ► V ■z •c='l# Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 /. f/. J M'i 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 i ^ ma = c = a constant. The equation of the line then becomes « c — 7na 1 = 0. Proceeding as before, we find the equation of the envelope to be (mx — yY — 2c(mx + y) -|- c' = 0, which is still the equation of a parabola. 6. To find the envelope of a line which cuts off intercepts subject to the condition -■4-- = l (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 Of = — ; cos a-—, a _ b The equation of the varying Lbj will then oecome d>(x, v) = — sin or + — cos or = 1. By differentiating with respect to a, we have d4e).-{a]r.+- Now, consider the case in which the curves meet at the point P, whose abscissa is x^. Then y, - ^0 = 0, v\ li I li I 182 THE DIFFSBENTIAL CALCULUS. and the intercept of the ordinate will be [(f),- (I-).]* + '«""" '" *■' «*"•' which, when h becomes infinitesimal, is an infinitesimal of the first order. « If we also have ldy\ _ ldy\ \dxi~ xdx); the ordinates will differ only by a quantity containing h* as a factor, and so of the second order. Hence: Wfien two curves are tangent to each other, they are sepa- rated only by quantities of at least the second order at an in- finitesimal distance from the point oftangency. 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 1 ^.'erential coefficients for the two curves are equal at the p > 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 contact ; in those of an odd order they do not. For, in contact of the ?ith order, the first term of y' — y (§ 104) which does not vanish contains A^^* as a factor. If n is odd, 7i + 1 is even, and y' — y has the same alge- braic sign whether we *^^ake h positively or negatively. Hence the curves do not intersect. If n is even, n-\-l\B odd, and the values oi y' — y havf» opposite signs on the two sides of the point of contact, thus showing that the curves intersect. i)i .J. CURVATURE; EV0LUTB8 AND INVOLUTES. 183 106. Eadius of Curifature. The curvature at anyiK>int is measured thus: We pass froi;:i the point P to a point P' in- finitesimally near it. The curyature is then measured by the ratio of the change in the direction of the tan- gent (or normal) to the distance PP*» Let us put a 7^ the angle which the tangent at P m^ikes with the axis of X. a -\- da = the same angle for the tangent at P\ ds E the infinitesimal dibtance PP\ Then, by definition. Curvature = -j-, ds Fio. 41. Now, because we have, by differentiation. tan a = -^, dx Bed' a da = -v^^dx. sec' Of = 1 + tan' a — 1 -{- Also, and da = y ll ■}- —-Adx, From these equations we readily derive dx* Curvature = —r ds (^+gr Now, draw normals to the curve at the points P and /", and let C be their point of intersection. Because they are perpendicular to the tangents, th^ angle POP' between theni will be da, and if we put p = PO, nil !i ' t 1 1 1 I ( if! i i* ■I livfM ■'if.; 184 THE DIFFERENTIAL CALCULUS. * we shall have ds Hence p — —-^ da curvature PP' =zds = pda. dx* The length p is called the radius of curvature at the point P, and O 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, p7 Fio. 48. 107. The Osculating Circle, If, on the normal PC io any curve at the point P, we take any point as the centre of a circle through Pf 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 Xf y, the co-ordinates of P; jo = ~- f or the curve at the point P; q = ^ for the curve at the point P. CURVATURE; E VOLUTES AND INVOLUTES. 185 These last two quantities are found by differentiating the equation of the curre. * Now, ~ and -i-^ must have these same vnlues at the point {x, y) in the case of the circle having contact of the second order (§ 104). Let the equation of this circle be (X - aY + (y - hy = r\ (a) By differentiation, we have (x - a)dx + (y - h)dy = 0, whence dy __ X — a _ dx ^ b — y = p. w Differentiating again, ^ _ _i_ , (^ ~(i)dy _ (y - by H- ( r - a)* dx'-b-y'^ (b-yydx - {y- by From (b) combined with (a) we find {x - ay _ r* w 1+y = l-f I _ {y - by (y - b) ty ■■■ (!+/)• = (y - *)■■ Dividing this by (c) gives ^ _ a+py r = , Q 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. ill ) ,; fi '\ \ n M ; lee THK DTFFKRKNTIAL CALCULUS. J f Cor. Tho osculating circle will, in general, intersect the curve at the point of contact, for it has contact of the second ord3r. This may also bo seen by reflecting that tho 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 v/ill either increase on both sides of this point or diminish on both sides^ whence the circle will not intersect the curve, \ 108. Radius of Curvature when the Abscissa is not taken as the Independent Variable. Suppose that, instead of x, some other variable, ti, 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 \dul (2) Also, we have fdy_Y fdxV /dyV 1 _L (^y\ — 1 _L ^^"^^^ ^ _ \C?^< / \du I . . "^ \d^} ~ "^ idxv ~ TdP\' ' ^"^^ \du I \du I These expressions being substituted in the expression for the radius of curvature, it becomes CURVATURE; E VOLUTES AND INVOLUTES. 187 * ) i P = - \ \du I ^ [iiu I \ (Py dx d^x dy du* du du' du (4) lOO. Radius of Curvature of a Curve referred to Polar Co-ordinates. Let the equation of the curve be given in the form 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 OS 9, y — r sm 6, regarding r as a function of 0, we find, when we put, for brevity, dr d'r — — r sin ^ -|- r' cos 6) dx Te II = (r" - r) cos e - 2r' sin ff; -^ = r cos 6' + r' sin 6*; at/ A ^ ^^f, _ ^.) gijj _^ 2r' cos e. By substituting these derivatives with respect to for those with respect to u in (4) and performing easy reductions, we find - _Jrl±i!!)L__ ^-r' - rr" + 2r" ~ (5) which is the required expression for the radius of curvature. 'i ' M 'P 11 :\ tl n 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 '' - 2px, whence we find y dx y ' cly_ dx^~ Then, by substituting in the expression for /), we find n? ^ _ ( y' + f) \ the negative sign being omitted, because we have no occasion to apply any sign to p. At the vertex y = 0, whence p=p. Hence, at the vertex, the radiis of curvature is equal to the semi-parameter, and the cent.'e o! cuivature is therefore twice as far from the vertex as the i'ocus is. 3. Show that the radius of curvature at any point {x, y) of an ellipse is ^ ~ a*b' 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 s^me in the case of the hyperbola as in that of the ellip?3e. 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 = i^\. !.;!l ) occasion I St^me m 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 (?), §80, of the cycloid, we find dx -^— = a — a cos w = y, du ^ d^x dy du du -~z = a cos ti. du (2) Then, by substituting in (4) and reducing, we find, for the radius of curvature, p = 2^a i^l — cos w = 4a 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 = ad) is a (l + n^ 8. The Logarithmic Spiral, The equation of the loga- rithmic spiral being 10 r = ae , show that the radius of curvature is p = r VT+T. Hence show that the line drawn from the centre of curva- ture of any point P of the spiral to the pole is j.>erpendicular 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 P = a «■ 3 l^cos 36* 3r- ! H. ■ ' I '11 II t I !-1 '■ 1 tf ! ' 4 1 'nm 4 ^ mi 190 THE DIFFERENTIAL CALCULUS. 110. Evolutes and Involutes. For every point of a curve there is a centre of curvature, found by the preceding for- mul8B. 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 fio. 43. C, we have 1 a; = a;^ — p sin a; y = yj + PCOB a. Substituting for p its value (§ 106), and for sin a and cos a their values fiom the equation we find tan^ = g. 1 + ^.y. dy; dx/ dx; 1 + y = y,+ dyl dx; d% 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. ; X. CURVATURE; EV0LUTE8 AND INVOLUTES. 191 111. Cane of ail Auxiliav^ 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 ~, ~J, etc. , as in § 108. Thus we find, instead of (1), (dxy du du" x= X, dy^ \du u I ^ \du I dud'y^dx^ d^x^dy/ du^ du du'' du IdxV iclyV _ dx^ \du / \dn I ^ ~ ^' du iv'y^ dx^ d'x^ dy^* dit* du du^ die m T which are the equations of the evolute in the same form. EXAMPLES OF EVOLUTES. 112. T7ie 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 , , V/ , 3 y; * ^ I) ^ ^ p We now have to eliminate y^ from these two equations, x^ having already been eliminated by the equation of the curve. They give 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 t "1 y =27 V W'^ '' 1111 If ■m 192 THS! DIFFERENTIAL CALCULUS. 113, Evolute of the Ellipse. From the equation of the ellipse, we find a'y/ dx; .1 •• By substituting in (1) and reducing, we find b*x;\ __ a*b' - a*y; - b,x/ X ' \ ci> y/J a'y// ' a*b^ Remarking that the equation of the ellipse gives a*b' - a*y; = a'b'x% and putting 6^ = a^ — b*, the preceding equation becomes 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 {b)f and then substitute them in the equation of the ellipse. From (a) and (b), we find which values are to be substituted in the equa- tion a' + J' " ■^• We thus find, for the equation of the evolute of the ellipse. The figure shows the form of the curve. The following T)roperties should be de- duced by the student. CURVATURE; EV0LUTE8 AND INVOLUTES. 193 (a) The evolute lies wholly within the ellipse, or cuts it (as in the figure), according as e' < i or e* > \. {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 formulae (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 _ d^ ^ — _ VI _ ^• dii^ du du* du ~ ^ '' X = x^ -[- 2a sin u = a(u -f sin ti); y = y^ — 2a{l — cos u) = — a{l — cos u). 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 = Tt we have X = aTt; y= -2a; giving a point 0, 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 x' and y'. We then have x' ■='X — an ■= a(d — tt -f sin 6)\ y' = y -|- 2a = a{l + cos 6). Fio. 45. If we now put these equations become e' = e-7r, I •'1 1 'i'i ' HI < < lii'ni 194 THE DIFFEBENTIAL CALCULUS, X* = a{B' - sin 6/'); y' = «(i - cos ey, which are the equations of another cycloid, equal to the original one, and similarly situated. The cycloid therefore posesses the remarkable property of being identical in form with its own evolute. 115. Fundamental Properties of the Evolute. Theorem I. The involute of a curve 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^, '>y definition, the locus of the point of intersection of consecutive normals. But this point has been shown to be the centre of curvature, whose locuc is, by definition, the evolute. Hence follows the theorem. Corollary. 