AN ELEMENTARY TEXT-BOOK ON THE DIFFERENTIAL AND INTEGRAL CALCULUS WILLIAM H. ECHOLS Professor of Mathematics in the University of Virginia. B It gfc mk 1 .;:;. NEW YORK HENRY HOLT AND COMPANY 1902 THF LIBRARY O? CONGRESS, t „t> CciPi^ RtcavED (fat XX_/q»i^ C«-»«* fl-'YXn No, Z4 W- X. U- Cj COPY B. Copyright, 1902, BY EENRY HOLT & CO. ROBERT lUII'MMOSR, PRINTER, NEW YORK. PREFACE. This text-book is designed with special reference to the needs of the undergraduate work in mathematics in American Colleges. The preparation for it consists in fairly good elementary courses in Algebra, Geometry, Trigonometry, and Analytical Geometry. The course is intended to cover about one year's work. Experi- ence has taught that it is best to confine the attention at first to func- tions of only one variable, and to subsequently introduce those of two or more. For this reason the text has been divided into two books. Great pains have been taken to develop the subject con- tinuously, and to make clear the transition from functions of one variable to those of more than one. The ideas which lie about the fundamental elements of the calculus have been dwelt upon with much care and frequent repetition. The change of intellectual climate which a student experiences in passing from the finite and discrete algebraic notions of his previous studies to the transcendental ideas of analysis in which are involved the concepts of infinites, infinitesimals, and limits is so marked that it is best to ignore, as far as possible on first reading, the abstruse features of those philosophical refinements on which repose the foun- dations of the transcendental analysis. The Calculus is essentially the science of numbers and is but an extension of Arithmetic. The inherent difficulties which lie about its beginning are not those of the Calculus, but those of Arithmetic and the fundamental notions of number. Our elementary algebras are beginning now to define more clearly the number system and the meaning of the number continuum. This permits a clearer presen- tation of the Calculus, than heretofore, to elementary students. As an introduction and a connecting link between Algebra and the Calculus, an Introduction has been given, presenting in review those essential features of Arithmetic and Algebra without which it is hopeless to undertake to teach the Calculus, and which are unfor- tunately too often omitted from elementary algebras. The introduction of a new symbolism is always objectionable. IV PREFACE. Nevertheless, the use of the " English pound " mark for the symbol of "passing to the limit" is so suggestive and characteristic that its convenience has induced me to employ it in the text, particularly as it has been frequently used for this purpose here and there in the mathematical journals. The use of the "parenthetical equality" sign ( = ) to mean " converging to " has appeared more convenient in writing and print- ing, more legible in board work, and more suggestive in meaning than the dotted equality, =, which has sometimes been used in American texts. An equation must express a relation between finite numbers. The differentials are denned in finite numbers according to the best mod- ern treatment. In order to make clear the distinction between the derivative and the differential-quotient, I have at first employed the symbol Df, after Arbogast, or the equivalent notation f of Lagrange exclusively, until the differential has been defined, and then only has Leibnitz's notation been introduced. After this, the symbols are used indifferently according to convenience without confusion. The word quantity is never used in this text where number is meant. True, numbers are quantities, but a special kind of quantity. Quantity does not necessarily mean number. The word ratio is not used as a relation between numbers. It is taken to mean what Euclid defined it to be, a certain relation between quantities. The corresponding relation between numbers is in this book called a quotient. The quotient of a by b is that ?iumber whose product by b is equal to a. In preparing this text I have read a number of books on the subject in English, French, German, and Italian. The matter pre- sented is the common property now of all mankind. The subject has been worked up afresh, and the attempt been made to present it t<> American students after the best modern methods of continental writers. I am especially indebted to the following authors from whose books the examples and exercises have. been chiefly selected : Tod- hunter, Williamson, Price, Courtenay, Osborne, Johnson, Murray, Boole, Laurent, Serret, and Frost. My thanks are due Dr. John E. Williams for great assistance in reading the proof and for working out all of the exercises. W. PL E. University of Virginia, October, 1902. CONTENTS . INTRODUCTION. FUNDAMENTAL ARITHMETICAL PRINCIPLES. PAGE Section I. On the Variable. i The absolute number. The integer, reciprocal integer, rational num- ber. The infinite and infinitesimal number. The real number system and the number continuum. Variable and constant. Limit of a vari- able. The Principle of Limits. Fundamental theorems on the limit. Section II. Function of a Variable 19 Definition of functionality. Explicit and implicit functions. Continu- ity of Function. Geometrical representation. Fundamental theorem of continuity. BOOK I. FUNCTIONS OF ONE VARIABLE. PART I. PRINCIPLES OF THE DIFFERENTIAL CALCULUS. Chapter I. On the Derivative of a Function 35 Difference of the variable. Difference of the function. Difference- quotient of the function. The derivative of the function. Ab initio differentiation. Chapter II. Rules for Elementary Differentiation 41 Derivatives of standard functions log x, x a , sin x. Derivative of sum, difference, product and quotient of functions in terms of the component functions and their derivatives. Derivative of function of a function. Derivative of the inverse function. Catechism of the standard derivatives. Chapter III. On the Differential of a Function 55 Definition of differential. Differential-quotient. Relation to difference. Relation to derivative. Chapter IV. On Successive Differentiation 62 The second derivative. Successive derivatives. Successive differentials. The differential-quotients, variable independent. Leibnitz's formula for the nth. derivative of the product of two functions. Successive derivatives of a function of a linear function of the variable. vi CONTENTS. PAGE Chapter V. On the Theorem of Mean Value 74 Increasing and decreasing functions. Rolle's theorem. Lagrange's form of the Theorem of Mean Value, or Law of the Mean. Cauchy's form of the Law of the Mean. Chapter VI. On the Expansion of Functions 82 The power-series. Taylor's series. Maclaurin's series. Expansion of the sine, logarithm, and exponential. Expansion of derivative and primi- tive from that of function. Chapter VII. On Undetermined Forms 92 Cauchy's theorem. Application to the illusory forms 0/0, 00 /oo , o x » , <» - w , o°, 00 °, i 00 . Chapter VIII. On Maximum and Minimum 103 Definition. Necessary condition. Sufficient condition. Study of a function at a point at which derivative is o. Conditions for maximum, minimum, and inflexion. PART II. APPLICATIONS TO GEOMETRY. Chapter IX. Tangent and Normal 112 Equation of tangent. Slope and direction of curve at a point. Equa- tion of normal. Tangent-length, normal-length, subtangent and subnor- mal. Rectangular and polar coordinates. Chapter X. Rectilinear Asymptotes 121 Definitions. Three methods of finding asymptotes to curves. Asymp- totes to polar curves. Chapter XI. Concavity, Convexity, and Inflexion 127 Contact of a curve and straight line. Concavity. Convexity. In- flexion, concavo-convex, convexo-concave. Conditions for form of curve near tangent. Chapter XII. Contact and Curvature . 130 Contact of two curves. Order of contact. Osculation. Osculating circle, circle of curvature, radius, and center. Chapter XIII. Envelopes 138 Definition of curve family. Arbitrary parameter. Enveloping curve of a family. Envelope tangent to each curve. Chapter XIV. Involute and Evolute 144 Definitions. Two methods of finding evolute. Chapter XV. Examples of Curve Tracing 147 Curve elements. Explicit and implicit equations. Tracings of simple curves. CONTENTS. vii PART III. PRINCIPLES OF THE INTEGRAL CALCULUS. PAGE Chapter XVI. On the Integral of a Function 165 Definition of element. Definition of integral. Limits of integration. Integration tentative. Primitive and derivative. A general theorem on integration. The indefinite integral. The fundamental integrals by ab initio integration. Chapter XVII. The Standard Integrals. Methods of Integration 175 The irreducible form / dn. The catechism of standard integrals. Prin- >J ciples of integration. Methods of integration. Substitution (transforma- tion, rationalization). Decomposition v parts, partial fractions). Chapter XVIII. Some General Integrals 193 Binomial differentials. Reduction by parts. Trigonometric integrals. Rational functions. Trigonometric transformations. Rationalization. Integration by series. Chapter XIX. On Definite Integration 215 Symbol of substitution. Interchange of limits. New limits for change of variable. Decomposition of limits. A theorem of mean value. Exten- sion of the Law of Mean Value. The Taylor-Lagrange formula with the terminal term a definite integral. Definite integrals evaluated by series. PART IV. APPLICATIONS OF INTEGRATION. Chapter XX. On the Areas of Plane Curves 226 Areas of curves, rectangular coordinates, polar coordinates. Area swept over by line segment. Elliott's extension of Holditch's theorem. Chapter XXI. On the Lengths of Curves 243 Definition of curve-length. Length of a curve, rectangular coordinates, polar coordinates. Length of arc of evolute. Intrinsic equation of curve. Chapter XXII. On the Volumes and Surfaces of Revolutes. 255 Definition of rotation. Revolute. Volume of revolute. Surface of re volute. Chapter XXIII. On the Volumes of Solids 264 Volume of solid as generated by plane sections parallel to a given plane. viii CONTENTS. BOOK II. FUNCTIONS OF MORE THAN ONE VARIABLE PART V. PRINCIPLES AND THEORY OF DIFFERENTIATION. PAGE Chapter XXIV. The Function of Two Variables 273 Definition. Geometrical representation. Function of independent variables. Function of dependent variables The implicit function of several variables. Contour lines. Continuity of a function of two vari- ables. The functional neighborhood. Chapter XXV. Partial Differentiation of a Function of Two Variables 282 On the partial derivatives. Successive partial differentiation. Theorem of the independence of the order of partial differentiation. Chapter XXVI. Total Differentiation 290 Total derivative defined. Total derivative in terms of partial deriva- tives. The linear derivative. Total differential. Differentiation of the implicit function. Chapter XXVII. Successive Total Differentiation 299 Second total derivative and differential of z = f(x, y). Second deriva- tive in an implicit function in terms of partial derivatives. Successive total linear derivatives. Chapter XXVIII. Differentiation of a Function of Three Variables 306 The total derivative. The second total derivative. Successive linear total differentiation. Chapter XXIX. Extension of the Law of Mean Value to Functions of Two and Three Variables 309 Chapter XXX. Maximum and Minimum. Functions of Sev- eral Variables 314 Definition. Conditions for maxima and minima values of f{x , y) and f{x. y, z). Maxima and minima values for the implicit function. Use of Lagrange's method of arbitrary-multipliers. Chapter XXXI. Application to Plane Curves 329 Definition of ordinary point. Equations of tangent and normal at an ordinary point. The inflectional tangent, points of inflexion. Singular point. Double point. Node, conjugate, cusp-conjugate conditions. Triple point. Equations of tangents at singular points. Homogeneous coordinates. Curve tracing. Newton's Analytical Polygon, for separat- ing the branches at a singular point. Envelopes of curves with several parameters subject to conditions. Use of arbitrary multipliers. CONTENTS. IX PART VI. APPLICATION TO SURFACES. PAGB Chapter XXXII. Study of the Form of a Surface at a Point. 347 Review of geometrical notions. General definition of a surface. Gen- eral equation of a surface. Tangent line to a surface. Tangent plane to a surface. Definition of ordinary point. Inflexional tangents at an ordinary point. Normal to a surface. Study of the form of a surface at an ordinary point, with respect to tangent plane, with respect to osculating conicoid. The indicatrix. Singular points on surfaces. Tangent cone. Singular tangent plane. Chapter XXXIII. Curvature of Surfaces 365 Normal sections. Radius of curvature. Principal radii of curvature. Meunier's theorem. Umbilics. Measures of curvature of a surface. Gauss' theorem. Chapter XXXIV. Curves in Space 375 General equations of curve. Tangent to a curve at a point. Oscu- lating plane. Equations of the principal normal. The binormal. Circle of curvature. Tortuosity, measure of twist. Spherical curvature. Chapter XXXV. Envelopes of Surfaces 385 Envelope of surface-family having one arbitrary parameter. The characteristic line. Envelope of surface-family having two independent arbitrary parameters. Use of arbitrary multipliers. PART VII. INTEGRATION FOR MORE THAN ONE VARIABLE. MULTIPLE INTEGRALS. Chapter XXXVI. Differentiation and Integration of In- tegrals 391 Differentiation under the integral sign for indefinite and definite inte- grals. Integration under the integral sign for indefinite and definite integrals. Chapter XXXVII. Application of Double and Triple Integrals 396 Plane Areas, double integration, rectangular and polar coordinates. Volumes of solids, double and triple integration, rectangular and polar coordinates. Mixed coordinates. Surface area of solids. Lengths of curves in space. Chapter XXXVIII. Differential Equations of First Order and Degree 409 Rules for solution. Exact and non-exact equations. Integrating factors. Chapter XXXIX. Examples of Differential Equations of the First Order and Second Degree 428 Rules for solution. Orthogonal trajectories. The singular solution. c- and /-discriminant relations. Redundant factors not solutions. Node, cusp, and tac- loci. CONTENTS. Chapter XL. Examples of Differential Equations of the Second Order and First Degree 439 The five degenerate forms. The linear equation, and homogeneous linear equation having second member o. APPENDIX. Note 1. Weierstrass's Example of a Continuous Function which has no deter- minate derivative , 45 1 Note 2. Geometrical Picture of a Function of a Function 453 Note 3. The nth Derivative of the Quotient of Two Functions 454 Note 4. The nth Derivative of a Function of a Function 455 Note 5. The Derivatives of a Function are infinite at the same points at which the Function is infinite. 456 Note 6. On the Expansion of Functions by Taylor's Series 457 Note 7. Supplement to Note 6. Complex Variable 465 Note 8. Supplement to Note 6. Pringsheim's Example of a Function for which the Maclaurin's series is absolutely convergent and yet the Function and series are different 467 Note 9. Riemann's Existence Theorem. Proof that a one-valued and con- tinuous function is integrable 468 Note 10. Reduction formulae for integrating the binomial differential 470 Note 11. Proof that a curve lies between the chord and tangent, when the chord is taken short enough 471 Note 12. Proof of the properties of Newton's analytical polygon for curve - tracing 47 l Index 475 INTRODUCTION TO THE CALCULUS. SECTION I. ON THE VARIABLE. 1. Calculus, like Arithmetic and Algebra, has for its object the investigation of the relations of Numbers. It is necessary to under- stand that the symbols employed in Analysis either represent numbers or operations performed on numbers. 2. The Symbols. i, 2, 3, 4, . . . (i) are symbols used to represent the groups of marks which we call inte- gers. Thus * 1=1, 2 = 1+1, 3 = 14-1 + 1, The system of integers (i) extends indefinitely toward the right, as indicated by the sign of continuation. This system is called the table of integers. Each integer has its assigned place, once and for all, in the table. Any integer in the table is, conventionally, said to be greater than any other integer to the left of it, and less than any integer to the right of it in the table (i). 