'- THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES FROM THE LIBRARY OF ERNEST CARROLL MOORE AN ELEMENTARY TREATISE ON THE CALCULUS. AN ELEMENTARY TREATISE ON THE CALCULUS WITH ILLUSTRATIONS FROM GEOMETRY, MECHANICS AND PHYSICS BY GEOEGE A. GIBSON, MA., F.R.S.E. PROFESSOR OP MATHEMATICS IN THE GLASGOW AND WEST OF SCOTLAND TECHNICAL COLLEGE MACMILLAN AND CO., LIMITED NEW YORK : THE MACMILLAN COMPANY 1903 All rights reserved First Edition, 1901. Reprinted, iy03. GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSK AND CO. Engineering I Mathematical Sciences Library 303 S-3S PREFACE. THE rapid growth in recent years of all branches of applied science and the consequent increasing claims on the time of students have given rise in various quarters to the demand for a change in the character of mathematical text-books. To meet" this demand several works have been published, addressed to particular classes of students and designed to supply them with the special kind and quantity of mathe- matics they are supposed to need. With many of the arguments urged in favour of the change I am in hearty sympathy, but it is as true now as it was of old that there is no royal road to mathematics, and that no really useful knowledge can be gained except by strenuous effort. It is sometimes alleged that a thorough knowledge of the derivatives and integrals of the simpler powers, of the exponential and the logarithmic functions, and perhaps of the sine and the cosine, is quite sufficient preparation in the Calculus for the engineer. This contention has a solid substratum of truth ; but a knowledge that goes beyond the mere ability to quote results is not to be obtained by the few lessons that are too often considered sufficient to expound these elementary rules. It may be possible to state and illustrate in a few lessons a sufficient amount of the special results of the Calculus to enable a student to follow with some intelligence the more vi AN ELEMENTARY TREATISE ON THE CALCULUS. elementary treatment of mechanical and physical problems ; but, though such a meagre course in the Calculus may not be without value, it is quite inadequate, both in kind and in quantity, as a preparation for the serious study of such practical subjects as Alternate Current Theory, Thermodynamics, Hydrodynamics, and the theory of Elas- ticity, and to a student so prepared much of the recent literature in Physics and Chemistry would be a sealed book. Besides, it should surely be the aim of every well- devised scheme of education to place the student in a position to undertake independent research in his own particular line of work, and the very complexity of the problems presented to modern science, with the vast accum- ulation of detail so characteristic of it, enhances in no small degree the value of a liberal training in mathematics. Subsequent specialisation makes it the more, not the less, necessary that the mathematical training in the earlier stages should be the same whether the student afterwards devotes himself to pure mathematics or to the more practical branches of science, especially as the processes of thought involved in any serious study of mechanical, physical, or chemical phenomena have much in common with those developed in the study of the Calculus. The early text-books on the Calculus, such as Maclaurin's or Simpson's, were not written for pure mathematicians alone, but drew their illustrations largely from Natural Philosophy ; the later text-books, probably in consequence of the ever-widening range of Physics, gradually dropped physical applications, and even tended to become treatises on Higher Geometry. In the present position of mathe- matical science, however, it is just as much out of place to make an elementary work on the Calculus a text-book of Higher Geometry as it would be to make it a text- book of Physics or of Engineering or of Chemistry. What PREFACE. vii may be reasonably required of an elementary work on the Calculus is that it should prepare the student for immediately applying its principles and processes in any department of his studies in which the Calculus is generally used. With this end in view, the subject should be illustrated from Geometry, Mechanics, and Physics while the peculiar difficulties of these branches are relegated for detailed treatment to special text-books, so that the illustrations may really serve their purpose of throwing light on general principles, and may not introduce rather than remove intellectual obscurity. As regards Chemistry, a sound knowledge of the Calculus is of special importance, since it is the properties of functions of more than one variable that are predominant in chemical investigations ; the lately published book of Van Laar, Lehrbuch der Mathematischen Chemie, is a sign of the times that cannot be mistaken. In this text-book an effort has been made to realise the aims just indicated. With respect to mathematical attainments, the reader is supposed to be familiar with Geometry, as represented by the parts of Euclid's Elements that are usually read, with Algebra up to the Binomial Theorem for positive integral indices, and with Plane Trigonometry as far as the Addition Theorem ; but no use is made of Complex (imaginary) number, nor is a knowledge of Infinite Series presupposed. The excessive refinements of - modern mathematics have been deliberately avoided, as being neither profitable nor even intelligible to the young student; constant appeal has been made to geometrical intuitions, while at the same time considerable attention has been paid to the logical development of the subject. The early chapters may seem to contain a great deal of matter that is foreign to the book : but the theory viii AN ELEMENTARY TREATISE ON THE CALCULUS. of graphs and of units is of such importance, and is as yet so imperfectly treated in elementary teaching, that some account of it appeared to be a necessity. After considerable hesitation I have included in my plan the elements of Coordinate Geometry, so far as these were likely to be of real service in elucidating fundamental principles or important applications ; but for many applications of the Calculus an extensive acquaintance with Coordinate Geometry is not necessary, and I hope that a sufficiently clear account of its principles has been given to meet the practical needs of many students. I have, however, excluded the discus- sion of the theory of Higher Plane Curves and of Surfaces as unsuitable for an elementary treatise. Another innovation is the chapter on the Theory of Equations ; the innovation seems to be justified, not merely as an arithmetical illustration of the Calculus, but also by the practical importance of the subject, and by the absence of elementary works that treat of transcendental equations. The general development is that which I have followed in class-teaching for several years. The somewhat lengthy discussion of the conceptions of a rate and a limit I have found in practice to be the simplest method of enabling a student to grapple with the special difficulties of the Calculus in its applications to mechanical or physical problems ; when these notions have been thoroughly grasped, subsequent progress is more certain and rapid. No rigid line is drawn between differentiation and inte- gration, and several important results requiring integration are obtained before that branch is taken up for detailed treatment. The discussion in Chapter X. of areas and of derived and integral curves is designed, not only to furnish a fairly satisfactory basis for the geometrical definition of the definite integral, but also to illustrate a method of graphical integration that is of some importance to PREFACE. ix engineers, and that may be of some value even in purely theoretical discussions. As in some of the more recent text-books, the discussion of Taylor's Theorem has been postponed ; the Mean Value Theorem is sufficient in the earlier stages, and the some- what abstract theorems on Convergence and Continuity of Series are most profitably treated towards the end of the course. The treatment, however, is such that teachers who prefer the usual order may at once pass from the Mean Value Theorem to Chapters XVII. and XVIII. Functions of more than one variable are treated in less detail than functions of one variable ; but I have tried to select such portions of the theory as are of most importance in physical applications. The book closes with a short chapter on Ordinary Differential Equations, designed to illustrate the types of equations most frequently met with in dynamics, physics, and mechanical and electrical engineering. Simple exercises are attached to many of the sections; in the formal sets will be found several theorems and results for which room could not be made in the text, and which are yet of sufficient importance to be explicitly stated. I have tried to exclude all examples that have nothing but their difficulty to recommend them ; and with the object of encouraging the student to put himself through the drill that is absolutely necessary for the acquisition of facility and confidence in applying the Calculus, I have freely given hints towards the solution of the more important examples. In the preparation of the book, I have consulted many treatises, and where I am conscious of having adopted a method of exposition that is peculiar to any writer, I have been careful to make due acknowledgment. It is difficult, however, when one has been teaching a subject for years to X AN ELEMENTARY TREATISE ON THE CALCULUS. recognise the sources of his knowledge, and it may well be that I have borrowed more largely than I am aware. I am greatly indebted to my friends Professor Andrew Gray, F.R.S.; Mr. John S. Mackay,LL.D.; Mr. Peter Bennett ; Mr. John Dougall, M.A.; and Mr. Peter Pinkerton, M.A., for help in the tedious task of the revision of proof-sheets and for useful criticism. In all matters bearing on Physics, Professor Gray's advice has been of the greatest service. To Mr. Dougall my obligations are specially great ; he has taken a lively interest in the work from its inception, and has read the whole of it in manuscript, placing at my dis- posal, in the most generous way, his great knowledge of the subject and the fruits of his experience as a teacher ; to him, too, I owe the verification of the examples. I desire to thank Professor R. A. Gregory for his constant and kindly advice on matters relating to the passage of the book through the press. I am also grateful to the printers for the excellence of their share of the work. GEOEGE A. GIBSON". GLASGOW, September, 1901. CONTENTS. CHAPTER I. COORDINATES. FUNCTIONS. ART. PAGE 1. Directed Segments or Steps, 1 2. Addition of Steps, 2 3. Symmetric Steps and Subtraction of Steps, 3 4. Abscissa of a Point. Fundamental Axiom, 4 5. Measure of a Step, ........ 5 6. Axes of Coordinates. Squared Paper, .... 6 7. Distance between two Points, 9 8. Polar Coordinates, 10 9. Variable. Continuity, . . . . 12 10. Geometrical Representation of Magnitudes, 13 11. Function. Dependent and Independent Variables, . 13 12. Notation for Functions, 16 13. Explicit and Implicit Functions, ..... 16 14. Multiple-valued and Inverse Functions, .... 17 Exercises I. , 19 CHAPTER II. GRAPHS. RATIONAL FUNCTIONS. 15. Object of the Calculus. Graphs, 20 16. Graph of x-, 20 17. Equation of a Curve. Symmetry. Turning Values, 23 18. Graph of ex 2 , 24 19. Scale Units, ......... 26 20. Coordinate Geometry, 27 Exercise* II., ..... 29 xii AN ELEMENTARY TREATISE ON THE CALCULUS. ART. 21. The Linear Function. Intercepts, . . . , 22. Gradient, PAGE . . 30 32 Exercises III., . 23. Rational Functions. Point of Inflexion, . 24. Asymptotes, Exercises IV., . 32 34 37 41 CHAPTER III. GRAPHS. ALGEBRAIC AND TRANSCENDENTAL FUNCTIONS. CONIC SECTIONS. 25. Algebraic Functions. Graph of Inverse Functions. Cusps, . 43 26. Conic Sections, 47 27. Change of Origin and of Axes, 52 Exercises V., 54 28. Transcendental Functions. Trigonometric Functions, . . 56 29. The Exponential Function and the Logarithmic Function, . 57 30. General Observations on Graphs, 59 Exercises VI., 60 CHAPTEE IV. RATES. LIMITS. 31. Rates, 65 32. Increments, 65 33. Uniform Variation. Measure of a Uniform Rate, ... 67 34. Dimensions of Magnitudes, ....... 68 35. Variable Rates, 71 36. Average Rate, 71 37. Measure of a Variable Rate, ....... 73 38. Limits, 74 39. Examples of Limits. Definition of Tangent, .... 74 40. General Explanation of a Limit, .'..... 79 41. Definition of a Limit. Notation. Distinction between Limit and Value, 79 42. Theorems on Limits, ......... 81 43. Examples. Mensuration of Cylinder and Cone, ... 83 CONTENTS. xiii CHAPTER V. CONTINUITY OF FUNCTIONS. SPECIAL LIMITS. ART. PAGE 44. Continuity of a Function, 87 45. Theorems on Continuous Functions, 89 46. Continuity of the Elementary Functions, 90 47. L(jc-a n )/(a;-a), 91 48. L fl + -^ m . The number e, 92 49. The Function e*, 96 50. Compound Interest Law, 97 Exercises VII. , 98 CHAPTER VI. DIFFERENTIATION. ALGEBRAIC FUNCTIONS. 51. Derivatives. Differentiation, ....... 101 52. Increasing and Decreasing Functions. Stationary Values, . 103 53. Geometrical Interpretation of a Derivative, . . . .105 54. Derivative as an Aid in Graphing a Function, . . . .106 55. Derivative not Definite, ........ 107 56. Fluxions. Velocity, 109 57. Derivative of a Power, Ill 58. General Theorems, Ill Exercises VIII 115 59. Derivative of a Function of a Function and of Inverse Functions, 116 Exercises IX., 119 60. Differentials, 120 61. Geometrical Applications. Tangent. Subtangent, etc. , . . 122 62. Derivative of Arc, 124 Exercises X., 125 CHAPTER VII. DIFFERENTIATION (contimted). TRANSCENDENTAL FUNCTIONS. HIGHER DERIVATIVES. 63. Derivatives of the Trigonometric Functions, .... 129 Exercises XL, 131 64. Inverse Trigonometric Functions, . . . . . . 1 33 Exercises XII., 134 xiv AN ELEMENTARY TREATISE ON THE CALCULUS. ART. PAGE 65. Exponential and Logarithmic Functions, ..... 135 Exercises Kill., 130 66. Hyperbolic Functions, 130 67. Higher Derivatives, ,142 68. Leibniz's Theorem. Examples, 144 Exercises XIV., 116 CHAPTER VIII. PHYSICAL APPLICATIONS. 69. Applications of Derivatives in Dynamics. Simple Harmonic Motion. Potential, 149 70. Coefficients of Elasticity and Expansion, 156 71. Conduction of Heat, 157 Exercises XV., 159 CHAPTER IX. MEAN VALUE THEOREMS. MAXIMA AND MINIMA. POINTS OF INFLEXION. 72. Rolle's Theorem and the Theorems of Mean Value, . . . 161 73. Other Forms of the Theorems of Mean Value, . . . .164 74. Maxima and Minima, . . . . . . . . 166 75. Examples. Graph of e' ax ain(bx + c), 168 76. Elementary Methods, 171 77. Variation near a Turning Value, ...... 174 Exercises X VI. a, X VI. b, X VI. c, . 176-180 78. Concavity and Convexity. Points of Inflexion, . . .180 Exercises XVII., 182 CHAPTER X. DERIVED AND INTEGRAL CURVES. INTEGRAL FUNCTION. DERIVATIVES OF AREA AND VOLUME OF A SURFACE OF REVOLUTION. POLAR FORMULAE. INFINITESIMALS. 79. Derived Curves, . . 183 80. Derivative of an Area, ........ 185 81. Interpretation of Area, 187 CONTENTS. xv ART. PAGE 82. Integral Function, 188 83. Integral Curve, 190 84. Graphical Integration, 192 85. Surfaces of Revolution, , . 193 86. Infinitesimals, . . 195 87. Fundamental Theorems 197 88. Polar Formulae, 200 Exercises XT II I., 201 CHAPTER XL PARTIAL DIFFERENTIATION. 89. Partial Differentiation. Continuity of a Function of two or more Independent Variables, 204 89 a. Coordinate Geometry of Three Dimensions. Direction Cosines. Equations of Line and Plane. Equation of Surface, . . 205 90. Total Derivatives. Complete Differentials, . . . .211 91. Geometrical Illustrations. Tangent Plane. Normal, . . 214 92. Rate of Variation in a given Direction. Note on Angles, . 218 93. Derivatives of Higher Orders. Commutative Property. Laplace's Equation, 220 94. Complete Differentials, 224 95. Application to Mechanics. Potential, ..... 225 96. Application to Thermodynamics, ...... 228 97. Four Thermodynamic Relations, 231 98. Change of Variable. Differentials of Higher Orders, . . 233 99. Transformation of V*u, 235 Exercises XIX., 238 CHAPTER XII. APPLICATIONS TO THE THEORY OF EQUATIONS. Rational Integral Functions. Zeroes, ..... 242 Any Continuous Function, 243 Newton's Method of approximating to the Roots of an Equation, 244 Tests for Degree of Approximation, ..... 245 Examples, .......... 246 Successive Approximations, 247 xvi AN ELEMENTARY TREATISE ON THE CALCULUS. ART. PAGE 106. Expansion of a Root in a Series. Reversion of Series, . . 249 107. The Equation x = tan x, 251 Exercises XX., ...... 253 108. Proportional Parts, 255 109. Small Corrections, . 258 Exercises XXI., 260 CHAPTER XIII. INTEGRATION. 110. Integration. Indefinite and Definite Integral. Constant of Integration, 262 111. Standard Forms, '264 112. Algebraic and Trigonometric Transformations, . . . 267 Exercises XXII. , 269 113. Change of Variable, 271 114. Examples of Change of Variable, 272 115. Quadratic Functions, 274 116. Trigonometric and Hyperbolic Substitutions, .... 277 117. Some Trigonometric Integrands, 278 Exercises XXIII. , 280 118. Integration by Parts, 281 /? 119. Successive Reduction. The Integral / sin m xcos"xdx, . . 284 Jo Exercises XXIV., 288 120. Partial Fractions, 290 121. Integration of Rational Functions, 292 122. Irrational Functions, 294 123. General Remarks, 295 Exercises XXV., 296 CHAPTER XIV. DEFINITE INTEGRALS. GEOMETRICAL APPLICATIONS. 124. Definite Integrals. Theorems, 298 125. Related Integrals, 301 126. Infinite Limits. Infinite Integrand, 304 Exercises XX VI., 306 CONTENTS. xvii ART. PAGE 1:27. Some Standard Areas and Volumes. Curve Tracing, . . 309 Exercises XXVII., 312 128. Area of Closed Curves, 316 129. Area Swept out by a Moving Line, 319 130. Planimeters, 321 Exercises XXV III., 322 CHAPTER XV. INTEGRAL AS LIMIT OF A SUM. DOUBLE INTEGRALS. 131. Integral as the Limit of a Sum, ...... 324 132. Examples, 327 133. Approximations. Simpson's Rule, 328 Exercises XXIX., 331 134. Mean Values, 333 135. Double integrals, 334 136. Notations for Double Integrals. Polar Elements. Triple Integrals, 337 137. Centres of Inertia, 341 138. Moments of Inertia, 343 139. Polar Element of Volume. Definition of Line Integral and of Surface Integral, 346 Exercises XXX., 347 Gamma and Beta Functions, 349 CHAPTER XVI. CURVATURE. ENVELOPES. 1 40. Curvature. Fundamental Formula, . . . . 352 141. Circle, Radius, and Centre of Curvature, .... 354 142. Various Formulae for the Curvature. Intrinsic Equation of a Curve, 355 Exercises XXXI. , 359 143. Evolute, Involute, Parallel Curves, 361 144. Envelopes, 364 145. Equation of Envelope. Contact Theorem, .... 365 146. Cycloids. Epicycloids. Hypocycloids, 368 . Exercises XXXII. 371 b xviii AN ELEMENTARY TREATISE ON THE CALCULUS. CHAPTER XVII. INFINITE SERIES. ART. PAGE 147. Infinite Series Convergent, Divergent, Oscillating, . . 375 148. Existence of a Limit. Theorems, 377 149. Tests of Convergence. Fundamental Test ; Comparison Test ; Test Ratio. Remainder 379 150. Absolute Convergence. Power Series, 382 151. Uniform Convergence. Continuity of Series, .... 385 Exercises XXX HI., .... 387 CHAPTER XVIII. TAYLOR'S THEOREM. 152. Taylor's Theorem. Maclaurin's Theorem. Remainder, . 390 153. Examples of Expansions: since, cosa:, e x , (l+x) m , log(l+.r). . 393 154. Calculation of the 7i th Derivative. Examples, . . . 397 155. Differentiation and Integration of Series, .... 399 156. Expansions. Approximations. Examples of Integration of Series, 401 Exercises XXXIV., 403 CHAPTER XIX. TAYLOR'S THEOREM FOR FUNCTIONS OF TWO OR MORE, VARIABLES. APPLICATIONS. 157. Taylor's Theorem for Functions of two or more Variables, . 408 158. Examples : Tangent Plane, Euler's Theorems of Homo- geneous Functions, . . . . . . . .411 159. Maxima and Minima of a Function of two or more Variables, 412 160. Examples. Undetermined Multipliers, . . . . . 414 Exercises XXXV., 416 161. Indeterminate Forms. Elementary Methods, . . . 418 162. Method of the Calculus, 419 Exercises XXX VI., . \ . . . 422 CONTENTS. xb CHAPTER XX. DIFFERENTIAL EQUATIONS. PAGE Differential Equations. Definitions. Examples, . . . 424 Complete Integral, 426 Exercises XXXV II., .... 427 Equations of the First Order and of the First Degree. Vari- ables Separable. Homogeneous Equations. Linear Equa- tions. Exact Equations, ....... 428 Equations of First Order, but not of First Degree. Clairaut's Equation. Singular Solutions, ...... 431 Equations of the Second Order. Simple Pendulum, . . 432 Linear Equations. General Property 433 The Complementary Function, ...... 434 The Particular Integral, 436 Simultaneous Equations. Example from Electric Circuits, . 437 Exercises XXX VI II., .... 439 ANSWERS, 442 INDEX, ...... 454 SUGGESTIONS FOR FIRST READING. The following order of reading the book may be adopted by students who are just beginning the study of the Calculus : Chapters I.-IV, V. 44-47, VI., VII. 67 (with Exercises XIV. 1-4, 11-14), IX. 74-76 (with Exercises XVIa., XVIb.), 78 (with Exercises XVII. 1-6). This course includes the fundamental properties of the algebraic functions, with several interesting applications. Chapters V. 48-50, VII., VIII., the rest of IX., X., XIII-XV. Chapters XI.-XIL, XVI.-XIX. may be read as the needs of the student demand at any time after Chapters I.-X. have been mastered, and Chapter XX. as soon as some progress has been made in integration. AN ELEMENTARY TREATISE ON THE CALCULUS. CHAPTER I. COORDINATES. FUNCTIONS. 1. Directed Segments or Steps. Let A, B (Fig. 1) be any two points on a straight line. In Elementary Geometry it is customary to denote the segment of the line between A and B by AB or by BA indifferently, the order of the letters being of no consequence. It is useful, however, for many purposes to distinguish the segment traced out by a point which moves along the line from A to B from that traced out by a point which moves from B to A. When this distinction is made, the segment is called a directed segment or vector or step, and the distinction is represented in the symbol for the segment by the order of the letters ; thus, AB denotes the segment traced out by a point which moves from A to B, w r hile BA denotes the segment traced out by a point which moves from B to A. The length of the step AB is the same as that of the step BA , but the steps have opposite directions. A B D 7 C 5 FIG. 1. Two steps AB, CD are defined to be equal if (1) they are on the same straight line or on parallel straight lines, (2) the lengths of AB and CD are equal, and (3) D is on the 2 AN ELEMENTARY TREATISE ON THE CALCULUS. same side of C as B is of A. Thus, if D' be at the same distance from C as D is, but on the opposite side, A B is not equal to CD' but to D'G. The step AB has the same length and the same direction as CD or DC, but though it has the same length as CD, it has not the same direction and is therefore not equal to CD in the sense in which " equal " has been defined for steps. 2. Addition of Steps. Let A, B, C be any three points on a straight line. Whatever be the relative position of the points A, B, C, a point which moves along the line from A to B, and then from B to C, will be at the same distance from A and on the same side of A as if it had moved directly from A to C. AC is therefore taken as the sum of the steps AB and BC, and the operation of addition of steps is defined by the equation AB+BC=AC. When B lies between A and C, the sum of the lengths of the steps AB and BC is equal to the length of the step AC, and therefore in this case addition of steps agrees with the usual geometrical meaning of addition of segments in which length alone is considered. But when B does not lie between A and C, the sum of the lengths of the steps AB and BC is not equal to the length of the step AC. It will be seen immediately that steps can be represented as positive or negative, and that addition of steps corresponds to algebraical addition. If D be any fourth point on the line and in the same way the sum of any number of steps may be defined. To find the sum of AB and CD when B and C are not coincident, take the step BE equal to the step CD ; then AB+CD = AB+BE=AE. If x be any positive number, xAB is a step in the same direction as the step AB, and of a length which is to the length of AB in the ratio of x to 1 ; thus, SAB is a step thrice as long, and in the same direction as the step AB; STEPS. LAWS OF OPERATION. 3 is a step five-thirds of the length of AB and in the same direction. The student will have no difficulty in showing that the commutative and associative laws for the addition of numbers hold for the addition of steps. 3. Symmetric Steps and Subtraction of Steps. If in the first case of the preceding Article the point C be supposed to coincide with A, the step AC becomes the zero-step A A., which is denoted by 0. Hence, in symbols, = AA=0. Similarly, AB+BC+CA=AO+CA = 0. In Algebra the negative number a is defined by the equation In the same way the negative step AB may be defined by the equation as being the step BA ; that is, the step AB is the step BA of the same length in the opposite direction. The symbol + may now be attached to a step AB, and +AB may be called a positive step. The two steps +AB and AB (or BA) are called symmetric steps. Obviously, if two steps are equal, so also are their symmetric steps. The operation of subtraction of a step is defined as the addition of the symmetric step ; in symbols, AC-BC=AC+CB = or, AB-CD = AB+DC=AC', if Precisely as in Algebra, the commutative and associative laws may be shown to hold for subtraction of steps, and there will be no confusion caused by the use of the symbols + and to indicate symmetric steps as well as the operations of addition and subtraction. By the definition of subtraction, if A, B be any two points on a line and any third point, 4 AN ELEMENTARY TREATISE ON THE CALCULUS. 4. Abscissa of a Point. Let be a fixed point on a line X'OX and P, P' two points on opposite sides of but at the same distance from it (Fig. 2); let U be another point on the line on the same side of as P is, say to the right of 0. The steps U, OP have the same sign ; the steps U, OP' have opposite signs. Let U be taken as a standard of length, say 1 inch, and as a standard of direction ; it may therefore be called the unit step. Steps measured like OU to the right will be called positive steps, while those measured to the left will be called negative. Thus OP, P'P are positive, OP', PP' negative steps. X' P' A' U' 6 U ~~A P X FIG. 2. If OP is equal to xO U, then OP'=-P'0=-OP=-xOU. The positive number x is called the abscissa of P with respect to the origin ; the negative number x is called the abscissa of P' with respect to the same origin, and the line X'OX is called the axis of abscissae. Every point of the line to the right of will have a positive number for abscissa, and every point to the left of a negative number ; the abscissa of itself is zero. Thus if OA =20U, the abscissa of A is 2 ; the abscissa of U is 1 ; the abscissae of U' and A', the points symmetric to U and A, are 1 and 2 respectively. As thus defined, the abscissa of a point P is the ratio of OP to the unit step U, taken with the positive or negative sign according as P is to the right or to the left of 0, U being supposed to be to the right of 0. When a point P has the abscissa x, it is convenient to say that the point P and the number x correspond to each other. Thus the point A and the number 2, the point U' and the number 1 , the point and the number correspond to each other. AXIOM. The fundamental axiom on which the application of Algebra to Geometry rests is that, when the origin and FUNDAMENTAL AXIOM. 5 the unit step OU have been fixed, there is a one-to-one correspondence between the points of the axis and the system of real numbers ; that is, to every point on the axis corresponds a definite number, namely the abscissa of the point, and to every number corresponds a point on the axis, namely the point which has the number for abscissa. When the ratio of OP to U is a rational number, that is, a positive or negative integer or fraction, P is determined by laying off OU, or a submultiple of U, a certain number of times along the axis, to the right or to the left, according as the number is positive or negative. Thus if the number be -, we lay off to the left a line equal to 7 times the third part of U. When, however, the ratio of OP to OU is an irrational number, such as ^2 or TT, the position of P may be determined in practice by taking a rational approximation to the irrational number. Thus for TT we may take 31 or 314 or 3142, etc., according to the size of the unit line. Of course, whatever size the unit line may be, a stage is soon reached when the closer approximations become indistinguishable in the diagram ; if the unit be 1 inch it would be difficult to distinguish the points whose abscissae are 314 and 3'142 from each other. Irrational numbers are, however, subject to the same laws of operation as rational numbers, and though in a diagram it may be impossible to distinguish the points corresponding say to TT and 3142 from each other, yet in our reasoning they are to be considered distinct, just as in reasoning about a straight line we consider it to have no breadth, although we cannot represent such a line in a diagram. Ex. 1. Mark the points whose abscissae are : Ex. 2. If x be the abscissa of a point, mark the points which are determined by the equations : = 0; ,r 2 -4=0; 3.r 2 - 4r - 1 = 0. 5. Measure of a Step. If the abscissae of A, B are a, b respectively, then AB=OB-OA = bO U-aOU=(b-a)OU. 6 AN ELEMENTARY TREATISE ON THE CALCULUS. The number b a may be taken as the measure of AB; the numerical value of b a gives the ratio of the length of AB to the length of the unit step U, and the sign of b a gives the direction of AB. Thus if OU be 1 inch, b = 5, a = 2, AB will be 3 inches and B will be to the right of A ; if b = 5, a= 2, AB will be 3 inches long, and since 5 + 2 is negative B will be to the left of A. The unit step OU is generally omitted, and AB is said to be equal to b a. By the definition of the expression " algebraically greater," b is algebraically greater than a when b a is positive; therefore when b is algebraically greater than a, B lies to the right of A. Similarly when 6 is algebraically less than a, B lies to the left of A. We have, therefore, the con- venient relation that the number 6 is algebraically greater or less than the number a, according as the point whose abscissa is 6 lies to the right or to the left of the point whose abscissa is a. Instead of the expression " the point whose abscissa is a" it will be more compact and equally clear to use the phrase " the point a." Ex. 1. Determine in sign and magnitude the step AB for the cases : a \i b = ar >d P the point dividing AB in the ratio of k to 1, show, as in 5, Ex. 3, that the coordinates of P are l+k ' l+k ' What is the sign of k (i) when P lies between A and B, (ii) when P does not lie between A and B ? 7. Distance between two points. Let P (Fig. 5) be the point (x v 2/1). Q the point (x z , y 2 ); draw PM, QN perpendi- cular to X'X, and let PR be drawn parallel to X'X to meet iVQ (or NQ produced) at R. 10 AN ELEMENTARY TREATISE ON THE CALCULUS. Whatever be the relative position of P and Q, we have for the measures of PR, RQ As regards magnitude we have, by Euclid I. 47, and whether the signs of X 2 x^ and y z y\ be positive or nega- tive the squares of these numbers will give the number of square units in the squares described on x PR and RQ. Hence FIG. 5. p(\\ and therefore the length of PQ is where the positive sign must be given to the root. If Q coincide with 0, x? and y z are both zero, and the length of OP is ^(xf + y*). The student should verify the result for different positions of P and Q. Ex. 1. Find the distance between the points (3, 7), (9, 6), the length of the unit being 1 inch. Let the distance be r inches ; then r 2 = (3 -9) 2 + (7-6) 2 = 37 ; r= v /37 = 6'083, so that the distance is 6'083 inches. Ex. 2. Find the distances between the following pairs of points : plot the points in each case. i. (1, 1), (3, 2). ii. ( - 1, 1), (3, 2). iii. ( - 1, 0), (0, 2). iv. (-2, -3), (2, 3). v. Or, -,r), -, Ex. 3. Show that if the point (x, y) be any point on the circle whose radius is 3, and whose centre is the point (2, 1 ), 2 - 4z - 2 - 4 = 0. 8. Polar Coordinates. The position of the point P (Fig. 6) would clearly be determined by the angle which OP makes with the fixed line OX, and by the length of the POLAR COORDINATES. 11 radius OP. We must be clear, however, as to the meaning of the word " angle." Following the usual convention in Trigonometry, we consider the radius OP to be always positive, and define the angle that OP makes with the positive direction of OX as the angle through which a line coinciding with OX (not with OX') has to be turned till it passes through the point P. The angle will be considered positive when the rotation is counter-clockwise. If OP be r units of length and the angle XOP 9 degrees or radians according to the unit of angle adopted, the two numbers r, 6 are called the polar coordinates of P, and P is described as the point (r, 0). (r, 0') ; 6' is negative. p' FIG. 6. Similarly, P' is the point With the usual system of rectangular axes in which OX has to be rotated counter-clockwise through 90 till it coincides with Y, the positive direction of the axis Y' Y, we see that the polar coordinates (r, 0) of P are connected with the rectangular coordinates (x, y) by the equations x = r cos 9, y = r sin 9. These equations, w r hen solved for r and 9 in terms of x and y, give r=+7(a; 2 +2/ 2 ), tan 6> = ^. It must be noted, however, that tan 9 does not definitely determine the angle 9. For if tan 9 be positive we can only infer that P lies in the first or third quadrant, while if tanO be negative that P lies in the second or fourth quadrant. We must consider also the signs of x and y or of cos 9 and sin 9. It is usually most convenient to suppose 9 to vary from 180 to +180 so that a point above the axis X'X has a positive angle, and a point below that axis a negative angle. 12 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 1. If P is the point ( - 3, 4), find its polar coordinates (r, 0). ) = 5; tan 6 =-^= - 1'3333 ; (9=126 52'. Since tan is negative, 6 is in the second or fourth quadrant ; but x or cos 6 is negative, and therefore is in the second. Ex. 2. If P is the point (3, 4), show that its polar coordinates are (5, -53 8'). 9. Variable. Continuity. Let A be a fixed point on a line, say, on the a;-axis X'X, and let a point P start from the position A and move steadily along the axis, say to the right, till it reaches another position B. The segment AB described by the point P is the most perfect type of a continuous magnitude ; there is no gap or break in it. As P moves from A to B, the step AP steadily increases ; AP is a continuously varying magnitude during the motion of P. If a, b are the abscissae of A, B, and x the abscissa of P at any stage of the motion, then, as P moves from A. to B, since AP = x a* x steadily increases (algebraically) from a to 6 ; a; is a continuously varying number or, more briefly stated, x is a continuous variable. Again, since P coincides in succession with every point lying between A and B, so x assumes in succession every value lying between a and b. If a be negative and b positive, A will be to the left and B to the right of the origin 0, and when P passes through 0, x will be zero so that as x passes from negative to positive values it passes through the value zero. Had P instead of moving always to the right moved sometimes forward, sometimes back- ward, then every time it passed through the value of x would have been zero, so that x would only change from negative to positive or from positive to negative by passing through zero. We will assume then, as characteristic of a continuous variable, that as it varies continuously from a value a to a value b it assumes once at least every value inter- *Here, and in similar cases, it is the measure of the step AP that is of importance ; it will cause no confusion to let AP stand for the step, and also for the measure of the step as is usually done in all applications of geometrical theorems. VARIABLE. CONTINUITY. 13 mediate to a and b ; if one of these values, say a, be negative and the other positive, one of the values the variable takes will be zero. 10. Geometrical Representation of Magnitudes. The measure x of any magnitude A is the ratio of A to another magnitude U of the same kind that is chosen as the unit. If then on any axis a unit step U is taken as representing the unit magnitude U, the step OM where OM is equal to xOU will represent the magnitude A. There is thus established a correspondence between the magnitudes of the particular kind considered and the points of the axis ; the point 1 corresponds to the unit magnitude U, the point 2 to the magnitude 2 U, and so on. Many of the magnitudes considered in Geometry and Physics, for example, lines, angles, velocities, forces, are often treated as directed magnitudes, and their measures may then be either positive or negative; when the meas- ures are negative, the points that correspond to the magni- tudes will lie on the opposite side of from that on which U lies. A variable magnitude P will be represented by a variable segment OP, and when the magnitude varies continuously the point P will trace out a continuous segment of the axis. For purposes of calculation it is the measure of the magnitude that is of importance, and, to avoid a tedious prolixity of statement, such an expression as " a velocity v " will often be used in the sense " a velocity whose measure is v units of velocity." Of course in all cases care should be taken to prevent ambiguity as to the units .employed. 11. Function. Dependent and Independent Variables. In any problem the magnitudes dealt with will usually be of two classes, namely, those that retain the same value all through the investigation and those that are supposed to take different values: the former are called constants, the latter variables. It has become customary to denote con- stants by the earlier letters of the alphabet, a, b, c, . . . , and variables by the later letters, z,y,x,.... Of course when there is any advantage in denoting a variable by a or a constant by z there is no reason against doing so. 14 AN ELEMENTARY TREATISE ON THE CALCULUS. Again, taking first the case of only two variables, it will usually happen that when one of the variables is given a series of values the other variable will take a series of definite values, one for each that the first is supposed to have been given. The second variable is then said to be a function of the first, or to be a variable dependent on the first, which is distinguished as the independent variable. Instead of the phrase " independent variable," the word argument is often used, and the dependent variable is then called a function of its argument. Thus, if. we consider a series of triangles, all of the same altitude, the area of any triangle is a function of its base. The distance travelled by a train which moves at a constant speed is a function of the time during which it has moved at that speed. The pressure of a given quantity of gas which is maintained at a constant temperature is a function of its volume. In these examples the independent variable or argument is the base, the time, the volume ; and the dependent variable or function is the area, the distance, the pressure respectively. It is usually a mere matter of convenience which of the two variables is considered as independent. Thus if the time at which the train passed certain stations on the railroad were the subject of inquiry, the distance would be taken as the independent variable and the time as the dependent. When there are more than two variables it may happen that when definite values are assigned to all but one of them the value of that one becomes determinate ; this one variable is then said to be a function of or to be dependent on the other variables which are called the independent variables of the problem. Thus the area of a triangle is a function of the base and of the altitude when both base and altitude vary. The pressure of a given quantity of gas is a function of the volume and of the temperature when both volume and temperature vary. Generally, a variable y is said to be a function of another variable x when to every value of x there corresponds a definite value of y; a, variable y is said to be a function of two or more ARGUMENT. FUNCTION. 15 variables, x, u, ..., when to each set of values of the variables x, u, ... there corresponds a definite value of y. While it is important to keep this general notion of functional dependence in mind, it will, however, be usually assumed that a function is defined by an equation (see 13, 26, 27, 28), and that it can be represented by a graph ( 16). This assumption implies (i) that as the argument varies continuously, in the sense explained in 9, from a value a to a value b, the function also varies continuously from a value, A say, to a value B ; (ii) that to a small change in the argument corresponds also a small change in the function. The assumption implies a good deal more than what is here stated, but at this stage the student is earnestly urged to pass lightly over the purely theoretical difficulties and to try to get a thorough grasp of the fundamental conceptions of variation and functional dependence by working out for himself the graphical exercises in the next chapter. He will find by trial that, except for special values of the argument, the property (ii) is actually found in all the ordinary functions ; the pro- perty (i), though apparently simpler, is really much harder to demonstrate mathematically. A mathematical definition of the continuity of a dependent variable will be given in Chapter V, 44. The student should notice the phrase " definite value " or " determinate value." It may happen that the analytical expression for a function ceases to have meaning for certain values of the argument ; for these values, therefore, the function is not defined. Thus the function (x 2 l)/(x 1) is defined for all values of x, except the value 1 ; because when x = \ the expression takes the form 0/0, which is absolutely meaningless. We should not get out of the difficulty by first dividing numerator and denominator by x 1 and then putting 1 for x ; because in dividing by x 1 we assume that x I is not zero, division by zero being excluded by the fundamental laws of algebra. Again, such a function as ^(lx 2 ) is only defined for values of x that are numerically less than or equal to 1 ; in this case we may say that the function is defined for values of the argument in the range from 1 to +1 inclusive. 16 AN ELEMENTARY TREATISE ON THE CALCULUS. It has always to be understood in reasoning about a function that only those values of the argument are to be considered for which the function has a definite value, or, in other words, for which the function is well-defined. 12. Notation for Functions. A function of a variable is often denoted by enclosing the variable in a bracket and prefixing a letter; thus, f(x\ F(x), are functional symbols, not multipliers ; the symbol f(x) must be taken as a whole, and means simply " some function of x," the context or some explicit statement determining which particular function is meant. For different functions occurring in the same investigation different functional symbols must of course be used. /(a) means " the value of the function /(a?) when x has the value a," or " the value of the function f(x) when x is replaced by a." Thus, if f(x) denote the function then /(0)=-1; /(l)=-3; f /(a; 2 ) = O 2 ) 2 - 3 2 - 1 = a 4 - So; 2 - 1. A similar notation is used for functions of two or more variables ; thus, f(x, y), F(p, v), (x, y, z) denote functions of x and y, of p and v, of x, y, and z respectively. If f(x, y) = So; 2 - 2xy -y z + 4>, then /(I, -l) = 3 + 2-l + 4 = 8; /(a, 6) = 3a 2 -2a6- The letters should be separated by a comma to indicate that there are two or more variables, and thus distinguish the function from one in which the argument is the product of two or more variables. Thus, f(xy) is a function whose argument is the product xy, and if f(x) be ax+b, then/(^-y) is axy + b. 13. Explicit and Implicit Functions. One variable is usually defined as a function of another by an equation. The dependent variable is called an explicit function of its argument, or is said to be given explicitly when the MULTIPLE- VALUED FUNCTIONS. 'l7 equation is solved for the dependent variable in terms of the argument. Thus y=f(x): 8 are equations which give y, s, p explicitly as functions of x, t, v respectively. When the equation is not solved, the dependent variable is called an implicit function of its argument, or is said to be given implicitly. Thus y is given as an implicit function of x by the equation axy + bx + cy + d = 0. This equation when solved for y in terms of x gives bx + d ax+c' and y is now an explicit function of x. 14. Multiple-valued and Inverse Functions. When a function is given implicitly by an equation, it may happen that to one value of the one variable there correspond two or more values of the other. The definition of a function given in 11 assumes that to each value of the argument there corresponds but one value of the function, and in reasoning about a function we must always suppose that it has but one value for each value of its argument; in other words, that the function is single-valued. When the defining equation gives more than one value of the one variable for one value of the other, we can usually consider the equation as defining a function that is made up of two or more functions each of which is single-valued; such a function is called a multiple-valued function. Thus, if y is given as a function of x by the equation then y=jcJ(2afi-I), and to each value of x there correspond two values of y ; y is a two- valued function of x. The equation really gives two functions of x, namely, 18 AN ELEMENTARY TREATISE ON THE CALCULUS. each of which is single-valued, and denned for those values of x for which 2# 2 is greater than or equal to 1. Again, the equation defines y as a single- valued function of x, but x as a two-valued function of y, namely x is either J(y 1) or >J(y 1). When the graphical representation of functions is considered, it will be seen that the separate functions represent different parts of the one curve (e.g. 20). The equation x 2 y + l = 0, as we have just seen, not only defines y as a function of x but also defines a? as a function of y. More generally, the equation y=f(x), which defines y explicitly as a function of x, also defines x implicitly as a function of y\ the two functions thus defined by the one equation are said to be inverse to each other. For example the equation y = x 3 when solved for x gives x = ^/y and thus defines two functions which are inverse to each other, namely the cube and the cube root. It is usual in English books to employ /~ l as the symbol of the function inverse to that denoted by the symbol / so that x =f~ l (y) when y =/() The student will be already familiar with this notation in the case of angles. Thus sin" 1 ?/ means, not l/siny but, the angle (within a certain range) whose sine is y ; and just as we have the identity, 8in(sin~ l y) = y, so we have f(f-\y}}=y or, as the identity is usually written, Again it may well happen that the inverse function is not single-valued. Thus, sin -1 a; may, unless some restric- tion be imposed, be any one of an infinite number of angles. To secure definiteness some restriction has in such cases to be placed on the range of the variable ; for example, sin -1 a; may be restricted to angles lying between Tr/2 and + 7T/2 (inclusive of 7r/2 and + 7r/2), and then sin~ ] a; is single-valued. For further information, see 25, 27, 28. INVERSE FUNCTIONS. EXERCISES I. 19 EXERCISES I. 1. If fi.r)=x s -x+l, find f(0), /I),/ - 1), and show that f(x + K) =f(x) + (3-r 2 1 ) h + 3xk 2 + A 3 . 2. If f(x)=x' i x-2, write down^a^+6). 3. If f(x) = x 2 - 5x + 1, write down /^ 2 ), /(x 5 ), f(s\n x). What is the value of/I sin^ ) ? 4. Iff(x) = log.r, show that 5. If i /^Jf)*aa!*+6af*4-cas*+dj show ihatf( x) is equal When f( x) =f(x\ the function f(x) is called an even function of its argument. 6. If f(x)=ox j + 1x^+03? + dx, show that/(-^r) is equal to f(x). When f( x) = f(x\ the function f(x) is called an odd function of its argument. 7. Show that sin #, cosec #, tan x, cot x are odd functions of x, and that cos .r, sec x are even functions of x. 8. Show that (e* e~*)/x is an even function of x. 9. If f(x, y) = ox 2 + bxy + c, write down fiy, x), f(x, x\ and jfy, y). 10. if == 11. If y=f(x)= ---- , show that xf(y). 12. If fa,y) = x*-y 2 , show that /cos 0, sin 0) = cos 20, and that sec0, tan0)=l. CHAPTER II. GRAPHS. RATIONAL FUNCTIONS. 15. Object of the Calculus. Graphs. Stated in the most general terms the object of the Calculus may be said to be the study of the changes of a continuously varying function. The investigation of the rate at which a given function is changing for any specified value of its argument belongs to the Differential Calculus ; the converse problem of deter- mining the amount by which a function changes for a specified change in its argument, when the rate of change of the function is known, belongs to the Integral Calculus. An almost indispensable aid to this study is furnished by the graphical representation of a function, and for the sake of those students who may have had little or no experience in graphical work a few hints will now be given that may be of service to them. At times the tracing of a graph involves a good deal of tedious calculation, but the student will be well repaid for his labour by the insight he will obtain into the fundamental conceptions of variation and continuity of a function. When he has made but a little progress in the differential calculus he will find several methods of reducing the necessary calculations. An ex- tremely good discussion of graphs from an elementary standpoint will be found in Professor Chrystal's Introduc- tion to Algebra. (London : A. & C. Black.) 16. Graph of x 2 . In geometry and physics we frequently find a function defined by an equation of the form y = ex 2 where c is a constant. Thus the area of a circle varies as the square on the radius; the distance that a body falls GRAPH OF x 3 . 21 from rest, the resistance of the air being neglected, varies as the square of the time of fall ; the heat generated by an electric current in a given time varies as the square of the current in the circuit and so on. These statements when expressed in the usual algebraical way all lead to an equation of the above form ; x denotes the number of units of the one kind of quantity, for example the number of feet, or the number of seconds, or the number of amperes; y denotes the number of units of the second kind, for example the number of square feet, or the number of linear feet, or the number of ergs (or other heat units). The number c is a constant, that is, does not change when x changes ; it is not, however, the same constant in the different problems ; thus for the area of the circle C = TT, for the falling body c = \g, for the electric circuit c depends on the resistance and on the heat unit. Suppose for simplicity that c = 1 ; the more general case can be deduced from this one. Let X'X, Y'Y be two rectangular axes (Fig. 7), OU, 0V unit segments on these axes. Give to x a series of values, and from the equation y = x 2 deduce the corresponding values of y. Associating each value of x with the corresponding value of y, we obtain a series of pairs of numbers, and each pair may be taken as the coordinates of a point in the plane of the diagram, the value of x being the abscissa and the corresponding value of y the 'ordinate of the point. If the values given to x form an increasing or a decreasing series of numbers, and if the difference between any two con- secutive values be small it will be found that the consecu- tive points determined on the diagram lie pretty close to each other ; the curve drawn through these points with a free hand is called the graph of the function x 2 . Tabulating values, we have x I 0, -1, -2, -3, ... 1,1-1 . y | 0, -01, -04, -09, ... 1, 1-21... x | -!, --2, --3, ... -1, -1-1 ... y | -01, -04, -09, ... 1, 1-21 ..." Take OU, 0V each, say, 1 inch and plot the points 22 AN ELEMENTARY TREATISE ON THE CALCULUS. (0,0), (-1, -01) ... (--I, -01), (--2, -04) ...; by drawing a curve through the points we get the graph of 2 (Fig. 7). -i- X FIG. 7. Of course only a comparatively small number of points can be plotted, but by actual calculation we find that a small change in x produces but a small change in y; we are there- fore warranted in concluding that an ordinate corresponding to a value of x that has not been used in plotting the points but that lies between two values that have been used can differ but little from the ordinate of the graph correspond- ing to that value of x. When there is any room for doubt, a few more values of y at closer intervals may be calculated. When x is at all large, y will be much larger and it becomes impossible to plot the points in the diagram ; we must then try to follow in imagination the course of the graph or if it be of importance to know the form of the graph for such values we may take the unit lines U, V smaller. See further 19. EQUATION OF A CURVE. 23 17. Equation of a Curve. Symmetry. Turning Values. Let us now consider the graph of a? 2 from the purely geo- metrical point of view. (i) Equation of the Curve. A point in the plane will or will not be on the graph of x 2 according as the ordinate of the point is or is not equal to the square of its abscissa ; in other words, the condition that a point should lie on the graph is that the coordinates of the point should satisfy the equation y=x 2 which states the law according to which the curve was constructed. This equation is generally called the equation of the curve, and the curve is said to be repre- sented by the equation ; the two expressions " the graph of the function x 2 " and " the curve whose equation is y = x 2 " (or " the curve represented by the equation y = x 2 ") mean the same thing. More generally, " the graph of the function f(x) " and "the curve whose equation is y =/(#)" mean the same thing, and the condition that a point should lie on the curve or graph is that its coordinates should satisfy the equation y=f(x). Thus the point ( |, \} does, and the point ( ^, ^-) does not, lie on the graph of x z ; the origin lies on the graph of x 2 but not on that of x z + l. (ii) Symmetry. The ordinate of the point on the graph of x 2 which has a for its abscissa is equal to the ordinate of the point which has a for its abscissa, since each ordinate is a 2 . If A is the point (a, a 2 ) and B the point ( a, a 2 ) AB will be perpendicular to OF and will be bisected by OY; that is, since a may be any number whatever, the graph is symmetrical about OY, or Y is an axis of sym- metry (cp. 6, ex. 3). In plotting the graph by points therefore, it would be sufficient to calculate y from positive values of x alone ; the part of the curve to the left of OY is simply the reflection in Y of the part to the right. We might imagine the plane of the diagram turned through two right angles about Y and the part of the curve origin- ally to the right of Y would after rotation form the part to the left of OF. The graph of a function f(x) is not, as a rule, sym-* metrical about the ^/-axis or about any other line ; but the function should always be examined for symmetry since 24 AN ELEMENTARY TREATISE ON THE CALCULUS. the presence of symmetry saves labour. The graph of f(x) will be symmetrical about OF if f(x) is an even function (Exer. I., ex. o), for in that case the ordinate/( a) of the point whose abscissa is a is equal in sign and in magni- tude to the ordinate f(a) of the point whose abscissa is a. (iii) Variation of the Function. Suppose a point to start from and move along the graph. At first the ordinate of th$ point increases very slowly ; as the point gets nearer to the point (1, 1) its ordinate grows more rapidly; when it has passed (1, 1) its ordinate grows still more rapidly. As x increases from to | the ordinate increases from to \ ; as x increases from | to 1 the ordinate increases from \ to 1 ; as x increases from 1 to f the ordinate increases from 1 to f . Thus for the same increase of ^ in x the ordinate increases by the amounts \, f , f respectively. The course of the graph shows very clearly that after a certain point has been reached the ordinate grows more rapidly than the abscissa while near the origin it grows less rapidly ; the graph thus gives a vivid picture of the variation of the function x z represented by the ordinate. (iv) Turning Values. If a point move along the graph from any position on the left of Y to any position on the right the ordinate of the point decreases till the point reaches and then increases. The point where the ordi- nate ceases to decrease and begins to increase is called a turning point of the graph, and by analogy the value of the function x 2 at 0, namely zero, is called a turning value of the function. The turning value is in this case a 'minimum value of the function or ordinate. In general those points on a graph at which the ordinate ceases to decrease and begins to increase, or else ceases to increase and begins to decrease are called turning points of the graph, and the corresponding values of the function turn- ing values ; the turning values are respectively minima and maxima values of the function, that is values respectively less and greater than any other values of the function in their neighbourhood. 18. Graph of ex 2 . We might by assigning values to x, and calculating the corresponding values of y from the SYMMETRY. TURNING VALUES. 25 equation y = cx 2 construct the graph of ex 2 ; it will be instructive to consider another method of deriving the graph. First let c be positive. Let any ordinate of the graph of x 2 be denoted by y l and the ordinate of the graph of ex 2 for the same value of x by y 2 ; then y z = cy l , because y l = x 2 , y z = cx 2 and x is the same number in both equations. The two ordinates may be called " corresponding ordinates." FIG. 8. Hence to obtain any ordinate of the graph of ex 2 we have only to multiply the corresponding ordinate of that of x 2 by c ; in other words, if MP is any ordinate of the graph of x 2 divide MP or MP produced at P so that MF 26 AN ELEMENTARY TREATISE ON THE CALCULUS. is to MP as c to 1 and P' will be a point on the graph of ,.2 car The upper dotted curve (Fig. 8) is the graph of 2x z , and is obtained by doubling each ordinate of the graph of x z (full curve). It will be noticed that the general character of the two graphs is the same ; the graph of 2x z however recedes more rapidly from X'X than does that of x z and is steeper. In general, the graph of cx z lies above or below that of x z according as c is greater or less than 1. Next let c be negative, say 2. The graph of 2 2 may be got from that of '2x z by reflection in X'X, or by rotating the graph of 2 2 through two right angles about X'X ; for the ordinates of the graph of 2cc 2 are simply those of the graph of 2x 2 with signs changed. The lower dotted curve is the graph of 2a; 2 ; is a turning point of the graph and zero a maximum value of the function 2x 2 , the value being taken algebraically. 19. Scale Units. Let us now consider the graph of ex 2 as the geometrical representation of the law of falling bodies ; c may be taken as 16 when the foot and the second are the units of space and time. The graph shows clearly how rapidly the distance fallen increases with the time, for the curve moves rapidly away from the axis OX ; in this case the part of the curve to the left of OF does not belong to the representation since negative values of x are not considered. But if OU and OF are, as has been supposed, of the same length it will be impossible to represent the connection between the distance fallen and the time of fall, even for values of a; up to 1, within the limits of an ordinary sheet unless U and F are both very small. The remedy is to choose these segments of different lengths. The foot and the second are magnitudes of different kinds and there is no necessity therefore that the segment which represents 1 second should be of the same length as that which repre- sents 1 foot, nor is it implied in the definition of the coordi- nates of a point that U and V should be of the same length. M being the foot of the perpendicular from P on X'X, the coordinates of P are x, y if OM = xOU, MP = yOV SCALE UNITS. 27 and P is definitely determined whether U, V are of the same length or not. In the case of y = IQx 2 we might therefore take OU equal to 1 inch and V equal, say to T Vth of an inch ; an abscissa 1 inch long would therefore represent 1 second while an ordinate 1 inch long would represent 1 6 feet ; an abscissa 2 inches long would represent 2 seconds, an ordinate 2 inches long would represent 32 feet and so on. A similar choice would in other cases bring the graph within manage- able size. But even when the two magnitudes whose connection is represented by a graph are of the same kind it is often advisable to have units of different lengths. The value of the graph will not be thereby impaired ; the purpose of the graph is to show to the eye how one magnitude changes as another with which it is connected changes, and the ratio of the two lines, say MP and NQ, which represent any two values of the first magnitude is independent of the size of the line which represents the unit magnitude. For where OF represents the unit magnitude and yflV, y 2 OV the two values considered. Thus, in a contour road map, if the heights were represented on the same scale as the horizontal distances, it would be difficult to trace the character of the road ; hence the heights are exaggerated by using a much larger unit for the vertical than for the horizontal distances. If the graph is to be used to determine actual heights, the scale of the drawing must of course be given. 20. Coordinate Geometry. Many of the properties of a curve can be most simply investigated by using the equation of the curve ; the study of curves from this point of view is the subject of coordinate geometry. On the one hand the curve may be defined by some geometrical property ; the law of the curve is then expressed in the equation of the curve. Thus the law of the circle is that every point on it is at the same distance from the centre. Now, taking rectangular axes, let be the centre of the circle, c its radius and P (x, y) any point 28 AN ELEMENTARY TREATISE ON THE CALCULUS. on it. Then ( 7) OP 2 is equal to x 2 + y~ ; also OP is equal to c. Hence t - # 2 -f y z = c 2 (1) and this equation is true for the abscissa and the ordinate of every point on the circle but of no other point. As P moves round the circle, x and y change in value, but always the sum of their squares is equal to c 2 . Equation (1) is therefore called the equation of the circle with radius c. On the other hand an equation between x and y defines y as a function of x, and the graph of this function may be plotted point by point ; numerous examples will be found in later articles. As a simple case we might consider the equation y = x z which gives the graph of 16 ; or we might take equation (1). In that case y is defined as a two- valued function of x, y= x /(c 2 a? 2 ), for values of x from x= c to x= +c; clearly if x is numerically greater than c y is imaginary. The graph will be symmetric about the axis X'X, and by considering the inverse function x = ^/(c z y 2 ), we see that the graph is also symmetric about YY. We might then plot points for which x and y are both positive and thus arrive at the form of the graph. The two functions + /v/(c 2 & 2 ) an d ^/(c 2 a; 2 ) are repre- sented respectively by the semicircles above and below the ic-axis. In later sections it will be seen how the geometrical properties of the graphs of the simpler functions can be deduced from the equations (see 26). If in plotting the graph of the function defined by equation (1) the units OU, OF are of different lengths the graph will seem to be not a circle, but an ellipse (Exer. V. 4) ; if V be, say, half of U, each ordinate will be only half the actual length of the ordinate of the circle. So long as OU, 0V are of the same length the shape will not be altered; a change in the size of the units, so long as the units remain of equal length, only enlarges or reduces the figure since all lines are altered in the same proportion. Even in studying the geometrical properties of curves, however, it is often necessary to choose units of different lengths in order to get the curve represented on a sheet of reasonable size; it must then be borne in mind, that the COORDINATE GEOMETRY. 29 graph will only show the ratios and not the actual lengths of the lines whose measures are the numbers taken as the ordinates. In all cases the units should be chosen so as to make the graph as large as possible ; a diminutive graph usually defeats the end of its existence. EXERCISES II. 1. Are the points A(\, 1), 5(J, ), C(-%, J), D(5, 100), E(3, 40) on the curve whose equation is y = 4.1? ? 2. Is the y-axis Y'OY an axis of symmetry for the graph of any of the functions (i) 2.r 2 -3A' 4 ; (ii) 2^ 2 -3^; (iii) a? n ; (iv) o? n +\n integral) ; (v) (#+!)/(#*+!); (vi) l/(.r 2 -t-l); (vii) a + bx* + cx* + dx*? Does the point (1, - 1) lie on any of the graphs ? What must be the value of a if the origin lies on the graph of (vii) ? 3. Trace the graphs of the following functions for values of x between 2 and + 2, and find the turning points of the graphs and the abscissae of the points where the graphs cross the axis of abscissae (i) .r 2 -! ; (ii) 2.^-1 ; (iii) -2s 8 +l ; (iv)ar-ar 2 ; (v) \-x-x z ; (vi) -l + 3x-2x 2 . How may the graphs (i), (ii), (iii) be derived without calculation from the graphs of x 2 , 2.r 2 , 2^- 2 respectively ? How may the graph of (iii) be derived from that of (ii), and the graph of (vi) from that of (iv) ? 4. Having given the graph of the function fix), show how to obtain the roots of the equation f(x) = Q. Illustrate from the graphs of ex. 3. [Let a be the abscissa of any point A on the graph ; by the nature of a graph the ordinate of A is f(a). Hence, if }^a) = 0, A must be on the axis of abscissae ; but if ^a)=0, then a is a root of the equation f(x) = 0. Therefore the roots of the equation f(x) = are the abscissae of the points where the graph of f(x) crosses the axis of abscissae.] 5. Trace the curve whose equation is y=x 5 . To every point P on the curve there corresponds another point P on the curve which is symmetric to P with respect to the origin ($ 6, ex. 3.) ; for if P is the point (a, b\ P' is the point ( - a, - 6), and when b = a 3 then also - b = ( - a) 3 . When, as in this case, the equation is not altered by replacing x and y by x and y respectively, the origin is called a centre of symmetry of the curve. 6. On which of the curves given by the following equations is the origin a centre of symmetry 4 3 (i) i/ = ax 3 + bx 5 ; (ifyy^x 3 ; (iii) y =x* ; (iv) -*i) r a=(ya-yi)/(*s-*i) and therefore the equation of the line through the points (x lt y^), (afc y 2 ) is 2 1 6. Find the equations of the lines through the following pairs of points (i) (1,2), (2,1); (ii)(-l,2),(2, -1); (iii) (0, 0), (1,- 1) ; (iv) (0, 3), (-2, 0). 7. Find the equation of the line with the gradient 2 passing through the point (3, 1). 8. Find the equation of the line with the gradient c passing through the point (a, b). 9. Find the coordinates of the point of intersection of the two lines given by the equations (i) ^+2y = 3; (ii) 3#+y = 4. Since the point of intersection lies on both lines, its coordinates must satisfy both equations (i) and (ii). Solving these as simultaneous Q.C. c 34 AN ELEMENTARY TREATISE ON THE CALCULUS. equations, we get for the required coordinates x= 1, y = 1. Verify the result by means of a diagram. 10. Draw on one diagram the curves whose equations are and find by measurement the coordinates of the points of intersection. Verify by solving the equations as simultaneous equations. 