2'he nor- mals io 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. Tf the os- culating circle move around the curve, the motion of its centre is along the line join- ing that centre to the ^oint of contact. This theorem will be made evident by a study of the figure. If the line P^C^ be one of the nor- mals from the point of contact P, to the centre, then, since ■ a. 46. CUli VA TUBE ; F.VOL JITE8 AND INVOL UTES. 1 95 • n [ual to the (1 therefore 5al in form wlope of its a straight ' upon the ive an en- sous of the this point locuc is, an, since this normal is tangent to the locus of the centre, it will bo the line abng which the centre is moving at the instant. TiiEOMEM III. The arc of the cvolute contained between any two points is equal to the difference of the radii of the osctdating circles whose centres are at these points. For» if we suppose the points C„ C,, etc., to approach in- finitesimally near each other, then, sinro the infinitesimal arcs Cf\, ^-fi^y ®^^'* ^^'^ 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 - «)« + (y ^ lY = P\ where a and h are the co-ordinates of the centre of curvature, and therefore of a point of the e •. olute. The complete differential of this equation gives {x — a) (dx — da) + (^ — 5) {dy — db) -• pdp. (a) If, in this equation, we suppose x and y to be the co-ordi- nates of the point 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. (b) Subtracting this equation from {a) and dividing by p, we find y -b c — a da ■+■ P -db = — J/o, (c) 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 /? for the angle which the normal radius makes with the axis of X, we have X a = cos /3', V -~ ^ = sin y^. (d) 1 ;i '; I t ;i 196 THE niFFEllENTIAL CALCULUS. i: \\ I I I S i But this same normal radius is a tarigent to the evolute. If wo call (T the arc of tho evolute, wo find by a simple con- struction da = cos pd(T; db — sin {ido\ Multiplying these equations by cos /? and sin /', respectively, and adding, we find da = cos ftda -\- sin ^dh. Comparing (c) and {d), we find d(T := — dp, or d{(T -\- p) = 0. Now, a quantity whose difierential is zero is a constant. Hence we always have a -\- p =: constant, or (T = constant — p, \ If wo represent by c, and a*, the arcs from any arbitrary point of the involute to the two chosen points, and by p, and /o, the values of p for these points, we have (T, = const. — p,; (T, = const. — Pj,. .•. (T, — (T^ = p, -p,, 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, as ^, ^, C or D, and the intercepted arc must then be taken as zero. 116. Invohites. The involute, of a curve is that curve which has C as its evolute. The fundamental property of the involute is this: The involute may be formed from tiie evolute by rolling a tangent CUBVATURUE; EV0LVTE8 AND INVOLUTES. 197 line upon the latter. A point P on the rolling tangent will then describe the involute. This will be been by reference to Fig. 4G. The rolling line, being tangent to the evolate, coincides with the radius /*,6'„ and as it rolls along the evolute into successive positions, /^,C„ P,C„ etc., the motion of the point P is continually normal to its direction. It will also bo seen that the radius of curvature of the in- volute at each point is equal to the distance PC from P to the 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. :;1 '•: i i .lii PART II. THE INTEGRAL CALCULUS. l: |: ■It ■ I I ii li I Mi THE 117. a functi( we may. which w< In the We have and the j entiated, Every i The pr The op called "i: function pression means: th F'{x)dx. ' t PART II. THE INTEGRAL CALCULUS, w CHAPTER I. THE ELEMENTARY FORMS OF INTEGRATION. 117. Definition of Integration, Whenever we have given a function of a variable x^ say u = F(x), we may, by differentiation, obtain another function of a;, <^W nil \ n = -^ (^)' which we call the derived function. In the integral calculus wo consider the reverse process. We have given a derived function and the problem is: What function or functions, token differ- entiated, 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 / , called " integral of,'' written before the product of the given function by the differential of the variable. Thus the ex- pression fF'(x)dx means: that function whose differential with respect to x is F'(x)dx. u I 1 ,i li 'W M\\ 202 THE INTEGRAL CALCULUS. ^ 111 i:H Calling u the required function, then if *ye have we must also have ^^^* T7// \ w As examples: Because we have Because we have d{x^) = 2xdx, ' 2xdx = x^. d{ax^ -{-Ix-^- c) = {2ax -j- b)dx, I {2ax + ^)dx = ax* -{- bx -{- c. And, in general, if, by differentiation, we have dF{z) = F'{x)dx, \ we shall have / F'{x)dx = F{x), 118. Arbitrary Constant of Integration, The following principle is a fundamental one of the integral calculus: If F{x) is the integral of any derived function of the va- riable X, then every function of the form F{x)-\-h h being any quantity whatever independent of Xy will also be an integral. This follows immediately from the fact that h will dis- 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 of view : Since the problem of differentiation is to find a func- tion which, being differentiated, will give a certain result, and since any quantity independent of the variable which may be added to the original function will have disappeared by differentiation, it follows that wc must, to have the most THE ELEMENTARY FORMS OF INTEGRATION. 203 Hi general expression for the inte^al, 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 the integration is performed. It follows from all this that the integral can never be com- pletely found from the differential equation alone, but that some 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 ist it be given that when X — a, then the integral = K, We must have, by this datum, F{a) -{-h = K, 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 positive 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. Inter/ration of Entire Fxmctions, Theorem I. The integral of any j)ower of a variable is the power higher by unity, divided by the increased exponent. In symbolic language, we have x^dx = , ^ -f- h. f n-\-l X n + l «' For, by differentiating the expression -|- h, we have ; 1 il IIP, 204 THE INTEGRAL CALCULUS. I;. I i\ Theorem II. Any constant factor of the given differen- tial may be written before the sign of integration. In symbolic language. faF\x)dx = aCF\x)dx, This is the converse of the Theorem of § 23. By that theorem we have d{aF{xy) = adF{x), 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) = - J*F'{x)dx. Hence the corollary: If the integral is preceded by the nega- tive sign we 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, /*(jr-{ r+ Z-\- . . ,)dx - f Xdx-\- J Ydx^ C Zdx^ . . 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"^ -\- ex' -\- » . ,)dx. By Theorem III., u= I aTf^dx -{- I bx^dx ■\- I cx'dx -f- o o • . I differeu' By that lows. [x)dx. y the nega- Ue derived I of several frals of the fzdx-\- . . 22. he integral function in 3 integral TEE ELEMENTARY FORMS OF INTEGRATION. 205 By Theorem II., / ax^dx = a I x^dx; eic* ere. y and by Theorem I., etc. w + 1 etc. By successive substitution we then have u = aa;"*+* . ia;"+* . cx^-*-^ --r + + -{■ ah^ -V Ih^ 4- cA, + . . . , m -\-l n -{-1 jo + 1 where /*,, 7i„ 7i,, etc., are the arbitrary constants added to tho separate integrals. Since the sum of the products of any number of constants by constant factors is itself a constant, we may represent the sum ah^ + M, -f ^'*a l>y the single symbol h. Thus we have y (aa;"* -f hx"" -f cx^ -\- . . .)dx ax m+l T + Ja;"+' ex p+i w + 1 ' n + 1 ' p \-l EXERCISES. -{- , . .-\-h. Form the integrals of the following expressions, multiplied by dx\ I. X \ -t .-s S. «a;'. 6. Ja;'. 7. ax~*, 8. Ja;~'. 9. rtiB + I. 10. «a;' — c. II. ax^ + c.r. 12. ax^ — ca;* 13. a;*. 14. xi. 15. a;-*. 16. ax-^. 17. «.T*— o.-c-*. 18. 7»a;* :. 10. — ., 20, a A — ■«. a* XX. X 120. ^Ae Logarithmic Function, An exceptional case of Theorem I. occurs when n — — 1, because then n -{-1 = 0, and the function becomes infinite in form. But since d'\os X = — = x~^dx, ° X ^I'r f. i I j I I, ■ j I ! I M 206 THE INTEaBAL CALCULUS. ■1.1 .1 it follows that we have for this special case / x-^dx = / — = log a; + ^*- («) Let c be the number of which h is the logarithm. We then have log x-\-h-= log X + log c = log ex. We may equally suppose 7i = — log c = log -. Then X log X -{-h^ log — . or Hence we may write either rdx , y-=log.:., /dx , X — = log -; X ° c ' (*) c being an arbitrary constant. We thus have the principle: The arbitrary constant added to a logarithm may he introduced hy multiplying or dividing hy an arbitrary constant the number whose logarithm is ex- pressed. 121. We may derive the integral (a) directly from Theo- rem I., thus: In the general form //pn + l X'^dx = —r + h w 4- 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 n + l n +1 + A = 0; h=~ a r + l W + 1 Thus the integral will become / x^'dx = X n+l a n + l 71 + 1 m^^' THE ELEMENTARY FORMS OF INTEGRATION. 207 in which a takes the place of the arbitrary constant. This expression becomes indeterminate for ?i = — 1. But in this case its limit is found by § 71, Ex. 5, to be log x — log a. Thus we have / x^^dz = log X — log a = log -, as before, log a being now the arbitrary constant. 122. Exponential Functions. Since we have e?(rt*) = log a . a^dxy it follows that we have f log a. a'dx = a* -f h. or, applying Th. II., § 119, to the first member and then di- viding by log a, «* + /* / a'dx = log a' which we may write in the form / a'dx = a' log a + h. h because z is itself a constant which we may represent by h, 123. 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 which 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. ,.,, «r ■ iil il! 1 ■HI III 208 THE INTEGRAL CALCULUS. M :i ?-f I ' ■1 ! i ! y » • • / y'^dy = _ y n + l W + 1 (1) % Jit J y = log y. (2) d'sin ij = cos ydy, . • .ycos ydy = sm y- (3) '.' d-coay = — sin ydy, • . • 6?- tan y = sec' ydy, '.'d-coty ^ - ^y /•■ t/ ■/ sin ydy = — cos y (4) • 2 * Sin y dy cos' w dy — tan ?/. (5) ^''f^. =-coty.(6) sm y . . 7 tan w , '.'d'secy = "d?/, cos y *^ •.wZ-sin<-'>v = — ^--, dy /tan ?/ ^•a» ^ = n^ log rt<^?y, dy ■f ^/;/ 1 + J/ -, =tan(-')f/.(10) ly =: ft" log a (11) d • sin h^~ ^h/ = — — " — Vf+1 dy ^?*cosh^~%= = sin h<-»)y = log (y + Vf-{- 1). (12) «/ ■/ i^f- 1 dy Vy" = = cos h^-')?/ = log (y 4- i/y»- 1). (13) c?*tanh^~*\y __ tanh^-"y = -- log 2 If y 1-y- (14) INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 209 I f CHAPTER II. INTEGRALS IMMEDIATELY. REDUCIBLE TO THE ELEMENTARY FORMS. 124. Integrals Reducible to the Form I y^dy. 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 4- xYdx. We might develop {a + 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 + ^)> we may write the integral thus: / {a -f xYd{a + x). It is now in the form (1), y being replaced hy a -\- x. Hence (1) In the same way, / {a — xYdx = — J {a— xYd{a — x) =h — ~Z\ — • To take another step, let us have to find f{a + IxYdx. We have 1 1 dx = ■j-d{hx) — jd{a -(- Ix), Hence, by applying Th. II., .1 :.:■ 7 I r 11' i; 210 THE INTEGRAL CALCULUS. i « 1 il We might also introduce a nev/ symbol, y =a -{-hx, and then we should have to integrate y"^/y with the result in § 123. Substituting for y its value in terms of x, we should then have the result (2).* These transformations apply equally whether n, a and b are entire or f ractional, positive or negative. EXERCISES. Find: i. / (« -f- xydx, 2. / d{a — xydx» 3. / (rt — 2xydx, 4. I {a-{- x)''*dx, $. I {a — x)~*dx, 6. / (a-{-mx)~^dx. y. 1 {a — mxydx. 8. / {a — mx)~^dx. /* dx f* dx p dx 9-y (« ^ ^y 'o y (^ _ ^)»- "-J (« _ 4a;)"^- ' 12. / (rt -}- xydx. 13. / (a -f nxydx. 