3. Definition of Infinite Integer. — When an integer is so great that its place in the table of integers cannot be assigned in such a manner that it can be uniquely distinguished from each and every other integer, that integer is said to be unassignably great or infinite. Mathematical infinity has no further or deeper meaning than this. 4. The Inverse Integer. — The reciprocals of the integers i»Uf ( H ) constitute an extension of the table (i) to the left of the integer 1, which number is its own reciprocal. As before, any number in this table is said to be greater than any number to the left of it, and less than any number to the right of it. Corresponding to each number in (i) there is a number in (ii), * The symbol = is to be read, ' ' is identical with, " or ' ' is the same as. ' ' 1 2 INTRODUCTION TO THE CALCULUS. [Sec. I. and conversely. Those numbers in (ii) which are the reciprocals of the infinite or unassignably great integers, are said to be infinitesimals or unassignably small.* 5. The Absolute Number. The Absolute-Number Continuum. When in the table of numbers . . . , |, 1 1, 2, 3, . . . (iii) the gap between each pair of consecutive numbers is filled in with all the rational (fractional) and irrational numbers that are greater than the lesser and less than the greater of the pair, we construct a table of numbers which is called the absolute-number continuum. Each number in this system has its assigned place. It is said to be greater than any number to the left of it and less than any number to the right of it. Each number in the absolute-number continuum is called an absolute number. Any and all numbers in the table that are greater than any integer that can be uniquely assigned, as in § 3, are said to be infinite or unas- signably great. In like manner any number in the table that is less than any reciprocal-integer that can be uniquely assigned a place in the table is infinitesimal or unassignably small. The absolute continuum is thus divided into two classes of num- bers : the uniquely assigned or simply the assigned numbers, which we call the finite numbers ; and the numbers which cannot be uniquely assigned or transfinite numbers. The transfinite numbers greater than 1 are called infinite, those less than 1 infinitesimal numbers. 6. Zero and Omega. — The absolute-number system, as con- structed in § 5, extends indefinitely both ways, in the direction of the indefinitely great and in that of the indefinitely small. In this sys- tem there is no number greater than all other numbers in' the system, nor is there any number that is less than all others in the system. The system is conventionally closed on the left by assigning in the table a number zero whose symbol is o, which shall be less than any number in the absolute system. Since now there is no number greater than 1 to correspond to the reciprocal of this number o, we design arbitrarily a number omega whose symbol is £1 as the reciprocal of o, and which is greater than any number in the absolute system. The number o is the familiar naught of Arithmetic. The num- ber XI is the ultimate number of the Theory of Functions, and with which we shall not be further concerned in this book. The number o is not an absolute number, but is the inferior boundary number of that system. In like manner the number CI is not an abso- lute number, but is the superior boundary number of the absolute system. * The words ' great ' and ' small ' have in no sense whatever a magnitude mean- ing when applied to numbers. They are mere conventional phrases and the words 1 right ' and ' left 'or ' in ' and ; out,' might just as well be employed. Art. 7.] ON THE VARIABLE. 3 7. The conventional symbol for the whole class of unassignably great or infinite numbers is 00 . There has been adopted no conven- tional symbol for the class of infinitesimals ; the symbol most com- monly used is the Greek letter iota, 1. 8. The Real-Number System. — When in the algebraic system of numbers — n, . . . , — 3, — 2, — 1, o, + 1, + 2, + 3, . . . , -f n, the gap between each consecutive pair is filled in with all the rational and irrational numbers that are greater than the lesser and less than the greater of the pair, the system thus constructed is called the real- number continuum. It is understood that any number in this table is greater than any number to the left of it and less than any number to the right of it. The modulus of any real number is its arithmetical or absolute value. Thus, the modulus or absolute value of -f- 3 or — 3 is the absolute number 3. If we employ the symbol x to represent any number in the real continuum, then its modulus or absolute value is represented by \x\ or mod x. In this book we shall be directly concerned only with real num- bers and their absolute values. Hereafter when we speak of a number, we mean a real number unless otherwise specially mentioned.* 9. Geometrical Picture of the Real-Number System. — We assume a cor- respondence between the points on a straight line and the numbers in the real continuum. -4 -3 x'-2 -1 o + i +% x±3 44 (-; h-. I . I , H 1 1 1 1 1 1 D C P B A. O A B P C D Fig. 1. Select any point O on a straight line. Choose arbitrarily any unit length ; with which construct a scale of equal parts, A, B, C, . . . starting at O proceeding * The real-number continuum is a closed system of numbers to all operations save that of the extraction of roots. When we consider the square root of a nega- tive number we introduce a new number. The complex or complete number of analysis is x + iy, where x and y are any two real numbers, and i is a conventional symbol represent- ing -f- V — 1. Corresponding to any real number y there are as many complex numbers as there are real numbers x ; and corresponding to any real value x there are as many complex numbers as there are real numbers y. The complex system is a double system. In the theory of functions of complex numbers, which includes that of real numbers as a special case, the ultimate number fl is conventionally a number common to all systems in the same way as is o. The student is already familiar with the impossibility of solving all questions in analysis with real numbers only. For example, in the theory of equations when seeking the roots of equations. All the more so is this true in the Calculus, for we cannot solve the fundamental problem of expanding functions in series with- out the use of complex numbers, except in a very few particular cases. If z is any complex number x + iy, its modulus or absolute value is 4 • INTRODUCTION TO THE CALCULUS. [Sec. I. toward the right, and A', £•', C' f . . . toward the left. Mark the points of divi- sion, o at the origin O, and -f- I, -j- 2, etc., toward the right ; — I, — 2, etc., toward the left. Then, corresponding to any real number x there is a point P oix the line to the right of O if x is positive ; and P' to the left of O if x is negative. The number x is the measure of the length OP with respect to the unit length chosen. Conversely, corresponding to each point Pon the line there is a number in the real-number system. io. Variable and Constant. — In the continuous number system, as designed in § 8, it is convenient to use letters as general symbols to represent temporarily the numbers in that system. Thus, we can think of a symbol x as representing any particular number in that system. Further, we can think of a symbol x as representing any particular number, say -\- 3, and then representing continuously in succession every number between -j- 3 and any other number, say -j- 5, and finally attaining the value + 5. We speak of such a symbol x, representing successively different numbers, as a number, and w.e speak of any particular number which it represents, as its value. Definition. — A number x is said to be variable or constant ac- cording as it does or does not change its value during an investigation concerning it. We shall frequently be concerned with symbols of numbers which are variable during part of an investigation and are constant during another part. Generally, variables are represented by the terminal letters u y v, w, x,j', z, etc., and constants by the initial letters a, b, c, etc., of the alphabet. This is not always the case, however, as the context will show. 11. Interval of a Variable. — We shall sometimes confine our attention to a portion of the number system. For example, we may wish to consider only those numbers between a and b. We shall employ the symbol (a, b), a being less than b, to represent the numbers a, b and all numbers between them. If we wish to exclude from this system b only, we write (a, b ( ; if a only, we write ) I, the sum of the series and the equivalent member on the right increase indefinitely with «, in absolute value, and can be made greater than any assigned number and therefore become infinite. Under these circumstances the series has no limit ; its value becomes indeterminately great. Geometry furnishes numerous illustrations of the limit. The most notable being : 4. The evaluation of the area of the circle as the limit to which converge the areas of the circumscribed and inscribed regular polygons as the number of sides is indefinitely increased. 5. The evaluation of the irrational and transcendental number n representing the ratio of the circumference of a circle to its diameter. Trigonometry furnishes an illustration of a limit which will be found useful later: 6. To evaluate the limit of the quotient sin x ~- x as x diminishes indefinitely \ in absolute value. Draw a circle with radius I. Draw MA = MB perpendicular to OT. Then Area quadrilateral OA TB = tan x, Area triangle OAMB = sin x, Area sector OANB = x, where x is, of course, the circular measure of Z AOT. Then, obviously, from geometrical consider- ations, sin x < x < tan x, x I or i < —. — < , I > sin x cos x > cos x. Art. 13.] ON THE VARIABLE. 7 When ,r diminishes" indefinitely in absolute value, cosjc becomes more and more nearly equal to I, and has the limit I as x converges to o. Consequently the quo- tient (sin x)/x converges to the limit I as .r converges to o. In our symbolism, /< )= 13. Definition. — When a symbol x, representing a variable num- ber, has become and subsequently remains always less, in absolute value, than any arbitrarily small assigned absolute number, x is said to be infinitesimal. When a variable becomes and remains greater, in absolute value, than any arbitrarily great assigned number, the variable is said, to be infinite. When a variable x is infinitesimal, we write* x( = )o. It follows from the definition that when a variable becomes infinitesimal it has the limit o, or assigns the number o. When x has the limit a, or £x — #, then by definition £{x - a) = o. When x — a is infinitesimal, we write x — a( = )o. This same relation we shall frequently express by the symbol x( = )a, . . , meaning that the absolute value of the difference between x and a is infinitesimal. When a is the limit of x, the symbol x( = )a is to be read, "as x converges to tf," or " x converges to a." We shall frequently use the symbol e (epsilon) to represent an arbitrarily small assigned absolute number. We then speak of the interval (a — e, a -j- e) as the neighborhood of an assigned number a. The symbol x(=)a means that " x is in the neighborhood of a." All numbers that are in the neighborhood of an assigned number are said to be consecutive numbers. When a variable x becomes infinite we write x = 00 . Such a variable has no limit, it simply becomes indeterminately great. The symbol x = 00 merely means that x is some number in the class of unassignably great numbers. 14. The Principle of Limits. I. A variable cannot simultaneously converge to two different limits. *The equality sign in parenthesis (=) may be read " parenthetically equal to," the word ' parenthetically ' carrying with it the explanation of the nature of the approximate equality. It is simply another way of saying that the difference between two numbers is infinitesimal. \x — a\ = 1 and x — a( = )o mean the same thing. The symbol = has been used for (=), but appears less con- venient, expressive, and explicit. 8 INTRODUCTION TO THE CALCULUS. [Sec. I. It is impossible for a (one-valued) variable x to converge to two unequal limits a and b. For, the differences \x — a\ and \x — b\ can- not each be less than the assigned constant number \ \ b — a | for the same value of x . The direct proof of this statement rests on this: The number x must be either greater than, equal to, or less than the number \{a -\- b), where say a < b. If x = ±(a + b), .'. x-a=-$(b-a). If x > \{a + t>), .-. x - a > \{b - a). If x J(* - a). II. If two variables x and jy are always equal and each converges to a limit, then the limits are equal. If £x = a, and £y = b, and x =y = z, then, by I, the variable z cannot converge to two unequal limits simultaneously. Therefore a-b. 15. Theorems on the Limit.* I. If the limit of x is o, then also the limit of ex is o, where c is finite and constant. For, whatever be the assigned constant absolute number e, we can by definition of a limit make and keep \x\ less than the constant I e/c I , and therefore ex less than e in absolute value. Consequently, by definition £ (ex) = o = c£(x). x(=)o II. If each of & finite f number of variables x lf x % , . . . , x nf has the limit o, then the algebraic sum of these variables has the limit o. Let x be the greatest, in absolute value, of the n variables. Then l*i + * 2 + • • • +Xn\^nx- Since n is finite, the limit of this sum is o, by I. III. If £x l = a x , £x 2 = a 2 , . . . , £x n = a H , then when n is a finite integer £(x x + x 2 + . . . + x H ) = £x x + £x % -f . . . -f £x u . For, put x x = a x -f- a y , . . . , x H = a n -\- a H . By definition, the limits of a x , . . . , a n are o. Hence * t + *,+ ... + *« = fo +...+«.) + K + .... + ««), by II, gives £(x, + ...+x n ) = a l + ...+* H . Therefore the limit of the sum of a finite number of variables is equal to the sum of their limits. * The theorems of this article are of such fundamental importance and so absolutely necessary for the foundation of the Calculus that it will, in general, be assumed hereafter that they are so well known as to require no further reference to them. f If the number of variables is not finite, this theorem does not hold in general. Art. 15.] ON THE VARIABLE. <) IV. The limit of the product of two variables x x and x 2 which have assigned limits a x and a 2 , is equal to the product of their limits. Let, as in III, x x = a x + a x , x 2 = a 2 -f a r . •. Xy x % = a x a 2 + a x a 2 -f- a 2 a x -f a x a r By III, we have £(*!**) = «A + «u£" a + * 2 ;£"i + ;£K",)- But, ^^ = o, ;£ar 2 = o, and a fortiori £(a x a 2 ) — o. Therefore £{x x x 2 ) = a x a 2 = (£x x )(£x 2 ). Cor. The limit of the product of & finite number of variables having assigned limits, is equal to the product of their limits. In symbols * £ n(x r ) = n£(x r ). r — 1 r = 1 V. The limit of the quotient, x x /x 2 , of two variables is equal to the quotient of their limits, provided the limit of the denominator is not o. With the same symbolism as in IV, *\ _ a x + a i __ a i 1 a i + a 2 a i x 2 <* 2 -{-<* 2 a 2 a 2 +a 2 a 9 * a 2 a A a t + <**)' By hypothesis, £a x = o, £a 2 = o, and a 2 =£ o. Therefore the denominator of the second term on the right is always finite, while, by III, the limit of the numerator is o. The limit of this term is o, by If fx i\ - a _i- i^L ~ — „ ~ /-„ * It a 2 £x 2 VI. If x and y are two variables and a is a constant, such that y always lies between x and a, then if £x = a, also £y = a. * As the symbol ~2 is used to indicate the sum, so U is used to indicate the product of a set of numbers. Thus, n 2x r = X x -f *„ + . . . + X n , IIx r rJ;,X^X--- X x n . 