11. Show that the roots of the equation 2x+x 2 -3 = are the abscissae of the points of intersection of the curves of ex. 1 0. 12. Show that the roots of the equation f(x) c are the abscissae of the points of intersection of the curves given by Compare Exer^ II. ex. 4. 23. Rational Functions. An expression of the form z +...+kx n -' L + lx n (1) where the coefficients a, b, c, ... are constants and the indices of the powers of x are all positive integers of which n is the greatest is called a Rational Integral Function of x of degree n. The quotient of two rational integral functions of x is called a Rational Fractional Function of x. It is known from the theory of equations that an expression of the form (1) will in general vanish for n values of x ; hence the graph of the function (1) will in general cross the ic-axis n times. (See Exer. II. ex. 4). Some of the values of x for which (1) vanishes may how- ever be imaginary and for such values of the abscissa there are no real points on the axis so that the graph may not have as many as n crossings. When two of the values of x for which (1) vanishes are equal, the student will find that the graph touches the aj-axis at the corresponding point. Graphs of the even powers. The graphs of the even powers of x, x z , x* ... are all of the same general character ; they touch the #-axis at and have the y-axis as an axis of symmetry. The greater the index however, the slower does the graph recede from the cc-axis near the origin ; on the other hand, the greater the index the more rapidly does the graphic point move upwards when x is greater than 1. The general shape of the graphs of ax 2 , ax*, . . . GRAPHS OF POWERS. INFLEXION. 35 can be seen by dividing the corresponding ordinates of the graphs of x 2 , x* ... in the ratio of a to 1 , as in 18. Graphs of the Odd Powers. The graphs of the odd powers higher than the first, 3?, of, ... touch the #-axis at the origin but they do not have the y-axis as an axis of symmetry. For these the origin is a centre of symmetry. (Exer. II. 5). For positive values of x the graphs resemble those of the even powers ; near the origin the graph of x* is flatter than that of x 2 , not so flat as that of a; 4 , while for values of x greater than 1 the graph of x 3 lies above that of x 2 , below that of * 4 . To construct the graph of x 3 for negative values of x, take a .point P on the graph of the posi- tive values of x, produce PO backwards its own length to P', and P will be the point on the graph symmetric to P (Fig. 10). The same construction holds for any curve that has the origin for a centre of symmetry. The graphs of the odd powers thus both touch and cross the cc-axis at 0, bending away from the axis in opposite directions on opposite sides of (Fig. 10). DEFINITION. A point such as where the curve crosses its tangent and bends away from it in opposite directions on opposite sides is called a Point of Inflexion, and the tangent at the point is called an Inflexional Tangent. The student should plot on the same diagram for values of x between 1 and +1, using a pretty large unit, the graphs of x 2 , x 3 , x*, x & . He will gain useful ideas of the relative magnitude of the powers of x when x is a proper fraction. He will also be able to deduce the general course of the graph of such a function as x% for values of x between and 1 ; the graph will lie below that of x 2 , but above that of x 3 . If x be negative x* is imaginary, and there is no part of the graph to the left of the y-axis. 36 AN ELEMENTARY TREATISE ON THE CALCULUS. In the same way by plotting the graphs of the same functions for values of x between 1 and 3, using a small unit, he will see how rapidly the higher powers of x increase when x is greater than 1. He can readily verify the important principle that the term of highest degree in a rational integral function will for sufficiently large values of x be numerically greater than the sum of all the other terms, and will therefore determine the sign of the function for large values of x. The construction of the graph of the general rational integral function is usually laborious ; when the student is able to differentiate a function he will find that the labour may be considerably reduced. As an example take the function f(x), where Write /(a) = ^1 _-+. \ X 2 CC 3 / Now, if x is numerically equal to or greater than 2 the expression within the bracket will be positive, as a little consideration shows. Hence if x is positive and equal to or greater than 2, f(x) will be positive ; if x is negative and numerically equal to or greater than 2, f(x) will be negative, since x 3 will be negative and the expression within the bracket positive. The graph must therefore cross the -axis once at least between the points on that axis at which x is 2 and 2 respectively. Examining further, we find /(-2)=-l; /(-!)= +3; /(!)=-!; /(2)=+3, and therefore the graph must cross thrice, namely, between the points 2 and 1, 1 and 1, 1 and 2; since the equation is of the third degree, the graph cannot cross more than thrice. There will thus be two turning points. Again, /(-1'9)= -'159, /(-l-8)=+-568, so that the graph crosses between 1*9 and 1'8. GRAPHS OF RATIONAL FUNCTIONS. 37 When x = -l'88, /() =-'005, so that the graph crosses very nearly where x= 1'88, and this value is an approximate root of the equation In the same way it may be found that the other two roots are approximately '35 and 1'58. The turning points occur where x= 1 and x = l, and the calculation of a few values of f(x) shows that the graph is of the form shown in Fig. 11. 24. Asymptotes. The simplest example of a rational fractional function is I/x. When x is small and positive, l/x is large and positive, and as x tends towards zero l/x becomes extremely large or, in the usual language, l/x tends toward infinity; thus when, x takes the values *1, '01, '001, ... \/x takes the values 10, 100, 1000, ... respectively. Hence as the point x moves from the right toward till it all but coincides with the graphic point moves upward and recedes to a very great distance from the o>axis while approaching very close to the y-axis ; when x is zero, that is when the point x coincides with 0, the graphic point may be said to be at infinity. In this case the graph is said to approach the 38 AN ELEMENTARY TREATISE ON THE CALCULUS. positive end of the 7/-axis asymptotically, or to have the ?/-axis as an asymptote. In the same way it may be seen that when x is very large and positive 1/x is very small and positive; the graph approaches the positive end of the cc-axis asymp- totically. The graph is obviously symmetrical with respect to the origin, and approaches both ends of both coordinate axes asymptotically (Fig. 12). DEFINITION. In general, when a curve has a branch extending to infinity, the branch is said to approach a straight line asymptotically, or to have the straight line for an asymptote, if as a point moves off to infinity along the branch the distance from the point to the straight line tends towards zero as a limit, that 'is, if as the point moves off to infinity the distance becomes and remains less than any given length. If x a be a factor of the denominator of a rational fractional function of x in its lowest terms, the function will tend towards infinity as x tends towards a and the line whose equation is x a will be an asymptote. If as x tends towards infinity the function y tends to a finite value ASYMPTOTES. 39 /3 then y = j3 will be the equation of an asymptote. These asymptotes are parallel to or coincident with the coordinate axes, as in the example just considered ; but there may be asymptotes that are not parallel to either axis, as in the following example : Here we may write If we denote by y the ordinate of the graph and by y z the corresponding ordinate of the straight line whose equa- tion is y = x-t-I, we see that Hence whether x be positive or negative y l is greater than ;y 2 and therefore the graph of the function is always above the straight line. Again when x is numerically very large I/a; 2 is very small, and the difference between y 1 and 2/ 2 will as the point x moves either to the extreme right or to the extreme left of the a;-axis become less than any given fraction ; hence the graph approaches both ends of the line whose equation is y = x + 1 asymptotically. The y-axis is also an asymptote ; y is positive when x is either a small positive or a small negative number and therefore the graph does not approach the negative end of the y-axis but it approaches the positive end both from the right and from the left. The graph will cross the oj-axis for those values of x which make the numerator x s + x 2 -\-~L zero; a few trials will show that the numerator vanishes only once, namely when x = T47 approximately. When x is algebraically less than T47, y is negative ; for all other values of x the ordinate is positive. When x=I,y = 3; when x = 2, y 3, and there is a turning point when x=l'3 approximately. 40 AN ELEMENTARY TREATISE ON THE CALCULUS. The graph is shown in Fig. 13. The unit for the abscissae is double that for the ordinates ; if the units were equal the portion ABC would be at a considerable distance above X'X and the diagram would have to be very large X' -a -v^^ 1 Y' FIG. 13. to show that part clearly. The curve approaches the asymptote GH very rapidly but the asymptote Y more slowly. In plotting the graph of a fractional function it will be frequently found convenient to split the function up into partial fractions as has been done above. Thus, if we can write -*r+-5- and we see that there are three asymptotes whose equations are v = l x=\ x=2 In this case the graph crosses the horizontal asymptote at the point whose abscissa is , because when i/ = l we have 1=7 For the equation #=, we should have * EXERCISES IV. 41 and there would again be three asymptotes, two of which are parallel to the y-axis while the third has for equation y=x+3, and this third asymptote cuts the graph again at the point whose abscissa is ^. EXERCISES IV. Graph the functions 1 6 : 1.3^-5^-1; 2.^ + 2^ + 2; 3. a?-x; , _ 2#-3 0< ; < ' x-l 7. Show that the roots of the equation x 3 - cue -6 = are the abscissae of the points of intersection of the graphs of 3? and of ax+b. 8. Find to two decimals the roots of the equations (1)^-7^ + 3=0; (ii) a?-*Jx+9 = 0. Graph the functions. 9. If f(x)=x*- 4x?- 4.^ + 1 6#+l, show that the equation fix) =0 has four real roots, and find these to two decimals. [Find the values off(x) for x equal to 2, 1,0, 3, 4 respectively. The ordinate f( - 2) is positive and the ordinate f( l) negative, so that the graph crosses the axis of abscissae between the point 2 and the point 1. Proceed in the same way with the other numbers.] 10. A point is moving in a plane and at time t seconds reckoned from a fixed instant, its coordinates with respect to two rectangular axes in the plane are x and y feet. Construct the path of the point in the following cases : (i) #--=* + !, ?/=2* ; (ii) x=a + bt, y = c+dt ; (iii) x = 2t, y = 8t* ; (i v; x = t, y = t 3 . [The position of the point at any instant may be found by calculating the values of x and y for the value of t at that instant ; having found the position of the point for a number of values of f, the graph can be drawn in the usual way. Or, the equation of the path may be found by eliminating t. Thus in (i) t may be considered a function of x, namely t=x 1 ; but y is always 2, and therefore y and x are always connected by the equation y 2(# 1). In this case therefore the path is a straight line. In (ii) the path is also a straight line. The equations of the paths in (iii), (iv) are y = 2^ 2 , y=3?. This method of representing the path of a point by means of two equations is of frequent occurrence both in Geometry and in Mechanics.] 42 AN ELEMENTARY TREATISE ON THE CALCULUS. 11. The angle 6 between the two straight lines whose equations are (i) y=mx + c, (ii) y = m'x + c' may be found from the equation a_ m m! l+mm' [Let (i) make the angle a, (ii) the angle ft with X'OX ; suppose a > ft, then 6 = a ft and n_ tan a -tan ft _ m m' 1 +tan a tan ft I +mm' If the numerical value of (m m') /(l+mm') be taken, the acute angle between the lines will be obtained whether a > ft or a < ft] 12. The angle between the lines given by ax+by + c=0 and a'x+b'y + c'=Q is given by tan 6 (ab 1 a'b)/(aa' + bb'). 13. Show that the lines of ex. 12 are (i) parallel if a/b = a'/b', (ii) perpendicular if aa' + bb'= 0, CHAPTER III. GRAPHS. ALGEBRAIC AND TRANSCENDENTAL FUNCTIONS. CONIC SECTIONS. 25. Algebraic Functions, y is called an Algebraic Function of x when it is determined by an equation of the form Ay+By n - l +...+Ky + L = Q, in which the indices of the powers of y are positive integers and the coefficients A , B, . . . K, L, are rational integral functions of x. Manifestly, rational functions are special cases of algebraic functions. y will usually be multiple-valued and its graphical repre- sentation is much more difficult than that of the rational function except in particular cases of which the following are of special importance l : i Type I. y n x = or y = x n . When n is an even integer, x must be positive and y will be two-valued ; when n is an odd integer, x may have any i value and y will be single-valued. The graph of x n is readily found from that of x n . Let QOP (Figs. 14, 15) be the graph of x n , and let PN i be perpendicular to FT; then ON=NP n , or NP=ON n . Hence if OF be taken as the axis of abscissae, that is, as the axis of the argument, and OX as the axis of ordi- 1 The beginner may find this article somewhat difficult ; he should work out the simple examples of the various cases that are set down at the end of the article and the discussion will become more definite. He need not however spend much time on this article at a first reading of the subject. 44 AN ELEMENTARY TREATISE ON THE CALCULUS. nates, that is, as the axis of the function, the curve QOP !_ will be the graph of the function ON n . It is desirable however to have OX as the axis of abscissae and Y as the axis of ordinates, that is, the figure has to be turned so that OF becomes horizontal and coincides with the present position of OX, while OX becomes vertical and coincides with the present position of OF. The simplest p.. X' X X' Y FIG. 14. way of securing this is to suppose the whole figure rotated through two right angles about the bisector BOA of the angle XOY as axis; NAP will thus come into the position N'AP, and QOP will come into the position Q'OP'. Q'OP' i i will be the graph of x n , because N'P'=ON' n , since N'P' = NP and ON'=ON. Fig. 14 is the graph when n is even and when, therefore, for one value of x there are two values of y ; on the other hand, when n is odd, as shown in Fig. 15, to one value of x there is but one value of y. Construction of Graph of an Inverse Function. The same transformation gives the graph of the function inverse to a given function. If y = x n and if x be taken as the argu- ment, QOP is the graph of x n ; the function inverse to y is i x where x = y n , and, when y is taken as the argument, QOP GRAPHS OF ALGEBRAIC FUNCTIONS. 45 !_ is the graph of y n , that is, of the function inverse to y or x n . It is convenient however to represent the argument in all cases by lines measured along X'OX and to denote the argument of the inverse function by the same letter as is used for the argument of the original function ; that is when the inverse function has been formed we then replace y by x and x by y, and the graph of the inverse function, when this replacement has' been made, will be the original graph rotated through two right angles about the bisector of the angle XOY. In this notation the graph of x n is QOP; the inverse i i function, which as first stated is y n , is now x n , and its graph is Q'OP. Again, when the graph of a function has been constructed, we see how to choose the range of the variables so that the inverse function may be single-valued. When n is even OF is the graph of +x" and OQ' that of -a"; that is, OP' and OQ' are the two branches of the two-valued function inverse to x n when n is even. Type II. y n x m = where m, n are unequal and not both even. If m, n were both even the equation would be equivalent n m n m to the two equations y 2 x* = 0, y*+x* = Q, and there would therefore be two graphs, each of which would come under one of the following groups. The student should notice the remark in 23 about the graph of such a function as x% ; it will be found useful in the discussion of the groups contained in the general equation. JYI (A) m > n ; y = x n where is an improper fraction. f fb (A X ) m, n both odd. The graph is of the form QOP (Fig. 15) ; is a point of inflexion and X'OX a tangent at 0. (A 2 ) m even, n odd. The graph is of the form QOP (Fig. 14) ; Y is an axis of symmetry and X'OX a tangent at 0. 46 AN ELEMENTARY TREATISE ON THE CALCULUS. (A 3 ) m odd, n even, y is imaginary when x is negative ; OX is an axis of symmetry, and both branches touch OX (and each other) at 0. (Fig. 16.) DEFINITION. A point on a curve such as 0, at which two branches OA, OB have the same tangent, but beyond which they do not pass, is called a Cusp. It must be observed that neither branch passes beyond ; a point moving from A along the curve to reverses its direction in order to proceed along the other branch OB. X' o FIG. 16. (B) m < n ; y = x n where is a proper fraction. n m, n both odd. The graph is of the form (Fig. 15); is a point of inflexion and Y'OY a tangent at 0. (B 2 ) m odd, n even, y is imaginary when x is negative ; OX is an axis of symmetry and Y'OY a tangent at 0. The graph is of the form Q'OF. (Fig. 14.) (B S ) m even, n odd. OF is an axis of symmetry and is a tangent at ; is a cusp. ("Fig. 17.) Thus if 971 = 2, n = 5, since -f lies between f or ^ and or ^, the graph of x* will, when x is positive, lie between those of x* and x$, each of which has the form OP' (Fig. 15). The branch OB is present because OF is an axis of symmetry. The student will have no difficulty in deducing the graphs when the equation is y n + x = ; they are deduced CONIC SECTIONS. 47 from those of y u x m = by rotation about one of the coordinate axes. Thus the graphs corresponding to (AJ) and (fi t ) are obtained by rotation about X'X. More gene- rally, the graphs of y n ax m = can be deduced by dividing the ordinates of y n x m = in the ratio of a n to 1. Ex. 1. Draw the graphs of the following cases of Type I. : (i) y'*=x ; (ii) y*=x ; (iii) # 2 = -x ; (iv) /= -x. Ex. 2. Draw the graphs of the following cases of Type II. (A) : (i) y 3 =x 5 ; (ii) y 3 =x t ; (iii) y*=o? ; (iv)y3=-^; . (v)^=-^; (v\)y*=-a?. Ex. 3. Draw the graphs of the following cases of Type II. (B) : (i) /=a^ ; (ii) y* = x* ; (iii) y*=x 3 ; (iv) y f '= -x 3 ; (v^ /= -x 2 ; (vi) y*= -x 3 . Ex. 4. Draw the graphs of (i) y z = Qx 3 ; (ii) y> = - 9a? ; (iii) y 3 = 27^. Ex. 5. Graph the functions (i) 1 ; - (ii)4_; (iii) 1. X z X s X$ 26. Conic Sections. For the sake of readers unfamiliar with the conic sections we give in this article the equations- of the conic sections and define the most frequently occur- ring technical terms connected with them. DEFINITION. A conic section is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line. The fixed point is called the focus, the constant ratio the eccentricity, and the fixed line the directrix. Let S (Fig. 18) be the focus, KN the directrix and SK perpendicular to KN. Let e be the eccentricity and on KS take A so that AS = eKA ; then A is a point on the conic. As axes of coordinates take KAS and the perpendicular through A to KAS. Let P be any point (x, y} on the 48 AN ELEMENTARY TREATISE ON THE CALCULUS. conic and draw PM perpendicular to KS; then x = AM, = MP. Let KA p ; then AS=ep. Now But SP = eNP by the definition of the conic : hence FIG. is. or so that, inserting the values of SM, MP, NP, we get (x - epf + y z = e\x + pf, or after reduction (l-e z )x*-2e(I+e)px+y z = () ................... (1). Every point whose coordinates satisfy equation (1) will be a point on the conic section ; for different values of the constants e, p there will be different conies. Evidently AS is an axis of symmetry. If AK were taken as the positive direction of the axis of abscissae, then in equation (1) we should have +2e(l +e)px, for the change in the direction of the axis is equivalent to writing x in place of x. Special Forms of the Conic Section. I. If e = l, the conic is called a parabola. In this case equation ( 1 ) reduces to y z = 4 \^) - 7 "> 6 2 o To a 2 o 2 and these may be considered the standard forms. From these equations we see that both curves are symmetrical about both axes. The origin C is a centre of symmetry; C is called the centre of the conies, and the ellipse and the hyperbola are called central conies. The parabola has no centre. The axis of ordinates meets the ellipse (Fig. 20) at two CE^ r TRAL CONICS. 51 points B, R ; BR is called the minor axis. From the equation a^/a 2 + y z /b* = I it is easy to see that x is never numerically greater than a nor y greater than 6. The ellipse is therefore a closed curve. The circle is the particular case of the ellipse in which b = a and e = 0. The axis of ordinates does not meet the hyperbola because when x = 0, y 2 = 6 2 and therefore y is imaginary. It will be seen further that y is imaginary if x is numeri- cally less than a, so that no part of the hyperbola lies between the lines through A, A' perpendicular to A A'. FIG. 21. The curve consists of two branches extending to infinity to the right of A and to the left of A' respectively. It will be a good exercise for the student to prove that the lines E'E, F'F whose equations are y = bx/a, y= bx/a, are asymptotes (Fig. 21). If b = a the hyperbola is said to be equilateral ; since the asymptotes are in that case at right angles the hyperbola is also said to be rectangular. From the symmetry of the central conies about the axis of ordinates through C it may be inferred that they have a second focus S' and a second directrix K'N' symmetrical to S and KN with respect to C; the curves might be con- 52 AN ELEMENTARY TREATISE ON THE CALCULUS. structed from S' and K'N' in the same way as from 8 and KN, the eccentricity being the same. Some useful properties of the Conic Sections will be found in Exercises V., VI. 27. Change of Origin and of Axes. The device of chang- ing the origin of coordinates is often useful in simplifying the equation of a curve and thus making the construction of the curve more simple. I. New Axes parallel to Old Axes. In Fig. 22 let B be the new origin, and let X\BX^ Y\BY l be parallel to X'OX, Y'OY respectively. Y Y, 3 x; B M' x, X' A M X Y,' Y' FIG. 22. OM= OA+AM= OA +BM' ; Let (a, 6), (x, y} be the coordinates of B and of any other point P with respect to the old axes X'OX, Y'OY; and let (x', y'} be the coordinates of P with respect to the new axes X\BX V Y\BY^. Then and and therefore x = a + x; y = b + y' ...................... (1) Conversely x' = x a; y' = y b ...................... (I/) When x and y have been replaced by a + x and l> + y' the accents may be dropped, it being remembered that the origin is then 5, so that x will mean not OM but BM' , and y not MP but M 'P, CHANGE OF AXES. 53 EXAMPLE. The equation y 2 4r - 2y 1 =0 may be written (y- l)* = 4(#+i). Put ^ + ^ = x', that is, .r = i + y and y l=y', that is, # = 1 +y', which means transferring the origin to the point ( , 1), and the equation becomes w' 2 = 4.r'. This equation, and therefore also the given one, represents a parabola with its vertex at the new origin and with the new axis of abscissae as its axis. The latus rectum is 4 ; the focus is the point (1, 0) with respect to the new axes, and therefore the point (, 1) with respect to the old because the coordinates of any point with respect to the old axes are equal to those with respect to the new increased by the coordinates of the new origin. II. The origin not changed, but the New Axes obtained by turning the Old Axes through a positive or negative angle 0. In Fig. 22a let P be the point (x, y) when referred to the old axes X'X, Y'Y, and the point (x, T/') when referred to the new axes X\X V Y\Y V so that x=OM, y = MP; x'=OM', y' = M'P; 'M' By elementary trigono- Y ' * * Vlf fir, metry, FlG - Wa " OM= OM' cos9-M'P8in9; MP = OM' sin 9 + M'P cos 9 ; that is, x = x' cos 9 y' sin 9 ; y = x ' si n 6~\~y' cos (2) Conversely, solving for x' and y' in terms of x and y, x' = xcos9+ysin9; y'= xsin9+ycos9 (2') It may be possible to choose 9, so that the new equation is simpler than the old or even is an equation of which the graph is known. EXAMPLE. By turning the axes through 45 the equation xy=c 1 becomes yf _^ j +y > Z . JJt = />2 or r '2 _ ,/2 _ 0,,2 / ^/2 * V 4C , since, by (2), 54 AN ELEMENTARY TREATISE ON THE CALCULUS. The new form shows that the curve is a rectangular hyperbola ; half the transverse axis, denoted in 26 by a, is >J^c. Hence the graph of c^lx is a rectangular hyperbola referred to its asymptotes as coordinate axes. III. The origin changed to (a, b) and the axes turned through an angle 0. Combining cases I., II. we get the more general transformation x = a + x' cos 6 y' sin ; y b + x sin 6 + y' cos (3) x' = (x a) cos 9 + (y b) sin 6 ; y'= (x a) sin + (y b) cos 6 (3') EXERCISES V. Unless otherwise stated the equations of the conic sections in this set of Exercises are supposed to be in the standard forms (P\ (C) of 26. 1. In the central conies prove CS=eCA^ CA=eCK. For the ellipse, AS : A K= e = A 'S : A'K, and therefore e=A'S-AS : A'K- AK=S'S : A'A = CS : CA, e = A'S+AS:A'K+AK=A'A : K'K=CA : CK. For the hyperbola, A'S - AS : A'K- AK= CA : CK, A'S+AS : A'K+AK=CS : CA. 2. In Fig. 20, S is the point ( - ea, 0), & the point (ea, 0). In Fig. 21, S is the point (ea, 0), S' the point ( ea, 0). In Fig. 19, S is the point (p, 0). 3. Show that the latus rectum (or parameter) of a central conic is 26 2 /a, 4. On A A' (Fig. 20) as diameter a circle is described; if MP is produced to meet the circle at Q, show that MP : MQ=b : a = constant. For MQ Z = CA*- CM 2 = a 2 - # 2 ; J/P 2 = - 2 (a 2 - ^ 2 ). The circle is called the Auxiliary Circle of the ellipse. The theorem shows that if the ordinate MQ of a circle to any diameter is divided internally at P, so that MP : MQ constant, the locus of /' is an ellipse whose major axis is the diameter of the circle. The student may prove that if P is in MQ produced, the locus is an ellipse whose minor axis is the diameter of the circle. EXERCISES V. 55 5. Show that the point (a cos 0, b sin 0) lies on the ellipse, whatever be the value of 6. For the equation of the ellipse is satisfied by #=acos 0, y = bsm 0. As 6 varies from to 360 the point travels round the ellipse. In the notation of ex. 4, if P is the point (a cos 0, b sin 0) is the angle A'CQ and is called the Eccentric Angle of P. 6. Show that the point (pt 2 , 2pt), p being a constant, lies on a parabola whatever be the value of t. 1. In Fig. 20, if CM=x, prove that SP=a+ex, S'P=a-ex, SP + S'P=2a. For SP=eNP=eKC+eCM= &P=e. PN' = e. CK' - SP, S'P are called the focal distances of P, and therefore in the ellipse the sum of the focal distances is constant, the constant being the major axis. 8. In Fig. 21, if C3f=x, prove SP=ex-a, SP=ex + a, S'P-SP=2a. Hence the difference of the focal distances of a point on a hyperbola is constant. 9. In the parabola (Fig. 19) prove SP= KA +A Af= AS+ A M=p +x, x being the abscissa of P. 10. On any of the conies (Figs. 19, 20, 21) a point Q is taken and the chord PQ (produced if necessary) meets the directrix KN at Z. Prove that SZ bisects the exterior angle PSQ, except when P and Q are on different branches of the hyperbola when SZ bisects the interior angle. Draw QR perpendicular to KN; then SP:PN=e = SQ:QR, therefore SP : SQ = PN : QR = PZ : QZ, and the theorem follows by Euc. vi. 3 or A. 11. Trace the conies given by the equations, (i) afi + 4y*=4-, (ii) 2*-3? 8 = 6, and find the eccentricity of each. In (i) a 2 = 4, 6 2 = 1, and 6 2 = a 2 (l-l. As x increases from N to +N, where N is a large positive number, a x will increase from a very small positive number a~ N through 1, the value of a x when x = 0, to a very large positive number a +N . (ii) a = l. In this case a x is always 1. (iii) a y * T^yj+dJ, where t is a variable, say the time. Show that the point describes the ellipse given by the equation -s - 2-4 cos 27ra +'4 = sin 2 27ra. a 2 ab b 2 11. The coordinates of a point are given by x = acos(2Trt/T), y = 6cos(47r x l the increment is positive, so that x has increased algebraically ; if a;, < x l the increment is negative, so that t'las decreased algebraically. In both cases the one word G.C. E 66 AN ELEMENTARY TREATISE ON THE CALCULUS. "increment" is used, so that a negative increment is an algebraic decrease. Since x 2 x l = Sx^ we have x z = x l -\- Sx v so that if x change from the value x l to another value and if the increment that x takes is Sx l that other value is x l + Sx l ; the student must accustom himself to this method of denoting the value to which x changes, for although x 1 + Sx^_ seems more cumbrous than x 2 its form is more suggestive and is really simpler in many investigations. Let y be a function of x, say 5x 3, and let x v y l be corresponding values of x and y. When x changes from x 1 to x l + Sx l let y take the increment Sy v so that the value of y corresponding to # t + Sx l is y l + Sy l ; then and therefore Sy l = 58x v If 2/ = 3# 2 + 7a: 2, we find, using the same notation, y 1 = S^ 2 + 7x l - 2 ; ^ + Sy 1 = 3^ + SxJ* + 7(x : + SxJ - 2, and therefore, by subtracting the left side of the first equation from the left side of the second, and the right side of the first from the right side of the second, In general, if y=f(x), we have <^i =/(*i + &i) -M) = M). The same notation is used whatever letters denote the variables, so that if s = (t), and so on. As this process of finding increments is of constant occurrence the student should make himself quite familiar with it. The following examples should be worked through. Ex. 1. If y=l show that Sy l= - 2 < f' l8 f' +( .f 'f. x* (x l Ex. 2. If frx)=x 3 -l, show that INCREMENTS. UNIFORM RATE. 67 Ex. 3. If y = log x, show that Ex. 4. If y=.r 3 , &/ 1 = (^ 1 + &r 1 ) 3 -ar 1 3 , calculate 8y l and S^/Sa^ when x=lO, 8^ = 1, -5, -1, -01, -001. Ex. 5. If y=sin^, show that 8y l = 8 sin x l = sin(^j + TJ) - sin a?j . From the Tables calculate 8^ and *-J for the following values O^Ti of #! and &r l5 the numbers denoting the value of the angles in degrees : (i)A = 30, &r 1 = l, "5, "2, -1; (ii) ^ = 60, &rj = l, -5, '2, '1. Ex. 6. If y = log 10 .r, find from the Tables the values of 8y, and jg-* OCFj (i) when ^ = 325 and &C! = 2, 1, '5, '1 ; (ii) when ^ = 72 and ^ = 2, 1, -1, '01. 33. Uniform Variation. When the argument of a func- tion takes a series of values x v x z , x z , x 4 ... the function takes a corresponding series of values y v y z , y s , y. When the increment of the function is in a constant ratio to the corresponding increment of the argument the function is said to vary uniformly or at a constant rate with respect to its argument. If the constant ratio is a, then _ . - U/ , < ' . j ^^ \ *^i ^^ 3 and y z -yi = a ( x z- x i)'^ 2/4-2/3 = ct (*4- a; 3)- If the increments (o; 2 o^) and (x^ x 3 ) of the argument are equal so are the corresponding increments (y z y\) an d (2/4 2/3) of the function. The increment (x z x^) may be either positive or negative and may be of any magnitude whatever; the corresponding increment of the function is a(x 2 x l ), and always, when the argument takes two equal increments so does the function. It follows from the definition that the uniformly varying function is a linear function of its argument. For when the argument changes from any value x l to any other 68 AN ELEMENTARY TREATISE ON THE CALCULUS. value x, let the function change from y l to y; then the increment of the argument is (xx-j), the increment of the function is and that is, 2/ = ax + 2/! ax t . But a^, 2^ are fixed values of the argument and function and the ratio a is constant, so that y is a linear function of x. It is easy to see conversely that if y is a linear function, ax + b say, of x, then y varies uniformly with respect to x. Measure of a Uniform Rate. The constant ratio a is taken as the measure of the rate at which the function varies with respect to its argument. Instead of saying that the ratio a measures the rate we shall generally use the briefer expression that a is the rate. When a is a positive number, y increases as x increases and decreases as x decreases ; when a is a negative number, y decreases as x increases and increases as x decreases. The particular case in which the function reduces to a constant, y = b, may be included in the general category of uniformly varying functions by saying that the function varies at the rate zero ; a = 0. Since the graph of ax + b is a straight line with the gradient a ( 22) the gradient of the line measures the rate at which the function varies with respect to its argument. It should be noticed that if in plotting the graph the unit for the ordinates is not of the same length as the unit for the abscissae the tangent of the angle shown on the diagram will not be equal to the rate a ; if the unit for abscissae is 1 inch and for ordinates, say '1 inch, then to an increment 1 of the abscissa the diagram will show an increment, not of a but of 'la of the ordinate, so that the real gradient or rate will be found by multiplying by 10 the tangent of the angle shown on the diagram. 34. Dimensions of Magnitudes. It is customary and convenient to use such expressions as "the area of a rectangle is the product of its base and its altitude," " the speed of a body which moves uniformly is the distance DIMENSIONS OF MAGNITUDES. 69 gone in a given time divided by the time," and these expressions are represented in the form of equations : ... , -. distance area = base x altitude ; speed = r-. . time When considered as equations in the sense commonly understood in algebra these must be interpreted as " the number of square feet (or square inches, etc.) in the area is equal to the product of the number of linear feet (or inches, etc.) in the base and in the altitude," " the number of units of speed is equal to the quotient of the number of units of length in the distance by the number of units of time." But the equations may be interpreted in a different manner. Let capital letters denote, not numbers but magnitudes ; L the straight line of unit length, T the interval of time taken as the unit. Taking as unit of area the square on the line L, and as unit of speed that of a body which moves uniformly a distance L in time T, the equations may be stated for the unit magnitudes in the form unit area = LxL; unit speed = -~, ', or, combining the symbols by the algebraic laws of indices, unit area = L z ', unit speed = LT~ \ These equations are usually called dimensional equations, and the indices are said to give the dimensions of the magnitudes ; thus the first equation states that the unit area is of 2 dimensions in L, the unit of length, and the second states that the unit of speed is of dimension 1 in L and of dimension 1 in T. Since all areas are magnitudes of the same kind as the unit area, area is said to be of 2 dimensions as to length and to have Z 2 as its dimensional formula. Similarly, the dimensional formula of speed is LT-\ If M denote the unit mass the dimensional formula of momentum will be MLT~ l , because momentum is the product of mass and velocity. It may happen that a magnitude has zero dimensions; thus angles when measured in radians have zero dimensions, 70 AN ELEMENTARY TREATISE ON THE CALCULUS. because the radian is " arc divided by radius," and its dimensional formula therefore is L/L, that is L. A notation that suggests the dimensional formula is sometimes used ; thus an area of 10 square feet is denoted by 10 ft. 2 , a speed of 10 feet per second by 10 ft./sec., a pressure of 14 pounds per square inch by 14 lb./in. 2 and so on. The characteristic word for expressing a rate, namely per, is represented by the symbol of division. When a function varies uniformly the number which has been denned as the rate of variation is quite inde- pendent of the magnitude of the increment which the argument takes; it is therefore possible to choose at pleasure the increment of the argument that shall be called unit increment. Thus we may speak of a speed of 30 miles per hour, although the motion may only last 5 minutes, or 1 minute or less; a rate of 30 miles per hour is the same thing as one of half a mile per minute, or of 44 feet per second. It is important to bear in mind this aspect of a rate when discussing non-uniform variation. Again, the statement that the speed of a moving body is 30 miles per hour is equivalent to the statement that the distance travelled varies with respect to the time at the rate 30, when it is understood that the units are the mile and the hour. The latter mode of expression is more simple in many cases. When the measure of a magnitude is interpreted as a rate the dimensional formula for the magnitude will be the quotient of the formula for the function by that for the argument. Thus force may be measured as the rate of change of momentum with respect to time ; its dimensional formula is therefore MLT~ l /T or MLT-\ It is important to bear in mind that the measure of one magnitude can often be interpreted as the rate of change of a second magnitude with respect to a third, because it is through this connection that the calculus is applied to the investigation of the numerical relations of magnitudes, and in all such interpretations the theory of dimensions is of great service. For a full treatment of that theory the student is referred to the books named below. 1 1 Everett's Units and Physical, Constants ; Gray's Absolute Measurements in Electricity and Magnetism ; Maclean's Physical Units. AVERAGE RATE. 71 35. Variable Bates. So far, only uniformly varying functions have been discussed. But it may happen that the increment of the function is not in a constant ratio to the corresponding increment of its argument, or in other words, that two equal increments of the argument do not always produce two equal increments of the function; in that case the function is said to vary non-uniformly , or at a variable rate, with respect to its argument. Let y = '3x z ; when x varies from x l to x l + h let y vary from y l to y^k, and when x varies from x 2 to x 2 + h let y vary from y 2 to y 2 + k'. Then therefore k/h = Qx l + 3/i ; and in the same way we find The two ratios k/h, k'/h are therefore unequal, so that y varies non-uniformly with respect to x. In this case the ratio k/h depends both on h and on x l ; the characteristic property of a uniformly varying function is that the ratio k/h depends neither on h nor on the value x l of x, from which the increment begins. To obtain the number which is taken as the measure of a variable rate we proceed as follows. 36. Average Bate. We first define an average rate, thus : The average rate at which a function varies with respect to its argument while that argument takes a given increment h is defined to be that uniform rate which would give the actual increment k taken by the function. The average rate is thus k/h. In the example of last article the average rate at which y varies with respect to x while x varies from x l to x l + h is the average rate at which y varies while x varies from x 2 to a:, + A is The average rate thus depends both on x and on h. Next, it agrees with our ordinary notions of a rate of change to suppose that the smaller h is the better will the average rate measure the rate at which the function varies 72 AN ELEMENTARY TREATISE ON THE CALCULUS. as x varies from x l to x^ + h. But as h is taken less and less the average rate 6cc 1 -f 3/i approximates more and more closely to the definite number 6x r The average rate will never be exactly 6x v because it would be absurd to suppose h actually zero ; that would amount to supposing that x had not changed from the value x l at all. On the other hand, however small h may be, provided it is not zero, the quotient k/h can be calculated and the average rate for that small increment determined. We may therefore suppose h to be, not zero, but so small that the difference between 6x t + 3h and 6x v namely 3h, shall be less than any fraction that may be named, however small that fraction may be, provided only it is not zero ; for example, the difference will be less than "001 if h be numerically less than one third of '001, say less than '0003. It is natural therefore to consider Qx i as measuring the rate at which y changes with respect to x as x increases or decreases from the value x v We therefore define Qx l as the rate at which the function y or 3# 2 varies with respect to its argument x for the value x l of the argument. In the same way 6x 2 is, by definition, the rate of change for the value x 2 and in general for any value a of the argument the rate is 6a, because the reasoning does not depend on the particular value x l ; the reasoning is the same whatever value of the argument be chosen. When x has the values 0, \, |, f, 1, f, 2 ...the rate is equal to 0, f , 3, f , 6, 9, 12 . . . respectively ; thus for the value 1 of x, y is increasing twice as fast as for the value J, for the value f thrice as fast, for the value 2 four times as fast and so on. The student should compare these statements with the information to be derived from an inspection of the graph of 3cc 2 . When x is positive the rate is positive, so that as the point x moves to the right the graphic point moves up ; on the other hand, when x is negative the rate is negative, so that as the point x moves to the right the graphic point moves down. It will be noticed that in stating a variable rate the phrase " for the value a^ of the argument " occurs ; the \ VARIABLE RATE. 73 phrase is needed because, unlike that of a uniformly vary- ing function, the rate is in this case itself a variable. If the number s of feet described by a moving body in t seconds be o 2 , the rate at which s varies with respect to t at time j seconds after motion begins is 6^, that is, the speed at time j is 6^ feet per second. 37. Measure of a Variable Rate. The method just given of defining a variable rate is of fundamental importance, and the student should make sure that he masters the reasoning on which the definition is based. The process consists of three steps : (i) We find the average rate k/h: the number kfh depends both on x l and on h. (ii) We assume as consistent with our notions of rate of change that the smaller h is the better will the quotient kfh measure the rate at which the function changes as the argument changes from x l to x^ + h. It usually happens that by taking h less and less the quotient k/h gets nearer and nearer to a definite number; h is not supposed to become zero, but in general we can take h so small that the difference between k/h and a definite number will become, and for smaller values of h will remain, less than any stated, non-zero, fraction. The number will depend on x v (iii) We then define this number as the rate at which the function changes with respect to the argument for the value x l of the argument. The more rigorous of the older mathematicians, such as Maclaurin, starting from definitions or axioms respecting variation at a greater or less rate, proved that 6^ is the " true measure " of the rate at which 3a; 2 varies with respect to x for the value x l ; but the reasoning on which we have based the definition seems sufficient to establish its correct- ness. Of course if the values considered were determined by measurement a stage of smallness for h would soon be reached at which it would become impossible to distinguish between Qx l and 6x l + '3h ; the average rate determined by the smallest available value of h would therefore coincide with that determined by the process and definition we have adopted. 74 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 1. If s=ligt 2 , find the rate at which s varies with respect to , when t has the values 0, ^, 1 , 2. Ex. 2. If p = - , find the rate at which p varies with respect to v when v = v v 38. Limits. It would seem at first sight as if the rate Qx l could be determined from the average rate tix l + '3h simply by putting h equal to 0. But the logic of such a step would be faulty, because the equation = 605, + 3k ti can only be established on the assumption that h is not zero ; in proving the laws of division in algebra the case in which the divisor is zero is expressly excluded. But further, if h = 0, so also is k, and the quotient k/h would appear in the form 0/0 a symbol which has absolutely no meaning whatever. The ground in common sense for defining Qx t as the rate of change for the value x l is that 6x l is the one definite number towards which the average rate k/h settles down as h is taken smaller and smaller. (See the values of Sy l /8x l in examples 4, 5, 6 ( 32) as an illustration of this settling down.) In mathematical language we are said, in determining the number towards which the quotient k/h settles down, to find the limit of k/h when h tends to zero as its limit ; in this process h is a variable number, positive or negative, and it may take any value except zero ; zero is so to speak a boundary to which it gets nearer and nearer, but which it never actually reaches. Before giving a formal definition of a limit we will consider a few typical cases ; by carefully studying these the student will gather the necessity for the introduction of the word and will see what it really means. 39. Examples of Limits, (i) Let AB (Fig. 24) be a chord of a circle whose centre is ; AT, BT the tangents &tA, B. Let OT cut the chord AB at M and the arc A B at N ; M and N will be the middle points of the chord and the arc respectively and OM will be perpendicular to AB. LIMITS. EXAMPLES. 75 The triangles OMA, OAT are equiangular; therefore MA ON AT OA (1) FIG. 24. Suppose now that the chord AB moves towards N, the point N remaining fixed and AB being always perpendicu- lar to ON: let A, B always denote the ends of the chord, M its mid point and T the point where the tangents at A and B meet. So long as A and B are not coincident, that is so long as AB is really a chord, equation (1) remains true. The ratio MA: AT is a function of OM, for as soon as OM is fixed every other line in the figure is fixed, and the ratio can be calculated. When OM is all but equal to ON both MA and A Twill be all but zero; nevertheless the ratio MA :AT will be all but equal to 1, because equation (1) remains true and OM is all but equal to ON which is equal to OA . Manifestly the nearer M gets to N the nearer does the ratio MA :AT get to unity. This behaviour of the ratio MA : AT is expressed in the words : as OM approaches ON as its limit the ratio MA :AT approaches 1 as its limit. Here again it has to be noted that the reasoning ceases to be just if OM becomes actually equal to ON, for the tri- angles will then have disappeared and the equation (1) on which the reasoning is based could not be established. We might equally well consider the ratio as a function, not of OM but, of the angle NO A ; if the angle NOA approaches zero as its limit the ratio approaches 1 as its limit. (ii) Suppose AB (Fig. 24) to be the side of a regular poly- gon of n sides (regular n-gon) inscribed in the circle ; then it is easy to prove that the side of the regular n-gon cir- cumscribed about the circle is equal to AT+BT or *2A T and that the angle NOA is 180 /n degrees. If p, P denote the 76 AN ELEMENTARY TREATISE ON THE CALCULUS. perimeters of the inscribed and of the circumscribed poly- gons respectively, then p = nAB = 2nMA; P = 2nAT, and p_MA_OM MN P~AT~OA~ (Li- Imagine a series of polygons constructed corresponding to greater and greater values of n. When n becomes very large the angle NO A will become very small ; AB and MN will also become small, and therefore the ratio p/P will become nearly equal to 1. Hence when the angle NO A approaches as its limit, the ratio p/P approaches 1 as its limit ; or again it may be put thus, when n becomes indefi- nitely large p/P approaches 1 as its limit. We may express the relation between p and P in a slightly different way. From equation (2) we get MN p fJ _ rf\ __ . JL . p ~ OA When n is greater than 4, P will be less than the peri- meter of the circumscribed square, that is less than 80A ; hence P-p<8MN. Kow let e be any line that is as small as we please, only not zero. By the geometry of the figure we see that we can take n so large that MN shall be less than any given line ; choose n therefore so large that MN is less than e/8. Then for this and for all greater values of n, 8MN will be less than e and therefore Pp less than e. It is here that the limit notion comes in ; no matter how large n may be P and p will never exactly coincide, but as n increases beyond all bounds the difference P p tends to zero as its limit, that is the perimeters P and p tend towards the same limit. S n FIG. 25. On the straight line FH (Fig. 25) mark off Fg n , FG n equal to the perimeters p, P respectively ; then clearly for every LIMITS. EXAMPLES. 77 value of n, Fy n is less than FG n . But, when n has been chosen as above, g n G n = Pp < e, and therefore n can be taken so large that g n G n shall be less than the line e. Hence the common limit of p and P is a line FG greater than every one of the lines Fy n , but less than every one of the lines FG n . Since the circumference C of the circle always lies between p and P, the circumference will be equal to the line FG ; the circumference may therefore be considered as the limit either of an inscribed or of a circumscribed regular polygon when the number of its sides increases indefinitely. (iii) Show that the area of a circle may be considered as the limit either of an inscribed or of a circumscribed regular n-gon ; and that an arc of a circle may be considered as the limit of the sum of n equal chords obtained by dividing the arc into n equal arcs. The polygons have been supposed regular, but it would not be difficult to show that the theorems hold even if they be not regular, provided that as n increases beyond all bounds the length of each side of the polygons approaches zero as its limit. (iv) Let 9 be the number of radians in the angle NO A, where the angle is supposed to be acute ; we have chord AB < arc AB < A T+ BT, and therefore MA < arc NA < AT. MA arcNA AT 'OA < OA < OA ] that is, sin < < tan 6. Divide by sin 6 ; therefore sin 9 cos 9 ' and therefore 1 > ^ > cos 9. u Thus the quotient sin 9/9 lies between 1 and cos 9. When 9 approaches as its limit cos 9 approaches 1 as its limit ; therefore also sin 9/9 approaches 1 as its limit. 78 AN ELEMENTARY TREATISE ON THE CALCULUS. From the nature of a limit, or from the last inequality, we see that the statement that sin 0/6 approaches 1 as its limit when 6 approaches as its limit may be put in the form : when is a small number sin is approximately equal to 6. The student should verify this statement from the Tables; thus, for L.NOA = l, 6 = -0174533; sin 9 = '0174524 ; for L NO A = 5, . = '08726 65; sin 0= "087 15 57. (v) Show that the limit of tan 6/6, as 6 approaches as its limit, is unity. (vi) Provided x is not equal to a, The equation holds true so long as x is not equal to a ; but we can take x so nearly equal to a that x + a shall differ from 2a by as little as we please. That is, the quotient can be brought as near to 2a as we please simply by taking x near enough to a. Hence although the quotient has no meaning whatever, no value, when x is equal to a, it has a definite limit, namely 2a, for x approaching a as its limit, (vii) Let SPT (Fig. 26) be the tangent to a circle at P : PQ a secant and PR a given length measured along the secant. Describe a circle with centre P and radius PR, cutting PT at R. Now let Q move along the arc PQ towards P; R will therefore move along the arc RR towards R. The nearer Q approaches P, the nearer does JL R come to R, and the smaller j 26 becomes the angle TPR. If we suppose Q to approach P as its limiting position, the secant PR will approach the tangent PT as its limiting position. If we suppose the secant drawn on the other side of P, as PQ', PS will be the limiting position of the secant as Q approaches P. Hence we may define a tangent thus : DEFINITION. A tangent to a curve at a point P is the DEFINITION. LIMIT AND VALUE. 79 limiting position of a chord PQ as Q approaches P as its limiting position. It is this definition of a tangent that will be subsequently used in the book. (viii) Show from the theorem in Exercises V. 10, that if T is a point on the directrix KN such that the angle PST is a right angle, the line PT will be the tangent to the conic at P. 40. General Explanation of a Limit. The special meaning of the word limit should now be fairly clear. In each of the examples there are two variables, one being a function of the other. One of these variables, the argument, is supposed to become all but equal to a definite number, for example to a. or or ON', or else it is supposed to increase beyond all bound. In the former case the definite number is called the Limit of the argument ; it is not a value that the argument actually takes ; thus in (iv) is not a value that 6 assumes. In the latter case the argument is generally said to have infinity for limit, though this mode of expression seems rather a contradiction in terms; the argument has infinity for limit if it is supposed to become greater than any number N, no matter how great N may be. Again, when the argument becomes nearly equal to its limit the function at the same time becomes nearly equal to a definite number; not only so, but we can make the argument differ so little from its limit that the function shall differ by as little as we please (except by the difference zero) from that definite number. This definite number therefore is called the limit of the function for the argument approaching its limit. We will now give a formal definition of a limit ; the first mode of statement is somewhat rough, the second is more definite, but in a first reading it may be found a little more difficult to grasp. 41. Definition of a Limit. Notation. Distinction between Limit and Value. DEFINITION 1. When it is possible to make the argu- ment of a given function so nearly equal to a definite number a that the function will differ from another definite 80 AN ELEMENTARY TREATISE ON THE CALCULUS. number A by as little as we please, that difference remaining as small as we please when the argument is taken still nearer to a, then A is called the limit of the function for the argu ment approaching (or converging to) a as its limit. DEFINITION 2. Given any positive number e that may be as small as we please, except that it must not be zero , given also two definite numbers a, A ; if it be possible to find a positive number r\ such that a given function shall differ from A by less than e for all values of its argument that differ from a by less than rj (the value a itself being excluded), then A is called the limit of the function for the argument approaching (or converging to) a as its limit. The modifications required when either a or A is infinite offer no difficulty. In general a variable is said to become infinite if it takes values that are numerically greater than any positive number N, no matter how large N may be ; if the variable is positive it converges to + oo , if negative to oo . The definite number A will be the limit of a function for its argument approaching +00 as its limit, provided that a positive number N can be found such that for every value of the argument greater than N the difference between the function and A shall be as small as we please. The notation for a limit is the letter L or the first three letters of the word limit, namely lim. To state that the function f(x) approaches A as its limit when x approaches a as its limit, the notation is L f(x) = A when L x = a, or, more usually, L f(x) = A ; x=o read " limit of f(x) for x equal to a is A." It must be remembered however that the more usual form is a con- traction for the first, and that a, A are not values that the variables are supposed actually to take. In this notation, if LNOA = 6 and OA=a, ex. (i) of 39 may be stated MA MA L = l or L = 1; ex. (ii): Ljo=LP=<7; , sin 6 T tan 6 ex. (iv), (v) : L -g- = 1, L r = 1. e=o v e =o v THEOREMS ON LIMITS. 81 The necessity for the introduction of the notion of a limit arose from the consideration of cases in which the function ceased to have meaning when a particular value was assigned to the argument ; but the notion of a limit is not, by the definition, restricted to such cases. Whether f(x) has or has not a definite value when x is equal to a the limit is found by considering the values of f(x) for values of x nearly equal to a ; the value a itself is not to be used in the process. It may of course happen that the limit A of the function coincides with the value /(a) ; still, even when A and /(a) coincide, the fact that they are deter- mined by different processes should not be forgotten. Instances frequently occur in which the limit A and the value /(a) are both definite and yet unequal. 42. Theorems on Limits. We now state the principal rules for working with limits. . In the following theorems the functions have the same argument, x say, and the limits spoken of are the limits for each function as the argument approaches a limit, say a, the limits of the functions being finite ; it will be sufficient therefore to use the letter L without the subscript " x = a." The number of functions is supposed to be finite ; the theorems are not necessarily true if the number be infinite. THEOREM I. The limit of the algebraic sum of any number of functions is equal to the like algebraic sum of the limits of the functions. THEOREM II. The limit of the product of any number of functions is equal to the product of the limits of the functions. THEOREM III. The limit of the quotient of two functions is equal to the quotient of the limits of the functions, pro- vided the limit of the divisor is not zero. The proof of these theorems is simple ; it depends on the particular cases that if the limit of each of a finite number of variables is zero, then the limit of their sum and of their product must be zero. Let h v h z , h 3 , for example, be three variables the limit of each of which is zero. To prove that the limit of their sum is zero we have to show that x can be taken so near a that that sum will be numerically less than any given positive G.C. F 82 AN ELEMENTARY TREATISE ON THE CALCULUS. number e. Now, since the limit of each is zero, we can take x so near a that each of the variables shall be numeri- cally less than e; hence we can take x so near a that their sum shall be less than e. The same reasoning holds if there be n variables ; each can be made less than e/n. It does not matter whether the variables be positive or negative or whether the sum contain negative terms since it is the numerical value alone that is concerned. Mani- festly the product will also have zero for limit. Again, if C be any finite constant, the limit of Ch^ will be zero ; we need only choose x so near to a that h^ shall be numerically less than e/G. Now, let u, v, w be functions of x whose limits are U, V, W. Then by the nature of a limit when x is nearly equal to a, u, v, w are nearly equal to U, V, W\ hence we may write u=U+h ^ v =V+h 2 , w=W + h s , where h v h 2 , h s are variables which have zero for limit. Then u + v-w=U+h 1 +V+h z -W-h s Hence, since the limit of each of the numbers h v h z , h s is zero ' L (u + v-w)= U+ V- W, = Lu + Lt> L m Again, uv so that L (uv) = UV=(Lu~)x(Lv). Again, L (uvw) = L (uv) x L it;, = LitxLvxL v, by applying twice the case for the product of two variables. U _U Vh.-Uh, ~ The limit of the second fraction is zero because the numerator can be made as small as we please, while the TRANSFORMATIONS. 83 denominator is not zero, since V is by hypothesis not zero ; it follows that u_U_Lu v~V~ITo We have for simplicity taken only three functions, but clearly the reasoning holds if there be more than three functions, the limits U, V, ... being of course all Unite and no denominator having zero for limit. If one or more of the functions be constant it is evident that the reasoning holds ; thus u might be a constant, and then we might consider Lu as being simply u itself, without in any way violating the conditions for a limit. 43. Examples. We will now give a number of examples in which the above principles come into play. In seeking the limit it is useful to bear in mind that any transforma- tion of the function which is legitimate when the argument is not equal to its limit may be applied as a help towards the solution. Thus l x+l-I 1 The division of x out of numerator and denominator is legitimate so long as x is not zero ; but in finding the limit for x = Q, x is not to become 0, and therefore the first and the last of the three fractions are equal for all values of x considered. Hence =L 1 = 1 2' In the same way we find We take it to be sufficiently evident that the first of these limits is i; by Def. 2, 41, we should be able to find ^, so that when x is numerically less than rj the difference be- tween the function and \ shall be less than any number we may name, say less than '001 . But the search for rj is usually very troublesome, and in such simple cases as we have to deal with we shall usually dispense with that part of the 84 AN ELEMENTARY TREATISE ON THE CALCULUS. investigation, as the nature of the processes involved will show that such a number can be found. Ex.1. n 1+2 + 3+ ... For the sum = - n , - ji 2 7i 2 2 2?i This example shows that Th. I., 42, is not necessarily true unless the number of functions is finite ; for although the limit of each term in the bracket is zero, the limit of the sum is not zero. F l 2 + 2 2 + 3 2 +... 4 a 1 Ex. 2. L - A - = g, 6' and therefore 'IT-g^o^I - + M2 = 1 + J_ + J_ n 3 3 2/t 6w 2 so that the limit is -. o Ex. 3. If r be a proper fraction and n a positive integer, L r*=0. For any positive proper fraction is of the form 1 ( 1 + a), where a is a positive number. Now, by the binomial theorem or otherwise we can readily show that (1 + a) n is greater than 1 +na. Hence, so far as numerical value is concerned, n= I 1 ~(l+a) n< l+na and since the limit of 1/(1 +na) for n= is zero, the finrit of r n is also zero. Ex. 4. Show that if r be a proper fraction and n a positive integer L nr"=0, L ?iV=0, etc., 1 for (l+a)">l + na + =', 2 1 Ex. 5. for the fraction and the limits of numerator and denominator are 2 and 3 respectively. EXAMPLES. 85 Ex. 61 for fi 2 9 First remove the common factor x+\; it is the presence of this factor that makes the fraction take the form 0/0 when we try to calculate its value for x= 1. Ex. 9. for Ex. 10. z^2\/(-+2) un 3a: 1 Ll u, x=0 sin 3x sin 3# ^ Q , -, sin 3o; X Ax 1=0 *'^ cot x L "" = 1. Put x = ~-y; then when # approaches - as its limit y approaches as its limit. Hence cotff tany_ iJ ^ m Tliis device of changing the variable is often useful ; for example : Ex. 11. Lfr+^-^sA*. A=0 2 Put ,r=^ 2 and x + h (y + Kf, so that when h approaches so does k; therefore (#+)*-#* ~ 3y 2 3 3 i _^L_ i/ _ ^2 ~2.y~2 y ~2^ Ex. 12. P, P' are the perimeters of two regular -gons circum- scribed about two circles whose radii are a, a' and circumferences C, C'; show that P:a = P':a' and C":a = C":a'. The constant ratio of circumference to radius is denoted by 2-x- ; TT is an irrational number, approximately equal to 3'14159. Ex. 13. Show that the area of a circle of radius a is Tra 2 , and that the area of a sector of the circle of angle radians is \Qa?. 86 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 14. Show that the volume of a right circular cylinder, the radius of the base being a and the altitude /;, is ircPh. Show that the area of the curved surface is 2irah. Ex. 15. If A is the base and h the altitude of a triangular pyramid, and if the pyramid be divided into n slices, each of height hjn, by planes parallel to the base ; show that the volume of the pyramid is less than n but greater than Hence show, by ex. 2, that the volume is \hA. Extend the result to any pyramid. (Let V be the vertex and DEF the base ; through the line in which a plane meets the face FjE^draw a plane parallel to VD to meet the two planes next above and next below the plane containing the line. Two sets of triangular prisms will be formed : the one set will lie within the pyramid, the other set will include the pyramid. The two sums are the volumes of the two sets ; the highest pyramid of the upper set is got by drawing a plane through the vertex parallel to the base.) Ex. 16. Taking a circular cone as the limit of a pyramid whose vertex is the vertex of the cone and whose base is a regular -gon inscribed in or described about the base of the cone, deduce from ex. 15 that the volume of a cone is $hA, h being its altitude and A its base. Ex. 17. Show that the volume of the frustum of a right circular cone is $h(A + \' ' AB + B) or ^-( a 2 + ab + b 2 J, where h is the height of the frustum, A, a and B, b the areas and the radii of the circular ends. Ex. 18. C and a are the circumference and the radius of the base, and I is the slant side of a right circular cone ; show that the area of the curved surface is \IC or irla. (The curved surface may be considered as the limit of the lateral surface of either of the pyramids of ex. 16.) Ex. 19. If the slant side of a frustum of a right circular cone is ?, and if the radii of the circular ends are a, b show that the area of the curved surface is Trl(a+b) ; if c, c' are the circumferences of the ends, the area is CHAPTER V. CONTINUITY OF FUNCTIONS. SPECIAL LIMITS. 44. Continuity of a Function. The conception of a limit enables us to put in arithmetical form the property that may be considered as most characteristic of a continuous function. The argument will be said to vary continuously from a to 6 when it takes once and once only every value lying between a and b ; when the argument is represented as an abscissa, the corresponding point will move along the axis from the point a to the point b as the argument varies continuously from a to b, and will coincide once and once .only with every point on that segment. In plotting the graphs of the elementary functions it was found that, except in the immediate neighbourhood of those values of the argument for which the function became infinite, a small change in the argument produced only a small change in the function. Now by the defini- tion of a limit, when x is nearly equal to a the function, f(x) say, is nearly equal to its limit A ; if therefore the limit A be identical with the value f(a) of the function, we see that when x either increases or decreases from the value a by a small amount the function f(x) will also change by a small amount from the value f(a). Hence the DEFINITION. A function f(x) is defined to be continuous for the value a of x, or more simply, continuous at a if (i) /(a.) is a definite (finite) number, and (ii) 88 AN ELEMENTARY TREATISE ON THE CALCULUS. For continuity therefore the value of f(x) for x = a and the limit of f(x) for x = a must coincide ; since infinity is not a value, in the sense that is required for the application of the laws of algebra, a function ceases to be continuous, that is it becomes discontinuous, for those values of the argument that make it infinite. Again it is implied in the definition that x may approach a either through values less than a or through values greater than a ; that is when /(a) is represented as an ordinate the point x may approach a either from the left or from the right and the limit must for both methods of approach be the same. It will sometimes happen, as for example when f(x) *J(a 2 x 2 ), that x can only approach a from one side, the function being undefined for values of x on the other side ; in such cases of course the condition that the limit must be the same from whichever side x approaches a has to be modified, but the modification offers no difficulty. To express that x is to approach its limit a through values less than a the notation is sometimes used, and in the same way the notation x = a-\-Q implies that x is to approach a through values greater than a ; but we shall as a rule use the ordinary notation and leave the student to modify it to suit special cases. The only other type of discontinuity that needs special mention is that represented in Fig. 27. As x varies from a value a little less than a to one a little greater the func- tion changes by the finite amount BC. Here the func- tion f(x) has not a definite value when x = a; as x ap- proaches a from the left f(x) A X approaches one definite limit FlG 27 AB, while as x approaches from the other side f(x) ap- proaches another definite limit AC. If a moving particle were at a certain instant to experience an impulse the THEOREMS ON CONTINUOUS FUNCTIONS. 