14. / (« + x^Yxdx. '^-/{^^ + 7' + ¥*)^^^- ^^•/(^^-• 19. / (a-{-bx -\- cx^){b -\- 2cx)dx. 20. / («-[-&« + ca;'')"(J -f 2cx)dx, 21 ■/ (a + /S'a; + crc*)* * The question whether to introduce a new symbol for a function whose difTerential is to he used must lie decided by the student in eacli 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. IMi. INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 211 125. A2)plication to the Case of a Falling Body. We have shown (§ 33) that if, at a time tj 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 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 tlio 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 , It = - ^*' («) 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\ (b ) Here the constant 7i 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 ia the distance through which the body has fallen. The equation {b) gives h — z = ^gt^. (c) Hence: The distance through which the body has fallen is proportio7ial 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 througn which it would move in the same period luith its acquired velocity at the end of the period. 1 If I 'Si ii fv < ■ ; i I ; 1: r Ji ■4\ m m 1 ;? ffll 't IH 212 riTi? INTEGltAL CALCULUS. 136. Reduction to the Logarithmic Form, Let ua havo to find /mdx ax + d* Since -I^^L + /.. (2) II. If q —p* is negative, that is, if 4ac — - J" is negative in (3), 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 {h) in the form ci{x-\-p) p'-q-{x-^py' The integral is now in the form (2) of § 128, and we have dx n d{x -f- p) s x^ + 2j3a: + q /' a\x - V^q- (^ + VT = A- = h- Vp' =rtanh<-« x-^p Vjf — q log ^V f-q + x + p ^ ^3^ ^ Vy — q ° ^P'— 1—{x-\-p) Making the same substitutions in these equations that we made in Case I., we find dx , 3 , , (_i) 2ca; -f ^ J a -\-bx-\- ex' = 7i - = h- 1 tan h^" \^b'' — ^ac i/l,^^iac-\-2cx-\-b ,,. , - log c— - -^—•(4) 4/6'-4r?c ^/b^-^ac-{2cxAry) III. If p^ — q=. 0, the expression to be integrated becomes dx T— j— -T5. We have already integrated this form and found dx ^ 1 n — f i^+Pf x-^p I. INTEQliALS REDUCIBLE TO ELEMENTARY FORMS. 217 r e have h h. (2) Degative in quantities. lat the ex- e, we write i we have -^+P. (3) IS that we 4iaG cx-{-b) i becomes d found EXERCISES. Integrate the following expressions: dx dx I. x^ - 2x - 4* dx 2. {x -a){x -fi)' ^' a-\- 2bx - a;'* dx dx dx 130. Inverse Sines and Cosines as Integrals. From what has already been shown (§ 123, (8) and (9)), it will be seen that we have the two following integral forms: — = sin ^~^> a; + A E ui Vl-x' == = cos ^~^^x-\- li' = u'\ where we have added h and W as arbitrary constants of in- tegration. Comparing the first members of these equations, we see that each is the negative of the other. The question may therefore be asked why we should not write the second equation in the form ^•* -lit _•__ t—\\ .. / \ u = -/- = li"— sin^'^^o;. yi-a;" as well as in the form (i). The answer is that no error would arise in doing so, because the forms (6) and (c) are equivalent. From {b) we derive X = cos {u' — V) — cos (/*' — u')\ (d) and from (c), X = sin {h" - u'). (e) Now, we always have sin (a + 90°) = cos a. Hence (d) and (e) become identical by putting A" = h' + 90°, which we may always do, because the value of A" is quite arbitrary. il :: > : I i m 1 1 r 218 THE IJSTEGBAL 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 ifi expressed in the form du dx Vl X i> we may express the relation between u and x themselves either in the form u = Bm^~^'^ X •\- h or in the form x = sin (u — h). So, also, in the form (1) of § 128 we may express the rela- tion between x and i(> either as it is there written or in the reverse form, a; = a tan a(u — h). 132. Integration of dx We have dx i/a" T x' d'^- X' sin(-»)- + A. a (1) In the same way f- 7^^ =cos^-'>-4-^ or A-sin(-^>-, (2) ^ Va' - a;' « a ^ ' We also have = log-(^ + ^^' + «').(3) a INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 219 I fact that I expressed be relation the form themselves 3 the rela- L or in the h. (1) ^)-. (2) + «').(3) r--t=. = C-^^^ = cos h <-> i + A = \og-(x+V'.^-a'). (4) EXERCISES. Integrate the differentials: dx I. Vc — x' ndy Va' - nY mdz II. 13. dy ' Via' 4- V dy Vb-{-~c(x-ay 2xdx Va*- 2. 4. 6. 8. 10, 12. 14. rf«/ l/4« dx Va' -{X- dz ay Via ' - m'z' mdx Va' 2 2 -- fU X dx Va* 4- wi'(a; dx -ay V(x - ay - ■ic' nx'' ^-'dx a;' ^^2u _ ^2» COS xdx 15. If du = — -===r-nrr then ein X = a cos (?< + h). Va — sm' « \ I / 16. 18. 20. e*^:c Vl -e ,2x 17. iill, 220 THE INTEGIUL CALCULUS. 183. Integration of clx Every differential of Va -\- bx ± ex'' 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 ax'. Case I. The term ex* is negative. Putting, as before. _1 h a we have V« + hx - ex' = Ve Vq + '^j^x - a;' = V^ ^F^~^H-J^-p)' Then, comparing with (1) of § 133, wc find dx 1 /* d{x — p) r dx - Jl /*. Vc*^ Vp"" -\- q — {x — pY = — ==sm (-1) X P _ V^ -zzBin (-1). 2ex (1) Vp' + q Vc Vb' + lae In order that this expression may be real, p'^ -\- q or b' -\- iae must be positive. If this quantity is negative the integral will be wholly imaginary, but may be reduced to an inverse hyperbolic sine multiplied by the imaginary unit. Casi: II. The term ex' is p)Ositive. We now have Va -f bx -{- ex' — Ve V{x -f- p)' -{- q ~ p"' dx _ 1 /* ^^(-^ + V\ ^bx + ex' ~ VcJ ^{x+pY -{-q - p' J Va = — log C{x -}-p + Vx' + 2px + q) re C = T log ;7i(3ca; + b-i- 2c* Va + bx + ex'). Because Cis an arbitrary quantity, the quotient of C by 2ei is equally an arbitrary quantity, and may be represented by the single symbol C. Thus we have / dx Va~j^bx + ^ = —log C{b-{-2ex-^2 Vc Va-{-bx-\-cx').{2) ex INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 221 M! (rential of IS of the 29. The term cx^. (fore, - ix-p)\ b_ " — • (1) rb' + 4:ac Q integral ,n inverse \-q) bx -\- cx^). of C by presented +.:.'). (2) Integrate: EXERCISES. dv I. V^a" + 4:by — f ydy _ cos Odd 7. sin d — sin' sin 6 cos Odd V4: - COS 26* - cos'' 20 2. 4. 6. 8. (?y |/(^ + ?/) (^» - y) dy VaY - by -{- y' cos OdO a sin ^rZ/9 Va' - Z>"(i - CCS 0) ■i' 134. Exponential Formfi. Using the form (11) of § 123, we may reduce and integrate the simplest exponential dif- ferentials as follows: a'^''dx = ~ / « «* d(mx) = —, h h. (1) jnj ^ m log a ^ ' Ca'+^'dx = fa^-^^dix -{- b) = ~- + h, (2) /I /» ^ ma; + 6 ^mx + 6^;^^ ± / a'»* + ^/(wX+3) =-^i [-h, (3) 7Wt/ ^ ' ?>ilog« ^ ' Ca-'^'dx ■= Ca-'^^'di^— mx) = -^;:77i;^— -• (4) in log « * Integrate: r. e^dx. 4. {a 4- b)e''dx. EXERCISES. 2. b^dy. 5. a^-^dy. 3. a^-'^dy. 6. a~'^dx. 7. («^ + rt-*)fZa;. 8. (a* — rt-'')^Zaj. 9. (rt + e'')r7a;. 10. (a^''—a~'^)dx. II. e^^f?.?: 1 -I- e*' 14. (1 + a'^ydx. 12. e^'^dx 1+7'^* 15. («"^ + «-""«)'t/a;. 16. Ce^^xdx. 17 •/ e^^'^xdx. [8. /*e - «(^' - %c?a;. I, < ; |i! i ! ^r ; ! li \i :il' in ::!K vim ■•it ii H!! r 11 ■M 222 2!ffiE? INTEGRAL CALCULUS. CHAPTER III. INTEGRATION BY RATIONAL TRANSFORMATIONS. 135. We haye 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. Integration of ^^^—dx. Any form of this kind, when m is entire, may be integrated by developing the numerator by the binomial theorem. We then have 1 {a + xY _ ^ I ma •" ~ ^« + e)«-^'+eto.. and each term can be integrated separately. \in < in •\- 2, and entire, one of the terms of the integral will contain log x. II. Integration of x^dx We may reduce this form to {a + bx)*"' the preceding, by introducing a new variable, z, defined by the equation z = a-{- bx, rrii . . z — a , dz This gives X = dx = b ' b' Substituting these values of x and dx in the expression to be integrated, it becomes {z — a)^dz wh'.ch may be integrated by the method of the last article. III. Integration of — -t-t — r — a* a "T* ox "jc, cx We reduce the denomi- 1ATI0NS. lich cannot in the last Al consider cind, when numerator INTEGRATION BY RATIONAL TRANSFORMATIONS. 223 nator to the form ± (y - q) ± {x + jw)' as in § 129. Then, putting, for brevity, i' =y - q, 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 b' ± z'" " ¥~±~? ~ F±7'' 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 + bx)dx N ±{^-pr be, < m + 2, tain log X. lis form to defined by EXERCISES. )ression to article, e denomi- Integrate: (x — ay dx 1 . 5 • X x*dx dx \a XI x^dx II. • (a' - xY xdx * '^'^{b-xf (y - b)dy 13 {y-f^y + {y + hr {x — a)dx 15 :c{pi, — b) ' z'^dz 2. (1 - 1)V. \a xj X x^dx ^' (1+^:7* , (x -\- a)dx o. 8. {a - xy • x^dx lO. (1 _ ir W x' I zdz 12. i6. {a + zY + (a - zy {z — c)dz a^ — az -\- z^' {y + a)dy 'a^-{y + br z^dz (1 - zf •' t lilt :i n I i I . ■ S I f ! ; Ui 224 THE INTEOHAL CALCULUS. 1 i il! 'm 136. Rcductio7i 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 q. + (7,^ -f q^^" + . . . + qn^"" ~ D ' If the degree w 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 n. Then, if we put Q for the entire part of the quotient, and R for the remainder, the fraction will be reduced to D ^^ D' If we have to integrate this expression, then, since Q is an entire function of Xy the differential Qdx can be integrated by § 119, leaving only the proper fraction -j-. 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- eqiial. Let these roots hQ a, ^, y . , . 6. Then, as shown in Algebra, we shall have D={x- a){x - /3){x -y) . . . (x- 6). "We then assume ^ A . B . C D X a + + . • • > X — fi X — y Ay By C, etc., being undetermined coefficients. To deter- mine them we reduce the fractions in the second member to the common denominator D, equate the sum of the numera- tors of the new fractions to Ry and then equate the co- eflficients of like powers of av iL A ra- lominator >rm is jgrea n of y the de- ) than 71. it, and M e ^ is an ntegrated w, such a mm of a provided le theory w by ex- es which and un- is shown deter- mber to nuraera- the co- INTEGRATION BY JIATIONAL TRANSFORM iTIONS. "22^) As an example, lot us take the fraction ■ , - - ax. X — X We readily find, by solving the equation re' — a; = 0, X .» Assume a; + 3 X = x{x - l){x + 1). X' =4+ B . C X X X — \ X ■\-\ _ {A J^B -\-C)x' ^{D- C)x -A — IT • X — X Equating the coefficients of poAvers of x, we have A-\- B^ C-0) B- C=l; A = -3; whence B = 2 and = 1. Hence a; -f • 3 _ 3 2 1 x" — x~ XX — Ix-^V and then, by § 120, *J X. — X d X t/ X — 1 d X -\-l = - 3 log a; + 2 log (x - 1) + log (a; + 1) + log C C{x + l){x - 1)' log Integrate: X' EXERCISES. I. 3- 5- 7 {x — Vjdx X? — a; — 6* xdx X^^x^' 'a;' 4- a; + 1 x^ -\-x^ — %x {x" + 2a;*)r?a; xdx 2. x" r • x" + 2:c' - 8a;* x^dx ^* a;' - {ci ^h)x + ab' 4 6. 