1 The advantage of such symbolism is in compactness of the formulae. t Notice particularly the provision that £x 2 =£ o. For, when £x 2 = o and £x x =± o, the quotient Xx/x? inci eases beyond all limit or becomes infinite as x { and x 7 converge to their limits. An infinite number cannot be a limit under the definition. Again, if £x 2 = o and also £x x = o, the quotient of the limits 0/0 is com- pletely indeterminate, while the quotient x,/x 2 = q mayor may not converge to a determinate limit. The value of this quotient as x, and x 2 converge to o depends on the law connecting the variables x } and x 2 as they converge to o. This case is one of profound importance and is the foundation of the Differential Calculus. io INTRODUCTION TO THE CALCULUS. [Sec. I. The truth of this is obvious, since \x — a\>\y — a\, and x — a has the limit o. In like manner, it follows that if x and z have the common limit a, and y is a third variable between x and z, then also must jQy = a. For, \y — a\ must at all times be less than one or the other of the differences \x — a\ and \z — a\, and each of these differences has the limit o. VII. If one of two variables is always positive and the other is always negative, and they have a common limit, that limit is o. Let a be the common limit of x and y, where x is always positive and y is always negative. Then -f|*| = fl + tf, and —\j>\ = a + /3, where £a = o, £(3 = o. Subtracting, \x\ + \y\ = <*-p. Since £(a — fi) = o, . •. a -f- a = 2a = o, and a, the com- mon limit of x and_y, is o. VIII. If a variable x continually increases and assumes a value a but is never greater than a given constant A, then there must exist a superior limit of x equal to or less than A. (1). No number such as a which x once attains can be a limit of x. For, since x continually increases, it must subsequently take some value a' > a, and it is never possible thereafter for x — a to be less than the constant a' — a. (2). The variable x cannot attain the number A, since if it did, x continually increasing must become greater than A, which is contrary to hypothesis. (3). Divide the interval^ — a = h into 10 equal parts. The vari- able x after attaining a must either attain a -f- -^h or remain always less than a -f- ^h. \{x attains a -f- ^h, it must either attain a -\- -f^h or remain always less than a -f- -f^h. We continue to reason thus until we find a digit p x such that x must attain a -| -k and remain always less than a -}- * + ma. (I) In fact, (i + af - I -f 2a -f- a 2 > I + 2a. The formula (i) is true when m = 2. Assume it to be true when m = n. Then ( i -\- a) n > I -j- na. Multiply both sides by I -f <*■- .-. (I + a)"+» > i -f (» -f i)a + «a 2 , > i + (»+i)a. (I) is true also for » -\- I. But, being true for m = 2, it is also true for m = 3, and therefore for m =■ 4, etc., and generally. Therefore, since ma and conse- quently (1 -\- a) m can be made greater than any assigned number, the proposition is demonstrated. 2. The successive powers of any assigned absolute number less than 1 diminish indefinitely and have o for limit. Any number less than I can be written as the quotient 1/(1 -f- a). By Ex. 1, 11 I (i-f- a) m i-\-ma ma' This can be made less than any assigned number e, by sufficiently increasing m. 3. The successive roots of an absolute number greater than 1 continually diminish ; those of an absolute number less than I continually increase ; and in either case have the limit I. Whatever be the absolute number a, - (*+ a n = a D_L_ f ^_J* +I Therefore, by Exs. 1, 2, whatever be the integer n, a n > a n - l , if a > 1 ; a n < a M+I , is a < I. Ua> 1, then a" > 1. Let a — 1 -f- «-, and a w ~ then (1 + a)« = 1 + A or (i4>a) = (i-(-/J)«> i+« /? < a/«, and we have -1+/? Hence ^ a « Art. 15.] EXERCISES. 13 Let a < 1, say a =1/(1 -f a). 1 1 Also, a n a*> i + a/n' which shows again that £ a« = i. 4. Show that when a is any assigned positive number, £«* = h whatever be the way in which x converges to o. (1). Let m, n, p, q be any positive integers. Then tn p m p a n a I, then a* > I t a n • q > a n f an( i Therefore a x continually increases as x increases by rational numbers. P_ If a < 1, then ai < I. — + -£- — — ^^L + i.^ _!!L . •. a* 1 < a n , and a V » ■//># ». Therefore a* continually diminishes as x increases by rational numbers. When \x\ is rational and less than I, there can always be assigned two con- secutive integers m and m -\~ 1 such that r— < 1*1 < -• tn -f- I tn The above results show that whether a be greater or less than 1, a* lies between a™ + l and a m . When m .— 00 , a m + l and a m converge to I, Ex. 3, and there- fore also does a x \ and £a x =r 1, when x(=)o. (2). When x is irrational there can always be assigned two rational numbers a and ft differing from each other as little as we choose, such that a < x < ft. The number a x is defined by its lying between a a and aP. Since x( = )o when a( =)ft(= )o, we have, as before, a x converging to I along with a a and aP. 5. Show that £a* = a £x = aP, if £x = ft. We have a& — a x = a&{ I — a x - &). Passing to limits, we have, by Ex. 4, aP — £a x — o. 6. If a and ft are positive numbers, and £x — ft, show that £ log* x = log a £(x) — log a /?. We have log a ft — log a x = log a — . The above exercises show that however x converges to ft, £ \og a {ft/x) = o. Therefore loga P ~ £ ^ga X = O. 14 INTRODUCTION TO THE CALCULUS. [Sec. I. 7. Utilize Ex. 6, to prove IV. V, from III, § 15. 8. Use Ex. 6, to show that £(y*) = (£)>) £x . where y has a positive limit, and the limit of x is determinate. 9. A set of numbers a t , a 2 , . . . , a r , . . . , arranged in order is called a se- quence. Any number of the sequence, a r , is called an element of the sequence ; the number r is called the order of the element a r . Any sequencers said to be known when each element is finite and known when its order is known. If a lt a 2 , . . . , a H , . . . be a sequence of numbers such that a r is finite when r is finite, then will £a ny when n = 00 , be o or 00 according as £\ is less or greater than 1, respectively. Let, when n — oo , £(&n + i/«») — k, and k > 1. Then, by the definition of a limit, we can always assign a number k' such that I < k? < k, whence corre- sponding to W we can find an integer m for which we have, for all values of «, On ±J1±1 > p a n + m ••• a,n + 1 > & a m , &m + 2 > k' a m + j > k a m , a m + n> k' n a m . By hypothesis, a m is finite. Since we can make k' n greater than any assigned number by sufficiently increasing n, we have £a n = 00 . In like manner, if £{a n + Ja r ?) = k < I, a m + H < k' n a m , which can be made less than any assigned number by increasing n, when as before I > k' < k. . \ £ci n = o, when n = 00 . In order that the element a n may have a finite-limit different from o, it is neces- sary that * I | = | I. /' The quotient, a n + 1 /a M> of each element by the preceding one will hereafter be called the convergency quotient of the sequence. This theorem is of importance and will be used later. 10. The series of numbers a i + a 2 +•..+««+•• • (i) is said to be absolutely convergent when the corresponding series of the absolute values of the terms is convergent. That is, when 5W=|«xl + l«al+ • • • +l*«l has a determinate limit when n = 00 . Show that (i) is absolutely convergent if / and if this limit is greater than 1, the sum of the series is 00 . * When the symbols | = |, |>|, |n+n < k' n a m . Hence the sum of the series after a m is less than k'a m + . . . + U«a m + . . . =a m (k>+ . . . + £'* + ■ ■ • ), k' This is finite, since k' ^ I. Therefore S x must be finite. Also, by Ex. 9, £ a m = o, when m = 00 . Consequently we can always assign an integer n such that for all values of m, where e is any assigned number. Hence S n has a determi- nate limit. Otherwise, the existence of the limit of S tt follows at once from VIII, §15. For ^continually increases, but can never exceed k' *! + *>+■'••+ a ™ + am YZTk>' Again, if £(a n +j/a n ) > 1, say equal to k > I. Then, as before, we can assign k' between k and 1, and have the sum of the series after a m greater than a m (#+...+#«+...), which is 00 . The number £(a n ±\[an) is called the convergency quotient of the series. 11. The arithmetical average, or mean value of a sequence of n numbers, a \y <*%, ' ' ' t a n, . is one nth. of their sum, or 1 H n Show that when the number of elements in a sequence increases indefinitely according to any given law, the mean value has a determinate limit, if all the ele- ments are finite. Since L < a n < M, where L and M are the least and greatest elements respectively, the mean value must remain finite. Also, I ** p n +P n +i n[n-\-p) i 16 INTRODUCTION TO THE CALCULUS. [Sec. I. But / ^ |^| ** G P G ■ G being an assigned number, than which no element can be greater in absolute value. Whatever be the assigned integer p, we can always assign an integer n that will make a n +f — oc n less than any assigned number e. The mean value therefore converges to a determinate limit. The value of this limit depends on the law by which the sequence is formed. 12. Find the limit of K)" when z becomes infinite in any way whatever. Divide both numerator and denominator in X m — where m is a positive integer, by x m — i. Whence results l-fi'»-f . . . +\x m ) .I-h— =r — — • x — I m m Hence *^_ x x x nt - i > - m (. + 4rr>K)" Therefore, the value of the expression continually increases with m, and is always greater than 2, by Ex. I. (2). If x '— i , each of the m terms in the denominator of the same m 4- i m 4- i fraction is greater than i. i i m + I X~~™~ — X < I + w Hence or I — X m + .* m f» + > < m* Art. 15.] EXERCISES. 17 / i \ * + 1 / 1 \ >» Therefore the expression continually diminishes as the positive integer m in- creases. (3). Whatever be the positive number x, we have X2 > x? — 1. ... x > x + ' Hence (, - I) % (, + I)' whatever positive value jr may have. (4). Also, if we put a: = y -f- !• These results (1), . . . , (4), show that I \z K* continually increases as z increases by positive integers, and continually de- creases as z decreases by negative integers, and that the latter set of numbers is always greater than the former, by (3). Also, these ascending and descending sequences have a common limit,* by (4). The value of this limit lies somewhere between (1 -f 1/6)6 = 2.521 ... and (1 — 1/6)- 6 = 2.985 . . . We represent it, conventionally, by the symbol /// + r . . . pi + m p L + z . , . . pi + m . . . has for its limit the rational number ' io'(io»« — i) where ^/= 0.^ .. .pi, and JV = pipi + , . . . pi + M , and pi + r = pi + gm + r , q being any integer, and r any integer less than or equal to q. *The evaluation here given is a modification of one due to Fort, Zeitschrift fur Mathematik, vii, p. 46 (1862). See also Chrystal's Algebra, Part II, p. 77. SECTION II. ON THE FUNCTION OF A VARIABLE. 16. Definition. — When two variables x and y are so related that corresponding to each value of one there is a value of the other they are said to he functions of each other. If we fix the attention on y as the function, then x is called the variable ; if on x as the function, then y is called the variable. Such functions as x and y defined above are not amenable to mathematical analysis until the law of connectivity between them can be expressed in mathematical language. Classification of Functions. Functions are classed as explicit or implicit functions according as the law of connectivity between the function and the variable is direct, explicit, or indirect, implied, implicit. 17. Explicit Functions. — The simplest form of a function of a variable x is any mathematical expression containing x. Such a function is called an explicit function of x, because it is expressed explicitly in terms of the variable. Our attention will be confined in Book I principally to explicit functions of one variable. The three standard or elementary functions, x a , sin x, \og a x, and their inverse functions, xr a , sin _I x, a x , represent the three fundamental classes of functions called algebraic, circular, and logarithmic or exponential. All the elementary explicit functions of analysis are formed by combining these standard functions by repetitions of the three fundamental laws of algebra, Addition, Multiplication, Involution, and their inverses, Subtraction, Division; Evolution. Explicit functions are classified as algebraic or transcendental accord- ing as the number of operations (including only addition, multiplication. involution, subtraction, division, evolution, by which the function is constructed from the variable), is finite or infinite. 19 20 INTRODUCTION TO THE CALCULUS. [Sec. II. 1 8. The Explicit Rational Functions. I. The Explicit Integral Rational Function. The function of the variable x t a + a x x +"*** + • • • +a n x n , where the numbers a , . . . , a H are independent of x, and n is a finite integer, is called an explicit integral rational function of x, or briefly a polynomial in x. This is the familiar function which is the subject of inquiry in the Theory of Equations. Its place and properties in the system of func- tions correspond in many respects to the place and properties of the integer in the system of numbers. It can advantageously be expressed by the compact symbolism n ^5 a r x r , r—o meaning the sum of terms of type a r x r from r=otor=». II. The Explicit Rational Function. The quotient of two explicit integral rational functions of a vari- able x, a o + io 1 - Z> io io where each p r (r = 1, 2, . . . , n) represents some one of the digits o, 1, . . . , 9. If f(x) is o for some one of the interpolated numbers obtained by continually subdividing (a, b), the theorem is proved; if not, then the. two numbers a n and b n , the former always 24 INTRODUCTION TO THE CALCULUS. [Sec. II. increasing, the latter always diminishing, converge to the. common limit Meanwhile /(a,) and /(b n ) converge to the common limit f(Z), by the definition of continuity. The first of these f(a n ) is always negative, the second /(b H ) is always positive. Also, since b n — a n = h/io n , we must have £\/{K) -/(«.)} = £\ |/(*.)l + l/(«.)l I. = 2\f(S)\. But this limit is o, by definition of continuity. .-. f(£) = o. In like manner we prove the theorem when /(a) is positive and f (b) is negative. II. The general theorem now follows immediately. For, what- ever be the numbers / (a) and /(b), if N lies between them, then /(*) - n must have contrary signs wheme = a. x = b. Therefore, by I, there must be a number B, in (a, b) at which f{Z)-N=o. The important fact demonstrated by this theorem is this : If a function f (pc) is uniform and continuous in an interval (a, b) of the variable, then as the variable x varies continuously through the inter- val (a, b), the function must vary continuously through the interval determined by the numbers /*(#) and/*(£). That is, the function f(x) must pass through every number between /(a) and /(b) at least once. 24. General Theorems. — The following general theorems result immediately from the theorems on the limit, § 15, and the definition of a continuous function. I. The sum of a finite number of continuous functions is a con- tinuous function throughout any common interval of continuity of these functions. If/j(jr), / 2 (x), . . . ,/ n (x), are continuous at x, then *(*) =/,(*) + ■ • • +/.(*) is a continuous function at x. For we have £ = /,(•') + • • • +/«(*) = 0(*). Art. 24. j ON THE FUNCTION OF A VARIABLE. 25 II. The product of a finite number of continuous functions is a continuous function in any common interval of continuity of these iunctions. if 0W-/W •/,(*) • • •/.(*), then f*(*)=£\SJP) ■ ■ •/.(*')]. = £AW) ■ ■ ■ £/.(*'), =£(£*') ■ ■ ■ A(£x'), =/iW • • • /«(*) = «(*)• Corollary. Any finite integral power of a continuous function is a continuous function in the same interval of continuity. III. The quotient of two continuous functions is a continuous function in their common interval of continuity, except at the values of the variable for which the denominator is zero. If f(x) = (p(x)/ip(x), then we can consider/^) as the product of (x) and i/(J:(x). The theorem is then true by trie reasoning of the preceding theorem. Otherwise, r fix'\- r* {x,) - £ ^ x,) provided ip(x) ^ o. If tp(x) = o and cp(x) =£ o, then/"(jr) = 00 and is not continuous at x. If ip{x) = o and also {x) log f(x), £**> = £&(*) kg A*)1, = £4>(x)-£\og/(x). ... log£y=(x) - tf>{y) = TP\x/y), F(x-y) = F(x)/F(y). 6. If f{x) = #x 2 — 3x -f- ). 10. If/ = log (x -)- fx 2 — 1), x is called the hyperbolic cosine of y and written cosh y. Find this as a function of y. 11. Show that I ? 14. Investigate x« W =00 for |x| £ 1. /x n 30 INTRODUCTION TO THE CALCULUS. [Sec. II. 15. The identity ab ta + b^ la— bV' shows that the geometrical mean, \/, <-) x 4 *; - /*. Then / -f- $r = b -f- r, and / a convenient and characteristic symbolism because it shows the association of the derivative f\x) with the primitive function f{x) from which it has been derived. Art. 30.] ON THE DERIVATIVE OF A FUNCTION. 