89 graph of its velocity would present a discontinuity of this kind for the value of the abscissa representing the instant. (See 69, ex. 6, for an example of discontinuity.) 45. Theorems on Continuous Functions. When f(x) is continuous at a it is merely stating the definition of continuity in another form to say that if x be nearly equal to a, f(x) is nearly equal to /(a) ; or again we may say that f(x) =/() + d, where d is a variable which converges to zero when x converges to a. A function is said to be continuous over the range from a to b if it is continuous for every value of its argument that lies between a and b ; the range is understood, unless the contrary is stated, to include its extremities a, b. A range which includes its extremities is sometimes called a closed range ; one which excludes its extremities an open range. The following theorems are of constant application : THEOREM I. If /(x) is continuous at a and if /(a) is not zero, then for values of x near a, /(x) has the same sign as /(a)- For if f(x)=f(a) + d, the sign of f(x) will be that of the numerically greater of the two numbers /() and d ; since x may be taken so near to a that d shall be less (numerically) than any given number, and therefore less (numerically) than /(a), the sign will be that of f(a). The meaning of the phrase " near a " and of similar phrases will be gathered from the proof. THEOREM II. ///(^) be continuous over the range from a to b. and *//(a) = A and /(b) = B, then /(x) will assume once at least every value lying between A and B as x ranges continuously from a to b ; in particular if A and B have opposite signs /(x) will become zero for at least one value of x lying between a and b. A mathematical proof of this theorem lies beyond our scope : so far as a function is adequately represented by a graph the theorem is geometrically evident. It is easy also to show by use of a graph that the converse theorem is not necessarily true. 90 AN ELEMENTARY TREATISE ON THE CALCULUS. 46. Continuity of the Elementary Functions. The theo- rems on limits stated in 42 enable us to prove that the elementary functions of a single variable are continuous for all values of the variable except those for which a function becomes infinite. When x varies continuously so does the product x n and the product ax n , n being any positive integer and a a con- stant. ( 42, Th. II.). Hence by Th. I. a rational integral function is continuous for all finite values of its argument ; and by Th. I. and Th. III. a rational fractional function is continuous for all finite values of its argument except such as make its denominator vanish. From the geometrical definition or by direct application of the limit test we see that the trigonometrical functions are continuous for all values of the variable except such as make the function infinite. The sine and the cosine are continuous for all values of the argument ; the tangent and the secant for all values except the odd multiples of ?r/2 ; the cotangent and the cosecant for all values except and multiples of TT. A full discussion of the continuity of a? would take us too far into abstract considerations ; we will therefore assume that a x is continuous for all finite values of x and that its inverse, log x, is continuous for all finite positive values of x but discontinuous for x = 0. When x is irra- tional we may in practice replace a x by a*' where x' is a rational approximation to x ; the simplest discussion is based on the exponential series. Function of a Function. When y is a function of u, say y = 0(u), and u a function of x, say u =f(x), then y is said to be a function of a function of x ; y is, thus given as a function of x mediately, through u. Functions of func- tions are of constant occurrence in the calculus, and there may be several intermediate variables such as u. If y is a continuous function of u, and u a continuous function of x, the student will have no difficulty in showing that y is a continuous function of x ; in the notation of 42 Again when a function is continuous so is its inverse THE POWER LIMIT. 91 function. Hence x 11 is continuous when n is fractional, positive or negative, except for x = when n is negative. In the same way we see that the inverse trigonometric functions are in general continuous. /v?i _ _ f*n 47. L - . The limits discussed in 47-49 are fundamental. T x n a n _ n _j rp ^. ft x=a * * n being any rational number. (i) Let n be a positive integer ; then x n a n _ n _ l4 _ _ n x a -no"' 1 , since the limit of each of the n terms is a"" 1 . (ii) Let n be a positive proper fraction p/q, where p, q are positive integers. Put y q for x and 6 ? for a ; then when x = a,y = b. Hence since x n = x p lv = y p and a n = b p . x-a 2/n ^_ ft n rp m __ ri wi y^ -^ n ^ \f x a xa x a x m a m ' 92 AN ELEMENTARY TREATISE ON THE CALCULUS. /x n -a n \ T ix m -a m \ .-. L (- - )=- L - - x L x=a \ x-a J x=0 x m a = ma m ~ l X-^= ma- m - 1 = na n - 1 , since the limit of the first factor is ma m ~ l by cases (i) and (ii). The student should be able to identify the theorem in whatever notation it may be presented ; thus T (x + h) n -x n , 1 / 1 1 f . i__ ' _ - ryi rfll - 1 J-J ; - 'wC 5 -L' n \ i n~ COR. If h be a small positive or negative number, + k) n is equal to x n + nhx n ~ l approximately. 8 48. L H . The number e. m=co\ m/ (i) Let m be a positive integer and expand by the Binomial Theorem ; then 4_J_Y' l -1 4-!^ 1 m(m-l) 1 + m) '' ^T'm 4 " ~2l~" m2 w(m-l)(m-2) 1_ 3! " m^ + ) / 3 In the expansion there are (m + 1) terms and every term after the second can be written in the form given to the 3rd and 4th terms; for example, the last or (m + l)th term is A IV, 2 \ /, m-l\ (1 -- Ml -- )(! -- -)-j-m! \ m/\ m/ \ m / Let n be any positive integer less than m and let n be kept fixed while m increases. Denote the first (n + 1) terms of the expansion by S' n+l and the remaining (m n) terms b' n . Then + = THE NUMBER e. 93 Now, " n-fl = 1 + H -- ~1 ' ~! ' . m The limit for m=oo of each of the factors (l ), / 2\ (1 )... is 1, and since there is a finite number of factors V m/ the limit of each numerator is 1. Denote by S n+l the limit for m = oo of S' n+l ; therefore S n+l = LS f n+l = l + ~+~+-, + - + - f We have now to consider the limit of R' n+l . The first term of R' n 4-, is and this term is a factor of every one that follows it. Hence R' n+l is the product of and / . >/ 71 + 2 (7l+2)(7l "* H to (m n) terms i Everywhere replace each of the factors fl -- J, fl -- J, J, which are all positive and less than 1, by the 1- m factor 1, and replace each of the factors (7i + 2), (71 + 8) ... m by (n + l); by so doing we shall increase JR' n + v which is therefore less than to w -" terms 94 AN ELEMENTARY TREATISE ON THE CALCULUS. But the series within the bracket is a geometrical pro- gression whose sum is (n+l) m - n ..rc + l/, __ L_ M.. 1 W 1 (71+1)-' 1 /' W+l and for every value of m greater than n, this sum is less than (n-j-l)/. Hence R' n+l is a positive number which for every value of m greater than n is less than R" n+l , where _1 -Ll 71+1 ~ (m I 1 \ I A n But I The first limit is $ n+1 , and the second limit is a positive number less than R" n+l ; therefore, inserting the values of S n+l and R" n+l we get, 11 1 but where R n+1 is less than -R" n+1 or -7~fv When n is even moderately large, R n +i is very small ; for example, when n = l2, l/n(n !) is less than 3xlO~ 10 ; so that the value of the limit may be obtained very approxi- mately by calctilating the series as far as 1/12 !. The calculations are very easy to effect, and the value will be found to be, for the nearest 7-figure approximation, 27182818. The limit is usually denoted by e ; e is really an irrational number. It is easy to see, by comparing S n+l with the sum 1+1+^ + 22+. +2^ r i' INFINITE SERIES FOR e. 95 which is greater than S n + l and equal to 3 1/2 W ~ 1 , that no matter how great n may be $ n+1 is certainly finite and less than 3. Since e $ w+1 is equal to -R w+1 , and since the limit for n = of J% n +i is zero, e may be considered as or, in the usual phraseology, e = 1 + 1 + Q-J + ^ + . . .to infinity. (ii) Next suppose that m proceeds to infinity through positive fractional values ; m will therefore always lie between two consecutive integers, say n and 7i + l. Hence 11 1 7i m Ti + 1 ' n/ \ m But L W and L n/ m=00 \ 7i/ n=00 n i \ n + 1 / i \ n + 1 = L(l+-4-) - M=QO \ n+1/ n= by case (i). Hence in this case also the limit is e, because as m becomes infinite so does n. (iii) Let m be negative, m n where n is positive but either integral or fractional. Then n n ~ / 1 V" / 1 \ n L (1+-) L(H ^1 -oo\ W/ n = ooV n I/ 71 I/ w=00 \ 71 = ex 1 by cases (i) and (ii). (l\ m IH ) =e, ^m/ 96 AN ELEMENTARY TREATISE ON THE CALCULUS. whether m proceeds to infinity through integral or frac- tional values. - COR. 7i 49. The Function e x . If &=| = 0, we see, by putting m = MX, that when m becomes infinite so does M ; hence Mx (fr\ m ( / ~\\M}x I / 1 \M~\x 1 + ^-) L-Ul+i) f =| Ml+i) i =e*, __ m/ jf.oolV T Jf/ J l^ooV ^J// J by 46 (Function of a Function). Since M may be positive or negative, integral or frac- tional, the result holds whether a; or m be positive or negative, integral or fractional. By exactly the same method as in 48, it may be shown that -- (X \ m 1 H J for positive integral values of m. It is easy to see that this series is a finite number no matter how great n may be ; for as soon as n is numerically greater than x, /y7l-f-l /y7l-f-2 /y71-J-3 l + (n + 2)l + (n-' OV1 + x n+l ( x x* I -. i it ' / ~t Cl\ / -_ i O\ I is numerically less than />. n+i where ! is the numerical value of x. The series in brackets is a geometrical progression with a common ratio numerically less than 1 ; hence if we write ................ (A) THE FUNCTION e*. 97 R n+ i will, for every value of n greater than x numerically, be less than x n+l ,x If x l = 1 , this gives the value of R n +i in 48. COR. L = 1 . i=0 X For if x be a positive proper fraction, we may put 1 for n in (A) ; therefore e x >l+.v, but e* 1 so that - > 1, but ^ - , x Z-x from which the result follows for positive values of x. If x be negative, x= h where h is positive, then T e*-l T e-*-l T e*-l 1 L -- = L - 7= L -j x-r = l z=0 # fc=0 ft=0 6" by the first case, so that the limit is the same whether x proceeds by positive or negative values towards its limit 0. 50. Compound Interest Law. When an exponential function is spoken of, the base is usually understood to be e ; where the base is any other number, say a, the func- tion a x can be written e tx , where & = lpga. The rate at which ae kx increases with respect to x when x = x l is kae kx i, that is, is proportional to the value of the function when x = x r For when x increases from x l to x-^+h the increment of the function is ae^+V _ ae^ _ a e kx i(e kh - 1 ), and the average rate is By 49, COR., the limit of this expression for h = is kae tx i. Many processes in nature follow this law ; the law is sometimes quoted as the compound interest law, since the simplest case of it is that of compound interest. For, suppose a principal of P pounds to earn interest at the' rate of p per cent, per annum ; let interest be calculated at n equal intervals in each year, and let it be added to the principal as soon as it is earned, so that the interest bears 98 AN ELEMENTARY TREATISE ON THE CALCULUS. interest. It is easy to see that at the end of t years the principal will amount to *-Y> ^iooW Let us suppose now that n increases indefinitely, that is, suppose that the interest is added on at shorter and shorter intervals ; we thus approach a condition in which the interest is added on continuously. Put n = ^r s that when n becomes infinite so does m. The limit of the above expression for n increasing indefinitely is Kl \ m~\ pt pt 1+) [io6 = PeI66 mJ ) Again, we see that if t increase in any arithmetical pro- gression, whose common difference is h, A will increase in a geometrical progression whose common ratio is e 100 ; for if p(t+h) ph t become t + h, A will become Pe 10 , that is, Ae 100 . Hence A is a quantity which is equally multiplied in equal times. The density of the air as we descend a hill is a quantity which is equally multiplied in equal distances of descent, for the increase in density per foot of descent is due to the weight of that layer which is itself proportional to the density. Many other instances may be found in physics. EXERCISES VII. 1. If J{x)=a.T n + bx n ~ 1 + ... + kx + l is a rational integral function of x, show that and therefore that when x is numerically large, where d is a variable whose limit is zero for x= oo . Use Th. I., 42. 2. Show that fix) in ex. 1 has the same sign as a when x is a large positive number, but has the same sign as ( - l)"a when x is numerically large but negative (that is, has the sign of + a or a according as n is even or odd). EXERCISES VII. 99 3. From the result of ex. 2, show, by applying Th. II. of 45, that every equation of odd degree has at least one real root, and that if it has more than one it must have an odd number. 4. If f(x) is a rational fractional function prove (i) fix) = -"'"(l + rf t ) if m > n ; A (ii) X* (iii) f( x ) where c? l5 \+x+\x 2 ; therefore # e -* = <_ r-jrp> that is > T " > - + l+ir a; and the limit of the fraction last written is zero. Next put x=e~"; then x\ogx=-ye~* and the limit for #=0 is equal to the limit for y = + oo , which is zero. 9. Prove L x n e-*=Q. 10. Prove that if n be positive L #"log#=0. For x" log X = - x" log (x n ) and the limit is zero by ex. 8 since the limit of x n is zero. 11. Prove L in xlog xj= 0, For sin x log x = ( sinx \ ( x log x). \ x I 12. If .' is any finite quantity, prove 100 AN ELEMENTARY TREATISE ON THE CALCULUS. Suppose x equal to or less than the integer //, ; then, numerically, x n x* x x x x* ( x But #/(/i + l) is a proper fraction. 13. If a be a constant or a function of x which is finite for every value of x, prove (i) L (dn2:?yi; (ii) L (iii) L (tan- : -V=l. x=\ X XI 14. If s n is a variable that (i) always increases as n increases but (ii) always remains less than some definite fixed number a, show that as n tends to infinity s n tends to a definite limit that is equal to or less than a. Take the values 15 2 , s 3 ... as the abscissae of points A^ A 2 , A 3 ... on the #-axis and let A be the point whose abscissa is a ; for every value of n, A n will be to the right of A n ^ 1 but to the left of A. As n increases the point A n will move further and further to the right, but will not for any finite value of n coincide with A. There must therefore be some point S to the left of A or coinciding with A to which A n may be made to approach as near as we please; if the abscissa of S is s then by the definition of a limit L *=*, n=oo and s is less than or equal to a. (Compare 39 (ii) and Fig. 25.) 15. If s n is a variable that (i) always decreases as n increases but (ii) always remains greater than some definite fixed number b, show that as n tends to infinity s n tends to a definite limit that is equal to or greater than b. 16. If ,, =l+ ^ + ^ + ... + l., show that s n converges, as n tends to infinity, to a number that is greater than 2 but less than 3. show that s n converges to a number that lies between 1 and 2. Let ,.-l + ^ + ^ + ... + ( ^ = 2 _I <2 , then for every value of n (greater than 1) <'< 2. 18. Apply the theorems of exs. 14, 15 to establish the results of -exs. (i), (ii), (iii) of 39 when the %-gons are not regular but are such that as n increases indefinitely the length of each side diminishes indefinitely. CHAPTER VI. DIFFERENTIATION. ALGEBRAIC FUNCTIONS. 51. Derivatives. Differentiation. The process of 36, 37 can now, by making use of the notion of the limit, be stated more compactly. The average rate at which the function 3ic 2 varies as x varies from a^ to x l + Sx l , where Sx^ may be either a positive or a negative increment, is by definition and the number which is taken as measuring the rate of change when x = x 1 is L ^(3x 1 2 ) = L ^ +3(J )==6 *i=o o#i a Xl=0 The reasoning does not depend on the particular value x 1 of the argument, and we therefore state the result in the form, " the function '3x z varies with respect to a; at the rate 6#," leaving it to be understood that when x = x l the rate is 63;^ when x = x z the rate is 6x 2 and so on. It will save multiplication of symbols to use x as the symbol for the argument in general and also as the symbol for some definite value of the argument, and the student will find that, as a rule, it causes no ambiguity to do so ; if he ever finds difficulty, let him choose a separate symbol as x l for the definite value at which the rate is measured. Now take the general case. Let f(x) be a continuous function of a;; as the argument varies from x to x + Sx, r here fix may be either a positive or a negative increment, 102 AN ELEMENTARY TREATISE ON THE CALCULUS. the function varies from f(x) to f(x + Sx). The average rate of change of the function when the argument changes by Sx is Sx ' Sx and the number which measures the rate of change is We shall find that for all the elementary functions this limit is a definite number, except, it may be, for particular values of x. In general the limit will depend on x, and a special name is given to it, namely, " the derivative of f(x) with respect to x." Instead of " derivative," the names " differential co- efficient," " derived function " are also used ; in certain connections also the word " gradient " or " slope " is used. ( 53.) The process of finding the derivative is called " differentiation " ; the name " differential coefficient " was formerly more frequently used than " derivative." Again, there are special notations for the derivative. A very convenient notation is obtained by accenting the functional letter, as f'(x) ; another is got by prefixing the letter D, with or without the suffix x, as D x f(x) or Df(x). If the function be denoted by a single letter, as y, the nota- tion for the derivative of y with respect to the argument x is similar, as y' x , D x y or y', Dy. As a rule the suffix is omitted when there is no ambiguity as to the argument. Finally, to denote the value of the derivative for a special value of x, say x v the following notations are used : f(x,}- [D x f(x)] x=Xl ; [y'] x=Xl . As a matter of fact, the derivative is really formed for such a definite value, but the functional character of the derivative is more prominent when that value is denoted by the same symbol x as represents the argument in general. To sum up then we have the defining equations : (x) = D x x= L M^= L 5.r=0 OX Sx=Q OX ' Dv L ty-- i^x -L< DERIVATIVES. NOTATION. 103 The function thus determined is called the derivative, or the differential coefficient, or the derived function of f(x) with respect to x and it measures, or briefly it is, the rate at which the function varies with respect to its argument for the particular value x. Of course other letters than x, f, y, may be used ; thus and 0'(0 is the derivative of 0() with respect to t. It will be convenient often to use such expressions as the x-derivative of f(x), or the time-rate of change of a func- tion, instead of the derivative with respect to x, or the rate of change ivith respect to the time. Ex. 1. D JC (W-lx + 3) = 6jc -4, for Now (c\ c - } = -j, (c constant), for If v = 0, the above process cannot be carried out. 52. Increasing and Decreasing Functions. By definition of a limit and of a derivative where a is a variable which is very small when Sx is very small and converges to when Sx converges to 0. For an illustration of the difference between 8f(x)/Bx and f(x\ see the results of examples 4, 5, 6, 32. Hence if f'(x) is not zero the sign of /'(#) + a and there- fore of df(x)/8x will be, for sufficiently small values of x, the 104 AN ELEMENTARY TREATISE ON THE CALCULUS. same as that of f(x) (compare 45 Th. I.) ; therefore the sign of 8f(x) will be that of f'(x)8x. Now suppose Sx a positive increment ; then Sf(x) will be positive or negative according as f(x) is positive or negative. But // \ / hence f(x + Sx) is algebraically greater or less than f(x) according as f'(x) is positive or negative. In other words, f(x) increases as x increases so long as f'(x) is positive, but f(x) decreases as x increases so long as f(x) is negative, increase and decrease being algebraical and not numerical increase or decrease. If we suppose 8x a negative increment then 8f(x) will be negative or positive according as f(x) is positive or negative; f(x) will decrease as x decreases so long as f(x) is positive but will increase as x decreases so long as f'(x) is negative. Hence the mere sign of f'(x) tells how the function changes as x changes; if f(x) = ax+b, f'(x) = a and the conclusions agree with the statements of 33 for the uniformly varying function. DEFINITION. A function which increases as its argument increases and decreases as its argument decreases is called an increasing function ; one which decreases as its argu- ment increases and increases as its argument decreases is called a decreasing function. Thus since D(3cc 2 ) = Qx, 3cc 2 is a decreasing function for all negative values of x and an increasing function for all posi- tive values of x. The function ceases to decrease and begins to increase as x passes through the value 0; hence when x = the function is a minimum (17, iv), and its value is then 0. It will be noticed that when x = the derivative is ; the rate of change is therefore zero for the minimum value. The derivative of Sx 2 4*4-3 is 6& 4; hence so long as Qx 4 is positive, that is, so long as Qx is greater than 4, that is, so long as x is greater than f , the function is an increasing one; on the other hand so long as x is (al- gebraically) less than f it is a decreasing function. When x = f the function is a minimum, the minimum value being -. Here again when x = % the derivative is zero, that is the rate of change of the function is zero. INCREASING AND DECREASING FUNCTIONS- 105 Stationary Values. The conclusions about increasing and decreasing cease to hold for those values of x for which f'(x) is zero. Since f(x) measures the rate of change of the function it is usual to class those values of the function for which f(x) is zero as stationary values. Ex. Show that the function 3? + 1 has a stationary value when A*=0, and that for all other finite values of x it is an increasing function. 53. Geometrical Interpretation of a Derivative. A specially useful interpretation of a derivative is obtained from the graphic representation of a function. Let ABP (Fig. 28 a, 6) be the graph of /(a?). Take a point P on the graj>h ; OM=x, MP = y=f(x). y FIG. 28 a. Let MN=Sx; then ON=x + Sx, NQ=y+fy=f(z+&cl From P draw PR parallel to the ic-axis to meet NQ (or NQ produced) at R ; then, both in sign and in magnitude, RQ = NQ- NR = NQ - MP =f(x + Sx) -f(x) = Sf(x), and When Sx converges to as its limit, the quotient Sf(x\/$x converges to f'(x) as its limit. But as Sx converges to 0, N tends towards coincidence with M and Q tends towards coincidence with P. Hence since tan RPQ converges to a 106 AN ELEMENTARY TREATISE ON THE CALCULUS. definite value, namely f(x), the angle RPQ converges to a definite angle and therefore the secant PQ8 tends to a definite limiting position PT. The line PT is, by defini- tion ( 39, ex. vii.) the tangent to the curve at P. Hence f'(x) is the trigonometrical tangent of the angle that the tangent to the graph at P, the point (x, f(x)), makes with the *-axis. From this property of the derivative, the name gradient is used (see 22). In Fig. 28 a, the tangent PT makes with the -axis the positive angle RPT or XLP; in Fig. 28 & it makes the negative angle RPT or XLP'. We will usually denote the angle by <, so that tan < =/'(#) FIG. 286. In the diagrams Sx is positive, but it is evident that the same conclusions can be drawn if Sx is negative, that is if Q is on the opposite side of P. In particular cases it may happen that P can only be approached from one side. Thus if f(x) = l ^/x 3 , x cannot take negative values; in finding /'(O) therefore Sx must be positive. Here 3 = L ^(&j) = 0, = L &x = Q OX Sx=Q and the tangent makes a zero angle with the #-axis ; since /(O) = 0, the x axis is itself the tangent at the origin. Ex. Find the gradient of the graph of Sx 2 4r+3 at the points whose abscissae are 1, 0, , 1, 2. 54. Derivative as an Aid in Graphing a Function. The conclusions drawn in 52 from the sign of the derivative are valuable as an aid to a mental representation of the GEOMETRICAL INTERPRETATION. 107 variation of a function ; those of 53 are equally valuable in helping us to graph the function. The diagrams of 53 may be considered the standard ones. We see that when the gradient f(x) is positive the graphic point moves upward as the point x moves to the right as along BPQ Fig. 28a, along HAB Fig. 286 ; when the gradient is negative the graphic point moves down as the point x moves to the right as along AB Fig. 28a, along BPQ Fig. 286. At B the gradient is 0, and the tangent is parallel to the cc-axis ; the graphic point is for the moment stationary. The student must not confuse moving upwards with motion away from the a>axis ; thus near H (Fig. 286) the graphic point in moving up gets nearer the axis. The graphic point moves up or down when the point x moves to the right according as NQ is algebraically greater or less than MP ; for NQ MP =f'(x)Sx approximately, and when f'(x) is positive, Sx being supposed also positive, NQ is algebraically greater than MP. If MP and NQ are both negative this implies that NQ is numerically less than MP. As an exercise, trace the graph of f(x)x* 3x+l, already shown in 23. Here it is easily found that (x- 1). So long as x is less than - 1, that is, so long as the point x is to the left of the point 1, both x+\ and x 1 are negative, and there- fore f(x) is positive. Hence, as the point x moves from the extreme left of the .r-axis to the point - 1, the graphic point moves steadily upwards. So long as x is greater than 1, but less than 1, x+\ is positive and x 1 negative, and therefore f'(x) is negative. Hence, as the point x moves from the point 1 to the point 1, the graphic point moves downwards. If x be greater than 1, f\x) is positive. Hence, as the point x moves from the point 1 to the extreme right, the graphic point moves steadily up. The turning points of the graph are found where x= 1 and where x= + 1 ; when x= 1, the function has a maximum value 3, and when x= + 1, it has a minimum value 1. 55. Derivative not definite. It may happen that the limit of 6f(x)/Sx is not a definite finite number. There are two chief cases. 108 AN ELEMENTARY TREATISE ON THE CALCULUS. (i) f(x) may be infinite for particular values of x. Thus if f(x) = 4/t then f(0)= L but, for all other positive finite values of x, We see that as x approaches the origin the gradient gets greater and greater, and when x coincides with the origin the tangent to the graph is perpendicular to the #-axis. In general the tangent at a point on the graph at which f(x) is infinite will be perpendicular to the cc-axis. When f(x) is infinite for a finite value of x as in the case of l/x for x = 0, it will usually be found that as x tends towards that value f(x) tends towards infinity; we may say, therefore, that the tangent which meets the graph at the infinitely distant point is perpendicular to the #-axis. Such a tangent is an asymptote. (See the graphs of 24.) (ii.) It may happen that at particular points of the graph there are two tangents, as at A, Fig. 29. Although the function is continuous when x = a, the gradient f'(x) is not. There is one gradient as we approach A from the left, another as we approach A from the right; as x increases through the value a, f'(x) changes sud- denly from tan XBA to tan XCD. It will be found that for all the ordinary func- tions the derivative f(x} is, except for particular values of x, a continuous function and therefore these func- tions can be appropriately discussed by means of their graph. FIG. 29. FLUXIONS. 109 56. Fluxions. Newton founded his treatment of the calculus on the conception of the growth of mathematical quantities by a continuous motion ; he called the time-rate of change of a variable its fluxion, the variable itself being called the fluent. He laid little stress on notations but sometimes denoted the fluxion of a variable, say x, by the symbol x, and this notation is still often used in works on mechanics to denote a time-rate of change. We may take one illustration of a time -rate of change. Suppose a particle to move along the path APQ (Fig. 28) and at time t seconds from a chosen instant let it be at P, the point (x, y\ where y=f(x) is the equation to the path. Let s be the length, in feet say, of the arc ABP measured from some fixed point A on the path, x, y, s are then all functions of t. Suppose that when the time increases from t to t + St the particle comes to Q (Fig. 28 a, 6) and denote the increments MN, RQ, arc PQ of x, y, s by Sx, Sy, Ss. By the usual definitions, the chord PQ is the displacement of the particle in time St and the quotient of the displacement by St is the average velocity of the particle during the interval, the direction of this velocity being given by the angle HPQ. To get the velocity at time t, find the limit of the average velocity for St approaching 0. Now the limit of the angle RPQ is RPT, so that the direction of the velocity at time t will be along the tangent PT. Again, to find the magnitude of the velocity, or the speed, as the magnitude is now usually called, we have to find L chord PQ =o St We will assume as an axiom that when the chord PQ is very small, the arc and the chord are nearly equal ; or, in the more definite language of limits, we will assume L /chord PQ\ t chord P=O V arc PQ / Now, chord PQ = chord PQ arc PQ = chord PQ & St '' arcPQ ' St '' arc PQ ' St 110 AN ELEMENTARY TREATISE ON THE CALCULUS. Hence, since Ss = when <$ = 0, we have , T chord PQ r /chord PQ\ T 8s speed= L -- ^ - = L ( - pn )X L -^ = s. pn arc PQ / This equation of course simply states that the speed is the time-rate of change of s, and may be considered merely as the symbolic statement of the definition of speed; but, however simple the conception of a rate is at bottom, it will be well for the student to recur again and again to the process by which the number is determined. Again, x is the rate of change of x, that is, x is the rate at which the point moves to the right, and in the same way y is the rate at which the point moves upward. These two rates are called the components of the velocity parallel to the coordinate axes. From the diagram we see that and therefore ^2 /chord PQ\ 2 /&\ 2 ~ V arc PQ ) ' \8t) ' Hence, taking the limit for 8t = 0, we get (So? _ \8tJ ^^StJ ~\ arcPQ a result that expresses the usual rule for determining the resultant velocity s, when the component velocities x, y are given. Ex. Suppose x=t, y=t z . For every value of t, y=x^ ; that is, the point P lies on the parabola whose equation is y =a?. The component velocities are i=l, y = 22, and the magnitude or the resultant velocity is s /(.aJ 2 +y 2 ) = v /(l + 4 2 ). The direction of the velocity is given by tan (f) = D x y = 2x = 2t. It will be observed that the path of the point is given by stating where at each instant the point is, because whenever the instant is named, that is, whenever the value of t is given, the coordinates x, y can at once be calculated. By eliminating t between the equations determining x and y, we find a relation that holds between the coordinates of every point on the path, that is, we find the equation of the path in the usual form. (See Exercises IV. 10, VI. 4, 6, 10, 11.) We will now show how to find the derivatives of the ordinary functions ; in the exercises examples will be found DERIVATIVE OF POWER. HI illustrating the geometrical and the mechanical applications of the derivative. After the student has gained some facility in differentiating, other examples will be considered. 57. Derivative of a Power. By definition , Sx=0 OX and, by 47, this limit is nx n ~ r ; that is, D(x n ) = n-x n - 1 . Hence the derivative of a power with respect to its base is got by multiplying by the index and then diminishing the index by 1. It is obvious that the derivative is a continuous function for all finite values of x, except for x = 0, and it is then discontinuous only when n 1 is negative ; that is, when n is less than 1 algebraically. COR. If a be a constant, D x (ax n ) = nax n ~\ = D( x% \ = I Ex. Ex. 2. Write down the derivatives with respect to t of *, % v, 4- V* Ex. 3. Write down a function of x which has 3? as its derivative. Reverse the process for obtaining a derivative, that is, increase the index by 1, ana then divide the result by the new index. Thus, one function whose ^-derivative is a? is ^x 3 , as may at once be tested by differentiation. Ex. 4. Write down for each of the following functions a function of which it is the derivative, , . , 2 1 V-r ' Jj x* * 5 * 58. General Theorems. The following theorems are of constant application. We suppose x to be the independent variable, so that the suffix may be omitted in indicating derivatives. 112 AN ELEMENTARY TREATISE ON THE CALCULUS. THEOREM I. An additive constant disappears in differentiation; or, two functions which differ only by a constant have the same derivative. For letf(x) = (x) + C, where C is a constant, that is, does not change as x changes ; f(x) and <(#) therefore differ only by the constant G, f(x + Sx) -f(x) = [)] = Sf(x) + 8F(x) - S(x). Therefore, dividing by Sx and taking the limit, we get The same proof holds for more than three functions ; the number of them, however, must be finite, for if there be an infinite number the theorem is not necessarily true, just as in the case of the corresponding theorem in limits ( 42, Th. I). Ex. D^-bx+\)=D(Z^)-D(?)x) Th. III. and I. = 3Z)(^) - 5D(x) Th. II. = 6.r-5. THEOREMS ON DIFFERENTIATION. 113 THEOREM IV. D(uv) = vDu+uDv, where u, v are func- tions of x. When x takes the increment x, let u, v take the incre- ments Su, Sv respectively, then S(uv) =(u + 8u\v + Sv) - uv = vSu + uSv + SuSv ; S(uv) Su , Sv , Su , --- -. Sx Sx Sx Sx When Sx converges to so does Sv; the limit of the last term is therefore 0, and we get D(uv) = vDu + u Dv. If there be more than two factors, say u, v, w, we may extend the theorem by applying it twice ; thus, first consider vw as forming one factor, we get D(uvw) = D(u . wf) = vwDu + uD(vw). But D(vw) = wDv + vDw ; If we divide both sides by uvw we get D(uvw) _ Du Dv Dw uvw u v w ' More generally, if there be n factors, u l} u z , ... u n , we have D(u l u 2 ...u n ) = Du 1 Du z u Logarithmic Differentiation. When the differentiation is carried out in the form last written, the process is usually called logarithmic differentiation. (See 65, ex. 3.) The student must particularly notice that the derivative of a product is not the product of the derivatives of its factors. Ex. = 3(Xr-29. The result may be verified by first distributing the product and then differentiating. G.C. H 114 AN ELEMENTARY TREATISE ON THE CALCULUS. THEOREM V. D\ } = - provided v is not zero for the values of x considered. Ju\ u+Su u vSuuSv EQY (x) we have D*y = D x [f(x} - #] =/'(*) - *'(aO = 0. Hence the gradient of the graph of y is zero for every value of x ; the graph must therefore be either the a?-axis or a straight line parallel to that axis. But the equation of every line parallel to the o>axis is y = const. = G ; the equation will represent the axis itself if (7=0. Therefore f(x) - $(x) = C or f(x) = (x) + 0. EXERCISES VIII. 115 Ex. If D^=x 2 - 1, determine the general value of y. The derivative of \s? x is x 2 1, as may be tested by differentia- tion ; therefore the derivatives of y and of J^ 3 - x are the same for every value of x. Hence y and \&-x can only differ by a constant, that is, y \y? x+C. This value is called the general value, because every function which has the same derivative as y will at most differ from ^x 3 - x by a constant, and C may be any constant. The particular function which has the value 2, say, when x has the value 1, will require a particular value of the constant C. But always therefore 2 = J-l + (7, /. (7= ; and y = ^x 3 #+f. It is to be observed that the derivatives must be equal for every value of the argument ; thus x 2 1 and x 3 1 are equal when x is or 1, yet the functions %o? x+C and \x* x + Cf, of which they are the derivatives, do not differ merely by a constant : they are different functions. EXERCISES VIII. Differentiate with respect to x, examples 1-10 : 2. 2. 3. (^-1X^+2X^-3). 4. 5. V- 6. 7. x+. 8. 9. 4^ + 2*i-2#~i+3r*. 10. . x z+x~ Differentiate with respect to t, examples 11-14 : 11. (at + V)l(ct+d). 12. 13. (at* + 2bt + c)l(At* + 2Bt + C). 14. 15. Give a geometrical interpretation of Th. I. 58. Deduce Th. V. from Th. IV. by putting -=w, so that Du = D(ww). 16. If at time t the adjacent sides of a rectangle are u and v feet respectively, where w, v are both functions of t, show that at time t the area is growing at the rate vu + uv. If at time t the three edges of a rectangular parallelepiped which meet at one corner are , v, w feet respectively, find the rate at which the volume is increasing. Show that these results give a geometrical interpretation of Th. IV., 58. 116 AN ELEMENTARY TREATISE ON THE CALCULUS. 17. Find the values of x for which the following functions are (i) increasing, (ii) decreasing, (iii) stationary. Apply the results to the graphing of the functions, and state the turning points. (a) 3-tf+# 2 ; (6) ^-3.r+2; (c) x*-2x 2 -l. 18. State the most general function which has as its derivative (i) 2x l ; (ii) 3# % ; (iii) oc 19. The gradient of a curve is aP x+l, and the curve passes through the point (1, ) ; find the equation of the curve. 20. If pv=p Q v where p , v are constants, show that vD v p p. 21. The speed of a particle at time t seconds from the beginning of its motion is Vgt feet per second; find how far it has moved in t seconds. . 59. Derivative of a Function of a Function and of Inverse Functions. The derivative of such a function as (x 2 x + l)^ cannot be found by immediate application of the rule for the deri- vative of a power. In a case like this we may proceed as follows: Denote (x 2 x+Vfi by y\ now put x 2 x-\-l=u. Then y = u^ where u = x 2 x + I; that is, y is a function of u where u is a function of x. In other words, y is a function of a function of x ( 46). When x takes the increment Sx let u take the increment Su', when u takes the increment Su let y take the incre- ment Sy. Hence when x takes the increment Sx, y takes the increment Sy, and when Sx converges to zero so do Su andfy. Now ' ^^ ^ Sx Su' Sx' therefore L^=L^ L^; S2=OO^ Su=QOU Sx=QOX that is, D x y = D u y x D x u. In the derivative D x y, y is supposed to be expressed as a function of x, while in the derivative D u y, y is supposed to be expressed as a function of u. That is, D a i(x 2 -x+ l)* = . 2(x z DERIVATIVE OF FUNCTION OF FUNCTION. 117 where, after the differentiation is effected, u is replaced by its value in terms of x, namely x* x+l. The reasoning is perfectly general, so that we have the theorem : If y =f(u) and u = (x), then y is a function of a function of x, and D x y = D u f(u)xD x (x) or D x y = D u yxD x u. If we had y=f(u), u = (v), v = \js(x), we should get in exactly the same way D x y = iy (u) x D v (v) x D x ^(x) or D x y = D u y x D v u x D x v. The same method shows how to obtain the derivative of an inverse function. Let y=f(x) so that x is the inde- pendent variable. The inverse function is x=f~ l (y) and y is now considered to be the independent variable. Let Sx and Sy be two corresponding increments of x and y, so that Sx and Sy vanish together. Then &-*_. Sx Sy therefore L ^ X L =- = 1 ; that is, D x y x D t/ x = 1. sx=oSx Sy =oSy The result is evident geometrically. In Fig. 28 ( 53) D x y is the tangent of the angle that PT makes with OX, DyX is the tangent of the angle that PT makes with Y, and since these two angles are complementary the product of their tangents is 1. This theorem is of great use in finding the derivatives of inverse functions ( 64, 65) ; meanwhile we note that and the theorem remains true even if one of the derivatives is zero. The student should carefully note the following ex- amples ; at all stages the rule for differentiating a function of a function has constantly to be used. Ex. 1. D 3 j(ax+b) n =na(ax+b) n - 1 . Put cuc+b = u, then D x (ax + 6)" = D u u n x D x u = nu"~ l x a = na(ax + 6)"- 1 . 118 AN ELEMENTARY TREATISE ON THE CALCULUS. With a little practice, the student will be able to dispense with the actual substitution of u. Thus he will write D x (ax + b) n = n(ax + b) n ~ l x a = na(ax + 6)"- 1 ; 3 Ex. 2. Ex. 3. If D$ = xj(xi - a 2 ) and u = x* - a?, find D u y. D u y = D x y x D u x = D x y/D x u=x ll J(x 2 - a?)/2x ; therefore D u y=^(j^-a z )=^ >k /u. Ex. 4. If y is a function of x, so is y 2 , 7/ 3 , ... ^y, y?y ..., and ) x D x y = *yD x y, and generally, using y 1 for D x y, =x m ~ l y n ~ l (nxy' + my). Conversely, ytf^ This transformation is specially useful in mechanical problems. Thus, t being the argumentj xx = Aft**) ; yy = D t (%y 2 ). Ex. 5. .If v=s, prove v=D s ( v = D t v= D,v xD,s = sD,v = vD,v = or, in words, the time-rate of change of v is equal to the space-rate of change of z> 2 (see 69). Ex. 6. If the coordinates of a point on a curve are given in the form x=f(f), y = (t), where, for example, t may denote the time, find D^. y is a function of t, and t may be supposed to be determined as a function of x by the first equation. Hence But D x t = \jDtX by the rule for inverse functions ; therefore v %y=i xy D t x x Thus, if x= at 2 , y=2at, 2al2a WORKED EXAMPLES. EXERCISES IX. 119 Ex. 7. When y is given as an implicit function of x by an equation of the form Ax n y n + Bx p y+... + Kx+Ly-irM=Q, ................... (a) we can find y by the method of ex. 4. For in whatever way x changes y must change so that the equation (a) always remains true ; therefore the rate at which the expression on the left side of (a) changes as x changes must always be zero ; that is, that is, Each term may now be differentiated and the equation solved for Dxy or y. For example, given then that is, and therefore i/= -- ^- ...................................... (y) y v/ ' To find the gradient of the ellipse represented by (y8) at particular points, we proceed as follows : When x=l, y 2 +#=0 ; that is, y=Q or 1 ; at the point (1, 0), / = - J= -2 ; 2 1 at the point (1, -1), y=- - = 1. To find where the tangent is parallel to the #-axis, we have to solve (/3) and the equation y' = 0, taking care that the values which make the numerator of y' vanish do not also make the denominator vanish. If this were to happen, then y 1 would for such values take the form 0/0, and y 1 might or might not be zero. In the above case we have to 1 2 solve (/5) and 2x+y=Q. The values are x=-^, y~~Ht-> an< ^ 1 2 * ' * x= j=, y = ^-; at these points the tangent is parallel to the .r-axis. In the same way we find where it is perpendicular to the #-axis by solving ((3) and .r + 2w = 0, which makes y 1 infinite. The points are EXERCISES IX. Differentiate as to x, examples 1-8 : 1. V(l-*)- 2. xU(l-x). 3. ^(2*4- IX*- 4. xlJ(a*-x*). 5. ^/V(a 2 +^). 6. 7. ^(s+l)/V(* 8 -l). 8. 120 AN ELEMENTARY TREATISE ON THE CALCULUS. In differentiating a quotient of the form (x+a) m /(x+b) n , it is often advisable to write the quotient as a product in the form (x+a) m (x+b)- n ; when simplified, the result will appear in its lowest terms. Differentiate in this way : 9. (#+l) 3 /(*-l) 4 . 10. (*+a)"Y(j?+&). 11. ljx\x- 1) 3 . 12. State in words the equation D t y=D x yx.DtX. /I 1\ k 13. If v 2 = 2k[ --- I, show that v= - 2 , s and v being functions of \ 5 a / s the time t. 14. If 2^ 2 +3?/ 2 = 5, find y'. Then find the gradient at the points : * (i) (1, 1), (ii) (-1, 1), (iii) (-1, -1), (iv) (1, -1). 15. If (x+yf - 5x+y = 1, find y'. Find the gradient at the point or points where the line whose equation is x+y = l cuts the graph. 16. If x=at, y = bt-%ct 2 , find the components parallel to the axes of the velocity of the point (#, y), and find the direction in which the point is moving at time t. (Compare Ex. VI. 4.) 17. Find D x y in the following cases : (i) (x - of + (y - 6) 2 = c 2 . (ii) y 2 = Ax + Bx\ (iii) xy = c i . (iv) x m y n = c m+n . 18. If D x y^=x^^(ax^+b) and u = ax 3 +b, find D u y. 19. If D x y =(x+ a)(^ 2 + 2ax + b) n and v = x* + Vox + b, find D u y. 20. If D x y=f(ax+b) and u = ax + b, find D u y. 60. Differentials. In Fig. 28 a, 6, 53, the value of f(x) or D x y is tan RPT, and Now, suppose that as x increases from OM to ON the ordinate y or f(x) increases uniformly at the rate f(x) or tan RPT ; then the point P will move, not along the arc PQ but along the tangent PT, and the increment that y on this supposition will take will be, not RQ but RT. This hypothetical increment of y is called the differential of the function y or f(x) and is denoted by dy or df(x). The actual increment of y, denoted by Sy or Sf(x), is not RT but -RQ. Writing as usual Sx for the increment MN of a; we have dy==RT =f(x)Sx Sy = RQ = (f'(x) + a)Sx, where a is used in the same meaning as in 52. DIFFERENTIALS. 121 If f(x) is the function x, then f(x) = 1 , and we have df(x) = dx = l.Sx, so that for the independent variable Sx and dx may be con- sidered to be the same thing. We may therefore write dy = RT= f(x)dx ; Sy = RQ = (f(x) + a)dx. The first of these equations gives a new notation for the derivative, namely This notation, which is perhaps the most common, has the advantage that its form recalls the process by which the derivative is obtained. Again, we have another advantage. For Sy-dy = (f(x) + a)dx - f(x)dx = adx, and (see 52) when dx or MN is very small a is also very small, and therefore Sy is very approximately equal to dy. The notation of differentials is due to Leibniz ; the above mode of defining a differential is usually attributed to Cauchy, but the differential is equivalent to Newton's " moment," which is explained in exactly the same way by Benjamin Robins (see his Mathematical Tracts, London, 1761). A reading of Robins' Tracts would well repay the student who is fortunate enough to get hold of a copy ; the book is now somewhat rare. The notation of differentials is practically a necessity in the integral calculus, and the student should accustom himself to it. In practical work dx and therefore dy are usually supposed to be very small quantities ; but it is only their ratio that is of importance. The symbol -^ is often written as -j-y, but when used in this way the symbol T- is to be taken as a whole and as meaning exactly the same thing as D x . Since du = D x u dx, dv = D x v dx, etc., when the independent variable is x, we have d(u + v w) = du + dv dw, d(uv) = vdu + udv, and so on. We may, in fact, replace D in the theorems of 58 by d. 122 AN ELEMENTARY TREATISE ON THE CALCULUS. du Again, since - T means D~u we have dx d(u+v w) d, ._dudv_dw dx dx dx dx dx' d(uv) du dv dx dx dx' and so on. Ex. 1. d Ex. 2. d. Ex. 3. Ex. 4. dy dy du dy I , -7- = -t 7 , -f- = -j (s dx du dx dx dx dy (x' i -\)dx=d(^-x). State in the form of differentials Ex. IX. 1-6. 61. Geometrical Applications. Let OM be the abscissa and MP the ordinate of the point P on the curve whose equation is y=f(x); and let the tangent at P meet the axes at L, K (Fig. 30). The line CPG drawn through P perpendicular to the tangent is called the normal to the curve at P. When the tangent and the normal are spoken of as finite segments the portions LP, GP, intercepted between P and the #-axis, are the seg- ments referred to. In the same way the projections of these segments on the ic-axis, namely LM and M G, are called the subtangent and the subnormal respectively. GEOMETRICAL APPLICATIONS. 123 These segments can be expressed in terms of the values of x, y, y' at P. Subtangent = LM=-r = -, ; Subnormal = MG = y tan ^ = yy'; Tangent = LP = y cosec = t/ Normal = GP= y sec = y OK= - OL tan 0= J-ij' =y- xy '. These expressions are true for all positions of P, provided the signs of the segments be attended to. Thus, if LM is expressed by a nega- tive number, L will be to the right of J/, since, in the above diagram which is taken as the standard, LM is positive when L is to the left of M. There is no need to commit these formulae to memory ; the values can at once be obtained in any given case by drawing a diagram. We may also find the equations of the tangent and normal. For this purpose let the values of x, y, y' at P be denoted by x 1} y v y{ in order to distinguish them from the coordinates (x, y) of a point on the tangent LP or the normal OP. The equation of the tangent is r y-yi=yi(x-Xi), since it is a straight line passing through (x v y^) and making an angle with the o;-axis. The acute angle that the normal makes with the #-axis is ^ and tan ( < ~ J = cot = 1/y/ ; hence the equa- tion of the normal is 124 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 1. Find the subtangent and the subnormal in the ellipse given by^+jj-l. If we suppose y to be positive, then subtangent =,= -- ; V x subnormal =yy' = -- p When x is positive, both these numbers are negative ; L therefore lies to the right of M and G to the left of M ; when x is negative, the positions are reversed. a 2 OL=x- subtangent = ; x a well-known property of the ellipse. is of course the centre of the ellipse, denoted in 26 by C. Ex. 2. If the equation of the curve is xy = c 2 , find the ratio of KP toLP. KP OM an/ c 2 c 2 Here - r ^=- n - = -^-= --- -, =-1. Lr LM y x x The ratio is given both in sign and in magnitude ; hence P lies between K and L, and KL is bisected at P. The curve is a hyperbola ( 27, II., ex.), and this is a well-known property. 62. Derivative of the Arc. Let s be the length of the arc AP measured from a fixed point A on the curve (Fig. 30) ; to find D& DyS. Proceeding exactly as in 56 we get the equation where 8s and Sy are the increments of the arc s and of the ordinate y due to the increment Sx of x ; Ss = arc PQ. The average rate of change of s with respect to x, namely Ss/8x, is determined by the equation V arc PQ ) \Sx, DERIVATIVE OF ARC. VELOCITIES. 125 Since the limit for &e = of the first factor on the right is 1 we get or In exactly the same way we obtain Again, T x T (Sx arcPQ \ dx cos0 = L ^TJ= L 1-5 --- r -j-i57sJ = Aa5=-T- FQ=oPQ PQ=O\$S chord PQ/ cfe T 8y T fSy arc P0 \ -. dy sm0= L ^= L (-/--r j-7j^) = Dsy = ^r- p 1/1 '*'1\ o ^ ^ & =72 1 1 2^ } SlnCe t ' = 9 6 2 \ a 2 / a 2 (See Exercises VI. 18.) 10. If P is the point (a cos 0, b sin #) show that the equations of the tangent and normal at P are (see Exercises V. 5) -cos0+f B in0=l; -^ a ^--i- a v = 2 -6 2 . a o cos sm fl 1 ' EXERCISES X. 127 11. If P is the point (at z , 2at) on the parabola y i =4ajc show that the equations of the tangent and normal at P are (see Exercises V. 6) 3C y = - + at; y = tx+2at+at 3 . t 12. From the result of ex. 3 or otherwise show that if the tangent at P to the parabola (Fig. 1 9) meets the axis at T TS=AS+AM=SP. If NP is produced to Q show that TP bisects the angle SPN and PG bisects the angle SPQ. Also that, if SN cuts the tangent at the vertex at Z, SZ is perpendicular to and bisects TP and SZ* = AS.SP. 13. In the notation of 61 show that for the curves af"y n =c m+n KP:LP=-m:n. Sketch the curve (i) if m = 7, = 5 ; (ii) if m = 10, 7i = 9. These are Adiabatic Curves. 14. Show that for the parabola y*=4ax ds 15. In the semi-cubical parabola ay z =x 3 show that Show also that ^ = J( 1 + }; dx \ \ 4a/ . and verify that if the arc s is measured from the origin 27 V 4/x / 27 16. Show that the tangents at the points where the straight line aje + hy=0 meets the ellipse are parallel to the .r-axis, and that the tangents at the points where the straight line kx+by = Q meets the ellipse are parallel to the y-axis. 17. Show that the tangents at the points where the parabola ay = x i meets the folium of Descartes, whose equation is (compare Exercises VI., 13) are parallel to the #-axis, and that the tangents at the points where the parabola y^ ax meets the folium are parallel to the y-axis. 128 AN ELEMENTARY TREATISE ON THE CALCULUS. The origin (0, 0) is one of the points, and the coordinate axes are tangents, though this is one of the exceptional cases referred to in 59, ex. 7. The other points are (a V2, a 2/4), (a S/4, a V2). 18. Show that for the ellipse x*[a?+y*/b 2 = l and that for the hyperbola z 2 /a 2 -y 2 /b 2 = 19. Show that for the curve y cx 20. Show that for the curve --a* * ~2 a ^ ' the arc being measured from the point (o, a). L_(2 7 \X CHAPTER VII. DIFFERENTIATION (continued). TRANSCENDENTAL FUNCTIONS. HIGHER DERIVATIVES. 63. Derivatives of the Trigonometric Functions. The fundamental limit is that proved in 39 (iv.), the angle being measured in radians, namely T sin0 fi ~ 0=0 C7 (i) DxSinx = cosx. since For D x smx= L- Sx=Q OX . SX ( , &K\ / , o \ " Sin -x- COSl X + -tr) ny-r sm(a; + Sx) sin a; _ Sx Sx * "2 2 The limit for ^05 = of the first factor is 1, and of the second factor is cos x. Hence D x sin x = cos x. (ii) D x cosx=-sinx. Sa:=0 003 - . &5 . / , fe\ and cos( + c) cos cc = 2 sin -^- sm ( as + -- ) The rest of the work is the same as in (i). o.c. i 130 AN ELEMENTARY TREATISE ON THE CALCULUS. (iii) D x tanx = - = sec 2 x. cos z x ,-, T tan(o; + oV) tana? For D x ia,nx= L - Sx=0 OX 1 sin Sx ^-; | j The result may, of course, be obtained by writing tan x in the form sin x/cos x, and applying the rule for differen- tiating a quotient. Directly from the definition or by applying the rule for differentiating a quotient we obtain (iv) D x cosec x = cosec x cot x ; (v) D x sec x = sec x tan x ; (vi) D x cot x = r- g = cosec 2 #. The knowledge of the derivatives makes it easier to graph the functions, and the student should test such graphs as he has already drawn by examining the gradient in the light of the derivative. The derivatives of the sine and cosine are continuous for all values of the argument. The derivatives of the other functions become discontinuous for the values for which the functions become discontinuous. The rule for differentiating a function of a function has often to be applied, for it is very seldom that the argument is x simply. The most important case is that in which the argument is a linear function ox+b. Put ax+b=u, and we have D x sin (ax + b)=D u sin u x D x (ax + b) = cos u x a=a cos (ax+b). In the same way we find and so on. In fact the student should from the first accustom himself to these forms. Again, to find the derivative of sin 2 (# + &), let sin (ax+ b) be denoted by u ; then DIRECT TRIGONOMETRIC FUNCTIONS. 131 DJ[sin 2 (ax + b)] = D u v? x D x sin (ax + b) = 2u x a cos (ax+b) = 2a sin (ax + b) cos (ax + b). With a little practice, and the application of common sense, even this substitution will not be necessary. NOTE. If the angle is measured in degrees, then D. t sin x is not cosa; but ^^cosa?, because x degrees make Trx/180 radians, and / j \ /THE T sm(a; deg.) = sm( rad. sin(a? deg. ) = D x sin( ^- rad. ) Mow / 7T / TTX -. l80 COS Vl80 rad EXERCISES XI. Differentiate with respect to x, ex. 1-9: 1. sin 3#+cos3.r. 2. sin (x+b). 3. sin mx cos nx. a 4. x sin x+ cos x. 5. sin x x cos x. 6. \x 7. ^+Jsin2j;. 8. |sin.r+^Vsin3x 9. Write down for each of the functions 10-15 a function of which it is the ^-derivative. 10. cos 3x sin 3j-. 11. cos(a.r+6). 12. sec 2 (o#+6). 13. cos 2 x. 14. sin 2 #. 15. sin 4x cos %x. Differentiate with respect to x, ex. 16-22. 16. cos 2 (ax + 6). 17. tan 2 (r+l). 18. 19. sinVcos 3 *. 20. - T 21. 1 ~ coa;g - . . 1 + COS X 1 + COS # 22. sip * 1+ tana; 23. Show that /) :r [tani^]=T ^ - , 2 J 1+cosjr and that /),[^ tan fcrl = J 132 AN ELEMENTARY TREATISE ON THE CALCULUS. 24. Show that - - steadily decreases as x increases from to _ x - ; graph the function from #=0 to X=TT (see also ex. 34). 2 To prove the theorem, show that fhe derivative of sin xjx is negative, and therefore smxjx a decreasing function. Since smx/x=2/Tr when #=7r/2 and sinxJ(a? x^)dx ; dxa cos 6 dO ; dy 30. If dy = >J(x' i + a?)dx and x = a tan 0, show that x 31. If dy= .. 2 - 2\ an d #=asin 0, show that dy = dO. y^Cti ~~ 3C J 32. If dy = -:.'' -, K and x=a(\ +sin 0), show that dy = d 33. If f(x)=\ \x z COS.T, show that when x is positive negative. Hence show that for positive values of x flx) is a decreasing function. Since f(.v) = when #=0, it must therefore be negative for every positive value of x. INVERSE TRIGONOMETRIC FUNCTIONS. 133 34. Show that when x is positive x \s? < sin x < x. Take fyx)=x-\y? -$ia\.x ; then (by ex. 33) '(z) is negative, since *'(*)=/*) 35. Prove in the same way that when x is positive 1 - -zx* < cos x < 1 - -x 2 + -T.X* ; 2 24! x ^x 3 < sin x < x -^ + >x. O . O . O : These inequalities may be carried out to any number of terms. 36. How should the inequalities of examples 33, 34, 35 be stated for negative values of x ? 37. Show that if x is positive and less than Tr/2 x < J tan #+ sin x. 64. Inverse Trigonometric Functions. The direct trigo- nometric functions are single-valued but the angle has to be restricted to a certain range in order that the inverse functions may be single-valued (see 28). The range is from 7T/2 to 7r/2 for the functions inverse to the sine, the cosecant, the tangent, and the cotangent, but from to ir for those inverse to the cosine and the secant. In finding the derivatives the theorem expressed in the equation D x y = l/DyX is used ( 59). Let y = sin ~ l x ; then x = sin y and DyX = cos y = + >v/(l "~ a;2 )> because cosy is positive, y lying between 7r/2 and Tr/2. Hence D^ sin ~ l x = D x y = ~ = (ii) I Let y = cos~ 1 a;; then x = cosy, and DyX= siny= [-f ^(1 a; 2 )], because sin y is positive, y lying between and TT. Hence 134 AN ELEMENTARY TREATISE ON THE CALCULUS. This result may also be obtained from the equation cos ~ x = - sn ~ x. ~ l - ~ 1 In the same way the following results are established : (iii) H.tM-fa-y-Lp; (iv)^ C ot-^=- T -^; (v) D c c~ l x= -(vi)D -%?. 1 \ ' X // ' O 1 \ ' V / * ,// ,v,2 1 Of the above results (i), (iii) are the most important. The root is a positive number, so that, for example, /^/(x 2 ) means +x when x is positive, but x when x is negative. The results (v), (vi) hold so long as x is positive ; when x is negative the sign of each must be changed. It is worth noting that the derivatives of the inverse trigonometric functions are not transcendental but are algebraic functions. The derivatives (i), (ii), (v), (vi) become discontinuous for 03= 1; (iii), (iv) are continuous for every finite value of x. In the case of the inverse functions also the student should accustom himself to the form in which the argument is not x but a linear function of x, specially x(a or Thus, if x/a = u i/^\ r i n /^A 1 1 a D^tan- 1 - ) = D M tan-% x ZU - ) = =- - a - = -5-: s ; W \a/ 1+w 2 a a, 2 +or / / \ i /fc r> _i| iv \ ri 1 " >s/' c /I O1T1~ A I I " // I O Tl ~ A ^* X//}. Olll / 7 I f.i n\ * -*-/!Z veil! 77 7 o EXERCISES XII. Differentiate with respect to x, ex. 1-6 : 1. Bin-ia*. 2. Bin-t(*nlY 3. \ 3 / 4. sin- J (l-^). 5. ^-sin- 1 ^. 6. EXPONENTIAL FUNCTION. 135 Write down for each of the functions 7-9 a function of which it is the .r-derivative : 7 -^y 'iff " 10. Prove that 11. Show that* . cosxj a + bcosx If a 2 is less than 6 2 the derivative is imaginary ; explain this. 12. Show that asin.tr b _ t fb+acosxX} _ (a 2 6 2 ) cos x ;/ J "~ 13. Show that 9 . 1-Fsin 2 ^ 14. Show that 15. If x=rcos6, y=rsind, and x, y, r, Q are all functions of t, prove (i) ,=rcos0-rsin 60. (ii) y = rsinO+rcoad 6. (iii) xy-yx^r 3 '!). 65. Exponential and Logarithmic Functions. The funda- mental limit is now that discussed in 48, namely, and that stated in the corollary to 49, which may be put in the form ^1 L ^^ = 1. &r=0 OX *The value of the derivative given in ex. 11 is only true if a is positive and x lies in the first or second positive quadrant. If a is negative, or if x lies in the first or second negative quadrant, the sign of the result must be changed. A similar remark applies to ex. 12. ii ex. 14 the result holds if a is positive and if x lies in the first positive or in the first negative quadrant. 136 AN ELEMENTARY TREATISE ON THE CALCULUS. a&X _ 1 _ For D x e x = L ~# L Sx = OX sx = O COR. D x a x = log a . a x . For if k log a, a x = e kx , so that putting kx u D x a x = D u e u x D x (kx) = e kx x k = log a x o* n. T-, n , T log (x+Sx) loga; T 1 , /, . (>x\ For D a; loga;= L - , ' = L log IH ). 5^=0 8X S x = fa V X) Sx 1 Put = , so that if x =1=0, as Sx converges to 0. m con- ic m verges to oo . Now 1 , / &c\ m, /_ , 1\ 1, f/, , IV"! -s-iogi i H )= logi IH )=-iog ( IH ) Sx s \ a? / x & V m/ a; & L\ m/ J and L lol+ = - L X 1 f / l\ m l = -log L (1 + -) x & L w =oo\ m/ J = - log e. x Since the base of the logarithms is supposed to be e the result is established. COR. D x \og 10 x = -log lo e. Ju Assuming the derivative of log x the derivative of e* may be obtained by the rule for the derivative of inverse func- tions ; and conversely that of log x may be obtained from that of e x . Thus, Let y = e x ; then x = log y, and DyX = l/y, Again, LOGARITHMIC FUNCTION. 137 For, put ax + b = u and we get D x log (ax+ &) = D u log u x D x (ax+b) = - x a = -- j-. 1l> (JL'JC ~7~ Since logo: is a real number only when x is positive, log ( x) will be real only if x is negative. The #- derivative of log( x) is however l/x, as may be seen by putting a= 1, 6 = 0. Hence the function whose ^-derivative is l/x is log x or log ( x), according as x is positive or negative. It will be noticed that the derivative of logo; is an algebraic function, discontinuous for x = like the function itself. Ex. 1. D x log (x + N/V 2 + ) = //J. ^y Let u=x-\-jJ(fl? + k) ; then D x log (x + */x* + ) = Dulog u x D x u = - x ^z* and A. and the result follows at once. The student should note that = ^-derivative of ... ,ov but -777 - 5\= ................... of sin" 1 ( -77) or of cos" 1 1 -77 ). J(k-jt?) \ x // \x/^/ These results are frequently required in the Integral Calculus. Ex. 2. Find the derivative of e 1 " sin (bx + c) and of c* cos (bx + c). These functions are of very frequent occurrence in certain branches of physics. D x {S=-V0 2 + 6 2 ), tan0=-. Cd Replacing a and 6 by R cos 6 and R sin 6, we get In the same way we find D x {^ cos (bx + c) } = .ffe"* cos (6^c + c + B), where R and 6 have the same meaning as before. 138 AN ELEMENTARY TREATISE ON THE CALCULUS. Some care is necessary however in making the transformation, because 6 is not uniquely determined by its tangent ; the quadrant in which 6 lies is determined by the signs of a and b. Thus, It being taken positive, if a and b are both positive, tan 6 is positive and 6 is in the first quadrant ; but if a and b are both negative, tan 6 is also positive, but 6 is now in the third quadrant. Similar observations hold when a and b have opposite signs. In practice it is usually simplest to choose R positive when a is positive, but negative when a is negative ; then to choose B as a positive or negative acute angle. When numbers are given it is best to work the example without reference to the general formula. Thus, ^{e-^cos (4^+ 1)} = -e-^fS cos (4r+ 1) + 4 sin (4j?+ 1)}. Choose .ft cos = 3, R sin = 4, and therefore R = 5, tan#=f. Now = tan53 8' and 53 8' = '9274 radian, so that I> x {e~^ cos (4x+ 1)} = -5e- 3x {cos -0726). Ex. 3. Find the .r-derivative of J(x - 1 ) (x - 2)/V(>~- 3~X# - 4). In this and in similar cases where the function is a product, it is often simplest first to take the logarithm of the function and then differentiate. Denote the function oy y ; then logy = log (ff-l)+ilogOF-2)-$log(a?-3)-ilog(j?-4). Now D z log y = D y logyx D x y = -D x y ; (x- A JL i '. In the same way, if the function be uvw/UVW, wliere u... W are all functions of .r, we should get, denoting the function by y and taking logarithms (see 58 Th. IV.), Dy_Du Dv Dw DU DV DW yir + T + w u v w Ex. 4. If u, v are both functions of x, we may find the derivative of u" as follows : Put y =u" and take logarithms ; then logy = v log u, For example, Dx* ^(log x + 1). WORKED EXAMPLES. EXERCISES XIII. 139 EXERCISES XIII. Differentiate with respect to #, examples 1-13: 1. xlog.r. 2. x n logx. 3. log sin x. 4. log cos x. c , , /l+sin^\ I 5. log tan 4*. 6. log , .- - . 7. log & 8. log^. Write down for each of the functions 14-18 a function of which it is the .r-derivative : i is. i [..LYJL-JLYl. oc" el/ L_ 2td \x a x H~ rt / _I 19. If y show that D x t/ Compare Exercises XII. 10. 20. If y = ^(a 8 + ^) show that D x y 21 If y-log , n show that D x y Compare Exercises XII. 11. X 22. In the exponential curve, the equation being y = cea, find the subtangent and the subnormal. 23. The curve whose equation is y=a(e^"+e~) is called a " catenary " ; find the subtangent, the subnormal, and the normal. Show that the perpendicular drawn from the foot of the ordinate at any point to the tangent at that point is of constant length. Graph the curve. 24. In the catenary, show that, the arc s being measured from .r=0, ds - -- a x ~- ~ and s=e^-e . 66. Hyperbolic Functions. In recent years certain func- tions called Hyperbolic Functions have been introduced ; these have many analogies with the trigonometric or circular functions, and in some respects have the same 140 AN ELEMENTARY TREATISE ON THE CALCULUS. relation to the rectangular hyperbola as the trigonometric functions to the circle. We shall not make much use of them, but it seems proper to define them, so that the student may not be altogether at a loss should he fall in with them in his reading. They are called the hyperbolic sine, cosine, etc., and are defined as follows, the symbol sink meaning hyperbolic sine of; cosh, hyperbolic cosine of, and so on. = |(e a; e~ x }\ cosh x = ^ sinh x -, . -^- cosh x e x + e~ x smh x -^j sinh x Identities. The following identities, similar to those for the trigonometric functions, -are readily established by sub- stituting the values of the functions in terms of x. (i) cosh 2 x sinh 2 x = 1 ; (ii) 1 tanh 2 x = sech 2 x ; (iii) coth 2 x 1 = cosech 2 x, where cosh 2 x means (cosh xf, etc. Addition Theorem. Again, corresponding to the addition theorem in trigonometry, we have (iv) sinh (xy) = sinh x cosh y cosh x sinh y ; (v) cosh (xy) = cosh x cosh y sinh x sinh y ; By putting y = x we get (vi) sinh 2x = 2 sinh x cosh x ; ( vii) cosh 2x = cosh 2 x + sinh 2 x ; = 2 cosh 2 x 1 = 1 + 2 sinh 2 x. In drawing the graphs of these functions it should be noted that the sine, the tangent, and their reciprocals are odd functions, but that the cosine and its reciprocal are even functions. The sine may take any value from oo to + oo ; the cosine is never less than 1 and is always positive; the tangent may take any value between 1 and 1, and the HYPERBOLIC FUNCTIONS. HI lines whose equations are y = 1 are asymptotes to the graph of tanh x. Derivatives. The derivatives are readily found : D x sinh x = cosh x ; D x cosh x = sinh x ; D x tanh x = sech 2 x ; D x coth x = cosech 2 x ; D x cosech x = cosech x coth x ; D x sech x = sech cc tanh a?. Inverse Functions. The inverse functions can be ex- pressed by means of the logarithm. If y = sinh~ l x, then x = sinh?