8. lO. {x + x^yix ' {x-l){xi-l){x-2){x-\-2y x^dx 9 a* X — a {x' + x')dx x{x - l){x -1- l)(a; - 2)' dx x' — {a + hyx" + ahx' ; ij - ! . n i ■: \ 1 '■ 11 m; '■■\^- ii h i V'M 226 THE INTEGRAL CALCULUS. Case II, Soine of the root a equal to mch other. Let the factor X — a appear in D to the nth. power. Then, if we followed the process of Case I., we should find ourselves with more equations than unknown quantities, because the n fractions X — a + B X — a + C X — a + ... would coalesce into one. To avoid this we write the assumed series of fractions in the form rn + B n-1 "T" • • • + F X a 4- H x—fi + etc.. {x — a)" {x — a) and then we proceed to reduce to a common denominator as before. The coefficients Ay B, etc., are now equal in num- ber to the terms of the equation Z^ = 0, so that we shall ha>e exactly conditions enough to determine them. As an example, let it be required to integrate a;' -5 ^ X — X — X -\-l We have x' - x' - x -{-1 - (x - 1)' (x + 1). We then assume x' - T) _ A B C {x-iy{x-{-i)~ {x-iy'^ x-i'^ x^i - (g + Oy + {A - '^C)x -{-A-B+C {x-iy{x-\-i) We find, by equating and solving, A = -2; B = +2; C=-l, Hence x'-6 - 2 i + {x - iy{x -^1)" {x-iy^x-i a: + r IN Tl - 2 I. hi \i Let the n, if we Ives with the w INTEOBATION BT RATIONAL TRANSFORMATIONS. ^27 The required integral is -./(.-i)-V. + ./^-|L^-/,^^ 2 + 2 log (.r - 1) - log (a: + 1) + log C x-l 2 , . C{x- 1)' ^ 1 . 1 1 1 (11 assumed - etc., nator as in num- all ha\e B+C EXERCISES. Integrate: I. dx x{x-]-iy' x^dx 2. dx x\x - ly' dx ^* {x - iy{x 4- ny {a -f x)dx ^' x\x - af " {x - a)\x - by 6. {a — x)dx x\x -f- ay{x - by 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 -{- ^i (where i E V — 1) is a root, then x = a — fti will be another root. To these roots correspond the product (x- a- /3i)(x - « + /3i) ={x- ay + yS'. By thus combining the imaginary factors the function D will be divided into factors all of which are real, but some of which, in the case of imaginary roots, will bo of the second degree. The assumed fraction corresponding to a pair of imaginary roots we place in the form A-\-Bx {x - ay + /3" II I i! .1 1 :■-:, ' s ^28 THE INTKGRAL CALCULm^. and then proceed to determine A and B as before by equa- tions of condition. Wo then divide the numerator A + Bx into tlio two parts A -}- Ba and B{x — a), the sum of which is vl -f- Bx. Thus we have to integrate r A-^Bcx r B(x - a )dx J (a; _ ay + /^' "^ J {x - ay + /i'* Tiio first term of (a) is, by methods already developed, A 4- Ba . , ..X — a ____ tan<->^-, and the second is ^Blog{{x-ay-\-^^). "We therefore have, for the complete integral, r A^-\-Bx J (x - i") (., _ ay + p^ A 4- Ba , , ..X — a -jr — tan ^~ ^' + i^ log! (2: -«)' + /?•}+ A. i ' EXERCISES. IM 'i M rx-\-^x\ [. / — T — -T-dx. «/ :/; — 1 r dx The real factors of the denominator in Ex. 1 are (a' -\- \){x -f- 1)(« — 1). We resolve the given fraction in the form A-\-Bx . G , D cc' + l 'aj + l'a;-l' X 1 \ and find it equal to r -| — -r— -\ . Then the integral is found to be \ log (a;2 + 1) + log (a;» - 1). The factors of the denominator in Ex. 2 are x—\ and a;' -j- * + ^ ~ (« + i)' + f. + 1' "* t-/ x^ - 2x + 4* r dx '' J x'+l' ' Note that a; + 2 is a factor of the denominator in (4). JNTbJG RATION BY UATWNAL THAN S FORMATIONS. i?*iO e by equa- -iv A-\- Bx tegrate (") loped, n+h. f 1)(« - 1). •al is found + «+! = 137. Integration bif Parts. Lot u aud v be any two functions of x. Wc liavo lUiir) (Iv , (111 ax dx ax By transposing iitid iutegniting wo have (Iv , /• (In (1) which is a general formula of the widest iippllcation, and should bo thoroughly memorized by the student. It shows us that whenever the dilTerential function to be integrated can be divided into two factors, one of which (,//.') can bo integrated by itself, the problem can be reduced to the inte- gration of some new expression [v-j—dA. The formula may bo written and memorized in the simpler form / udv — iiv — / vdUj (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 the new expression will be any easier to integrate than the original one; aud 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 formula) (1) trnd (•■i) is this: Express the given differential as the product of o7iof unction into the differential of a second function. Then its integral toill be the product of these tivo functions, minus the integral of the second fimction into the differential of the first. I ' ; ( « '\ lit m 230 TEE INTEGRAL CALCULUS. EXAMPLES AND EXERCISES IN INTEGRATION BY PARTS. JT I. To integrate x cos xdx. Wo have cos xdx = d'&m x. Therefore in (2) we have u^x\ v = sin x; and the formula becomes / X cos xdx = I xd'Bm x = x o'lux — I mi xdx = 2; sin a; + cos a: -|- h, which is the required expressi^ a, as we may readily prove by differentiation. Show in ^he same way that — r ■ ^ 2. I X sin xdx = — x cos x -\- sin x -f- h. 3. j X sec' xdx = X tan x — (what ?). 4. j x sin X cos xdx ■=. — \x cos "%% + i sin 2.c + A. 5. / log xdx -~ X log X — I xd' log x~x log a; — a; -f- ^-^^ 6. The process in question may be applied any number of times in succession. For example, I X* cos xdx = / re' c?" sin a; = x* sin x — % j x sin xdx. Then, by integrating the last term by parts, which we hiive already done, / x* cos xdx = a:' sin x -\-'^x cos a; — 2 sin a; + ^• 7. In the same way, / a:' cos xdx = / a:*c?*sin a; = a:* sin a; — 3 / a:' sin x^/j; / x' sin a;f?a; = — / a;'c?' cos a; = — a;' cos x-\-% I x cos a;(?x. Y PARTS. xdx proTe by 1u umber of II xdx. we have sin xdx; cog xdx. INTEGRATION BT RATIONAL TRANSFORMATIONS. 231 Then, by substitution, / x* cos xdx = {x* — Qx) sin J + (3a;' — 6) cos x + h, 8. In general, / a;** cos xdx = / x'^d'Bm x = x^BiD.x — n / x"*'^ sin xdx; — I x^~^ sin xdx = / x'*~^d'GOB x — a;"-* cos x — (n — 1) I a;""^ cos xdx; — I a;""' cos xdx =— aj^'^sin a; +(w — 2) / a;**"' sin a;(?a;; /a;'*"' sin a;t?a; =— a^^-'cosa; -{-(/j — 3) /a;""* cos a;(/a;. etc. etc. etc. Then, by successive substitution, W'9 find, for the required integral, \x''-n{n -l)x''-^-^7i{n-l){n-%){n-^)x''-^— . . . } sin a; + |wa;"-^ — n(n — l){n — 2)a;"-'+ . . .} cosa;. 9. In the same way, show that / x^ sin xdx = I -a:»+w(w-l)a;*-=*-w(n-l)(w-2)(w-3)a;'»-*+. . . { cosa: -}- {^ia;*-^ — 7i(n — 1) (?i — 3)a;'*-^ +• • • | sin x. /I p x^ "*"* a;** log xdx = / log a:c?- (3;" + ^) = - log a; 1 /»x ~ n + ij ~ n+l -c?a; = X n + l w + 1 log a; — X n+l II., Cxe-'^dx-- /*la:^-(e-««) = _^^f— I'+l Ce-'^dx, J J a ^ ' a aj Now, we have ie'^^dx = ■ ,— ax. Hence / xe~'^dx = — xe — ax a a s • i'l '11 ; ( 232 THE INTEGBAL CALCULUS. • •% til -u /■', 12. To integrate x'^e~'^dx when m 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, y.mn — x^ -j- h 1fl i: \ ill :l: • H iii i '< ¥ n m «! t- i 234 THE INTEGRAL CALCULUS. If the fractional exponent belongs to a function of x of the first degree, that is, of the form ax -\- J, we apply the same method by substituting the new variable for the proper root of this function. Example. To integrate {a + hx)^dx 1 + (a + bx)' We put (a -f- boc)^ = z; a -{-bx = z^; 2zdz dx = b ' {^' - ?^)' The expression to be integrated now becomes 2z^dz _2 b{l -{-z')~ b of which the integral is ~(z- tan<-»>2 + A) = ^ I {a + Jo;) *- tan <"»>(« + 5^^)*+/* I • EXERCISES. Integrate: x^dx I. 1-i-x (a — x)^dx l-\-a — x (x + c)* , 7. ^^ — ■ — '—dx. (x^cf 1 + (^ - c)\ l + (z- c)^ 2, - *^a x^ax y 5. 8. 1 + a;^ {a — x)^dx I -(a- xf {x - cf (x — c)i 3. xdx. 6. dx. l±±^^dx. (a - x)^ (2x — a)^dx 1 + (2a; - a) 1* 10. dz. II. 12. —ax, 1- 1 + (a; + «)' =^* + IS ^3. 14 ^y -{x- «)* a:*- «?a;. »' {x-ay-\-{x-a)^ dx. INTEOBATION OF IRRATIONAL DIFFERENTIALS. 235 3f X of the the same roper root rxY+h \ . -xdx. t — X , x)^ — ayclx 2x — a) i' 139. Cases when the Given Differential contains an Irrational Quantity of the Form Va-{-hx -\- ex*. It is a fundamental theorem of the Integral Calculus that if we put R = any quadratic function of x, then every ex- pression of the form F{xy VR)dx, (F{x, VTl) being a ratio7inl 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 integ :al can, in general, be expressed only in terms oi . ertain 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-\- ex* = Vc(x + z)\ , a -\- hx -{• cx^ = ex* + '^cxz -f- cz*. This giv33 X = cz — a 2cz' dx = — 2c a — Iz -\- cz* (b - 2c'zy dz; V a + iz + cz ' = - iTe^L^hp^. — 2cz («) By substituting the values givsn by (a), (b) and (c) for the radical, x, and dx, the expression to be integrated will become rational. Second Case : a positive and c negative. If the term in x* is negative while a is positive, we put Va + bx We thus derive X = , ^.^.a — 4/^ _|. xz^ b -2Va z («) t 11 I ; III ii.. I m f i 236 TEE INTEGRAL CALCULUS. t I 'i ¥!3 dx = _ 2( Vaz^ -Vac- hz) Va-i-hx — cx^ = — Vaz' - Va. ac Iz z' + c w io) The substitution of Lliese expressions will render the equa- tion to he integrated rational. Third Case : a and c both negative. If the extreme terms of the trinomial are both negative, we find the roots of the quadratic equation — a -{■ bx — ex* = 0, which roots we call a and /3. We then have — a-\-bx — ex* = c(a — x) {x — /?), and we introduce the new variable z by the condition V— a-{- bx — ex' = Vc(a — x) {x — fi) — Ve(x -— a)z, which gives az*-\- 6 ^ "";?' + 1 ' a^ _ 2(^ - ^ )zdz. V2 ) V—a-\-bx- cx V^j-a)z^ z*-^l "' substitutions which will render the equation rational. (a) (*) 140. We have already integrated one expression of tlie dx form just considered, namely, - without ration- Va + bx + ex* alization. There is yet another expression which admits of being integrated by a very simple transformation, namely, d0 = — =J?_. r Var* -}- br — 1 This is the polar equation of the orbit of a planet around the sun. To integrate it d;.'ectly, we put (*) the equa- me terms )ts of the n — a)z, (a) ? W (c) 1. )n of tlie it ration- idmits of imely. it around INTEQItATION OF IRRATIONAL DIFFERENTIALS. 237 1 , dx X = -: dr = -„. r X We thus reduce the expression to — dec -\-})X — x^ Proceeding as in §133, Case I., we find the value of the integral to bo / dr — cos^~ ^^ 2:r - I r— r>nH(~l) hr r Var^ -{- b?' — 1 Thus, e-7r = cos(-^> 2 - br cos' r Via 4- b' r Via + b' 7t being an arbitrary constant. Hence 2 — br , „ . = cos {6 — 7t). r Via + b' Solving with respect to r, we have, for the polar equation of the required curve, 2 ** ~ 5 + |/(4« + b') cos (^ - 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, 2 r = ^(^ - ^') = 1 + « cos 6^ 2 + 2e «(1 - e') ' «(1 - e') Comparing with (a), we have • a{l — e^) = J- = parameter of ellipse =p; cos 6 2e P or — jL\ — JH — -_ — eccentricity of ellipse. m « • H :li t i »■, 238 TBB INTEGRAL CALOVLVS. > ill I'-, if'. • Irrational Binomial Forms. 141. Oeneral Theory. An irrational binomial differen- tial is one in the form (a-\-hx''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, be T -; and let us put y = (a + Ix^y. (3) This will give, when raised to the rth power and multi- plied by x^dx, 1 (a + Ix'^Yx^dx — x'^y^dx, (3) We readily find, from (2), Ix"" = y» -^a; (a) _sy'-'dy ^ dx = bnx n-l > 'K y^iTdx— r~x ^ bn TO — n + la/r + «— 1 r 'dy; or, substituting for x its value from (a), x^y-dx = -iJy^-^Y ""^ y^'^'-'^dy. (4) This last differential will be rational if -— is an in- n teger, which will be the case if — ^t_ ig 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 a;" and multiplying the factor outside of it by a;"^, which will leave its value unchanged. It will then be 1 differen- (1) while p is luced to a fc terms, be (2) md multi- I (3) (a) (4) 1. - IS an m- We shall tionalized, ng the bi- it by a;"^, be INTEGRATION OF IRRATIONAL DIFFERENTIALS. 