37 We shall also use sometimes another symbolism, to represent the operation by which this limit is derived, instead of the cumbersome one employed above representing the limit of the difference-quotient. We use the characteristic letter D as a symbol to represent the operation gone through of dividing the difference of the function by the corresponding difference of the variable, and determining the limit of this difference-quotient when the arbitrary value of the variable converges to the particular value of the variable as a limit. In compact symbols, we write x'( = )x But we have already agreed that this limit, the derivative, shall be represented by f\x). Hence we have the equivalent symbolism D/(x) =/'(x). Or, the operation D performed on the function f(x) results in the derivative f'(x). This operation is called differentiatio?i. 30. Observations on the Derivative. — We observe that in order that a function f{x) may be differentiate (have a derivative), it must be continuous. For, unless we have x'( = )x x'( = )x as is required by the definition of a continuous function, then, since we do have £{ x ' - *) = °, xf( = )x the value of the corresponding difference-quotient would be 00 , or no limit exists. Hence the Differential Calculus deals directly with none but continuous functions. The converse of the above statement is not true, i.e., a function that is uniform and continuous is not always different! able. There exist functions that are uniform and continuous and yet the limit of the difference-quotient is completely indeterminate for all values of the variable in certain finite intervals. * We shall not have occasion to meet any of these highly transcendental functions in this book, and the functions with which we deal will, in general, be differentia- ble. Only for isolated values of the variable will the derivatives of these functions be found indeterminate. Such values are singular values and receive treatment in their appropriate places. The evaluation of the derivative of a function falls under the case specially excepted in § 15, V. Here, the limit of the numerator * See Appendix, note I. 38 FRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. I. (the difference of the function, 4/)> an< ^ tne limit of the denomi- nator, Ax, are each o. The quotient of the limits o/o is always indeterminate. We are not concerned, in evaluating the derivative, with the quotient of the limits, but only with the limit of the quotient. We are not concerned or interested in the differetice-ratio but with the difference-quotient. This is a variable number which does or does not have a limit according as the function is or is not differentiate at the particular value of the variable considered. The derivative of any con stant is necessarily o by the definition. For, the quotient of differences is constantly o and remains o for Ax( — )o. EXAMPLES. 1. Differentiate the function x 2 . We have the difference-quotient, x ' — x = X -\- X. X — X The limit of this number when x'(=)x is 2x. . •. Dx 2 = 2X. 2. Differentiate the function x". We have x' h - x l _ I x' — x y 2 -\- x* the limit oi which is x */2 when x'(=.)x. '... Zte* 3. If f(x) ~ s i n x \ show that D sin x = cos x. We have, by Trigonometry, x' — sin x = 2 cos l(x' -j- x) sin ^(x' — x). sin x' — sin x sin l(x' — x) ; = cos Ux 1 -4- x) , -\ — '. x' -x * v ^ ' i(^ _ jr) sin ■^r =I ' / sin *(x' *'( = )* . \ f'{x) = cos jr. 4. Show that the derivative of any constant is zero. If A is any constant, it keeps its value unchanged whatever be the value of jr. Therefore the difference-quotient is A - A . = o x x — X for all values of x x =£ jt and when jr x ( = )x. Consequently DA = o. 5. Show that the derivative of the product of a constant and any function of x is equal to the product of the constant and the derivative of the function. Art. 31.] ON THE DERIVATIVE OF A FUNCTION. 39 Let a be constant and y a function of x. Let y take the value y y when X takes the value x v The difference-quotient of «y is log .v = -. According to common usage, when the base of the logarithm employed is e we omit writing the base and put log x for log,, x. 35. Derivative of x a . — Let a = p/q, where p and q are positive integers. Dividing the numerator and denominator of the difference- quotient L ± x,t — x -\ Rule: Multiply by the exponent and diminish the exponent by 1. 36. Derivative of sin x, cos x. — It has been shown in Chapter I, § 30, Ex. 3, that D sin x = cos x. The derivatives of all the other circular functions can and should be deduced in like manner. They can, however, as we shall see, all be deduced from that of the sin x. For immediate use we have, from Trigonometry, cos x' — cos x = — 2 sin \(x' -f- x) sin \(x' — x). cos x' — cos x •-,/,, x sin H x ' — x ) Hence, on passing to the limit, D cos x = — sin x. Rules for Differentiation. 37. We proceed to establish rules for the derivative of the (1) sum, (2) product, (3) quotient, (4) inverse function, and (5) function of a function, in terms of the derivatives of the functions involved. These are the general rules for the differentiation of all functions with which we shall be concerned. It is necessary to know them perfectly, for they are the tools with which the Differential Calculus works. 38. Derivative of an Algebraic Sum. Let y = u -f- v -f- w, where u, v, w are differentiate functions of x. Let the differences of these functions be Ay, Au, Av, Aw, respectively, corresponding to the difference Ax of the variable x. Then, if y, u, v, w take the values^, u x , v lf w 1 when x takes the value x lt we have y x -y = 4y> • • • a -y + 4y> and so for «, v, w. )\ — «i T" »! + U\ , y x -y = (u x -u) + (V x -V)+ (W 1 - W), or Ay — Au -j- Av + Aw. Ay _Au Av Aw Ax ~ Ax Ax Ax ' The student should observe the detail with which the difference- quotient is worked out here, as this detail will be omitted hereafter and he will be expected to supply it. 44 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II. Since the limit of a sum is equal to the sum of the limits, we have for Ax( = )o, on passing to limits, By = Du + Dv + Dw, (I) or D(u -\- v ~{-w) = Du -f- Dv -\- Dw. Corollary. What has been proved for three functions here is equally true for any finite number of functions k, . . . u n , and it can be proved in the same way that £>2u r = %Du r ; hence the rule : The derivative of the algebraic sum of a finite number of differ- entiable functions is equal to the sum of their derivatives. In all cases in which we pass from an equation in difference- quotients to one in derivatives, the student is required to quote the corresponding theorem of limits, §15, which justifies the equality. EXAMPLES. 1 . The derivative of any polynomial in x, a -f a x x + a 2 x* -{- . . . -f- a M x", is a x -\- 2a 2 x -j- 3a 3 Jf 2 -}-... -\-na n x n - 1 . This can be expressed in the following rule : Strike out every term independent of x, since its derivative is zero, and multiply each remaining term by the exponent of that term and diminish that exponent by I. 2 . If y = 2x5 -j- log x b — 3 sin x, show that Dy = 5^ -f- 5/x — 3 cos x. 3 . If /(*) =E CXl + b * + a , show that f\x) = c — ax-2. 4. Make use of the identity sin (a -j- x) = sin a cos x -{- cos a sin x, to show that D sin (a -\- x) — cos (a -f- x). 39. Derivative of a Product of Functions. Let y — uv. Then, with notation as in §40, we have Ay = (u -\- Aii){v -j- Av) — uv, = v Au -j- u Av -f- du • Av. Ay Au Av Au-Av ... Since, by hypothesis, Art. 40.] RULES FOR ELEMENTARY DIFFERENTIATION. 45 are finite, the last term on the right of (i) has the limit o when Ax( = )o ; for it can be written either A i AV M — "© or (£') A, > and Au( = )o, z/z>( = )o, when zlv(=)o, the functions being con- tinuous. Therefore, in the limit, (i) becomes D(uv) = vDu + u Dv. (II) In particular, \iv is constant, v = a. then Da = o, and D(au) = aDu. Corollary. Show that D(uvw) = uv Dw -j- uw Dv -\- vw Du, and, in general, that the derivative of the product of a finite number of functions is equal to the sum of the derivative of each function multi- plied by the product of the others. EXAMPLES. 1. Show that D{x n sin x) = nx n ~ l sin x -f- x n cos x. 2. D{x* log x) = x*-* (log x a -f- 1). 3. Show that £ {D log x sin x _ cos x i og x } — It x(=)o 4. Show that D sin 2 x = sin 2x. 2 5. If y = (log x) 2 , show that Dy = log (-*")*. 6. If /(*) EE log x 2 , then /'(*) = 2/r. 7. Show that D sin 2x = 2 cos 2jt. 8. Show that Z> cos 2jc = — 2 sin 2x. Use cos 2jt = (cos x -f- sin jc)(cos x — sin*). 9. Show that D (log Jt:*) = log x -\- 1. 40. Derivative of a Quotient. Let v = — . Thenj', «, z;, become y -f- z7>', « -j- z7w, z; -p- ^0, when x becomes x -\- Ax, and we have « -{- ^« u y ~ v -f~ Z/z> "" Z> ' z;z7# — #z7z> z>(z> -(- /fa) Au Av Ay _ Ax Ax Ax ~ v(v -f- Av) 46 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ctt. II. Since Av( = )o when Ax( = )o, we have, provided v j£ o, on l)assing to limits In particular, if u = a, any constant, then Du = o, and Dv (IV) *o ^ EXAMPLES. 1. Show that D tan .r = sec 2 x. ___ , _ _ sin x We have i? tan x = D , cos JC cos 2 jr -)- sin 2 •* cos 2 x = sec 2 x. 2. Show that Z> cot x = — esc 2 x, using both cot .r = cos x/sin x and cot x = i/tan x. 3. Show that D sec jt = sec x tan .r, using both sec x = i/cos x and sec x = tan x/sin x. 4. Show that D esc jc =r — esc x cot x. 5. Show that Z> vers x = sin x. 6. Show that D , ~ = -. * + * (* + *)* '% 8. Show that i log x log a x log ax " (log axf 41. Derivative of the Inverse Function. — If y is a continuous function of x, we must have Ay( = )o when J.v( = )o, by the defini- tion of continuity. Therefore for any particular value of x at which y is a continuous function of x we can always make Ay converge to o continuously in any manner we choose, such that simultaneously we have Ax — o. Also, for corresponding differences Ay and Ax, we have Ay Ax _ Ax Ay ~ If we represent the derivative of v with respect to x, by D x y, and the derivative of x with respect to y, by D y x, then whenever y is a differentiate function of x and D x y ^ o, we shall have x a differ- entiate function of ;', and the relation D. x y.D y x = 1 always exists. Therefore, if^ and x are functions of each other and the deriva- Art. 41.] RULES FOR ELEMENTARY DIFFERENTIATION. 47 rive of the first with respect to the second can be found, then the derivative of the second with respect to the first is the reciprocal or inverse of the first derivative. If v = f(x), then x = 0(.r), obtained by solving y = ftx) for x, is the inverse function of/"(~v). Geometrical Illustr vtiox. If the curve AB represents the function y = f(x), and we con« sider x as the function and y as the variable, we have x = (f)(y) Fig. 7. represented by the same curve, except that now Oy is the axis of the variable and Ox the axis of the function. For a particular x, the point X represents f\x) and {y). Again, if 6, are the angles made by the tangent to AB at X, with Ox, Oy respectively, measured according to the conventions of Cartesian Geometry, we have D x y =/ / (x) = tan 0, D r x = cf)'(y) = tan 0. But, since we always have tan 6 tan = 1, '• D x \ ''DyX = I. EXAMPLES. 1. If y = ** + 2ax + A then DyX = 2(x + a). 1 2(x + a) It we solve for x we get the x = inverse — a ± function Va? _l_ y _ b a function which we do not yet know how to differentiate, but we know its deriva- tive must be the value D y x obtained above. 2. \iy = -T5, find D x y. D y x, and verify DyDx = 1. 48 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II- 3. Differentiate sin - l x. My = sin-'jr, then x — sin y .-. DyX = cosy — vi _ x 2 . Hence D x y — — . Vl - x* We know from Trigonometry that the angle whose sine is x, sin - *x, is multiple-valued and that sin [nit -f- (~i)"6] = sin 9, where n is any integer. In the derivative of sin -1 .* above, the radical shows its value is ambiguous as to sign. But if we agree to take sin— l x to mean that angle between — \n and -f \it whose sine is x, there is no ambiguity, since then cos y is positive. Then we have D sin— l x = . -f Vl — x* 4. Show in like manner that D cos— x x = — Vl — x 1 where o < cos— 1 x < n. This can also be shown immediately by differentiating the identity sin— l x -f- cos -1 .*- = \it. 5. Show that D tan- 1 .* = \ * „ . I -\- x 2 Put y = tan— 1 x, then x = tan y, and DyX = sec 2 y = i -f x 2 . Ex. I, § 4] .-. Z>ta.n-*x±=. +* where we take tan— l x to be that number such that — \it < tan- 1 * < -f \it. 6. Show in like manner that D cot-Jx = 1 4- x 2 ' where o < cot -1 * < it. Also, by § 38, from tan- 1 * -f- cot- x .r = \it. + 1 7. Show that D sec -1 * = x y'x 2 — 1 If y = sec _I Jr, then x = sec y, and D y x = secy tan/ = x yx 2 — 1. Ex. 3, §41. x y'x 2 - I 8. Show, as in Ex. 7, that D esc -1 * = x \/x 2 - I Also, by § 38, using the identity csc-'-r -f- sec- 1 * = lit. Art. 42.] RULES FOR ELEMENTARY DIFFERENTIATION. 49 9. Differentiate a x . Put v = a*, then x = log a- y. Therefore, by § 34, we have DyX = - log a e. . -. D x y — = a* log, a. loga e In particular, if a — e, then Da x = a* log a becomes De* = ', and finally ;' a function of ^, then the difference- relation Au _ Au Av Aw Az Ay Ax ~~ Av Aw Az Ay Ax leads to the derivative D x u = D v u-D w v-D z w- D^-D x y whenever the derivatives on the right are determinate. Hence the following rule: The derivative of a function of a function, etc., is equal to the product of the derivatives of the functions, each derivative taken with respect to its particular variable. EXAMPLES. 1. Differentiate jr*, when x is positive and a irrational. Put j = x*, then taking the logarithm or, as we shall say, " logarating,"f we have logj' - a log x. *For a geometrical picture of a function of a function, see Appendix, Note 2. ■f-The term "taking the logarithm" is the meaning of an operation so frequently used that it seems to deserve a verb "to logarate." 50 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II. Differentiate with respect to x. We have -Dy = -. y x , \ Dy = a — = ax*-i, the same formula as when a is rational. 2. Differentiate (a -f /;x) a . Put _A .' t = .)' a , where ;' = <7 -f- far. .-. f x (y) = ay«->Dy, and Z>j/ = b. .-. D(a -f bx)* = ba{a -\- bx)*-i. 3. To find D cos x from D sin x = cos x. We have cos x = sin (-J-7T — x). D cos x = D sin (,V7T — .v). = COS (^7T — X) D(±7t — X), = — sin x. 4. Deduce in like manner D cot x, D esc x, given the derivatives of tan x and sec x. 5. If j = cos— J x, then x = cos;'. Differentiate Loth sides with respect to x. i = — siny Dy. .-. D cos— l x, as before. 6. Find in like manner D cot- 1 ^, D esc— l x, from D tan x, D sec x. 7. If y — a x , then log;' = x log tf. Differentiating with respect to x, we have D y log j • D x y = log a, or - D x y — log a. .-. D x y = a-* log a, as before 8. Differentiate \ a 1 — x 1 . Put u — a 1 — x 2 . ... Z>*«* = D u ii\D x u, = l«-2(- 2*), Va 2 - x 2 ' 9. As an example of the differentiation of a complicated function of functions. differentiate log sin e cos(a-£*) 3 t Let y — a — bx. z = {a - bxf = y\ u = cos (a — bxf = cos z, w = sin e cos («—*•*) = sin z/, Therefore the required derivative is the function — vV' sin z cos v, w which can be expressed as a function of x directly. D x v = - />. ZV« = 3 f 2 - D z ii — — sin z. D H V rrr *«. Z>-,7<7 = COS Z\ Art. 43-] RULES FOR ELEMENTARY DIFFERENTIATION. 51 43, Examples of Logarithmic Differentiation. — The differentia- tion of products, quotients, and exponential functions are frequently simplified by taking the logarithm before differentiation. EXAMPLES. 1. Show that D(uv^) _ Du Dv 1lV±i U V ' the upper signs going together and lower signs going together. Put^' = uv± l , then taking the logarithm, log y = log u ± log v. Dy _ Du , Dv y ~ it v ' This expresses compactly the formulae for differentiating the product and the quotient of two functions. 2. Show that if u x u 2 . . . un is the product of n functions of x, the derivative of the product is given by DltfUr _ V^Dllr II«U r /j Ur % I 3. Differentiate u v , where u and v are functions of x. Put y — uv and take the logarithm. ••• log }' = v log «• Differentiating, Dy n . , Du = Dv • log u 4- v . y u Du* = K» (log u Dv -J Du]. 