/, just as when y = sin~ l x, x=smy. To find the logarithmic form of y we have to solve the equation x = k(ey-e-y) or e 2 v 2o^-l=0, which gives e* = x ^/(x 2 + 1 ). Since & is always positive the + sign can alone be taken; therefore e y = x + ^/(x 2 + l), and In the same way we find cosh ~ l x = log (x Since (x /Jx z 1) = l/(x + njx 2 1) we have log (x - *Jx z -\} = - log (x + x/ 2 -l). In this case the inverse function is not single- valued ; to each value of x greater than 1 there are two values of cosh" 1 ^, equal numerically but of opposite sign. The graph of cosh x is in general appearance like that of 1 + x z ; by rotating the graph of 1+x 2 about the bisector of the angle XOY we should get a curve resembling that of cosh' 1 ^, and the curve would be symmetrical about the aj-axis as the graph of cosh a; is symmetrical about the y-axis. If x if x z >l, 142 AN ELEMENTARY TREATISE ON THE CALCULUS. Derivatives of Inverse Functions. The derivatives of the inverse functions, taking for greater convenience x/a instead of x, are - D a; cosh- 1 -= a a x a / For the positive ordinate of cosh' 1 - the -j- sign must be taken. It should be noticed that sinh ~ l - = log = log (x + v a; 2 + a 2 ) log a, a/ a so that the derivative of sinh* 1 - is the same as that of a log(fl3 + x/# 2 +cO> the constant log a disappearing in the differentiation. The occurrence of the divisor a in the logarithmic form of sinh' 1 - has to be borne in mind when comparing the same result expressed in logarithms and in inverse hyperbolic sines (or cosines). 67. Higher Derivatives. The derivative of f(x) is usually itself a function of x and may therefore be differ- entiated with respect to x. Thus the derivative of x 3 is 3o; 2 and the derivative of 3x z is Qx. Qx is therefore called the second derivative of x 3 , while 3# 2 , which has hitherto been called simply the derivative of x 3 , may be called for distinction the first derivative of a? 3 . The notation for derivatives higher than the first is modelled on the analogy of indices. Thus the first o;-derivative of y is D x y, the second D m (D m y) written D/y, the third D.(D*y) D x *y, 4-V>p /ftth T) f T) n-I<,i\ T) n. W*" ' i ^ L>iA U~ y ) ,, J-f f^ny && generally D x n y becomes -7 . y or -r-&. dx n 9 dx n The accent notation is also used ; thus f"(x), f"\x), / iv (ce)... /<">() mean the 2nd, 3rd, 4th... nth derivatives of /(a;), n being enclosed in brackets to distinguish the nth derivative from the w th power. In the same way y", y'" ... x, x ... are used, but the notation is rather cumbrous when more than two accents are used. Ex.1. If f(x) = ax* + bx 3 + ca? + dx + e, find /"(.*:), f"(x), f*(x). f'(x) = 4ax? + 3kr 2 + 2 (x\ where (f> (x) is a rational integral function that does not vanish when x=a, show that /(a)=0, /'()=0, /"(a) = 24>(a). 19. If f(x) = (x (i) r $>(x) where r is a positive integer and <(.r) as in Ex. 18, show that /()=0, /=0, ... /<""(a)=0, /<(a)=r! (x) = x - log (1 + x) ; then see Exercises XI. 33. 21. If x is positive and less than 1, show that log (1 x) >x. 22. Show that the limit for n = oo of * - log n, where is a finite quantity (called Euler's Constant) lying between and 1. From the inequalities of examples 20, 21, r log {/(%- Hence log { (n - !)/( - 2)}>^->log {n/(n - 1)}, By addition, therefore 1 > s n - log n > log < 1 + - j- , from which the result follows at once. The value of the constant is 57721566490 23. If x=at\ y=2at, find D^y in terms of t. 148 AN ELEMENTARY TREATISE ON THE CALCULUS. 24. If x=acost, y = bsint, find DJy in terms of t. 25. If v*=a?+y*, show that in the notation of 62 26. If a^ + 2A^ + % 2 =l, show that D^=(h 2 -ab)/(k 27. If car 2 + 2hxy + by 2 + tyx +*2fy + c = 0, show that where A = abc + 2fgh -af^-bg 2 - ck 2 . 28. If a?+y*-3axy=Q, show that 29. If u is a function of x, show that 30. If y=tan~X show that (i) Dy = cos 2 y ; (ii) D 2 y = cos [ 2y + - ) cos 2 y ; (iii) IPy = 2 cos 3y + 2 cos 3 .y ; (iv) D n y = (n - 1) ! cos 7ty+~ 7r cos"y. CHAPTER VIII. PHYSICAL APPLICATIONS. 69. Applications of Derivatives in Dynamics. We give in this chapter a few simple examples of the use of derivatives in physical problems. Take first the case of the rectilinear motion of a particle and let the units of time, length, and mass be the second, the foot, and the pound respectively, and the units of force and work the poundal and the foot-poundal. At time t, that is, t seconds from some chosen instant, let the particle be at P, distant x feet from a fixed point on the line of motion and let the mass of the particle be m pounds. Denote the velocity at time t by v, the accelera- tion by a, the momentum by M, the force by F, the kinetic energy by E; these quantities may be expressed in terms of t, x, m. ; 1 1 1 X O P Q X FIG. 31. When t increases by St let x increase by Sx = PQ', then the average velocity during the interval St in the direction in which x increases, namely, in the direction OX, is Sx/St, and the velocity at time t is the limit of this quotient for St = Q. Therefore T Sx dx v= L = -j- = x. st=o oi ai v is in general a function of t. The average acceleration during an interval St in the direction in which x increases is Sv/St, where Sv is the increment of v in time St; the 150 AN ELEMENTARY TREATISE ON THE CALCULUS. acceleration at time t is the limit of this quotient for St = 0. Hence Sv_dv_d L dx__cPx_.. a ~st=o St ~ dt~dt dt ~ dt z ~ X ' The momentum in the direction in which x increases is M = mv mx. By the second law of motion the force F in the direction in which x increases is the time-rate of change of the momentum in that direction. Hence j-, dM F= -rr = mv = mx. dt We may express F in another form, by considering v as a function of x, and x as a function of t, so that (see 59, Ex - 5 ) du^dv fa =: dv^ = dt~dx dt ~dx ~ Now E=%mv* and therefore ,, dv d ,. 9 . dE F ~ m di=dxU mv)= dS- Hence the force may be defined either as the time-rate of change of momentum dM/dt or as the space-rate of change of kinetic energy dE/dx. Let W denote the work done on the particle by the force F in moving it from some standard position, say from the position at which x = a, to the position P; SW the work done in moving it from P to Q. At Q the force is F+ SF ; hence when <5te is small the work done will lie between FSx and (F+8F)Sx. For FSx is the work done on the supposition that the force is constant over PQ and equal to its value at P, while (F+8F)Sx is the work done on the supposition that the force is constant over PQ and equal to its value at Q; evidently the work will lie between these two values. Hence SW/Sx lies between Pand F+SF and therefore j ^p- dx Since dE/dx is also equal to F, E and W differ only by a constant. DYNAMICAL FORMULAE. 151 Again, the time-rate at which the force works is dW/dt, and W may be considered as a function of x and x, a func- tion of t Therefore . at dx dt The student should note the dimensional formula for these magnitudes ( 34). If x is the measure of a length so is dx, and the dimensional formula for v or dx/dt is LT~ l ] similar observations hold for the other quantities. Ex. 1. Suppose F constant ; then the acceleration will be constant, equal to f say. Hence =/, and therefore v=ft + const. Let the motion be such that when t=0, v= V and x=a; these are called the initial conditions. The constant in the value of v is therefore V. We can now find x ; for x=v=ft+ V; x=%ft z + Vt + const. as may be tested by differentiation. The constant is a, since when t=0, x=a, so that finally Vt + a. To get E in terms of t we have E=fynv 2 =$m(ft+ V) z . Using the value found for x and putting E for %m V 2 we get This form may be obtained at once from the energy equation dEjdx = F. Finally since dW\dx=F we have W=F(x a), W being zero when x=a. Hence EE = W; that is, the gain in kinetic energy is equal to the work done by the force. Ex. 2. Suppose F to be an attraction proportional to the distance of the particle from 0. Let the intensity of the attraction, that is, the force on unit mass at unit distance from (?, be p.. If x be positive, that is, if the particle be to the right of 0, the force towards is \i.mx ; if x be negative, that is, if the particle be to the left of 0, the force towards is mfj.( x). In both cases therefore the force in the direction in which x increases is fj.mx. But the force in the direction in which x increases is always mx. Hence or This equation is called the differential equation of the motion of the particle, the word "differential" being used because the equation contains the differential coefficient x. 152 AN ELEMENTARY TREATISE ON THE CALCULUS. The student will easily verify that the equation will be satisfied (see Exercises XIV., 16) by x = A cos Jut + B sin ^[it, where A, B are any constants whatever. The motion becomes definite when in addition to the law of force we are told the position and the velocity of the particle at any one instant. Suppose for example that when t=0, x = a, v=0. Putting =0 and x=a in the equation for x we find A = a. Again v is found by differentiating x with respect to t ; therefore But when t=Q, v=Q ; therefore we get = ^5, that is, .6=0, and we find that x = a cos *J[j.t. Simple Harmonic Motion. When the law of force is that stated in the example the motion is called simple harmonic motion, and the form x = a cos i^/ /j.t is the simplest way of stating the relation between x and t. Obviously the motion is periodic, the period being 2ir/J/jL because while t increases from a value ^ to the value t l + %-irMiL both x and x go through their complete range of values, a is called the amplitude of the motion. The student may show that if x=c, v= Fwhen =0, then V x=c cos *Jfj.t + r- sin *J[Lt = a cos (*Jfj.t 6), If V 2 \ V where a=A/(c 2 H I, acos#=c, a sin 9=r~- \ \ p> / V/A a is again the amplitude and 27T/V/* the period. Ex. 3. A rod is stretched from its natural length a to the length a+x : assuming Hooke's Law to hold, find the work done. The ratio x/a is called the extension, and by Hooke's Law the force required to produce that extension is proportional to it. Denoting this force by F, we have F=Exja, where E is a constant. When the extension is (x + 8x)/a, the force will be F+8F=E(x + 8x)/a. If the work done in producing the extension x/a is W, and if 8 W is the work done in producing the further extension, then 8 W will lie between F8x and (F+8F)8.v, so that 8W/8x will lie between F and F+8F. Taking the limit for 8x converging to zero, we get dx ~ a' Hence W=$E+ const. cc Since W= when x = 0, the constant is zero, so that EXAMPLES FROM DYNAMICS. 153 Ex. 4. A fluid is in communication with a cylinder in which a piston is free to slide, the cross section of the cylinder being S, a constant. Let W be the work done by the fluid in pushing out the piston a distance x, and let the intensity of pressure on the piston be p. Show that d Wjdx=pS. The force on the piston due to pressure is pS ; when the piston is pushed out the further distance 8x, let the intensity of pressure be p + Sp so that the force on the piston is (p + 8p)S. The work 8 W done in pushing out the piston through the distance 8x will lie between pSSx and (p+Sp)S8a?, and therefore 8W/8x will lie between pS and pS + 8pS. Hence d Wjdx =pS. The result may be put in another form. If v be the volume of the fluid, then SSx is the increment of volume which may be called 8v. Hence 8 W/8v lies between p and p + 8p, and we get dW_ dv~ p - Ex. 5. A body is rotating about an axis ; a line fixed in the body and perpendicular to the axis makes at time t an angle d with another line fixed in space and perpendicular to the axis. What do 6 and 6 measure ? Q is the time-rate of increase of 0, that is, 6 is the angular velocity of the body about the axis. In the same way we see that 6 is the angular acceleration. If a point P is moving in a plane, and if Q is the angle which the line joining the point P to a fixed point in the plane makes with a fixed line through 0, 6 and $ are sometimes called the angular velocity and the angular acceleration of the point P about 0. Ex. 6. A positive charge m of electricity is concentrated at a point ; the repulsion on unit charge at P (Fig. 31) is m/x 2 where x=OP. Find the work done as unit charge moves from A to B where OA= a, OB = b. Let W be the work done from A to P ; then dW m j Tir m , -j = -r, and W f- const. dx a? x When x=a, TF=0, and the constant is therefore mja. Hence at P the work is W= --. a x The work in moving from A to B is therefore W - m m Wl ~~a b' Potential. When B is so far off that m/b is negligible in comparison with m/a then TF 1 = m/a. Hence in this case the work done as unit charge moves from A out of the 154 AN ELEMENTARY TREATISE ON THE CALCULUS. field is m/OA. This function m/OA is called the potential of the charge m at A. At P the potential is m/OP. Denoting it by V we have V dV_m x ' dx~ x z * so that the force at P is the space-rate of diminution of the potential V at P, and the direction of the force is from that of higher to that of lower potential. For gravitational forces the attraction between two particles, m, m' (grammes) at a distance from each other of x cm. is kmm'/x z dynes where k is the constant of gravitation (equal to 6 '6 x 10 ~ 8 ). See Gray's Treatise on Physics, 195. [London: J. & A. Churchill.] The potential V of m at the point x is km/x and the attraction towards m is D X V ; the force outwards from m is -\-D x V. It is proved in works on Dynamics (e.g. Gray, 484 ; see also Exercises XXX., 24) that the potential at the point x of a sphere of radius a and uniform density p is V= / 27rkp(a 2 %x z ) for an internal point (xa) .............. (ii) > x Since the field is symmetrical the force is radial at every point and the attraction at the point x is therefore The functions V and D X V have each different analytical expressions according as x is less or greater than a, but they are each continuous functions near x = a ; for we see from (i) and (ii) that whether x tends to a through values less or through values greater than a, V tends to 4:Trkpa 2 /3 and D X V to QnrkpajS, and these are the values of V and D x V when x = a. On the other hand the function D X Z V is discontinuous at a; for when x tends to a through values less than a we find from (i) that D^V tends to 4x&/o/3 and when x tends to a through values greater than a we find from (ii) that D*V tends to +8dfe/>/3. The function D X Z V has POTENTIAL. 155 therefore no value when x = a, but has one definite limit for x approaching a from one side, and another definite limit for x approaching a from the other side (see 44). To graph the functions F, D X V, DV suppose for simplicity a=l, 4?r/>/3 = l; the graphs for other values can be derived in the usual way (Fig. 32). FIG. 32. A BCD is the graph of F; the part AB is a parabola, the part BCD a rectangular hyperbola. The dotted curve OEF is the graph of D X V] the part OE is straight. The graph of DV is the straight line GE parallel to OX and the curve HGK. The parts to the left of the vertical dotted line represent the functions for xa. 156 AN ELEMENTARY TREATISE ON THE CALCULUS. 70. Coefficients of Elasticity and Expansion. Let p be the intensity of pressure and v the volume of unit mass of fluid, p being a definite function of v. When p increases by Sp let v increase by Sv ; if we suppose Sp positive then Sv will be negative. The quotient Sv/v, that is, the ratio of the diminution of the volume to the volume at pressure p, is called the compression or the mean compression, and the limit of the increment of pressure, Sp, to the compression produced, Sv/v, is called the coefficient of the elasticity of volume, or simply the elasticity of volume, or sometimes the coefficient of the resilience of volume. Hence the elasticity of volume is ... Sp dp L --= -V-T- SV=Q Sv dv V For a gas expanding at constant temperature pv = k, a constant, so that the elasticity of volume is d(kjv) -k _ /? \ / / ^^ 4. / V 7 ~" V a IJ- dv v 2 For a gas expanding adiabatically pv Y =c, a constant, and in this case the elasticity is yp. A rod whose length at a standard temperature, say at 0G, is the unit of length expands when heated to a temperature 6 so that its length becomes 1+/(0); denote l+/(0) by x, and when the temperature becomes 6> + SO let the length become x + Sx. The quotient Sx/SO is called the mean coefficient of linear expansion as the temperature increases from 6 to 6 + SO, and dx/dO is called the coefficient of linear expansion at the temperature 0. Usually /(O) is of the form aO or aO + bO" where a, b are very small constants. When x = l + aO, the coefficient dx/dO is a and is independent of 0; if f(0) = aO + bO z and x = l+aO + bO*, the coefficient is a +260 and depends on 0. If a solid expand equally in all directions, the area and the volume which are unity at 0C. would become 2/ = (l+/(0)) 2 and z = (l+/(0)) 3 at temperature 0. The numbers dy/dO, dz/dO are called the coefficients of super- ELASTICITY. EXPANSION. 157 ficial and of cubical expansion respectively at temperature 9. If /(0)- 00, then Clu Since a is very small a 2 and a 3 will be much smaller and the coefficients will be very approximately 2a and 3a. Ex. The volume at temperature 6 of the water which occupies unit volume at 4 is approximately I+a(0 4) 2 where = 8'38 x 10~ 6 ; find the coefficients of cubical expansion at temperatures and 10. 71. Conduction of Heat. A slab of thickness d whose opposite faces are parallel planes has one face maintained at constant temperature v and the opposite face at constant temperature v x (v>v l ); the quantity Q of heat which in tkne t crosses an area A forming a part of a section parallel to the faces and lying between them is Q=kA(v-v l )t/d, where fc is a constant, called the conductivity, depending on the material of the slab. This equation expresses the law of steady flow of heat in a conducting solid and is a result of experiment. If the temperature v of a solid vary from point to point of the body at the same instant, and from one instant to another at the same point in the body, v will be a func- tion of more than one variable, namely of t and of the coordinates of the point. A t a given point in the solid the time-rate of change of v is In forming this derivative the coordinates of the point do not change; v changes through lapse of time at a given point. On the other hand, let P be a point in the body whose distance from a fixed plane is MP = s, and R a point in MP produced such that PR = Ss ; then at the same instant the temperature v at P will be different from that at R, which 158 AN ELEMENTARY TREATISE ON THE CALCULUS. may be denoted by v + Sv. At the time t the space-rate of variation of v at the point P in the direction PR will be T Sv n L T = DiV ' Si=QOS Let us assume that at any given time t the temperature is the same at every point in any plane perpendicular to MP though different for different planes. We may assume therefore that the heat flows in straight lines parallel to MP ; let v be the temperature at P, v + Sv the temperature at R where PR = Ss, and let SQ be the amount of heat which in time St crosses unit area of a plane perpendicular to PR and lying between P and R. The formula given above for Q is assumed to give the average value of the amount of heat crossing a section when the flow is not steady, St and Ss being small. In that formula, therefore, put SQ for Q, 1 for A, v + Sv for v v St for t, Ss for d, and we get SQ = k{v-(v+Sv)}8t/d8, SQ jSv and St = - k Ss' Take the limit for St and Ss converging to zero, and we get D t Q=-kD s v; in words, the time-rate at which heat crosses the section of unit area at P is A; times the space-rate of diminution of temperature in the direction perpendicular to the area. D t Q or its equal kD s v is called the flux in the direction in which s increases ; obviously the flux is from places of higher to places of lower temperature, and this is shown by the form kD g v since if v decreases as s increases D s v is negative and D s v is positive. to Ex. v= Ve~ c sin x where F, c are constants. When #=ir/2, D t Q=Q whatever t may be; that is, there is no flow of heat across this plane ; when #ir/Z it is towards the right, the positive direction of x being towards the right. This problem gives an example of a function of more than one variable ; such functions will be taken up later. CONDUCTION OF HEAT. EXERCISES XV. 159 EXERCISES XV. 1. A point P moves with uniform velocity V along a straight line AB ; OA is perpendicular to AB and equal to a. Find the angular velocity of P about 0. 2. A point P moves with uniform velocity u along a straight line A B, and another point Q with uniform velocity v along an intersecting straight line A C. Find the rate at which the distance between P and Q increases. 3. If p is the density and p the intensity of pressure of the atmo- sphere at a height of x feet above sea-level, express in symbols the statement that the rate of increase of pressure per unit of length downwards is equal to the density multiplied by the acceleration due to gravity. Assuming that p = kp where k is a constant, and that at sea -level p=p , show that 4. If N be the number of lines of force passing through a circuit, state in words the meaning of dNjdt. 5. Express in symbols the statement that the electromotive force E is the sum of two terms of which the first is the product of the resist- ance R and the current C, and the second is the product of the self- inductance L and the time-rate of increase of (7. 6. Express in symbols the statement that the force X acting on a magnetic shell in the direction x is equal to the space-rate of diminu- tion in that direction of the energy E. 1. If in ex. 4, 69, W l is the work done as the fluid expands from volume v l to volume v%, find W l (i) if pv = k, (ii) if pv7=k, k being constant. 8. The potential of a long uniform rod of linear density o- at a point P whose distance PC from the rod is x is V= 2&r log (cjx). Show that the attraction of the rod on a unit particle at P is towards C and equal to %kcrjx. 9. The potential of a thin circular disc, of surface density cr, at a point P on the normal to the disc through its centre is where a is the radius of the disc and OP=x. Show that the attraction on unit mass at P is Show that if x is small compared with a, the attraction is approximately. 160 AN ELEMENTARY TREATISE ON THE CALCULUS. 10. The coordinates of a point at time t are given by x=acos(2nt a), y = bcosnt. Show that the equation of the path of the point is The ^-coordinate is a simple harmonic function of amplitude a and period ir/n, while the y-coordinate is a simple harmonic function of amplitude b and period 27r/w, double that of the ^-coordinate. The motion is therefore said to be compounded of two simple harmonic motions in rectangular directions and of periods in the ratio 1 : 2. When a = 0, the path is a parabola. Figures of the curves for different values of a will be found in Gray's Physics, Vol. I., p. 70, and in various other books. 11. Show that two simple harmonic motions of the same period and in the same straight line compound into a simple harmonic motion of the same period and in the same straight line. 12. If in ex. 1 1 the motions are in rectangular directions, show that the curve compounded of the motions will be an ellipse. CHAPTER IX. MEAN VALUE THEOREMS. MAXIMA AND MINIMA. POINTS OF INFLEXION. 72. Rolle's Theorem and the Theorems of Mean Value. The following theorems are of constant application. THEOREM I. // F(x) and F'(x) are continuous as x varies from, a to b, and if F(x) is zero when x = a and when x = b, then F'(x) will be zero for at least one value of x between a and b. (Rolle's Theorem.) In geometrical language, the theorem simply states that at one point at least on the graph of F(x) the tangent is parallel to the ic-axis. There may be more points than one ; if there are more than one there must be an odd FIG. 33. number of such points, as C, D, E (Fig. 33). The student should show by a graph that the theorem is not necessarily true if either F(x) or F'(x) becomes discontinuous at a point in the range from a to b. 162 AN ELEMENTARY TREATISE ON THE CALCULUS. The theorem is otherwise obvious, because F(x) cannot either always increase or always decrease as x increases from a to b, since F(a) = and F(b) = 0. Hence for at least one value of x between a and b, F(x) must cease increasing and begin to decrease, or else cease decreasing and begin to increase ; for that value of x, F'(x) will be zero. Obviously a may be either less or greater than 6. THEOREM II. If f(x) and f (x) are continuous as x varies from a to b, then there is at least one value of x, ajj say, between a and b such that ) ...... (1) (Theorem of Mean Value). In Fig. 34 let A be the point (a, f(a)), B the point (b, /(&)) ; the gradient of the chord AB is /(&)-/(<*) b-a and the theorem simply asserts that there is at least one point, as P, on the graph between A and B such that the tangent at P is parallel to the chord AB. If the abscissa of P is x l the gradient at P is /'(^i) an d the equation is established. The student should draw graphs to show that there may be more than one point such as P, and that on the other hand the theorem may not be true if either f(x) or f'(x) becomes discontinuous for a value of x between a and 6. The theorem may however be deduced from Th. I., and the method of deduction is important as it leads to the theorem known as Taylor's Theorem, one of the most far reaching in the Calculus', -indeed the present theorem is only a special case of Taylor's. Consider the quantity Q defined by the equation _ ( fr_ a)Q = ......... (2) M FIG. 34. THEOREMS OF MEAN VALUE. 163 Let F(x) denote the function f(x)-f(a}-(x-a}Q formed by replacing 6 by a; in the expression f(b)-f(a)-(b-a)Q. By (2) F(b) is zero; also F(a) is zero. Hence the con- ditions of Th. I. hold for F(x) since F(x), F'(x) are continuous. Therefore F'(x) will be zero for at least one value of x, x l say, between a and b. But F\x}=f(x)-Q; and therefore f(x l ) Q = or Q=f'(x l ) so that the theorem is established. THEOREM III. If f(x), f(x), f"(x) are continuous as x varies from a to b, then there is at least one value of x, x z say, between a and b such that f(b) =f(a) + (b-a) f'(a) + 1(& - a) 2 /"(^)- This theorem is an extension of Th. II. To prove it consider the quantity R defined by the equation f(b)-f(a)-(b-a)f(a)-$(b-a?R = .......... (3) As before, take the function F(x), such that F(x) = f(x) - /(a) - (x - a) /(a) - \(x - a) z R. Here, F(a) = 0, F(b) = (by (3)), and F(x) satisfies the con- ditions of Th. I. Now, F'(x)=f(x)-f'(a)-(x-a)R, and therefore for at least one value of x, x 1 say, between a and b F'(x l )=^f(x l )-f(a)-(x 1 -a')R = G. Hence F'(x) vanishes when x = x l ; obviously it also vanishes when x = a; the conditions of Th. I. apply therefore to F'(x) so that its derivative must vanish for at least one value of x, x 2 say, between a and x v and therefore between a and b. But the derivative of F\x) is F"(x) and F"(x)=f(x)-R, and therefore F"(x z }=f"(x^-R = Q; orR=f"(x,) and we get which establishes the theorem. 164 AN ELEMENTARY TREATISE ON THE CALCULUS. The theorem has the following geometrical interpretation. If the tangent at A (Fig. 34) meet DB at R, then DR=f(a)+(b-a)f(a); DB=f(b), and therefore, both in sign and in magnitude, RB=DB-DR= $(b - a) 2 f"(x z ). Hence the deviation of the curve at B from the tangent at A, that deviation being measured along the ordinate at B, is equal to %(b of f"(x^). 73. Other Forms of the Theorems of Mean Value. The following forms may be given to Theorems II., III. If x be any number lying between a and b, then x a and 6 a are of the same sign whether a is less or greater than 6 ; therefore (x a)/(b a) is a positive proper fraction, 9 say, and we can write x = a-\-6(b a), so that any number between a and b is of the form a + 0(6 a) where 6 is a positive proper fraction. Now let b = a + h, b a = h; Th. II. will become f(a + h)=f(a)+hf(a + 0h) (Ha) and Th. III. will become f(a+h)=f(a)+hf(a)+^f(a + B l h} (Ilia) The 6 of Th. III. is not necessarily the same as the 6 of Th. II. and 6 l is used for distinction: All that is known of 6 is that it is a positive proper fraction ; it depends in feneral both on a and h. In special cases its value may be 3und. Thus, if f(x} = x 2 f(x) = 2x ; f(a + Oh) = 2 (a + Oh). But (a + hy* = a 2 + 2ah +h? = a? + h.2(a + $h), and (a + h) z = f(a) + hf(a + 0h) = a z +h.2(a+ Oh), so that in this case $=|. In Fig. 34 if APB is an arc of a parabola, M is the mid point of CD, and MP bisects the chord AB. If we replace aby x the above forms become OTHER FORMS OF THEOREMS. 165 If we make a zero and then put x for h we get Theorem II. affords another proof, though really at bottom it is not different, of Theorem VI. 58. For if f(x) is zero for every value of x, then /'(^i) is zero, and we get f(fy = f(a), that is, any two values /(a), /(&) of f(x) are equal ; in other words f(x) is a constant. Hence if '(x) F\x) is zero, the function 0() F(x) is a constant. Ex. 1. If x is positive, show that log(l+.r) is less than x but greater than x \x: = logl=0; By Th. II.c, log(l +*)= By Th. IIl.c, log (1 +x) = /(0) + Ex. 2. Show that cos x is greater than 1 - \x"*. fix) = cos x ; f(x) = - sin a; ; /'(#) = - cos x ; ^0) = 1; /(0)=0; /'(^)=-cos(^). By Th. III.c, cos x = 1 - ^ cos (0j.r) > 1 - fcc*, since cos (^j^) is numerically less than unity. It is easy to deduce that, cos.r=l - Qx 2 where 6 is a positive proper fraction less than . Ex. 3. The student may try to prove by assuming that if f(x) and its first three derivatives are continuous, will be equal to f"(.r 3 ), where x 3 lies between a and b. By putting for a and x for b we should get where 3 is a positive proper fraction. Ex. 4. By using the theorem of ex. 3, show that if x lies between and 7T/2 ^>sin.r>.r- Jar 5 ; tan x > x + Jtf 3 . How would the inequalities be stated if x lay between - ir/'2 and ? Ex. 5. If /(>) = (#- 1)5- 1, f(x) is zero when x=0 and when #=2. Does f(x) vanish for any value of x between and 2 ? 166 AN ELEMENTARY TREATISE ON THE CALCULUS. 74. Maxima and Minima. In 17, 52 attention has been called to the turning values of a function ; a turning value may be either a maximum or a minimum value of the function. A formal definition of such values may be given. DEFINITION, /(a) is defined to be a maximum value of f(x) if f(a) is (algebraically) greater both than f(a h) and than f(a + h) for every positive value of h less than a small but finite positive quantity >;. f(a) is defined to be a minimum value of f(x) if f(a) is (algebraically) less both than f(a fi) and than f(a + h) for every positive value of h less than q. It is to be noticed that a maximum value is not necessarily the greatest value the function can have nor a minimum the least ; f(a) is a maximum if it be greater than any other value of f(x) near /(a) and on either side of it. The condition for a maximum or a minimum value is easily obtained. If /(a) is a maximum value of /(#), then as x increases from a h to a, f(x) is increasing, and therefore /'() is positive ( 52) ; on the other hand as x increases from a to a + h, f(x) is decreasing, and therefore f(x) is negative. Hence as x increases through a, f(x) must change from a positive to a negative value. Con- versely, if as x increases through a, f'(x) changes from a positive to a negative value, /(a) will be a maximum value of f(x). Hence /(a) will be a maximum value of f(x) if and only if f(x) changes from a positive to a negative value as x increases through a. In the same way it will be seen that /(a) will be a minimum value of f(x) if and only if f'(x) changes from a negative to a positive value as x increases through a. This condition may be called the fundamental condition or test. For ordinary cases a simpler form may be given to the condition. Usually f'(x) will be continuous ; now a con- tinuous function can only change sign by passing through the value zero ( 45, Th. II.). Therefore, if f(a) is a turning value of f(x), /'() will be zero. .MAXIMA AND MINIMA. 167 Again, if f(a) is a maximum value of f(x),f'(x) changes from a positive to a negative value as x increases through a; therefore near a, f(x) is a decreasing function, and therefore its derivative, namely f'(x), must be negative near a. But if /"(*) is not zero, then near a the sign of f"(x) is that of / "(a). Hence f"(a), if it is not zero, will be negative when f(a) is a maximum value of f(x). In the same way we see that f"(a), if it is not zero, will be positive when /(a) is a minimum value of f(x). Conversely, /(a) will be a maximum or a minimum value of f(x) according as f"(a) is negative or positive. Hence the rule for determining the maxima and minima values of f(x) when f(x), f'(x) are continuous : The roots of the equation /'(x) = are, in general, the values of x which make /(x) a maximum or a minimum. Let a be a root of /'(x) = ; then /(a) will be a maximum value o//(x) // (a) is negative but a minimum iff"(&) is positive. When f"(a) is zero this rule for testing whether f(a) is a turning value fails ; f'(a) may be zero and yet /(a) neither a maximum nor a minimum. When /'(a) = and also f"(a) = 0, recourse may be had to the fundamental test that f(x) must change sign. It will be seen in 78 that, in general, the point on the graph of f(x) for which both f\x) and f"(x) are zero is a point of inflexion. We leave it as an exercise to the student to show that maxima and minima values occur alternately. Thus in Fig. 33, 72, which is the graph of F(x), the function is a maximum at C, then a minimum at D, then a maximum at E. At F and H on that graph the function turns though F'(x) is not zero at these points ; however F'(x) has opposite signs on opposite sides of F and H. Again at G, F'(x) is zero, yet the graph has no turning point there ; F\x} has the same sign on opposite sides of G, and G is a point of inflexion. The above conclusions, when f(x) and its derivatives are continuous at a, may also be deduced from the Theorem of Mean Value. For if /(a) is a turning value of f(x) the differences must have the same sign for small values of h : the negative sign 168 AN ELEMENTARY TREATISE ON THE CALCULUS. when f(a) is a maximum, but the positive sign when f(a) is a minimum. Now, by 73 (Ilia), '(a + Oh)}, hf(a- &h)}. When h is a very small positive number the signs of D l and Z* 2 will, if /'() is not zero, be the same as the signs of f'(a) and f(a) respectively (compare 45, Th. I.) ; therefore D l and D. 2 cannot have the same sign, and therefore f(a) cannot be a turning value unless /'() = 0. _ Again, if /"(a) is not zero the sign of f"(a + Oh) and of f'(a O'h) is the same as that of f'(a) ; therefore if f(a) = both D l and D 2 will be negative when f'(a) is negative, but positive when f'(a) is positive. We thus get the same rule as before. By Taylor's Theorem (chapter xvm.) D l and D 2 can be expressed in a series of ascending powers of h ; the same line of argument as that just followed leads to the conclusion that if f'(a),f"(a),...,f (n ~ 1} (a) all vanish, but f (n \a) does not vanish, then /(a) will be a turning value of f(x\ provided that n (the order of the first of the derivatives that is not zero) is an even integer, but not a turning value when n is an odd integer : the turning value will be a maximum or a minimum according as f (n \a) is negative or positive. It will be a good exercise to deduce this conclusion by examining the signs of the derivatives near a ; for example, show that if f'(a) and f"(a) are zero but f"(a) not zero, f'(x) changes sign, and therefore f(x) does not change sign as x increases through a, but that if f'"(a) is zero and f lv (a) not zero f"(x) does, f'(x) does not and f(x) does change sign as x increases through a. 75. Examples. Ex. 1. Find the turning values of S^ 4 - 4^ + 1. Denote the function by f(x) ; then f(x) = 1 2^ - 1 2^ ; f'(x) = 36-r 2 - 24.r . Now/(#) = 12.r 2 (.r- 1), and is therefore zero if #=0 or 1. /"(I) = 36 - 24 = 1 2 = positive number. Since /"(I) is positive, /(I) =0 is a minimum value of f(x). Again /"(0) = ; in this case consider the sign of f'(x) near 0. Let h be a small positive number ; then where only the sign of each factor and of the product needs to be written. f(x) is therefore negative both when x is a little less and when x is a little greater than ; that is, f(x) decreases as x increases from -A to and continues to decrease as x increases from to h. WORKED EXAMPLES. 169 Hence y(0) is not a turning value of f(x) ; on the graph of f(x) there is a point of inflexion where .r=0. We may prove otherwise that /(O) is not a turning value ; for /'"(0)= 24, that is, the first of the derivatives which does not vanish when #=0 is of odd order. As x increases from oo to 1, f(x) is negative, and therefore /(x) is a decreasing function ; as x increases from 1 to + oc , f(x) is positive, and therefore f(x) is an increasing function. Hence f(l) is not only a minimum value of f(x), but it is also the least value fix) can take for any value of x ; f(x) is not negative for any value of x. The student should graph the function. Ex. 2. Given the total surface, 2?ra 2 , of a right/ circular cylinder, find the cylinder of maximum volume. Denote the radius of the base by x, and the height by y ; then volume = Trx^y ; surface = 2irxy + 2irx 2 = lira?. From the second equation xy=a z x 2 ; therefore denoting the volume byA^X we get fix) = irx.xy= ir(a-x - x 3 ). Therefore /(*) = 7r(a 2 - 3* 2 ) ; f(x) = -TTX; f(x) = if x = a/^/3 ; the negative root may be discarded as irrelevant. Now f(a/J'3) is negative, and therefore f(a/\/3) is a maximum ; the maximum volume is 27ra 3 /3V3. The height is given by y = (a? x 2 )/x, and when #=a/ v /3, y = 2a/ x /3, so that the height of the cylinder of maximum volume is equal to the diameter of its base. The student should observe how the given condition enables us to express rrx 2 y as a function of the one argument x. Ex. 3. If r=acos 2 04->sin 2 0, find the maximum and minimum values of r, where a, b are positive constants. Examples of this type are most simply solved without the use of derivatives. Thus, r=|a(l + cos 20) + i6(l -cos 20)=(a + 6) + (a- 6) cos 20. Now obviously r will be a maximum or a minimum according as ^{a-ft)cos20 is so. If a>6, the greatest and least values of f(a - b) cos 20 are |(a - 6) and - ^(a - 6), so that the greatest and least values of r are a and b. These values are reversed if <&, since in that case the greatest and least values of (a- b) cos 20 are - i(a - b) and (a - b). In a similar way we can find the maximum and minimum values of x^+y 1 when .r and y are connected by the equation ax 2 +2hxy + by 2 =}. For put ,i - =rcos 0, y=rsin 0, and then x' z +y 2 becomes r 2 , where r~(a cos 2 + 2h sin cos 0+6 sin 2 0) = 1 . Now )~ will be a maximum or minimum according as l/r 1 ' is a 170 AN ELEMENTARY TREATISE ON THE CALCULUS. minimum or maximum, and we may write, from the equation between r and 6, l/r* = a cos 2 6 + Zh sin cos + b sin' 6 where R cos B' = v(a b), R sin & = h, and R = + The maximum and minimum values of 1/r 2 are given by 1/r 2 = \(a + b) ^v/{ (a 5) 2 + 4A 2 }. Geometrically, this example is the problem of finding the semi-axes of the conic whose equation is ax' 2 + 2hxy + by* = l. The values of B that give the axes are determined by cos(2(9-#')=l, 2<9=(9' or Tr + ft, 6 = %& or ^Tr + ^B 1 , so that the two axes are at right angles. The value' of O 1 is uniquely determined by the two equations R cos B' = \(a b) and R sin ff = h. The solution of problems of this kind by use of derivatives is much more tedious. Ex. 4. If f(x) = e~ ax sin (bx+c) where a, b are positive, find the turning values of f(x). f(x) = - e~ ax { a sin (bx +c) b cos (bx + c)} = -Re~ ax sin (bx+c -B), where Rcos 6 = a, RsinO = b, R= + J(a 2 + b 2 ), f(x) = R 2 e~ ax sin (bx + c - 2(9). Since e~ ax is not zero for any finite value of x, the roots of /'(#) = are those of sin (bx+c $)=0 ; therefore /'(#) is zero when bx+c-B=mr (n=0, 1, 2,...). Denoting by x n the value of x corresponding to any n, we have f"(x n ) = R^e~ ax n sin (bx n + cBB) = R 2 e~ ax n sin (mr B). Now sin (mr - B) i = - cos mr sin B ; and sin B and /{ 2 e~ ax are positive, so that the sign of f'(x n ) is the same as that of COSTITT, that is, of (-l) w+1 . Hence f(x) is a maximum for n Q, 2, 4 ..., but a minimum for = 1, 3, 5 ..., limiting consideration to zero and positive values of n. ^ _ mr c + 6 _ TT and f(%n) = e~ ax n sin (bx n + c) = e~ ax n sin (mr + B), which may be put in the form mra ac-aQ f(x^)=(-l) n e~ >> .e b .sin ft Thus the values of x for which f(x) turns form an arithmetic progression with common difference irfb ; the values of x for which f(x) is a maximum (or a minimum) have the common difference 2ir/b. If e~ ax be called the amplitude of f(x n ), the amplitudes of the maxima WORKED EXAMPLES. 171 and minima values of f(x) form a geometric progression with common _2jra ratio e b . Since D x e~ ax = -ae~ ax , the gradient of e~"" is equal to that of f(x) for those values of x, for which -ae~ ax = -Re~ ax sm(bx+c-6), that is, for which that is, for which bx + c-d=2mir + ^. + Oor (2? + l)7r-f -0 (w=0, 1,2 ...). 2t & Now when bx + c 9 (2m + 1 )TT - = - 0, sin (bx + c) = 1 , and therefore I for these values of x, e~ ax =f(x). Therefore when bx + c=2mir+Tr/2, e*"* and^^r) have the same value and the same gradient, and therefore their graphs touch at the points whose abscissae are given by these values of x. The discussion of e~ OJ: cos(6^+c') can be reduced to that of e~ oz sin(u; + c) by putting d equal to c--. FIG. 35. Fig. 35 shows the graph for a='l, 6=1, c=0. The dotted line is the graph of e ~ io* . 76. Elementary Methods. Certain types of problems can be solved very simply by elementary algebra or trigonometry. The discussion of the quadratic function or the quotient of two quadratic functions will be found in any book" on algebra ; the turning values of y where y = (ax 2 + bx + c)/(Ax 2 172 AN ELEMENTARY TREATISE ON THE CALCULUS. are found by writing the equation in the form (Ay a) x 2 + (By-b) x + Cy-c = 0, and determining the values of y that make the discriminant, that is, vanish. A little consideration distinguishes the maximum from the minimum if there are two values of y, and shows whether the solution is a maximum or a minimum when there is only one. A more general case occurs when there are more variables than one and these are connected by a relation, all the quantities being positive. For two variables the 5th, 8th, and 9th propositions of Euclid's second book or their algebraic equivalents are fundamental. (ii) (m) When the sum (x + y) of two quantities is given, we see by (i) that their product is greatest and by (iii) that the sum of their squares is least when the two quantities are equal. When the product xy of two quantities is given we see by (ii) that their sum is least when they are equal. These theorems may easily be extended. Thus let x, y, z, w . . . be n positive quantities and let their sum (a) be given ; then their product xyzw . . . will be greatest when they are all equal. For let x v y v z v w v . . . be a set of simul- taneous values of these variables ; then if any two of these, say x v y v are unequal it will be possible, without altering the sum of the n quantities, to get a greater product than x iyi z i w r" by replacing both x l and y l by K^i + ^/i) anc ^ leaving z v w v . . . unaltered, because the product of the two equal quantities K^i + S/iX that is K^i + S/i) 2 ? i g greater than x ii/r ^0 long, therefore, as any two are unequal the pro- duct has not reached its greatest value. When they are all equal each is equal to a/n, so that ELEMENTARY METHODS. 173 xyzw ... is less than ( ) . , /x + y + z + w ...\ n or xyzw ... is less than ( t, unless x = y = z = w ... = ajn. If we suppose p of the quantities equal to x, q of them equal to y, r of them equal to z, where p + q + r = n then we may write the last inequality except when x = y = z, and then the inequality becomes an equality. It is easy to see that this inequality is true even if p, q, r are positive fractions. In the same way it may be seen that when the sum of the quantities is given the sum of their squares will be least when they are all equal, and when the product of the quantities is given their sum will be least when they are all equal. These theorems may be again extended. For suppose x,y,z,w... connected by the linear equation, .. k, a constant, the quantities being all positive. Then we may put _(ax)(by)(cz)(dw)... JU U & lAJ . . . - - i - T - abed . . . and xyzw . . . will be greatest when the numerator of the fraction is greatest. But if we put x' for ax, y' for by ... we reduce the case to that in which x' + y' + z' + iv'+ ...is given. Hence the product is greatest when x', y', z', w' ... are all equal, that is when ax = by= ... =k/n. By means of the above theorems a large number of the simpler problems of maxima and minima of functions of more than one variable may be solved. For a full dis- cussion of- the algebraic treatment, see Chrystal's Alyebra, Vol. II. chap. xxiv. 174 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 1. The equilateral triangle has maximum area for given perimeter. In the usual notation for triangles, A = where x=s a, y=s b, z=s-c, 2s being constant. Now Hence xyz, and therefore sxyz, and therefore /\ is greatest when x=y=z, that is, when a = b c. Ex. 2. From the identity deduce (i) a% 2 + 6 2 y 2 + c 2 z 2 = minimum, if for + my + 712= const., (ii) lv + my+nz = maximum, if aW + 6 2 i/ 2 + c 2 z 2 = const., when 2 .r/Z=& 2 y/m=c 2 z/%. These results are obvious. We might write A, , C instead of a 2 , 6 2 , c 2 , but A, B, C must be positive. The student may prove a / \ / 1 2 m? n 2 rp \ similar theorem for ( % 2 + 6 2 y 2 + c% 2 +rf% 2 j( ~2 + Tj + ~2+^2 ), and extend to any number of variables. 77. Variation near a Turning Value. When a function f(x) and its first two derivatives are continuous near a we have f"(a + 0h) will be nearly equal to /"() when h is small, and we may write as a very approximate equation Hence when /(a) is a turning value, so that /'(a) = 0, we have Thus when f(a) is a turning value the change f(a + h) /() as x changes from a to a + h is, approximately, proportional to h 2 ; if /(a) is not a turning value the change is, approxi- mately, proportional to h. Near a turning value therefore a function changes much more slowly than near a value VARIATION NEAR A TURNING VALUE. 175 for which it does not turn, since when h is small h z is much smaller than h. If therefore in a physical application of the theory of maxima and minima it is not possible to make the arrange- ment that which corresponds to the exact solution, there will frequently be no great disadvantage in a slight departure from the theoretically best arrangement. Thus when a battery of mn cells is joined up so that m rows of n cells each, connected in series, are joined in parallel, the current y is mnE where E is the electromotive force of each cell, r the internal resistance of each cell and R the external resist- ance. Since mn is constant y will be a maximum when mR + nr is a minimum, that is, when mR = nr or when R = nr/m, that is, when the total external resistance is equal to the internal resistance of the battery. It may not be possible to join up the battery so as exactly to satisfy this condition ; but if the condition be very nearly satisfied the current will not fall far short of the maximum. In any case the nearer the arrangement can be brought to satisfy the condition the stronger will be the current. Again in applying the theory of maxima and minima to physical problems great care is necessary in drawing conclusions ; an arrangement that best satisfies one set of conditions may conflict with that which best satisfies another set of equally important conditions. Thus the above arrangement of cells gives the highest rate of working in the external part of the circuit but it is not the most economical. The student may with advantage read the remarks on pp. 85, 86 and chap. ix. of Gray's Absolute Measurements in Electricity and Magnetism. (London : Macmillan.) The theory of maxima and minima is of great value as a guide in all such investigations, but has to be applied with caution and not blindly. 176 AN ELEMENTARY TREATISE ON THE CALCULUS. EXERCISES XVI. a. The cones and cylinders referred to in the examples are right circular cones and cylinders. For the mensuration of various solids see chap. iv. Investigate the maxima and minima values of the functions in examples 1-12. 1. 2.zr 3 -3.r J - 12.Z + 2. 2. x*-5x* + 5x?-l. 3. #V+1) 3 . 4. x(a+xf(a-xf. 5. ^(l+tf 2 ). 6. (1 7. x/(ax* + 2bx+a). 8. x(x 2 +l)/(x*-x* + l). 9. 10. x/(l+xrf. 11. a + b(x-cf. 12. a + b(x+cf. 13. Find the maximum value of x m y n if x+y=k, a constant, the quantities being all positive. Hence show that \ except when a = b. 14. From the inequality in example 13 deduce that (l + l/z) z constantly increases when z is positive and increases, but decreases when z is negative and increases numerically. Hence show that the limit of (1 + 1/0)' for z QO is a finite number greater than 2'5 but less than 3. Puta = H , 6 = 1; then a=l, b = l [_ m n J 15. If a/x+b/y = e, find the least value of ax + by, the quantities being all positive. Find also the minimum value of xy. 16. For what value of x is a minimum, w l5 m 2 ,...m n being all positive. In the following examples the methods of 76 may be used; the quantities are understood to be all positive. 17. The equilateral triangle has minimum perimeter for given area. 18. The cube is the rectangular parallelepiped of maximum volume for given surface and of minimum surface for given volume. 19. Find the minimum value of bcx+cay + abz if xyz=abc. 20. Find the maximum value of xyz when and the minimum value of x?/a?+y 2 jb z + z 2 /c 2 when xyz=d?. 21. If xyz = a 2 (x+y + z\ then yz + zx+xy is a minimum when x=y = z=a*j3. 22. The electric time-constant of a cylindric coil of wire is approximately u = mxyz/(ax+by + cz) where x is the mean radius, y the difference between the internal and external radii, z the axial MAXIMA AND MINIMA. EXERCISES XVI. a, b. ]J7 length, and m. a, 6, c known constants; the volume of the coil is nxyz. Show that when the volume V is given, u will be a maximum when cu: = by = cz *J(abcVjn). 23. If u = (a^ 2 +b^)j^(a z ^ + b 2 y-) where # 2 +2/ 2 = l show that the minimum value of u is 2J(ab)/(a + b). 24. If P is a point within a triangle ABC such that AP Z + EP 2 + CP* is a minimum, show that P is the centroid. 25. In any triangle the maximum value of cos A cos B cos C is \. 26. Find the greatest rectangle that can be inscribed in an ellipse whose semi-axes are a, 6. EXERCISES XVI. b. 1. ABCD is a rectangle, and APQ meets BC in P and DC produced in Q. Find the position of APQ when the sum of the areas ABP, PCty is a minimum. 2. Given one of the two parallel sides (a) and the two non-parallel sides (b) of an isosceles trapezium, find the length of the fourth side so that the area of the trapezium may be a maximum. 3. From a rectangular sheet of tin, the sides being a and 6, equal squares are cut off at each corner, and a box with open top formed by turning up the sides. Find the side of the square so that the box may have maximum content. 4. An open tank is to be constructed with a square base and vertical sides to hold a given quantity of water ; show that the expense of lining the tank with lead will be least if the depth be half the width. If the tank be cylindrical show that the depth will be equal to the radius of the base. If the section of the cylinder is not circular but if its shape is given show that the curved surface will be twice the base. 5. Show that the altitude of the cone of maximum volume that can be inscribed in a sphere of radius R is 4/2/3. Show that the curved surface of the cone is a maximum for the same value of R. 6. A cone is circumscribed about a sphere of radius R ; show that when the volume of the cone is a minimum its altitude is 4/2 and its semi- vertical angle sin" 1 ^. 7. Show that the altitude of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2/?/ N /3. Show that when the curved surface is a maximum the altitude is RJ2. Show that when the whole surface is a maximum that surface is to the surface of the sphere in the ratio of N /o + 1 to 4. G.C. M 178 AN ELEMENTARY TREATISE ON THE CALCULUS. 8. A cylinder is inscribed in a cone ; show that its volume is a maximum when its altitude is one-third that of the cone. Show that the curved surface is a maximum when the altitude is half that of the cone. Show also that the total surface cannot have a maximum unless the semi-vertical angle of the cone is less than tan- 1 !- 9. Given the total surface of a cone show that when the volume of the cone is a maximum the semi-vertical angle will be sin" 1 ^. Given the volume of the cone show that the total surface will be a minimum for the same value of the semi-vertical angle. 10. PP' is a double ordinate of the ellipse whose equation is # 2 /a 2 +y 2 /& 2 =l and A is one end of the major axis; find when the triangle APP' has maximum area. Find also when the cone formed by the revolution of the triangle about the major axis has maximum volume. 11. The strength of a rectangular beam varies as the product of the breadth and the square of the depth ; find the breadth and the depth of the strongest rectangular beam that can be cut from a cylindrical log, the diameter of the cross-section being d inches. 12. The stiffness of a rectangular beam varies as the product of the breadth and the cube of the depth ; find the breadth and the depth of the stiffest rectangular beam that can be cut from a cylindrical log, the diameter of the cross-section being d inches. 13. A person in a boat a miles from the nearest point A of the beach wishes to reach in the shortest time a place b miles from A along the beach ; if he can row at u miles an hour and walk at v miles an hour (u = c, show that a cos d+bcos^ is a minimum when = , a, 6, c being positive and the angles 6, acute. (Com- pare ex. 15a.) 6. Given the length (a) of an arc of a circle, show that the segment of which a is the arc will be a maximum when a is the diameter of the circle. 7. A circular sector has a given perimeter ; show that when the area is a maximum the arc is double the radius, and that the maximum area is equal to the square on the radius. 8. From a given circular sheet of metal it is required to cut out a sector so that the remainder can be formed into a conical vessel of maximum capacity ; show that the angle removed must be 2(1 - JV^) 71 " radians (66 4'). 9. Draw a line through the vertex of a given triangle such that the sum of the projections upon it of the two sides which meet at that vertex may be a maximum. 10. The lower corner of a leaf, whose width is a, is folded over so as just to reach the inner edge of the page ; find the width of the part folded over when the length of the crease is a minimum. 11. A ship sails from a given place A in a given direction AB at the same time that a boat sails from a given place C ' ; supposing the speed of the ship to be u and that of the boat v (u, v constant), find in what direction the boat must sail so as to meet the ship. Discuss the condition that it shall be possible to meet. The course of the boat is understood to be rectilinear. 12. The distance between the centres of two spheres of radii a, b respectively is c ; find at what point P on the line of centres AB the greatest amount of spherical surface is visible. Note. The superficial area of a segment of height h is Zirah. a being the radius of the sphere ( 85, ex. 2). 13. A straight line is drawn through a fixed point (a, ft), meeting the axis OX at P and the axis OF at Q, the axes being rectangular and a, b positive ; if the angle OPQ is equal to 0, find B, (i) when PQ, (ii) when OP+OQ, (iii) when OP. OQ is a minimum, 180 AN ELEMENTARY TREATISE ON THE CALCULUS. 14. A tangent is drawn to an ellipse, whose axes are 2a, 26, such that the part intercepted between the axes is a minimum ; show that its length is a + b. 15. If D= (f> and sin<=/Asin<' where ^ is greater than 1, and , ' not greater than ir/2, show that D increases as < increases. Show also that the second and third derivatives of D with respect to < are positive. 16. A ray of light travels in a plane perpendicular to the edge of a prism of angle i ; if the angle of incidence is < and the angle of emergence <', show that the deviation i is a minimum when 17. Find the maximum value of xe~ x and graph the function. 18. Find the minimum value of x log x and graph the function. 19. For what value of x is the ratio of log x to x greatest ? 20. Find the maximum value of x 2 log -. X 21. If a, b are positive and aJl}. From ,v= - QC to #=J(2- v /7), and again from #=J(2 + v /7) to #=+oo, the graph is concave upwards; from # = (2-^7) to #=(2+^7) it is convex upwards. Ex. 3. -2 3 = #-4. 182 AN ELEMENTARY TREATISE ON THE CALCULUS. When #=4, ?/ = 2, but y 1 and ?/" are both infinite. When # = 4 A, ?/' is positive, but when x=4 + h, y" is negative. We may conclude therefore that the tangent at (4, 2) is perpendicular to the .r-axis, that to the left of (4, 2) the curve is concave upwards, and that to the right of (4, 2) it is convex upwards. The point (4, 2) therefore must be considered a point of inflexion. EXERCISES XVII. 1. Determine the points of inflexion of the graphs of the following functions, and state for what range of values of x they are concave upwards or convex upwards. (i) a 3 ; (ii) a 4 ; (iii) ^; (iv) ar n+l ; (v) sr n \ n being a positive integer. 2. Find the points of inflexion on the curve whose equation is y = (y& l) 2 . Graph the curve. 3. Find the points of inflexion and graph the functions 4. Show that the curve whose equation is ,y(;c 2 + a 2 ) = a 2 (a-.r) has three points of inflexion which lie on a straight line. 5. Find the points of inflexion on the curve whose equation is a?y 2 =x 2 (a 2 x 2 ), and trace the curve. 6. Show that the curve whose equation is (a .v)y 2 =x 3 has no point of inflexion, and trace the curve. 7. Find the points of inflexion for values of x between and 2ir (0 included, 2rr excluded) on the graphs of (i) sin#; (ii) COS.T; (iii) tana;. 8. Show that the graphs of e* and of logx have no points of inflexion. 9. Find the points of inflexion on the graphs of (i) xe~ x , (ii) e~ x ". Trace the graph of e~ x . 10. Find the point of inflexion on the graph of e~ ax -e~ bx where a, b are positive and a less than b. 11. Find the points of inflexion on the graph of e~ ax sin(bx + c). 12. When the equation of a curve is given in the form *=/(*), y=KO show that the points of inflexion will be determined by the equation xy-yx=. Show that the curve whose equations are x=a(t sint), y=a(l-cost) is everywhere convex upwards. (See 68, ex. 2.) 13. Show that no conic section can have a point of inflexion. CHAPTER X. DERIVED AND INTEGRAL CURVES. INTEGRAL FUNCTION. DERIVATIVES OF AREA AND VOLUME OF A SURFACE OF REVOLUTION. POLAR FORMULAE. INFINITESIMALS. 79. Derived Curves. It is of some service in tracing the variation of a function f(x) to draw the graph of the derived function /'() The graph of f(x) may be called the derived graph or curve of f(x), while in relation to the graph of /'() that of f(x) may be called the primitive curve or for a reason given in 83, the integral curve of /'(#) It is usually most convenient to take a common axis of ordinates for the two curves, but to take the axis of abscissae of the derived curve at a convenient distance below that of the primitive curve. Assuming the unit seg- ment for abscissae to be the same for both curves, but that for the ordinates to be the same or different as is most con- venient, we may call those points and ordinates of the two curves which have the same abscissa " corresponding points and ordinates." Corresponding points on the two graphs may be denoted by unaccented and accented letters. The student will easily prove that in general the follow- ing theorems hold : (i) To the turning points (T) of the primitive curve cor- respond points (T) at which the derived curve not only meets but crosses its axis of abscissae ; and conversely. (ii) To the points of inflexion (7) of the primitive curve correspond turning points (/') of the derived curve; and conversely. 184 AN ELEMENTARY TREATISE ON THE CALCULUS. The following geometrical construction may be given for the graphing of the derived curve when the functions f(x), f(x) are not analytically expressed. Let U (Fig. 38) be a point to the left of O l on the axis 0-^K' of the derived curve, and draw Up parallel to the tangent PT meeting the common axis of ordinates at p. Draw PR and RT parallel and perpendicular respectively to OX ; the triangles PRT, UO^p will be similar. Hence P' where OM=O l M' = x, MP=f(x); therefore O lP =f(x).UO r Draw pP' parallel to UO-^X' to meet M'P at P'; then M'P'=O lP =f(x). U0 r If we take the unit segment for the ordin- ates of the derived curve equal to UO^ we shall have M'P'=f(x\ so that P' is the point corresponding to P. Take any other point Q; draw Uq parallel to the tangent QS, and qQ' parallel to 1 JT' to meet the ordinate through Q at Q'. Q' will correspond to Q, and in the same way any number of points may be found. If the unit segment for the ordinates be not equal to U0 l the ordinates will still be proportional to /'(#) To the turning points B, C correspond B', C'; to the point of inflexion 1 corresponds /' which is a turning point of the derived curve. At D the derivative f'(x) is discontinuous. As a point moves along the primitive curve from C to D the corre- FIG. 38. DERIVED CURVES. 185 Spending point moves from (7 to D' ; as the first point, however, changes from CD to DE the corresponding point passes abruptly from D' to D". As x increases through the value OL or 0-^L', f'(x) suddenly changes from L'D' to L'D". LD is a maximum value of f(x), but owing to the dis- continuity of f'(x) the derived curve does not (as at B' or C') meet the #-axis. In a similar way the derived curve of f'(x), that is the second derived curve of f(x), may be formed, and so on. 80. Derivative of an Area. Let CPD (Fig. 39) be the graph of a continuous function of x, F(x) ; OA = a, AC=F(a) ; OM = x, MP = F(x). (i) Suppose the ordinates positive and A C to the left of MP. Let A G be fixed, MP variable, and let z be the measure of the area A MPC. We may consider the area as generated by a variable ordinate setting out from AC and moving to the right; z will be a function of x which is zero when x a. Let us find the a'-derivative of z, that is the x-r&te of change of the area. Let x take the increment Sx or MN\ an increment Sz, the area MNQP. M N FIG. 39. z therefore will take Complete the rect- angles MNRP, MNQS; the area MNQP will be greater than MNRP but less than MNQS, therefore MP.Sx, show that Ex. 2. If CE, PF (Figs. 39, 39a) are perpendicular to OY, and if w is the measure of the area EFPC, snow that D y w = -FP=-OM; dw=- xdy, the sign of w being positive or negative according as the area is to the left or to the right when its boundary is described in the order EFPCE. Consider the cases in which the abscissa is negative, and also the cases in which the fixed abscissa is on the opposite side of FP from that in the figures. 81. Interpretation of Area. The interpretation of the number z considered as the measure of an area will depend on the unit segments chosen for the abscissa and the ordinate. If the value 1 of # represents say 6 inches and the value 1 of y say 10 inches, then the value 1 of z will represent 60 square inches ; if on the graph the value 1 of x is half an inch and the value 1 of y quarter of an inch, these representing 6 and 10 inches respectively, an area on the graph of one-eighth of a square inch will represent the area of 60 square inches. The physical interpretation of the area will depend on the nature of the quantities represented by abscissa and ordinate. Suppose that the ordinate represents the speed of a moving point and the corresponding abscissa the time at which the point has that speed ; the graph is then the speed -curve of the motion. The speed is the time-rate of change of the distance, and the ordinate (which represents the speed) is the rate of change of the area with respect to the abscissa (which represents the time); hence the area AM PC will represent the distance gone in the time repre- sented by AM. If the value 1 of a; represents 2 seconds and the value 1 of y a speed of 16 feet per second, then the value 1 of will represent a distance of 32 feet. If the ordinate represents a force that acts in a constant direction, and if the abscissa represents the distance through which the force has acted, the area AM PC will represent 188 AN ELEMENTARY TREATISE ON THE CALCULUS. the work done by the force acting through the distance represented by AM. If the force is not constant in direc- tion the result holds provided the ordinate represents the component of the force along the tangent to the path of its point of application. Ex. 1. If the ordinate represents acceleration and the abscissa time, what does the area represent ? Ex. 2. If the ordinate represents the intensity of pressure of a gas, and the abscissa the volume, what does the area represent ? 82. Integral Function. The fact that z in 80 is a function of x which has F(x), the ordinate of the curve CPD, as its derivative at once suggests the problem of finding a function which has a given continuous function as its derivative. Now, if the derivative of f(x) is F(x) so is the derivative of f(x) + G where G is any constant; further ( 58, Th. VI.), every function which has F(x) as its derivative must be of the form f(x) + G. The problem, therefore, as stated above, is indeterminate, since its solution involves a constant G which may have any value whatever; it becomes deter- minate, however, when stated in the form : To find a function of x which shall have a given continuous function F(x) as its derivative and which shall take a given value A when x has a given value a. The solution is as follows : Let CPD ( 80) be the graph of F(x), and let z be the measure of the area A MFC where OAcb. z therefore is zero when x = a, and z has F(x) as its derivative; the function z-\-A gives the solution. We may, if we please, consider the constant A as the measure of an area. It does not follow, however, that we can find an analytical expression for z in terms of known functions ; thus, if F(x) = ^(\ +.Z 3 ), we cannot find in the ordinary algebraic or transcendental functions one which has F\x) as its derivative. The geometrical discussion shows, however, that in so far as we consider functions as being adequately represented by graphs, there always exists a function which is the solution of the problem, and it is possible by methods of approximation to get an analytical expression, for example, in the form of a series, that may be considered as a solution. Or, again, it may be possible by mechanical methods to get an approximate value of the area AM PC. INTEGRAL FUNCTION. 189 Any function f(x) which has F(x) for its derivative is called an Integral Function or simply an Integral of F(x). If f(x) is one integral, f(x) + C is called the General Integral, being called the constant of integration. To find f(x) when F(x) is given we fall back on the known results of differentiation. In the Integral Calculus the search for integral functions is systematically carried out, but from the nature of the case the process is largely tentative. The fundamental test that f(x) should be an integral of F(x) is that D x f(x) should be equal to F(x). Just as sin" 1 a; means a function whose sine is x so we may for the present use the symbol D x ~ l F(x) or D~ l F(x) as meaning a function whose derivative is F(x), that is D x ~ l F(x) is an integral of F(x). We will suppose that D~ l F(x) contains no constant of integration, so that the general integral is D~ l F(x)+C* We may now express the area AM PC in the new notation. Since D" 1 F(x) is an integral of F(x), the area z or AM PC is a function of x of the form z=D- l F(x) + C. Now when x = a, z=Q; denote by [D" 1 F(xJ] a the value of the integral when x = a ; therefore, and z = D- l F(x)-[D- l F(x)] a . The area A BCD is the value of z when x=b; therefore area ABCD = [D~ l F(x)] b -[D- 1 F(x)] a . This symbol is usually contracted into and this last symbol means " replace x by 6, then replace x by a and subtract the second result from the former. " In the same way the function whose derivative is F(x) and which is equal to A when x is equal to a is denoted by For the ordinary notation for an integral see 110, 190 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 1. Find the area between the graph of # 2 -3.r + 2, the /r-axis, and the ordinates at x and ,v=\. F(x) = x"- - Zx + 2 ; D~ l F(x) = A^ 3 - as may be tested by differentiation. Hence the required area is Suppose the ordinates to be those at x=\ and ,r=2; then the area is The reason for this apparently strange result is that from x=\ to x=\ the ordinates are positive while from x = \ to x = 2 they are negative. From x=\ to #=2 the measure of the area is negative; numerically this area is equal to that for which the ordinates are positive. Ex. 2. The area between the .r-axis and the graph of sin x between the points #=0, XTT at which it crosses the axis is [D~ 1 sinxY=[-cosx] 7r = -cos7r-(-cosO) = +1 + 1 = 2. Ex. 3. A point moves on a straight line so that its velocity at time t is Fcos nt ft./sec. ; show that the space described from time =0 till it first comes to rest is V/n ft. Let x ft. be the distance described in time t seconds ; then y When t=0, x=0 and therefore #=0. The point first comes to rest when t has increased from to ir/2n because cos nt is first zero when nt = ir/2. Hence we get for the distance required V . IT V sm- = n 2 n 83. Integral Curve. The graph of an integral function is called an integral curve. Since any two integral func- tions of F(x) differ only by a constant C, the graph of the integral function f(x)+C may be obtained from that of f(x) by shifting the latter parallel to the i/-axis through the distance C. A geometrical construction may be given for graphing an integral curve based on that for the graphing of the derived curve ( 79). Divide O^X' (Fig. 40) into equal short segments at the points lj, 2 1? 