239 (h-\-ax-''yx'^^'*'dXy {!') which is another differentia^ of the same form in which n is changed into — n and m into m + np. Hence, by Case I., this form can be made rational whenever — ^ is an n integer; that is, when — "^ \-p^'& such. n We have, therefore, two cases of integrability, namely: w + 1 Case I. : when n = an integer. Case II. : when — — — |- jo = an integer. Bemark. It will be seen that all differentials of the form r (a + hx^Yx^dx must belong to one of these classes, because — ^ — IS an integer when m is odd, and — ^ h 9- is such when m is even. In this statement we assume r to be odd, because if it is even the original expression is rational. 142. 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 — «*•+•-! hrt y dy, the integral of which, after replacing y by its value in (2), be- comes /(a + J.T-"-.<^.= (|+^' + c. (5) Again, if the integer in Case II. is — 1, we have m-\-l-\- np = — n, or m -\- np = — n — 1. The expression (1) reduced to the form (!') will then be {b -f ax-'^yx-^'-^dx = — (5 + ax-'')^ — d{b + aa;""). ( i s ; 1/ ■' i m m 240 THE INTEGRAL CALCVLIS. I which is immediately integrablc, and gives by simpV reduc- tions Ha + hx^'Yx - («p + " ^ ^)dx = c - (tn{2) -^ l)x''^^ ' ^^' (C) 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 bo reduced to one of the forms just considered. The objects of these transformations are: I. To replace m. by m -f- n or in — w; II. To replace ;; by ;> -f- 1 or ;> — 1. 144. Firstly, to replace m by m -f- n. Lot us write, for brevity, which will give dX =: bnx^~hlx, and the given differential will bo XPx^'dx, which, again, is equal to X m — n + 1 XPdX = X m — n + 1 bn bn{p-{- 1) Integrating by parts, we have r YP ^1 - a:"'-" + ^J^^+^ _ 7n-n + l J X X ax - ^^^^^ ^ ^^ ^^^^^^ ^^^ rf(X^+»). ^-.JXt'^H'^-^dx. {a) Since X^ + ^ = XP{a + bx^) = aXP + bX^x^ the last integral in the above equation is the same as a fx^x '^-''dx-\-b Cx^x'^dxy of which the second integral is the same as the original one. Making this substitution in («), and then solving the equa- tiol wh| de] do INTEGRATION OF IRli A TTONAL DIFFKIiENTlALS. 241 tion BO as to obtain tlio value of / X^x'^^dz, we fiud / X'x'^dx = y. ; -,-T - 77^ — ■ — ^-.{ / X^x'^-^'dx. {A) J b{ni}-\- m -\-X) h{)ii)\-m-\-\)J ^ Thus the given integral is made to depend npon another in which the exponent of x is clianged from m to m — n. By reversing the equation we malfe the given integral depend on one in which the exponent is increased by n. To do this we change m into m + 1^ all through the equation (A), thus getting j-p + i^m + i a{m-\-l) f^ XPx^^\lx=z X^x'^dx. l(np-\-m-\~n-{-\) b{;np-\-m^n-\-iy Solving with respect to the last integral, we find / X^x'^dx = —. —r.- ^-^ — ~ —-- ^ / .1 Px'^^^'dx. IB) V a{m + 1) rt(?^i + 1) *^ The repeated application of {A) and (/>') enables us to make the value of the given integral depend upon other in- tegrals of the same form, in which w is replaced by m-\-n}, m-\-%}i\ etc.; or by m ~ n\ m — 2h; etc. 145. Next, to obtain forms in which p is increased or diminished by unity, we express the given differential in the form X^x^^dx = X^d X m f-1 \m. -f- h Integrating by parts and substituting for dX its value bnx^~^dx, we have f X^x^dx = m -f- 1 m 4- ■^rx^-'x^''+\lx. (b) Now, we have X m + n hffi^ti X'^X _ x'^(X — a) _ Xx"^ ytfti ax^ ~b* I ■i i ;, '! !il i .1 n * 242 THE INTEORAL CALCULUS. and therefore, by multiplying by JL^~^dXf Substituting this value in (d), and solving as before with respect to / X^x^dx^ we shall find ^^P -fx^-^x-dx, (C) fX'x^dx = --r~-T^ + np -\- m -\- 1 np -\-m -\- in which p is diminished by unity. If we write jo + 1 for j» in this equation, the last integral will become the given one. Doing this, and then solving with respect to the last integral, wo find X^x'^dz = 7 — r-Tv + , . il / X^^x'^dxAD) By the repeated application of the formula (0) or (D) we change p into jo — 1, p — 2, p — d, etc., or p into p -{- 1, p -{- 2, p -\- 3, etc. 146. To see the effect of these transformations, let us put, in the ciiteria of Cases I. and II., § 141: m-\-l I. n = i, an integer. II. — — — [-p = i', an integer. n Then when we apply formula (A) or (B), since we replace mhj m — n ov m -\- rif^e have, for the new integers: m T w + 1 I. n = i T 1. II. "l^i^-i+^ = z'Tl. n It is also clear that by (C) and (/)) we change II. by unity. Thus, every time we apply formulae (A), (B), (C) or {D) 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 Reduction. If, in an integral of Case II., i' is positive, we eannot change it from zero to — 1 by the formula (-4) or ((7), because, when — — \- p = Q, we have m -{• \ •\- 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 We see that if we diminish the exponent | by unity, we shall reduce the integral to a known elementary form of § 135J. So we apply (C), putting m = 0; w = 2; p -=• \', a = a*; b = ± 1, Then (C) becomes We therefore have, from § 133, f(a' + x'Ydx = 1 I x(a' + x^)^ + a' log ^(a;+(a'+ a:')*) | • /*(«' - x')^dx = i I x{a' - cc')* + a' sin<-« j + ^ | • Deduce the following equations: 2. / (c' — x^) xdx =h — i(c' — ic')'. 36''a;' r dx , (c' -h xy if ill if il iiii 244 I. ^; m ' ysp ill 6. /- a, imaginary quantities will enter into the integrals, although the latter are real. If, in the first form, the denominator is m^ sin'' x — n^ cos' x, we shall have, instead of (ll^i), the integral r dt_ ^ 1 /' iU _ 1 p_jit _ J mH^ — n'^ "ZnJ mt — n "ZnJ mt -\- n^^ ' 1 , w^ \ n , y i^ ■ - n ''Zmn Hence, corresponding to (Ib^ tvc L>'vo thu result (Ix /» (ix J vi^ bin* x — if cos' x 1 m tan .r + u . ^ — log — . - - — ' f h. (IG) 2niu m tan x — n ^ ' If, now, in § 15:^, h > a, we write (14) in the form _ o r I'i^^ ^J (b — (() sin' hj — [a -\- b) cos' ^y* and instead of (15) we have the result dy / , , 1 , Vb—ai'd.\\hi-\-Vb-\-ii ,^^. a+b cos y Vb'-a' ^b-atan {y - Vb -^ a 154. Intvyrdtion of sin mx cos nxclx. 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 -}- ^ sin {m — n)x. Hence /• . , cos (ni 4- n)x cos ()n — n)x , ^ I sni mx cos nxdx = 77-^ — ; — { z— r— + k J 2{ni + fi) 2{m — 71) We find in the same way /sin (m 4- 7i)x , sin (m — 7i)x , , cos mx cos 7ixax = ~-y — ; — r- H — ~ ^^ — h «; 2{)n -\- 71) 2{m — 71) /sin im -\- n)x , sin (m. — 7i)x , , sm nix sm 7ixdx = ^> — ,- — r- -\ — ttt r — h «. 2(7)1 -\- 7i) 2(m — n) INTEGRATION OF TRANSCENDENT FUNCTIONS. 253 ! opposite titles will il. If, in cos" X, we 13G) f h. (10) m ±.1 (17) substitiit- •Gssions in le angles. n)x. ~ + h %)x + A; 'f+k. 155. Integration hij 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. EXAMPLES AND EXERCISES. I. We may find / sin xdx as follows: We know that sm X — x q"! " 5 f 7 f ' • * * ^ . • . /sin xdx = h -\. — __-|-_ — etc., which we recognize as the development of — cos x with . -i arbitrary constant h ■\-l added to it. Of course we may find / cos xdx in the same way. dx 2. To integrate 1 l + o;* = (1 + a:)-^ = 1 - a; + a:« -a;='+ . . . ; l-\-x /dx T , x^ , x^ X , (a) IS -^x " ' ~ 2 ' 3 4 /dx ' = log (1 + a;). Hence {a) i the development of log (1 + ^), 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: (^ + 1) -^ = a;-i X -8 ■\-x~^-x-^^ • . . . Hence we derive, when a; > 1, log (a: -f 1) = log a; +--—, + 5773 - j::;^ + 'i X 2x^ ' 3a;' 4a;* J>54 TUE INTEQliAL CALCULUS. ilt 1 The arbitrary constant is zero because, when x is infinite, Jog (.c + 1) ~ log -^ is infinitesimal. 3. To find / ' — - = sin <~ *> x in a series. J Vl-x^ (i-x')-» =1+^.:' + x^ + j;f:^^"+ . . . . Hence / dx vr X ,3 2-4 • (-1) _ ,1 ^' I 1'3 a:"* 1-3-5 x' 2 3 ' 2-4 5 ' 2-4-6 7 T rp The arbitrary constant is zero by the condition sin^~*^0 = 0. This series could be used for computing n by putting x = 71 \y because \ = sin 30° = sin -. But its convergence would be much slower than that of some other series which give the value of TT. 4. From the equation y - rive the expansion dx Vl-{-x' log (re + 1^1 + x') de- iog(a;+4/i4-^')=^-^.3-+|;||'-|;|;|.y+...> dx 5. By expanding , = ^s> • • • ^'n« Then we shall have ^u. rr U, — «^; /lu^ z= u. — w.; ^u^ r= 1^ — «,; • • • • • • ~y" • • • r -^W„_l = Un — Un-x. Taking the sum of all these equations, we have A^l^ + An, + Av^ + . . . + J?*n_i = ?«„ - t^„; That is, the difference hetivcen the two extreme valnes of a variable is equal to the sum of all the successive increments ly which it passes from one 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 Au^ is evidently Au^ + Au^ + . . . + Au^, ^ tto «i «a "3 «4 «5 Since the proposition is true how small soever the incre- ments, it remains true when they are infinitesimal. ! 256 THE INTEGRAL CALCULU3. if. i: 1 '. : I 1^ U If ! Fig. 47 157. Differential of an Area, Lot P^PP' bo any curvo whatever, aud lot us iuvostigate the differential of the area swept over by the ordinate XP. Let us suppose the foot of the ordinate to start from the position X^y and move to the position X, During this motion XP swct'ps over the area X^PJ^Xy the magnitude of which will depend upon the distance OX, and will therefore be a function of Xy which represents this distance. Let us put u = the area swept over; y E 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 y' the new ordinate X'P', It is evident that we may always take the increment XX' ~ Ax so small that the area XPP'X' shall be greater than yAx and less than y'Ax or vice versa. That is, if y' > y, as in the figure, we shall have yAx < An < 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, the area u is such a function ofx that its differen- tial is ydx, and its derivative with respect to x is y. From this it follows by integration that u =fydx 4- 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 il any curve \ the area X X s distance. the corre- Lct us call nay always I XPP'X' vice versa. approach iiality y^x tnitj (1) s differen- (2) from any i» 158. The Conception of a Definite Integral, Suppose the area X^ 1\ P X ee u to bo 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 oi triangles at the top of *'»o- 48. the several rectangles. That is, using the notation of § 15C, we have u = y,Ax, -f- y,Ax^ + yjx^ + . . . -f ?/„_,^:r„_i + T, T being the sum of the areas of the triangles; or, using the notation of sums, u : :S y,Ax, + T. Now, let each of the increments Axf become infinitesimal. Then each of the small triangles which make up T will be- come an infinitesimal of the second order, and their sum T will become an infinitesimal of the first order. We may therefore write, for the area w, x-OX x=OX u = lim. 2 yAx = 2 ydx. x=OXo x—OXa That is, u is the limit of the sum of all the infinitesimal products ydx, as the foot )f the ordinate XP moves from X^ to JTby 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 OXe x^, are called the limits of integration. The infinitesimal increments ydx, whose sum makes up the definite integTal, are called its elements. 17 li t ■ <; 258 THE INTEGRAL CALCULUS. 'II I * liij ^lir ] I i^ |;:|ii^ 11^ |- 159. Fundamental Theorem. The definite integral of a continuous function is equal to the difference bctioeen the vahies of the indefinite integral corresponding to the limits of integration. To show this let js write t})(x) for y, and let us put, for the indefinite integral, a; for example, k = 2a. Then the theorem would give 2 /» •« _ _ 1 _ 1 - Jq y ~ a a ~ a a negative finite quu.