4. Differentiate log„ u. Put r = log„ u, then v y = u. Logarate this with respect to the base e, and we have y log v = log u. Differentiating with respect to x, y „ Du log v Dy 4- — Dv = . D log„ u (Du log 11 Dv \ u log v v J log v 44. For general reference in differentiation a table or catechism of the standard rules and elementary derivatives is compiled and should be memorized. In this table u or v is any differentiate function of a variable with respect to which the differentiation is performed. 52 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cu. IT. The Derivative Catechism. i. D(cu) = c Du. 2. D(u + v) = Du + Dv. 3. D(uv) — u Dv -j- z>Z>#. z> Du — u Dv <-) v 4 /i \ Z>z> ^t) =--*■ 6. Z># a = tf« a ~ : Du. Du , 7. Z> log.w = — log a *. Z>« 8. DXogu — . u 9. Da u = # M log tfZ>w. 10. Zte M = e u Du. 11. Z>^ = «" log uDv -{- vu v ~ l Du. 12. Z> sin z/ = + cos uDu. 13. D cos zz = — sin #Z>#. 14. D tan # = + sec2 uDu. 15. Z> cot u = — esc 2 z/ Z?#. 16. Z> sec « = -J- sec z/ tan ?/ Z)z/. 17. D esc z/ = — esc u cot //Z>«. ^" 18. D sin -I « — -f- 19. D cos~ l u = — 20. D tan _I « = -f~ 21. Z> cot _I w = — 22. D sqc~ 1 u = + 23. Z> csc _r z/ = — 24. Z> vers _t z* Vi — z/ Du Vi — « 5 Du + 2" Du 1 -f- «' r Z>/< z* 4/zz 2 - Z>/z 7/ V/r - 1 Du V211 — tr ART. 44-] EXERCISES. 53 EXERCISES. 1. Differentiate by the ab initio process, and check by the catechism, the follow- ing functions : (1), x. {2), ex. (3), 2*\ U),cx*. (5),*-'. (6),*r-4. (7), x* - 2X. (8), s.r 3 - 4_r -f 7. (9), l/(ax+ b). (10), x* - $x - 2x~*. (il), (x - l)( 3 x + 2). (12). (x - 3)/(.r + 5>. (13), *i (14), j'. (15), x~i. The solution in each case depends on the fact that a n — b n is divisible by a — b when n is an integer. x (16), cos-. (17), sin ax. (18), tan ax. (19), esc ax. 2. Draw the curve y = 3X 2 and find the slope of the tangent where x = 2. 3. Draw the curve;' — x 2 -f 2x — 3, and find the angle at which it crosses the Ox axis. 4. Use the relation of the derivatives of inverse functions to find the derivatives of xK -ri x~K x~n , and check the results by the rule for differentiating a function of a function. 5. Show that the equation to the tangent to any curve y = f(x) is y=f { a) + (x-a)f'(a), the point representing/^) being the point of contact. 6. Differentiate Va* — x 2 , Vx 2 — a 2 , Va 2 -\- 2bx. Ans. - x(a 2 - x 2 )~i, x(x 2 - a 2 )~l , b(a 2 -+- 2bx)~i. (1) D.a -f xy = c{a -f xy-i. (2) D(a + x 2 ) 3 = 6x(a -f- a: 2 ) 2 . (3) D{c -\- bx 3 )* ■= \2bx\c 4. ^x 3 ) 3 . (4) Z>(ax 2 4- £x 4- cf — $(ax 2 4- bx + ^(2ax 4- 3). (5) Z>(a» - x 2 ) 5 = - \ox{d 2 — x 2 f. (6) D(a 2 x 4- &* 2 ) 7 = 7(« 2 x 4- bx 2 )\a 2 4- 2^). (7) D(b 4- or"*)* = mncx™-\b -f ftp«)»-i. (8) Z>(i -f ax 2 )' 1 = - ax(i 4- ax 2 )~K (g) D{a 2 - x*y = - \x\a 2 - x*)~K (10) D sin* ax = — D cos 2 ax = a sin 2ax. (11) Z> sin M ax — na sin^-^ax cos ax. (12) Z? sin (sin x) = cos x cos (sin x). 7. Show that the equation to the tangent at x = a, y = ft, for the curve (1) x 2 4- y* = a 2 is xa 4"//? = fl2 - . . x 2 y 2 . xa yft (2) - ± ^ = 1 is — ± <-' - = 1. v ' a 2 b 2 a 1 b l (3) f 2 - \P X is J/ 3 = 2 /(* + a). 8. Given sin 3X = 3 sin x — 4 sin 3 x, find cos 3*. 9. Given cos 5X = 16 cos 3 x — 20 cos 3 x-f* 5 cos •*"> find sm S x - 10. Verify cos x = 1 — 2 sin 2 |x, by differentiating. 11. Obtain new identities by differentiating sin 3a 4" s i n 2a — sin a = 4 sin a cos |a cos fa, sin <5 sin (£# — b) sin (J7T 4~ ^) = i sin 3^> a and ^ being variables. 54 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. II. 12. Differentiate the identity- cos 3 2x — 3 cos ix — 4(cos 6 x — sin 6 x). 13. Differentiate x 3 sin x, x' 2 \/a -)- bx, (ax -\- by. 14. D[{x + l) 5 (2* - i) :! ] = (\6x + i)(x -f l) 4 (2* - I) 2 . 15. /ft* + !)(*■ - *)*] - 7 2 j/x 3 — Jf 16. Z?j(i - 2j-f 3x 2 - 4x 3 )(i + xfl = - 20jt- 3 (i 4- x). 17. Z>{(i - 3JC 2 + 6**)(i + -r 2 ) 3 } = 6o^(i + a-)-'. 18. Show that i + x _ i _ D x -f- 3 3 - 6x - jc- 2 \ i - x (l _ x) ^ _ ^ x 2 + 3 (x 2 4- 3)-' x n nx n - 1 n x __3 V 1 + x2 (i -f x 2 ) ? ' 3 + x* (3 + X 2 f ^ x \/a -\- x + \/a — x __ a 2 -\- a \/ a 2 — x 2 \fa -\- x — 4/ a — x x l \/a 2 — x 2 19. Show that x I _ . 3 + 2jc i JD tan-i ■ = — ; Z>sin-i">— i 4/1 — x 2 4/1 — * 2 4/13 |/l - 3* — x 2 2?. Differentiate sin— l (x/a), tan— '(ax -|« ^)> cos -1 - — , sec — 1 (a/x), 4/a 2 -f- ■* 2 sec- J (jr + ex 2 ). 21. Z> log sin x = cot x; D log cos x = ? 22. Differentiate e 2x , e~ x , e nx , e sinx , Jogx. 23. Differentiate a cx , a s ™ ax , a l0 &* a****. 24. Differentiate log x^, log (a -)- x), log (ax 4- /;), x M ^, a*ex. 2*, e* log (x + a), log (.*• -f- e x ), ^°§ (■*'*)> s * n (**) * g ■*» *" cos * leg (cos x), log a tan x, $ lo e x , $ sinax , \og x 2 (cos ax). 25. Z> sin [cos (ax -f- b) n ] = — na(a -f &r)*-i sin (ax -f £)» cos [cos(ax + b)"] . 26. If j = £(** - *-*), show that ^ = log ( y + 4/i +/>), and that jC^j Z^j: = 1. 27. In Ex. 1, § 41, differentiate xasa function of>' and check the result there given. CHAPTER III. ON THE DIFFERENTIAL OF A FUNCTION. 45. Definition. — The differential of a function is denned to be the product of the derivative of the function and an arbitrary differ- ence of the variable. If f{x) represents any function of x, and x\ — x any difference of the variable, then (-v, - *y(*) is the differential otf(x) at x. The value of the differential at a particular value x depends on the value assigned to the arbitrary number x v We use, after Leibnitz, the characteristic letter d to represent the differential, and write d/[x) to represent the differential of the func- tion J\x) at x. Thus df(x) = (x 1 - x)f\x\ =f'(x)Ax. 46. Theorem. — The differential of a function is equal to the product of the derivative of the function into the differential of the variable. For, \ttf(x) = x, theny^jr) = 1, and dx = Dx • Ax = Ax. Therefore we can write dx for Ax, and have #T») =/'(•*) dx - The differential of the variable is then any arbitrary difference or increment of the variable we choose to assign. In writing the differential of a variable we choose to assign to it always a finite number as its value. In fact we cannot assign to it any other value. 47. The Differential-Quotient of a Function.* — Since the differ- ential of the variable is a finite number we can divide by it, and have ^-m * By some writers the derivative f\x) is called the differential-coefficient of the function /(jc), because of its relation to the differentials in the equation dffx) =/'(x)dx. 55 56 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. III. or, the differential-quotient of a function, which is the quotient of the differential of the function by the differential of the variable, is equal to the derivative of the function. This furnishes another notation, due to Leibnitz, for the value of the derivative, and expresses that number as the quotient of two numbers. The advantages of this notation will appear continually in the sequel, in the symmetry of the equations, and in the analogy and relation of differentials to differences. We frequently abbreviate the differential-quotient into df f , dv /-=/', or — , dx dx where y =zf(x). Also, \xhenf(x) is a complicated function we fre- quently write df(x) d dx s z*» 48. Geometrical Illustration. — We have seen, § 31, that if y =/(x) is represented by a curve PP X , then the derivative/" (x) or J / y ^ p> M s$\ a a h X Fig. Dy is represented by tan 0, where 6 is the angle made by the tangent PT to the curve at P with the axis Ox. Assign any arbitrary number x xt and let P x represent f(x x ), and T the corresponding point on the tangent to the curve at P. Then we have PM —x x — x — Ax — dx. df(x) = (x x - x)f[x) 9 = PM tSLTi MP1\ = MT. MT therefore represents the differential of the function f(x) at x corresponding to x v While MP, =/(.v,) -/(.v) = Af(x). rf/"and Af are more nearly equal when Ax or dx is a small number. f(x) = tan 6 £ Art. 49.] ON THE DIFFERENTIAL OF A FUNCTION 57 Observe that for a particular x the differential-quotient d/{x) dx is constant for all values of x v 49. Relation of Differentials to Differences. — Since the differ- ence-quotient has the derivative for its limit, we can put + ., where a( = )o, when Ax(=)o. Therefore AJ\x) —f\x)Ax + a Ax, — f\x) dx -\- a Ax. . 4M -, I a ' ' d/(x) + f'( X y Hence, when _/*'(.*•) 7^ o, we have 4A*) _ . This substantiates the remark made in § 48 that the difference and the differential of a function are more nearly equal the smaller we take dx. 50. Differentiation with Differentials.— Observe that all the formulae in the derivative table, § 44, are immediately true in differ- entials when we change D into d. For we need only multiply such derivative equation through by dx in order to make it read differen- tials instead of derivatives. We have d/[u) — D x /(u) dx. For, by definition, df{u) = DJ(u) du, = D u /(u).D x udx, = D x /{u) dx, since D x f(u) = f (u) D x u, and du = D x udx. .-. D u f(«)du = JD x f(u)dx, or the first differential of a function is the same whatever be the variable. More generally, let u, v, and w be functions of x. Distinguish- ing differentials like derivatives by subscripts, we have dj\u) = D v /(u) dv = D u f(u) D v u dv, — D u f{u) du = dj\u). In like manner, d w f(u) = d u f{u). Therefore 58 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [ Ch - iil or the differentia] of a function is independent of the variable employed. It is not necessary, therefore, to indicate the variable by subscripts or in any other way; in fact the variable need not be specified. It is due to this that frequently the use of differentials has marked advantages over that of derivatives. 51. We add a further list of exercises in differentiation, using in- differently the notations of derivatives, differential-quotients, and differentials in order to insure familiarity in their use. The sequel will show the advantage of each in its appropriate place, Art. 51.] EXERCISES. 59 EXERCISES. 1. If x, y, are the coordinates of a point on a curve, show that or ( V — y ) dx — {X — x) dy is the equation of the tangent at x, y, where X, Fare the current coordinates on the tangent. This equation can also be written Y - v X - x dy dx 2. Show, with the above notation, that (F — y) dy -j- {X — x) dx = o is the equation of the normal at x, y. 3. Show that d(x s logx) = x 2 (log x 3 -f- 1) dx. 4. d(cos mx cos nx) =. — ;;/ cos nx sin mx dx — n cos mx sin nx dx. ~ d . 5. -7- sin M x = n sin"— 1 x cos x. dx 6. d sin (1 -(- x 2 ) = 2 x cos (1 -\- x 2 ) dx. 7. If y = sin'" x sin «, show that <#/ sin 2 x — = m sin w + J x sin(/;z 4- i)x. ox 8. D{a sin 2 x -|- <$ cos 2 x) M = n{a — ^) sin 2x (a sin 2 x -f- ^ cos2 •*) M— *« 9. d sin(sin x) = cos x cos(sin x) dx. 10. f{x) = sin-i(jc w ) ( show/'(x) = «x m -i(i — x 2 *)-*. 11. ^sin-i(i — x 2 )i = — (I —x*f^dx. d 3 4- a cos x ^a 2 " — b 2 12. -r- COS- dx a -\- b cos x ~~ ' a -f- ^ cos x 13. d sec M x = n sec M x tan x dx. idx 14. a sec - J (x 2 ) = ■■ ■ . x Vx* — 1 15. d(a 2 -f x 2 )i = x(a 2 + x 2 )-*/" r ' __ rf sin"' sin'"- 1 40. — 7- « = 7~( m cos 6 + « sin 0)- rt'O cos" cos"+i v T ' 41. 2 (a + /;x -f fjr 2 ) = B{b + 2rx), = 2C. 2. Z> cos <7jr = — a sin a.r, D 2 cos fl-Jtr = — aD sin rt.r = — a 2 cos tf.r. 3. Z? log ax = - log ax = — fl/jf 2 . 4. D \fa 2 - x 2 = — x(a 2 - x 2 )~*, £T- tfa 2 — x* = — (a 2 - x 2 )-> -f x 2 :a 2 — x 2 )~%. 53. Successive Differentiation. — The second derivative like the first is, in general, a differentiate function. Its derivative is called the third derivative of the function, and written /»,.,, ££& /"(*) Xl( = )X 62 Art. 54-] ON SUCCESSIVE DIFFERENTIATION. 63 In general, if the operation of differentiation be repeated n times on a function /[x), we call the result the «th derivative of the func- tion. We write the »th derivative in either of the equivalent symbols D"f(x) =/»(*). It is customary to omit the parenthesis in f [u) {x), including the index of the order of the derivative attached to the functional symbol /"when there is no danger of mistaking it for a power, and write D«/{x) =/■(*). The index of either D or f in D" , f n denotes merely the order of the derivative and number of times the operation is performed. 54. Successive Differentials. — In defining the first differential of a function, the differential of the independent variable was taken to be an arbitrary number. In repeating this operation it is con- venient to take the same value of the differential of the independent variable in the second operation as that in the first. In other words, we make the differential of the independent variable constant during the successive differentiations. Thus the second differential oif(x) is = d[/\x) dx], = d[f'(x)].dx, (i) since dx is constant. But, by the definition of the differential, «*[/»] = J3[/\x)l dx, =f"(x) dx. (ii) Substituting in (i), we have for the second differential d*f(x) =f"(x)(dx)\ or the second differential of a function is equal to the product of the second derivative into the square of the differential of the variable. It is customary to write the square of the differential of the variable in the conventional form dx 2 instead of [dx) 2 , whenever there is no danger of confounding dx 2 = (dx)* with d(x) 2 , the differential of the square of x. We shall write then dV(x)=f\x)dx\ In like manner for the third differential oi/[x) d\dy(x)-]=d[f'\x)dx*l ' = d[f"(x)].dx*, since dx is constant; and since by definition 4T(*)] = D[/"(*V\ **, =/'"(x) dx, 64 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV. we have for the third differential <^/(-v) =/'"(•*) JA and so on. In general, the nth differential of a function is equal to the product of the nth derivative of the function into the nth power of the differential of the independent variable. In symbols d n /{x) — f M (x)dx n f where it is always to be remembered that dx n means (dx) n , and d n , f n indicate the number of operations and order of the derivative respectively. EXAMPLES. 1. We have d sin x = cos x dx. and d l sin x = fl\cos x dx) — d(cos x)-dx =. — sin x dx*. 2. d*{a + bx 2 ) = d[2bx).dx — 2b dx 2 . 3.d 2 \o g x=d^.dx = -^. 55. The Differential-Quotients. — The nth differential-quotient of a function is the quotient of the nth differential of the function by the nth power of the differential of the independent variable. In symbols we have, from § 54, This symbol is also written, for convenience, in the forms ill of which notations are equivalent to either of /3"/(.v)=/»(.v), and are used indifferently according to convenience. 56. Observations on Successive Differentiation. — In practice or in the applications of the Calculus we require, in general, only the first few derivatives of a function for solving the ordinary problems that are proposed. But, in the theory of the subject, i.e., the theory of functions, we are required to deal with the general or nth derivative of a function in order to know all the properties of the function. The formation of the nth derivative of a given function presents no theoretical difficulty, but owino: to the fact that differentiation, in general, produces a function of more complicated form (owing to the introduction of more terms) than the primitive function from which it was derived, the successive derivatives soon become so Art. 56.] ON SUCCESSIVE DIFFERENTIATION. 65 complicated that the practical limitations (of our ability to handle them) are soon reached. The Differential Calculus as an instrument for investigating func- tions finds its limitations fixed by the complexity of the general or nth derivative of the function whose properties we wish to investigate. There are a few functions whose nth derivatives can be obtained in simple form, as will be shown below. We are aided in forming the «th derivatives of functions by the following: (1). The nth derivative of the sum of a finite number of functions is equal to the sum of their «th derivatives. (2). The «th derivative of the product of a finite number of func- tions can be determined by a formula due to Leibnitz, which we shall deduce presently. (3). The nth derivative of the quotient of two functions can be expressed in the form of a determinant and in a recurrence formula, directly from Leibnitz's formula. This is done in the Appendix, Note 3. (4). The nth derivative of a function of a function can be expressed in terms of the successive derivatives of the functions involved. This is also given in the Appendix, Note 4. In the application of the Calculus to the solution of ordinary geometrical questions, we need the first, frequently the second, and but rarely the third derivative of a function. When the function is given explicitly in terms of the variable, these derivatives are found by the direct processes as heretofore applied. If the derivatives are to be found from an implicit relation, such as -f 2 {D e P f -pDlp ~ (cos 9 D e p — p sin S) 3 ' In which D e p and D\p must be determined from the polar equation (p(p, 0) = o.