3 X , ... and draw the ordinates through these points. Let the ordinates at 2j, 4 J5 ... meet the graph of INTEGRAL CURVE. 191 F(x) at 2', 4', ... and let 2", 4", ... be the projections of these points on 0-^Y, I" being the point where the graph cuts O^Y. Let us take the integral curve that passes through 0. Then the tangent at is parallel to UV. Let this tangent be drawn and let it cut the ordinate drawn from l x at 1. The tangent at the point on the integral curve corre- sponding to 2' is parallel to VI". Draw 13 parallel to U2" cutting 2 X 2' at 2, and the ordinate drawn from S l at 3 ; 2 is the point corre- sponding to 2'. In the same way draw 35 parallel to 74" cutting 4 X 4' at 4 ; 4 is the point corre- sponding to 4'. The construction may be repeated and we get a series of lines 01, 13, 35, ... which may be considered as, approxi- mately, the tangents at 0, 2, 4, ...to the integral curve. That curve may now be drawn with a free hand through the points 0, 2, 4 ____ The point from which the con- struction begins is, of course, arbitrary, but when that is fixed the integral curve is determinate. The position of the other points 2, 4, . . . is approximate ; the nature of the approximation and the justification of the construction may be seen thus. Let f(x) be the integral function ; the equations of the tangents at the points on the graph of f(x) at which x is equal to a and b respectively, are FIG. 40. The abscissa of the point of intersection of these tangents is given by 192 AN ELEMENTARY TREATISE ON THE CALCULUS. Now, by the theorems of mean value, if b = a + k, we have where x v x z are each greater than a but less than a+h. Substituting these values in the equation for x and reducing we get Assuming the derivatives continuous, then if h is small /"(i) and f"(x 2 ) will differ very little from each other and from f"(a). Therefore approximately x = a + ^h; that is, the abscissa of the point of intersection of the tangents is very nearly that of a point half way between a and 6. Hence, in the figure the tangent at the point on the integral curve corresponding to 2' passes through 1 ; the point 2, which must lie on the ordinate 2 X 2', is therefore got as the intersection of the line through 1 parallel to C/"2". Similarly for the other points. It may be noticed that if F(x) is of the first and, there- fore, f(x) of the second degree, the construction is exact since f"(x) is constant. 84. Graphical Integration. The area between O^X', T Y, the graph of F(x) and the ordinate M'P' (Fig. 40) is equal to f(x) where f(x) is that integral of F(x) which is zero when x = 0. But the ordinate MP of the integral curve is f(x). Hence the area 0-^M'P'Y is equal to the ordinate of the integral curve at the point corresponding to P r . We thus have a graphical method of finding the measure of an area and also of constructing an integral function even when the analytical form of the function F(x) is not assigned. The integral curve can be drawn with considerably greater accuracy than the derived curve. It is also possible to trace out the integral curve corresponding to a given curve by means of an instrument called an Inte- graph. For detailed description of the Integraph the reader is referred to the work of M. Abdank-Abakanowicz, Les Intfyraphes ; la courbe integrate et ses applications (Paris: Gauthier-Villars), or to the German translation by GRAPHICAL INTEGRATION. 193 Bitterli, Die Integraphen (Leipzig : Teubner). The con- structions given above are taken from this work ; the notes of Bitterli contain several investigations on the properties of the integral curve and also numerous references to original memoirs. An article by Prof. W. F. Durand in the Sibley Journal of Engineering, January, 1897, will also be found serviceable. 85. Surfaces of Revolution. Let V be the volume of the surface traced out by the revolution of the arc CP (Fig. 41) about OX ; OM = x,MP = F(x) = y. To find D x V. When x increases by MN or Sx, V increases by SV, the volume traced out by MNQP. Clearly, when Sx is small, SV is greater than the cylinder of height MN and base the circle of radius MP, but less than the cylinder of height MN and base the circle of radius NQ ; therefore Hence taking the limit for Sx = Let 8 be the area of the surface traced out by the arc GP, and let CP be s. To find D X S. On the tangent at P take a length PT equal to the arc PQ, and let L be the foot of the ordinate to T. When x increases by MN or Sx, CP increases by the arc PQ or Ss; we may assume that the area traced out by the arc PQ is, when Sx is small, greater than that traced out by the chord PQ but less than that traced out by the tangent PT. If the arc PQ lies below the chord PQ the inequalities will be reversed. The curved surface of the conical frustum having MP, NQ as the radii of its circular ends and the chord PQ for slant side is Tr(MP+NQ)PQ ; the surface traced out by PT is similarly Tr(MP+LT)PT. Hence M ML X FIG. 41. 194 AN ELEMENTARY TREATISE ON THE CALCULUS. The limit for Sx = Q ot PQ/Sx and of PT/Sx is D^s ( 62) and the limit of MP+NQ and of MP+LT is 2A1P; hence D = 1-jrMP DJ = Z or dS=2-7ry--j- The volume V is that integral of -jry z which is zero when x= OA, and the surface S that integral of 'l-jry ds/dx, which is zero when x = OA. l Ex. 1. If the curve revolves about OY show that dV=Trx*dy; Ex. 2. Show that the volume of a spherical cap of height h is irA 2 (./2 \h) and that the area of the surface of the cap is R being the radius of the sphere. The equation of CPQ is y=J(B?-a?); hence V=0 when x=OA=R h, and therefore C= - * (%& -Rh 2 + JA 3 ) ; F= TT(X& - ^ - 7r(/P - Rh* + A 3 ). The volume required is the value of V for x=R and is therefore xA'(fl-JA). Again therefore So that S=2irR(x+h-R\ and when x=R, S=2irRh. Ex. 3. If the area of a section of a surface by a plane perpendicular to OX is a known function of x, F(x), and if I" is the volume contained between a fixed plane perpendicular to OX and the plane which gives the section of area F(x), show that SURFACES OF REVOLUTION. 195 86. Infinitesimals. The student will doubtless have noticed that in finding derivatives a good deal of work would have been saved had it been possible to reject at the outset those parts of an expression that had zero for limit. Thus in 80 Sz consists of the rectangle MNRP and the curvilinear triangle PRQ, which is less than the rectangle PRQS. Sz/Sx is therefore the sum of MP and of a line which is less than RQ. Since RQ converges to zero with Sx we may, so far as the limit is concerned, throw aside RQ from the outset ; we should thus at once obtain MP as the limit of Sz/Sx. Now that the student has had so much practice in finding derivatives and limits generally, he will be ready to grasp the method which enables us to reject, at any stage, a quantity which we can see will not occur in the limit ; the method is that of Infinitesimals. DEFINITION. A variable quantity whose limit is zero is called an infinitesimal A constant, however small, is not an infinitesimal in the sense now defined ; an infinitesimal is a variable quantity. Let a, /3, y . . . , be infinitesimals, and let (3, y . . . , be such that when a converges to zero /3, y ... also converge to zero ; /3, y ... are dependent on a and we can compare them with a and with one another. When a is taken as the standard of comparison a is usually called the principal infinitesimal. ft is said to be an infinitesimal of the same order as a when a=0 where k is a finite number not zero. When k is zero /8 is said to be an infinitesimal of a higher order than a ; when k is infinite ft is said to be an infinitesimal of a lower order than a. When the limit of /3/a is infinite j8 is sometimes called an infinite with respect to a. In practice one infinitesimal is chosen as principal infini- tesimal and the other infinitesimals are said to be of a certain order, first, second, etc., the principal infinitesimal being either explicitly stated or sufficiently indicated by the context. 196 AN ELEMENTARY TREATISE ON THE CALCULUS. /3 is defined to be an infinitesimal of order n with respect to a, n being positive but not necessarily integral, when where k is a finite number, not zero. By the definition of a limit we may write a n = k + a> or 3 = k where &> is a variable that converges to zero with a, that is to is an infinitesimal. The difference (3 ka n or aia n is an infinitesimal of a higher order than a 11 because the limit of (j$a n /a n , that is of a), is zero. ka n is called the principal part of /3; manifestly the ratio of an infinitesimal to its principal part has unity as its limit. If L{3 a = k, n, but an infinite of order n m if m < n. For and in the same way the quotient theorem may be proved. Ex. I. sin a, 1 cos a, sin a (1 cos a) are of the 1st, 2nd, 3rd order respectively with respect to a. For T sin a , T 1 - cos a i T sin a (1 - cos a) , Li -- = 1; Li --- 5 - = ; JU ^ 3 ~ => a=0 a a =0 or a=0 and their principal parts are a, |a 2 , ^a 3 respectively. Ex. 2. If /S = V(9ci-2a 2 +3a 3 ), ft is of order and its principal part is 3,/a. For L ^-= L v /(9-2a + 3a 2 ) = 3. a=0 * a =0 Ex. 3. tan a - sin a is of the 3rd order and its principal part is a 3 . This follows at once from ex. 1. INFINITESIMALS. 197 87. Fundamental Theorems. The value of the explicit discussion of infinitesimals depends on the principle that so far as the limit of an expression is concerned we need only in general attend to the principal part ; the other terms of the infinitesimal being of a higher order than the principal part will have to that part a ratio whose limit is zero, and may therefore be discarded from the outset. If an expression contain a finite constant term A and infinitesimals a, (3, y . . . , then so far as the limit is con- cerned we may, in general, at once replace A + a + /3 + y+ ... by A. The essential thing is to find out the order of the expression ; in comparison with infinitesimals the principal part alone need be retained, while in comparison with finite quantities no infinitesimal need be retained. The order of an infinitesimal /3 + y + ... is, of course, that of its principal part. Care must, however, be exercised in applying this prin- ciple. Thus 1 cos a + sin a contains the constant term 1 ; but 1 cos a is of the second order, sin a of the first. Hence the whole expression is an infinitesimal of the first order, its principal part being a. The following are the two fundamental theorems. THEOREM I. The limit of the quotient of two infini- tesimals is not altered by replacing each infinitesimal by another having the same principal part. Let /3, y be two infinitesimals. In order that their quotient should have a finite limit, not zero, each must be of the same order. We may therefore write, the order being 7i, /3 = fca" + o)a w ; y = k'a n + co'a n . 1^ &> 7i be two other infinitesimals having the same principal parts as ft y respectively ; then ft = /.'a n + ftjja 71 ; y! = k'a n + w/a", where a> 1( a>/ are infinitesimals different from w, a/. Now, T ft T k + ( g l k T /3 Li = Li jj- - , r, = Lt a = 07l a = (A +&>! * a = 0y The reasoning would clearly hold if (3 were of higher order than y, for the limit both of /3/y and of ft/yj would be zero. If /3 were of lower order the theorem would hold in 198 AN ELEMENTARY TREATISE ON THE CALCULUS. the sense that the limit both of /3/y and of fi l ly l would be infinite. _, T sin ax T ax a Ex. L - - = L j = j-. z-otan ox X-QOX o From its great use in the differential calculus this theorem is often called the fundamental theorem of the differential calculus. THEOREM II. The limit for n infinite of the sum of n infinitesimals is not altered by replacing each infinitesimal by another having the same principal part, provided all the infinitesimals are of the same sign. The theorem is not necessarily true if the infinitesimals are not all of the same sign. Let % = & + &+ ... + /3 n ; V n = y 1 + y 2 + ... + y n , where /3 l has the same principal part as y 1? /3 2 as y 2 . . . . The principal infinitesimal, previously denoted by a, is here 1/n, and therefore the limit of each of the quotients /i^/y^ P^/yz . . . for a = or 71 = oo is unity. Of course the principal parts of /3 V /3 2 , /3 3 . . . are not necessarily the same. It is a known theorem of algebra that when the quanti- ties (3 V yi ? /3 2 , y 2 > are all of the same sign the fraction u n /v n lies in value between the greatest and the least of the fractions /^/y^ /3 2 /y 2 Hence, for every value of n the fraction ujv n lies between two fractions each of which has the same limit, unitjr. Therefore, and therefore if v n converges to a limit, u n will converge to the same limit, that is L u n = L v n . H = 00 71 00 Ex. Let /3p=n/(n+p) 2 , y p =n/(n+p)(n+p + l); then the limit of (3 p /y p for ?i = oo is unity for every integral value of p. But y p = n/(n +p) - n/(n +p + 1) ; _/ n n \ f n n \ I n n \ n ~\n+l n + 2/ \n + 2 n + 3; \n+n n + n + 1/ n n THEOREMS. WORKED EXAMPLES. 199 From its use in integration this theorem is often called the fundamental theorem of the integral calculus. Ex. 1. the When dx is the principal infinitesimal, then ( 60) principal part of 8f(x) is df(x)=f(x)djc, and that of 8f(x) is df'(x) = f"(x)dx. If y(tf) = tan< then the principal part of 8 tan< is c?tan =f"(x)dx. Ex. 2. Let PQ be the graph of f(x) ; PT, TQ the tangents at P, Q. Let h or PR be the principal infinitesimal RS, PS, PQ are of the first order. Let f"(x) be finite, not zero, from y x=a to x=a + h; then by Th. III., 72, FIG. 42. Hence SQ is of the second order. 8(f> is of the first order and its principal part is h cos 2 and also (ex. 1) to hf"(a\ so that Again, a and (3 are of the first order. For sinPSR=co8, and sina_ sin a sinPSR_SQ cos h ~sinPSR' h~'~P' h ' sin a, and therefore a, is thus of the first order ; the principal part of a is \h cos 2 . Since f3 = 8(f> a its principal part is equal to that of a, that is, to half that of 8(j>. D/7T ain /3 ^YD J^^P so that PT, TQ are of first order. Also PT+ TQ-PQ = PT(l -cosa)+TQ(l- cos /?) ; so that the difference between PT+ TQ and PQ is of the third order, since PT and TQ are of the first and (1 - cos a) and (1 - cos ft) of the second. Hence the difference between PT+ TQ and the arc PQ is at least of the third order since the arc PQ is greater than PQ. The fact that the limit of PT/PQ is 1/2 is sometimes expressed in the words " PT is ultimately equal to \PQ " or " PT is in the limit equal to $PQ." Similarly it is said that "the triangle PTQ is ultimately isosceles." This phraseology, though occasionally con- venient, is apt to lead beginners astray. If f'(a)=Q, SQ is of a higher order than the second, and 8, a, ft are also of higher order than the first, and PT+TQ-PQ of higher than the third. 200 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 3. In Fig. 24, 39, if MA be principal infinitesimal, prove (i) AT, arc AN of first order, (ii) MN, NT, MT of second order. Draw MC perpendicular to A T ; then prove (iii) MC of second order, (iv) CT of third order. Ex. 4. Show that (Fig. 42) if arc PQ = 8s ds s=o8s 88. Polar Formulae. Let APQ (Fig. 43) be a curve whose polar equation is r=f(6); letLXOP = 6, LPOQ = 0; OP = r, OQ = r+Sr. Draw QR perpendicular to OP. FIG. 43. We will consider the arc PQ positive when the angle POQ is described by a positive rotation of the radius vector OP; the tangent PT is to be drawn towards the positive direction of PQ and by the angle, \fs say, between the tangent PT and the radius OP is meant the angle RPT between the outward drawn radius OP and the tangent PT. (i) To find tan ^, tan RPO - * - ^~- Sr cos SO - r ( I - cos 86) POLAR FORMULAE. 201 If 86 be principal infinitesimal, Sr is of first order since dr/d6 is in general finite ; therefore Sr sin 66 and (1 cos 86) are of second order. We may, therefore, omit the quantities of the second order and put 66 for sin 66 and 1 for cos 66. Hence tan \lr = L tan RPQ = L r^- = r-r- Sr dr (ii) To find the derivative of the arc. Let AP = s, arc PQ = 6s; then retaining only infinitesi- mals of the lowest order and remembering that PQ and arc PQ are of the same order we get _ ~ \d6 ~ I (66)* or ds = J{dr z + r 2 d6 2 }, and sin -^ = rd6/ds, cos \js = dr/ds. (iii) To find the derivative of the area. Let sector AOP = z, sector POQ = 6z; then 8z is inter- mediate to the circular sectors of angle 86 and radii OP, OQ respectively. Hence 8z/86 lies between |r 2 and |(r+={ 1 +(/ 2 F//- If SQ is produced to meet the circle at Q 1 , show that the limit of SQf is 2 sec 2 ;r or sometimes ( -5 )> I -j- ) ?>x dy \dx/ \dy/ PARTIAL DIFFERENTIATION. 205 the form "d instead of d, or the bracket, indicating that the derivative is partial. Notations analogous to f'(x),fx(x) are also in use. Thus /*'(> 2/)> /*' OB. y\ MX, y), f y (x, y\ u x , u y , denote partial derivatives of the functions f(x, y) and u. There is no notation, however, that is in itself quite free from ambiguity ; the reader must usually infer from the context whether a derivative is partial or not. The formal definition of "dufdx, 'du/'dy where u=f(x, y) is, therefore, ^ f(x + Sx, y)-f(x, y) . ^= L -f(x, y\ Ex. 1. If u=i <3w/3.r=i Ex. 2. If w=sii 'du/'dx = a cos (ax + by + c) ; c). II* 89a. Coordinate Geometry of Three Dimensions. A knowledge of coordinate geometry of three dimensions will greatly assist the reader in obtaining a clear conception of partial derivatives ; we will therefore give in this article a few fundamental theorems regarding the representation of points, lines, and surfaces by means of three coordinates. In many cases the extension from two to three co- ordinates is extremely simple. (i) Coordinate Planes and Axes. Coordinates of a Point. Through a point let three planes YOZ, ZOX, XOY be drawn, the angle be- tween each pair of planes being 90, and suppose the planes to be pro- ducea indefinitely, their intersections being the lines X'OX, TOY, Z'OZ; these lines will be mutu- ally at right angles. We will suppose Y'OY and Z'OZ to lie in the plane of the paper and the portion OX to be drawn upwards .towards the reader (Fig. 44). 206 AN ELEMENTARY TREATISE ON THE CALCULUS. From any point P draw PN perpendicular to the plane XOT and NM perpendicular to the line X'OX; complete the parallelepiped. The position of P will be determined by the segments OM or M'P. OL'-or LP, ON' or NP. The three planes YOZ, ZOX, XOT are called the coordinate planes, the three lines X'OX, Y'OY, Z'OZ the coordinate axes and the three segments OM, OL', ON' the coordinates of P ; is the origin of coordinates. The positive directions of the axes and therefore of the segments or coordinates are from to X, from to 7, from to Z respectively. P may be denoted the point (x, y, z) where x=OM=L'N=M'P; y = OL'=MN=LP ; z = ON' = L'M'=NP. The coordinate planes divide space into eight portions (octants) and there will be eight arrangements of the signs +, corresponding to the octant in which the point is situated. Thus when the signs are ( + , +, +) P lies in the space bounded by YOZ, ZOX, XOT; when they are (-, -, +) P lies in that bounded by T'OZ, ZOX', X'OY', and so on. (ii) Distance between two Points. The geometry of Fig. 44 shows that OP 2 =OM 2 + MN* + NP 2 ; OP=,J(x 2 +y 2 +z*) ............ (1) If P l is the point (x^ y-^ Zj) and P 2 the point (x z , y^ z 2 ) draw through P v , P 2 planes parallel to the coordinate planes (Fig. 45) forming the parallelepiped P^X^T^ZJP^ ; then -M-A 2 = ^2 ~ x li and FIG. 45. If we suppose the point P in Fig. 44 to vary its position, but always to remain at the same distance, a say, from 0, it will lie upon a sphere ; the coordinates of P will by (1) always satisfy the equation COORDINATE GEOMETRY OF THREE DIMENSIONS. 207 which is therefore called the equation of the sphere. Similarly we see from (!') that the equation of the sphere with centre Pi(x lt y : , z^) and radius a is (^-a?,) 8 +(y-y 1 ) 2 + (*- 1 ) 8 =a 8 (2) (iii) Direction Cosines of a Line. Let OP (Fig. 44) be any line through ; on the line take one direction, say, the direction from to P as positive. The position of the line will be definitely fixed when the angles that the positive direction of the line makes with the positive directions of the coordinate axes are known. These angles, namely XOP, YOP, ZOP, are called the direction angles of the line, and the cosines of these angles are called the direction cosines of the line. Each of these angles may be taken as lying between and 180 inclusive. Thus the direction angles of OX are (0, 90, 90), of OX' (180, 90, 90), and the direction cosines are (1, 0, 0), ( 1, 0, 0) respectively. If a, (3, y are the direction angles of OP, then cos(3=OL'/OP, cos y = ON' /OP, and cos 2 a+cos 2 /? + cos 2 y = OM z +OL" i +ON'* OP 2 = 1. If we write I, m, n in place of cos a, cos ft, cos y, we see that the direction cosines (I, m, n) of a line are connected by the identical relation P+m 2 + w a = l (3) When the line does not pass through the origin, draw a line through parallel to the direction on the line that is taken as positive ; the direction cosines of the line so drawn are those of the given line. If the distance between P l and P 2 is r, r being considered positive, the direction cosines of the segment P^P 2 are are l - z 2 )/r. (3') and those of the segment (iv) Cosine of the Angle between two Lines. Let (Z 1} m^ n^), (1 2 , m 2 , n 2 ) be the direction cosines of the lines, and draw OP, OQ (Fig. 46) parallel to the positive direction of the lines. Let OQ be the projection of OP on OQ, and let 2 } N be perpendicular to the plane XO T and NM perpendicular to OX. Then OM=1 V OP, MN=m l OP, NP= n^OP, OQ = OP cos 0, where is the angle between the lines OP, OQ. By the fundamental principles of projection, the projection of OP on OQ is equal to the sum of the projections of OM, MN, NP on OQ FIG. 46. 208 AN ELEMENTARY TREATISE ON THE CALCULUS. But the projection of OM on OQ is 1 2 OM, of MN is m z MN, and of NP is n 2 NP, since 1 2 is the cosine of the angle between OM and OQ, etc. Hence OP cos = Z 2 0fl/ + ra 2 MN+ n z NP and therefore cos 0= 1 / 2 4-7H 1 m 2 + 1 w 2 ................................. (4) Since sin 2 0= 1 - cos 2 and I l 2 + m l 2 + n 1 2 = l, 2 2 + m 2 ' 2 +% 2 = l, we have sin 2 = (Ij 2 + m^ + nf)(l, d/D respectively,- the sign of the root being chosen the same as that of d so that p or d/D may be positive. The quantities ajD, b/D, c/D are direction cosines since the sum of their squares is unity which is the condition required by (3) for direction cosines. These quantities are the direction cosines of the normal to the plane. The direction cosines of the normal to the plane x = are (1, 0, 0); of the normal to the plane yax+b, that is, ax+y=b are ( a/ v /(a 2 + l), l/ x /(a 2 + l), 0), ana so on. (vii) Equation of a Surface. Equations of a Curve. In general an equation of the form z=f(x, y) or F(x, y, z) = represents a surface. Thus by (ii) the equation ,r 2 -t-y 2 +z 2 -a 2 = represents a sphere of radius a. Again, when the coordinates of a point satisfy two equations F(x, y, z)=0, ; then /, $, < are called the polar coordinates of P. The relations between the rectangular coordinates (x, y, z) and the polar coordinates (r, 0, <) of the same point P are easily seen to be #=rsin 0cos (/>, y r sin 6 sin , z = rcos#. (ix) Cylindrical coordinates. In Fig. 44, let ON=p, LXON=^ NP=z ; then p, <, z are called the cylindrical coordinates of P. Evidently p=rsmO; x=pcos, y=psin are the plane polar coordinates of N, the projection of P on the plane XOY ; r, $, < are sometimes called spherical polar coordinates. Ex. 1. Find the equation of the plane through the three points (1, 0, 0), (0, 2, 0), (0, 0, 3). Let the equation be ax + by + cz=d; the coordinates of each point must satisfy the equation. Hence, to find a, b, c, d, we have a = d; 2b = d; 3c = d, that is, a/d=l, bjd=^ c/d=^; and the required equation is x+yft +2/3 = 1. It will be noticed that only the ratios of a, b, c, d are required ; the equation of the plane thus contains only three independent constants just as that of the straight line in Plane Geometry contains only two. Ex. 2. The equation of the plane through (a, 0, 0), (0, 6, 0), (0, 0, c), is x/a+y/b + z/c = l ; a, b, c are the intercepts made by the plane on the coordinate axes. Ex. 3. The equation of the plane through the three points (2, 0, 3), (-1,5, 2), (3, -4, -2) is 29,r+16y-73=37. Ex. 4. The equations of the line through (# 1( y,, 2,), (# 2 , y 2 , z 2 ) are x \~ x i y\~yu dy / A \ ~di~^x dt + Vy dt" When t takes the increment St let x, y, u take the incre- ments Sx, Sy, &u respectively ; then Su=f(x + 8x, y + Sy)-f(x, y), and this equation may be written Su = [f(x+Sx, y+Sy)-f(oc, y + Sy)] +[/(*. y+W-ffr, y)] ....................... (i) By the mean value theorem 72 f(x + Sx, y + Sy)-f(x, y + Sy)=f x (x + e i Sx, y + Sy) to.. .(2) /(*, y+Sy)-f(x, y)=f y (x, y + 9 2 Sy) Sy ......... (3) where 6 V 9 2 are proper fractions. The coefficient of Sx in (2) is the ic-derivative of f(x, y + Sy) taken on the supposi- tion that y + 8y does not vary and with x replaced by x + O^x; the coefficient of Sy in (3) is the ^/-derivative of f(x, y) taken on the supposition that x does not vary and with y replaced by y + O.^Sy. Hence 212 AN ELEMENTARY TREATISE ON THE CALCULUS. xr T 3u _du T Sx _dx j Sy _dy st=o St dt st=oSt (It st=o dt dt L f x (x + e^x, y + Sy) = f x (x, y) ; st=o L f y (x, y + 6 2 Sy)=f y (x, y) ; St=0 since Sx, Sy converge to zero with St, and the functions are all supposed to be continuous. Writing du/dx, du/dy in place of f x (x, y), f y (x, y} we get equation (A). In the same way if u=f(x, y, z) and x, y, z are all func- tions of a variable t we get du _ du dx du dy du dz / B \ dt~dx ~dt + ty ~dt + ?te dt" and so on for any number of variables. In (A) we may suppose t to be the variable x ; y is then a function of x, and u is really a function of the one vari- able x. Equation (A) becomes in that case du_c)u ~du dy / A '\ dx ~ "dx dy dx and in the same way, from (B) du_?>u ?>u dy dii dz . B /\ dx *dx 'dy dx 15z dx In these equations 'du/'dx and du/dx have quite different meanings. The derivative du/'dx is formed on the supposi- tion that an explicitly named variable x alone varies ; on the other hand du/dx is the limit of Su/Sx where Su is the change in u, due (i) to the change Sx in the explicitly named variable x, and (ii) to the changes Sy, Sz, which are them- selves due to the change Sx. du/dx, du/dt are called total derivatives with respect to x and t respectively. Ex. If u=3?+y 2 , then dufdx = 2# ; BM/^?/ = 2y. But if y is a function of x, say y = ax+b, then TOTAL DERIVATIVES 213 or we may use (A 1 ) ; then du _"du "du dy _ ~" + ~ since dyjdx=a, and we get the same result as before. If x, y are independent, and if Su be the change in u, due to the independent changes Sx, Sy, equation (1) may be written Su = f x (x + e.Sx, y + Sy) Sx+fax, y + Q z Sy) Sy = [/*(#, y) + wj Sx + [fy(x, y) + o> 2 ] Sy where ci^, &> 2 converge to zero with Sx and Sy. Hence if we take Sx, Sy as independent principal infinitesimals and write dx, dy in place of Sx, Sy the products w^dx, w 2 dy will be of order higher than the first and the principal part du of Su will be given by ~ " (c) Similarly for three (or more) independent variables , "du , dz .................. (D) du is called a total differential or a complete differential 'du i 9u 7 ~du -, and ^- rtic, ^- dy, rf0 Sa; ST/ ^ are called partial differentials. These partial differentials are sometimes written d x u, d y u, d z u. If x, y, z are not independent but functions of t then, since dx = (dx/dt)dt ... , we should get equations of the same form as (c), (D) by multiplying (A), (B) by dt. These equations (A) . . . (D) have important applications in geometry and mechanics. For plane geometry the equation (A') is very useful; the reader should study the following examples carefully. Ex. 1. Let u=az 2 +by*-I ; then, x and y being independent, Consider now the equation u = 0. The variables x, y are no longer independent; the point (ar, y) must lie on the conic ?=0 and y may be considered a function of x, namely an ordinate of the conic. Since 214 AN ELEMENTARY TREATISE ON THE CALCULUS. u is now always zero for the admissible variations of x and >/, the total ^-derivative of u (not the partial) must be always zero. Hence by (A 1 ) ~du 'du dy , dti ~du /ou ax ^- + -5-7r= and ;r = -5^7 5T= -IT- oar qy a*' ate cm oy 6y This equation gives the gradient at the point (x, y) on the conic. Ex. 2. Let u be any function f(x, y) of .r and y ; the equation =0, that is f(x, y)=0 defines y as a function of ,r, namely y is the ordinate of the curve /(.r, y)0- As in ex. 1, the total ^-derivative of u is zero and the gradient at the point (x, y) is given by dy _ _~du fdu _ _ "df fdf dx 'dxl 'dy oxf, 'dy where / is written for brevity instead of f(x, y}. Ex. 3. If f(x, y)=x 3 +y 3 3axy, the gradient of the curve whose equation is f(x, y)=0 is dy __ ZX* - 3a?/ _ ay - x 2 dx 3y 2 Sax y 2 ax Ex. 4. If pv = kQ (k constant) find dp in terms of dv, dd. ??_ _k_ _P "dp k_p ~~~~ ~ -. ov 06 v 6 Ex. 5. If u = tan~ l (yfx) prove that du (xdy - ydx)/(x* +y 2 ). Ex. 6. If #=rcos#, y = rsin^, r and 6 independent, show that dx = cos Qdr - r sin ddd, dy = sin Qdr + r cos 6dO xdy ydx = r z d6. Ex. 7. Let u=f(x,y)-z, then 3w_9/ 3r<_3/ 3__i ?)x~'dx > ~dy 3y' ~dz The equation uQ defines a surface, and now z may be considered a function of two independent variables x, y, namely z =f(x, y), 'dz^_'df _'du ~dz _3/ _'du 'dx 'dx 'dx ~dy 'dy 'dy 91. Geometrical Illustrations. Let P be the point (x, y. z) on the surface given by z=f(x, y}, and let APB, DPF be sections made by planes through P parallel to the planes YOZ, ZOX respectively (Fig. 48). GEOMETRICAL ILLUSTRATIONS. 215 For points on the curve DPF, y is constant. Hence 'dzj'dx or 'df/'dx is the gradient at P of the curve DPF. Similarly 'dzfdy is the gradient at P of the curve APB. If the equation of the surface is u = 0, where u is the function F(x, y, z}, the equation u = defines z as a function of two inde- pendent variables x and y. Along the curve DPF, y is constant. Hence along that curve the total ^-derivative of u or F(x, y, 0) must be zero, u being for that curve a function of x and z which is always zero. Therefore, as in 90, ex. 1, 2, FIG. 48. ,1 * . II 3z 'dF fdF _ __ _ / _ . 'dx 'dx/ "dz (1) and 'dz/'dx is the gradient at P of the curve DPF. Similarly, the gradient at P of the curve APB is These expressions reduce to those first given if we put u=f(x, y) z. (Compare 90, ex. 7.) Tangent Plane. In Fig. 49 let APP 2 , BPP l be sections of the surface by planes parallel to YOZ, ZOX respectively. Let P be the point (x, y, z), MM l = Sx, MM 2 = Sy, M s the point (x + Sx, y + 8y, 0) and P 3 the point on the surface (x+fa, y + Sy, z + Sz). Let PT V PT, be the tangents at P to BPP V APP 2 , T l lying on MJ>~ produced and T z on M*P 2 produced ; P r m- i m z r m ti is a rectangle parallel and con- gruent to MM^M^M* ; PjPg, P 2 P 3 are the curves in which the planes M^m^ M z m 3 cut the surface, and T^y T 2 T S the straight lines in which the same planes cut the plane through P7\7 7 2 . 216 AN ELEMENTARY TREATISE ON THE CALCULUS. Since the gradient at P of the curve BPP l is 'dz/'dx and of the curve APP 2 is 'dz/'dy we have , 22 . Also the geometry of the figure shows that so that (2) But is the principal part of Sz ( 90, c) ; therefore, when Pm v Pm 2 represent Sx, Sy the line m 3 T 3 represents the principal part of 8z. Again, if the plane PMM 3 P & cut the surface in the curve PP^ the gradient at P of PP 3 is the limit of Sz/MM 3 or Sz/Pm y But by the principles of infinitesimals the limit of &?/Pm 3 is the same as the limit of m 3 !ZyPm 3 , since wi s T 3 is the principal part of Sz. Hence the gradient at P of PP 3 is m 3 7yPm 3 and therefore PT 3 is the tangent at P to the arc PP -LJT S . The plane PT^T t is completely determined by the two lines PT V PT 2 , that is, by the point (x, y, z) and the derivatives 'dz/'dx, 'dz/'dy. By proper choice of the independent increments Sx, Sy we could get any point Q on the surface near P and the tangent to the arc PQ would lie in the plane PT^T Z . This plane is therefore called the tangent plane to the surface at P, and the line through P perpendicular to the tangent plane is called the normal to the surface at P. To find the equation of the tangent plane suppose T 3 to> TANGENT PLANE AND NORMAL. 217 be any point on it and let its coordinates be (X, Y, Z) those of P being (x, y, z) ; then and therefore by (2) which is the equation of the tangent plane at the point (x, y, z) on the surface, X, Y, Z being the current coordinates of any point on the plane. When the equation of the surface is F(x, y, 0) = we get by substituting the values of "dzfdx, 'dz/'dy from (1) and (!') The direction cosines of the normal are (89a (vi)) pro- portional to the coefficients of the current coordinates X, Y, Z and therefore the equations of the normal are = -(Z-z) ...... (4) Ex. I. The equation F(x, y, z)=x 2 +y 2 +z*-a?=0 represents a sphere of radius a. 2.y --=. = 2y, -^-r- = 22. ox oy Oz Hence the tangent plane at (x, y, z) is (X - x)2x + ( F- y)2y + (Z- z)2z = 0, or or x+y+z=x+y+z=a, since (x, y t z) is on the sphere. If we take x, y, z as current coordi- nates and (#1, y lt z t ) as the point of contact, the equation is #iX + yiy+ZiS=a 2 - The equations of the normal are (X-x)l2x=(Y-y)h2y=(Z-z)l2z, or Xlx=Yly = Z\z. With (#, y, z) as current coordinates the equations are xl-ri=ylyi=z/Zi- The normal clearly passes through the origin which is the centre of e shere. the sphere. 218 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 2. The equation ax? + fry 2 + cz 2 1 = represents a surface called a central conicoid (a plane section is in general a central conic). Find the equations of the tangent plane and the normal at (x v y^ Zj). Ex. 3. The equation bi/ 2 + cz 2 2x=Q represents a non-central conicoid. Find the equations of the tangent plane and the normal at (a?,, y v zj. Ex. 4. The equation aa? + by z + &P=0, where a, b, c are not all of the same sign, represents a cone with its vertex at the origin. Find the equations of the tangent plane and the normal at (x^ y n 2,). If F\x, y, 2) = o.r 2 + % 2 + cz 2 , the derivatives 'dF/'dx, 'dFj'dy, "dPfdt are all zero when x=yz=Q. Every tangent plane to the cone goes through the origin, and there is no definite normal at the origin ; the equations of the tangent plane and normal are illusory if formed for the origin. At special points on a surface it may happen that the three partial derivatives are all zero ; in that case there is no definite tangent plane or normal at the point. Such points are usually called conical points, the vertex of a cone being the simplest case. 92. Rate of Variation in a given Direction. It is often necessary to find the rate at which a function of the coordinates of a point varies in a given direction. Thus at a point in a cooling solid the rate of diminution of tem- perature will usually be different along different lines issuing from the point. (i) Let u be a function f(x, y) of two variables, and let u p , U Q denote the values of u at P(x, y} and at Q(x + Sx, y + Sy) respectively, where PR = Sx, RQ = PS = Sy (Fig. 50). _ Then x u p =f(x, y), FIG - > u q =f(x + Sx,y + Sy\ The average rate of increase of u in the direction PQ is (u q Up)/PQ which may be written u 9 u P _u K u p PR U Q U K RQ ~PQ~ ~PR~ PQ + ~RQ~ ' PQ As in 89a (iii) let the direction PQ be distinguished from that of PQ', and let PQ make with OX the angle (see note at end of this article) ; then PR/PQ = cos 0, RQ/PQ = sin 0. VARIATION IN A GIVEN DIRECTION. 219 Exactly as in 90 it may be seen that the limits for PQ = of (u K u P )/PR and (u Q u a )/RQ are 9w/9a; and 'du/'dy respectively. If the element PQ be denoted by Ss (where s may represent the length of a line, straight or curved, measured from some point up to P) then the average rate of increase of u is (u'^ Up^/Ss or Su p /Ss and the rate of increase of u in the direction PQ is then 9u ~du c>u . :r- = COS0 + smd> .................... (1) 9s cte oy If the rate of increase of u in the direction PT perpen- dicular to PQ is denoted by 9w,/9s', PT making the angle + 7T/2 with OX 9it "du . 9u ^-7= ^ sm< + cos0 .................. (2) 9s 9o; cty (ii) If u be a function f(x, y, z) of three variables the rate of increase 9it/9s in the direction PQ may be proved in exactly the same way to be 9u 7 9w "du "du /o\ ^r = l^r + n^ hw ..................... (J) 9s da; 97/92 where (I, m, ri) are the direction cosines of PQ. If (1) and (2) be solved for 'du/'dx, 'da/'dy we get ~du . "du , n ,. = ^- sin + ^-7 cos -TT) with OX. With this new convention cos may have a negative value. For lines in space the determination specified in 89a (iii) is always sufficient. 93. Derivatives of Higher Orders. The derivatives of u =f(x, y) will usually be functions of x and y, and will therefore have derivatives. Hence we have 2nd, 3rd,... partial derivatives. The notation for these is similar to that for functions of one variable : U yvy '" The brackets and the letters within them are usually omitted and the last pair are written f xx , f yyv . Again, the ^/-derivative of 'diij'dx is _9_ ' ' 'by 'dx 'dy'dx while the as-derivative of "dufdy is .r ;r or When all the functions in question are continuous these two derivatives are equal (see below). For example, let u = ax m y n ; then < , ^ = nax m y n ~ 1 ; ?>y so that oyox oxoy when u = ax m y n . In other words the order of differentiating is indifferent ; the operations of differentiating as to x and as to y are commutative. Ex. Verify that these two derivatives are equal when y (i) u=xsmy+ysmx ; (ii) u=x\ogy ; (iii) w = tan~ ! -. HIGHER PARTIAL DERIVATIVES. 221 The symbol means that u is to be differentiated first three times as to y then twice as to x ; while the symbol means that u is to be differentiated first twice as to a? then thrice as to y. Similar meanings and notations hold for the higher derivatives of a function of any number of variables. A sound proof of the commutative property is somewhat difficult. Consider the expression f(x +h,y+k) -/(*, y + 1) -f(x +h,y) +/fo y) m hk " A ' By the definition of a derivative Hence the limit of (1) for A = is {f x (x,y + k)-f x (x,y)}/k ............................ (2) Again the limit of (2) for k = is the ^-derivative of / z (.r, y\ that is /,,. By interchanging the second and third terms in the numerator of (1) and finding first the limit for =0 and then the limit for A=0 we should get f xy . Thus f yx and f x!l are both derived as limits from the same expression. But the assumption that the limits will be the same in whatever order we make h and k tend to zero is equivalent to assuming the theorem to be proved. A simple example will show that the order of taking the limits is not necessarily indifferent. Take the function (A + 2)/(A + ) 2k ( Li j- 52, T > i i i~ T- i k t=0 U=oA-(-/; J k = h r / T A + 2*\ _ A T ' LrT 1J , = Li -f = 1. A *=oU Of course neither ih this expression nor in (1) must h or k become zero ; zero is the limit not a value of A and k. Assuming all the functions in question to be continuous we may proceed as follows. Let, for brevity, 222 AN ELEMENTARY TREATISE ON THE CALCULUS. then the numerator of (1) is F(x + K)-F(x). By the Mean Value Theorem Q ^ < ^ < 1 , or, returning to the function f(x, y), P(x + h)-F(x) so that (1) becomes (2') Now apply the Mean Value Theorem to the function of y in (2') ; MX + 6ih, y + k) -f x (x + OJi, y) = kf yx (x + 0^ y + BJe), 0< #,< 1 , and (1) becomes /?*(# + #A y + O.Jc) ............................... (3) Again, taking (y + k) (f)(y). Apply the Mean Value Theorem and proceed as before. We thus find that (1) is equal to . ...... (4) The two expressions (3), (4) are therefore equal. Since the functions are continuous the limits are therefore equal in whatever way h and k tend to zero, that is f yx f x y The commutative property may be easily extended by induction to higher derivatives, the functions being sup- posed all continuous. Thus, since 'dxdy ~ 'dy'dx fdu u\_ x) ~ 'dx'dy'dx ~ dxdy \dx ~ didx \dx In general, "dP+9+r-U, as may be readily shown by induction. Ex. 1. In Fig. 48, 91, let Fbe the volume bounded by the surface APDC, the coordinate planes and the planes MP, LP. Prove (i) l = area MNP A ; = NP. ox oyox (ii) |?= oy */ t7 If V be taken as the function f(x, y), we get a geometrical proof of the commutative property. COMMUTATIVE PROPERTY OF DERIVATIVES. 223 Ex. 2. If u = log r where r 2 = (x of + (y 6) a and (x - a), (y b) are not simultaneously zero, show that - . ox ox ox ox r "on _dlogr "dr _1 x a_x a f 'dx dr ~dx r r ~ r* '' Similarly _ y- y ' "dy*~r* r* ' and therefore by addition, since r 2 = (4; - a) 2 + (y - 6) 2 , the result follows. Ex. 3. If ?t = l/r where r 2 =(^-a) 2 + (y-6) 2 + ( 2 -c) 2 and (tf-a), (y 6), (z c) are not simultaneously zero, prove that A charge 7?i of electricity concentrated at (a, 6, c) has at (#, y, 2) the potential m/r. The potential F therefore satisfies the equation last written, usually called Laplace's Equation. If charges i x , w 2 , ... are concentrated at (j, fej, Cj), (a 2 , 6 2 > C 2)> ^ ne potential Fat (j;, y, z) of these charges is 2(7n/r) where so that the potential at any point (x, y, z) not coincident with any of the masses also satisfies the same equation. Ex. 4. If u=f(x, y) and x, y are functions of t find d 2 u/dt z , We have ^=^ ^+^ f , .. ...(i) we 0^0.^ rf 2 M_3w d_ a f ^ &y ^(dx dt 2 ~dx dt 2 'dy dt' 2 + *dx 2 \dt 9 ^u_ dx dy ?^(dy\ 2 + 'dx'd dt dt + * \dt) ' 224 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 5. If f(x, y)=0, show that d 2 y/dx* is given by the equation dy , _ n This may be obtained directly ; or in Ex. 4 put t = x and note that u=f(x, y)=0 for every value of x and _y, and therefore dujdt and d^u/dt 2 are both zero, while dx/dt = l, d 2 .r/dt'* = 0. Deduce in this way the results of examples 26, 27, 28 of ExercisesXIV* /.v w u ,..,. ?t w (i) ~-=a~-; (u) o- = a ; 5~?- Oic oy aa^ o/ 2 Ex. 6. If u=f(y + ax), prove /.v 3w ~du (i) ~-=a~- Oic oy Ex. 7. If u = f(x+at) + (j>(x-at\ prove Verify for w = A cos (^ + at) + Bsm(x at). 94. Complete Differentials. If u is a function of the two independent variables x and y the complete differential of u is ( 90) u 3u dy ........................ (1) Now the question arises; given two functions (x, y), \}s(x, y) of two independent variables x, y, is there always another function u which has , y)dy .................... (2) as its differential ? If x and y are not independent, say if y is a function f(x) of x, we may replace y by f(x) and dy by /'(#) cfce. The expression (2) will thus become of the form F(x) dx and in this case ( 82) there is a function which has F(x) as its x-derivative or F(x) dx as its differential. But if x and y are independent the case is altered. For suppose the expression (2) to be the complete differential of a function u; then the expressions (1) and (2) must be equal for all values of dx and dy. Since dx and dy are independent we may put dy = Q, cfe+0 and we get and in the same way i/r(oj, y) = 'du/'dy COMPLETE DIFFERENTIALS. 225 for all values of x and y. Therefore 3 3% c)y ~ 'dy'dx ~ 'dx'dy ~ ?>x Hence the expression (2) cannot be a complete differential unless 30/Sy = 3\/r/3a;. Condition (3) is therefore a necessary condition ; it is also a sufficient condition, but for the proof of sufficiency we refer to treatises on Differential Equations. If P, Q, R are functions of three independent variables x, y, z the necessary and sufficient conditions that Pdx + Qdy + Rdz should be a complete differential, that is, that there should be a function u of x, y, z such that du = Pdx + Qdy + Rdz 9Q -dP 'dR "dQ 3P 3R are that ^ = ^ > ^ = -=-> -=- = ~ 3aj 9y &y dz ^>z ?>x The student may show that these conditions are necessary. Ex. 1. (&C 2 - 4.vy)dx+(3y* - 2^)dy is a complete differential for and u=& Ex. 2. If P=yz(2x+y+z), Q=zx(x+2y+z), R=xy(x+y + 2z) show that du = Pdx + Qdy + Rdz where u=x*yz+y' i zx+z z xy. 95. Application to Mechanics. Let the plane curve A PQ be the path of a particle which is acted on by a force F, making an angle e with the tangent PT, F and e being functions of the coordinates x, y of P. Let W be the work done from the position A (a, 6) FIG. 51. up to the position P, and let the arc AP be denoted by s. To the first order of infinitesimals the work done over the distance ds is dW=Fco8eds. o.c. p 226 AN ELEMENTARY TREATISE ON THE CALCULUS. Let PT, PF make the angles 0, \fs with the #-axis ; then cos (ft = dx/ds, sin < = dy/ds, and since cos e = cos ($ \fs) dx , r, . . dy v dx , ^ T dy - T - + Fsm\L-^- = X- r +Y-f, ds Y ds ds ds where X = Fcos\fs, Y=Fsin\fs, the components of F parallel to the axes. We thus get dW=(X~+Yj-)ds (1) \ as ds/ Suppose now that Xdx+ Ydy is the complete differential of a single-valued function f(x, y). Therefore X = 'df/'dx and Y='df/dy, so that ,^ _ f j _jf dxds 'dy ds) ~ ds Hence, as the particle moves along the curve, the rate dW/ds at which W changes is equal to the rate df/ds at which the function f(x, y) changes, and any change d W in W is equal to the corresponding change df in the function f(x, y). As the particle moves from A to P the work done is therefore equal to the change in f(x, y), so that W=f(x,y)-f(a,b) ...................... (2) If W is the work from A to P when the particle moves along a different path of length s', we have as before so thSat W'=f(x, y) -/(a, 6) = W. In this case, therefore, the work done by the force is independent of the path between A and P, and, when A is fixed and P variable, is a function simply of the coordinates of P. When P coincides with A, that is, when the path is a closed curve, the work done is zero (see Ex. 2 for an illustration in which f(x, y) is multiple-valued). Suppose on the other hand that Xdx + Yd y is not a complete differential. In this case the coefficient of ds in (1) is not the total derivative of a f unction f(x, y). To find the work from A to P we must express y in terms of x by APPLICATION TO MECHANICS. 227 using the equation of the path. Equation (1) will then become and the coefficient of dx in (!') is a function of x alone. For different paths the function X + Y (dy/dx) will have different values, and therefore W will depend not merely on the coordinates of P but also on the path from A to P. (See Ex. 3). If APQ is not a plane curve it is easy to prove by the same method as that of finding dxfds, dy/ds for a plane curve ( 62) that the direction cosines of the tangent PT are ( 89a, iii. (3')) dx/ds, dyfds, dz/ds. If I, m, n are the direction cosines of PF ,dx , dy , dz l^r + m-T- -- ds as and dW-x+7+gd. ...... (3) \ ds as dsJ where X = IF, Y=mF, Z=nF are the components of F parallel to the axes. Exactly as before we see that if Xdx+ Ydy + Zdz is the complete differential of a single-valued function f(x, y, z) dW=df and W=f(x, y, z)-f(a, b, c) where A is the point (a, b, c). In this case W is independ- ent of the particular path from A to P. If however Xdx + Ydy + Zdz is not a complete differential it will be necessary to use the equations of the path and W will depend not merely on the coordinates of A and P but also on the particular path from A to P. When Xdx + Ydy + Zdz is a complete differential the force F is said to be conservative ; the components are the derivatives of a force function u or a potential V, X=^ or X=-^; dW=du or dW=-dV. 228 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 1. If F=mjr 2 where r 2 =x 2 +y 2 + z 2 = OP 2 , and the direction of F is from to P, then F is a conservative force. For X--F A f and dW= or dW=-^dr=d( - j, since xdx+ydy + zdz=rdr. Hence W= m/r+ const, and if V=mjr, x= - a vfdx, r= - a v^y, z=-?> r/a*. The work from position P to position Q is and is independent of the path between P and Q. Ex. 2. Let ^T= -y/r 2 , Y=.r/r 2 , where r z =z 2 +y*. In this case, putting y/.r = tan 6, dW= (xdy - ydx)jr 2 = d . ta,n~\ylx) = dO, and therefore W 6 + constant. If the point P sets out from A and, after describing a closed curve within which the origin lies, returns to A, the angle # and therefore W will increase by ZTT. The work done is not zero, although d W is a complete differential dO ; the function 6 is multiple-valued. If, however, the path is a closed curve within which the origin does not lie, the work done over that path will be zero. Ex. 3. Let X= -y, T=x. In this case xdy -ydx is not a complete differential. Let A coincide with the origin 0, and let the path be the parabola y = cx 2 . Then, by (!'), d W= ( ex 2 + x . 2cx)dx = cx 2 dx ; W= ^cx 3 = $.ry. If the path is y=cx 3 , we find W= \cx* = \xy, the work being different for different paths. 96. Applications to Thermodynamics. The condition of a given mass of thermodynamic substance, say unit mass, is completely defined by three variables p, v, 6 the intensity of pressure, the volume and the absolute temperature, p, v, 9 are connected by an equation, the characteristic equation of the substance, f(p, v, 6) = ; for a perfect gas the equa- tion is pv = k6, k being a constant. Of the three variables, therefore, only two are independent. Since f(p, v, 0) = its total differential is zero ; therefore ^o ..................... (i) APPLICATION TO THERMODYNAMICS. 229 If p be constant, and v, 6 vary, then dp = and we have -dv_ _f>/\_3/ W ~ W '' *dv Forming in the same way W/'dp, "dpfdv and multiplying ~dv W 'dp we get ^^-^r=- 1 ......................... ( 2 ) ^f^ ^ Clearly It must be remembered that in all these expressions the derivative of one of the variables p, v, 6 with respect to a second is formed on the supposition that the third variable is constant. If a small quantity SQ of heat be communicated to the substance and change p, v, 6 by Sp, Sv, SO respectively, then SQ can be expressed in terms of any two of these increments. To the first order of infinitesimals we may write, with 9, v as the variables, dQ=MdO+Ndv ........................ (4) It is to be most carefully noticed that dQ, dv are any arbitrary small changes of temperature and volume. Ihe three differentials dQ, dv, dp are subject merely to the restriction expressed in equation (1), and any two of them may have values chosen at will. The specific heat at constant volume (K v ) is the limit for SO = of SQ/S6 on the supposition that the volume does not change when increases by SO, that is, on the supposition that dv = 0. But if dv = equation (4) gives dQ/dO = M so that K V = M. The specific heat at constant pressure (K p ) is the limit for S0 = of SQ/SO on the supposition that p is constant, that is, that dp = 0. To find K p equation (4) must be trans- formed so that and p shall be the independent variables. Since v is a function of and p we have 'dv "dv and (4) becomes 230 AN ELEMENTARY TREATISE ON THE CALCULUS. Therefore K = M+N=K V +N ............. (C) The elasticity of the substance is vdp/dv ( 70). Let E 6 denote the elasticity when the substance expands at constant temperature; therefore. E e = v ;*- 3v where 'dp/'dv is taken subject to the condition that is constant. Let E^ denote the elasticity when the substance expands adiabatically, that is, so that heat neither enters nor escapes. We must distinguish the ^-derivative of p in the two cases. For the present denote the -y-derivative of p for adiabatic expansion by ('dp/'dv)^ and let 'dpf'dv retain its previous meaning. Therefore, To find Cdpfdv}^ we must transform (4) so that p and v shall be independent variables. Now i* 90, 30 d6 = ^- dp and therefore dQ = Mdp + M + Ndv ............ (8) is the value deduced from (8) on the supposition thatcQ = 0. Therefore "* N "* -^ -\S\ 3p dp W The numerator last written is K p and M = K V ; therefore -K IK *-K IK bv (to K **ap w ** by * Hence * = ................ ............... (9) FOUR THERMODYNAMIC RELATIONS. 231 For a perfect gas K P \K V is a constant, y; also for a perfect gas pv = kO, and therefore Hence for adiabatic expansion, by (7) and (9), that is, pv y = constant. The results (1)...(3), (5)... (9), are merely formal con- sequences of the definitions and the two equations f(p, v , 0) = and dQ=Md6+Ndv. 97. Four Thermodynamic Relations. dQ in the previous article is not a complete differential ; we cannot express Q in the form F(6, v) F(0 , v ) without assuming some further relation between and v. Physically, Q is not a function of 6 and v; heat may be given to the substance and 0, v go through a range of values and return to their initial values, while the heat absorbed in the process is not equal to that given out. Compare 95 when. dW is a complete differential ; when x, y return to their initial values a, b, W = 0, that is, the work done by the force F is equal to that done against it. It is shown in treatises on thermodynamics that if we put dQ = Qd(f> where is the entropy we can replace (4) by dE=Od-pdv ........................ (10) E is the intrinsic energy and pdv the work done in the infinitesimal expansion dv. dE is a complete differential ; that is, E is a function of the variables that define the state of the substance. There are now four variables p, v, 0, (f>, but of these only two are independent. If v, are chosen as independent the symbol "dp/W is now not sufficiently clear; it means the ^-derivative of p when v is constant. But if 0, were the independent variables, 'dp/W would mean the 6- derivative when is constant. To avoid confusion we will, when there is doubt, enclose the derivative in a bracket and affix the independent variable which is supposed to be constant. 232 AN ELEMENTARY TREATISE ON THE CALCULUS. Thus (dp/W) v means that v, 6 are the independent variables, and that v is constant in forming ~dp/W. Since dE is a complete differential we have (94 (3)) from equation (10) Let now y, be the independent variables, then since (10) becomes dE=e ou \ ov and therefore . or since the two derivatives of second order are equal. In the same way, by taking p, as independent variables, fdv\ fdO we get I) = (5- V30/P \3p and by taking p, 6 as independent variables Equations (I'), (2'), (3'), (4') are those numbered (1), (2), (3), (4) in Maxwell's Heat, p. 169. In effecting the differentiations it must be borne in mind that for example when v, 6 are the independent variables 'dOj'dv is zero. The careful working out of these four rela- tions will give much information as to the meaning of partial derivatives ; it is necessary at each step to attend to the meaning of the operations rather than to the notation. CHANGE OF INDEPENDENT VARIABLE. 233 Ex. 1. d For a perfect gas Kp, K v are constant, and by 96, K, = M and K 9 - K,**ffCdv[dff\- But for a perfect gas (dvlW)r> = vld. Hence =K-K.v fnd -K=d . log as may be tested by differentiation. Therefore = const, or pv*t= const., as in 96. Ex. 2. The gain in energy dE due to a supply dQ of heat is given by dE=dQ-pdv = ( Show that if dE is a complete differential, dQ is not. Since dE is a complete differential, we have that is, *dNfdO, "dJtffdv are not equal and the result follows. Ex. 3. Prove that _ = __i ~ '' ~ Ex. 4. Show that equation (10) may be written in the forms dE= d(6) - dO -pdv ; dE= - d(pv) + Bd^> + vdp ; dE=d(6$)- d(pv) - dO + vdp, and then prove (!'), (2'), (3')- Ex. 5. It is shown in works on Thermodynamics that d is a complete differential. Prove that 3 98. Change of Variable. Differentials of Higher Orders. When the independent variable x of a function y is changed by a substitution, x = (t) say, to a new independent variable t the ^-derivatives of y, D x y, Dly..., must be expressed in terms of the ^-derivatives of y, y,y... . We have found ( 68, ex. 2) that (1) 234 AN ELEMENTARY TREATISE ON THE CALCULUS. and it is easy to find D*y, D*y... when these are required. Since 0() is supposed to be given, the values of x, x..., can be calculated, and the substitution of <() for x and the above values for Dy, D 2 y changes any expression containing x, y, Dy, D 2 y... into one containing t, y, y, y... . If we wish to make y the independent variable and x the dependent, then and so on. Again if we change from rectangular to polar coordinates, an equation f(x, y) = becomes an equation between r and 9 and we may express D x y, D^y..., in terms of D e r, D 2 r, ...9 being the independent variable. For x = r cos 9, y = r sin 9 and we can differentiate the products r cos 0, rsinfl with respect to 9, r being a function of 9, dx a dr . . -ST. = cos 9-j^ r sm 9 d9 d9 d 2 x ,.d 2 r . dr d9 2 = C 8e d9 2 - Sme d9 with similar expressions for dy/d9, d 2 y/d9 2 . In equations (1) we may suppose t replaced by 9, since of course t may represent any variable; x would be replaced by dx/dv, x by d 2 x/d9 2 and so on. We should thus express Dy, D z y in terms of r, 9, drjdQ, d 2 r/d9 2 . In geometry and mechanics differentials of order higher than the first are often required. When x is the inde- pendent variable, dy = y'dx ( 60). The second differential of y is denoted by d 2 y and is defined by the equation and in general the nth differential of y is denoted by d n y and is defined by the equation d n y = yWdx n = (D?y)dx n where dx n means (dx) n . If dx is an infinitesimal of the first order d n y is, in general, of the -nth order. DIFFERENTIALS OF HIGHER ORDERS. 235 In the second of equations (1) multiply the numerator and denominator of the fraction on the right by dt s . Since t is the independent variable we have dx = xdt, d?x = xdt 2 , dy = ydt, d z y = ydt 2 , and therefore D^y = (dx d 2 y dy d 2 x)/dx 3 .................. (4) D%y is thus expressed as a quotient of differentials; the independent variable for the differentials is not x but t (or any other variable of which x and y are functions). If x is the independent variable, then by definition d z x = (D*x)dx* = x dx* = 0, and similarly we see that d 3 x, d*x, . . . are zero. In other words, the differential of the independent variable is constant. From (4) we may easily derive (2). Take y as the independent variable ; then d 2 y = 0, dx = (DyX)dy, d z x and (4) becomes For more than one independent variable the trans- formations are complicated. We will consider only one case that is of great importance in mathematical physics. 99. Transformation of V 2 u. Let u be a function of two independent variables x, y, and let x, y be changed to polar coordinates* r, 6] we wish to express u x , Uxx... in terms of u r , Urr Of course a derivative u r implies that x, y have been replaced in the function u by r cos 6, r sin 0. 'duj'dr is the rate of variation of u in the direction in which r increases, 6 being constant. In 92 put = 6, s = rand we find 'duf'ds' in 92 is the rate of variation of u in the direction + 7T/2. Let = 6 so that PT is perpendicular to OP and ,n8Q 'du_ y $ u _ T ' ~ ' ~ Ss' ~ S =0 f tan S6 r TT 1 "du n "bu , r .'du Hence -= -sm0_- +cos0 236 AN ELEMENTARY TREATISE ON THE CALCULUS. The element ~ds' is replaced by rW ; rW is the element in the direction perpendicular to r just as ?>r is that in the direction of r. Equations (1), (2) are so important that we give another proof of them. By 90 (A), taking x and y as functions of r, being kept constant, we get by putting r for t _ ~ ~ Here 3u/3# means (dti>/3)^ and 'dx/'dr means (3aj/3r) e in the notation of 97. Also dx and the substitution of these values in (!') gives (1). In the same way 'du_'du/'dx\ 'du/'dy\ / 9 /\ W ~fa\W/ <\dd/' ' and ^z OC7/ from which equation (2) follows. Solving (1) and (2) for 'du./'dx, "dufdy we get smfl du /g\ ^ J . = cost7;5 r ov (4) r cty 3r r W mu * 4.- ^^ , ZPu , &u The function ^+^ + ^ is of very frequent occurrence in Physics and is usually denoted by V*u. It is often necessary to transform V 2 u so that other variables shall be the independent variables. First, let u be a function of the two variables x, y so that the third term is absent, and transform it so that r, 6, polar coordinates, shall be independent variables. Denote 'du/'dx, 'du/'dy by u x , u y ; then we can find 9 2 w/3# 2 in terms of r, 6 by writing u x in place of u in (3). We must calculate 'du x /'dr, "du x /W. Now, TRANSFORMATION OF V 2 U. 237 du x _ "^r ~ x wk_ v/ ^x or or I 01 t sin 3w,l r 30 J n ^ , sin 3it sin = 0080;^ + ^ 3^3; 3 2 u . -3u sin 03% cos -^cosO - z sm0- -- S0 dr30 3r r 30 2 r Hence from (3) sin "du _ a; _ ,, a; ~dx 2 ~dx c)r r W and when the above values of Bw^/dr, 'du x jW are substituted we get, after an easy reduction, 2 sin 6 cos 3% sin 2 9 _ ~ , sin 2 3% . 2 sin cos 3rt /-\ _| _1_ Wr r 3r + r 2 30 In a similar way we find TPu . ^u . 2 sin cos ?Pu cos 2 3 2 u T = sm 2 0^-^+- r cos 2 3u _ 2 sin cos 3u /^\ ~~r~~ 3r~ ~^ 30'" Adding (5) and (6) we get transform V z u from x, y, z to cylindrical coordinates Here is not changed ; we have merely to write p,