<>tity; whereas, in reality, the area is an infinite '"lUf.ntitv. The theorerr- fp.'Js bec-iuse, when x — a, y becomes infinite, 80 that ydx is not ihan necessarily an infinitesimal, as is pre- supposed in the demonstration. II becomes i the two arily hold and finite >le of the uch limits a, we put Thus (5) it, so that 11 increase well as by Then the area is an es infinite, , as is pre- ^^.% DEFINITE INTEQBALa. } *< 263 163. Change of Variable m Dejiuite Integrals. When, in order to integrate an expression, wo introduce a new vari- able, we must assign to the limits of integration the values of the new variable which correspoad to the limiting values of the old one. Some examples will make this clear. Ex. 1. Let the definite integral be ^ dx I fQ a-\-x Proceeding in the usual way, we find the indefinite integral to be log (a -f- a;), whence we conclude /";Ffi = '"S2«-log« = log2. 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 lim't, x = a, we have y = 2a. Hence the transformed integral is ^dt/ which we find to have the same value, log 2. Ex. 2. «« = / 2 sin x{l — cos x)dx. We may write the indefinite integral in the form / sin xdx + / cos xd{co^ x). In the first term x is still the independent variable. But, as the second i? written, cos x is the independent variable. Now, for and for x = 0, cos X = l'y cos X = 0. _ ^ A/ Henc^, writirg ^0 rr "^' '^os X, the value of u is in xdx -f- / ydy l-^^-h. T^. ill ii 264 fik I'i i I I' :'! J f 4^! 2!ffl2? INTEGRAL CALCULUS. Remabk. The variable ivith respect to xoliich 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 y, we shall always have f (p(x)dx = f (f>{y)dy = f y»C pd pfl I (/){x)dx + / {x)dx + . . . -f / 0 /»a /-«« / (f)(x)dx = I (p(x)dx + / (p(x)dx = 2 / (J){x)dx, J -a J -a t/0 t/0 Theorem II. If^(x) is an odd function of x, then, what- ever be a, ^{x)dx =■ 0. a 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. ill /»8 /,!/ l\n I, Show that / e-'x^dx = / (log -J dz. substitute x = log -. ?. Show that whatever be the function 0, we have pin pin I 0(sin z)dz = I 0(cos xdx). As an example of this theorem. / i^rt + 5 cos" X a — b cos'* X dx i'^rt -f- b sin" X b sin" X t/o a — dx. The truth of this theorem may be seen by sliowing that to each cle- ment of the one integral corresponds an equal element of tlie other. Draw two quadrants; draw a sine in one and an e(iual cosine in the otiier. Any function (p of the sine is equal to the corresponding function of the cosine. We may lill 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. (i llii 1 1 1! i I ^ ■•; ; ^; I :■;:; : M; '■■ Ig.i:, Ij !$ 266 T/r£? INTEGRAL CALCULUS. To express this proof analytically, we replace « by a new variable y = i7t — X, Avhieh gives sin x — cos y; dx — — dy\ and tben we invert the limits of the trausfiirnicd integral, and change y into a in accordance Willi the remark of the last article. IT 3. Show that r /(sin ?)dx = 2 r'^f(Em x)dx, 4. Show that / 0(sin a:) 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 through Integration ly 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, kt us put (uv)^ and (tiv)^, the values of tiv for the upper and lower limits of integration, respectively; udv and / vdu, the values of the two indefinite in- tegrals for the upper limit, a;,; ', the values of the integrals for the lower Jf udv and / i limit f -■■0' We then have, by the rule of § 161, udv Jf udv = / udv — / ^ = (w^)i - f '^^w - (uv), +J* fJxa I w variable y 3n we invert a accordance )ns, odd or of an even = 0. )y Parts. — ral. To ex- r and lower idefinite in- 'or the lower vdu DEFINITE INTEGUALS. 267 In order to assimilate the form of this expression to that of a definite integral, it is common to write EXAMPLES AND EXERCISES. 1. We have found the indefinite integral / log xdx = xlog X — j dx. If we take this integral between the limits x = and x = l, the term x log x will vanish at both limits, so that (x log x)^ - {x log x\ = 0. Hence / log xdx =— / dx = — 1 -\- = — 1, t/o t'o 2. To find the definite integral, / sin*" xdx. In the equation (11), § 150, the first term of the second member vanishes at both the limits x = and x= n. Hence / sin"* xdx = m — 1 n"" . m t/o ^xjx"^-^ xdx. Writing m — 2 for wi, and repeating the process, we have Jr sin*"~^irc?2; = -^ / sin"*~*a;^a;; m — 2e/o f sin*""-* «/o "^xdx m m — 5 m 7 / sin'^'^xdx: — 4t/o etc. etc. If m is even, we shall at length reach the form / dx = 7t — = 7t. Then, hy successive substitution, wc shall havQ vu ' h:, ■ i ♦ n s^ ;i ..*.-. f ^pp ii-i: r^ I i ; li'i if" ■( ■ : ■ ' .... I IM '11 ;. ^68 THE INTEOltAL CALCULW. r ^m-xdx = 0^^ - 1) (^ - 3)( m - 5) ... 1 ^ t/o 'm(7/i — ;ii)(«i - 4) . . . Jj If m is odd, tho last integral will bo / sin xdx = + 2, and wo shall liavo Jo m{7ii — 2) (m — 4) ... 3 3. From the oquiition (6) of § 140 wo have, by forming the definite integral and dividing by m + w, / sin"* X cos" xdx =( sin"'^^ a:cos"~^a;\'' + n — I m + 1 /»" sin** a; cos**"* xdx. Since sin ;r = sin = 0, the first term of the second member vanislics between tho limits, and wo have Jr siu"* x cos** xd ■ = — ; — / sin"* x cos""" xdx. m + 71 Jo Writing 71 — 2, and then 71 — 4, etc., in place of 71, this formula becomes /*^ 71/ —~~ 3 z*"^ / sin"* X cos""* xdx — — ; / sin"* x cos"~* xdx\ t/o 711 ■\-7l — 2,/o ' / sin"* X cos""* xdx = — ; / sin"* x cos""* xdx: t/o m + ?j — 4t/o ' etc. etc. If 71 is odd, the successive applications of this substitution will at length lead us to the form / sin"* X cos xdx 771 -|- 1 (sin"* + ' ;r-sin'" + ^0) = 0; and thus, by successive substitution, tegrals to bo zero, we shall find all the in- A ' . We also have . 1 2)1 — 3 2{?i - 1) ~ 2(u - 1)' ) i ! IMAGE EVALUATION TEST TARGET (MT-3) A €// ;/^« 1.0 :f« I.I 1.25 2.8 12.5 22 1.8 1.4 IIIIII.6 Photographic Sciences Corporation S •^ s. 4- ^ :\^ V \ ,"^- <5 % ^^ 6^ 33 WEST MAIN STREET WEBSTER, NY 14580 (716) 873-4S03 4^ «p 4 270 THE INTEGRAL CALCULUS. Hence we have the formula of reduction dx 2n — 3 /»* dx /■'^ ax _ zn — 6 ff" Jo {x' + a'Y " 2(ir^)7i'Jo {oF + xY-'' ^^^ We can thus diminish the exponent by successive steps until it reaches 2. The formula (b) will then give dx 1 /*'*' dx n Jq {x' + ay ~ 2^e/o ^ + ^^ Then, by successive substitution in the form (/>), we shall have n"^ dx _ {2n - S){2n - 5 ) ... 1 _;r Jo {x'-{-aY~{2>i-2){2n-^)T772'2a' ^v {c) If in (c) we suppose a = 1, and write the second member in reverse order, we have /»« dx 1-3-5 . . . {2n — 3) tt Jo (l + a:'')»~2-4-6. . . (2w-2)"2* »* x'^dx 5. To find / — = Jo Vl-x' y m* Vl-x- Let us apply to the indefinite integral the formula (A), 144. We have in this case a = 1; J = — ij n = 2; j» = — ^. The f^TTiula then becomes /- x^dx X' .-i|/l_ m — 1 Px^'hlx VI -x' In the same way m X' , m — 1 /* u. ^v.. , . 7n Vl^ X A m — 8 dx X m -Wl-x Vl-X" in- — a;' m. - 3 /*jk 2 " ~^ m — 2*/ ^ m — 4 dx V\-x' Continuing the process, we shall reduce the exponent of x to 1 if m is odd, or to if m is even. Then we shall have /xdx /. ,v. f* dx — = = — (1 - xy or / -—= ~ = sin<~*\T. % Vl -X- Taking the several integrals between the limits and 1, we DEFINITE INTEGRALS. 271 sive steps , we shall in- v (0) d member mula (A), ~hlx l-X' m-4 • («) dx onent of x all have i<-»>.r. {h) ) and 1, we note that in (a) the first term of the second member vanishes at both limits, while {b) gixres J/»^ xdx _ 1 . p^ i/r^r^~ ' J dx Vi Vin Vl — x- 1/ r 1 — a; We thus have, by successive substitution, _ r^x^^'^dx _ 27i(2u - 2) (2?i - 4). 1 2 |/lZr^' (2w+l)(2>i-l)(2w-3) _ p^dx_ _ {2n-l){27i-3){2n-5) ^^"^ -Jo |/fZ:"^« ~ 2n{27i - 2) {271 - 4). .' 2 . .3' . . 1 7t y (c) t t /i lil Let us now consider the limit toward which the ratio of two values of y^ approaches as m increases to infinity. We find, from («), Vm _ m — 1 ym-2 fn ' a ratio of which unity is the limit. Next we find, by taking the quotient of the equations (r), Tt_ I2-4- 6 . . . (27i — 2)'27i}' y^ 2 - I3-5-7T. . (2/i - l)r(2^Tl)'2^* Since, when n becomes infinite, the ratio y^n * .Van • i ap- proaches unity as its limit, we conclude that \7t may be ex- pressed in the form of an infinite product, thus: 2 ~3*3'5*5'7'7-9*9'ii* ' * t7ijimt2im. This is a celebrated expression for tt, known as Wallis's formula. It cannot practically be used for computing 7t, owing to the great number of factors which would have to be included. il !. A ';\ 272 TUia INTEGRAL CALCULW. CHAPTER VII. SUCCESSIVE INTEGRATION. 166. Differentiation under the Sign of Integration. Let us have an indefinite intesral of the form u = I (p((x, x)dx = F(a, x). (1) a being any quantity whatever independent of x. It is evi- dent that u will in general be a function of a. We have now to find the differential of u with respect to or. The differentiation of (1) gives ^ = 0(«, ^); dhi _ d(f){a, x) dadx T, d u rv du Because -^ — 7- = Da-^r- dadx dx da Dxi-i we have, when we consider da du -T— as a function of x (cf. § 51), \da I dxda aa Then, by integrating with respect to x, £ =/^fe'.., (.) in which the second member is the same as (1), except that 0(0-, x) is replaced by its derivative with respect to a. Hence we have the theorem: The derivative of an integral with respect to any quantity which enters into it is expressed by differ entiati7tg with re- spect to that quantity under the sign of integration. SUCCESSIVE! INTEGRATION. 273 n. Let (1) t is evi- ^e have consider (2) apt that Hence \uantity with re- 16 1. 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 a;, and a;,, and put u^ and w, for the corresponding values of u, we have, for the definite integral, r^{a,x) ^^ ^ dF{a,x) ^ J da da * whence, if x^ and x^ are not functions of a, (3) d^{a,x) ^^ ^ dF{a, x,) _ dF{a, x,) da da da (*) Hence from (3) we have the general theorem Da I (p(a, x)dx = / Da(p{a, x)dx. That is, the symbols of differentiation and integration with respect to two independent quantities may he interchanged in a definite inicgralj 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 u„' with respect to a (§ 41), du' fdu^^ da \ da ) By § 160 we have du^^ dx^ du^^ dx^ dx^ da dx^ da' dii^ dx^ - (oty X,), I > 18 274 THE INTEGRAL CALCULUS. Thus from (3) and (4) we have du '0 da ' /»*» d. Hence 'jy. , dx ^ a r^" e--'^ dv =1 r «/-oo '^ aj- e dx = . <» a By differentiating with respect to a, and simple reductions, we find and from this. ""yV--»V, = ^; 00 2a' -j- 00 y ♦.-"'J/'././ — dy^T 3 Vn 00 etc. 4 a" etc. -' « Hi J il^ \<\m' II 276 TffE INTEGRAL CALCULUS. 1 . - sin ax, a EXERCISES. 1. By differentiating the integrals / cos axdx /sin axdx = cos ax, a twice with respect to a, prove the formulas I x* cos axdx =1 -A sin ax -\ — 5 cos ax*, / a; sm aa:aa; = -, 1 cos ax A — -. sm ax, J W aJ 'a' Thence show that we have Jy' cos ydy = (y' - 2) sin y-{-2y cos y; Jy^ sin ydy = (2 - y') cos ?/ + 2«/ sin y. 2. Prove the formulae: /o 1 /»o 1 e^'^dx =-; (^) / a;e"*Ja; = ^; x'e'^dx = -,; (^) / x^'e^dx = (- 1)"-^. 