* EXAMPLES. 1. Thewth derivative of x a , a being constant, (i). Let a = m be a positive integer. Then Dx m — mx m ~ I , D 2 x m = m{m — i)x™- 2 , J)n x m — m ( m _ i) . . . ( w _ w _j_ i)x>«-*, for all values of n < m. If n = m, then D m x m = m(m — i)... 3.2.1= m\ This being a constant, all higher derivatives are o. .-. D m +/>x m = o for all positive integers p. Also, when x = o, D n x™ — o, n < m. (2). Let the constant a be not a positive integer. Then, as before, D n x a = a(a — 1) . . . {a — n -f- i)jr a - w . Whatever be the assigned constant a, we can continue the process until n > a, when the exponent of x will be negative and continue negative for all higher deriv- atives. Consequently, when x = o, D n x* = 0, n < a. D*x* =oo, n > a. * The differentiation of an implicit function (p(x, y) = o is, properly speaking, the differentiation of a function of two variables, and a simpler treatment will be given in Book II. It will be shown in Book II that the derivative of y with respect to x, when 0(x, y) = o, is dep dy _ dx dx = ~ djp ' by where -^ means the derivative of >' — ax y — °> 90 , , dip then ~ = 6x 2 - ay ; — - = - o^ 2 - ax. Ox oy Therefore, as in the text, dy _ 6x 2 - ay dx ~ oy 2 -+- ax ' Art. 56.] ON SUCCESSIVE DIFFERENTIATION. 67 2. Deduce the binomial formula for (1 -j- x ) u t when the exponent n is a posi- tive integer. We have (1 + x)(i -f x) = (I -f af = i + 2x + **, (1 + x)(i + *)« = (f + *)P = 1 + 3* + 3 *a 4- «•; By an easy induction we see that (1 -f .r) w must be a polynomial in x of degree n. It is our object to find the numerical coefficients of the various powers of x in this function. Let (I + x)» = a -f- a x x -f a^a + • • • + ««*«. Differentiating this r times with respect to x, we have »(«— I) . . . («— r-(-l)(i+jr)»- r =r! a r -f- . . . -f- n[n— 1) . . . (n—r-\-i)a n x H ~ r . This equation is true for all assigned values of x and r, and when x = o, «(» — 1) . . . (« — r -j- i\ /Z_ — - /. a number which it is customary to represent conventionally by either of the symbols P> This number is of frequent occurrence in analysis. In Algebra, when n is an integer, it represents the number of combinations of n things taken r at a time. Hence we have the binomial formula of Newton, (I +*)« = J C n xr. (I) r = o Corollary. If we wish the corresponding expression for (a -\- y) n , then {a+yy = a* (1 + ^V- Put j/a for x in (1), and multiply both sides by a n . ... {a+y)" = 2 C M r a«-rxr. r = This can be written more symmetrically thus: (a -f j/)» _ * a«-r a:'' « ! — ^ (n — r , ! r f o V ' 3. The wth derivative of log x. We have D log x — — = .r-i. Therefore, by Ex. 1, Z>» log jr = (— i) w -i(» — 1) ! — . 4. The «th derivative of a*. We have Da x = a x log a. . •. D^a* -- a x (log a) n . In particular, Zte* = e x \ D n e x — ^*. This remarkable function is not changed by differentiation. 68 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. IV. 5. The nth derivative of sin x and cos x. We observe that D sin x = -j- cos x\ D cos x = — sin x; D 2 sin j: = — sin x\ D 1 cos .r = — cos x; D* sin x = — cos x; Z> 3 cos x = -f- sin x\ D k sin x = + sin ;r; Z> 1 cos jr = -f- cos jr. Thus four differentiations reproduce the original functions and therefore the higher derivatives repeat in the same order, so that D±n-\ sin x = (— i)*- 1 cos x; D 2H ~ l cos x = (— I) M sin jr; D 2n sin a: = (— l)« sin jc; Z> 2M cos # = (— ij" cos x. In virtue of the relations cos x = sin [lit -\- x), sin jt = — cos (\it -\- x), these formulae can be expressed in the compact forms D n sin x = sin [ # -| # Z>" cos x = cos f jc -| 6. Given ^ +4 = I > find ^7,^7- n \ Differentiating with respect to x, Differentiating again, x y ay |_ <- jL — o. V- x — — , since — = - , ay 6 dx a 1 y Differentiating again, we can find d' A y -$b*x dx 3 ~ a*y* 7. If y* = 4 ax y — °? dy _ x- — aV d 2 y dx ~ y 2 — ax ' dx* " ~ (j 2 — ax) 3 ' 10. If sec x cos j = a, show that dy _ tan x dy _ tan 2 v — tan dx tan 7 ' dx 2 ~ tan 3 j w yf dy 3 a 2 x dx*~ (y- *f show that zcPxy Art. 57. J ON SUCCESSIVE DIFFERENTIATION. 69 57. Leibnitz's Formula for the nth Derivative of the Product of Two Functions. — Let //, v be any two functions of .v. For sake of brevity, let us represent the successive derivatives of u and v by these letters with indices, thus : u , u , «,..., w v', v" ', v'" 9 . . . , v" Then D{uv) = u'v -j- v'u, D 2 (uv) = u"v -f- u'v' -f- u'v' -f- v"u, — 2i"v -J- 2«V -f- 7/z>". In like manner, differentiating again this sum of products, we find on simplification B 3 (uv) — u'"v -j- 3«'V -f 3«V + «z>'". Observing, when we use indices to indicate the derivatives, the symbols LPu, f°(x), v°, mean that no differentiation has been performed and the function itself is unchanged, . • . JDPu = u° = u, and f°(x) =/(x). In the above successive derivatives of uv we observe that the indices representing differentiation follow the law of the powers of u -f- v when expanded by the binomial formula, and the numerical coefficients are the same as those in the corresponding formula of that expansion In order to find if this law is generally true, let us assume it true for the nth derivative and then differentiate again to see if it be true, in consequence of that assumption, for n -j- 1. Assume that (see Ex. 2, § 56) n D n (uv) = 2C Mtr u n ~ r v% r = o = u n v +C HtT u n ~ l v'+ ...+C Mtr u n ~ r v r + . . . + uv n . Differentiating this, we have &*+ 1 (uv)=u n+I v+C„ tJ «V+...-fC M)r u n - r+l v r +...-j-u / v" _j_ u n v , +...-\-C njr _ 1 u n - r+l v r -\-...+nu'v n +uv n+ \ = t^^v^C n+ltl u-v'-\-...-\-C n+lr( X ) n\ ^ (n - r)\ r\ ' 58. Function of a Function. — A formula for the «th derivative of a function of a function will be deduced in the supplementary notes.* However, the simple case of a function of a linear function of the independent variable is so useful and of such frequent occur- rence that we give it here. Let u = ax-\-b, and f{u) be any differentiable function of u. Then D x f{u)=f u { U )D x u, = dfju). Dl/(u) = aD x [f u {u)] i = af' u \u) Du, and generally 1. Show that 2. D"e ax = a n e ax 3. Show that I) x /(u) = *"/». EXAMPLES. D n sin ax = a M sin (ax -\ 7t\ £>" cos ax = a n co. c [ax -I 7t\. \ 2 / D n (-±-\ - W! "" \x - a) ~ (x - «)»+i Appendix, Note 4. _, Art. 58. 1 EXERCISES. 71 EXERCISES. Show that \x) x*+i \x») K ' x n + r 3. DA = r\ — - . \C - x) \C - X)r+i 4. D« log (1+*)= (- i)-« |* ~^ . 5. Z> 4 (x 3 log jr) = 6.X-1. 6. /^(jc 4 -(- M (* log *) = (- i) n (n — 2)! jr-*+i. 11. £> 2 .r* - ^(1 -f log Jf) 2 + #»-i. 12. Z> 3 log (sin .r) =: 2 cos x esc 3 .r. 13. Z^JcMog.r?) = 2±.r-i. 14. Z?*a« = a* (log a c ) M . ./ ! { ac -\- b ac — b 15 X* - C* K ' 2C \ {X — C)n+i (X + ,)«+ Observe that by the method of partial fractions we can write ax -J- b I $ ac -\- b ac (x — c)(x -\- c) 2c I x — c x -\- c ax -\- b _ ^ 1 l ap + £ _ ^_+_^\ _ . (jf - /)(* - q) p -q\x—p x - q) 17. Make use of the method of partial fractions, to find the «th derivative of ax i 4. bx + c _ I t af+pb-\-c aq 2 + bq -\- c \ (x - p){x - q) = / - ? ^ x - p x - q J + d _ _, , I d\*2x 2 — 4_r— 6 , «! 18. Show that — ] — ^ = - 6 — - . \y.r J x 2 — $x -f- 6 (2 — x) M + I 19. If j = fl(l + x 2 )-i, show that (1 + x 2 )y( n ) 4- 2nxy( H -V 4- «(» — i)_y(*- 2 ) = 20. If f(x) = a cos (log x) 4- b sin (log x), show that x 2 f"{x) + xf\x)+f(x) = o. 21. Show in 20 that the following equation is true : x 2 /«+2 4- x(2n 4- i)/«+i 4- (« 2 4- i)/ n = O. 22. If ;/=*<* sin -1 * show that (1 — jc 2 )"+ 2 , y n+1 , y n - 72 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cn. IV. 23. If y = (x + IV 2 - i)"\ show that (x 2 — i)d 2 y + xt"(pt — il>tt" dt _ (4>tf dx , d 2 y dx dy d 2 x ~di 2 ~di ~ "dt ~dt 2 m ' 29. If x = : sin 3/, y = cos 3/, show that Dly = - y~3. 30. Z>» tan- ... , sin (n tan- 1 .*— - J x — (— 1 )«-x(» 1) ! v . ; /{a) = /(/?), the function must increase between a and x' and decrease between x' and /?, in order to pass from /"(a:) to the greater value/j^r'), and from/^') to the lesser value /[/3). Also, if /(•*') in (x 1 , x 2 ), and therefore in (a, /?), at which we have /'{£) = o. In particular, if /{a) = o and/(/?) = o, then there is a number B, between a and fi at which f'(S) = o. Rolle's Theorem is usually enunciated : If a function vanishes for two values of the variable, its derivative vanishes for some value of the variable between the two. Or, the derivative has a root between each pair of roots of the function. The figure in § 60 illustrates the theorem. 62. Particular Theorem of Mean Value. — If f(x) is a one- valued differentiate function having a continuous derivative in [a, /?), and if a and b are any two values of x in [a, /?), then fyl) -/(a) = (b- a)/'{S), where £ is some number in [a, b). 7 6 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V. The truth of this theorem follows immediately from Rolle's Theorem. Let k represent the difference-quotient A") k=m Then or b-a A*) -A«) = (* - «)*> f{b)-kb=f(a)-ka. (i) The function f{x) — kx is equal to the number on the left of (i) when x = b, and to the number on the right when x = a. Therefore, by Rolle's Theorem, having equal values when x = a and when x = b, its derivative must vanish for x = B, between a and b. Differentiating, f{x) — kx, k being independent of x, we have at £, /'(g) -k = o, which proves the theorem. Another way of establishing the result is to observe that the function (a - b)/(x) - (x - b)f(a) + (.v - a)/(i) vanishes when x = a, also when x = b. Therefore its derivative must vanish for some value of x, sav £,, between a and b. .-. (a - i)f'{£) -f(a) +f(i) = o. Geometrical Illustration. y ? i M %^ B' A' X' ~*^\ c i i V ( ) Fig. io. Each of these processes admits of geometrical illustration. (i). k is the trigonometrical tangent of the angle which the secant AB makes with Ox. Draw OA'B' parallel to AB. Then BB' = /(/;) - kb = A A' - /[a) - ka. XX' = f{x) — kx is equal to AA' when x = a, and to BB' when* == 6. The theorem asserts that there is a point E on the curve y ~ J\x) having abscissa % at which /'(£) = k, or the tangent at E is parallel to the chord AB. (2). The function (a - d)/(x) -(x- b)f{a) + (x - a) /(b) is nothing more than the determinant /(->-). x, I /(a), a, I /(/')• b, i Art. 63.] ON THE THEOREM OF MEAN VALUE. 77 which is the well-known formula in Analytical Geometry for twice the area ol the triangle AXB, in terms of the coordinates of its corners. This vanishes when X coincides with A or B. It attains a maximum when the distance oJ X from the base AB is greatest, or when X is at E, where the tangent is parallel to the chord. This theorem amounts to nothing more than Rolle's Theorem when the axi - of coordinates are changed. 63. Lemma. — Ex. 39, § 58, forms the basis of the most important theorem in the Differential Calculus, i.e., the Theorem of Mean Value for a function of one variable. On account of its usefulness, we inter- polate its solution here. The starting point of the Differential Calculus is the difference- quotient. On that is based the derivative of the function. We shall now use it in presenting the Theorem of Mean Value. Let/(x) be a one-valued successively differentiable function of x in a given interval (a, (3). Let x represent any arbitrary value of the variable, and y some particular value of the variable at which the derivatives of/~are known. (1). Consider the difference-quotient A*)-M x — y If we hold x constant while we differentiate this n times with respect to the variable y by Leibnitz's Formula, § 57, and then multiply both sides by (•* ->r +1 n\ we shall obtain /(-v) -Ay) - (* -y)f\y) ----- ^^>W = {*-y) a+1 (*\" (A*)-Ay) \ For, we have *>;IA*) -Aril = -fh')> D;- r (x -y)- 1 = (n - r)\(x - y y-<"+», which values substituted in the form of Leibnitz's Formula in Ex. 3, § 57, give the result. (2). On account of the importance of this formula we give another deduction which does not use Leibnitz's Formula directly. Let A*) -Ay) = Q x-y Then A*)-Ay)±(*-s)Q- 78 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V. To introduce the known derivatives at y, let x be constant and differentiate this last equation successively with respect to_>'. Thus -/X>') =(*-jOg_a (2) -f'\y) = (* ->')«' - *& (3) -/«0) = (* -j^)(2t M) - «^- 1} . (»+ 1) Multiply (2) by (a- -jy), (3) by i (* - j^ . . . , and (a + 1) by — (x — y) n , and add the n -f- 1 equations. There results /(-v) -/w - (*-j-)/*w-. • • - ^r-/"0) = ~^' m =/(*) -A») -(*-My (*)-...- i-j^/'W - '' n ,' Q in which, as in § 63, does not contain z and is constant with respect to z. Observe that this function F(z) is o when z = x, because the first two terms cancel and all the others vanish when z = x. Also, F(z) is o when z =y, by reason of the identity (g). Consequently, by Rolle's Theorem, § 61, the derivative F'(z) must be o for some value B, of z between x and y. Differentiating with respect to z, and observing that the terms on the right, after differentiation, cancel except the last two, we have *'(«) = - {jL ^r 1 f ,+ \*) + (» + ^r 1 ^- Hence, when z = £,, at which F'(£) = o, V * ~~ n + 1 * Art. 65.] ON THE THEOREM OF MEAN VALUE. 79 Substituting this value in (q), we have Lagrange's form of the Theorem of Mean value,* /(•v)=A.)+(.v-t')/'0')+- • .+ (£ =r ! /"W + { -^^f M ( s )' Sr-^+V+V 7 " 1 ^- (L) =1 65. Theorem of Mean Value. Cauchy's Form. — Cauchy has given another form to the evaluation of the difference r = o which for some purposes is more useful than that of Lagrange. Its deduction is somewhat simpler. Let x be constant and z a variable. Consider the function F(z) =/{z) + (x- z)f\z) + ... + (X ~ Z) "/ "(z). (i) By the Theorem of Mean Value, § 62, F(x) - F(a) = (x- a)F'(S), (ii) where B, is some number between x and a. When z = x, we have from (i) F(x) =/{x). When z = a, then from (i) F(a) =/(a) +(x- d)/\d) + ... + {X ~ a) " / "(")■ Differentiating (i), and F,'{*) = £-^/»H>>), F'(S) = ^—fil/^S). Substituting in (ii), we have Cauchy's form n /{x) = J^ (f-=^L r{a) +{x - a) { -^±y^(£). (C) r = o * In order that this result shall be true, it is necessary that the function f{x) and its first n -f- 1 derivatives shall be finite and determinate at x and at^', and also for all values of the variable between x and^. This important formula will be presented in another form in the Integral Calculus, Chapter XIX, § 152. For a proof of the Theorem : If a function becomes 00 at a given value of the variable, then all its derivatives are 00 there, and also the quotient of the deriva- tive by the function is 00 , see Appendix, Note 5. 80 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. V. The numbers represented by B, in (C) and in (L) are not equal numbers. All we know about S, in either case is that it is some number between certain limits. 66. Observations on the Theorem of Mean Value. — The formula (L) or (C) is a generalization of the theorem of mean value stated in § 62; that theorem corresponds to the particular value n = o. The Theorem of the Mean is the basis of the expansion of a function in positive integral powers of the variable. When this expansion in an infinite series is possible, it solves the problem : Given the value of a function and of its derivatives at any one par- ticular value of the variable, to compute the value of the function and of its derivatives at another given value of the variable. The Theorem of Mean Value is the basis of the application of the Differential Calculus to Geometry in the study of curves and of sur- faces, as will be amply illustrated in the sequel. It solves the problem : To find a polynomial in the variable which shall have the same value and the same first n derivatives at a given value of the variable as a given function. This polynomial, therefore, has the same properties as the given function at the given value of the variable, so far as those properties are dependent on the first n derivatives. This is a most important and valuable property of the formula, for it enables us to study a proposed function by aid of the polynomial, and we know more about the polynomial than about any other function. 