2 '" It is sometimes useful to write the first two terms of (11) in the equivalent forms 1 3 2 (ru) r 2 3r \ 3r/ r 3r 2 ' and we may transform (11) to 2 (12) 2 "^ 1 3r 2 30 2 30 3 2 (ru) 3 2 u since Q/ =^5g etc - EXERCISES XIX. 1. If ^r=rcos0, y=rsin^, show that 'dx\ fdr\ .... Cdx\ (r'de\ The equation (i) is not in conflict with the theorem that when x is a function of the single variable r, the product of dxfdr and dr/dx is unity. The student should prove the equations by using a diagram, and he will see their meaning much more clearly. EXERCISES XIX. 239 2. If x=rcosQ, y = rsinO, prove (i) D 3 *y={r*+2(D g r) 2 -rD;~r}/(cos 8D e r-rsin Of ; (ii) {1 + (JQ*H*/i*> = { r* + (ZVfP A*- 2 + 2(ZV) 2 - rDjr}. Deduce from (i) the condition for a point of inflexion on the curve given by the polar equation r=f(B). 3. If x=a(l cos i), y = a(nt + sin t), express Dly in terms of t. 4. If x r cos 6, y=rsin d, and x, y, r, 6 functions of t, prove (i) xcos#+#sin# = r ; (ii) -xsin + #cos B=rd ; (in) xcosO+ysin6=r-rd 2 ; (iv) -#sin 6+y cos$r&+2rO. If P is the point (x, y\ equations (i) and (ii) give the velocity of P along and perpendicular to the radius vector while equations (iii) and (iv) give the acceleration of P in the same directions. It is easy to see that 5. If s is the arc of a curve measured from a fixed point on it up to the point P(x, y, z\ prove, using accents to denote s-derivatives and dots to denote ^-derivatives, (i) x r x"+y'y"+s?2f'=Q ; (ii) x=o/s ; (iii) x=x's (iv) xfx +y'y + z'z = i (v) & +y* +&= s 2 + i V ; where l/p*=x"*+y" 2 + z" 2 . Equation (i) is obtained by differentiating as to s the identity this relation holding since a/, y', / are direction cosines. These results are important in Mechanics. Thus (iv) gives the tangential velocity, (v) the total acceleration. 6. If the axes are turned through an angle a, the old coordinates (#, y) of any point are connected with the new coordinates (f, rj) of that point by the equations ( 27) prove that a ,-, 'du ~du ~dx 'du "dy 'du "du . * or > ^ = ^^t+^^-^ cosa+ 3^ 81 ~du ^ am t* T ^ < ox oy "bu 'du'dx 'du'dy 'du . 'du ^^=^-^- + 0-0= --=r-sma+~r.cosa. 240 AN ELEMENTARY TREATISE ON THE CALCULUS. Solving for 'du/'dx, 3w/3y, we find 3it "ou ~du . ~ou "du . "bu . . ^--^r ^ sma, ; ^ = ~- sin a + ^~ cos a : ox O| 077 3# Of Or) 3 2 M 'dux "ou x ~du x . --3= ~ =-~TT- cos a - ^ sin a, etc. O^r ox 3| 097 A similar equation to (i) holds for three variables .r, y, s. 8. If in 99 (12) $ be changed to p. where fjL cosO, show that w becomes a 9. -P, P' are the points (#, y, 2), (a/,y\ /) and PP'=r, a positive number; PQ = ds, P'Q'=ds' ; the direction cosines of PQ,P'Q' are (, TO, w), (', ?i', %'), and the angles QPP', Q'P'P are ^, ^, while e is the angle between the directions of PQ and P'Q'. Prove (i) "drfds = - cos 6 ; (ii) Br/3s' = - cos ff ; ..... 3V 3r3r ,. . 3C7- 1 ) cos^ (111) r ^-^~, + ^-^r, = -cose ; (iv) W-^ = ^- ; v x 2 _ 2cos e+ 3 cos (9 cos ^ \/r 3s 3s' r 2 In 92 (3) put M=r ; then since r*=(x' -xf + (y > -y^ + y -z)~, 3r/3# =(af x)/r, 'drj'dxf =(xf ai)/r, etc. and the ^-direction cosine of PP' is (x 1 -x)jr ; of P'P, (x-x')lr. ^DT O?* 9?* O?* I -^xf ~~ & I Then ^-^ ^ m ^ Hw^-= - \F- -- \- . .. + ... f = -cos^. OS ox oy oz I r J In finding 3V/3s 3s' by differentiating 3r/3s as to s', it is to be noted that I, m, n and x, y, z are independent of s' ; so are I', m', n' and #', y, s' of s. 3_ < ^(^-a;) = ^3y l(of-x)^rjl' l x'-x 1 3r ? 3s' r r 3s' r 2 3s' r r r 3s' since l'='dz'fdJ. Finding the derivatives of m(y' - y)/r, n(^ - z)/r and adding 3V + r + m? * + n- \ r r cos_3r !_ 3r r 3s r 3s'' which gives (iii). Also 3(r~*)/3s= r~ 2 orfds = cos Ofr 2 , which is (iv). EXERCISES XIX. 241 10. Let QP in ex. 9 be produced backward to Q lt making PQ l = QP. Let u = l/PP' = l/r, and let u q , u^ denote 1/QP', 1/&P'. Show (as in 92) that ~OU r Un Un, ~OU 11. With the notation of ex. 9, let P be the centre of an elementary magnet of moment M, whose axis is in the direction PQ ; show that the potential V at P' of the magnet is V- ,f At Q, Qi (see ex. 10) let quantities 772, - m of magnetism be placed ; the potential at P' of these quantities is mu q - Let Q$ tend to zero while the product mQ^Q remains constant and equal to M ; then V is the limit of the fraction just written, which by ex. 10 is M ~- = ~- = -5. os -r 12. The components of the magnetic force at P' (ex. 11) are -"dVfdaf, -'dVfdy', -3F/3/; show that _ 'dV_3M(g/-x)cosO Ml ^~ i* r 3 ' with similar expressions for the other two components. 13. If an elementary magnet of moment M' is placed with its centre at P' and its axis along P'ty, show that the mutual potential energy W of the magnets is TV- nr^ V - nmr^( r ~ 1 ') JUf(co& c + 3 cos cos ff) rr JJL ~^~~7 JH Ja ~^ *^. ,~ - - 5 OS OS OS 1T Apply the method of ex. 11, taking Fin place of u or l/r. Q.C. CHAPTER XII. APPLICATIONS TO THE THEORY OF EQUATIONS. 100. Rational Integral Functions. If f(x) is a rational integral function of x of degree n, it is proved in treatises on the theory of equations that in general there are n values of x which make f(x) zero ; these values are called the roots of the equation f(x) 0, or the zeroes of the function f(x). These values are not, however, necessarily real numbers, nor are they necessarily all different. Thus, if f(x) = (x l) 2 (a; 2)(a? 2 + l), f(x) is of the 5th degree; two of the roots of f(x} = are equal to 1, one root is 2, and there are two imaginary roots J( l). a is called an r-ple root of f(x) = 0, or an r-ple zero of f(x) if f(x) contain (x a) r , but no higher power of (x a). In this case f(x) is of the form (x a) r 0(#), and (a) is not zero; if (a) were zero, then by the Remainder Theorem proved in Algebra 0(a?) would contain x a, and therefore f(x) would contain a higher power than (x a) r . When f(x) = (x a.) r <}>(x) it is obvious that the 1st, 2nd ... (r l)th derivatives of f(x) will contain a; a as a factor, and will therefore vanish when x = a. We leave it as an exercise to the student to show that the necessary and sufficient conditions that a should be an ?*-ple zero of f(x) are that f(x) and its first (r 1) derivatives should vanish when 05 = a, but that the r th derivative should not vanish when x = a. Also that the multiple roots of f(x) = are roots of /'() = 0, and may therefore be obtained as the zeroes of the G.C.M. of f(x) and f'(x). Manifestly the graph of (x a) r (x) will or will not cross the ic-axis according as r is an odd or an even integer ; if THEORY OF EQUATIONS. 243 r>l the ai-axis will be a tangent at the point (a, 0), since in that case /'(a) will be zero. Ex. 1. Show that 2 is a triple root of the equation 3#* - 1 6.T 3 + 24# 2 - 1 6 = 0. /(2), /'(2), /"(2) are zero, but/"(2) is not zero ; /(2) is (x - 2) 3 (3.r + 2) so that 2 is a triple root, and 2/3 is the remaining root. Ex. 2. Find what relation must hold between q and r that the equation x 3 + qx + r = should have a double root. If the root be a, then./(a) = 0,/(a)=0,/"(a)=f=0 ; therefore a? + qa + r=Q (i) ; 3a 2 +g-=0 (ii) ; 6a=K> (iii). From (ii) a 2 = -q/3, and therefore by (i) 2g r a/3 + r = 0. Hence a 2 = -q/3, and a 2 =9^/4^, so that 27~ + 4 and when y 0, x = a v Hence a, = a -/()//'() ........................ (1) Let the line through B parallel to the tangent at A cut the cc-axis at the point 6 X ; the equation of the line is y =f(b)+(x-b)f(a). Hence b, == 6 -/(&)//'() ........................... (!') 246 AN ELEMENTARY TREATISE ON THE CALCULUS. and 6 X lies between b and a, so that b l is a better approxi- mation than 6, though not necessarily better than a. Now b,-a,= - { f(b)-f(a)-(b-a)f(a)}/f(a), which by the Mean Value Theorem may be written where x l lies in the interval (a, 6). Let d be the numerical value of (6 a), d l that of (b l a^), and let denote the greatest value of f"(x), g the smallest value of f(x) in the interval (a, b) ; then or d^d^k, k=G/2g. Since a a x is numerically less than b l a v we have a a x numerically less than d l or d 2 k, so that the error in taking a x instead of the root a is less than d 2 k. Similarly the error in 6j is less than d 2 k. We may repeat the process with a v b^ instead of a, b ; we should find, using a similar notation, d^d^k, that is, d s ^d'J, and the error in taking 2 or 6 2 is less than cZ 2 or (^A^. The process may be repeated. As soon as a, b are such that dk is less than 1, the approximation to a becomes very rapid. There is, as a rule, no need to calculate, 6 a , 6 2 ... The student will see by examining figures that if a is not chosen as stated, the value of a x or b l may be further from a than a or b. 104. Examples. Ex. 1. If f(x) = 3.T 3 - 4# + 5, find the roots of j(x) = 0. - )(*-); f'(x)=l8x; /"(*) = 18. /(-) = 6f is a maximum value of fix) ; /(|) = 3| is a minimum. The point (0, 5) is a point of inflexion. It is easy to see that the graph of f(x) crosses the #-axis once only, so that there is only one real root. /( 2)= 11, /(-I) =+6, so that the root lies between -2 and 1 ; as /( 2) and /( 1) are large, we seek a closer approximation before choosing a, 6. Now /( - 1 '6) = - '888, /( - 1 -5) = + '875. Since f'(x) is negative when x is negative, we take a= - 1'6, b= 1'5. WORKED EXAMPLES. 247 G= numerically greatest value of f'(x) in interval ( 1'6, - 1 '5) = 28 "8. g = numerically smallest value of f(x) in interval = 16'25. k=G/2g = 14-4/16-25 < 1 ; d='I ; cM < -01. CTI = a -/(a) //(a) = - 1 '6 + '04 = - 1 '56, and aj differs from a by less than '01. o s =a 1 -/(a 1 )//(o 1 )= - 1-56 + -0083= - 1-6517, and 2 differs from a by less than d*^ 3 or '0001. The values "04 and '0083 are of course approximations. Care must be taken that we do not go beyond the root. Thus but if we take '05 as the value, thus making a,= 1'55, we find f( I "55) to be positive. The reasoning, however, depends on having f(cii) of the same sign as /"(#), that is in this case negative. A closer approximation is a 3 =- 1-551 608 12, and the error is less than a unit of the last decimal place. Ex.2. Solve the equation x + sinx -^=0. o If A is a point on the circumference of a circle, and if AB, AC are two chords which trisect the area of the circle, then the angle between AB and the diameter through A is \x radians. ^ ; f(x) = l+cosx ; f"(x) = - sin x. o It is easily found that x lies between 30 and 31, or in radians 5236, and '5411. X'5236) = - '0236 ; /('5236) = 1-8660 ; /(5411)=+-0089; /( -5411) = ! '8572 ; d= -0175 < -02 ; k=G/2g < '2, d*k < '00008. Since /"(#) is negative, we take a = '5236, osj = a -/(a)//(a) = '5236 + '0126 = '5362, and the error is less than a unit of the fourth place. The next approximation gives 0.5=^ -/(!)//(!>= '5362 + -0000674 = -5362674, and the error is less than a unit of the last figure. In degrees the angle is 30 43' 33"'0. 105. Successive Approximations. Suppose the equation to be of the form x = (a) = h(f>(a + Oh). 248 AN ELEMENTARY TREATISE ON THE CALCULUS. Using the terms " greater " and " less " to mean numeri- cally " greater " and " less," we see that if, for every value of x that is nearer to a than a is, (j>(x) is less than a proper fraction m, the difference between a and (oC) is less than that between a and a. Hence '(a + 6h) and therefore less than mh. We find in the same way a (o(a l ) = a 2 is a closer approximation than a r The upper limits of the errors km, km 2 usually decrease pretty rapidly as m is, in the cases to which the method applies, often a small fraction. We may proceed, of course, with a z and so on. It is essential for the success of the method that (f>'(x) be, near the root, a proper fraction. It may be proved that Newton's method is a particular case of that of Successive Approximations, and unless m be pretty small the latter method has no advantage over Newton's. Ex. Solve the equation 10* =3456 *Jx. Take logarithms to the base 10, and we get x=\ log .r + 3-538 5737 = ^). If we draw the graph of \ log x and of x - 3-538 5737 we see that they intersect for a value of x near 4 and also for a very small value of x. Take first a = 4, now .,, x M '4343 ,, < ^^ = 2x = ~zH' t M=l S e > so that when x is nearly 4, '(#) is a proper fraction. Take 4-figure logarithms for the first approximations, a 1 = <(4) =3-5386 + '3010 = 3-8396; 2 = <(!) = 3-5386 + -2921 = 3'8307 ; a s = cj!>(a 2 ) = 3-5386 + '2916 = 3-8302. When .r=a 2 > x - ^(x) = -0005, so that a, is a fairly close approxima- tion. Take now 7-figure logarithms, and we find 4 = <(a 3 ) = 3-538 5737 + -291 6107 = 3'830 1844 ; a 5 = ^>(a 4 ) = 3-830 1835; e = '(z) is greater than 1. But since x is very small we get a good approxima- tion by taking the value of a-, which satisfies <(.r)=0. Therefore log.r= - 7-077 1474=8-922 8526 ; and x= -000 0000 8372 45. 106. Expansion of a Root in a Series. Reversion of Series. Let the equation x = (x) be or x where u 2 , u 3 ... are of the 2nd, 3rd... degree in x and y. If y is a small quantity one root will be approximately Ay, for this value of x makes u. 2 of the second order in y, u s of the third... . Call this approximation a. Clearly for small values of x we may suppose \x) a proper fraction. The next approximation is a l = (a) = (Ay). To the 2nd order in y we may neglect u 3 , u. . . and take = Ay+B l y 2 sa,y. The next approximation is a z = = tan ~ J ( - ) Hence x \x/ putting c for TI-TT + Tr/2 we have x = c tan ~ l ( - } \x/ It is shown in a later chapter that -J-...!..-!. x Sec 3 5x 5 Ix 1 so that x = c 1-?;-* 7r-? + ~- 7 x Sx 3 5o? 6 7a; 7 The equation may be solved by the method of last article, since x is, even for n = 2, greater than 7'5, and therefore l/x fairly small. 1st. App. x = c; 2nd. App. x = c ; 3rd. App. x = c EXERCISES XX. 253 / 1 2 V 1 1 / 1\~ 3 1 4th. App. x = c-c--- - 5C 5 ' 12 13 C ~~ ~~ ^~ ^ o "~~ ^T~i P I c 3c 3 loc 5 1 2 13 146 5th. App. x = c--- s -TR-6-: -_-_- For 7i = 2, 3, 4,..., this last approximation is amply sufficient for all practical purposes. The student may show that X/TT has the values 1*4303, 2-4590, 3*4709, 4'4747, 5-4818, 6-4844, for n = l, 2, 3, 4, 5, 6. [Rayleigh's Sound, I., p. 334 (2nd Ed.).] Many equations involving trigonometric and exponential functions were discussed by Euler, and the general solution of the equation x = tan x is due to him. EXERCISES XX. In the following examples it will usually be sufficient to calculate the root to 3 or 4 decimal places ; in some cases the results are given to more figures. 1. Find the real root of 3^ + 5^-40=0. 2. A sphere of radius 1 is divided by a plane into two parts whose volumes are in the ratio of 1 to 2 ; the distance x of the plane from the centre of the sphere is a root of the equation Sx 3 9#+2 = 0. Find x. 3. Find the root of ^-4^-7^+24=0 that lies between 2 and 3. 4. If (!+#)* = 27 -34, find.*:. 5. If 10* =20^, find.r. 6. The chord AB of a circle, centre (7, bisects the sector ACB; if the angle A CB is x radians, show that x = 2 sin x and find x. 7. Solve the equation x=cosx. 8. The equation 2.r=tan;3; has one root between and ?r/2 and another between TT and 3:r/2 ; find both roots. 9. Show how to solve the equation / . _i\ t I=a(e 2a -e "*) for a when I and c are given, I being not much greater than c ; for example, c=100, = 105. The value of a determines the catenary 254 AN ELEMENTARY TREATISE ON THE CALCULUS. assumed by a string of length I hanging from two points in a hori- zontal line distant c from each other. 10. Find the least roots of (i) (e x +e-*)cos^-2 = 0; (ii) (e*+e- x )cosx+2 = 0. Obviously zero is a root of (i) ; find the next smallest root. 11. Solve x a sin x =b where a = '245316, 6 = 5-755067. 12. Show that the approximations to the root a of x=(x) given by the method of 105 are alternately greater and less than a if '( a ) is negative. 13. If /(#, y) = and f(x,y) = have as an approximate pair of solutions x=a, y = b show that in general the values a + h, b + k will be closer approximations if h, k satisfy the equations /(a, *)+A + *=O, F(a y where in the derivatives x, y are replaced by a, 6. If f(x, y)=x 3 + 3xy z -y-U, find closer approximations to the roots near #=2, y = 1. 14. If (y-x)(y- < 2 l x)=x z + '2a?y+x*i/* show that when x is small there are two values of y given, as far as terms of the third order in #, by the equations =x 3? x z and = Show that the curve given by the equation has two branches that pass through the origin and that the tangents at the origin arey=# and y = 2#. Sketch the curve for small values of x. [Write y=x-\ ' r \ -- g-H -- - and proceed as in 106; then y tX y 2iX y 2>x write y = 2x+3?f(y x) + , etc.] 15. If a 2 (y 2 -x 2 )=x*+y*, show that for small values of x there are two values of y given by y=x+x*la 2 and y= x x*la z . Show also that near (o, a) the shape of the curve is given by Graph the curve. 16. If (y-x 2 = x 3 +x 2 +x* show that for small values of x Graph the curve near the origin. RULE OF PROPORTIONAL PARTS. 255 108. Proportional Parts. In the use of Logarithmic and similar Tables it is often necessary to find the value of the function for a value of the argument not given exactly in the Tables. It becomes necessary, therefore, to interpolate, and the ordinary rule is based on the assumption that the difference in the function is proportional to the difference in the argument. We will now examine the assumption. Let h and z be small quantities having the same sign, but z being numerically less than h] then by the Mean Value Theorem, f(x), f'(x), f"(x) being assumed continuous, the following equations are approximately correct. ............... (1) ............... (2) Let D=f(a + h)f(a) and eliminate /'(a) ; therefore ') ............. (3) Equation (3) is approximate, but by following the lines of the proof of the Mean Value Theorem we can show it to be exact if in place of f"(a) we write f'(a + 6h) where 6 is a proper fraction. Forlet (A) and let F(x) =f(x) -f(a) - ^-j^D -%(x-a)(x-a- h)P. Now F(a) = identically; F(a + z) = Q by (A); F(a+h) = identically, remembering the value of D. Hence F'(x) must vanish for a value of x between a and a + z, and again for a value of x between a + z and a+h; therefore F"(x) must vanish for a value of x between these two values, and therefore between a and a + h. But JH() -/*() -P, and therefore P =f'(a + 6h). Hence, instead of (3) we get the exact equation, (4) where D=f(a+h)-f(a). 256 AN ELEMENTARY TREATISE ON THE CALCULUS. In the figure (Fig. 53), OA=a, AC=z, V US=f(a+z)-f(a). UT=zD/h. C B error committed in replacing the arc PSQ by the chord PTQ is measured by ST. z is by hypothesis less than h, and the numerically greatest value of z(hz) is \fi 2 . Hence, if G be the numerically greatest value of f"(x) in the interval (a, a-\-h), the numerically greatest value of ST or of will be IWG. Suppose now that f(x) is tabulated for a series of equi- distant values of x, the difference between successive values being h. Let a + z be a value of x between a and a + h, and therefore not given in the Table. The ordinary rule is to calculate /(a + 0) from (4), neglecting the second term on the right ; that is, For a given value of a, the amount by which f(a) is in- creased to find f(a + z), namely zD/h, is therefore proportional to z ; the error in following the rule is therefore not greater than h*Q/8. Exceptions to the application of the rule occur in the following cases : I. G may be such that h 2 G/8 can not be neglected in comparison with zD/h ; in this case the difference D is said to be irregular. II. D may be so small that it vanishes to the number of figures in the Table ; in this case the difference is said to be insensible. The difference will be insensible when f'(a) is very small, since D =/(a + h) -f(a} = hf(a) + U 2 /"( + Oh). EXCEPTIONS TO RULE. 257 Example. f(x) = Iog 10 sin x. Let M = log lo e = "434 2945; then f'(x) = Mcotx; f"(x)=Mcosec 2 x. If x is small, f"(x) is large, and the differences are irregular; since cot* is not large the differences are not insensible. If x is nearly 90, cot x is small, and the differences are insensible; though f"(x) is not large the ratio f"(x)/f(x) = 2/sin 2x is numerically large and therefore h?G/8 can not be neglected in comparison with zD/h. Near 90 therefore the differences are both insensible and irregular. For tables that proceed at differences of V, h is 1' or in radians h = -000 2909, and MW = -000 0000046. To find when %Mh 2 cosec?x would affect the seventh figure we may put im 2 cosec 2 a = 5 x 10" 8 , and we find from this equation that x is about 18. Hence, apart altogether from errors due to neglected figures in carrying out the numerical work which may easily amount to more than a unit in the seventh place, the error due to neglecting the term h 2 G/S will amount to half a unit in the seventh place for angles less than 18. If h is equal to 10" the student may show that the seventh figure will not be affected by the neglect of h?G/8 till the angle is about 3. The student may with advantage consult Hobson's Trigonometry, Chap. 9. The advanced student will find a thorough discussion of all the principles involved in numerical approximations and the use of tables in Liiroth's Vorlesungen uber numerisches Hechnen (Leipzig: Teubner, 1900). Ex. 1. Show that for log cos x the differences are insensible and irregular when x is small, and irregular when x is near 90. Ex. 2. Show that for log tan x the differences are irregular when x is small and when x is near 90. Show also that the maximum error is least when x is near 45. o.c. R 258 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 3. In a 7-figure table of the logarithms of numbers, show that the term h' 2 G/8 is most important when the number is 10000, and that the greatest error arising from the neglect of that term is about 5'5 x 10~ 10 , and is therefore negligible for these tables. 109. Small Corrections. In practice all measurements are subject to errors, and it is therefore of importance to determine the influence on the result of a calculation when the argument or arguments of the calculated function are given by measurements whose errors are approximately known. Let a quantity x be determined by measurement and let y be a function f(x) of x. Suppose that the value x given by the measurement differs from the true value by Sx, then the true value of y is f(x + Sx) and the error Sy is Sy =f(x + Sx) -f(x) =f(x + eSx) Sx or Sy=f'(x)Sx approximately. The relative error Sy/y is, approximately, --, r . y /() As a rule it is the relative error that is important; of the 1 two factors Sx and f(x)/f(x) the first depends solely on the accuracy of the measurements while the second is conditioned by the general arrangements of the inves- tigation. If there are two or more variables, x, y, z say, then the error Su in the function u=f(x, y, z) is as far as quantities of the first order in Sx, Sy, Sz. Since the value of Su is of the first degree in Sx, Sy, Sz the joint effect of the individual errors Sx, etc., is obtained by addition of the effects due to each separately. This principle of " the superposition of small errors " is of great importance in practice. Ex. 1. The side a and the angles B, C of a triangle ABC are measured ; if these be liable to the errors 8a, SB, SC, to find the error in the calculated value of the area S. SMALL CORRECTIONS. 959 Denote by (&S) the error in S due to the error Sa taken by itself, and use a similar notation for the other errors. In finding the derivative of S it is most convenient to differentiate logarithmically. =a 2 sin B sin <7/sin (B + C), (8) a /=2Sa/a, (8S) B /S={cot B - cot(B+ C)}8B, (8)<7/={cot C-cot(B+C)}8C. The total error SS is got by adding these separate errors. As an example, let a = 250 (feet), ^=27 12', (7=45 18', 8a='25, 85=10', 8(7=20'. The percentage errors in a, B, C are 100^ = -!; 100^='6; 100^=7. a B C It is sufficient therefore to use 5-figure logarithms. We find ~ = -002 + -00474 + -00392 = '01066. o 5=10646; 85=113-49; 1 ^^=1'1. o The calculation of S from the values a + 8a, B+8B, (7+ 8(7 gives, if S' be the new value of S, ' = 10760; '-=114. Since 6 = a sin B/ sin (B +C), we have for the error in b 8b/b=8a/a+(cotB-cot(B+C))8B-cot(B+C)8C, so that 86/6 = "00390, 10086/6 = '4, 86 = -5 nearly, and in the same way Sc/c=-0040, 100Sc/c = -4, Sc='75. Ex. 2. The sides a, 6, c of a triangle ABC are measured ; to find the error 8A in A due to errors Sa, 86, 8c in a, 6, c. We may take the value of cos A given by and differentiate ; but the result may be obtained more quickly, thus : a b cos C+ c cos B ; therefore Sa = cos C 86 + cos B 8c - (b sin C 8C+ c sin B 8B) = cosC8b+cosB8c-bsmC(8C+8B) = cosC8b + cosB8c+bsinC8A, since 6 sin C= c sin B and A + B + C= 1 80, so that 8A+8B + 8C is zero. Hence 84 = (Sa - cos C 86 - cos B 8c)/b sin (7, and the trigonometrical functions may easily be expressed in terms of the sides if required. .260 AN ELEMENTARY TREATISE ON THE CALCULUS. EXERCISES XXI. 1. The area S of a triangle ABC is determined by a, b, C ; show that the relative error in the area is given by 8f^ = 8a 8b Sab Show that the error in the side c is given by 8c = cos B 8a + cos A 8b + a sin B 8C. 2. At a distance of 120 feet from the foot of a tower the elevation of its top is 40 16' ; if the distance and the elevation are measured to within 1 inch and 1 minute, find the greatest error in the calculated height. 3. If the density (p) of a body be inferred from its weights W, w in air and in water respectively, show that the relative error in p due to errors 8 W, 8w in W, w is 8p = -w 8W 8w p ~ W-w W + W~io 4. The side a and the opposite angle A of a triangle ABC remain constant ; show that when the other sides and angles are slightly varied, 5 , 9 oo be y. == Q COS B COS C 5. If a triangle A BC be slightly varied but so as to remain inscribed in the same circle, show that 8a 8b 8c ' H r5~r cos A cos B cos C 6. In a tangent galvanometer the tangent of the deflection of the needle is proportional to the current ; show that the relative error in the value of the current due to an error in the reading of the deflection is least when the deflection is 45. 7. If ordinates which differ by less than one-hundredth of the unit line are considered to be equal, show that the parabola y=x + 2x* will coincide with the graph of for values of x between '14 and -I- '14. 8. Show that the curve x s +y 3 = 3axy has two branches which pass through the origin and that the equations of these branches near the origin are *2 = 3ay, y* = 3a*. Show that closer approximations are given by EXERCISES XXI. 261 9. Show that, near the points stated, the curve X s + y 3 = 2aa? is given by the respective equations, a being positive. Near (o, o) y 3 = 2ax 2 ; near (2a, 0) if= 4a?(x-2a); At infinity y = -z + 2a/3 + 4a 2 /9x.' Show that y is a maximum when x = 4a/3 and graph the curve. 10. Show that for the curve xy z x 3 6x 2 +zy+y 2 =0 the following approximations hold : Near (o, o) y = 2x-%x 2 and y = S-r+f-r 2 . At infinity x+l = l/y, y=z+2-5/z, y=-x-3 + 6(x. Show that the asymptote #+1=0 crosses the curve at (-1, -5), the asymptote y = x + 2 crosses at ( , f ) and the asymptote y = -x-3 crosses at ( - f , - f ). Graph the curve. 11. Show that the curves have each a cusp at the origin but that both branches of (ii) lie above the .r-axis near the origin. Graph the curves. In case (ii) the cusp is called a cusp of the second kind or a ramphoid cusp while the ordinary cusp is called for distinction a cusp of the first kind or a ceratoid cusp. CHAPTER XIII. INTEGRATION. 110. Integration. In 82 the general problem of the Integral Calculus has been stated, namely : Given a con- tinuous function F(x), to find another function which (i) has F(x) as its derivative and (ii) takes a given value A when x takes a given value a. When condition (i) alone is given there is an indefinite number of solutions. These solutions, however, differ only by a constant; any one of them is called an indefinite integral of F(x) and the constant is called the constant of integration. This constant is sometimes called an arbitrary constant since it may have any value whatever. If f(x) is an indefinite integral, f(x} + G is called the general in- tegral, G being an arbitrary constant. Instead of the notation of inverse functions D x ~ l F(x) it is customary to denote the indefinite integral of F(x) by the symbol /* \F(x)dx- (1) read, " the integral of F(x) with respect to x," or " integral of F(x)dx." The differential dx indicates the variable of integration, namely x, and the joint symbol I . . . dx means " integral of . . , with respect to x." F(x} is called the integrand. What was in 82 denoted by [D~ l F(x$ is now denoted by r& F(x)dx- (2) J read, " the integral from a to b of F(x) dx." The function INTEGRALS. NOTATION. TERMINOLOGY. 263 denoted by the symbol is called a definite integral, and a, b are called the limits of the integral, a being the lower limit and b the upper. (The word " limit " in this use of it means merely " value of the variable of integration at one end of its range," " end- value " ; this use of the word must not be confused with the technical sense employed in other con- nections.) The interval (6 a) is called the range of integration. Geometrically, the symbol (2) denotes the area, in sign and in magnitude, swept out by an ordinate of the graph of F(x) as x varies from the lower limit a to the upper limit b. If /() is an indefinite integral of F(x) then as in 82 =f(b)-f(a) .......... (3) We may, if we please, use the general integral f(x) instead of f(x) ; the result will be the same since C, being a constant, will disappear in the subtraction. It follows at once from the geometrical meaning or from (3) that [*F(x) dx=- [ b F(x) dx = f(a) - /(&) ............ (4) Jb Ja that is, the limits a, b may be interchanged if at the same time the sign of the integral is changed. Again, the form /(&)/(), or the geometrical meaning, shows that the definite integral is a function of its limits, |*6 not of the variable of integration. Thus I F(u)du has rb Ja precisely the same value as I F(x) dx. Ja From the point of view of a rate, F(x) when it is the derivative of f(x) measures the rate at which f(x) increases with respect to x~, the amount, positive or negative, by which f(x) increases as x varies from a to 6 is f(b)f(a). Hence the definite integral (3) measures the amount by which a function f(x) increases for a given change (6 a) of its argument when the. rate of change, F(x), of the function is known. The function which has F(x) as its derivative, and which is equal to A when x is equal to a, is ( 82) 264 AN ELEMENTARY TREATISE ON THE CALCULUS. and is, in the present notation, represented by {*F(x)dx + A or by ^F(u)du + A (5) Ja Ja Here the upper limit x denotes the particular value of the argument for which the function is calculated. In the geometrical representation of 82 the upper limit x is the abscissa OM of the point P. From the point of view of rates the symbol (5) denotes the function which is equal to A when its argument is equal to a and which increases at the rate F(x). The subject of definite integrals will be more fully con- sidered in Chapter XIV. ; enough, however, has been given in this article and in Chapter X. to enable the student to solve the simple examples on areas, etc., which are given in the exercises of this chapter. 111. Standard Forms. Integration from the point of view from which it is now being considered is simply the inverse of differentiation and the first requisite for the calculation of an integral, definite or indefinite, is a table of known integrals; the table will be formed from an examination of the known results of differentiation. Various methods will then be given for reducing, if possible, an integrand not found in the table to a form that may be integrated by means of the standard forms. In all cases of indefinite integrals the test to be applied is that the derivative of the integral must be equal to the integrand. In symbols if so that the equation that defines an integral is dx\_j J Considered as symbols of operation d/dx and \...dx are inverse to each other. J In the language of differentials F(x)dx is the differential STANDARD FORMS OF INTEGRALS. 265 of f(x) when f(x) is the integral of F(x); f(x) is often called the integral of the differential F(x)dx. Since F(x)dx = df(x) = d\\ F(x)dx\ the operators d and I are inverse to each other. The following table contains what may be called the fundamental standard forms; other important forms will be given later. Most of the forms are given twice ; the argument occurs so often in the combination ax+b that the student should from the outset make himself familiar with the corresponding integral. The results should of course be tested by differentiation. 1. If TC =h-l. 2. If %= -1. \-dx = \ogx: \- jx Jax - - T ax + b a f a ]e*dx = e*; 4. I sin # efoe = cos a; ; \sm(ax + b)dx= cos(o;4-&)- J J ft f f * J J ft 6. sec 2 a; da; = tan a? ; sec 2 (00* + 6) das = - tan (ax + b). 7. I cosec 2 o; dx = cot x ; f 1 I cosec 2 (a# + 6) dx - cot (ax + b). J ft f dx . ,/x\ = sm" 1 a;; 177-5 5r = sin~ 1 ( -) : f(fl^sp\ \f1 / J w v 1 * <**/ dx 8. f x or =cos~ i x: or = cos \a. 266 AN ELEMENTARY TREATISE ON THE CALCULUS. 9 - 1n dx f dx 1 x f dx f * IT-. s tan- 1 ^; 1 jl+x 2 j a?+x 2 a 1 , l /x\ or = cot' 1 ^; or = cot' 1 !-)- a \a/ 11. -- z =- log if 2 >a 2 ( 112, Ex. 2), o; 2 a 2 2a & Since sin" 1 ^ ( cos" 1 .?;) is equal to ir/2, both sin" 1 ^ and cos" 1 ^ are integrals of l/*J(l x 2 ) ; a similar observation holds for the integral of 1/(1+^ 2 ). An indefinite integral may often be expressed in different forms, any two of which must however differ only by a constant. Particular care is required in dealing with the inverse trigonometiic functions since these are many-valued ; the restriction on the range of the angle ( 28, 64) must always be attended to. If x is negative, the integral of I/A- is not log^p but log( x) ; if x is less than a, the integral of l/(x-a) is log(a-.r). Form 11 is inserted for the sake of comparison with 10 ; for a similar reason forms 8 and 9 are brought together. Again, if x is negative, it may be verified that the integral of cosh- 1 - Instead of the logarithms in form 9 inverse hyperbolic functions may be used ( 66). dor /&\ [ dsc / 2 . 2\ = s i nn ~ 1 ( - )' // 2 - o\ = ^ + a ) \ a ' J*J( X a) and it should be remembered that cosh ~ l x is two- valued. The forms tanh^o;, coth' 1 ^, are of less importance. Ex. 1. Integrate with respect to x Ex. 2. Evaluate fw [, /-I ff r C^dr f-^dx 1*7 7 \Jv a 2 , because then, and only then, is (x- a)l(x+a) positive ; if x 2 < a 2 the integral is 1 , a x because in that case the integral of l/(x - a) is log (-#). The transformation is a particular case of the method of partial fractions, and the student should refer to some text-book of algebra for an account of the method ; see also 120. 3^-5 Smce - x-l x-V Ex. 3. The forms ~, -jf- a + bx* v '(a If a, b are both positive, we have f dx I r dx l f b\ a ^ If a is negative, 6 positive, we reduce the integrand to the form of ex. 2 ; thus dx 1 ' fcg 1 / In a similar way l/*J(a+bx 2 ) may be treated ; thus dx I f dx 1 ,/a\/2\ _ _ I _ __ Qiri~'M -^ 1. 3-2^~2'-^ 2 x/2 \N/3/ EXERCISES XXII. 269 "With a little practice the student should be able to do many of the steps mentally ; the full process for the first case is a + a .2 b Ja l^/a T* 7i" \ 77" Ex.4. sin n .r, cos"x, smmxcosnx. When n is a small positive integer, sin n .r, cos n x may, without any difficulty, be expressed in terms of sines or cosines of multiples of x ; for other values of n it is best to take the method of successive reduc- tion ( 119) or the method of ex. 4, 114, sin 2 # = (1 cos 2.r) ; sin 3 .r = f sin x - \ sin %x ; I sin?xdx=&x-%ain2x ; I sin?xdx= -f In the same way powers of cos x may be treated. Again, a product of a sine and a cosine, or of two sines or of two cosines, may be expressed as a sum or a difference of sines or cosines and then integrated. Thus sin mx cos nx = i { sin (m + n }x + sin(m ri)x } ; hence, if #t=j but if m = n, then the integral is J_ 4m EXERCISES XXII. Integrate, with respect to x, examples 1-15. 1. - ' ; 2. - ; 3. 4. ar ~ 7 ; 5. L_; 6. 1 . o q rrt a2 r . in pni 3 r' 7/ii ^-ov > o- .,* . 75^ cos * * cus m - 11. cos 2 (a^+6) ; 12. sin 4 ^; 13. sin3^sin4ar; 14. sin(3^ + 2)cos(4r-f 3) ; 15. cos x cos 2x cos 3x. Find the value of the integrals in examples 16-21. /"' r* , f z dx 16. / cos 2 x dx ; 17. / sin 2 2x dx ; 18. / Jo Jo Jo ' 19. ; 20. tJ 21. 270 AN ELEMENTARY TREATISE ON THE CALCULUS. 22. If m, n are unequal positive integei-s, prove that fir Cir I co$mxcosnxdx=Q = I sin mx sin nx dx, Jo Jo and find the value of each integral when m, n are equal positive integers. 23. Show by considering the graphs of the integrands that the following equations are true : / f% (i) I cos n xdx= / sm n xdx where n is positive, Jo Jo fir /? (ii) I sin n xdx=2 / s\n n xdx where n is positive, Jo Jo rir r% (iii) I cos n xdx2 I cos n xdx if n is an even integer, Jo Jo but =0 if n is an odd integer. 24. The area bounded by the parabola y*=4ax, and the double ordinate through the point (b, c) on it is |6c. 25. If , b are positive and a(u) ; dy dy dx . . dx n N then -T L = ^T L ^-=F(x)^- ................... C 1 ) du dx du ' du In (1) let dx/du be found from the equation x = (u) and then express the new integrand F(x) dx/du in terms of u by means of the same equation. Equation (1) will now be free from x and we shall have 2/=KW^7' It may happen that the new integrand is, as in the above example, a standard form ; if not, it may perhaps be more easily reduced to one than the old integrand F(x). Expressing y as an integral with respect to x, and equating it to the value given by (2), we have x}du .................. (3) The simple rule then for changing the variable is: Replace dx by (dx/du) du and by means of the equation 272 AN ELEMENTARY TREATISE ON THE CALCULUS. between x and u express the new integrand F(x) dx/du in terms of u ; the integral will then be a function of the new variable u. When the integration has been effected the integral should be expressed in terms of the old variable. If when x = a, u = a, and when x = b, u = {3, the relation being such that as x varies continuously from a to b, u also varies continuously from a to /3, then (x)du ................ (4) In this case, of course, there is no need for returning to the old variable. In applying the transformations (3) and (4) it is essential that to each value of x there should correspond one and only one value of u, and to each value of u one and only one value of x, within the ranges b a, /3 a of integration. When the equation between x and u gives u as a multiple- valued function of x, or # as a multiple-valued function of u, care must be taken to choose the proper value. (See 117, Ex. 3, 123.) 114. Examples of Change of Variable. Ex. 1. F(x) of the form -^(ax+b). Let u=ax+b; duadx^ dx=-du I TJS (ax + b)dx = - I ^r(u) du. This type constantly occurs. Thus if =#- 1/4, dx , f dx so that the integral is 1 4 dx if ^-x + ir^h{(*- A constant factor, like 2, can be taken outside the integral sign when necessary ; similarly a constant factor may be introduced, as in ex. 3. WORKED EXAMPLES. 273 Ex. 2. F(x) of the form \ls(x n )x n ~ l . Let u = x" ; du = nx n ~ l dx. x n ~ l dx-du n I ^(x"}x n ~ l dx - / \lf(u)du. Thus, if u = x\ f l f J 2iJ and the given integral is The integral may also be found by putting u = ax 1 + b, or by putting u 2 = ax 2 + b. The last substitution gives if \f u 3 xdx=-udu ; / *J(ax'* + b)xdx=- I u?du = ~ , leading to the same value as before. Ex. 3. F(x) of the form OK- r )]"V / (#)- Let u = -^r(x) ; du = ^r'(x)dx ; F(x)dx=u n du, and the integral is a power or a logarithm according as n is different from or equal to 1. We have (3a) From (3ft) we see that when the integrand is a fraction whose numerator is the derivative of the denominator, the integral is the logarithm of the denominator. The introduction of a factor is sometimes needed to make the integrand of the form 3. Thus (iii) I tan x dx = - I - dx = - log cos x. J COS X (i v) / tau 3 .r dx = / tan x (sec 2 .r - 1 ) dx = I tan x sec 2 .r dx - I tan x dx, and therefore = ^ tan 2 .r + log cos x. G.C. s 274 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 4. F(x) = a\ (i) When either m or n is an odd positive integer the integration can be effected by substituting u for cos.r when m is odd, but u for sin x when n is odd. For example, take F(x} = sin^r cos 5 .r. = (l - w 2 ) 2 / sin* x cos 5 x dx=\ (u% - 2u? + T ) du s = sin*dr(f - * sin 2 .r + 1\ s /"~snArcos 5 #cfa= f (^-2*+^)e&/= JO JO Again, if = cos #, rfw = - sin x dx I sin 6 x dx 1(1 u?) 2 du = - (u - f M 3 + iw 5 ), and / sin 5 ^c?^= cos^ + cos 3 ^ \ cos 5 x. (ii) When m + n is an even negative integer, let w = tan.r (or cota?) ; the new integrand can be expanded by the Binomial Theorem. Thus, dx l+M 24 du l 33 _ffl ~J U 6 and the integral is readily found in terms of x. Ex. 5. If F(x) is a rational function of x and of *J(ax + b), the sub- stitution ax + b = u z will make the new integrand a rational function of u. Thus, if x+ 1 =w 2 , = 2 (u 2 - l)Ww = 2(X - f M and after a little reduction we get for the integral The forms just given include many of the most important cases in elementary work, and the student should at once try the earlier examples in Exercises XXIII. Only through practice will he gain facility in making the transformations. 115. Quadratic Functions. If R = ax z + bx + c and if f(x) is rational and integral, the fraction f(x)/R can be expressed as the sum of an integral function and a proper fraction (Ax + B){R. We will now consider the forms and QUADRATIC FUNCTIONS. 275 For beginners the simplest method is to write R in the + : when a is positive we may take it as equal to +1, and when negative as equal to 1 : there is no loss of generality in so doing since a constant factor may always be taken outside the integral sign. If 4ac 6 2 is positive the factors of R are imaginary ; R is then of the form (i) If 4ac 6 2 is negative the factors of R are real, and for a= + 1, R = (x + a) 2 -p* ........................ (ii) for a= -1, R = pP-(x + af ....................... (iii) I. (Ax + B)/R. (i) If the factors of R are real resolve the fraction into partial fractions as in 112, Ex. 2. (ii) If the factors of R are imaginary then R = (x + a) 2 + /S 2 and we can transform the fraction so that the substitutions of Ex. 3 and Ex. 1 of 114 can be used. Choose A and /u. Ax+B . Hence -^ = A Jti . and the first integral being a case of 114, Ex. 3, the second of 114, Ex. 1. II. (i) Let R be either (x + af + pP or (a; + a) 2 -)8 2 . Make the same transformation of Ax + B ; then f( \ (Ax+B)dx f (2x + '2a)dx f dx " 276 AN ELEMENTARY TREATISE ON THE CALCULUS. (ii) Let J? = /3 2 -O + a) 2 ; then -(2x+2a)dx dx _ when A=0, X = and the integrand is of the type 114, Ex. 1. In working numerical examples it is best to find first the derivative of R; it is then easy to write A x -\-B in the required form. Ex. 1. dx Integral = - Ex. 2. Integral- -7 /-^ The types 1 1 x f j(ax z + bx + c)' (m + n) ^/(ax* + bx + c)' can be reduced to the cases just discussed by the substitu- tions x = l/u, r mx + n = l/a respectively. These give by logarithmic differentiation dx du dx 1 da x u mx + n m u The substitution of I/it for x is effective in other cases ; f dx f udu 1 thus -= I- - = , J (a 2 + x^ J (a 2 u z + 1 ) f a 2 (a 2 w 2 + 1 )* which, expressed in terms of x, is x/a z (a 2 +x z )*. TRIGONOMETRIC SUBSTITUTIONS. 277 The more general form l/(ax z + bx + cy can be treated in a similar way after expressing the quadratic in the form given at the beginning of this article. 116. Trigonometric and Hyperbolic Substitutions. Another method of treating the quadratic function is to transform it by a trigonometric or hyperbolic substitution. The particular transformation is suggested by the form of the quadratic. a 2 a; 2 , ^/(a 2 # 2 ); x = asinO or x = acosO; x + a 2 , x /( 2 + a 2 ); # = atan$ or x = asinh6', ft 2 a 2 , \/( x<2 & 2 ) 5 x = a sec 6 or x = a cosh 6 ; ^{/^-O+a) 2 }; x + a = (3smO; etc. Ex. 1. If x = asin.d; dx=acosddO. [v/(a 2 -. 2 )rfa;=a 2 (cosW0=^Y0 + sm 0cos 0\ and therefore / v /(a' 2 -^ i2 )^ i =^ N /(a 2 -^ 2 ) + ^r sin" 1 -. J 2i cu Ex. 2. If x = asinhO ; dx = acQshdd6. W(&+ dx = 1+U 2 Examples 1-3 may almost be reckoned among the standard forms ; the substitution is for each u = tan| x. Ex. 1. / . -= / = loe: u = \os tanA#. J sin x J u -n, [ dx [ 2du l+u , 1 + tani.r Ex.2. / - = /- - = locr- -- = log,- i-. J cos x J \-u i 1-u 1 tan \x The integral can be put in several forms as x 7r\ , , 1+sin.r The substitution v = ^-x or vx -^ will reduce the integral of t 1/cosx to that of 1/sinx. F o f dx _f _ 2du _ _ T du ~ ~ Let a + b be positive ; then there are three cases according as b is numerically less than or greater than or equal to a. (i) 6 2 < a 2 and therefore 6 < a, numerically / dx 2 _,/ la-b\ i T 77 o ro\"tan \ u\l 7 /, u i J a + b cos x v ( a ~~ b*) \ \ a + b/ (ii) 6 2 > a 2 and therefore 6 a positive, f dx 1 i N/5 + a ^a + 6cosa; \/6 2 -a 2 \ f b+a-ujb-a (iii) 6 2 =a 2 , /7 1 f J I ^ _ _ an 1^ . / ^^ _ _ C Q^ 1^ a+acos.u a J a acosx a Case (ii) is of less importance than (i). A more easily remembered form of the integral (i) is obtained by writing TRIGONOMETRIC INTEGRANDS. 279 whence cos = (a cos x + b)/(a + b cos #), or (a - b cos 6)(a + b cos x) = a 2 - 6 2 . a- 1 is the true and $ the eccentric anomaly in an ellipse of eccentricity b/a (Godf ray's Astronomy, 186 ; Gray's Physics, 520.) a + bcosx goes through its complete range of values if x varies from to TT or again if x varies from TT through negative values to 0. If x lies between and TT, is positive and lies between and TT ; but if x lies between - TT and 0, 9 is negative and lies between - TT and 0. Hence bearing in mind the restriction on the inverse cosine ( 28, 64). a+bcosx but dx j. , / u vva-ru \ rv<,_< II V./ X 71. There is no ambiguity when the integral is expressed in terms of the inverse tangent. See also Examples 11, 12, p. 135. ci . f dx integral = f J The If 6 2 ; 3i < ; 37 , 40. i 38. 1 + tan ^ ' 41. ~ sin x+ 2 cos x EXERCISES XXIII. 281 42. Find the value of the integrals Jo 5 + 3 cos x ' Jo 5 + 3cosx ' J- ; 5 + 3sin.r ' 5 + 3cos# Jo 5 + 3cosx J-~5 + 3sin.r 7(rH=l); (v) /" (00, and find the area of the loop. 46. Trace the curve given by 6>0, and find the area enclosed by it. 48. By transferring to polar coordinates, find the area of the ellipse whose equation is fo= WG* 8 a 2 ) |a 2 log (x+ Compare 116, ex. 2. The algebraic transformation used above is often useful ; a similar transformation occurs in integrating circular functions ( 119, 2, 3). The quadratic ^(a^+bx+c) can be integrated by expressing it as in 115, and putting 284 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 5. Find llogxdx. \\ogx dx=x\ogx- I x-dx=x\ogx-x. J J 3C 119. Successive Reduction. Ex. 1 . Let u n = I x n e*'dx ; then, integrating by parts, u n = \x n e x dx = x n e?- I nx n ~ l e* dx = x n e* - n I x n - l e?dx, that is, u n = x n e? nu n -\. Writing n - 1 in place of n, we find M n _i =x n ~ 1 e* -(n- l)M n _ 2 , that is, u n = x n e x nx n ~ l e x + n(n 1 )M n _o. Proceeding in this way, we see that if n is a positive integer, u n may be made to depend on u n , that is, / (fdx or e*. If n is not a positive integer but is still positive, u n may be made to depend on an integral in which the integrand contains x with a positive proper fraction as index. The integral cannot in that case be expressed in finite terms by means of known functions, but it is reduced to the most convenient form for studying. The above method of making an integral depend on another of the same form is called that of Successive Reduction. The integrals of x n sin x, x n cos x may be treated in the same way. Ex. 2. u n \sa\ n xdx. u n = I sin"^7 dx = I sin"" 1 ^ . sin x dx = sin"- 1 #( - cos x) - I (n-l) sin n ~ 2 .i' ( - cos 2 .r) dx = sin"" 1 ^ cos x+(n l)l sin"~ 2 .r cos 2 .# . dx. Now cos 2 # = 1 - sin 2 .r ; sin n ~ 2 .r cos'x = sin"~ 2 .r sin".r. Hence u n = - sin"" 1 .? cos x + (n 1 ) n _ 2 - (n - 1 ) u n , , , , , sin"" 1 ^ cos x n 1 and therefore u n = - n n The index n has thus been reduced by 2. Writing n - 2 in place of n, we e t sin"~ 3 ^ cos x n - 3 tt - 2= ^2 +^2"- and therefore sin"* 1 ^ cos x n1 sin n ~ 3 j? cos x (n I)(n 3) Un= ~^T "H ^2~ (-2) Un ~* If 7i is a positive integer, we can repeat the reduction until the index is 1 if n be odd, or if n be even ; M Z = -COS.T and u =x, since SUCCESSIVE REDUCTION. 285 sinj?=l. If is positive but not integral, u n may be reduced to depend on an integral in which the index is either a positive or a negative proper fraction. For negative values of n see ex. 4. The most useful case of the formula (i) is that in which n is a positive integer and the integral is taken between the limits and Tr/2. In this case (i) becomes r * sin" *c cos ^?7~1 ft T f"" / s,\n n xdx = \ u n \ = \ -- + M n _ 2 Jo LJoL n JowLJ _7l 1 P . n _ 2 , / sin "x cLx* n Jo since the integrated term vanishes at both limits. When n is odd, the last term of u n is but when n is even, Hence / sin n zdx= / n\ '-" o 1 ( n odd integer) ; Jo n(7i-2)...5.3 f f . (n-l)(n -3) ...3. ITT , / sin n xdx=- , ^ H ( even integer). ./o (?i-2) ...4.2 2 x If v n I cos*xdz, then and it is easy to prove from the formula or, better, directly from the meaning of the definite integral that rri ""Jo S A simple inspection of the graphs of sin".r and cos"^ will show that /T /-f Jo Sm ^ Jo 81 / cos"#efo=2/ cos"^c?j7 (t even integer), ^0 ^0 but =0 (n odd integer). In a similar way such results as rz-* [2* p I sin 3 .rtr=0; / cos 6 j70?a-=4/ Jo Jo Jo are readily proved. See also the rule given in ex. 3. 286 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 3. /(m, n)= lsin m .r cos n x dx. In f(m, n) the first letter is the index of sin.r, the second that of cos x. For brevity denote sin x by s, cos x by c. Then /(m, ri)= I s m c n dx I s m c . c"" 1 dx. Since c is the derivative of s, the integral of s m c is s m+J /(m+l)- Thus, m + i 71-1 - (i) But The first term is the integrand of /(TO, 2), and the second that of f(m, n). Substitute in (i), transfer f(m, n) to the left side, and then multiply by (m + l)/(m + ri). Therefore S m ^^C n ~^ n _ 1 f(m,ri) = \-- -- f(m, n-2) ....................... (A) yv ' ' v ' The integral thus depends on another of the same form with m unchanged but the other index reduced by 2. Had we begun by writing s m ~ l . sc" and integrating the cosine, we should have got ,, , s m -V +1 , m-1 ,., . f(m.n)= -- --- /(m-2.). ................... (B) J ^ ' ' j ^ and now m is reduced by 2, n unchanged. We will continue the reduction for the case in which m, n are positive integers, so as to obtain the definite integral from to TT/%. If n is odd, (A) makes f(m, n) depend on f(m, i) ; (B) then makes /(?, 1) depend on /(I, 1) or on/(0, 1) according as m is odd or even. If n is even, (A) makes /(TO, n) depend on f(m, 0) ; but /(m, 0) is the integral of ex. 2, with m in place of n. Thus, by ex. 2 (i), /(*, 0) depends on /(I, 0) or on /(O, 0) according as m is odd or even. Thus, /(?, n) may be reduced to depend on one of the four /(I, 1)= / scdx=^sin?x ; /(O, 1)= \ cdx = sva. x ; /(I, 0)= Isdx=-co8x\ /(O, Q)=\ldx=x. When the integral is taken between and Tr/2 the values of these are 1/2, 1, 1, 7r/2 respectively. The student may now show that the following rule is correct : / Jo .. xa, THE INTEGRAL / ~sin m xcos n xdx. 287 where a = l except when m and n are both even integers, in which case a = 7r/2 ; each of the three series of factors is to be continued so long as the factors are positive. It will be noticed that the factors of each series decrease by 2. The rule includes the integral of ex. 2, putting m (or n) zero and omitting negative factors. 1 X 3 . 1 7T 7T P L 6.4.2x4.2 7.5.3.1 7T 357T The great importance of the results of ex. 2 and 3 arises from the fact that many integrals are, by a proper substitution, easily reduced to these forms. For example, if we put x=asinO, so that when #=0, 0=0, and when xa, 0=7r/2, we get sin 2 cos" 8dd=^ oZ If we put #=asin 2 0, then fa s 9 /*? / x*(a xy dx=%a- I sin 5 6 cos*0d6 = Jn Ja 35 Ex. 4. If 72. is negative the index of w n _ 2 is numerically greater than that of n. In ex. 2 (i) let T& = m, where m is positive ; then [dx _ c m + l f dx J s m ~ms m+1 m J 8 m+ ' 2 ' therefore / -J^ = - -. . C . _ +l -\ ^j -|. Now put m + 2 = n, where n is positive, and we get f dx Cesar n 2f dx J sin n x ~~ (n - 1 ) sin"" 1 ^ n - 1 J sin"~% In many cases the integration will be simplified by writing sin 2 .r+cos 2 .r_ 1 cos:r sin'.r sin x sm *x sin x But these integrals are of small importance for elementary work. The key to the transformations is that after one integration by parts the new and the old indices differ by 2 ; when an index is negative it is simpler to begin by integrating the integrand with the reduced index. 288 . AN ELEMENTARY TREATISE ON THE CALCULUS. Ex.5. w n = / tan n xdx, u n = I tan"~ 3 x (sec 2 x 1 ) dx = I tan"~ 2 x . sec 2 x dx u n ^i so that u n = tan"" 1 x - _<>. n-l Other examples of reduction formulae will be found in the Exercises, but in many cases a trigonometric substitution will reduce the in- tegral to one of the forms just discussed. EXERCISES XXIV. Integrate with respect to x examples 1-24. 1. xe~ x ; 2.->/j * & t/ ? 3. x sin x ; 4. x cos x ; 5. ^ sin x cos #; 6. .r 2 sin x ; 7. x n log x (n=f=-l) ; 8. - log x ; 9. -Bin*; 10 *** X X . u?6 ^ 12. sin" 1 .? ; /I _i_ \2 J 13. tan" 1 ^ ; tr . ^*/ Sill u/ . 15. artan-'*; 1 A /f** 4- 9 7 1 ^\ * \J, ^ I O T^ AT**/ */ ^ j 17. V(3 + 2^r+^); 18. Jax-x iy. /y/^ow7"T*^r^^ i on x 21. x+sinx //I 2\ ' I + cos x ' 22. e- 3 'cos4r; 23. cosh x cos # ; 24. si nh x sin 25. Find the value of the integrals (i) / cos s xdx; (ii) / siifixdx; Jo Jo (iii) / sin e .r cos 4 .r dx ; (iv) / sin 3 . Jo Jo f 4 " T^ (v) / sin 8 o7 cos*x dx ; (vi) / Jo Jo 26. Find by a trigonometric substitution the value of (i) [*{C*i/(a*-st*)dx; (ii) [ ^x^(2ax - x 2 ) dx ; Ju Jo (iii) I x**J(2ax-a?)dx. Jo f a v*J(a z r 2 ) 27. Integrate f ' T, 2 \-/dx by the substitution .r 2 =q 2 cos20. EXERCISES XXIV. 289 28. Iff(m, n) = \x m (\ x~) n dx show that J f(m, 70=^ Hence, or by the substitution #=sin 2 0, find the value of r\ / .r m (l -z)"dx, Jo m, n being positive integers. 29. If u,,= ldxj(d*+x z ) n , prove that _ # _ _ 2T&-3 Un ~ (2 - 2) a*(a* + tf 2 )"- 1 + (27i - 2) a 2 *"- 1 ' 30. If ?( = / .^"^/(a 2 .r 2 ) rf.r, prove that ar-W-a n-l , 31. If u n = I.r"j(2ax-x 2 )dx, show that Write = .r-Ma - (a - #) where R = 2ax - x z , and then integrate by parts. 32. If u n = ^dxIJClax-x*), show that _ 71 7i 33. If TH, n are positive integers find the value of r\ i I (\-x) m dx. Jo 34. Find the value of (i) r*U(<* -*)<*; (ii) r^^ax- Jo Jo 35. OJ/" is the abscissa and MP the ordinate at the point P (, i?) on the hyperbola # 2 /a 2 -y 2 /& 2 = l, ^, 77 being both positive. If A is the vertex nearest P show that the area AMP is equal to |$,-JdHog(| and that the area of the sector OA P is 36. Trace the curve given by y 2 =(.r- l)(;r-3) 2 and find the area of the loop. 290 AN ELEMENTARY TREATISE ON THE CALCULUS. 37. Trace the curve given by a 2 y 2 =x 3 ('2a-x), a being positive; find the whole area enclosed by it. 38. Find the length of an arc of the catenary y=\a(e<^ + e' <*) measured from the point C where x=0. Show that the area between the two axes, the curve and the ordinate at a point P is a times the arc CP. 39. Find the length of an arc of the cardioid r = a(l cos 6), the arc being measured from the origin. 40. Find the length of an arc of the spiral r=aO, taking s = 0, when f-0. 41. Find the length of an arc of the spiral r = ae flcota , taking s = 0, when = 0. 120. Partial Fractions. The method of resolving a rational fraction into partial fractions is now found in most text-books of Algebra. We will therefore refer the student to Chrystal's Algebra, Vol. I, Chap, viii., for a full discussion of the theory, and will merely work out a few examples. The fraction will be supposed to be a proper fraction, that is to have the degree of its numerator in the variable x less than that of its denominator, and to be at its lowest terms. Let the fraction be F(x)/f(x) where F(x) and f(x) are rational integral functions of x. f(x) can be resolved into a product of real prime factors, each of which is a linear or else a quadratic function of x, but a factor, linear or quadratic, may be repeated several times. f\x)lf(x) can be resolved in one and in only one way into a sum of proper partial fractions ; these partial fractions are of the following types : (i) To every non-repeated linear factor x- a. of f(x) corresponds a partial fraction of the form A/(x-a). (ii) To every r-fold linear factor (x f$) r of f(x) correspond r partial fractions of the form (iii) To every non-repeated quadratic factor x 2 + yx+8 of f(x) corresponds a partial fraction of the form (Cx + D)/(x 2 + y.v + 5). (iv) To every r-fold quadratic factor (x 2 +yx+8) r of f(x) correspond r partial fractions of the form CfX+D r C r ^iX+D r -i C v v+D l The method of determining the coefficients A, B, ... will be learned from the examples. PARTIAL FRACTIONS. 291 Ex. 1. No factor of the denominator is repeated ; therefore x? ABC (# -!)(#- 2) x+1 x-1 x-2 Clear of fractions ; therefore This equation being an identity, we may give to x any value we please. Put x+ 1 =0, that is, x 1, and the terms in B and C vanish, and we get ,_., , 1V . ^ A -MR L A{ i i^ i &) or Ji i/o. Similarly, by putting x = \ we get B= 1/2, and by putting .r=2 we get (7=4/3 and x 2 _1 _1 1 1 4 1 (^-l)(j?-2)~6 ' ^+1 2' 07-1 3 '#-2* Or, to find J, multiply both sides by its denominator .r+1 and then put + !=() ; r r 2 J- In the same way, if .r a is a non- repeated factor of f(x) and Af(x - a) the corresponding partial fraction A = f'( r \ Aj[ T \ i / ' r (i\(t,'( r\ ftrtf\ so that A = Ex. 2. (-r J +;r+2)/(:r-l) 2 (.r 2 -.r+l). The repeated factor (xl) z gives two fractions, and the factor a?-x+l, since it has no real linear factors, gives a fraction of the type (iii) ; hence _A . B . Cx+D (a. Clearing of fractions, we get Putting #=1, we get A =4. Now bring the term in A to the left side and reduce after putting 4 for A. The right side will contain (x- 1) as a factor, and therefore, since the equation is an identity, the left side must also contain (x 1) as a factor. If it does not, there is* an error in the work. We get 292 AN ELEMENTARY TREATISE ON THE CALCULUS. Divide by (#- 1), and then by putting x=l we find B= - 1. Now take the term in B to the left, and again divide by (x- 1). Then so that, since the equation is an identity, (7=1, D= -3, and therefore 4 1 x-3 Ex. 3. (^ By (iv) and (iii), since there are no real linear factors of the denominator, ^-2 Ax+B Cx+D Ex+F Clearing of fractions, Put .r 2 +.r+2=0 and reduce x 2 and # 3 to linear functions by means of this equation. It gives -1 ; #2= -#-2, x* = -.r 2 -2.r= - and therefore x = Ax B, so that A=l, 5=0. Take the term in A and B to the left and divide by # 2 +# + 2 which must be a factor. Hence Put x 2 +x+2=Q and proceed as before. We get (7=0, D=\. Hence, after dividing by # 2 +#+2, -l=Ex+F; E=0,F=-I, and the fraction is equal to x 1 1 These examples show sufficiently the method of deter- mining the coefficients ; other methods will suggest them- selves to the student, and he will find full details in the chapter of Chrystal's Algebra referred to above. 121. Integration of Rational Functions. If F(x)/f(x) is not a proper fraction it may by division be expressed as the sum of a rational integral function and of a rational proper fraction. The integral of a rational integral function is a rational integral function. RATIONAL FUNCTIONS. 293 The integral of A/(x a) is A \og(x a). The integral of B/(x /3) r where r is different from unity is -/(?- l)(-/3) r - 1 . The integral of (Cx + D)/(x 2 + yx + S) has been discussed in 115 and is of the form We have, therefore, only to consider (Cx-\-D)l(x z -{-yx-{-8) r . Writing the quadratic in the form R = (x + af + (3' 2 the integral is In practice it is usually simplest to integrate \jR r by the substitution x + a = /8 tan ', but it is of some theoretical interest to get a formula of reduction. If we differentiate a)/R r - 1 we find d (x + a\_ 1 __ 2(r - dx^fr-i)"!?- 1 sr -(2r-3) 2(r -i ' by putting (a; + a) 2 = R (P. Integrating and rearranging (dx_ x+a 2r 3 f dx ~'- 1 + - 2r1 ' Hence the integral of (Cx + D)/ R r can be made to depend on that of l/R, which is an inverse trigonometric function. Thus the integral of any Rational Function of x can be expressed in terms of rational functions, logarithms and inverse circular functions. There is always a considerable amount of labour in inte- grating by the method of partial fractions. The student should, before resolving into partial fractions, examine whether the integral may be simplified by a substitution. rp,, f x^dx , f udu 2 Thus, ^ 2-rT = *l-2 -- u=x Jx* x z + l ~Ju z and the fraction in u is easier to handle than that in x. 294 AN ELEMENTARY TREATISE ON THE CALCULUS. 122. Irrational Functions. We consider one or two cases in which the integrand is an irrational function. (i) When the integrand contains only fractional powers of x let n be the L.C.D. of the fractions ; then the substitu- tion x = u n will make the new integrand rational in u. Thus, if x=u 6 r dx n [ u e du = 6 (}u? - w 5 + |w 3 - u + tan" 1 n) (ii) When the integrand contains *J(ax + 6) but no other irrationality the substitution ax-\-b = u 2 will make the new integrand rational in u. (iii) When the integrand contains ^(ax^ + bx + c) but no other irrationality the integral may be reduced to that of a rational function as follows : First, let a be positive arid write the root in the form y = *Ja tj^+px+q), p = b/a, q = c/a. Let > /(* 2 + ^>x + g) = u x so that, squaring and solving f or a _-, ~ du The new integrand will clearly be rational in u. Second, let a be negative. In order that y may be real the linear factors of ax 2 + bx + c must be real; if they were not real the quadratic would be negative for every real value of x and therefore y would be imaginary. We may therefore write, since ( a) is positive, y = * - For definiteness suppose (3>a (algebraically) and let -W{(-)/0-$ Then, u* = (x-a)/((3-x); (/3-cc)u 2 /3-a a /y* . __ V/ /< ^_ /v - ' _ /y* - ' P ~ du~ (1+u 2 ) 2 The new integrand will clearly be rational in u. IRRATIONAL FUNCTIONS. 295 In (ii), (iii) we may suppose all the roots to be positive. (See 123, end.) The above analysis shows that if y be either tJ(ax-\- 6) or ^/(ax 2 + bx + c), and if the integrand be a rational func- tion f(x, y) of x and of y, the integration of f(x, y) can always be reduced to that of a rational function, and therefore ( 121) requires for its integration only rational functions, logarithms, or inverse circular functions. (iv) Let the integrand be x m (a + bx n )P. (a) If p is a positive integer expand (a + bx n )P. (6) Try the substitution u = a + bx n which gives 1 7 * dx (u a)" 1 x = -,(u-a) n ; j- = - - r > fc du nb k and the integral becomes If, ,"+! -m+i\uP(u a) n du, no n J so that if (m+l)/7i is a positive integer the binomial may be expanded and the integral obtained in finite terms. (c) If (m + l)/n is not a positive integer let x=l/v and the integral becomes _ - m - np - 2 Instead of m we have now (m + np + 2) and therefore by (6) if (ra + np + 1 )/n be a positive integer, that is, if (m + l)/w+p be a negative integer the integral may be got in finite terms. The substitution is u = b + av n = b + ax~ n . 123. General Remarks. From the discussion now given it will be seen that integration is a somewhat haphazard process. The only general results obtained are those of 121, 122 ; in most cases the integration, when it is possible at all, has to be effected by reducing the given integrand by various methods to a few standard forms. Even for the cases discussed in 122 it is frequently simpler to take a special method for a given case than to apply the general theorem. 296 AN ELEMENTARY TREATISE ON THE CALCULUS. Much of the difficulty beginners find in integration is due to a deficiency in power of algebraic and trigonometric manipulations. When the standard forms have been com- mitted to memory the next step is to master the two principles of change of variable and of integration by parts ; but the student who has not a thorough mastery of elementary algebraic and trigonometric transformations will often fail to see the reasons that suggest the particular devices adopted and will have to struggle with difficulties that are due, not to the nature of the calculus but to his own deficient algebraic training. Integral dependent on the range of the variable. Another source of difficulty requires special notice, namely that the integral may have one form for one range of the variable and a different form for another range. Thus the integral of 1/x is logcc or log( x) according as x is positive or negative ; in this case the integral may be written Jlog (x 2 ), a form which covers both cases. See 117, Ex. 3, for another case. Again, difficulty may arise from the ambiguity of the square root ; in that ambiguity the explanation of the two forms for the integral of l/(a + bcosx) is to be found when the inverse cosine is derived from the inverse tangent. Thus, if it be agreed that the root is always to be taken with the positive sign, the transformation P^Q = t^/(P 2 Q) would only be correct if P were positive; if P were negative we should have P^/Q = EXERCISES XXV. Integrate with respect to x examples 1-24. 30J7 5 J ; 2 O. /TI _\/ 7 \ / _ ~\ 9 4. 7. EXERCISES XXV. 1 - 297 16 + 9 sin 2 .r' 20. 23. 25. Transform the integral [ dv ] (x-a) m (x-b) n by the substitution u = (x-a)j(x b) ; find its value when i = 3, n = 2. Integrate with 'respect to x examples 26-37. OR ^ X 97 9R 1 4- r ' lx+ S /Jx /(x\) 29 - 32. 33. CHAPTER XIV. DEFINITE INTEGRALS. GEOMETRICAL APPLICATIONS. 124. Definite Integrals. In this and the two following articles we will state a few of the more important theorems respecting definite integrals. . THEOREM I. A definite integral is a function of its limits, not of the variable of integration. This theorem is obvious from the geometrical meaning of the integral ; so long as the symbol F denotes the same function the graph of F(x) with x for abscissa is the same as that of F(u) with u for abscissa, and therefore \ F(x)dx={ F(u)du. Ja Ja Or, again, if F(x) = D x f(x\ then F(u) = D u f(u) and each symbol represents /(&) /() THEOREM II. JV() dx= - (V() dx. See 1 10. THEOREM III. // a) dx = \F(x) dx + fV(a?) dx + \F(x) dx, Ja Ja Jc Jg and so on for any number of subdivisions of the interval (a, b). Of course one or more of the numbers c, g, . . . , may be greater than the greater or less than the smaller of the two numbers a, b, provided F(x) is continuous for all the values considered. THEOREM V. If aL(b a). Ja For G F(x) and F(x) L are positive ; hence by Th. III. the integrals [\G - F(x)] dx and f \F(x) - L] dx, Ja Ja rb n rb rb that is I G dx \ F(x) dx and I F(x) dx \ L dx, Jo Ja Ja Jo or G(b-a)- \ b F(x) dx and fV(s) dx-L(b- a), Ja Ja are both positive, so that the integral is less than G(b a) but greater than L(b a). The integral will be equal to H(b a) where H is a number less than G but greater than L ; but since F(x) is 300 AN ELEMENTARY TREATISE ON THE CALCULUS. continuous it must, for at least one value x l of x between a and b, be equal to H. The value x 1 is of the form a + (b - a) where < 6 < 1 ( 73). Hence, The theorem is evident from the figure; for the area ABDEG is less than the rectangle G . AB, greater than the rectangle L.AB, equal to the rectangle H . AB or MP . AB where MP is an ordinate less than G but greater than L. The value H or F(x l ) is sometimes called the Mean Value or the Average Value of the function F(x) over the range (6 -a). (See 134.) FIG. 55. THEOREM VI. If a< b and if for every value of x in the interval (a, b), F(x) is (algebraically) less than (x) but (algebraically) greater than \[s(x), then I F(x)dx< I d>(x)dx but > ! \ Ja Ja Ja Proved in the same way as Th. V. since (j>(x)F(x) and F(X) \IS(X) are positive. For geometrical proof see the figure (Fig. 55). THEOREM VII. If a(x), \fs(x), one of which, (p(x), is positive for every value of x in the interval (a, b), then Cb Cb Cl> I L\ Ja Ja Ja INTEGRAL THEOREM OF MEAN VALUE. 301 where G, L are the (algebraically) greatest and least values of ^(x) in tJie interval (a, b). Proved in the same way as Th. V., since G \js(x), \fs(x) L, and therefore (G \ls(x))(x) and (\fs(x) L)(f>(x) are positive. If (x) is negative for every value of x in the interval (a, b) we shall have f 6 f 6 f 6 1 G\ (j>(x)dx but (x)dx. Ja J a Jo. In both cases, the function "^(x) being continuous, we may as in Th. V. write p f 6 I (f)(x) \l/(x) dx = \js(x l )\ (f)(x)dx ............... (A) Ja Ja where a < x l < b. The theorem expressed in equation (A) is called Tlie First (Integral) Theorem of Mean Value. (See Exercises XXVI., 29-31.) Ex. Show that if n > 2, the integral dx / Jo is greater than '5 but less than '524. For every value of x within the range of integration, the value excepted, so that the integral is less than but greater than f"w -5 Jo 125. Related Integrals. Ca fa THEOREM I. I F(x)dx= I F(a x)dx. Jo Jo Let x = a u ; then dx = du, and when x = 0, u = a, when x = a, u = Q, so that f *F(x) dx=-[F(a-u)du= \F(a - u) du, Jo J Jo 302 AN ELEMENTARY TREATISE ON THE CALCULUS. and in the integral last written we may put x for u ( 124 Th. 1). A useful case is T It IT ^2 |"2 f 2" I /(sin x) dx = \ /(sin[- x])dx = \ f(cosx)dx. Jo Jo Jo THEOREM II. ' F(x) dx = { "{ F( - x) + F(x) } dx. J -o. Jo For * F(x)dx=\ F(x)dx+^F(x)dx. J -a J -a Jo In the first integral let x = u and it becomes - (V( - u) du = f V( -u)du= f V( - x) dx, Ja Jo Jo from which the result follows. Hence 1 F(x) dx = 2 \F(x) dx, if F(-x}-= F(x\ J -a Jo = 0,if F(-x)=-F(x). Y X'- X' x Y' FIG. 566. FIG. 56a. The last results are evident geometrically from the figures. fa f* THEOREM III. F(x) dx=\ { F(x) +F(a-x)} dx, Jo Jo so that \F(x}dx=^{" F(x) dx, if F(a - x) = F(x\ Jo Jo = 0, if F(a-x)=-F(x). The proof is the same as for Th. II. ; divide the interval into (0, Ja) and (|a, a), and in the second integral put x = a u. RELATED INTEGRALS. 303 As a particular case /(sin x) dx 2 1 /(sin x) dx. Jo Jo THEOREM IV. // F(x) is periodic, with the period a, that is, if F(x-fna) is equal to F(x) for every integral value of n P<1 F(x) dx =p I V() dx, Jo Jo where p is any positive integer. A B FIG. 57. C X If 0-4 =a = J.5 = 5(7= ... then from the nature of the graph the areas OAKH, ABLK, BGML, ... are all equal, so that if OC= p . OA the area OCMH is p times OAKH. Or divide the range pa into p parts each equal to a, then Cpa fa f(*+l)a F(x)dx= \ F(x)dx+... + \ F(x)dx Jo Jo Jka F(x) dx. Cpa + . . . + J(P In the integral having ka, (k+l)a for limits let x = then dx = du, and when x = ka, u = 0, when x = (k+l)a u = a, so that f(*+l)a fa fo fa F(x)dx=\ F(u + ka)du=\ F(u)du=\ F(x)dx, Jia Jo Jo Jo since F(u + ka) = F(u). Thus each of the p integrals has the same value and the result follows. Similar reasoning shows that the theorem is also true when p is a negative integer. 304 AN ELEMENTARY TREATISE ON THE CALCULUS. As a particular case rf2ir /(sin x) dx = p \ /(sin x) dx. Jo These theorems are of great service in the evaluation of integrals. 126. Infinite Limits. Infinite Integrand. Up to this point the limits of the integral have been assumed to be finite, and the integrand has been supposed continuous and therefore finite for every value of the variable within the range of integration. It is, however, possible in certain cases to remove these restrictions by the use of limits. A. Infinite Limits. An integral with one of its limits infinite is defined as follows : f J F(x)dx = L \F(af}dai', F(x)dx = L F(x)dx, provided the limits for b = oo and for a = oo are definite quantities. Fdx T f b dx , /. n Ex. 1. / -o= L / -s= L I 1-7 1 = 1. h X* 6=00^ or j =a) \ bJ rdx T r"dx Ex. 2. = L / = L log 6. Jl X J =00 /l X 4=00 In this case the limit of log b is not a definite number, and the integral is therefore a meaningless symbol. /"oo Ex. 3. / e~ x cosxdx. /" . 3. / Jo By 118, ex. 3, the indefinite integral is ^e~ x ( cosx+sinx\ and we have to find the limit for 6 = 00 of + %e~\ cos 6 + sin 6). Now cos 6, sin& are each never greater than 1, and the limit of e~ b is zero so that the integral is equal to ^. The limit for #=oo of x n e~ ax , where a is positive, is often needed in dealing with these integrals. It is easy, by 49, to see that L x n e~ ax =0. 1=00 See also Exercises VII. ex. 9. B. Infinite Integrand. If F(x) is continuous for all values of x between a and b except for x = a when it is infinite, then the integral of F(x) between a and b is defined thus, a being less than b and e being positive, 305 INFINITE LIMITS. INFINITE INTEGRAND. [ b F(x)dx = L P F(x)dx, Ja e=oJa+e provided the limit is a definite quantity. If F(x) is continuous except at b then, e being positive, \F(x)dx= L P *F(x)dx. Ja e=(V . 1. / j^= L I -,-= L(2-2 v /e) = 2. Ex Ex. 2. = L - the limit is obviously sin~ : 1 or 7r/2. Ex.3. In this case there is no definite limit and the integral therefore does not exist. If a < c < b and if F(x) is continuous except when #=c, then the integral between a and b is defined thus, e, e' being positive, l b Ja F(x)dx= L ~F(z)dx+ L f = oJa e' = (> provided each limit is separately a definite quantity. Ex. 4. P^- 2 = L(-3V + 3)+ L (3-3 VO- J-l*JJr t =Q '=0 Here the first limit is 3, the second is also 3, and the integral is 6. Ex.5. /* ! _L(!-iUL (-1+1). Jo(jr-l) 2 ,=0\ / e' = 0\ . '/ In this case there is no definite limit and the integral does not exist. A change of variable will often remove the difficulty of an infinite integrand or an infinite limit ; thus, in ex. 2, we might put x = a sin 0. The change of variable is specially useful for the forms given in 116. These exceptional cases of integrals may be illustrated by consideration of the graph of F(x). Let F(x)= l/x n where n is positive ; then the #-axis is an asymptote and the area ABDG is \ b dx = 1 /I 1 \ )ax n ~n-l\a n - 1 b n ~ 1 )' 306 AN ELEMENTARY TREATISE ON THE CALCULUS. Hence, if n>l, the area ABDC tends to the value l/(n l)^"- 1 as b tends to oo ; while if 0<7il the area tends to oo since e 1 ~ n , that is, l/e n " l tends to infinity as e tends to zero. If n = l the area is log{(6 a)/e} and therefore tends to oo as e tends to zero. It is easy to show by the use of Th. VII., 124, that if near a, F(x) FlG 59 is of the form (x)/(x a) n , where (j)(x) is continuous, the area EBDF and the corresponding integral tend to a finite limit if n is a positive proper fraction, but that when 0) ; 2. / e~ ax sin bx dx (a > 0) ; Jo Jo /fj rv* [ ^ ; 4 / oo^ 2 + 2A' + 2 " Jo dx xdx EXERCISES XXVI. 307 8. *-<.)*-,&; 9. 