00 a t/— 00 fit 3. Show that the preceding formulae are true only when a is positive, and find the follov/ing corresponding forms when a has the negative sign: / e~°^dx = — ; / xe'^^'^dx = — „; Jo (I «/o a r x'e-'"'dx= -A C x'e-'^dx = ~ ; etc. Jo a' Jo (I* ' 4. By differentiating the form of § 132, namely, dx J \a' - xy with respect to a, show that /a. dx = sin(-i> X X a {a' - xy a'{n' - xY SUCCESSIVE INTEOiiAriON. 277 170. Dotible Integrals. The procoding results may ho Bummed up and proved thus: Let us have an integral of the form u = J(}>(x, y)dx, (1) and let us consider the integral J^iidy or f^ fmmon- with a onstant )re any may be entirely The student iihould 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 (f){x, y, z)dxdydz fff^ means the result obtained by integrating 0(;r, ;/, z) with re- spect to Xf 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'P{x, y, z) .. . dx dydz = ^('^' y^ ^)> and the problem is to find F(x, y, z) from this equation when (p{Xf 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 y. For, let us represent any three such functions by the symbols [y, z], [z, a;], [x, y], and let us find the third derivative of F{x, y, z) 4- [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 dx du _ dF{x, y, z) d[z, x] d[x,y] ^ dx dx dx ^"^ ■ d^u _ d^'Fi^x, y, z) d'[x,y']^ dxdy dxdy dxdy ' d\t d*F{xyy yz)^ dxdydz * dxdydz an equation from which the three arbitrary functions have entirely disappeared. ! in JH: 280 THE INTRORAL CALCULUS. i It is to be romarkcd that ono or both of tho variables may disappear from any of these arbitrary functions without chang- ing their character. The arbitrary function of y and z, being any quantity wliatever that does not contain x, may or may not contain y or z, and so with tho others. As an example, let it be required to find u •= I I I {x — a)(y ^ b){z — c)(lxdydz. Integrating with respect to z^ and omitting the arbitrary function, we have / / i(^ — «)^y — h)(z — cydxdy. Then integrating with respect to y, tl =f'^^'' - ") ^y - *)' <' - "■>'' which gives, by integrating with respect to x, and adding the arbitrary functions, u = i{x- ay(y - l)\z - c)' + [y, z\ + \z, x\ + [a;, 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 tho determination of a function of a single variable when the derivative of the func- tion of any order is given. EXERCISES. I. r j-^dxdy, 2. r r(x — a)(y—lYdxdy, 3. / / jxy^z^dxdydz. 4. j j j^dxdydz. 5. fff{^ - (^Y(y - *)(^ - cydxdydz. 6. ff{x " aydx\ 7. fff{^ + ^0'^^'- Ans. (6). ^g(.r — «)* -\- Cx -\- C, Cand C being arbitrary constants. 8UCGESSI VE INTEUHA TION. 281 lo8 may chang- z, being or may rbitrary ling oho mi\\ re- egrals. ion of a le f unc- lydxdy. iz. dz\ arbitrary 173. Defuiile Double Integrals, Lot U bo any function ppoBing y ot X and »y Integration with respect to x constant, we may form a deliuite integral sill _ Udx = U\ From what has been shown in § 103, Rem., U' will be a function of y, x^ and .r,. VV^o may thiuefore form a seeonii definite integral by integrating IJ'dy between two limits //„ and y^. Thus wo lind an expression f VUly = / / Udxdy, which is a definite double integral. The limii/S 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. 174. Definite Triple and Miiltiplc Integrals. A definite integral of any order may be formed on the plan just described. For example, in the definite triple integral ' / / ^\^y Vf z)dxdydz the limits x^ and x^ of the first integration may be functions of y and z; while y^ and y, may be 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. I i 283 THE INTEGRAL CALCULUS. EXAMPLES AND EXERCISES. t I'.r 1. Find the values of / / xy^dxdy and / / X'fdxdy. It will bo seen that in the first form the limits of x are con,5tants, and in the second, functions of y. First integrating with respect to x^ we have for the indefi- nite integral fxy^dx = ^xY, and for the two definite integrals Ixy'dx^ ^ay, r xy^dx = ^y\ y Then, integrating these two functions with respect to y, we have Jo Jy ^y'^^^^y = Vo ^ ^^ ^ ^* • Let us now see the effect of reversing the order of the in- tegrations. First integrating with respect tc y, we have / xy^dy = ^xb' = U. Then integrating with respect to x, we have / Udx = / / xy dydx = ^a''b*, the same result as when we integraicd in the reverse order between the same constant limits. 2. Deduce / ^ / cos (re -f l/W^^^y = — Ji 8 UCCESSI VE IN TEG It A TION. 283 f X are I indefi- jt to y, the in- lave se order 3. Deduce / / cos {x — y)dxdy = -|- 4. 4. Deduce / / {x — (i)(y — b)dxdy = ia^b^. 5. Deduce /*"/*\;i; - a){y - b)d>dy = \(2ab-d')(2ab-b'), 6. Deduce / " / ^ {x—a){y — b)dxdy = iCb — \aW— \a\ lis. Product of Two Definite Integrals. Theorem,, The product of the two definite integrals f'^'Xdx and f^Ydy is equal to the double integral Jf*yi P'^^xYdxdy, provided that neither integral contains Wn *JXa the variable of the other. For, by hypotliesis, the integral / Sdx= f/" does not con- C/*X/Q tain y. Therefore U f Ydy = f UYdy = / / XYdxdy, as was to be proved. 176. The Definite Integral f e~ ^"^ dx. This integral, «/ — CO which v/e have already mentioned, is a fundamental one in tlie method of least squares, and may be obtained by the ap- plication of the preceding theorem. Let us put /,=. r^^e-^\lx = ^ r^^e-'\lx = ^ r'^^e-'\ly.{%U^) J- 00 t/o t/o . + 00 ^4-co _ Then, by the theorem, ,/o t/o t/o t/o Let us now substitute for y a new variable t, determined by the condition y=ztx. '1 If 284 THE IJSTEQRAL 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 t/0 t/0 Since the limits are constants, the order of integration is indifferent. Let us then first integrate with respect to x. Since xdx = i^d'x" = 2n^i^\ ^' (^ + ^'')^^ the integral with respect to x is 2(1 + nJo - (1 + <«)«« d-{l-\-t')x' = 2(1 + n- Then, integrating with respect to t, '" dt Hjnce k' = -{-f = TT, /e "^dx = Vic, BECTIFICATION OF CURVES. 285 CHAPTER VIII. 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 Fia. 50. As = PP'. If PP' becomes infinitesimal, it has already been shown (§ 79) that we have, in rectangular co-ordinates, ds = 4WTW = /l + i^£jdx = /l + [^)dy, (1) and, in polar co-ordinates, If both co-ordinates, x and y, are expressed in terms of a third variable w, we have ds = |/( + (ciyy dn ) \du ) The- length of any part of the curve is then expressed by 1 f '/ iff) ;i lii'i- 286 THE INTEGRAL CALCULUS. the integral of any of these expressions taken between the proper limits. Thus we have or (3) In order to effect tlie integration it is necessary that the second members of (3) shall be so reduced as to contain no other variable than that whose differencial is written; that is, we must have ds = f{x)(lx; f(y)dy; f(6)(Wy or f{u)du. 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 \> (p' + .vT + .v (*) 179. Rectification of che Ellipse. The formulse 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.) x = acoB u; y = b sin u. We then have dx — — a sin udw, dy — h cos udu. Then if e is the eccentricity, so that a'e' = a' — J', ds — (a' sin' u -J- &' cos' \ifdu = «(! — e' cos' ufdu\ s = rt/(l— e' cos' ufdu. 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 1*1 (1 — e' cos' u)^ = 1 — ;r6' cos' u — ;z—.e* cos* u 2 2-4 1-1-3 2-4-G e cos u etc. 288 THE INTEQUAL CALCULUS. Ill i H II 1 HI 9.^ » > ,-■'■' : The terms in the second member may he separately in- tegi'ated by the formulae (0), § 140, by putting w = and 7i = 2, 4, C, etc. We thus find 2 / cos' ndu = sin n cos u •\- w, 4 / cos* luht = sin ?«(cos' ti -\- 1 cos n) -f fw; etc. etc. etc. Since at one end of the major axis we have u = and at the other end u = rr, we find the length of one half of the ellipse by integrating between the limits and 7t. Since sin u vanishes at both limits, we have / cos' udtt = ^n; t/o <* '■^ 1-3 cos* ud2C = :r— .?r; 2-4 '^ , 1-3-6 cos" U = K--i-^7t. 2 • 4 • 6 We thus find by substitution that the semi-circumference of the ellipse may be developed in powers of the eccentricity with the result _ /'i 1 ^ 3 , J';5_ , _ \ 180. TIte Cycloid. The co-ordinates x and p 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 ?^); y = a{l — cos It). Hence dfs' = dx"" + dy^ = rt'f (1 - cos n)' + sin' u\du'' = 2«'(1 — cos 'ii)du^ = ^a^ sin' ^u.du*. By extracting the root and integrating, 5 = ^ — 4rt cos ^u. If meei have and Tl one that HECTlFlCATtON OP CURVm. 280 If we measure tno arc generated from the poiut where it meets the axis of abscissa^ that is, where n = 0, we must have s = f or 7t = 0. This gives h = 4a and s = 4a(l — cos ^«) = 8^ sin' \u. This gives, for the entire length of the arc generated by one revolution of the generating circle, s = Set; 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 + ff'fde. Then the indefinite integral is (§ 147, Ex. 1) 5 = 1 1 ^(1 + ^')' 4- log q^ +(1 + ^^))* [ . If we measure from the origin we must determine the value of by the condition that when ^ = 0, then s = 0. This gives log C =0; .' . C =1. If instead of 6 we express the length in terms of r, the radius vector of the terminal point of the arc, we shall have , = ^ -{a- + r-y + ^ log 5^ -' . 183. The Logarithmic Spiral. The equation of this spiral (§ 83) gives Hence -- = ale^^ = Ir. au dfs = (1 + rfrdS. To integrate this differential with respect to 6 we should first substitute for r its value in terms of 6. But it will be 19 290 THE INTEGRAL CALCULUS. better to adopt the inverse course, and express dd in terms of dr. We thus have ds = ; ar; I whence _ (1 + ly I r + 5, 0* 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 hg.ve Between any two points of the curve whose radii-vectors are r„ and i\ we have s = (?*, — i\) sec y. Hence the length of an arc of the logarithmic spiral is pro- portional to the difference betioeen the radii-vectors of the ex- tremities of the arc. EXERCISE. 1. Show that the differential of the arc of the lemniscato is ds = add Vl-'H sin' e (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 terms of le radius i we hpve [ii-vectors al is pro- of the ex- iiiscato is )tic func- )ometrical are equal adrature. L algebraic equation reover, in id by the light line Fig. 51. must intersect it an -^ven number of times. The simplest case is that in wliich a line^ paral- lel to the uxis of Y cuts the bound- ary in two points. Tlien for every value of x the equation of the curve will give two values of y corresponding to ordinates termi- nating at P and Q. Let these values be //, and y^. Then, the infinitesinitil area in- cluded between two ordinates infinitely near each other will be O/i - yo¥^ = ^<^- The area given by integrating this expression will be in which the limits of integration are the extreme values of x corresponding to the points X^ and JT,, 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 If the curve is referred to polar co-ordinates, let S and T be two neighboring points of the curve, and let us put r^OS) r'^OT', ^0 = angle SOT. If wo draw a chord from S to T, 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. i>! Hi" ^ J 29r) Tllh: TNTKORAL CAICVLUB. and the radii vectors will be ^rr' sin A 6. Now let A 6 be- come infinitesimal. OS '• "' then approach ?