67. In Chapters I, . . . , IV, we may be said to have designed the tools of the Differential Calculus, for functions of one variable, in the derivatives on which the properties of functions depend. In the present chapter this design may be said to have culminated in the presentation of the Theorem of Mean Value. The subject has been developed continuously and harmoniouslv from the difference-quotient. The difference-quotient is the founda- tion-stone from which the derivatives have been evaluated, and by successive differentiation of the difference-quotient we have been led to the Theorem of Mean Value. It is not necessary to add here any exercises or examples of the application of the Theorem of the Mean, since it will be employed so frequently in what follows. We merely notice other forms under which the formula may be expressed. 68. Forms of the Theorem of Mean Value. (1). It is customary to write R n as a symbol of the difference between the functions n f{x) and £<*^*T/'(a), Art. 68.] ON THE THEOREM OF MEAN VALUE. Si so that n o Or, more briefly, where S H represents the 2 function. (2). In particular, if a = o, and f(x) is differentiable, n -\- 1 times at o and in (o, x), we have /(-*) =/(o) + .v/'(o) + . . . + J/»(o) + ie. , where, using Lagrange's form, R « = (£pi)/" +, <*>' * in (°> *>• or, using Cauchy's form, R = X ( X ~^' % /»+U£), £ in (o, x). (3). If we write the difference x — y — h, so that x =zy -\- h y /O + h) =/0) + a/'O) + • • • + J/*O0 + *«■ (4). Again, since h is arbitrary we can put h = dy. Then /ty + 60 =f(y) + d/(y) + ... + ^M +J fi>„, or „=*+%+._..+%+*. EXERCISES. 1. If f(x) = o when .# = «j , . . . , x '— a n , where a x < a 2 < . . . < a H , and/^jr) and its first n derivatives are continuous in {a x , #„), show that /(x) = (^_a 1 ). ..(x-anY-^, where £ is some number between the greatest and the least of the numbers x, a x , . , . , <*„. 2. In particular, if a x — a 2 = . . . = a n = a, then /w =^/.«), where | lies between .* and a. CHAPTER VI. ON THE EXPANSION OF FUNCTIONS. 69. The Power-Series. — To expand a proposed function, in general, means to express its value in terms of a series of given func- tions. This series has, in general, an infinite number of terms, and when so must be convergent. We confine our attention here to the expansion of a proposed function in a series of positive integral powers of the variable, based on the Theorem of Mean Value. The problem of the expansion of a proposed function in an infinite series of positive integral powers of the variable does not admit of complete solution in general, when we are restricted to real values of the variable, for the reason that the values of the variable at which the function becomes infinite enter into the problem, whether these values of the variable be real or imaginary. In the present chapter we shall confine the attention to those simple func- tions whose expansions can be readily demonstrated in real variables, relegating to the Appendix * a more complete discussion of the gen- eral problem. 70. Taylor's Series. — If in the formula of the Theorem of Mean Value, n f(x) =£ (x - a) y «{a) + R„ , (I) r-o the derivatives f r (a), r — 1, 2, . . . , at a, are such that the series r = o has a finite limit when n = 00 , and we also have £ *n = O, n= 00 then for the values of x and a involved we have A*) = A") + C* - «)/*(«) + —r^-/"^) + • • • (T) This is called Taylor s formula or series. * See Appendix, Notes 6, 7, 8. 82 Art. 7 i.] ON THE EXPANSION OF FUNCTIONS. ■ s 3 We may use any of the different forms of R n we choose in show- ing £R n = o. 71. Maclaurin's Series.— Under the same conditions as in § 70 if a = o, A*) =/(°) + ■*/» + ~/"(o) + . . . ( M ) This is called Maclaurin's formula* or series. The series (M) generally admits calculation more readily than does Taylor's (T), because usually the derivatives at o are of simpler form than those at an arbitrarily selected value of the variable a. EXAMPLES. 1. Any rational integral function or polynomial /(.*•) can always be expressed as /(a) -f (x - a)f\a) + . . . + ( * ~ ** /»(«), where n is the degree of the polynomial/^). For, since/ is of the nth degree, all derivatives of order higher than /« are o. Consequently the theorem of mean value gives r = o whatever values be assigned to x and a. In particular, we may put a = o, and have /(*) = /(o) + xf{o) + . . . + ~f-{o\ and this must be the polynomial considered when arranged according to the powers of x. 2. We may define as a transcendental integral function one such that all of its derivatives remain determinate and non-infinite for any assigned value of the variable. Any such function can be calculated by either Taylor's or Maclaurin's series for any finite value of the variable, whatever. For if / be such a function, then, whatever be the assigned number a, we have ' ( x — a) n -ri £ since / M+I (?) is finite for any £■ between x and a, for all values of n. Also. (x — a) n +*/(n -\- 1)! has the limit o when n = 00 (see § 15, Ex. 9). Moreover, the series is absolutely convergent (Introd., § 15, Ex. 10), since r=o r=o where Mis a finite absolute number not less than the absolute value of any deriva- tive of/ at a. The series on the right is absolutely convergent, since V - a\ £' n = co see § 15, Ex. 10. !M]<\i, * This formula is really due to Stirling. 84 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. Therefore, if f(x) be any transcendental integral function, we have for any assigned value of x or a f{x) =/(«) + (x - «)/'(«) + K 2SL*tf\d) -f- • • • Also, Ax) =/(0) + x/'(0) + ^y/"(0) + Such functions are sin x, cos x, e x . 3. Show that if f{x) is any transcendental integral function as defined in Ex. 2, then f(#x -{- q) can be expanded in Taylor's series for any assigned values of/, q, x and a. This follows immediately from 2, since \db?A* x + q) = P"f n U x + 9)' 4. To expand e x by Maclaurin's formula. We have D r e x = e x for all values of r. At o we have D r e* — e° = 1. Also, £x J = o. (» + 1) Hence, substituting in Maclaurin's series, we have x l x' A e* = I + * + -j + --, + ■ • r= o In particular, when x = I, ,= . + 1 + ^+1 + ^, + ..., which gives a simple and easy method of computing e to any degree of approxima- tion we choose. 5. To compute sin x, given x, by Maclaurin's formula. sin = 0, D 2 "- 1 sin o = (— I)"- 1 , and D 2n sin = 0, by Ex. 5, § 56. Therefore x 3 x 5 x 1 , 6. To compute in the same way cos x, given x. By Ex. 5, § 56, cos 0=1, Z* 2 "- 1 cos — 0, Z> 2 « cos o = (— i)». r* 2 x 4 x 6 ... cos*=i-- + _-- + ... The derivatives of sin x and cos x being always finite, these functions are trans- cendental integral functions and it is unnecessary to examine the terminal term R n . The limit of R H , however, is very readily seen to beo, since we have respectively x M +i / n \ *•= (^+i)i ain («+?*> for *«+' I. n \ — - — ■ -cos £-f--7r , for (« -4- ij! v 2 y Art. 71. J ON THE EXPANSION OF FUNCTIONS. 85 7. The binomial formula for any real exponent. Consider the expansion of (1 -\- x) a by Maclaurin's series, when a is any assigned real number. We have D>'[1 -f x)* = a (a — 1) . . . (a - n -f i)(i -f x)*-". .: [D*(i + x)*] x=0 z=a(a - 1) . . . (a - n + l). Substituting in Maclaurin's series, we have ala — I) a(a — i)(a — 2) 1 + ax + ,, V + J £ ' *" + ' ' ' The quotient of convergency, § 15, Ex. 9, of this series is |X|. (I) /\a - n I^TT- Therefore the series is absolutely convergent when \x\ < 1, or for all values of x in ) _ It _|_ i(. For !x| > 1, the series is 00 . Also, by (C), §.65, or § 68, (2), (*-€)» «(«-!)..■(«-») ^«-* ■»!.-■ (l + |)«+x-a • ( 2) Whatever be the value of | between x and o, so long as \x\ < 1 we have *a — n x — £ f For this limit is the same as "l-r-g / which is less than 1 when o I. 8. Expand log (1 -f- x) by Maclaurin's series. Let fix) = log (1 + x). and /*(o) = (- 1 )«+i(« -1)!. Substituting in Maclaurin's series, we get The convergency quotient of this is n £ -x\ = \x\. S6 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. The series is therefore absolutely convergent for \x\ < I, and is oo for \x\ > I. Also, we have, by (C), § 65, Whatever may be £ between x and o when |jr| < 1, we have, as in Ex. 7, x— £ 1 / < I. Therefore £Rn = o, and log (I + x) = x - $x* + $x* - ix* + . . . (I) This series converges too slowly for convenience, that is, too many terms have to be calculated to get a close approximation to the value of log (I -f- x). By changing the sign of x, log (1 - x) = - X - .&* - & - . . . (2) By subtracting (2) from (1), log L±^ = 2{x + **• + |*» + . . .) (3) If w and w are any positive numbers, put I + x n 4- m m — ! — = , then 2n -\- m Substituting in (3), I n -\- m \ _ t m ^_\_ nf> 1 m b \ ° g \^~) ~ 2 \^n~+^>i ' 3 &n + ™? + 5 {an + mf + ' ' 'J' a series which converges rapidly when « > w, and gives the logarithm of w -f « when log w is known. The logarithms thus computed are of course calculated to the base e. To find the logarithm to any other base, we have logaj =,— • log, a 72. Observations on the Expansion of Functions by Taylor's Series. — The expansion of a given function by the law of the mean is rendered difficult, in general, because of the complicated character of the flth derivative which it is necessary to know in order to get the law of the series and test of its convergency. Still more difficult is the investigation of the limit of R n . This latter investigation is usually more troublesome than the question of convergency of the series because of the uncertainty regarding the value of the number £. The only information we have with regard to £> is that it is some number which lies between two given num- bers. Moreover, we know that £ is a function of n and in general changes its value with n. It is therefore necessary that we should show that jQR„ = o for all values of <<; between x and a, in order to be sure that £R n is o for the partici^ar value £ involved in the Inw of the mean whatever may be that number t c ; between x and a. In the deduction of the form R n in the Integral Calculus, Chapter XIX, Art. 73-] ON THE EXPANSION OF FUNCTIONS. 87 § 152, it is there shown that not only is it sufficient that we should consider all values of t, in the interval (a, x), but it is also necessary. The equality of the function and the series depends on R n vanishing for all values of B in (a, x)* It is desirable therefore, that we should have such general laws with regard to the expansion of functions as will enable us, as far as it is possible, to avoid the formation of the ;/th derivative and the investigation of the remainder term R n , and which will permit us to state for certain classes of functions determined by general properties that the equivalence of Taylor's or Maclaurin's series with the func- tion is true for a certain definite interval of the variable. The general discussion of this subject is too extensive for this course. We give in the next article some observations which will be of assist- ance in simplifying the problem. In the Appendix a more general treatment of the question is discussed. 73. Consider a function f{x) and its derivative f'(x). We can state certain relations between a primitive and its derivative, with regard to the corresponding power series as follows: Cauchy's form of the law of the mean value applied to each of the functions f{x) and/" 7 ^) gives /(•v) =A«) + (* - *>/'(«) + • ■ • + ^—/"i") + *„, ( 1 ) /»=/» + (.v - a)/"{a) + ...+ { * a }"f "(a)+IK , (2) where («-i)! (x - £)* R n = {x _ a) v _,'/ »"(£), (3) 71 R - = (* - ")^^yr/* +1 (£')- (4) I. We observe that the quotients of convergency of (i) and (2), as obtained by taking the limit of the quotient of the {n -j- i)th term to the «th term, have the same value, for /x -a f n+1 (a) £ x — a f n +\a) r x-af'»(a) t 1 W X » + ' /"(") [ n + i) ■ « = oo * In the theorem of the mean, (I), § 70, the series " (x — «)" -fn {a ) may be absolutely convergent and yet not equal to the function f(x). For Prings- heim's example, see Appendix, Note 8. PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. Therefore, if / n+^L\ = R (5) is a finite determinate number, then the two series /(a) + {x - a)f\a) + fc=^)_/>) + . . . and /'(a) + (* - a)f"(a) + <£Z^l/'"( a ) + . . . are absolutely convergent in the common interval )a — R, a -f- R( , and are both oo for any value of x outside of this interval. The number a is called the base of the expansion, or the centre of the interval of convergence. The number R is called the radius of convergence. II. We observe that if, for all values of B, between x and a, we have ■*' w <" tan-ia-] = (- l)*-i(» — 1) ! sin(^«7T). Also, sin 2« —J = o, sin (2t?i -4- 1) — = (~ i)«. Therefore the Maclaurin's series for tan— ^ is x - \x* + Ax* - . . . , which has the interval of absolute convergence )— 1, -f- i(. For R n , in Lagrange's form, we have x n sin In tan- 1 :*: -1 ) Rn — 5 ^_± n (I + ?)i» the limit of which, for « = oo , is o when \x\ < 1. In particular, if x — tan \it — 1/^3, then 2V3 33 5 .5 7j which can be used to compute the number 7t. A better method, however, is given below 3. For all values of \x\ < I we have shown that (1 - **)-* = 1 + K + — ** + ^1 «•+... ' \ i \ 2# 4 2.4.6 ' But a primitive of (1 — x 2 )—* is sin— *x, and since sin— : o = o, we have, by §73, sin- 1 * = x4--^4— -^4 . . . 1 2 3 T 2-4S for x in )— I, -f- !(• In particular, since \it — sin -1 -£, we have 6 2'2-3 2 3 "'2.4.5 2 5 ~ r ' from which 7T can be computed rapidly. 4. Determine the Maclaurin's series for cos— x jt, cot -1 .*-, sec— l x, esc— r x. In each case determine the interval for which the function is equal to the series. 74. We can find the nth. derivative of sin _I x without difficulty, but it would be difficult to evaluate the corresponding limit of R n by the direct processes of Maclaurin's formula. Observe that the coefficients in the power series for sin -I „r can be determined from Ex. 38, § 58, where we have (1 — x 2 )D n+2 sin-^v — (2tt -4- i)xD H+1 sin -1 * — n 2 D n s\nr x x — o. .-. D n+2 sjn _1 o = n 2 D n sin _I o. When we have found D sin _I o, D 2 sin _I o, the other derivatives at o can be found directly, and the interval of the convergence of the series established. The interval of equivalence of the function and the series by evaluating £R n is a matter of considerable difficulty. 90 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VI. In the text we go no further into this matter of the expansion of functions by Taylor's formula. We have made use of it to show how the tables of the ordinary functions and of logarithms can be com- puted, and the numbers e and n evaluated. We add a few exercises in the application of the formula. The cases in which the remainder term R n is inserted are those for which we have not established either the convergence of the infinite series or its equivalence with the function ; they may be regarded as exercises in differentiation or as applications of the Law of Mean Value. Some of these results will be useful later in the evaluation of indeterminate forms and approximate calculations. We observe that for the purpose of approximate calculations, if M be the greatest and m the least absolute value of the (n -f- i)th deriv- ative in the interval (a, x), the error committed in taking m = y ^->(«) lies in absolute value between — '—m and ] ~ ~^M, («+i)! («+•!)! by Lagrange's form of R n . When we know the nth. derivative of the function to be calculated, we can thus determine beforehand how many terms of the series will have to be taken in order that the error shall not exceed a given number. EXERCISES. 1. If c is the chord of a circular arc «, and b the chord of half the arc, show- that the error in taking a = 1 -(Sd-c) is less than — =- where a < radius of the arc. 7680' 2. If d is the distance between the middle points of the chord c and arc a, in Ex. 1, show that the error in taking _ 8 d* ~ a ~ J V 32 d* is less than — — . 3 * 3. The series 1 -\~ x -\-'x 7 -\- ... is convergent for \x\ < 1. It is infinite when x ^ I, and also 00 when x < — 1. Show that we can make .r converge to — 1 in such a way as to make the sum of the series equal to any assigned number we choose. Let x = 1, where a is any assigned number. Then we have for n -f- 1 the sum of n -f- 1 terms of the series Art. 74. J OX THE EXPANSION OF FUNCTIONS. 