10 f* dx 11 ' Jo a 2 ' " in Pcos-x sin .r cfcr " Jo ; '' 14. / tonxdx; 15. / log-refo;; 16. / a?\ogxdx. J-* Jo Jo 17. Prove that if m and n are positive, / '^"(l -x) n dx= I V(l -;r) m rf.r. Jo .'o 18. Prove that if n is positive, /OO /-50 / e~ z x n dx=nl e~ x x n ~ l dx. Jo Jo Find the value of the integral if n is a positive integer. ,_ T . [*x sin x dx 19. If = / T Jo l+ ,, . prove that u - - H , l+cos 2 a; and then find the value of u. 20. If w= f , X X . where 0 I sin n+1 ^-o^r. Jo Jo Hence, show that 7r/2 lies between 2.2.4.4.6.6.. 2.2n and the fraction obtained by omitting the last factor in numerator and denominator. (This is often quoted as Wallis's value of ir.) 308 AN ELEMENTARY TREATISE ON THE CALCULUS. 23. Prove that, n being .a positive integer, / ^^sin oc I -- dv=u -u 1 + u 2 JQ X [*si where u k = I - Jo U + /br ' Show that , MJ, w 2 ...are positive, and that if k is equal to or greater than 1, u k is less than \jk. Interpret these results by con- sidering the graph of sinx/x, and show that the integral has a finite limit for n = oo . The limit is Tr/2 but the proof can not be given here. 24. Prove /* //. ^ , ^>l but <. Jo v/(4-^ 2 +.^ 3 ) 2 6 f 1 dx f 1 dx , [ l & 25. Prove / r,- 5 -. | - Jo ^(l-Sx+x?) Jo ^/(4-a^) Jo n-. ^ N -~r^ ^/(4-a^) Jo 16 that is, 19/32. f 2 fjf 26. Prove __^__>-573, but < -595. /i /4-3^+^ 3 Put ^=1+^6 ; then replace 3 + 3w 2 + 2 by 4 2 +2 and by 3w 2 - 27. If a and ^> are positive acute angles, prove dx f" Jb but < Jo ^/(l - sin 2 a sin 2 ^) ^/(l - sin 2 a sin 2 =7r/6, show that the integral lies between '523 and '541. More accurate methods give '52943 as an approximate value of the integral. 28. Prove (i) { er+dxxf xe-*dx ; (ii) j[ e~^dx(x)-(lf(tf)dx = (f>(a) I i/r(.r)c?.r, where a<(x) is a positive increasing function, Cb f (ii) / ^(x)-^f(x}dx=<^(b) I dx where 31. If <(?') increases (algebraically) as x increases from a to b, show that in ex. 30 (i) we may put () <(??) ' n place of <^>(^), while if STANDARD AREAS AND VOLUMES. 309 4>(x) decreases (algebraically) we may in ex. 30 (ii) put (a) - <(#) in place of (x). Show that, when these substitutions are made, both (i) and (ii) become In this case <(.?) may be either positive or negative. The theorem expressed by the equation is called The Second (Integral) Theorem of Mean Value ; it is true even if VK*') take both positive and negative values, though the illustration would require more careful elaboration to show this. Illustrate by an area when ^r(x) = \. 127. Some Standard Areas and Volumes. In this article we collect some of the more important results already obtained or easily proved. 1. The right Circular Cylinder. Let the radius of the base be a and the height h. volume = Tra 2 h ; curved surface = 2?raA. 2. The right Circular Cone. Let the radius of the base be a, the height h, and the slant side l = J(a' z + h 2 ). volume = \ira 2 h ; curved surface = -rral. For a frustum of height h, slant side I, and with radii of ends a, 6, volume = 3?r(a 2 + ab + b 2 )h ; curved surface = ir(a + b)l. Let A be the base, h the height, and X the section parallel to the base at distance x from the vertex of any cone ; then since parallel sections are similar figures. Let V be the volume of the portion having X for base and height x ; then to the first order of infinitesimals o V= X&x, and D x V is equal to X. Hence the volume of the whole cone is A /" Jo For a frustum of height h, the areas of its ends being A and B, the 3. The Sphere. Let the radius be R ; then, by 85, ex. 2, the volume of a spherical cap of height h is and the curved surface of the cap is ZirRh. By putting h=2R we get for the volume and the surface of the sphere $irl$ and AirR 2 respectively. It will be noticed that the surface of the cap is equal to the curved surface of a cylinder of the same height whose base is equal to a great circle of the sphere. 310 AN ELEMENTARY TREATISE ON THE CALCULUS. To find the volume of a spherical sector, add to the volume of the cap that of the cone whose vertex is at the centre of the sphere and whose height is R h. The result is where S is the surface of the cap. The result is more easily obtained by supposing the surface of the cap divided into a large number of small areas ; the sector may then be considered as made up of a large number of cones having the same height R, and the volume of the sector will therefore be 4. The Ellipse. The area of an ellipse whose axes are 2a, 26 is fa 4 fa 4 / ydx / J(a? - x 2 )dx = irab. J y a Jo * The volume of the spheroid generated by the revolution of the ellipse about its major axis 2a is This spheroid is called " prolate." When the axis of revolution is the minor axis 26, the spheroid is called " oblate." The volume of the oblate spheroid is 2jf *me% = 27r^/V -f)dy = f 7r 2 6. The surface of the prolate spheroid is ds where (ds\* \dj + \dx Let be the eccentricity of the ellipse ; then a 2 e 2 =a 2 -6 2 , and the integral may be written, since b = a*/(l e 2 ), and the value is easily found to be The limit of this expression for e=0 is 4?ra 2 , which gives the surface of the sphere of radius a. For the oblate spheroid the student will readily prove that the surface is 1-e 2 CURVE TRACING. 311 Since we find e = = 2; ( 48, COR.) so that the limit for e=0 of this area is also 4ira?. 5. The Ellipsoid x 2 /a z +y 2 The traces of this surface on the coordinate planes are ellipses ; the section MPQ by a plane parallel to the plane YOZ is an ellipse. If OM=x then and the area JT of the quarter- ellipse MPQ is If F is the volume bounded by the coordinate planes, the surface BCQP and the section MPQ, then to the first order of infinitesimals 8F= JT& and D X V=X. Hence the volume of the octant OABC is y IG< / TT&C /* I Xdx=-r-z I (a- x z )dx Jo 4a?J irabc so that the volume of the ellipsoid is 4;ra6c/3. The method of finding the volume illustrated in examples 2 and 5 is obviously applicable whenever the area of a section perpendicular to the .r-axis is a known function F(x) of x ; the volume is simply the integral of F(x) between proper limits. (See ex. 3, 85.) The modification needed when the axes are not rectangular is plain. Curve Tracing. Before proceeding to the next set of Exercises the student should read over carefully the hints given in the earlier chapters for tracing curves; these, with the additional help furnished by the first and second derivatives, should enable him to graph the more elementary curves. In general he should proceed in some such way as the following : (i) Examine the equation for symmetry, (ii) Find where the curve crosses the axes. (iii) Find the finite values of x (or of y) that make y (or x) infinite; these values usually show the asymptotes that are parallel to the axes. Asymptotes inclined to the 312 AN ELEMENTARY TREATISE ON THE CALCULUS. axes may in the simpler cases be found as in 24 or by the method of 106; but such cases lie outside elementary work. (iv) Find the values of the one coordinate that make those of the other coordinate imaginary. (v) Find the gradient (see 54) ; note the turning points. (vi) Find the second derivative; it determines the con- vexity or concavity of the arc and the points of inflexion. It is often laborious, however, to find the second derivative, and general considerations will frequently show the course of the curve without its use. For polar coordinates the procedure is similar. It is often convenient, however, to suppose that the radius vector may take negative values; thus the point ( 1, 1) in the third quadrant may be given in polar coordinates as (^2, 57T/4) or as ( ^2, 7r/4). In the second form (-J2, 7T/4), if LXOP is Tr/4 and OP equal to ^/2, produce PO beyond to P so that OP' = PO and P is the point (- */2, x/4). See Exer. XXVII, ex. 23. The general course of the curve should always be found before attempting to find an area, or arc, etc. In evaluating the integrals substitutions will usually be necessary, and the student will find that sometimes a considerable amount of labour will be saved by choosing a good substitution. Even though the curve is given in rectangular coordinates a change to polars will sometimes simplify the integrations. EXERCISES XXVII. 1. The parabola ?/ 2 =4a^ revolves about the #-axis ; find the volume and the surface of the segment cut off from the solid by a plane perpendicular to the #-axis through the point where x=h. 2. Find the volume cut off from the paraboloid y 2 /6+2 2 /c=2.r, by a plane perpendicular to the .r-axis through the point where x = h. 3. Find the area enclosed by the curve (Fig. 61) Symmetry about both axes; X ^^=-L y lllclX* OI V ^ O I & Find also the volume of the solid generated by the revolution of the curve about the .r-axis. EXERCISES XXVII. 313 4. Find the area enclosed by the curve c 2 y*=x 2 (x-a)(b-x\ where 6>a>0. If x is less than a or greater than 6, y is imaginary except when x=0 and then y = 0. The curve is therefore a closed curve symmetrical about the .r-axis ; the origin is called an -isolated point because its coordinates satisfy the equation, while there is no other point nearer the origin than (a, 0) which lies on the curve. 5. Find the area of the curve Change to polar coordinates. The origin is an isolated point. 6. Trace the curve by 2 =x(x a)(2a x) where a and b are positive.. y is imaginary (i) if #>2a ; (ii) if The curve consists of an infinite branch and an oval as in Fig. 62. 7. Find the area of the loop of the curve lGa?y 2 =b 2 x 2 (a 2.r) where a, b are positive. 8. Trace the curve ky 2 = (x a)(x b)(x c) where c>6>a>0, Consider the forms for which (i) a = b ; (ii) b = c ; (iii) a=b = c. The general form consists of an oval and an infinite branch like Ex. 6, only the oval lies to the left of the infinite branch. When a = b the oval shrinks up to an isolated point at (a, 0) ; when a=b = c the curve is the semi-cubical parabola, the point (a, 0) being a cusp. The area of the oval in the general case, a, 6, c unequal, cannot be expressed in terms of the elementary integrals. 9. Trace the curve y z (a x)=x 2 (a+x); find (i) the area of the loop, (ii) the area between the curve and the asymptote (Fig. 63). Here the gradient is zero when x is (lj5)a/2, but the value (1 + v /5) a/2 makes y imaginary. 10. The "cissoid" is the curve given by the equation y z (2a-x)=a?\ find the whole area between the curve and its asymptote (Fig. 64). Find also the volume of the solid generated by the revolution of the cissoid about its asymptote. If PM is perpendicular to the asymptote the volume is 2 r irPM 2 dy = 27rT (2a - xf dy. To integrate let x = 2a sin 2 #, then X y = 2a sin 3 #/cos 0, and the limits for 6 are and Tr/2. 11. Find the area between the curve and its asymptote ; also the volume of the solid generated by the revolution of the curve about FIG. 64. its asymptote. 12. Find the area of a loop of the curve =x\a l 13. The figure bounded by a quadrant of a circle of radius a, and the tangents at its ends revolves about one of these tangents ; find the volume of the solid. 14. An arc of a circle of radius a revolves about its chord ; if the length of the arc is 2aa show that the volume of the solid is 27ra 3 (sin a - J sin 3 a - a cos a), and that the surface of the solid is 47r 2 (sin a a cos a). 15. If s is an arc of the curve a n ~ l y=z n show that Show that the arc can be expressed by means of the elementary functions when n is of either of the forms (2 + l)/2& or 2/(2&-l) where k is any integer, positive or negative. x and. the .r-axis. 16. Find the area between the graph of 17. Find the whole area enclosed by the curve Put x=asm 3 0, then y = bcos 3 0, and the area is 4 ^ y dx = 12 ab Psin 2 cos 4 <9 dO = %ira1>. Jo Jo EXERCISES XXVII. 315 18. The cycloid is the curve given by the equations ( 146) x=a(B sin 6) ; y = a(Icosd). Find (i) the area between the .r-axis and one arch of the curve ; (ii) the length of the arch from 8=0 to 6 = a ; (iii) the volume of the solid generated by the revolution of the arch about the .r-axis ; (iv) the volume of the solid generated by the revolution of the arch about the tangent at the highest point (or vertex) of the arch, namely, where O = TT. r r rJ = a l and z is simply equal to (7, so that C=ls (6) where s is the total normal dis- placement of P. For, clearly, the integral (5) gives the area ABDGH diminished by the area ABKGH. In this case s is independent of a, that is of the position of P on A B. FIG. 70. FIG. 71. FIG. 72. (ii) Suppose that while B makes a complete circuit of C A travels round a closed curve C'. If C' is outside C THEORY OF THE PLANIMETER. 321 (Fig. 71) % and a 2 will be equal ; (5) will be Is but the area swept out by AB will be CC', so that C-Cr = l8 (7) If, however, C completely encloses (7 (Fig. 72) then a 2 a x will be 2-7T and we shall have C-C' = ls+^(^-al) (8) The signs of the numbers C, C' are supposed to be deter- mined by the convention of 128 (7). 130. Planimeters. The investigations in the last two articles contain the theory of several instruments that have been devised for mechanically evaluating the area of a closed curve ; the best known of these is Amsler's Polar-Planimeter. Essentially the polar-planimeter consists of two bars OA, AB freely jointed at A, the bar OA rotating about a fixed point 0. If B is made to describe a closed curve, A will move along the circumference of a circle. When A merely oscillates along the circumference, not making a complete revolution, the area enclosed by the curve which B describes is, by (6) of 129, Is. In this case s is independent of the position of P on the bar AB. To find s a wheel with axis parallel to A B is attached to AB; the wheel, as B describes its curve, partly slides and partly The sliding and the rolling motions are independent, and the sliding motion has no effect in the way of "^ / \ turning the wheel. The normal / ) displacement of P is therefore equal I to the circumference, 2?rr say, of ' the wheel multiplied by n, the number of turns made by the wheel while B describes its curve ; that is, s = 2-Trrn. A counter is provided that registers n ; n of course may be integral or fractional. If we suppose the curve C so large that the circle of radius OA lies wholly inside it then, by (8) of 129, es its rolls f^Z^N 3tions \^^ J liding <^&^* that is, 0=2 G.C. 322 AN ELEMENTARY TREATISE ON THE CALCULUS. since s = 2-Trrn. All the numbers except n are constants of the instrument. For information on Planimeters the student is referred to Henrici's " Report on Planimeters," Brit. Ass. Rep. 1894. The method of proof followed in 128, 129, is essentially that given by Appell in his Elements d Analyse Mathe'- matique. EXERCISES XXVIII 1. Show that in polar coordinates the area of a closed curve is given by the integral taken round the curve. Prove the result (i) by use of the polar formula for area ; (ii) by transformation of the last integral in (7), 128, by putting z=rcos 0, y = rsin 9. (See Exer. XII., ex. 15.) 2. If the coordinates of the vertices of the triangle OAB are, when taken in the order 0, A, B, (0, 0), (#, y), (x+Sx, y + &y) respectively, prove geometrically that the area of the triangle is ^(x8y y$x\ in sign and in magnitude. Apply the result to establish the theorem of ex. 1. 3. Find the area common to the two parabolas y 2 = 4ax, x 2 = 4ay. 4. Find the area between the asymptote y = a, the y-axis and the branch of the curve 3/ 2 (a 2 +# 2 ) = a 2 # 2 that lies in the first quadrant. The area is equal to Find the area by integrating with respect to y. 5. The " tore " or the " anchor-ring " is the solid formed by the revolution of a circle about a straight line in its plane. Let a be the radius of the circle, the y-axis the axis of revolution, and let the centre of the circle be on the a;-axis at a distance c from the origin. The coordinates of any point on. the circle may be taken as If V is the volume and S the surface of the tore, then, when c^a, prove f 8T (i) V= TT I (c+a cos ffa cos tdt 27r 2 a 2 c = A L ; Jo rtw (ii)5=27r/ (c + aco8t)adt = 47T 2 ac = CL, Jo where A is the area and C the perimeter of the circle, and L is the circumference 2irc of the circle described by the centre of the revolving circle. EXERCISES XXVIII. 323 6. The curve r=3-f 2cos consists of a single oval ; trace the curve and find its area. 7. The curve r=2 + 3cos# consists of two ovals (Fig. 74); if cos a = - (0 < a < ;r), show that the area of the large oval is A = V-a + 12 sin a + sin a cos a, and of the small oval is fs j. Show also that the integral of ^r 2 from 6=0 to 0=27r gives the sum of these two areas. Examples 6, 7 show the nature of the curve r = a 4- b cos Q for a > b and a < b respectively. FIG. 74. 8. How may the curve given by the equation f(mx, ny) = 0, where m and n are constants, be deduced from that given by f(x, #)=0 ? If the second curve is closed, show that the first is also closed and that the area off(mx, ny) = Q is equal to that of /(#, y) = divided by mn. Let mx=x', ny=y', and therefore x'dy' = mnxdy. Now apply (7), 128 ; the integral of x'dy' round the curve f(x', y') = 0, (which is the same thing as the integral of xdy round the curve /(#, y) = 0), will be equal to the integral of mnxdy round the curve f(mx, ny)-0, that is, to mn times the area enclosed by that curve (since mn is constant and the integral of xdy is the area). 9. Apply the method of ex. 8 to deduce from Exer. XXVII., ex. 5, the area of the curve 10. When AB ( 129) describes one complete revolution, show that P describes a curve which encloses an area C" given by (i) C" = (aC+bC')/(a+b)-irab, where PB=b and a, (7, C' denote the same quantities as in 129. Show also that if the ends A, B move on a closed oval curve C (ii) C-C" = irab. (Holditch's Theorem^ Use equation (8), 129. Put l=a+b and we get CC" ; then put l=a and we get C" -C'. The elimination of s gives (i). To find (ii), consider the areas swept out by AP and BP. CHAPTER XV. INTEGRAL AS LIMIT OF A SUM. DOUBLE INTEGRALS. 131. Integral as the Limit of a Sum. It is instructive and for some applications necessary to consider an integral as the limit of a sum. F(x) is, as usual, understood to be continuous. In the first place, suppose a b and F(x) positive, each of the differences (x l a), (x z X}), ... is negative and the limit gives the area with negative sign. Lastly, if F(x) is negative the limit is still the area if the appropriate sign be chosen as in 80. In regard to the sub -intervals we may if we please suppose them all equal, each therefore being (b a)/n; the only restriction on the sub-intervals is that each must have zero for limit when n tends to infinity as limit. We have supposed F(x) in the sum (1) to have its value at the beginning of each interval ; but the limit will be the same if we take the value at the end or at any inter- mediate point of each interval, as may be proved by 87, Th. II. For, restricting attention to the case a < b, F(x) positive, since the others can be easily deduced from this, if a', x[, x^ ... are values of x within or at the end of the inter- vals (x 1 a), (x z Xj), (x 3 x z )... respectively, we may take ft^flOfo-aX ^ = F(x^(x 2 -x l ) > ... 7l = F(a)(x l -a), /2 = F(x l )(x 2 -x l ), ... and the conditions of that theorem apply since, F(x) being continuous, the limit for n= of /Sj/yp /3. 2 /y 2 --- is unity. Having proved that the limit of (1) is the area z, we can now show, as in 80, that the derivative of that limit with respect to 6 is BD, that is -^(6), and therefore we can apply EXAMPLES. 327 all the theorems respecting integrals to the limit of the sum (1). The origin of the ordinary notation for integrals is also obvious, the I being a form of the initial letter of the word " sum " ; it will be remembered, however, that the integral is not a sum but the limit of a sum. (See 132, ex. 2). 132. Examples. Ex. 1. Evaluate / x*dx. Ju Divide the interval b into n equal parts ; in the notation of 131, 0/1=0, OA^b/n, OA 2 = 2b/n, ..., OA n ^=(n-l)b/n. The sum (1) becomes + + n \n/ n \n / n I n ) n and the limit is clearly 6 3 /3. Ex. 2. Show that if in 131, (2), we put F(x)=f'(x), the limit will be f(b) -f(a). By the definition of a derivative, f(x + ^l~ f(X ' } =f(x') + a ; f(x + 8x)-f(x)=f(x)8x+aSx, ..... (A) where a vanishes with 8x. Give successively to x and 8x in (A) the values in 131 ; a will not usually have the same value for all values of x, and we therefore use suffixes. Hence Add : /. /(&) -/(a) = 2/(#) Sx + R, where /? = a 1 Sa + a 2 S.r 1 + ... + a n &r n _i. Let a' be the greatest, numerically, of the quantities a x , Og, ... ; then, numerically, / < a'(8a + &Cj + . . . + 8x n -i) or a'(6 a). Since every a, and therefore a', has zero for limit, R will have zero for limit and the result follows. 328 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 3. Find the limit for n = oo of I T I ~C\ ~T i ~P I" ~ * n+l n + 2 n + 3 'zn We may write this sum 1 1 1 1 1 1 1 1 rj3' i+' i+?'^ + + i+ n '~ n n n n n Consider the function F(x) = l/x; in 131, let each difference be l/n, let a=l, 6 = 2, and the above sum will be the same as (1), 131, if we suppose the values of F(x) to be those at the end of each interval. Hence the required limit is 133. Approximations. The method of evaluating an integral by first finding the function of which the integrand is the derivative would fail if we could not find such a function. An important case in which that method can not be used is that in which the integrand is given only by its graph, as often happens in physical applications. Methods have therefore been devised for determining approximately the value of the integral when only a limited number of values of the integrand are known ; it is assumed that the integrand may be treated as a continuous function, though if only a limited number of values of the integrand are known, the analytical expression for the function can not be given. The rules now to be stated can be applied even when the analytical form of the function is known, though in general more powerful methods are available in that case, in particular the method of expansion in series. Let AB be divided into n equal parts, each part being equal to h, and suppose the (n + l) ordinates at A, B and the points of division to be known; let these be y lt y z> y 3 , .. .. The calculation of the integral \ b F(x)dx (1) Ja is then equivalent to finding the area ABDC (Fig. 76). APPROXIMATIONS. The most obvious method is to replace the graph by the inscribed polygon CCfi^.... The area of the first trapezium is ^h(y l -\-y 2 ), and this area may be assumed to differ but little from that of the corresponding strip of ABDC. Adding together all the trapeziums, we get, as an approxi- mation to the area, and therefore to the integral (1) ^ *^ "^ "s c*/\ s^ c / % c / / / 4 y* / 4 % y* / 1} (2) If the graph is, as in the figure, convex upwards through- out the value A l is in defect; if the graph is concave upwards, A l is in excess. Through the ends of the even ordinates y z , y ... let tangents be drawn and pro- duced to meet the adjacent odd ordin- ates; if the number of ordinates is odd, 2n + 1 say, we shall get n trapeziums FIG. 76. whose sum exceeds ABDC in area when the graph is convex upwards through- out. The area of the first trapezium is 2% 2 , of the second 2/17/4, an d so on. Hence we get another approximation A 2 = 2A(2/ 2 + 7/ 4 -}- . . . -f 2/2n) (3) The value of the integral (1) always lies between A l and A 2 when there is no point of inflexion on the arc CD, and the difference (A l A z ) gives a measure of the error involved in either approximation. The formula (2) is usually referred to as the Trapezoidal Rule. A formula that is in practice more accurate than (2) or (3) is got as follows : By 72 we may write A A| A 2 330 AN ELEMENTARY TREATISE ON THE CALCULUS. If x c is small we may assume that F'^xJ differs but little from F"(c) ; if F(x) were of the second degree F"(x 1 ) would be simply F"(c). The equation y = F(c) + (x-c)F'(c) + %(x-c}*F"(c) ............ (4) represents a parabola ; we therefore replace a short length of the graph of F(x) by this parabola. Now consider the double strip AA Z C 2 C; for convenience let OA 1 = c, OA=c h, OA 2 = c + h; then using (4) as the value of F(x) along the arc GC^C^ we find for the area of A ^4.2^2 h F(x)dx= I* F(x'+c)dx'=2hF(c) + $h?F"(c) ...... (5) c-h J -h where, to integrate, we put x = x' + c. We can now express (5) in terms of h and y v y 2 , y 3 , assuming F(x) to be given by (4). For F(c) = y z and 2/ x = F(c -h) = F(c) - hF'(c) By addition h*F"(c) = and (5) becomes Suppose now ABDC divided into an even number, 2n, of strips by an odd number, 2n + l, of equidistant ordinates. The formula (6) may be applied in succession to the n double strips ; the sum of the n expressions is, the terms being rearranged, 4$ = iM2/i + 2/2n+i + 2 (2/3 + & + + 2/2 - 1) + 4(2/ 2 + 2' 4 +.-.+2/2)} ......... (7) Formula (7) is known as Simpson's Rule, which may be stated thus : Let the area be divided into an even number of strips by equidistant ordinates ; find (i) the sum of the extreme ordinates, (ii) twice the sum of the other odd ordinates, (iii) four times the sum of the even ordinates ; add the three sums thus obtained and multiply this total sum by one-third of the common distance between the ordinates. SIMPSON'S RULE!. 331 Let u then in terms of h, u, v, w, we find and therefore A 3 = %A 1 + ^A 2 .......................... (8) Suppose the graflh convex upwards and the ordinates positive, so that J 4 1 (t)dt. Ja JO 6. Show that / log sin x dx = - I x cot x dx, Jo Jo value of the integral by Sirnpsoi al is -^Trlog 2. For let the inti n /"i r~% u= I \og8mxdx=l logcos#ete=|/ (log sin x + log cos x) dx, Jo Jo Jo and calculate the value of the integral by Simpson's rule. The exact value of the integral is - \TT log 2. For let the integral be u ; then MEAN VALUES. 333 so that fV K f* %u = I log (I sin 2#) dx= - log f + / log sin 2.r er ; JO t Jo rf /-T /? also I logsin 2j;o?^=i/ logsinscfo= / logsinzefo=tt, Jo Jo Jo from which the result follows. 7. Show that / logtan#dr=0. (No integration is necessary.) Jo 8. Show that the limit when n is oc of r=n-l 1 is 7T/2. 9. Show that the limit when n is oo of r -"- ] __ r=0 * + * 18 7T/4. 134. Mean Values. The arithmetic mean of n quantities y v 2/2, , 2/n IS (2/1 + 2/2 + + 2/n)/ 71 ' -L^ y\, 2/2' " ' 2/n k the values of ^(x) for x equal to a, a-\-h, ... , b h, the interval 6 a being divided into n parts each equal to h ; the limit for 71 = 00 of the arithmetic mean of y v y z , ... , y n is called the mean value of the function F(x) over the range b a. The mean value may be expressed as an integral ; for (y l + 2/2 + . . . + 2/n)/?i = (yji + yji + . . . + y n h)/(b a). . . .(1) The numerator of the fraction on the right is F(a)h + F(a + h)h+... + F(l-h)h, and the limit of it for n = oo (and therefore h = 0) is and the Mean Value is Ex. 1. The mean value of the ordinate of a semicircle of radius a is a2 " * 2 dx=a= ' 7854a ' In this case the diameter is divided into n equal parts. If, however, the semi-circumference is divided into n equal parts, so that the inde- pendent variable of the function is the arc aO from one end of the 334 AN ELEMENTARY TREATISE ON THE CALCULUS. diameter to the point from which the ordinate is drawn, the mean value is, since the ordinate is a sin 6, I 'a sin add = 2 a= ' irajo TT '6366a. In speaking of mean values, therefore, it is essential that the inde- pendent variable should be clearly indicated. Ex. 2. For the harmonic curve y = asin#, find (i) the mean ordi- nate, (ii) the square root of the mean of the square of the ordinate for the range from #=0 to X = TT. if* 2 (i) mean ord. = - / a sin x dx = -a = - 6366a. irJo TT In case (ii) the function is y 2 , and the mean value of y 2 is I 2 sin 2 .refo?=ia 2 , Wo and the square root of this mean is a/J'2 or '707 la. In the theory of alternating currents the important mean is not (i), but (ii) ; the latter is sometimes called tlie mean-square value of the ordinate. If the interval b a is divided into n sub-intervals h v h 2 , ... , and if y v y 2 , ... are the values of F(x) at any point of the intervals h v h 2 , ... respectively, the limit for n infinite (and each sub-interval h v h z , ... zero) of (yA + vA + + y n hn)/(b - ) is still given by (2). Y The integral (2) may be taken as the gene- ral definition of the mean value of F(x). 135. Double In- tegrals. Let EFGH (Fig. 77) be a plane curve, and let f(x, y) be a single-valued continuous function of x and y for all points within or on the curve. Let AH, BF, and CE, DO be the tangents parallel to the axes; we suppose that no straight line cuts the curve B X FIG. 77. DOUBLE INTEGRALS. 335 in more than two points ; any curve that does not satisfy this condition may be divided into partial areas, each of which satisfies it. Let AB be divided into m and CD into n sub-intervals, and through the points of division let parallels be drawn to the axes. The area bounded by EFGH will thus be divided into partial areas ; these areas are rectangles, though near the boundary EFGH the rectangles will contain points that lie outside the curve. Let x r , x r +Sx r be the abscissae of two consecutive points of division on AB and y s , y, + dy t the ordinates of two consecutive points of division on CD ; and let S, S' be the points (x r , y,), (x r +Sx n y,+Sy s ). Multiply Sx r Sy g , the area of the rectangle SS', by f(x r , y g ), the value of f(x, y) at S, and form the sum /<<*. y.) terfy. ......................... (i) for all points such as S within or on the boundary of EFGH. Geometrically, z =f(x, y) represents a surface ; the typical term /"(.r r , .y,)&r r Sy, of the sum (1) is the volume of a parallelepiped whose base is the rectangle &r r 8y n and height the 2-coordinate /(# y,) of the point in which the normal from S to the rectangle meets the surface ; the sum (1) is therefore approximately equal to the volume of the solid bounded by the surface, the plane XOT and the cylinder formed by a straight line which moves round the boundary EFGff, remaining always perpendicular to the plane XOT. (Compare Figs. 48, 49.) We wish to find the limit of (1) for m and n each in- creasing indefinitely, each element Sx r , Sy x , and therefore each area Sx r Sy t at the same time diminishing indefinitely. Seeing that there are two sets of increments we may appro- priately represent (1) as a double summation SS/<*y-)4M0. ...................... (2) the one Z referring to Sy s and the other to Sx r . First, keep x r and Sx r constant, that is, find the limit for C )4yH J MP 1 ( x r,y)dy ............... (3) MP by the definition of the integral of a function of one vari- able y. The integral (3) will contain x n MP, MP' ; MP and MP' are functions of OM or x r determined by the 336 AN ELEMENTARY TREATISE ON THE CALCULUS. equation of the curve EFGH. Hence (3) is a function of XT and may be denoted by $(x r }. Geometrically, (x r ) ?>x r is, to the first order of infinitesimals, the volume of the slice of the solid of thickness far. Next find the limit for m = oo . We get COB L 2*^^) I *(*)* .................. (4) m=oo J OA Hence, finally, the limit of (1) is expressed by (4) and that limit is the volume of the solid already mentioned. Since z ^ dxd y dz or \f( x >y' 2 ) dv ( ]1 ) dx dy dz or dv may be taken as an element of volume, and f(x, y, 0) might, for example, denote the density at (x, y, z). Integration with respect to 0, keeping x, y constant, would give the mass of the column standing on the base dx dy ; then the y -integration, keeping x constant, would give the mass of a slice of thickness dx perpendicular to the a>axis, and lastly the ^-integration would give the total mass. TRIPLE INTEGRALS. EXAMPLES. 339 Ex. 1. Find the volume of the tetrahedron bounded by the coordinate planes and the plane Z JC where a, b, c are positive. The curve EFGH is in this case the triangle OAB ; the equation of AB is and y = b(\-xla) ' = b(l-x/a), while MP in 135 is here zero. SR=f(x,y)=z =c(l-xfa-ylb). Hence using (5), the volume is [ a s I*** f a j / dx\ zdy=c\ dx Jo Jo * Jo T. -- I dx=- B abc. a) 6 Obviously \bc(l -xjaf is the area of the triangle LMP'. Ex. 2. Find the value of / x 2 dv taken throughout the volume of the ellipsoid x?/a z +y 2 /b 2 +z z /(?=l. = Ix^dx ffdydz = [" x*dx since, in integrating as to y and 0, x is constant and / \dydz is the area of the section perpendicular to the #-axis. Integrate now as to x ; the result is 47ra*6c/15. The mean value of the function x 2 throughout the volume of the ellipsoid is the above value divided by the volume, that is, 2 /5. In general, the mean value of a function /(#, y) over an area EFGH (Fig. 77) is the value of the integral (5) or (6) divided by the area ; and a similar definition holds for the mean value throughout a volume. If, in the example, x 2 is the density at (x, y, z) of a mass occupying the volume of the ellipsoid, then a 2 /5 is the mean density of the mass. 340 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 3. If f(x, y) is the product of a function <$>(x) of x alone, and of a function ^r(y) of y alone, it follows at once from 135 that the integral of the product <$>(x) ^s(y) taken over the rectangle A 1 S 1 C 1 D 1 (Fig. 77) is equal to the product of the integrals ft rv I (x)dx and | Now let ^(x)=e~ 1? , = e-, and U= [\-*dx= (* Jo Jo (i) It follows that Z7 2 , the product of these two integrals, is equal to the integral , (ii) taken over the square OABC of side OA = a (Fig. 79). Draw the arcs ADC, EBF from the centre with the radii OA a, OB = aj2. The integral (ii) is greater than the integral of the same func- tion over the area OADG and less than that over the area OEBF. These two integrals can be found by changing to polar coordinates ; dxdy ~X is replaced by rdrdd and e-<* 2+ * 2 > by e-^ and (ii) be- comes, for the area OADC, FIG. 79. since the integral of e~^r is \e~^. When the area is OEBF, the integral is j(l e" 2 ""). U* lies between these two values ; but when a tends to infinity both values tend to ir/4 ; and therefore also U 2 tends to Tr/4, and U to TT. Hence / This example is a particular case of an integral of great importance (see Ex. XXX. 21), and the transformation is worthy of careful attention. CENTRES OF INERTIA. 341 137. Centres of Inertia. It is shown in works on mechanics that the coordinates (x, y, z) of the centre of inertia of a set of n particles of masses m 1? m 2 , ... , m n situated at the points (a^, y v z^, 2 > y z , z 2 ), ... , (x n> y n , z n ) are given by the equations ^_ ll z ... nn _ l . , ~" with similar expressions for y, z. For a continuous distribution of matter the volume density p at the point (x, y, z) is the limit for #y = of Sm/Sv where Sm is the mass of the volume Sv surrounding the point ; hence to the first order of infinitesimals Sm = pSv. When the mass is supposed concentrated in a surface or in a line we have in a similar way Sm = a-8S, Sm = \Ss where or and A are the surface density and the line density at a point and SS and Ss elements of area and of length including the point. A continuous mass may be supposed to be divided into n elements 8m ; if (x, y, z) are the coordinates of any point in the element Sm then the coordinates of the centre of inertia of the mass are given by _ T _ C ~"~ " with similar expressions for y, z. The integrations in (2) are to be extended through the total mass. For volume, surface, and line distributions equations (2) take the forms \xpdv \xa-dS \x\ds ,^ Ipdv' \ we readily find J/"= MOMENTS OF INERTIA. 343 Again, f f f a f oir JUx = I I x . kxy dxdy=k\ x z dx \ ydy = ^ka?&, J J Jo Jo and therefore x=^~a. Similarly, y = ^b. Ex. 3. A circular sector of uniform density. Take the notation of ex. 1. We may take as element the small sector OPQ. The centre of inertia of OPQ may be taken as the point ( a > 0)> an d the moment about Y of the element is $a cos 6 . constant, let r become r + 6V; the area SS will describe an element of volume S V equal approximately to SS x Sr, that is, equal to r 2 sin Sr SO S. The limit of SV is the polar element of volume, so that dV=r 2 sinOdrdOd. The element of the surface of a sphere of radius r is If r =/(0) is the polar equation of a curve lying in the plane ZOX, the initial line being OZ, we find by integrating dV from = to = 2?r and then from r = to r=f(Q) that the polar element of volume of a surface of revolution about the initial line is f irr^smOdO, where r now means /($). Let P be the point (x, y, z) on a surface and let the rectangular parallelepiped standing on the rectangle SxSy as base cut out of the surface the element of area 6V, and out of the tangent plane at P the element &/. If the normal to the tangent plane at P make with OZ the angle y we have S(r' cos y = Sx Sy, So-' = Sx Sy sec y. POLAR ELEMENT OF VOLUME. 347 If we assume that the limit of 6V/oV is unity we find d 0, the integral [* e-'aS^dx Jo has a definite value ; the integral is a function of n, usually called the Gamma-function, and denoted by T(n). Show, by integrating by parts, that !=("- 1)I>-1), .............................. (i) and that when n is an integer, T(n) (n- 1)!, F(l) = l. If n is not an integer, let p be the integer next below n so that (n -p) is a proper fraction, then (i) shows that r()=(n-l)(-2)...(n-.p)r(-p) ..................... (ii) 16. Prove (ii) r(m + |) = ^ 1 ^^ . ... KN/T) - ( integral). Equation (i) follows from 136, ex. 3, by putting x=^z for * = f "e-^dx = | (Vz - 'A = *IX). z Jo .'o Then (ii) follows from ex. 15 (ii). 17. Prove f e- a *x n - l dx=^& (a positive). cr 350 AN ELEMENTARY TREATISE ON THE CALCULUS. 18. By the given substitutions, prove other formulae for T(n) : -.... ...(i) dz ........................ (ii) 19. When m and n are both positive, the integral P x m ~\l - x) n ~ l dx Jo has a definite value ; it is a function of m and n usually called the Beta-function, and denoted by B(TO, n). Show that B(TO, %) = B(?z, m). 20. By the given substitutions, prove other formulae for B(TO, n) : (i) .(ii) ~ l+y' 21. Using form (i) of ex. 18, write r(m) = 2 f" e~^^ m - l dx, T(n) = 2 and then show, as in 136, ex. 3, that r(m) x I = and therefore, by exs. 18 (i) and 20 (i), T (m) F() = T(m + w) B(wi, > Thus the Beta-function can be expressed in terms of the Gamma- function. 22. Let 2m - 1 =p, 2w - 1 = q ; then, from exs. 20 (i) and 21, g + 2\ where, since TO > 0, n > 0, we have ^(^o + 1) and \(q + 1) > 0, or p and 5- each greater than - 1 . The student may test that this result includes the rule given in 119. Tables of logF(n) for 1= = 2 have been calculated (no wider range for n is necessary by ex. 15, (ii)), and many integrals can be expressed in terms of Gamma-functions. BETA-FUNCTION. POTENTIAL. 351 23. Find the potential V at a point Q of a mass M distributed uniformly (density cr) over the surface of a sphere of radius a. Take 0, the centre of the sphere, as origin and OQ as 0-axis ; let dS be the surface element at P, and denote PQ by R and OQ by c. Then dS=a 2 sin6d6d<}>; R^a^+^-^accosO. The limits for < are and 27r, for they are and TT ; in integrating as to (, the other variable $, and therefore in this case also PQ or A! (which is a function of 6 and not of <) is to be kept constant. Hence Now change the variable from 6 to .ff ; we have RdR^acsin BdO. When 0=0, R (a c); .ft is a positive number, so that if Q is out- side the sphere R=c a, and if Q is inside R = a-c. When O=TT, R = a + c in both cases. Hence (# outside). acJ c = 47rcra ($ inside) ...................... (ii) Thus V=Mjc when Q is outside, but F= Mja = constant when Q is inside the sphere. 24. Same problem as in Ex. 23 for a solid sphere (density p = constant). Take as element of mass the shell bounded by radii r and r+dr, and use the results of Ex. 23, putting pdr for cr, and r for a. If is outside, the result (i) gives a F ,_ ^-ri (iii) If Q is inside, V consists of two parts, F 1? F 2 . Fj is the potential due to the sphere of radius c, and by the result just found F 2 is that due to the shell of radii c and a : by the result (ii) of ex. 23 F = Hence F= F t + F 2 =2*-/>(a 2 - Jc 2 ) .............................. (iv) When c=a the values given by (iii) and (iv) coincide. CHAPTER XVI. CURVATURE. ENVELOPES. 140. Curvature. Let P and Q be two points on a plane curve, and + }ds, is called the curvature of the curve at P. For a circle of radius R, Ss = RS and therefore 50_J L ^0_J_ (1) 8s ~R> ds~R" that is, the average curvature of any arc of a circle is equal to the curvature at any point of that circle. In other words, a circle is a curve of constant curvature and its curvature is equal to the reciprocal of its radius. Curvature is thus a magnitude of dimension 1 in length. The curvature may be expressed in terms of the first and second derivatives of the ordinate at the point. For, since dy dx tan = -~, cos = -j-> we get, by differentiating the first equation with respect to s, d . tan d _ d (dy\ dx d> ds dx\dx/ds' CURVATURE OF PLANE CURVES. 353 , . d d?y that is, sec-0 ~f- = 7 -*-aee ; r dfi ax* and therefore. -r^ = T-^-=-sec 3 ......................... (2) as aor Hence, since see 2 =l + (dy/dx) 2 we find Formula (A) may be considered fundamental. COR. When the gradient dyfdx is so small that for all values of x within the range considered its square may be neglected, the curvature is approximately d-y/dx 2 . This approximate value is often used in Mechanics ; for example, in the theory of the bending of beams. Ex. 1. The parabola y* = lax. 4a 2 dx y ' dx 2 y 2 dx y 3 d_-4a? . J 4a 2 \| = -4a 2 If the normal at P(x, y) meet the axis at G, cilld j~ :=i The meaning of the negative sign will be referred to in 141. Ex. 2. The ellipse x 9 -/ = _. = dx aty ' dx 3 a*y a?y z \ since 6 2 o; 2 + a 2 i/ 2 = a 2 6 2 by the equation of the ellipse. Hence If p is the perpendicular from the centre on the tangent at (x, y), p - W and ^__^.. P (bW + a*y^ d*~ M* If PG is the normal at P (x, y), , bW + a y , d b* PO Z = - j^ and -^=--ODT- a 4 ds a?PGP A similar result holds for the hyperbola. Thus the curvature of a conic section varies inversely as the cube of the normal. G.C. z 354 AN ELEMENTARY TREATISE ON THE CALCULUS. 141. Circle, Radius, and Centre of Curvature. Let the normals at P and Q (Fig. 82) intersect at C"; when Q tends to P as its limiting position, C' will tend to a point C on the normal at P as its limiting position such that PC is equal to ds/d(f>. For PC sinPQC' chord PQ smPC'Q chord PQ arcPQ (5s ' FIG. 82. The limit of PQG' is 90 and the limits of the three fractions last written are 1, ds/d, 1 respectively; hence the limit of PC" is dsfd^, as was to be proved. The circle with centre C and radius PC has therefore the same tangent and the same curvature as the curve has at P. This circle is called the circle of curvature, its radius PC or ds/d^ the radius of curvature, and its centre C the centre of curvature at P. If any line through P meet the circle again at R, PR is called a chord of curvature. If (x, y) are the coordinates of P, ( >/) those of C and p the radius of curvature PC or ds/d(J> it is easy to prove s) - PQ cos QPC'. 142. Other Formulae for the Curvature. Formula (A) is not very convenient unless the equation of a curve is in the form yf(x) or unless, as in the examples worked in 140, the values of the derivatives can be easily calculated. We will therefore give one or two other formulae ; the question of the sign of p usually needs special consideration. (i) Equation of form x=f(t), y = F(t). The variable t need not, of course, represent time but we will for brevity use the fluxional notation. Substitute in (A) the values of Dy, DPy in terms of #> &> y> y, as given in 98 ; we find - = (xy-yx)/(a?+f-)* (B) Since D x *y = (xy yx)/d?, we can determine the sign of p when necessary in accordance with the convention of 141. 356 AN ELEMENTARY TREATISE ON THE CALCULUS. (ii) Polar Equations. In (A) substitute the values of Dy, D 2 y in terms of D 6 r, D^r and we get _ r ^_ dO/ dQ-] I \dd Formula (c) is cumbrous. It is often simpler to find what is called the p, r equation, that is the relation between the perpendicular OZ from the origin on the tangent at P (Fig. 82) and the radius OP (see ex. 2), and then to apply a formula we will now deduce. In Fig. 82 we have, OZ=p, OP = r, ocr* = OP 2 + PC" 2 - 20P . per cos OPCT since p=OPcosOPC'=rsin\{s where \/r is, as usual, the angle between the tangent and the radius vector. If OQ = r-f <5r and if p + 8p = perpendicular from on tangent at Q we find in the same way Equating the two values of 0(7 2 we get 2rSr + (Sr) z + (QC r - P(7) (QCT + PC" - 2p) - ZQffSp = 0. But (QCT-PCT) and (Srf are of order higher than the first, and therefore Formula (D) may also be proved thus : since p = r sin and = 6 + \js we have (see 88) dp . , d\!s dO . d\ls dd> -r- = smur + rcosur ~^- = r-^ \-r -y-=T-f-, dr r dr ds ds ds and therefore ds/dw d6 dud8~~ u 2 dO' and therefore (i) becomes 1 . fdu\* Hence, differentiating with respect to u, p*du \dO/d6 2 du ,d z u But =.; ==-, du dr u* p r dr du and now by substitution in (iii), using (ii), we get 2 * The root being taken positive, p will be positive or negative according as the arc is concave or convex to the origin. (iii) Intrinsic Equation. Let s denote the arc of a curve measured from a fixed point on it up to the point P, and (f> the angle which the tangent at P makes with a fixed tangent ; the equation which expresses the relation between s and ^ is called the intrinsic equation of the curve. This 358 AN ELEMENTARY TREATISE ON THE CALCULUS. equation does not depend on any lines of reference outside the curve, such as the ordinary rectangular axes ; hence the name. When the intrinsic equation is given p is found at once by differentiation. In elementary work, however, the intrinsic equation is of comparatively small importance ; it has usually to be deduced by integration from the ordinary equation, one of the coordinate axes being taken as the fixed tangent. The angle < must not in this case be restricted to acute angles. Ex. 1. x%+y% = a?. Let x=acos 3 t, y asiu. 3 t, and use formula (E). y =3a sin" t cos t ; y = 3asin(2 cos-t sin 2 ) ; 5 ~f~ y ^ = \jGL Sill t COS t 9 *^y **~ y ~ = y^" 1 Sltl t COS" fc J p = - 3a sin t cos t = - 3 (axy)~*. In this case D x 2 y=l/3asin tcos*t and p, if determined by the con- vention of 141, will be + 3a sin cos . Ex. 2. r m = a m cosmd. Form the p, r equation and use formula (D). rd0_ '~W~ We will take \jr=md+7r/2 ; then p =r sin \fr = r cos mO = r m+1 /a m , and therefore P = -i = 7 . i\ m r dp (m + \)r By giving different values to m, we get several well-known equations. See Exercises XXXI. 10. Ex. 3. Find the centre of curvature and the locus of the centre of curvature of an ellipse. It is easy to show, with the notation of 140, ex. 2, that sin (f>= px/a 2 , cos <^> = py/b 2 , p = Let 6 be the eccentric angle of P(x, y) and these values become bq= -(a 2 -6 2 )sin 3 0. EXERCISES XXXI. 359 To find the locus of the centre of curvature eliminate 6 ; thus (aF + (br))%=(a*-brf, or, taking now x and y as current coordinates, The curve is shown in Fig. 83, 143. Ex. 4. Show that the normal acceleration at a point P on a curve is v-/p where v is the tangential velocity and l/p the curvature at P. At Q (Fig. 82) let the tangential velocity be v+8v ; the components in the direction PC' of the velocity at P and at Q are and ( + &;) sin 8 respectively. Hence the normal acceleration at P is (v + 8v) sin Sd> dd> dd> ds 1 r j _ _ I_ - nj _ L_ - vi _ 1 _ - -y _ *y 5< _ 8( dt ds dt ' p ' as was to be proved. EXERCISES XXXI. 1. The equation of any conic may be put in the form y 1 'ZAx-\-Bx 2 ', where the .r-axis is the focal axis and 2A is the latus rectum. If the normal at P meet the j--axis in G and if a is the angle between PG and the focal distance SP prove that p = - PG 3 /A*= - PG/cos*a. Note that the projection of PG on SP is equal to the semi-latus rectum. 2. From the value of p in terms of a (ex. 1) prove the following construction for the centre of curvature K of any conic : Draw GH perpendicular to PG to meet SP at H, then draw HK perpendicular to RP to meet PG at K ; K will be the centre of curvature. 3. For the rectangular hyperbola xy c 2 show that 4. C is the centre of an ellipse, CD is a semi-diameter parallel to the tangent at P and 6 is the eccentric angle of P ; show that, numerically, p = (a 2 sin 2 8 + & 2 cos 2 /ab It may be shown that the eccentric angle of D is O + ^TT or ^ir. CP, CD are called conjugate semi-diameters since, as may be readily proved, each diameter bisects all chords parallel to the other. 5. If r is the central radius of a point P on an ellipse, and p the perpendicular from the centre on the tangent at /', prove a? + 6 2 - r 2 = aW/f? ; p = For a hyperbola prove, with similar notation, r 2 a 2 + 6 2 = a-bP/p? ; p = 360 AN ELEMENTARY TREATISE ON THE CALCULUS. 6. For the curve a 2 y=x 3 show that p = (a 4 + 9x*)^/6a*x and for the 1 3 curve ay 2 =* 3 , p=ar(4a + 9x)'/6a. 7. At the origin on the curve ay = bx z + Zcxy +gy 2 + u 3 + u t + . . . + w n , where u n is of the wth degree and homogeneous in x and y t show that % = 0, Z> 2 ?/ = 26/a, p = a/26. 8. At the origin on the curve y = 2z + 3x 2 - %xy +y*, the radius of curvature is 5 N /5/6 9. Prove that the radius of curvature of the catenary a is y 2 /a, and that of the catenary of uniform strength y=c\ogsec(x/c) is csec(.r/c). 10. Verify the general results given in ex. 2 142 for the particular cases : (i) Lemniscate ? <2 = a 2 cos 26 ; r s =a 2 p ; p = a 2 /3r. (ii) Equilateral hyperbola r 2 cos 20= a 2 ; pr = a 2 ; p=r 3 /a 2 . 3 / 1 (iii) Parabola r(l+ cos 0) = 2a ; ar=p 2 ; pr v /a. (iv) Cardioid r=a(l +cos ^) ; r 3 =2p 2 ; p = 4ap/3r. For the parabola m= -1/2 ; for the cardioid m = l/2, and 2a takes the place of a. 11. Show that the chord of curvature through the origin is Zpdr/dp ; for the curve r m = a"*cosTO0, this chord is 2r/(m + l). 12. Show that for the equiangular spiral r=ae 0cota the radius of curvature is r cosec a ; show also that the radius of curvature subtends a right angle at the origin. 13. If i^ is the angle between the focal radius of a conic and the tangent at P and a the angle between the focal radius and the normal, show by formula (E) that p = J/sin 3 ^ = /cos 3 a, the equation of the conic being lu=l +ecos 6. Show also that if r and >' are the focal distances rr' cos 2 = b 2 = al, and that p= & = ^^. ab a cos a CONTACT OF CURVES. 361 14. If accents denote differentiation as to the arc s, show by differentiating the equations cos <=.', sin=y that, (, ?;) being the centre of curvature, l/p= -z"W=y"/ and 15. Show from formula (E) that the condition for a point of inflexion is 16. The circle (x - a) 2 + (y - /?) 2 = R 2 and the curve y=f(x) intersect at the point P(a, b). If at P the values of Dy and D-tj are the same for the circle and the curve show that the circle is the circle of curvature at P. The circle and the curve have the same tangent at P because P lies on both circle and curve, and the gradient of the circle at P is equal to that of the curve at P. Again, differentiate the equation of the circle twice and after differentiation put a, b (or /(a)), /"(a), /"(a), for x, y, Dy, D' l y respectively ; we get (a-)+(ft-/8y-JP. ..... (i); (a-a)+(6-j8)/()=0> ..... (ii); l+{/()P+(6-/3)/'() = ........................ (iii) From (ii) and (iii) we find and therefore by substituting these values in (i) Yf+f'(a) .......................... (iv) But R as given by (iv) is the radius of curvature at P and (a, /3) is the centre of curvature at /'. DEFINITION. Two curves y=F(x), y=f(%) which intersect at the point P (a, 6) are said to have contact of the 71 th order with each other at P if f(a)=f(a\ F"(a)=f'(a\ ...... >W(a)-/N(a) but J**+(a) not equal to/ (n+1| (a). The circle of curvature has thus in general contact of the second order with the curve. From Taylor's Theorem ( 152) it will be seen that when the curves have contact of the w th order at (a, b) the difference F(x) -f(x) between corresponding ordinates near (a, 6) is an infinitesimal of order n + 1 when x -a is principal infinitesimal; for where R is zero when x=a. 143. Evolute. Involute. Parallel Curves. DEFINITION. The locus of the centre of curvature of a given curve is called the evolute of that curve. 362 AN ELEMENTARY TREATISE ON THE CALCULUS. The coordinates ( rf} of the centre of curvature C corre- sponding to the point P (x, y) are given by g=x ps\n (1) The four quantities x, y, <, p can all be expressed in terms of one quantity, for example x or s or t ; the elimination of that quantity between the equations (1) will give a relation between g and / which will be the equation of the evolute. The evolute of the ellipse is ( 142, ex. 3) given by (ax? and is shown in Fig. 83. FIG. 83. E, E', F, F' are the centres of curvature corresponding to the vertices A, A', B, B' ; and EA = A'E' = b 2 /a, FB = B'F' = a*/b. It is obvious how the radius of curvature may be utilised for graphing the curve. The following are important properties of the evolute : (i) The normal at P to the given curve is the tangent at C to the evolute. (ii) The length of an arc of the evolute is equal to the difference between the radii of curvature of KVOLUTE AND INVOLUTE. 363 the given curve at the points corresponding to the ends of the arc. (i) In equations (1) take s, an arc of the given curve, as independent variable ; then d dx dd> .dp -^ = ~ -- P cos $ 7 sin -p- = sin r ^ ^ rfs l/p = d/ds. -r- = COS c as dn dr ds # < 2 > - ds dp -r-' ds .(3) ds ds since dxfds = cos Similarly Therefore ^ = -r / ^ fe = cot Ps ' then (5) gives arc C l C. 2 = p 2 + const. arc l3 = p 3 p v FIG. 84. which proves the required result. If a thread were wrapped round the curve G-^C^ and one end fixed at C 3 , the length of the thread being equal to p s , 364 AN ELEMENTARY TREATISE ON THE CALCULUS. % it is clear that when the thread is unwound, and kept stretched in the process, the free end will describe the curve P^P^Py It is from this property that the e volute is named. The curve P]P% is said to be an involute of C-fi y Ob- viously any point on the thread will describe an involute, so that a given curve has an infinite number of involutes while it has but one evolute. The two involutes P-f^ P\P% are called parallel curves, since the distance between them measured along their common normals is constant. 144. Envelopes. The equation y = ax + a/a (1) where a, a are constants, represents a straight line. If we give a different constant value to a, say a v the equation will become y = a l x + a/a l (2) and will represent a different straight line. The coordinates of the point of intersection of (I) and (2) are x = a/aa v y = a/a + a/c^ (3) Suppose now that a x is taken closer and closer to *; the line (2) will therefore come closer and closer to the line (1), but the values (3) show that the point of intersection tends to a definite position when a t tends to a as its limit. The coordinates of the limiting position of the point of inter- section are x a/a z , y = 2a/a (4) If we eliminate a between the two equations (4) we get y z = 4 From (iii) a/#= - (3/y, and therefore by (ii) ajx= -)8/y=c/ N /(^ 2 -y 2 ). Substitute in (i) for a and /? and 1 reduce ; we then get which is the equation of a lemniscate. It will be evident that the procedure is the same as that of finding maxima and minima. 368 AN ELEMENTARY TREATISE ON THE CALCULUS. 146. Cycloids. As the cycloid is of some importance in dynamics, we will very briefly investigate its chief properties. DEFINITION. The cycloid is the curve traced out by a point in the circumference of a circle (the generating circle) which rolls without slipping along a fixed straight line (the base). Let OD (Fig. 85) be the base, P the tracing point on the generating circle LPI, and 9 the angle between the radius SP and the radius SI, I being the point of contact with the base. Suppose P to be at when the circle begins to roll ; draw PM perpendicular to OD, and let OM=x, MP = y. Then if a is the radius, we have 0/=arcP/=a0, x = 01- SP sin = a (9 - sin 0) J ISL being the diameter through /. Equations (1) are those of the cycloid. THE CYCLOID. 369 When = TT, x = tra = OB and P is at A, the greatest distance it can be from the base. A is called a vertex. When e = '2ir, x = 2ira=OD, and P is at D. The arch OA D is symmetrical about BA, and BA is called an axis. If the circle were to continue rolling, P would trace out a series of arches congruent to OAD; when the cycloid is spoken of, it is usual to confine it to the one arch, and A, BA are then the vertex, the axis. Properties. The following are easily established : (i) tan = D x y = cot |0 = tan (|TT - W) = tan PIL, and therefore = | T 10 = ^ PIL, so that PL is the tangent and PI the normal at P. (ii) s = arc OP = 4a(l cos AO); &rcOA=4>a. (iii) p = PC= 4a sin |0 = 2PJ, numerically. If the tangent AT and the normal AB are taken as axes, and PN drawn perpendicular to AT, we put ff = .LSP=Tr- 9 ; then (io) $ = - (iia) s = arc^P=4asin*^; 2 =8a. NP=8ay. The coordinates of (7, the centre of curvature, are = 03f + 4a sin J cos = a (0 + sin 0), >7= 1C sin i# = a(l cos#). Hence, by equations (I'), the e volute of the cycloid OA D consists of the halves OCB', B'D of an equal cycloid. In (!') the positive direction of y is downwards, but when is origin the positive direction is upwards, so that r\ is negative. B is a cusp on the evolute ; 0, D are cusps on the original cycloid and vertices of the evolute. Epicycloids and Hypocycloids. The curve traced out by a point on the circumference of a circle which rolls without slipping on the circumference of a fixed circle is called an epicycloid or a hypocydoid according as the rolling circle is outside or inside the fixed circle. When the rolling circle surrounds the fixed one the epicycloid is sometimes called a pericycloid. G.C, 2 4 370 AN ELEMENTARY TREATISE ON THE CALCULUS. Let Figure 86 represent the generation of an epicycloid, P being the tracing point and C the starting point. Let a and b be the radii of the fixed and of the rolling circles, 6 and ff the angles CAI and IBP; AM=x, MP=y. Then arc P/=arc (77, so that, b& a$. = (a + b)cos6-b cos [(a + b) (9/6] FIG. 86. When the circles are on the same side of the tangent at /, that is, for the hypocycloids (6a), it is only necessary to change the sign of b. Hence, the equations of the hypo- cycloid are of the form x x = (a - 6) cos 6 + b cos [(a - b) 6/b] y=(a-b)8m6-bsm[(a-b)6/b] (3) When the ratio of b to a is a commensurab'e number the tracing point P will return to C after the circle B has rolled once or oftener round the fixed circle ; when the ratio of 6 to a is incommensurable P will not return to (7. Trocfioids. If the tracing point P is not on the circumference but on a radius or on a radius produced, the curve it describes is a trochoid or an epitrochoid or a hypotrochoid. If the distance of P from the centre of the circle is to the radius in the ratio of X to 1, the equations of the trochoid are got from equations (1) by multiplying sin 6 and cos 6 by A, while the equations of the epitrochoid and the hypotrochoid are got from equations (2) and (3) respectively by multiplying the coefficient b of the second term by A, as the student will easily prove. EPICYCLOIDS. HYPO-CYCLOIDS. 371 EXEECISES XXXII. 1. Show that for the parabola y i = lax, p == - 2a cosec 3 <, = 2a + 3a cot'- , r)= -2a cot 3 <, and then find the equation of the evolute. 2. Show that for the hyperbola # 2 /a 2 -# 2 /6 2 = l, * = (a 2 + &)*?, 6 4 )? = - (a 2 and that the equation of the evolute is 3. Show that for the rectangular hyperbola xy=c 2 , ,3 y 3 3 a* t~I*+& ^ and that the equation of the evolute is * ? 4. Show that for the curve xr+y' J =a s (see 142, ex. 1), = a cos 3 1 + 3a cos Z sin 2 , ?? = a sin 3 1 + 3a sin cos 2 1, and that the equation of the evolute is C*+y)*+(-jr)?-SB*. 5. Prove that the envelope of the family of straight lines (i) when a/3=a 2 , is the hyperbola, \xy = cP ; (ii) when a + /3=a, is the parabola >Jx+^Jy=^/a ; 222 (iii) when a 2 + /^= a 2 , is the curve #* +y Tr =a 5 . State the geometrical meaning of the conditions to which the para- meters a, /3 are subject. ' 6. Prove that the envelope of the family of ellipses (i) when a/3 = a 2 is the two hyperbolas 2xy= a-. 2 .2 2 (ii) when a + /3 = a is the curve ar+y s =a . State the geometrical meaning of the conditions to which the parameters a, /3 are subject. 7. The envelope of the circles described on the double ordinates of a parabola as diameters is an equal parabola. 8. If P, y cos (f> =p (i) ; x cos +y sin (/> = |3hr (ii), and show, from (ii), that ZP = Consider the curve as the envelope of its tangents. 15. With the same notation as in ex. 14, show that the coordinates (> J f) f the centre of curvature are given by ^ dp f. . , , d 2 p cos d> + 1] sm

+ ?? cos = -rj; 2 , (7

d(ff d(f> - 16. With the same notation as in the last two examples, show that the projection of OC, where C is the centre of curvature, on PC, is and that EXERCISES XXXII. 373 17. Show that the radius of curvature of the evolute of a curve is pdp/ds, where p is the radius of curvature at the corresponding point on the given curve. Use 143, (ii) ; d(f> is the same for curve and evolute. 18. If A is the area between a curve, its evolute and two radii of curvature, show that d.r. 2 dx 2\ \dx) / ' dx l 19. ABC is an arc of a circle whose centre is and radius a ; CP is the tangent at C and AP a part of an involute of the circle. Taking OA as the j?-axis and putting < for the angle AOC, show that the coordinates (#, y) of P are #= a cos < + ( sin <, y = a sin < a^> cos $, and that the intrinsic equation of the involute is All the involutes of a circle are identically equal, so that we may speak of the involute of a circle. 20. Show that the p, r equation of the involute of a circle is r 2 p 1 + a' 2 . 21. The total length of the evolute of an ellipse is 4(a?-b 3 )/ab. 22. The intrinsic equation of the cycloid, when the vertex A is the origin of s and the tangent A T the fixed tangent (Fig. 85) is s=4a sin . 23. Show that for the epicycloid (Fig. 86) PL is the tangent and PI the normal at P. v j. dy cos 6 - cos(0 + ff) For, tancA = -/ = . - , v r -' = and PL makes with the #-axis the angle 0+^ff. Similar results hold for the hypocycloid. 24. If s is the arc CP of an epicycloid, Fig. 86, show that ,= and that the length of CPD is 8b(a + b)/a. 25. The intrinsic equation of an epicycloid is ad\ -cos -J, and the radius of curvature is _4b(a+b) . Similar results hold for the hypocycloid. the sign of b being changed. 374 AN ELEMENTARY TREATISE ON THE CALCULUS. 26. If 6= a/4, show that the hypocycloid lias four cusps, arid that its equations are -, a -, a x=acos- i v, y = asm j v. Eliminating 0, we get x^+y^=a^. 27. Show that if b = a/2 the hypocycloid becomes a diameter of the fixed circle. 28. Show that if b = a the epicycloid becomes the cardioid the origin being at the point C ; that is, rcos0=x a, rsinO=y. 29. In ex. 25 put that is, measure the arc from the middle point V of CPD (the vertex), and we get , ,/ . , x ,, ,_46(q + 6) . a(f> S Sill ^rr a a+26 Show that the equation s = I sin .< will represent an epicycloid if is less than unity, a hypocycloid if n is greater than unity. 30. If s, a- are corresponding arcs of a curve and its evolute , ds a = -TT + const. d