■ as its limit; the ratio of sin AS to A6 it will approach unity, and the area of the triangle will approach that of the sector. Thus wo shall have, for the dillerential of area, If the polo is within the area enclosed by the curve, the total area will bo found by integrating this exprcs-sion be- tween the limits O'' and 300'. Thus we have, for the total area. - 4- h. 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, 0- = «» 8in<-« (+ 1) - rt' sin(-« (- 1) = na\ The Elli])se. From the equation of the ellipse referred to its centre and axes, namely. \ OX, and dling this •wo thirds we find The entire area will be , + a V = ± - Vrt' - x\ + a (Vi - y.¥^ = ^- v^' - ^f^^ = ^«*- ■ a '■x/—a 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 of the major 2U4 TUK l^TEURAL CALCULUS. Ifl axis. Tho oquivtion of the hyperbolu referred to its centre autl axes gives, for the value of ij ill terms of x, h y=z a S/x? a\ Fio. 64. If wo put a:, for the vahio of the abscissa M, tlicn, since OA = a, tlie area AMI* will bo equal to the integral /(.•-a')«^ = l.(.'-„-)»-|'l„gg+g-l)']; and for tho definite integral between the limits a and x, Area.iP^=l|(..-.').-flo«g + g-l)'] = |^y-flogg + g-l)*]. Now, \xy is the area of the triangle OPM; we therefore conclude that the second term of the expression is the area included between OA, OP and the hyperbolic arc A P. 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 ah 2 sin a' a CI out] i» y M X n I X, -)'] therefore the area P. urve, one QUADRATUllE OF PLA^E FIUUliES. 296 a being the angle between thu axes. We readily see that the difforential of the area is ydx x sin a instead of ydx simply. Hence for the area we have ab ab /J />ao , no , y sin adx — I - ax = -- log ex. If we take the area between the limits OM=x^ and OM ''» ab , ab x^, the result will bo J/»-*^» ao J ^(to . 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 Lemniscatc. The equation of this ourve in polar co-ordinates is (§ 81) ?•' = a' cos 3^. It will bo noted that r becomt^ imaginary when 6 is con- tained between 45° and 135°, or between 225" and 315°. The integral expression for the area is ^fr'de = ia'f COS 20de = ia' sin 26. To find the area of the right-hand loop of the curve we must take this integral between the limits 6 = — 45° and = + 45°, for which sin 26/ = - 1 and + 1. Hence Half area = !«'; Total area = «'. 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 dx =z a(l — cos u)du. Hence I l;^ 3 y,H 1 J mwM m 296 TUB INTEGRAL CALCULUS. / ydx =a^ I (1— cos uYdu—a'^ I (| — 2 cos w + i cos 2u)du. The indefinite integral is |?A — 2 sin w + i sin 2i/. To find the whole area wc take the definite integral between the limits and 27r, Thus we find Area of cycloid = 3;ra', 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 = Ixy. Differentiating both members, we have ydx = Ixdy + lydx ; , • = ^ ' ' y x' Then, integrating both members, log y"^ = \ogcx', .' . y"^ = ex, 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 y^ = cx^~^. CUBATURE OF VOLUMES, 297 OS %u)du. I between f the pa- be in order abola whose > of whose I must be CHAPTER IX. THE CUBATURE OF VOLUMES. 189. General Formulce for' Cubature. In the ancient Geometry to C2cbe 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 plane of YZ, and let u 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 u 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. 5G. and dv = udx, V — I udx, t/Xa (1) x^ and x^ being the distances of the catting planes from the origin 0, 298 TUB INTEGRAL CALCULUS. m i'i r 'H i« 1 t I 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 t* 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 I^ or of Z as well as of X. 190. The Sphere. The equation of a sphere referred to its centre as the origin is If we cut the sphere by a plane FMQ parallel to the plane of YZ, and having the abscissa OM = X, the equation of the circle of intersection will be ?/" ^ z' = a' — x'; that is, the radius MP of the circle will be Va"* — x\ and its area will be 7r{a' - x^). Hence the differential of the vol- ume of the sphere will be dv = 7t(a^ — x^)dx, and the indefinite integral will be V = n{a^x — \x^) -f 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', Fig. 57. X CtJBATURE OF VOLUMES. ■ for any total vol- function terms of I parallel n of the by planes '299 ', and its the vol- 191. Volume of Pyramid. Let the pyramid bo placed with its vertex at the ori-' gin, and its base parallel to the plane of XY, Let us also put h = OZ its alti- 'ude; 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 EFtm^AB, 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 =: OIJ : 0Z\ Putting Area ABCB = a, OL = z and OZ = h, Fig. 58. ArGB,EFGII = az The volume of the pyramid is therefore '^nz'dz 1 , 7?- = a"*- That is, one third the altitude into the base. The same formula} apply to the cone. 193. Tr'e Ellipsoid. The equation of the ellipsoid re- ferred to its centre and axes is + - = 1 a, b and c being the principal semi-axes. If we cut the ellipsoid by the plane whose equation is X = a;', the equation of the section will be c a 300 THE INTEGRAL CALCULUS. i )i' (:1 This is the equation of an ellipse whose semi-axes are a Va' - X /a and - Va^ — X n a Hence its area is 7tlc{a' - x") a Fio. 59. Then, by integration between the limits —a and -{-a, we find Volume of ellipsoid = ^mihc. From the known expression for the area of an ellipse (Ttah) it is readily found that the volume of an elliptic cylinder cir- cumscribing any ellipsoid is 27tabc. Hence we conclude: 2^ he volutiie of an ellipsoid is two thirds that of any rigJit elliptic cylinder circumscribed about it. 193. Volume of any Solid of devolution. In erder 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 M, 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 = xhe 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 ?/, whose area will therefore be Try^. Hence the volume contained between two planes at distance dx will be Tty^dx, and the volume between two sections whose abscissas are x„ and X. will be V= I 'tthMx. (1) / TTt/^dx. iti|:t CUBATUBE OF VOLUMES. 301 ,re , we find se (Ttah) ider cir- ide: ^ly rigJit ier that it must |R ifinitely Dlution. ace the md let , and 3 of ra- listanco are x„ (1) If the two co-ordinates are expressed in terms of a third variable w by the equation*- we have dx =■ ff)'{ii)du. Putting u^ and «, for the values of u corresponding to x^ and a;,, the expression (1) for the volume will become. V=7r I [>p{u)Y(p'(7i)du. tJuo (2) The equations (1) and (2) give the volume AA'B'B gen- erated by the revolution of any arc AB of 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 = x„ and OW = x^. To find the entire volume generated we must extend these limits to the points (if any) at which the curve intersects the axis of revolution. 104. The Paraboloid of Revolution. The equation of the parabola being y' = ^px, we readily find from (1) a result leading to the following theorem, which the student should prove for himself: Theorem. The volume of a para- boloid of revolution is one half that of the circumscribed cylinder. 195. The Volume Generated by the Revolution of a Cycloid around Us Base, From the equations of the cycloid in terms of FiQ. 61. 302 TBE INTEGRAL CALCULUS. lii i*j :! i 1' i , * i'l' 1- !'1 r i ink 1 the ajgle through which the generating circle has moved, we find the element of the volume to be dV= 7ra\l — cos ttydu. fience V = Tca' / (1 — 3 coa ?* + 3 cos' u — cos' ii)dti. By the method of §§ 149, 150, with simple reductions, we find / cos' udu — \u -f- \ sin 3?*; / cos' udv> = / (1 — sin' u)d.mi w = sin w — ^^ sin' u = I sin w -j- ^ sin 3w. We thus find, for the indefinite integral, V — na^{^\'i(, — Y sin w + 1 sin %u — -^ sin 3w). The total volume formed by the revolution of one arc of the cycloid is found by taking the integral between the limits w = and %i = 2;r. The volume thus becomes F==5;r'«', from which follows the theorem: The volume generated hy the revolution of a cycloid around its base is five eighths that of the circumscrihed cylinder. 196. The Hyperboloid of Revolution of Two Nappes. This figure is formed by the revolution of an hyperbola about its transverse axis. The general expression for the volume is found to be F=g(a;'-3ri'.r + ;0, 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 tha be wil CUBATURE OF VOLUMES. 303 that V shall vanish when x = a, because then the plane will be a tangent at the vertex o' the hyperboloid, and the volume will become zero. This condition gives h = 3a' - a' = 2a\ Thus wo have V = —(x^ - Sa'x + 2a') = -^,(x - ay(x + 2a). (1) By the same revolution whereby the hyperbola describes an hyperboloid of revolution the asymptotes will 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 = bx, we find for the volume of the cone F' = TT nV P^'-'M' (^^) 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 V" = 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 nappes 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. f 304 THE INTEGRAL CALCULUS. My m :i V i < i y, = ^P- Fig. 62. The points P and Q will describe two circles which will contain between them the sectional area Taking two ordinates at the infinitesimal distance dx, the corresponding infinitesimal element of volume will be dV=7t{y-^-y,yh'. (1) The integral V=7t r\y,' - y,')dx = Tt r\y, + y^) (y, - y)dz will express the volume of that part of the solid contained be- tween the two planes whose respective abscissas 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 AB\)Q9> 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, + 2/, = 2^*; y.-y. = ^ ^^"- (^ - «r; w CUBATURE OF VOLUMES. 305 any com- i an axis y Rrhich will ce dx, the be (1) - y)dz itained be- x^ and ajj. me points [id. the figure distance h V= inh r'[c' -{x- ayfdx. The limits of integration for the whole volume are x^ = a — c and x^ = a -\- c. If we put z = x the total volume will become a, V=^7tl)f_y{c' -Z')\lz. By substituting the known value of the definite integral, we have V='Zn'bc\ The area of the generating circle is Ttc^, and the circumfer- ence of the circle described by its centre is 2;rZ>. The product of these two quantities is %7t%c^. Hence: The volume of a circular ring is equal to the product of the area of its cross-section into the circumference of its central 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 Ti"^ : 8, or 5 : 4, nearly. /?. The area of the semi-ellipse is greater than that of the cycloid in the ratio tt : 3. y. The volume of the ellipsoid of revolution around the axis OX is greater than that generated by the revolution of the cycloid iu the ratio 16 : 15. 20 i! m It; •If! * !'• li. 4:f 306 THE INTFAHtAL CALCULUS. Q. o 4v Fio. 63. ''r' 190. Quadrahirc of Surfaces of Revolution, Let us put ^5 E a small arc PQ oi a curve re- volving round an axis 0X\ y E the distance of /* from the ' axis 0X\ y' = the distance of Q from the axis OX, Considering Js as a straight line, the surface generated by it will bo the curved surface of the frustum of a cone. If we put J(T = the area of this curved surface, we have, by Geometry, Jo- = 7r{y -f y')^s. Now let J.