we may omit the subscript "71=00." THEOREM I. // s n is a function of n that (i) always increases as n increases, but (ii) always remains less than 378 AN ELEMENTARY TREATISE ON THE CALCULUS. a definite quantity a, then as n increases indefinitely s n ivill tend to a definite limit that is less than or equal to a. THEOREM II. If s n is a function of n that (i) always decreases as n increases, but (ii) always remains greater than a definite quantity b, then as n increases indefinitely s n will tend to a definite limit that is greater than or equal tob. THEOREM III. The necessary and sufficient condition that s n should, as n increases indefinitely, tend to a definite limit is that the limit for n infinite of (s n+p s n ) should be zero for every value of the integer p ; or, in other words, given an arbitrarily small positive quantity e it must be possible to choose n, say n = m, such that when n>m the difference (s n+p s n ) shall be numerically less than e, what- ever value the integer p may have. We do not propose to prove these theorems ; the first and second have been given as exercises (Exer. VII., 14, 15), and the geometrical illustration there given affords some justification for assuming them. As to the third theorem it is easy to see that the condition stated is necessary. For, if s n has a definite limit s, then since Sn+p ~ S n = (n+p -) + (*- *), we have L (s n+p - s n ) = L (*+ - s) + L (s - s,,) = 0. To illustrate the sufficiency of the condition, take on the ^7-axis the points J.j, A 2 , A 3 , ..., which have s v 2 , 3 , ... as abscissae. In this case A n+l may be either to the right or to the left of A n , since s n does not necessarily either always increase or always decrease as n increases. But, by hypothesis, if n^.m, s n + P -s n \4 + 4 r 2' 6 + 8 + 7+P8 r 2' and so on. Thus, the sum of 2 m terms is greater than U+xJ+H + 2+ ...to m terms; that is, greater than l+m/2. We can therefore take n so large that s n shall exceed any assigned number ; that is, the series is divergent. TESTS OF CONVERGENCE. 381 Ex. 2. The series , , , is convergent if a>l, divergent if al. Group as in ex. 1, beginning with the second term. 11114 / 1 \ 2 and so on. Hence the series is less than v3 which is a G.P. with common ratio less than 1 and therefore convergent. The given series is therefore also convergent. (ii) a^l. The case a = l is that of ex. 1. When a\. The test fails to discriminate if p = 1. (i) /)m the ratio u n+ i/u n shall differ from p by as little as we please and therefore shall be less than a proper fraction r. If m be so chosen we have u m+l < u m r ; u m+2 < u m+l r < u m r- u m+3 < u m i* and so on. Hence, after the term u m , each term of the series is less than the corresponding term of the G.P. Since r is less than 1 the G.P. and therefore also the given series is convergent. (ii) p>1. In the same way the series may be proved divergent when ^>1. COR. The remainder r m of the given series is less than u m r + u m r- + ... , that is, u m r/(l r). 0? 3? Ex. 3. 1+ x + + + . . . (.r positive). z o u n +i x* x"- 1 n-l IT1 _ _ _ . rt - /y. ^^ *c- . p J'. 11 n n-l n Hence the series is convergent if .rl ; if .r=l the series is the harmonic series and therefore divergent, 382 AN ELEMENTARY TREATISE ON THE CALCULUS Ex. 4. l+#+f, + 5 T +... (.r positive). The series (the exponential series, 49) is therefore convergent for every positive value of x. It will be seen immediately that we may suppose x to be either positive or negative. 150. Absolute Convergence. Power Series. THEOREM I. If a. series which contains both positive and negative terms is convergent when all the negative terms have their signs changed, it is convergent as it stands. For the effect of restoring the negative signs is to diminish both | s n \ and p r n \. DEFINITION 1. A series is said to be absolutely or uncon- ditionally convergent when the series formed from it by making all its terms positive is convergent ; that is, u^ -f u z + . . . is absolutely convergent when u v \ + [ u 2 \ + . . . is convergent. Any other convergent series is said to be conditionally convergent (sometimes semi-convergent). The converse of Theorem I. is not true ; the series Uj + u 2 + . . . may be convergent, and the series % | + 1 u 2 \ + . . . divergent (see ex. 1). COR. A series is absolutely convergent if the limit of u n +i/u n is numerically equal to a proper fraction. Absolutely convergent series are of special importance ; no rearrange- ment of the terms affects the sum. It is possible, however, so to rearrange the terms of a conditionally convergent series that the series thus arising shall be convergent, but shall converge to a different value or even shall be divergent. Hence the words " conditional " and " unconditional." (See ChrystaFs Algebra, vol. 2, chap. 26, 13). THEOREM II. If u 1( u 2 , u 3 , ... are all positive, and each less than (or equal to) that which precedes it ; if, further, the limit of u n is zero, the series is convergent. This series is called the A Iternating Series. We may write the sum of an even number of terms in the two forms ABSOLUTE CONVERGENCE. POWER SERIES. 383 8zn = Ol - The first form shows that s 2n is positive and increases with n, while the second form shows that S 2n is less than u lt because each difference is positive. Hence s^ n converges to a limit, s say. Again, s 2n+ i = s Zn + u< i n+\, and therefore, since L tt 2n+ i is zero, s-2n+\ and s- 2n have the same limit ; the series is there- fore convergent. COK. r n \ is less than u n+ \. Ex. 1. 1 The series satisfies all the conditions, and is therefore convergent, as was shown previously ( 148); but the series 1 + 2 + 3 + i + ... is divergent. THEOREM III. // the series u 1 + u 2 +... is absolutely convergent, and if each of the quantities v x , v 2 , ... is numerically less than a finite quantity c, the series U 1 v 1 4-u 2 v 2 +... is absolutely convergent. For, the terms of |u 1 v 1 | + |u 2 v 2 |+... are less than the corresponding terms of or Hence |u 1 i> 1 | + |ivy 2 |+... is convergent, and therefore v l + uv z + ... is absolutely convergent. ,-, sin4r ~T~ ~22~' ~3*~ ~42~ + "' The series - -_ + --^+... is absolutely convergent, and no sine is greater than 1 ; thus the series is absolutely convergent for every value of x. DEFINITION 2. A series of ascending integral powers of a variable, x say, of the form a + a l x + a#?+...+a n x n +..., ............... (P) where the coefficients are constants, is called a Power Series in x. It is with Power Series we chiefly have to deal ; the following theorems are important. 384 AN ELEMENTARY TREATISE ON THE CALCULUS. THEOREM IV. If the limit of a 11+ i/a n is numerically equal to 1/R, the Power Series (p) converges absolutely ivhen x is numerically less than R, hut diverges when x is numerically greater than R ; it may or may not converge when x is equal to R. For, disregarding the first term a , we have numerically u n+ i_a n+ i T u n+ i T a n+ i x X , lj --- Xi-i v n a n u n a n Jt' and the result follows from Theorem I, Cor. The following is a more general theorem : THEOREM V. // when x = R none of the terms of the series (P) exceeds numerically a finite quantity c, the series (p) will be absolutely convergent so long as x is numerically less than R. For, if we write (P) in the form a + ^R (x/R) + a 2 R 2 (x/I,') 2 + . . . we see that the terms of (P) are numerically not greater than the corresponding terms of the geometrical progression and therefore the series is absolutely convergent so long as x/R is numerically less than unity. The series (P) may or may not converge when x = R; if it does converge each term must, when x = R, be finite, and therefore it will converge absolutely when x is less than R numerically. Interval of Convergence. When a series whose terms are functions of x is convergent when a b, we may speak of (a, b) as the interval of convergence. Ex. 3. The series x - - + - r --- - + . . . 234 converges (conditionally) when x=\ ; therefore absolutely when 1 < x < 1 . It diverges when x = 1 and when | x \ > 1 . Ex. 4. The series ^~ 22+02-72+ converges absolutely when 1 ==#= 1, diverges when \x\ > 1. For both series ( 1, 1) is the interval of convergence, UNIFORM CONVERGENCE. 385 151. Uniform Convergence. When the terms of a series are functions of a variable x and the series converges within a certain interval it will be possible, for a given value of x within the interval, to choose n so that the remainder r n will be less than a given quantity. For different values of x, however, different values of n will usually be required to make the remainder less than the given quantity. Hence the DEFINITION. A series, whose terms are functions of a variable x, is said to converge uniformly within an interval if it is possible to choose n, say n = m, so that for every value of n equal to or greater than m and for every value of x within the interval the remainder r n shall be less than any given positive quantity e. We will indicate the variable by the notation u n (x), s n (x), r n (x), s(x). THEOREM I. If the series u 1 (x)+u 2 (x)+ ... is uniformly convergent when a^x^b, and if each term is a continuous function of x for the same range, the sum s(x) is also a continuous function for that range. Let x and x l be two values of the variable within the range ; we have to show that, given e, it is possible to take x l so near to x that the difference \s(x l ) s(x)\ shall be less than e. With the usual notation we have sfo) - s(x) = s n (ic 1 ) - s n (x) + r n (x^ - r n (x), and therefore | s(x l ) - s(x) | <| s^xj - 8 n (x) | + 1 r n (x^\ + 1 r n (x) |. First, since the series is uniformly convergent, we can choose m so that if n^m both \r n (x^)\ and j r n (x) \ shall be less than e/3. Suppose m so chosen. Next, s m (x) is the sum of a finite number of con- tinuous functions and therefore we can take x l so near x that Sn^xJ s m (x)\ shall be less than e/3. Combining the two results, we can take x l so near x that s(x l ) s(x)\ shall be less than three times e/3, that is less than e. G.C. 2 B 386 AN ELEMENTARY TREATISE ON THE CALCULUS. The theorem is thus established when x is within the interval ; the slight modifications required when x = a or b may be left to the reader. THEOREM II. A Power Series a, +a ll K + a, 2 x 2 + ... repre- sents a continuous function within its interval of con- vergence ( R, R) ; the function may, however, become discontinuous at an end of the interval. We will show that if Rnp n | + 1 Wip n+1 1 + shall be less than e, and therefore for this m we shall have |r n (o?)| less than e. But this is the condition for uniform convergency. The proof requires x to be within the interval. We refer to Chrystal's Algebra, vol. 2, chap. 26, 20, for the proof of the theorem (Abel's Theorem) that if the series is convergent when .v=R (or - R), the function represented by the series is continuous up to and including the value R (or R) ; in other words, the value of the function when x = R is the same as that of the series when x=R. The method by which the uniform convergency of the power series was established is easily extended to prove THEOREM III. If the terms of a series are continuous functions of x when a^x^b, and if they are numerically less than the corresponding terms of an absolutely con- vergent series, whose terms do not contain x, the series will be uniformly convergent for the same range. The student must not mix up uniform and absolute convergence ; a series may be uniformly and yet not absolutely convergent, though such series are rather beyond our limits. The theorems contained in Examples 9, 10, 11 of the following Exercises should be specially noted. CONTINUITY OF SERIES. 387 EXERCISES XXXIII. L Show that the following series are convergent (i) l+2- 2 + 3- 3 + 4- 4 +...; (ii) (iii) ]/(a+l)+l/(+2) a + l/(a + 2. Show that the following series are divergent : (i) *+* + *+-; (ii) (iv) 2(+ l)/(n+ 1) ; (v) 2 ( 3. If prove (i) + 2 + + ... =ic; (ii) _ + _+. The value of c is 7r 2 /6 (Exercises XXXIV., 22). 4. Show that the series (the Binomial Series) is absolutely convergent for every value of m when |#|<1, but divergent when | x \ > 1. For tt^ 5. Show that if f(n) is a rational integral function of n, the series /(w)# n is absolutely convergent when |#|<1, but divergent when ~Letf(n)=an r + bn r ~ 1 + ..., the degree of /() being r ; then 6. If the series 2a, 26 are absolutely convergent, show that the series (ii) 6isin.r+6 2 8in 2# + 6 3 sin 3x+... are absolutely convergent for every value of x, and represent continuous functions. It follows that if (i) [or (ii)] represents a discontinuous function, 2a (or 26) cannot be absolutely convergent. 7. Show that if x>0, the series e~*cos (x a t ) + e-^cos (2x - 04) + e'^cos (3x - a 3 ) + . . . represents a continuous function. 8. Show that if x^O, and if 2a is absolutely convergent, the series a 1 e~ x cos (x - aj + a 2 e~ 2z cos (2x 04) + a 3 e~ 3 *cos (3x - a 3 ) + . . . represents a continuous function. 388 AN ELEMENTARY TREATISE ON THE CALCULUS. 9. If the power series a + a l x+a^sc 2 +... is zero for every value of x in the interval ( ft, R), show that every coefficient is zero. When .r = 0, the series reduces to the term ; therefore a = 0. We now have =.v(a 1 + a 2 x + ...)=xf l (x) say. Hence, either x=0 or f l (x) = Q. Suppose .r=|=0 ; therefore /j(.^)=0. But/i(jt) is a continuous function, and therefore the limit of f^x) for x=0 is equal to the value of f^(x) for # = 0. Hence 0^ = 0. Similarly a 2 = 0, 3=0, and so on. 10. Theorem of Identical Equality. If the two power series a + a 1 x + a 2 x 2 + ..., b + b l x + b 2 x* + ... are equal for every value of x in the interval ( R, R), show that a n = 6 , j = 6 t , .... For we have Q = (a -b ) + (a 1 -b 1 )x+(a 2 b 2 )x 2 +..., and the results follow from ex. 9. 11. Multiplication of Series. Suppose the two series to contain only positive terms, and to be convergent when x^R ; let w n = a b Q + (a &! + a 1 b )x where the terms of w n are formed by multiplying s n and t n , no term of degree higher than n - 1 being placed in w n . Show that the limit of w n is *, the product of the two given series. A little consideration shows that The inequalities show that w 2 n or, what amounts to the same thing, that w n converges to st. Next, let s and t contain both positive and negative terms, and let them be absolutely convergent when \x\^R. Let {a + Q(x - a)}. . . .(8) If we suppose n to increase indefinitely the sum on the right of (6) becomes an infinite series, and if the limit of R n (x) is zero the series is convergent. Since f(x) and its first n derivatives are by hypothesis continuous, every derivative must remain continuous in order that it may be possible to suppose n to become infinite. We therefore have the THEOREM. If f (x) and all its derivatives are continuous for the range considered and if the limit of R n (x) is zero, the infinite series (9) derived from (6) by making n infinite, is convergent and represents the function f (x), that is, converges to the value f(x).* The series (9) is called Taylor's Series for f(x) ; when it is necessary to draw a distinction between (6) and (9) the former may be called Taylor's Formula. Of course all that has been said about Taylor's series applies to the particular case of it, Maclaurin's series * (10) The value of R n (x) given by (8) is called Lagrange's form of the remainder in Taylor's series. Another useful form of the remainder is obtained by writing (b a)Q * Cases may be constructed in which the series (9) is convergent and yet does not converge to the value f(x) ; such cases, however, may be safely assumed not to occur in ordinary work. REMAINDER IN TAYLOR'S THEOREM. 393 instead of (b a)' l Q in equation (1). The last term of equation (3) becomes simply Q and (b a)Q becomes Hence This form is called Cauchy's form of the remainder. If we put (b a)pQ instead of (6 a) n Q in (1) we get called the Schlomilch- Roche form of the remainder', p gives Lagrange's form and p = 1 gives Cauchy's. In (5) put x for a and x + h for 6 ; we get a value of f(x + h) that is often useful. We will now apply these theorems to the expansion of functions, and will usually employ Maclaurin's Theorem; the two forms of remainder to be used are the first being Lagrange's and the second Cauchy's. 153. Examples. 1. sinx. f(x) = sin x ; f(x) = cos x f'(x) = sin x ; f"(x) = cos x ; /'">(#) = sin x ; /""(#) = s Hence /(0) = 0; /(0) = 1; /"(0)=0; /" /'"> (0) = sin ^ ; /<> (^) = Since sin (mr/2) is or 1 according as n is even or odd, the coefficients of the even powers of x will be zero, and only odd powers of x will occur, the terms being alternately positive and negative. Thus x 1 x" . ( a mr\ T/ 394 AN ELEMENTARY TREATISE ON THE CALCULUS. Again, R n (x) = ~s and therefore is not greater numerically than x n jn !, which has zero for limit. We thus get the series _a?_ x 6 y?_ '~3! + 5l~7l + '"' which is absolutely convergent for every finite value of x. 2. cosx. In the same way cosx=l .+ + ..., the series being absolutely convergent for every finite value of x. 3. e x . f(x) = e*; fW(x) = e*; /(0) = 1 ; />(0) = 1 for every n. x 2 . x 3 . the series being absolutely convergeut for every finite value of x. 4. (l+x). f(x)=(\+x) m ; /->(j?) = m(ro-l). ..(i Hence fa -1)1. If m is a positive integer the series stops with the (m + l) th term, since f n) (x) = when n>m ; if m is not a positive integer we have to consider R n (x\ We take Cauchy's form, The infinite series m(m- 1) \ / i . converges absolutely if |ar|1 (Exer. xxxiii., 4) ; we therefore need only consider values of x such that |#| = 1. (A) \x\ <1. R n (x) may be written as the product of the three factors, (n-l)! The first factor is finite for every n since (1 + 0x) m ~ l lies between 1 and (1 +x) m ~ l . The second factor cannot exceed unity. The third factor has zero for limit, since it is the n th term of the convergent EXAMPLES OF EXPANSIONS. 395 Hence the limit of R n (x) is zero, and the infinite series converges to (1 +x) m for every value of m so long as - 1 0, but conditionally if 0>m> - 1 ; oscillating if m= - 1 ; divergent if m< - 1. x= - 1 ; series absolutely convergent if ra>0 ; divergent if m<0. If a-^b the binomial (a + b) m may be written a m (l + b/a) m or b m (l +a/b) m and then x put for b/a or for a/b according as b is less or greater than a numerically. 5. iogr(i+x). It is not possible to expand log.r by Maclaurin's Theorem since logo? is infinite when x = 0. We may expand log.r in powers of (x a\ if a is positive, using Taylor's Theorem, but it is simpler to take /(0)=0; log(l +*)=*- + -+ ...+^*)i The infinite series diverges if |#|>1 and if x= -1 ; we therefore consider the remainder for -1 <#^S1. For x positive, Lagrange's form shows that the limit is zero, since {x/(l + 0x)}" is never greater than unity and the limit of 1 /n is zero. For x negative, Cauchy's form shows that when | x | < 1 the limit is zero ; for the limit of x n is zero and the other factors are finite for every value of n. Hence where -Kx^l ; the series is conditionally convergent when x=\. We may note that, putting x=\, we get 6. Calculation of Logarithms. The series just found is too slowly convergent for purposes of calculation ; a more rapidly convergent series is got as follows. We have r L x 3 x* -*-+:-+..., ...................... 0) 396 AN ELEMENTARY TREATISE ON THE CALCULUS. and by writing - x in place of x *)--*---- ...................... (2) By subtraction we find, since log (1+^)- log (1 -.r) = log{(] +#)/(! -A 1 )}, Suppose x positive and let (1 + #)/(!- x) = Equation (3) becomes from which log (?/ + !) is found when logy is known. It may be noticed that (4) is not a power series in y. With very little labour the logarithms of the prime numbers 2, 3, 5, 7,..., may be found: thus Then log 4 = 2 log 2 ; log 5 is obtained by putting 4 for y ; log 6 = log 2 + log 3 ; and so on. Series (4) converges rapidly even when / = 2. For particular numbers special artifices may be used. Thus, if y = 49 equation (4) would give log? when log 2 and log 5 are known, the series being very rapidly convergent. The student is referred to Chrystal's Algebra, vol. 2, chap. 28, 11, for further information and references. 7. Huyghens' Rule for the Length of a Circular Arc. If a is the chord of the whole arc and b the chord of half the arc, then the length (1) of the arc is, approximately, (86 - a)/3. Let the arc subtend at the centre of the circle an angle of 6 radians, and let the radius of the circle be r ; then l=r& and ...} ................ (i) ...} ............... (n) Multiply (ii) by 8 and then subtract (i) ; we thus eliminate (P. Therefore f nf& , ....) 120x2^ Hence, neglecting the fourth and higher powers of 0, we find l = (8b a)/3. It may be shown that for an angle of 30 the relative error is less than 1 in 100,000, for an angle of 45 less than 1 in 20,000, and for an angle of 60 less than 1 in 6000. DIFFERENTIAL EQUATION FOR n* DERIVATIVE. 397 154. Calculation of the n* 11 Derivative. The practical difficulty in finding a power series by Maclaurin's Theorem lies in the calculation of f^ n) (x) ; indeed, there are few cases besides those already treated in which the n th derivative can be expressed in a manageable form. The discussion of the remainder, R n (x), is impossible unless we know / (n) (#) ; in special cases, however, we can find / (n) (0), and the infinite series, if it converges, will (in general) represent f(x) within its range of convergence. In this connection Leibniz's Theorem ( 68) will be found very serviceable. As an example consider f(x) = sin (a sin" 1 x). It would be difficult to calculate f (n) (x) directly ; we will, therefore, first calculate f'(x) and f"(x), and will then form a differ- ential equation to which Leibniz's Theorem may be applied and which will lead to the value of / (n) (0). f(x) = sin (a sin" 1 ;?) ; /(or^acosCasin- 1 ^)/^!-^ 2 ) ......................................... (i) 3 f'(x) - a 2 sin (a sin" 1 ^)^! -x?)+acos(a$in- l x). xj(\ -x 2 )*, = -a?f(x)l(\-x*) + xf(x)l(\-x*) ............................. (ii) and therefore (\-3r)f'(x)-xf(x) + d*f(x) = Q ................................ (iii) By making x zero in f(x), f(x\ f'(x\ we find /(0)=0; /(0)=;/'(0) = 0. The function on the left of (iii) is always zero and therefore its 7i th derivative is always zero. The function, being a sum of pro- ducts, may be differentiated n times by applying Leibniz's Theorem to each of its terms and then adding the results. For the first term let f'(x)=u, (\-3r)=v. Every derivative of v above the second is zero; the n* derivative of f'(x) is / n+2 '(#), the (n-l) A is / (n+1) (x), and so on. Thus, =(1 - In the same way D"{ xf(x) } = xf n ^\x) + /<>(*). Also, 2){a*f(x)} = a*f(*(x). ^ Adding we find, after a slight reduction, (1 - ,r 2 )/(+ 2 >(jc) - (2n + l)^/("+ 1 '(a:) - (n 2 - 2 )/""(.r) = ......... (iv) and therefore when x0 ) .............................. (v) 398 AN ELEMENTARY TREATISE ON THE CALCULUS. From (v) we find in succession all the derivatives above the second for x 0, since we know the first two. /W(0) = (2 2 -a 2 )/"(0)=0; and so on ; thus every even derivative is zero. Again, /<">(0) = (1 2 - 2 )/(0)=(l 2 - 2 )a ; /W(0) =(3 2 - 2 )/< s >(0)=(3 2 - 2 )(1 2 - a 2 )a, and so on, the general value being /<2-i>(0) = a(l 2 - 2 )(3 2 -a 2 )...{(2-3) 2 -a 2 }. Hence, a(! 2 - 2 )(3 2 -a 2 ) The series (vi) will terminate if a is an odd integer ; in all other cases it will not terminate. The ratio of the term in y? n + l to the term before it is and since the limit of this ratio is a? the series (vi) is absolutely convergent so long as For many purposes only a few terms of the development of a function are required, and the calculation of a small number of derivatives may always be effected with more or less labour. Thus, the first three or four derivatives of log(l+sina;) are easily calculated and the first three terms of the expansion obtained, x It is usually simpler, however, in cases like this to proceed as follows : suppose Substitute for y in the series b + b t y + b 2 y 2 + ... its value in terms of x and rearrange in powers of x ; the series obtained will be convergent for sufficiently small values of x. For example, y = sinx=x-- + ... ; log(l +?/)=# -- + ^+ , The proof of the method cannot be gone into here. DIFFERENTIATION AND INTEGRATION OF SERIES. 399 155. Differentiation and Integration of Series. The pro- perties of a function are often most simply investigated by using an infinite series which represents the function ; we must, therefore, see under what conditions a series may be differentiated or integrated term by term. The rules for differentiating and integrating a sum have been proved with the express limitation that the number of terms is finite ; their extension to infinite series requires justification. We begin with the theorem in integration ; e denotes as usual a given arbitrarily small positive quantity. THEOREM I. // the series u 1 (x)+u 2 (x)+ ... is uniformly convergent from x = a to x = b and converges to f(x), then the series f* f* I u l (x)dx + \ u z (x)dx + ..., where a^c _ _ > provided the series u 1 '(x) + u 2 '(x) + . . . is uniformly convergent from x = a to x = b. Let F(x) = u l '(x) + u z '(x) +...; then by Theorem I., since u^(x} + iL^(x)- i f ... is uniformly convergent, Cx Cx C \ F(x)dx=\ Ui(x)dx+\ Jc Jc J = {UI(X)-UI(G)} = {u l (x) + u 2 (x) = f(x) constant. Therefore f(x}- that is, F(x)=f(x). By 151 Theorem II. we see that a power series may be integrated term by term if x is within the interval of convergence. We will now show that the series obtained by differ- entiating the power series is uniformly convergent when, in the notation of 151, Theorem II., R The series (A : ) is called Gregory's (sometimes Leibniz's) series for TT ; it is too slowly convergent, however, to be suitable for calculation. A better series is got by using Machines formula, namely It will be a good exercise to calculate IT from this formula by using the expansion (A); the series for tan -1 (l/5) and tan -1 (l/239) converge rapidly and give TT to 5 or 6 decimals with little labour. 2. sin x x. If 1 <#< 1, we get by the binomial expansion 1 ,_i 1,1.3, 1.3.5 -^-^ = (1-**) ^1 + -^ + ^ + ^ and therefore, integrating from to x, ,1s 3 .!. 3 a* . 1 . 3 . 5 a* . 1 *~* + 2~3 + 2T4 5 + 2.4.6 7 + "" Q.c, 20 402 AN ELEMENTARY TREATISE ON THE CALCULUS. The following example shows how we may obtain an approximate value of an integral by means of a series : 3. The time of a complete oscillation of a simple pendulum of length I, oscillating through an angle a on each side of the vertical is 4K*J(l/g) where ft d$ , - 1 = I -77= jT . 2 ,;, = sm;ra ; Jo ^/(1-Fsm 2 ^))' 2 K to find a series for K. Expand (1 -Fsin 2 <)~* by the binomial theorem, and then integrate term by term. We have 2 The integrals of sin 2 <, sin 4 , ... are given on page 285 ; therefore When a is so small that 2 , &*, ... may be neglected, jfiT=7r/2 and the period is 4. To evaluate \ _ co&rxdx ( r a positive, integer) Jo l-2acos#+a 2 v If | a | < 1 we have by ex. 13 Exer. XXXIII. 1-a 2 Also / cos^cosr^7^=0 if n=\=r Jo Therefore, when the aeries is multiplied by cosrx and integrated, every term will vanish except that arising from 2a r cos rx cos rx ; we thus get f* cosrxdx _ ira r ... Jo 1 2acos#+ 2 1 a 2 ' If | a | > 1 we have 1 2a cos x + a 2 a 2 . .1 1 1 -2-cos.rH - a or and we may expand in powers of I/a ; or, we may write I/a for a in (i) and then multiply by I/a 2 . We find for the integral 7ra- r /(a 2 -l). APPROXIMATIONS BY INTEGRATION OF SERIES. 403 EXERCISES XXXIV. 1. Prove that the following expansions hold for every finite value of x : y3. y3 (i) sin(.r+a) = sina+.rcosa- sin a- cosa+ ... o \ .... 2x* 2 2 ^ , mr x" , (u) e*cosx=l+x- --_-. ..+2* cos + ..., (...,_. , . *-*/ ^ *> . *. . mr x . in) e c smx=x+x*+j --^ -... + 2* sin f+---, _^2 ^,3 (iv) e rcosa cos(.sina) = l +a;cosa + cos2a+^- cos3a + .... 2t \ o 1 Show that />"e a:cosa cos(j7sina)=e ICO0 cos(j;sina + 7za). 2. From 1 (ii), (iii) derive the expansions of cosh x cos x, sinh JP sin x, cosh # sin .r, sinh x cos #. 3. Prove that if |a?|l, the limits of the functions "for # = being taken as the values for 404 AN ELEMENTARY TREATISE ON THE CALCULUS. 7 Tf X 1 ir I - * r 2 2 r* I 3 r6 gZ _ 1 - ^-t-^l 4! 61 show that #1 = 1/6, 5 2 = 1/30, .83=1/42.... The numbers B v JB 2 , . . . are called Bernoulli's numbers (see Chrvstal's Alg., vol. 2, 6hap. 28, 6). 8. Show that 2 v 9. If /(^) = (sin- 1 ^)/ x /(l -tf 2 ), show that and that if |#|<1, sin- 1 ^ 2 3 2.4 , 2.4.6 //I ,>'^\ ' ^ * ^ F\ ft R *y ^ I A ~~ */ I O O.t/ O v * I 10. Show from ex. 9 that (i) 0=si z f 2^ 2 24/2 2 \ 2 1 Put.^=sin0, tan 0=z. 11. Deduce from ex. 9 by integration that, if |#|<1, 2* * ft 51 ^. ft ^^ t O ^t O . O O 12. Show that cos (a sin" 1 x) satisfies equation (iii), 154, and prove the expansion (|#|<1), . _j v_ 1 _o 2 ^ 2 a 2 (2 2 -a 2 ) , a 2 (2 2 -a 2 )(4 2 -a 2 ) . 2!^ 4! 6! 13. Prove from the series for sin(asin~ 1 ^) and cos(asin~ 1 .r) that d_,v,o,'., /J_^(^!_- 12 ) Q ;, -/-2_12\/^,2_Q2\ (H) 4 ! Series for cos mO/coa and sin mO/cos 6 ma} 7 be obtained by differ- entiating sin(asin -1 ^) and cos(asin~ 1 j7). 14. Show that if x \ < 1, l *? 1-3 & 1.3.5. r' EXERCISES XXXIV. 405 15. Prove that if \x\ <1, (ii) To prove convergency, note that both in (i) and in (ii) the series formed of the odd terms and the series formed of the even terms are separately convergent or divergent according as \x\ is less than or greater than unity. 16. Show that, with the usual notation, the perimeter of an ellipse is ft 4a/ 4/(l-*Bnty>ty /_ (l\ 2 e? /1.3\V /1.3.5\e \ -*(!- (jj T-UTiJ s-vroJ i / 17. Prove (i) the perimeter of an ellipse of small eccentricity e exceeds that of a circle of the same area in the ratio l + 3 4 /64 approximately ; (ii) the surface of an ellipsoid of revolution (either prolate or oblate) of small eccentricity e exceeds that of a sphere of equal volume by the fraction 2e 4 /25 of itself. 18. Show by integrating (cos +#)/(! + 2#cos0+.r 2 ) first with respect to x, and next with respect to 6 see Exer. XXXIII., 12), that if \x\ 1 ; x&mxdx TT (iii) I cos rx log ( 1 2a cos x + a 2 ) efa- = - 7ra r /' t if a < 1 Jo = -Tra~ r /r if a>l ; r* sin .r sin rx dx ir , , .. - = -a r - J if a#> z) put for x, y, z the values just written, and expand by Taylor's Theorem ; therefore 0=/(A, A, l)+ r (\f k + K f t + v f t ) + Ai* + .................. (ii) But f(h, , = 0j since the point P is on the surface ; therefore one value of r given by (ii) is zero. The other roots of (ii) are the distances from P to the several points in which the line (i) meets the surface. Let r t = PQ ; then (ii) becomes, since r is not zero, = A/+/*/* + v /i + ^r 1 + ............................. (iii) As r, tends to zero the line (i) tends towards the position of a tangent line ; but (iii) shows that as r^ tends to zero, so does Hence the line (i) will be a 'tangent line if A, /i, v satisfy the equation V + M/H-v/,=0 ................................. (iv) If we eliminate A, /z, v from equations (i) and (iv), we shall obtain an equation which is true for the coordinates of any point on any tangent line through P. The result of the elimination is (*-*x+(r-J)/ft+(>-2),/t- It is easy to see that c) 3 \ ra and that the theorems may be extended to homogeneous functions of three or more variables. For example, xu x +yu y -\-zu z nu .............................. (iii) Ex. Let M=tan~ 1 (y/^); then u is of zero-degree. 1 y '_ y x = ~~ = ' _ 0. 159. Maxima and Minima of a function of two or more Variables. DEFINITION, /(a, 6) is said to be a maximum value of f(x, y} iff(a + h, b + k) is less than /(a, 6) for all values of h and k, positive or negative, that lie between zero and certain finite values however small ; /(a, 6) is said to be a- minimum value of f(x, y) if f(a+h, b + k) is greater than f(a, 6) for all such values of h and Tc. Similar definitions hold for functions of more than two variables. MAXIMA AND MINIMA. 413 We will assume the continuity of the functions and their derivatives for all values of the independent variables considered. A necessary condition that f(a, b) should be a maximum or a minimum (a turning value) is that both f x and f y should be zero when x = a, y = b. For f(a, b) cannot be a turning value of f(x, y) unless it is a turning value of the function f(x, b) of x alone when x = a and also a turning value of the function /(a, y) of y alone when y = b', there- fore f x (x, 6) vanishes when x = a and f y (a, y*) when y = b. To investigate sufficient conditions expand f(a + h, b+k); we get f(a+h, b+k)-f(a, b)=l;(h*f aa +2hkf ab +k% l> ) + R...(l) where the terms hf a , kf b are omitted since f a = Q,f b = Q when /(a, 6) is a turning value. If /(, 6) is a turning value the expression on the right of (1) must retain the same sign for all small values of h and k, the negative sign for a maximum value and the positive sign for a minimum value. Now R contains h and k in the third degree, if we suppose R to be the remainder in Taylor's Theorem ; it seems natural therefore to assume that, for sufficiently small values of h and k, the sign will be that of the quadratic expression in h and k. Yet this assumption is not sound as the following example, given by Peano, will show. Let f(x, ) = ar 2 -6^/ 2 +y*; tnen a = 0, 6 = 0, /(a, &) = 0, and equation (1) becomes f(h, ) = 8A 2 + (-6AF + ^) ......................... (2) Here we have R exactly, R=-6hk 2 + k i . The terms of second degree reduce to 8A 2 , and are therefore positive so long as h is not zero. Yet f(h, k) is not of the same sign for all small values of h and k. For let k=J(Xh) y and we find Hence f(h, k) is positive or negative according as A does not or does lie between 2 and 4. In other words, /(O, 0) is not a minimum value of /(#, y\ even though the terms of second degree are positive except when h 0. The difficulty just noticed would require a fuller con- sideration of the remainder in Taylor's Theorem than we 414 AN ELEMENTARY TREATISE ON THE CALCULUS. have room to give. We therefore simply state that f(a, 6) will be a turning value if faafbb > (fab) Z , and the value will be a maximum if f aa (or / 66 ) is negative, a minimum if / aa (or f bb ) is positive. It may be seen that a necessary condition that /(a, 6, c) should be a turning value of f(x, y, 0) is that f x ,f y ,fz, should all vanish when x = a, y = b, z = c. In many cases it is known that a turning value of a function must exist ; it is usual to assume without further proof that the values of the variables that make the first derivatives vanish are those that give the turning value. 160. Examples. The most important cases are those in which the function whose turning values are required is given as a function of two or three or more variables, the variables being connected by one or more equations of condition. The best method of proceeding in such cases is usually the following. Let the function be u and let there be, say, four variables with two equations of condition, u=f(x,y,z, w)(l); 0(o;, y, *,?)= 0(2); \js(x,y,z,w) = Q (3). Suppose for the moment that z and w are found from (2) and (3) in terms of x, y, and that these values are sub- stituted in (1) which thus becomes a function of two independent variables x, y; let D x ii, D y u denote the first derivatives on the supposition that the substitutions have been made. For a turning value D x u and D y u must both be zero. Now ^ ,, j, "dz * d>w .. A*-/.*/^*/.^, .................... (4) and dz/'dx, 'dw/'dx are found by differentiating (2), (3) ; thus 'dz n /KX = 0. ..(5); Instead of solving (5), (6) for 'dz/'dx and 'dw/'dx, multiply (5) by X, (6) by /u. and add to (4) ; therefore UNDETERMINED MULTIPLIERS. 415 D X U =f x 7)2 ?)W ^ + (f~ In exactly the same way we find ........ (8) It will be noticed that the coefficients of 'dz/'dx and 'dwj'dx in (7) are respectively equal to those of 'dz/'dy and 'dw/'dy in (8); therefore choose the multipliers X, /j. (and this is in general possible) so that these coefficients are zero, and the values of D x u, D y u will reduce to the first three terms of (7), (8) respectively. For the turning values of u the derivatives D x u, D y u are zero ; therefore for the turning values we have the four equations, y=Q,} ..... w = 0, J and these four equations together with equations (2), (3) are just sufficient to determine A, /u. and the values of x, y, z, ^u that give the turning values of u. The equations (9) are symmetrical in x, y, z, w, and this method, called the method of undetermined multipliers, is specially simple when the functions /, , i/r are homo- geneous. We have taken four variables and two equations of condition, but it is clear that the reasoning is quite general. We may state the rule for writing down the equations (9) thus : Form df-\-\d(f>+fji.d\ls and equate to zero the coefficients of dx, dy, dz, dw. Of course df means f x dx+f y dy+fzdz+f w dw and d, d\[s have like meanings. Ex. 1. M=# 2 +y 2 +z 2 (l); (J3 = ax+by+cz-k=0 (2) Clearly u has a minimum value ; for, by (2), x, y, z cannot be simultaneously zero and u is always positive. Now dv + (2y + Xb)dy + (2z + \c)dz, 416 AN ELEMENTARY TREATISE ON THE CALCULUS. and therefore, equating to zero the coefficients of dx, dy, dz, we find for the values of x, y, z that make u a minimum x/a = - A/2 =y/b = zjc. By (2) each of these fractions is equal to /(a 2 +6 2 + c 2 ), and then by substitution for x, y, z in (1) we see that the minimum value of u is The student, may also solve the example by replacing z in (1) by its value (k ax- by)/c deduced from (2) ; he must be on his guard against confounding the value of u x in this method with the value of u x in the first method. Ex. 2. Find the turning values of u when u=a?x*+bY+cW, ................................. (1) and x 2 +y 2 + z* = l ; ....................................... (2) lx+my+nz = ..................................... (3) In this case u is really a function of only one variable, but the method of undetermined multipliers is equally applicable. To get rid of the factor 2 we take A, 2/ut, as the multipliers ; then we readily find = ........ (4) Multiply the first of equations (4) by x, the second by y, the third by z, and add ; then taking note of (2), (3),. we find a?x 2 + b*i/* + c*z 2 +X=Q, that is, A= -u, where u is now a turning value, since the values of x, y, z that satisfy (4) are those that determine the turning values. Put u for A in (4), and we get x = fj2l(u - a 2 ), y= fj.m/(u - 6 2 ), z = ^nj(u - c 2 ). If we now put these values of x, y, z in (3) the factor /JL divides out and we get a quadratic equation for u, 2 /(w-a 2 ) + m 2 /(-& 2 ) + 2 /(w-c 2 ) = ................... (5) One root of (5) will be the maximum value of u, and the other the minimum. EXERCISES XXXV. 1. Verify Euler's theorem on Homogeneous Functions (taking first derivatives only) in the following cases. (i) ax 2 + 2b.vy+cy*; (ii) ax 3 + by 3 + cz 3 ; (iii) (iv) (x+y)l(a*+f); (v) (x+y+ (vi) tan- 1 (/) where r=J(x 2 +y 2 + z 2 ); (vii) 1/r. 2. If u is homogeneous of degree n, prove (i) xu xx +yu xy = (n-\')u x ; (ii) x 3. Show that if a is positive ^n.vy x^y z is a maximum when x=a, y = a, but neither a maximum nor a minimum when ,v=0, y=0. EXEECLSES XXXV. 417 4. The function 3?y 2 (Q xy) is a maximum when ,r=3, ?/ = 2, but is neither a maximum nor a minimum when x=Q, y = 0. 5. Show that if a, b, c are positive, and if a\x + b/y + c/z = 1 , the sum x+y+z is a minimum, when Show also that if p, q, r are positive, the product x p y q z r is a minimum, when px/a = g'y/ft = rzjc p + q + r. 6. If u=x?+y z , and if o# 2 + 2A#y + 6y 2 =l, find the maximum and the minimum value of u^ and interpret the result geometrically. 7. If u=a?+y*+&, and if x 2 /a 2 +y 2 jb 2 + z 2 /(? = l, and lx + my + nz=0, find the maximum and the minimum value of u, and interpret the result geometrically. 8. If u=x* + xf + ...+x n \ and if a^x^ + a^x z + . . . + a n x n k, show that the minimum value of u is k^j(a^ + a 2 2 + . . . + a n 2 ). 9. If *', y, 2 are the perpendiculars from any point P on the sides a, 6, c of a triangle of area A, show that the minimum value of is 10. Show that the minimum value of (a r r + b 1 y + c 1 ) 9 - + (a& + b z y + c 2 ) 2 + . . . + (a n x + b n y + c n ) 2 is given by the values of x and y which satisfy the equations ,% + (26,0,) = 0. 11. Show that the centroid of n given points is the point, the sum of the squares of whose distances from the n points is a minimum. 12. Apply the method of undetermined multipliers to find the evolute of an ellipse considered as the envelope of the normals. The normal is cPxja. - b 2 y//3 =a?-b* where a Hence - + A=0, + X-=0, cr a 2 p 2 b 2 and therefore A = K 2 -& 2 ), a 3 =a%/(a 2 -6 2 ), etc. 13. Show that the envelope of xa m +yf} n =a m+l where a" IS o.c. 418 AN ELEMENTARY TREATISE ON THE CALCULUS. 161. Indeterminate Forms. A function f(x), that is in general well defined for a certain range of values of its argument, may for a particular value, a say, of its argument take a form (such as 0/0) that has no meaning. It is possible, however, that f(x) may have a definite limit A when x converges to a. Although f(x) is really undefined, has no value that can be calculated by the ordinary rules of algebra, when x a, yet it has become the established practice to call f(a) in such a case an indeterminate form, and to define A as the value of f(x) when x = a. The value thus assigned by the definition is usually called the true value of f(x) when x = a. If it be clearly understood that this " true value " is assigned by definition and is therefore arbitrary, there is a certain advantage from the procedure, namely, /(a?) becomes continuous up to and including the value a, it being sup- posed that f(x) is in general continuous. The typical indeterminate forms are 0/0; oo /oo; oo-oo; Ox*>; 0; 00; 1". We have already had some important cases of such forms; the derivative of f(x) is a case of 0/0. X oo is seen in x log x when x = ; the true value is zero. x n e~ x or x n /e x when x= + oo gives x x or oo /oo and the limit is zero. (See Exercises VII., 8, 9.) It is easy to see that the result holds whether n is integral or fractional. i 1 is the case of (l+x) x when # = 0; in this case the limit or true value is e ( 48 COR.). In many cases the limits are found most simply by algebraical transformations and the use of series. We will take one* or two examples before indicating the general theorems. Ex. 1. -~ when x= i . form Divide numerator and denominator by (x* - 1) ; we see at once that the limit is -3/2. The "true value" of the fraction when x=\ is therefore 3/2. Ex. 2. (sin" 1 a; x)jx? when #=0 ; form 0/0. Expand sin~ l x(=x+x 3 /6+ ...); x cancels in numerator, and after dividing numerator and denominator by .r 3 we get 1/6 as the limit. INDETERMINATE FORMS. 419 Ex. 3. sec .r/sec 3.r when # = 7r/2 ; form oo/oo. Let#=i7r u ; then T = Li - --- =-3. u =o sinu Ex. 4. cot 2 .r when #=0 ; form oc oo . 1 /, x \( x \ /sin A* - # cos ,r> y-COt*JC=(l +-. COS#)( ) 5 .'- \ sin^p /\sm^/\ x* ) The limit of the first factor is 2 and of the second factor 1 ; also 6 T r ... x\l _ -f: ...) = -^-+ . .. , \ & / o so that the limit of the third factor is 1/3. Hence the limit or true value is 2/3. Ex. 5. x*whenx=0; f orm 0. Let =.*; then logu=x\ogx. The limit of #log.r or logw is 0, as we have just seen, so that the limit of v or x* is 1. Ex. 6. (l/^) tana: when^ = 0; form 00. The logarithm of the function is tan x . , , - tan x log x= -- (x log x) 3C and has therefore for limit ; the limit of the function is therefore 1. 162. Method of the Calculus. We will now prove the general theorem for the evaluation of indeterminate forms, the continuity of the functions near the critical values being assumed. THEOREM. If 0(a) and t/r(a) are either both zero or both infinite, and if / ( x )/ 1 A''( x ) converges to a limit when x converges to a, then 0(x)/i/r(x) converges to the same limit. It will save repetition to observe at once that if '(x)l\l/(x) is indeterminate when x = a, the theorem shows that if d>"(x)l\ls"(x) converges to a limit then $'(x)/\f/(sB) and there- fore also < f>'( x )> ^( x ), ty'( x ) are continuous for the range a^x^b and if \js'(x) is not zero so long as a(x) = '(x l ) l- ^(x) ^'(xj l-0 Now, let c be taken so large that ^(^iV^X^i) differs from its limit J. by less than e x and let c be then kept fixed; 0(c), ^-(c) will, though large, be finite. Then let a; be taken so large (and this choice is possible since (x), ^r(x} tend to infinity) that the second fraction on the right shall differ from 1 by less than e 2 . The fraction (x)l\js(x) is now the product of two factors, the first of which differs from A by less than e l and the second of which differs from 1 by EVALUATION BY METHOD OF CALCULUS. 421 less than e 2 where e v e 2 may be as small as we please. Hence the limit of (x) x ^(J(x 2 2ax) when x = oo . 4. y{(x + a 1 )(x+a 2 )...(x + a n )}-x when #=oo. Put ^7=1/2 and expand by the binomial theorem. 5. (1 + l/xf and (1 + 1 Ix^f when a? = oc . _ log x , 6 - and , tan x x , tan nx n tan x , 7. -. and : : when x.=Q. x sin x n sin x sin nx 8. (^ 3Mtan# and x tan x - - sec x when#=^. 9. log(l +cw?)/log(l +bx) and (e *- -"*)/ log (l+bji) when x=Q. 10. - - - 2 log (1 +x) when x=Q. OC 30 11. (a x b x )/(c* g x ) when x = 0. log tan ax , log tan ax - log tan bx , ft j.ti. : i 7 diAvl ~i ; i 7 w iicii i*/ \/ log tan ox log sin a^; log sin bx 10 ,o... n , 13. ( - - when #=0. , cosher cos x , 14. , - and -- s - when x$. or x* 15. (cos a#) coi!eol!6:e and (cos ax) cose ^ bx when x=Q. 16. If the equation of a curve is u 2 +u s +u i + ... = where M 2 , %> ^4, , are homogeneous of degrees 2, 3, 4, . . . , in the coordinates, show that when the factors of u 2 are real, the equation 2 = gives the tangents at the origin. Put #=rcos#, y = rsinO, and let u 2 , n 3 , ..., become r z v 2 , r 3 z? 3 , ... ; then two values of r are zero since r 2 is a factor of 2 + u 3 + If, then, 9 be chosen so that v 2 tends to zero, another value of r will tend to zero. The equation v 2 =0 is a quadratic for tan 6, and therefore when its roots are real and different we get two gradients ; when they are real and equal we get one gradient ; when they are imaginary the values of tan 6 are imaginary, and the origin is then an isolated point. EXERCISES XXXVI. 423 DEFINITION. A point on a curve at which there are two distinct tangents is called a node. At a node two branches of the curve cross each other, intersecting at a finite angle. In Fig. 61, p. 312, and in Fig. 63, p. 313, the origin is a node. 17. If a 7 + 2x G y-\-5x 3 +a?x 2 6 2 y 2 =0, find the value of dy\dx when x=0,y=Q. 18. If, when x becomes infinite, $(x) converges to zero, show that when x becomes infinite <'(?), if it converges to a finite limit at all, will converge to zero. Suppose (f>'(z) has the limit A different from zero ; the equation shows that (x) must tend to infinity, because the term {x c)<^'(x^) tends to (x c)A, that is, to infinity. But this is contrary to the hypothesis that (x) tends to zero, so that if A is finite, it must be zero. 19. Show that the series (log 2) (log 3)* (log 4) is divergent for every positive value of a. Compare with 1/2 + 1/3 + 1/4 + ... ; the limit for n = 00 of 11 { -/ \ a : , that is, of \n a /lognj (log) a n is infinite ( 1.62, ex. 1). Hence the given series is divergent since the harmonic series is divergent. The series is obviously divergent when a is negative. CHAPTER XX. DIFFERENTIAL EQUATIONS. 163. Differential Equations. We propose in this chapter to discuss a few differential equations that occur in elemen- tary work. Nothing beyond the merest outline can be given ; the student will find ample treatment in Forsyth's Differential Equations (Macmillan) or Murray's Differential Equations (Longmans). An ordinary differential equation is an equation between one independent variable, one dependent variable and one or more derivatives of the dependent variable. A partial differential equation is an equation between two or more independent variables, one dependent variable and partial derivatives of the dependent variable. We deal only with ordinary differential equations. The order of a differential equation is that of the highest derivative contained in it ; the degree is that of the highest derivative when the equation is cleared of fractions and the powers of the derivatives are positive integers. Thus the equation is of the second order and of the first degree. The equation is of the first order and of the second degree. By the theory of elimination explained in algebra we can eliminate one quantity from two equations, two quantities from three equations, n quantities from (n+1) equations. Hence if an equation containing x, y and constants is differentiated once the new equation will contain x, y,y' and constants, and from the two equations one constant may be ORDER AND DEGREE OF EQUATION. 425 eliminated ; the resulting equation will be a differential equation of the first order and will contain one, constant fewer than the given equation. Similarly, if the given equation is differentiated twice, we shall have three equations from which two constants may be eliminated ; the resulting equation will be of the second order and will contain two constants fewer than the given equation ; and so on. The given equation is in each case called the complete primitive of the resulting differential equation and we see that the complete primitive contains one, two . . . constants that do not occur in the differential equation when that equation is of the first, second ... order. In the process of elimination no account is taken of the particular value of the constants; these constants may therefore be called arbitrary. Ex. 1. Let the given equation be y = Ax*+B, (1) and differentiate twice ; we find Dy=2Ax; (2) D 2 y = 2A (3) The first differentiation eliminates B ; we can eliminate A from (2) and (3), getting ar/fy-ZtyM) (4) Whatever be the value of .5, equation (1) represents a parabola with a given latus rectum I /A, and with its axis lying along the ;y-axis ; hence (2) is the differential equation of all such parabolas. Equation (4) again is the differential equation of all parabolas whose axes lie along the y-axis. Ex. 2. Let the given equation be (x-a? + (y-V)* = ^-a)+jo^sin(jp^-a)}. it T~^ -' As f .increases, the term Ce~ RtlL becomes of less and less importance ; the other term gives the steady oscillation. The steady oscillation may be put in the form where tan c^ =pZ/.ft. The quantity ^/(R 2 +p 2 L 2 ) is called the im- pedance of the circuit. LINEAR EQUATIONS. EXACT EQUATIONS. 431 Type IV. Exact Equations. The equation where M, N are functions of x and y, is- called an exact equation if Mdx-}- Ndy is a complete differential, that is, if 'dmfdy is equal to 'dN/'dx ( 94). In this case there exists a function u such that du = Mdx + Ndy ^ and, obviously, the integral is u = constant. Here M=2xy-y 2 + %x, N=x 2 2xy + 2y, and DM/'dy = 2x 2y = 'dN/'dx, so that the equation is exact. Knowing that the equation is exact, we can readily arrange Mdx + Ndy as a sum of complete differentials ; Qxydx + x 2 dy) - (y 2 dx + Ixydy) + Zxdx + 2ydy, that is, d(x 2 y) - d(xy 2 ) + d(x 2 ) + d(y*\ so that u = x 2 y xy 2 + x 2 +y 2 , and the integral is x*y - xy 2 -\-x 2 +y 2 =C. Ex. 6. x 3 2y 2 + 2xyDy = 0. This equation is not exact, but it becomes exact when multiplied by 1 1 a?. We find ^,^t 2 y and the integral is (x 3 +y 2 )/x 2 =C, or, The factor l/x 3 which makes the equation exact is called an inte- grating tfactor ; when an equation is not exact it may be possible to guess an integrating factor and thus integrate it. 166. Equations of First Order but not of First Degree. Let Dy be denoted by p; the equation, when of the n th degree, will have the form Ap n +Bp n - l + ... +Kp + L = Q (1) where A, B, ... are functions of x and y (or constants). If possible, solve for p; there will be in general -n^values p=p 1 ,p=p 2 ,... and each of these equations when integrated will give a relation between x and y that will satisfy (1). Ex. 1. xyp 2 -(x 2 +y 2 )p+xy=0. Therefore P=yl x or P= x ly-> and these equations have as integrals 432 AN ELEMENTARY TREATISE ON THE CALCULUS. Ex. 2. Clairaufs Equation, y=xp+f(p) ..................................... (i) This equation is of a special form and is integrated thus : Differentiate (i) as to x, and we find Hence either dp/dx = 0, that is, p = constant = C ; or #+/(p) = ................................... (iii) The substitution of C for p in (i) gives the complete integral '. ............ (iv) On the other hand, if p is eliminated between (i) and (iii), we shall get a relation between x and y that will satisfy (i). This relation is not obtained by assigning a particular constant value to C in (iv), and is called a Singular Solution. The singular solution is in fact the envelope of the family of lines (iv) ; for if we eliminate (7 between (iv) and x + f'(C) = Q, we clearly must get the same equation as that called the Singular Solution (we have simply interchanged C and p). As we have seen ( 145), the gradient of the envelope is the same as that of the family (iv) at their points of meeting. For example, the complete integral of y=xp + afp is and the Singular Solution is given by y 2 =4ax. 167. Equations of the Second Order. Type I. D 2 y=f(x), a function of x alone. Integrate twice with respect to x ; two constants will be introduced. Type II. D z y =f(y\ a function of y alone. Multiply by Dy; then since Dy D*y = It may now be possible to integrate this equation of the first order. Ex. 1. The equation of motion of a simple pendulum of length I is lQ= -#sin 8. To integrate, multiply by 0, then When t = 0, let = o, 0=0 ; then C= gcosa TYPES OF EQUATIONS OF SECOND ORDER. 433 and the negative sign being taken because 8 decreases as t increases. If we put sin|0=siniasin , we get after reduction dt = I(L >~ " \ \ g The integration cannot be carried further by means of the elementary functions, but t may be expressed by an infinite series. The value of t for the quarter period is KJ(ljg) [ 156, ex. 3]. In general, <=.. /I - II - d $ Type III. D z y =f(Dy), a function of Dy alone. Let Dy = v and we get Dv =f(v) and it may be possible to find v, and then y. Ex. 2. The equation cIPy= {1 +(%) 2 }t gives (p. 276) .r = cvj(\ + z> 2 ) + a (constant). Then Dy = v= (x-a)/J{(?-(x-a) z }, y= T V{ 2 - ( x a ) 2 } + b (constant), 168. Linear Equations. The typical equation of the second order is D*y+PDy + Qy = R ..................... (1) where P, Q, R are functions of x alone (or constants). The complete integral of all linear equations is the sum of two functions : I. The Complementary Function (C.F.) which is the complete integral of the equation when R (or in general the term independent of y and its derivatives) is zero. This function will contain two . (when the equation is of the 7i th order, n) arbitrary constants. II. The Particular Integral (P.I.) which is any solution whatever of the equation as it stands. This function contains no arbitrary constant. We prove the proposition for equations of the second order, but it is easy to see that the reasoning is general ; for the equation of the ?t' h order there will be n functions like , v, and n constants. If y = u and y = v satisfy D*y + PD V + Qy = 0, ................................ (2) O.C. 2 E 434 AN ELEMENTARY TREATISE ON THE CALCULUS. so does y = Au + Bv where A t B are constants. For if then also D\Au + Bv) + PD(Au + Bv) + Q(Au + Bv) = Q, and therefore Au + Bv satisfies (2), and since it contains two constants it is the complete integral of (2). Next, if yw is the particular integral, that is, if w verifies equation (1) and if Au + Bv is the complementary function, then Au + Bv + w will satisfy (1). For when y = Au + Bv + w, D*y + PDy + Qy = D\Au+ Bv) + PD(A u + Bv) + Q(Au + Bv) The first line on the right is zero, and, since w satisfies (1), the second line is equal to R. This value of y therefore satisfies (1), and since it contains two constants it is the complete integral of (1). The only equations we consider are those in which P, Q are constants. 169. Complementary Function. The equation to be integrated is D*y + aDy + by = ...................... (3) I. Let y = e* x (X constant) ; then (A 2 + aX + 6)e A * = 0. If therefore A is a root of the equation (the auxiliary equation) X 8 + o\ + 6 = ....................... (4) e* x will satisfy (3). The two roots \ v X 2 of (4) are and e* lX , e^ are two solutions of (3) Hence the complete integral of (3) is y = Ae^ x + BeW=e-& x (Ae nx +Be~ nx ) ............ (5) where n = *J(^a 2 6). We must however consider special cases. II. If a 2 = 46 equation (4) has two equal roots, namely X 1 = X 2 = ^a. In this case (5) becomes and there is only one distinct constant, for we might obviously replace A +B by C. When a 2 = 46 let y e-^u and (3) becomes, after reject- ing the factor e'^, jyz u _ Q of which the complete integral is u = A + Bx, THE COMPLEMENTARY FUNCTION. 435 Hence the complete integral of (3) when the auxiliary equation has two equal roots, each = \a, is y = (A+Bx)e-* ax (6) III. If a 2 < 46 the roots of (4) are imaginary. Again, let y = e -iax u anc j equation (3) becomes D 2 u+m 2 u = (7) where \cf b= m 2 and m is real. Now (7) is satisfied by u = cos nix, u = sin mx ; its complete integral is thus u = A cos mx + B sin mx, and therefore the complete integral of (3) when 2 <46 is y _ e -\ax u _ e -$ax(A cos mx + B sin mx) (8) We shall now show how to write down (5) and (8) when the roots of (4) are known. Let i denote as usual x /( 1). When the roots of (4) are real let ^o, 2 6 = n 2 ; the roots then are and the solution is y = e-l ax (Ae nx +Be-). When the roots of (4) are imaginary let \a z b= ri 2 ; the roots then are \a + ni, \a ni, and the solution is y = e - i*( A cos nx + B sin nx), so that instead of e nix , e~ nix we have eosnx, sin we. It should be noticed that the auxiliary equation is ob- tained by replacing D by X and rejecting y. Ex. 1. Py + 7Dy 8y=Q. Aux. Eq. A 2 + 7A 8 = 0; A 1 = l, A^ 8. Solution y = Ae*+Be- s *. Ex. 2. D s y + 2Dy + IOy = Q. Aux. Eq. A 2 + 2A + 10 = 0; A 1 =-l+3^, A 2 =-l-3z. Solution y = e~ x (A cos Zx + B sin &r). Ex. 3. D*y - ZlPy + 5D 2 y - 8Dy + 4y=0. Aux. Eq. A 4 -- 2 A 3 + 5 A 2 - 8A + 4 = ; The equal roots \ lt A 2 give (A +Bx)e t ; the imaginary roots 2i, - 2i give ^cos 2x + Fsin 2.r. Hence the Solution y = (A+Bx)e x +Ecos2x + Fsin2x. 436 AN ELEMENTARY TREATISE ON THE CALCULUS. 170. Particular Integral. The most important practical cases are those in which R is a sum of terms of the form Le*, L sin ax, L cos ax ; the simplest method of finding a particular solution is by substitution. Equation ( 1 ) is now D 2 y+aDy + by = R ........................ (9) I. R = Le *. Let y = Ce** and try to find C so that equa- tion (9) shall be verified. We find 0(a 2 + aa + 6) e* = Le**, and C#* will satisfy (9) if C = L/(c? + aa + b). There are exceptional cases, however. I. (a). If a is a root of the auxiliary equation (4) then and the value of G is infinite. In this case try Cxe"* or Cx^e * according as a is a single or a double root of the auxiliary equation. Ex.1. D 2 y Aux. Eq. A 2 -2A. + 1=0; A = l twice. To find P.I. take e* and e 2 * separately ; that is, since the coefficient of x in e* is 1, and 1 is a double root of Aux. Eq., try <7.rV, for P.I. corre- sponding to M' 2 , the roots of (vii) have the same sign ; both are negative. If we call them AH \2 and take the constants as A^ A 2 and B^ B 2 we get for the solutions of (iii), (iv), z=A l e-W + A 2 e-^ t , y = B l e-W + B 2 e~ ) * t ............. (viii) B l is connected with A lt and B. 2 with A 2 by equation (v) or (vi), that is, If P, Q are constants, the particular integrals are clearly x=P\R, y^QIS, and these added to (viii) give the complete integrals of (i), (ii). The only other important case is that in which P= E cos(nt - a) , (ii) v/15. 14. (i) * ; (ii) * 11. , v. . 15. The equation is equivalent to y+ 1= ^(x -3). Set VI., p. 60. 12. (i) an ellipse, (3, -4), 2a = 6, 2ft = 4; (ii) a hyperbola, (-11, 5), 2a = 4, 20. (i) a&/ v /(a 2 sin 2 + 6 2 cos 2 0) which may be written bljtl -e 2 cos 2 0) when (ii) (1 - (iii) 6(l + ecos(9)/ x /(l - CHAPTER IV. 32, p. 67. 4. The values of 5^/5^ are in order 331, 315-25, 303-01, 300-3001, 300-030001. 5. The values of 8y 1 /5x l are (i) -015038, -015077, '015100, -015107 ; (ii) -008594, -008661, '008701, '008713. 6. The values of dy l /Sx l are (i) '001332, '001334, -001335, "001336 ; (ii) -005950, '005990, '006028, '00603. 37, p. 74. 1. 0, \g, g, 2g. 2. -a/*, 2 CHAPTER VI. 53, p. 106. -10, -4,0, 2, 8. 57, p. 111. 2. 5K 4 , - 4/< 3 , 3/2^, -'*. 4. yf, %x%, ^Jx, - 1/x, - 3/4a^. Set VUL, p. 115. 2. 112Z-10. 3. 3a? 2 -4a;-5. 7. ^(a;"- 1 -^-"- 1 ). 8. m(ax m - 1 -bx-- 1 ). 9. 8a^+|"*-.J"*-fr*. 10. 2 11. (od-6cc + c? 2 . 12. - 13. 2{(a J B 6(( 2 2) - 17. Abscissae of turning points (a) ^, (fc) 1, (c) 0, 1. 18. (i)a: 2 -a:+C'; (ii) %x* + - + C; (Hi) ^ax 3 + ^bx* + ex + C. K 21. R 444 AN ELEMENTARY TREATISE ON THE CALCULUS. , grarl. = 1. 16. a, 6 - c< ; tan = (6 - c)/a. 18. /. 19. i 20. \f(u). Set X., p. 125. 3. yy l -2a (x + x l ); CHAPTER VII. Set XI., p. 131. OTT 2?r 1. 3(cos3x-sin3a;). 2. cos (x + 6). 3. m cos rax cos nx - n sin rax sin nx. 4. a: cos x. 5. a; sin x. 6. sin 2 .r. 7. cos 2 o;. 8. cos 3 x 0. sin 3 o:. 1O. |- (sin Sx + cos 3x). 11. -sin (arc + 6). 12. -tan (ax + b). 13. ^a; + ^sm2x. Ctf ft 14. Y# - sin 2#. ls - ~ T2 cos 6x - T cos 2x. 16. -2acos(ax + &)sin(ax + i). 17. tan(-|a:+l)sec 2 (ia;+ 1). 18. cos 2x/*J sin 2x. 19. sin x (3 - cos 2 x)/cos 4 a;. 2O. sin cc/( 1 + cos x)' 2 . 21. 2 sin x/( 1 + cos a?) 2 . 22. ( cos x - sin a; tan 2 a;)/( 1 + tan x ) 2 . 25. (i) t = -{(N+^)TT + e}, s = 0; (ii) t = -(Nw + e), s= a where N is any integer. 26. - a sin , 6 cos t ; tan = - (b/a) cot /. 28. fttan(x/6); (a 2 /26)sin(2x/6). 36. For cosa; the inequalities are not changed ; for sin x put > in place of <^. ANSWERS. 445 Set XII., p. 134. 3 5-! x 6. tan" J a; + 1 + * 2 " (-o). 8. -stan-M-To). 9. sin- 1 ! ^- }. \*\** / V \ V* ** / \*"/ Set XIII., P . 139. 1. l+logar. 2. a; B " 1 (l+nloga:). 3. cota-. 4. -tana;. 5. I/sin x. 6. 2/cosx. 7. 2/sinar. 8allfi^ -r\ Iv O ^ 1O l-r4-\\f x . ui\u ~jsjii]jc, v. ^ // o /j2\* *\*. \*L~ri)e . xe* 11. x"" 1 (a: + ra)e a: . 12. -2e' x smx. 1C ...... - . x-a 1 , 2a;-3 1 17. los(x + \'x~+ 1). 18. -e"*. 21. fi Set XIV., p. 146. 1. 28a^ - Gar 2 ; 84a? - 12a; ; 168x - 12 ; 168. 3. (.r 2 +])~ T . 3. 12a?-l2ax + 2a? ; 12(2ar-a), 4. i/=-(a:-l)- ! y() = ( - l)(w !)(x- l)-"- 1 + two similar terms. 6. y>= -2"- 1 cos(2a; + 7iir/2). 6. y(")=:a; 2 costf + 2vi:csin0-n(n- I)cos0 where = x + nir/2. 1O. e r {a: 2 + 27la: + Jl(7^-l)}. 12. Ex. 1. x = ^; Ex. 2. ar = 0. 13. y" = when x = - l/ x /3 or + l/ v /3. 23. - =^-v 24. 446 AN ELEMENTARY TREATISE ON THE CALCULUS. CHAPTER VIII. 70, p. 157. - 67 x 10- 6 , 50 x 10- 6 . Set XV., p. 159. 1. aF/OP 2 . 3. -dp/dx = gp. 4. - dNjdt is the time-rate of decrease of the number of lines that pass through the circuit ; or, the time-rate at which lines are withdrawn from the circuit. 6. E=RG+LdCldt. 6.X=-dE/dx. 7. (i) klogiVv/V}) ; (ii) (v 2 1 -' y - V v )/( 1 ~7) r (P\i ~ P-P^Ky- 1). CHAPTER IX. Set XVI. a., p. 176. 1. x -1, max.; x2, min. 2. x=l, max.; x = S, min. 3. x = 0, min.; x= -4/7 max. 4. 0;= -a, max.; a;= --jra, min.; x ^a, max. 5. a;= - 1, min.; x=l, max. 6. x - 1, max.; x=-%, min. 7. 0:= -1, min.; x=l, max. if a>0. 8. x= -1 min.; a;=l max. 9. x = -%a, max. 1 11. x=c, min. if 6>0. 12. No max. or min. 13. m m ti n \kj(m + n)\ m+n . 15. 16. (m 1 x 1 + m.2X z + ...)/(m l + m., + ...). 19. 20. abc/3J3; 3d?/(alc)% 2O. 26. Set XVI. b., p. 177. 1. ta.nQAB = blaJ2. 2. 3. 11. y fd; *fd. 12. f; ^-d. 13. 14. 0/^2. 15. AP:PB = a:b. Set XVI. c., p. 179. 1O. ~7~a. 12. 13. (i) tan0= A/( ~ ) ; (ii)A./(~); (^) ~* 17> ~* 18 - "' 19. e. 20. 5-. 21. ANSWERS. 447 Set XVII., p. 18-2. 1. Origin a point of inflexion on (i), (iii), (iv). a. x= r- *J3 3. x for points of inflexion (i) ia/^3; (ii) 0, ia^/3; (iii) ia/^/3; (iv)(), a x '3. 5. The origin. 7. (i) 0, IT ; (ii) ^, ^ ; (iii) 0, IT. 9. (i)* = 2.(ii)* = -L. 10. ^ r ~l 11. bx + c-20=nir (75, Ex. 4). CHAPTER X. Set XVIII., p. 201. 9. (l)$W-*i 3 ); (2)la 2 (-^); (4) V tan a (e 2fl2 cot a - e 2 * 1 cot ) ; (5) |a 2 [60 - 8 sin + sin 20]J. 10. |a 2 . CHAPTER XI. 91. p. 218. 8. oa^ + fcyiy + cz^l; (x- 3. 6y 1 y + cz,z = o; + a;, ; -(x -x 1 ) = (y-y l )lb 4. ax^ + fyiy + cz^O; (x-Xi)/ax l = (y- Set XIX., P. 239. a 3. - CHAPTER XII. Set XX., P. 253. 1. 2-137812. a. -226074. 3. 2-188920. 4. 2-588968. 5. -057014; 1-46765. 6. 1-895494. 7. -739085. 8. 1-1656; 4'6042. 9. 91 '964. 10. (i) 4-73004; (ii) 1-8751. 11. 5 '600 257, or in deg., 320 52' 16". 13. x =1-996, y=-909. Set XXL, p. 260. 2. 1-57 in. CHAPTER XIII. 111, p. 266. 1. **; ZJx; |(3x-4)^; %J(3x-4); sin' 1 -^. 2. 2 0; 1 ; log(6 2 /a 2 ); -log 3. 448 AN ELEMENTARY TREATISE ON THE CALCULUS. Set XXII., p. 269. ^/-a? + 51x + 1501og(o;-3). 2. x + log (2x - 1 ). 3. log (a*-3x + 2). 4. log\Z*/. 5. ^ log (X 1) 4\/4l 6. -757 tan" 1 I x,\- }. 1. isin 8. -5- log (2x + v4x 2 + 3). 9. -30; + -j- sin 2x. l O. i > sin ox T~ "r" sin :' '. 1 1 12. a; - j sin 2x + -$- sin 4o;. 13. ^ sin x - -^ sin 1x. 16. ir/ 4 - 1<7 - V 4 - la - T/8- 19. 2O. T lo g(D- 21 - W 6 - 22 - 7r / 2 - 28 - (iii) 4?ra6c/3. Set XXIII., p. 280. 1. -rrrtan'M ' ) 2. sin-M- -} 3. log (x - ^a + \/x 2 - ax). x /^o \ r,Ja I \ a, / 4. sin-M- 5 ) . I log (a 2 + a: 2 ). 6. Jla* + a*). 7. irlogl -5 -r )' 8. -r^tan' 1 /., 9. 1O. logsinx. 11. log ( 1 + sin x). 12. log (x + sin #). 13. ^ tan 3 a; - tan x + x. 14. J cot 4 a; + | cot 2 x + log sin x. 16. -rtan" 1 ! - tana; ) 16. - cos x + cos 3 x - cos 8 x + 7- cos 7 x, 17. - -5 cos 5 x + f- cos 7 a; - y- cos 9 a;. 18. tanx-cotx. 19. jsec 4 ^. 20. 2 x /(a-x){i(a-a;) 2 -|a(a-a;). 21. -Ji(x + 2a)J(a-x). 22. log (x + \/^T) - - /9 tan - J ( -- ~. ^ + * V <1 \ V fi 23. (i) 8/15; (ii) 8/315; (iii) w/ab ; (iv)log3; (v) | log 2; (vi) Tr/3^/3; (vii) 27r/3 x /3 ; (viii) ir/2. 24. ^-Iog(x 2 + a;+l)+ ,6tan- J P 1 -/^- 25. x-2ta,n- l x. v^ \ v rf / 26. ^(a:-l) 3 + 21og(x 2 + 2o; + 3). 27. f log (x 2 - 2) + 1- log (a^ + 2). 28. tt + 41o(z-l)-4(a:-l). 29. sin^o;- /l -x 2 . 3O. N /(x 2 - I) + log(x + \'x 2 - 1). 31. 32. v /(aa:-* 2 ) + ^sin- 1 /' 2 ^-^Y 33. ANSWERS. 449 4O. Y x + Ylg( s i na; + cosa:; )- 41 - fa: -i log (sin a: + 2 cos a:). 42. (i) 7T/4; (ii) ar/2; (iii) ir/ 4 5 U v ) "/(I-** 2 ) or ^/(r 8 -!) according as r 2 < 1 or r 2 > 1 ; ( v) a/sin a ; (vi) ir/2J( I - 1) ; (vii) y log 3. 43. (i)T/2;(ii) - 7r/2. 45. 8a 2 /15. 46. Each = 2a 2 /3. 47. |(2a 2 + 6 2 )T. Set XXIV., p. 288. 1. -(x+l)e-*. 2. -(^ + 3^ + 6^ + 6)6-*. 3. sin x - x cos x. 4. a; sin a; + cos x. 5. - ^ a: cos 2a; + 1 sin 2ar. 6. - a- 2 cos x + 2x sin x + 2 cos x. 7. -a 8 ). 19. ^(x + a) "25" (4 sin 4x ~ 3 cos 4x ^ 33. ^ (cosh a; sin x + sinh x cos x). 24. ^ (cosh a: sin x - sinh a; cos x). 35. 357T/256, 57T/16, 3ir/256, 4/35, 7ir/256, 13/15 - ir/4. 26. 7ra 4 /16, 7ra 3 /2, 57ra 4 /8. 27. (ir-2)a 2 /4. 28. ml n\l(m + n + \) !. 33. ml nil (m + n)\. 34. n-a 6 /32, (2l7r/32-28/15)a 6 . 36.32^/2/15. 37. iro 2 . 39. 4a(l-cos3. Homogeneous functions, Euler's theorems on, 412. Huyghens' rule for circular arc, 396. Hyperbola, definition and simpler properties of, 50, 54, 61. area of sector of, 289. curvature of, 359. evolute of, 371. rectangular, referred to asymp- totes, 54. tangent properties of, 124-128. Hyperbolic functions, 139-142. Hypocycloid, 369, 373. Hypotrochoid, 370. Identical Equality, theorem of, 388. Impedance, 430. Implicit function, 17. differentiation of, 119, 214. Increasing function, 104. Increment, 65. Indeterminate forms, 418. Inductance, 159, 430, 438. Inertia, centre of, 341. moment of, 343. The numbers refer to pages. INDEX. 457 Infinite, 60, 80, 195. series, see 'Series.' Infinitesimals, 195-200. Inflexion, point of, 35, 180, 239. Inflexional tangent, 35. Integral curve, 190. function, 188. Integral, complete, 426. definite, 263, 298-309. double, 334. general, 189, 262. geometrical representation of, 188, 263. indefinite, 262. limit of a sum, 324. line, 347. particular, 426, 433. related, 301. standard forms, 265, 278. surface, 347. triple, 338. see ' Approximations. ' Integrand, 262. infinite, 304. Integraph, 192. Integrating factor, 431. Integration, 262, 295. by algebraic and trigonometric transformations, 267. by change 1 of variable, 271, 340. by partial fractions, 268, 290. by parts, 281. by successive reduction, 284. of quadratic functions, 274. of trigonometric functions, 278. of irrational functions, 294. of rational functions, 292. of series, 399. along a curve, 318, 347. over an area, 337. through a volume, ;?38. Intercept, 31, 33. Intrinsic equation, 357. Inverse function, 17. differentiation of, 116. graph of, 44. Involute, 361. Isolated point, 313. Lagrange's remainder, 392, 409. Lamb's Calculus, 348. Laplace's Equation, 223, 235. Leibniz, 121. series for IT, 401. theorem on derivative of pro- duct, 144. Limits, 74-86. distinction between limit and value, 81, 405. theorems on existence of, 100, 377. of a definite integral, 263. Line integral, 347. Linear differential equations, 429, 433. function, 31. Lituus, 202. Lodge's Mensuration, 331. Logarithmic differentiation, 113. function, 57. series, 395. Logarithms, calculation of, 395. derivative of, 136. graph of 58. Liiroth, 257. Maclaurin's Theorem, 391, 411. Maclean's Physical Units, 70. Magnitu des dimensions of, 68. directed, 13. geometrical representation of, 13. Mass-centre, 341. Maxima and Minima, 166. elementary methods, 171. of functions of several variables, 412. Maxwell's Heat, 232. Mean- Value Theorems Derivative, 162, 419. Integral, 300. 309. Mean value of a function, 332, 339. Mechanics, see "Dynamics." Minima, see "Maxima." Moment of differential, 121. Moment of inertia, 343. Momentum, 150. Multipliers, undetermined, 415. Multiple-valued function, 17. Murray's Differential Equations, 424. The number* refer to pa/jes. 458 AN ELEMENTARY TREATISE ON THE CALCULUS. Napier's base, 59, 92. Newton, 109. his method of approximating to the roots of equations, 244. Node. 423. Normal, 123, 201, 216. ^"Number e, 92. r, 85, 401. Order of differential equation, 424. of infinitesimals, 195. Ordinate, 7. Origin of coordinates, 8. change of, 52. Oscillating series, 375. Osgood on Infinite Series, 375. Pappus' Theorems, 348. Parabola, definition and simpler properties of, 48, 54, 61. arc of, 127, 314. curvature of, 353, evolute of, 367, 371. semi-cubical, 127. tangent properties of, 124-128. Parallel curves, 361. Parameter, 365. Partial Derivatives, see " Deriva- tives, partial." Peano, 413, 421. Pendulum, period of oscillation of, 402, 432. Pericycloid, 369. Period of a function, 56, 303. Perpendicular, length of, 63. Plane, equation of, 209. tangent, 215, 411. Planimeter, 321. Plotting of points, 9. Points, conical, 218. distance between two, 9, 206. isolated, 313. turning, 24, 167. Polar formulae, 200. tangent, normal, etc., 201. Potential, 153, 223, 351. Power, fundamental limit, 91. derivative of, 111. Power series, 383. continuity of, 386. differentiation and integration of, 400. Primitive of differential equation, 425. Prismoid, 332. Proportional parts, 255. Radius of curvature, 35 \ . of gyration, 344. Rates, 65-73, 101. Rational fractions, integration of, 290. Rational function, 34. integration of, 292. Rediiction, successive, 284. Remainder in Taylor's and Maclaurin's Theorems, 392, 409. Ring, see "Anchor-ring." Robin's Tracts, 121. Rolle's Theorem, 161. Roots, nee ' Equations.' Schlomilch-Roche's form of re- mainder, 393. Segments, directed, 1. addition and subtraction of, 2,3. measure of, 5, 12. symmetric, 3. Series, infinite, 375. alternating, 382. differentiation of, 400. integration of, 399. multiplication of, 388. semi-convergent, 382. See ' Convergence of series,' ' Power-series. ' Sign of area, 186. Simpson's Rules, 330, 332. Simultaneous differential equa- tions, 437. ainx, sin' 1 ^, expansion of, 393, 401. Slope, 102. Solution of a differential equation, 426. singular, 432. Space-rate of change, 103, 150. Sphere, surface and volume of, 194, 309. Spheroid, oblate and prolate, 310. surface and volume of, 310. Spiral, of Archimedes, 201. The numbers refer to pages. INDEX. 459 Spiral, equiangular, 202, 360. reciprocal, 202. Stationary value, 105. Step, see ' Segments.' Subnormal, 123, 201. Subtangent, 123, 201. Surface, equation of, 209. of revolution, 193. areas and volumes of, 309, 312-315. integral, 347. Symmetry, 9, 23. centre of, 29. Tan' 1 .*-, expansion of, 401. Tangent, definition of, 78. length of, 123, 201. inflexional, 35. plane, 216, 411. Taylor's Theorem and Series for function of one variable, 390-398. for function of several variables, 408-412. Thermodynamics, 228-233. Time-rate of change, 103. Tore, 322, 349. Total derivative, 211. differential, 213, 224. Trapezoidal rule, 329. Trigonometric functions, direct and inverse, 56. differentiation of, 129, 133. integration of, 265, 278, 284. Trochoid, 370. True value, 418. Turning value, 24. 166. Ultimately equal, 199. Uniform convergence. 385. Units, 26, 28. Value, stationary, 105. true, 418. turning, 24, 166. Variable, dependent and indepen- dent, 12. change of, 233, 271. Variation, near a turning value, 174. in a given direction, 218. Velocity, 149. angular, 153. components of, 110. Volumes, 193, 309, 331, 335. polar element of, 346. Wallis's value of TT, 307. Work, 150, 225. The numbers refer to pages. OLASOOW : PRINTED AT THE UNIVERSITY" PRESS BV ROBERT MACLEHOSE AND CO. University of California SOUTHERN REGIONAL LIBRARY FACILITY 405 Hilgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which it was borrowed. ,he DUE Plea ~ SRLF CHANGE MAR Jt/L S SEP '< FEB Rft JUN JOI 30m-8,'6 TE