91 I — x a If « = 2w or 2;« -j- 1. and m = zc , this sum is respectively equal to 1(1 + *-«) or i(i - *-*), one or the other of which can be made equal to any given number by properly assigning a. Show that 4. tan .t- = x -f- \x* 4- fgX? -4- R v 5. sec * = 1 4- §x* 4- &** + fax* 4- tf 8 . 6. log (1 4- sin x) =x - %x* 4- ^ _ ^ + ^ 7. ** sec x = 1 4- * 4- x* 4- f ** 4- *** 4- -V" + ^ 6 - 8. Show that for |jt| < I we have / / ;\ 1 ^ 1 • ^ x 5 & V ^ 2 3 2.4 5 Hint. D fcg (x 4- |/rp) = (14- ^)-l 9. Expand sin— 1 and tan -1 — , in powers of x, determin- 1 + -*' 2 \/l - x- ing the intervals of equivalence. §§ 72, 75. 10. Expand x\/x- -\- a 2 -\- a 2 log (x -j- ^x 2 -\- a 2 ), in powers of x and deter- mine the interval of equivalence. Hint. The derivative is 2 4/a 2 4- x" 1 . 11. Expand in like manner 1 , 1 4- x \/~2~\- x 2 , 1 x Jz _ log — L L__E j _ tan-i ▼ 44/2 I — -r 4/2 -j- x 2 2 |/2 I — * by using its derivative (1 4- x*)-*. 12. Show that the nth derivative of (x 2 -f 6x 4- 8)-* at o is 2K+2I 2"+iJ i)*»! -II — Expand the function in integral powers of x and determine the interval of equivalence. 13. Show by Maclaurin's formula that 1 (1 +x)*=e{l -** + &*■- A**}+* 4 . - , , logfi4-jr) Hint. If j = (1 + x)*, then log/ = — 2 — '. .: y = e*(*\ 0{x) = I - \x + \x 2 - \x* 4- . . . , and the first few derivatives can be found. 14. Compute the following numbers to six decimal places: e, it, log 2, log e 10, sin io°. CHAPTER VII. ON UNDETERMINED FORMS. 75. When u and v are functions of x, they are also functions of each other. If, when x(=.)a, we have ti(=z)o and z>(=)o, the quotient u v will in general have a determinate limit when x( = )a. This limit will depend on the law of connectivity between u and v. The evalua- tion of the derivative is but a particular and simple case of the evaluation of the limit of the quotient of two functions which have a common root as the variable converges to that root. For, in the derivative, we are evaluating the limit of the quotient x — a when f(x) — /(a)( — )o and x — a\ = )o. The evaluation of the quotient u/v when x converges to the common root a of u and v, is but a generalization of the idea involved in the evaluation of the derivative. For, let '(x) and i//(x), then 0'(tf) = o and ^'(tf) = o. In this case we shall require a further investigation in order to evaluate the quotient (p/tp. For this pur- pose we require the following theorems: 76. A Theorem due to Cauchy. — Let cp(x) and ip(x) be two functions which vanish at a, as also do their first n derivatives, but the {n + i)th derivatives of both (*)0 M+I (£), where B, is some number between x and a. Let z be a variable in the interval determined by the two fixed numbers x and a. Then the function /(*) = 0(*) *(*) - «*) 0(*)> = o when z =. a, also when 2 = ^. By the law of the mean, § 62, J'{z) = o for some number g = £ between jr and «. But, in virtue of the fact that / (#) = i/>'(a) = o, we have /'(#) = o. Consequently J"(z) = o for some number B, 2 between B, x and a. In like manner J"\z) = o for z = B> % between Bi and a, and so on until finally we have /H-i(£) = 0«+'(£) 0(*) - #**(£) 0(*) = O, where B, is some number between x and #. If tp n+1 (z) is not o between x and a, we can divide by it. Hence *(«) f**(«) • This theorem is of great generality and usefulness. For example, the functions (x — a) n +*/(n-\- i)\ and J\x)^f{x)- ^LZ^Lfr^) r= o are such that they and their first n derivatives vanish at x = a, while the (n -f- i,th derivative of the first function is 1. Therefore, by the theorem just proved, we have (x — a) H + l which is Lagrange's formula for the law of the mean. 94 rRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. This theorem can be utilized for finding many of the different forms of the remainder in the law of the mean. It has, however, its chief application in : 77. The Theorem of l'Hdpital. — If (■*) r 0" +i (g) _ r ,+, (-y) In*) X r + \t) X *■*(*)' For, by Cauchy's theorem, § 76, 0(*) = 0* + '(£) where B, lies between .a? and a. Hence, since B, and x converge to a together, we have for x(=)a f 0(a) 0"+-(a) 2 *(«) r + '(*)' Moreover, Cauchy's theorem shows that the quotients (.r 2 — 1) = 2x, = 2, when x = - 3_ _l_ 2JC 2 /jr 2 — 3-r 4- 2 _ /Vv 2. Show that Art. 78.] ON UNDETERMINED FORMS. 95 3. Evaluate, when .v( = )o, the following: /t' x — e~ x I x sin x : = 2 ; / . — = o. sin x / .i — 2 sin x ( = )o *(=)o r x( = )o, we have f t*-*-*-™ = 2 . / > + .-*- _* = 2 . / x — sin x I vers x e, when x(=z)o, /x — sin-'x _ _ I £ (i x — b x _ lo a Aa n .r — x _ "sin 3 x ~ 6 ' ^J a- g * ' ^J x - sin # -r( = )o *(=)o 4. Show that for x( = )o, we have 5. Evaluate, when x(=)o, 6. Find the limits, when x( = )o. /x — sin * _ I /" sin 3^ _ _ 3 /" ^3 — 6 ' 7" x — I sin 2x 2 ' J^ 1 cos mx ;«' 78. The Illusory Forms. — When u and v are two functions of x, which are such that the functions u/v, uv, u — V, u v , tend to take any of the forms 0/0, 00 /oo , o X CO , CO — 00 , o°, 00 °, I*, as x converges to a ; then when these functions have determinate limits for x( = )a, the theorem of l'Hopital will evaluate these limits. All these forms can be reduced to the evaluation of the first, 0/0, as follows : (1). 00 /oo and o X 00 reduce directly to 0/0. For, if u a = 00 , v a = 00 , then ^ _ 00_ _ I/Z^ __ O Va ~~ «> 1 / U a O' and we evaluate I/O* i/u x ' If « a = o, v a = co , then 1/V a o and we evaluate i/v x (2). In like manner, if u a = 00 , # a = 00 , then v a \ 1 — vju a o «„ — »„ = «„ I — provided £{v x /u x ) = 1, otherwise this form has no determinate finite limit and is 00 . 96 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. This illusory form can also be reduced to the evaluation of the form o/o when x( = )a, thus : e~ v which takes the form o/o when x = a. Therefore, if £e u ~ v = e c , £{u-v) =c, for x( = )a. (3). The last three forms, o°, co °, i°°, arise from the function u v , which can be reduced to 0/0, thus : Since u = e l0 ^ u , .-. u v — e vloeu . In each of the cases o°, co °, i°°, the function v log u takes the form o X 00 , which can be turned directly into 0/0 and evaluated as in (1). Examples of 00 /oo and 0/00 . The evaluation of u/v, when u = 00 , v = 00 . for x = a, is carried out in the same way as for 0/0. For we have /0(*)_ £ */*{* ) _ f -ft*)/W*)7 when x(=)a. If now w Therefore, for x( = )a, when /and take the logarithm, log (x — 1) ... \ ogy=a -^-. '. log sin 71 x 1 /log (x — 1) f x — r 1 /'sin 7tx -r-— . - = f = - / sec nx. log sin nx Jb n cos nx n J^ x — 1 sin nx 98 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. But ;£sec rex = — i, and /sin7r^"_ Ctt cos tcx * — i ~ X ~~~ i - • '• £ log ^ = ] og ^J' = «• Hence ; £j/ = e a , when #(=)i. Frequently the evaluation can be simplified by substituting for the functions involved their values in terms of the law of the mean. For example, evaluate for x( — )o, i (i + *Y- e Differentiating numerator and denominator, we have x- (i + or) log (i + x) (i + x)- x\i+x) jQ(i _|_ x) x = e, and the limit of the other factor is, by the ordinary process, readily found to be — £. Hence the limit is — e/2. Otherwise, put for (1 -j- x) x i\& value, Ex 13, Chap. VI, x 11 « 7 ^ ) 2 ■ 24 16 ' * J and the result appears immediately without differentiation. Geometrical Illustrations. (1). If/(«) = o, 0(a) = o, f'(a) ?* o, '{a), or /'(a) = tan B v 0\a) = tan 6 2 . tan X tan The limit of the quotient/ (x)/0(x) is represented by the quotient of the slopes of these curves at their common point of intersection with Ox. Art. 79.] OX UNDETERMINED FORMS. 99 (2). Consider the functions x and y in Differentiate with respect to .r and solve for Dy. 2JT 3 -\- 2xy 2 - - Dv = Dy takes the form 0/0 when x _ this, differentiate the numerator and denominator with respect to x 2x' z y -j- 2y' A -J- d l y ' o, for then also y — o by (i). - £ Dy 21/ \xy Dy + (2^2 _|_ 6 y2 + d i^ Dy » (i) To evaluate ■•• {£&)'?= I, or £Z* = ± 1. This means that the curve whose equation is (i) in Cartesian coordinates has two branches passing through the origin x = o, y — o, which is a singular point. There the slopes of the two branches to Ox are -f- 1 and — 1. The curve is the lemniscate. Fig. 12. We can find Dy at x = o. y = o for the curve (i), without indetermination by differentiating the equation (i) twice with respect to x. Thus (2a 2 - I2x 2 - 4j 2 ) = i6xy Dy-\-(vc i +i2y 1 + 2a 1 )(Dyy-\-($x' i yJ r 4y* +2a 2 y)D 2 y, which gives, as before, Dy =. ± 1, when x = o, y — o. (3). We know from trigonometry, that the radius p of the circle circumscrib- ing a triangle ABC with sides a, l>, c having area S, is abc Also, from Analytical Geometry, we have 2S = I x, y, 1 ?i> .Ti> x I ^2' ^2' I where jc, >'; x l . y x ' } x v y 2 , are the coordinates of the corners of ABC. Show that if A, B, C are three points on a curve y=f(x), then the radius of the circle through these three points, when x l ( = )x, x. 2 ( — )x, is [l + (Dyff We have D 2 y L.cfC. ' 2 = K-*) 2 + (-n-/) 2 , <$* = (*, _ xf + (^ - y)*, a*= (x 2 - Xl) 2+(y 2 -y x f. ioo PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. Also, x y I = xy x —yx x + y 2 (x x - x) - x 2 (y x - y). x x y x i Substitute these values in the expression for p. Observe, when x x ( = )x, p is of the form o/o. Divide the numerator and denominator by x x — x and let x^zz: )x. Fig. To evaluate xy x 13- for x x ( = )x, differentiate the numerator and denominator with respect toXj and then let x 1 (=)x. "xy x —yx x £ £(xDy x -y) = xDy-y. x 1 ( = )x Therefore, when B{=.)A, 2 y-> xf [x 2 - x) [I + (Z*)*]* y - (x 2 - x)Dy { \x 2 — x) The first factor takes the form o/o when x 2 = x. To evaluate it, differentiate the numerator and denominator with respect to x 2t and we have I 2(x 2 — x) _ 1 Dy 2 - Df this is again o/o when x 2 — x. To find its limit when x 2 ( = )x. differentiate the numerator and denominator with respect to x 2 , and there results which has the limit i/D 2 y when x 2 ( = )x. Therefore when the points B and C converge to A along the curve, the circle ABC converges to a fixed circle passing through A which has the radius \ _ , (dyy ) l R = -Ml d'-y dx> T liis circle is called the circle of curvature of the curve jy = f(x) at the point x, v. and A* is called the radius of curvature. Observe that when x,(=).i and x., ?£ x, the circle and curve have a common tangent at A, or, as we say, are tangent at A. When this is the case the curve and circle both lie on the same side of the tangent at A. Also the circle lies on the same side of the curve in the neigh- Art. 7.).] ON UNDETERMINED FORMS. 101 borhood of A. But when also x 3 (=)x the circle crosses over the curve at ./. The circle of curvature is said to cut a curve in three coincident points at the point of contact, in the same sense that a tangent straight line to a curve is said to cut the curve in two coincident points at the point of contact. Remembering that all points in the same neighborhood are consecutive, the above statement has definite meaning. Much shorter ways of finding the expression for the radius of curvature will be given hereafter, but none more instructive. EXERCISES. 1. Evaluate, when x(=)o, sin 2 jc — 2 sin x / e x _ 2cos x -\- e~ x _ P sin 2x -f- 2 1 x sin x T cos x 2. Also, for the same limit of x, f sin 4x cot x /'sin i-x cos 2x = 4- / sin 4-r cot x _ /'sin ±x vers 2x cot 2 2x ' T vers x en x(=)o, r m sin x — sin mx m P ta J x(cos x — cos mx) 3 ' I n cot X 3. Show, when x(=)o tan «jr — « tan x sin x — sin nx — 2. 4. If x{ = )o, then /(x — 2)ex + x + 2 I r TtX 2 T~Z —vT~ — = -z J ./ (1 - ^) tan — = -. {e x — if T 2 % / a \* I a\^ I aV°e* { C05 x) ' ( COS x) ' ( C0S ^j ' *( = ) 5. Evaluate for x = 00 , 6. Find the limits, when Jt(=)o, of tan a: / I \ sin x , ( — J , (sin x) sin * (sin .r) tan * 7. Find the radius of curvature of the parabola y 2 = \px at any point x, y, and show that at the origin it is equal to 2p. 8. Evaluate / ( fl 2 _ 02y* -f (a -. Q)8 _ |/i^ ( fl »_e3)i +(a _ e) i~ 1 + ^4/3"' #sin * ^ 9. -= : — —( = )a log a. when x(=z)iie, log sin .* V_ 4 + ^-f 2 cos x _ I #* 6' *( = )0 11. ^ jtc* = 00 j: 2 24 102 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VII. H £d l ax*+bx + c \ a ' £ dx \ px + q j p 15. Evaluate, when x = oo , \/x -f a — \/x -f- b, \/x 2 -j- ax — x, ax sin (cja*). 16. Find where the quadratrix TCX y = x cot — y ia crosses the y axis. t /Y— « r )=f- 2^ 2x tan ^ / 6 *( = )o CHAPTER VIII. ON MAXIMUM AND MINIMUM. 80. Definition. — A function f{x) is said to have a maximum value at x = a when the value of the function, /(«), at a is greater than the values of the function corresponding to all other values of x in the neighborhood of a. The function is said to have a minimum value at a when /(a) is less than_/~(:v) for all values of x in the neighborhood of a. In symbols, /(x) is a maximum or a minimum at # according as /(.r) -/(«) is negative or positive, respectively, for all values of x ^ « in (# — e, a -j- e) the neighborhood of a. 81. Theorem — At a value a of the variable for which the func- tion f(x) is differentiate and has a maximum or a minimum, the derivatives^) is o. At a value a at which /"(**") is a maximum or a minimum, by defini- tion the differences /(x')-f(a) and f{x")-f(a), where x' < a < jv", have the same sign. Consequently the difference-quotients a*-)-a<) and ? „ = /k;)-/(«) x — a x — a have opposite signs for all values of x' and x" in the neighborhood of a, since x' — a is negative and x" — a is positive. Therefore, since q' and q" have the common limit /'(a) when .*:'( = )a and x"( — )a, we have = 2 1/» 1 =0. Hence /" '(a) = o. Notice that at a maximum value of the function the derivative is o, and since, by definition, the function must increase *up to its maxi- mum value and then decrease as x increases through the neighbor- hood of a, the derivative on the inferior side of a is positive and on the superior side is negative, § 60. Hence, at a maximum, a, the derivative, f'(a), is o and J\x) changes irom positive to negative as x increases through a. 103 104 PRINCIPLES OF THE DIFFERENTIAL CALCULI'S. [Ch. VIII. In like manner, at a minimum, x = a, the derivative, /'(a), is o, and_/"'(.v) changes from negative to positive as x increases through a. Conversely, whenever these conditions hold, then the function has a maximum or a minimum value at a, accordingly. For example: 1. Let f(x) = x 2 — 2x -j- 3. ... /(*)= 2(X- I). We have /'(I) = o. Also for x < 1, we have f'(x) negative, and for x > 1, f'{x) positive. Hence /(i) = 2 is a minimum value of f{x). 2. Let f(x) = — 2X 1 -f Sx — 9. .-. f(x) = 4{2-x). We have f{z) = o, /'(2 - e) = +, /'(2 -f e) = — . .-. /(2) = - I is a maximum. 82. The condition /'(a) = o is necessary, but it is not sufficient, in order that the function f{x) shall have a maximum or a minimum value at a. For the derivative f'(pc) may not change sign as x increases through a. It may continue positive, in which case J\x) continues to increase as x increases through a; or f\x) may be negative throughout the neighborhood of a, in which case the func- tion continually diminishes as x increases through a. These condi- tions can be illustrated geometrically thus: Geometrical Illustration. Represent j =/(x) by the curve ABCDE. Then f'(x) is represented by the slope of the tangent to the curve to the x-axis. At a maximum or a minimum, f'(x) = o or the tangent to the curve is parallel to Ox. In the neighborhood of Fig. 14. a maximum point, such as A or C, the curve lies below the tangent, and the ordinate there is greater than any other ordinate in its neighborhood. In like manner at a minimum point, such as B or D, the points fi, D are the lowest points in their respective neighborhoods. At a point E the tangent is parallel to <9.v, and f\x) = o, but the curve crosses over the tangent and is an increasing function at E, also the derivative f'(x) is positive for all values of x in the neighborhood. It will frequently be impracticable to examine the signs of the derivative in the neighborhood of a value of x at which f'(pc) = o. A more general and satisfactory investigation is required to discrimi- nate as to maximum and minimum at such a point. Art. 83.] ON MAXIMUM AM) MINIMUM. 105 83. Study of a Function at a Value of the Variable at which the First n Derivatives are Zero. (1). Let /(a) be a function such that /'{a) ^ o. Then by the law of the mean, §§ 62, 64, f{x) -f{a) = (x - a)f\B). By hypothesis, /'{a) ^ o is the limit of f\x) and of /'(£) as .r( = )<7, since B, lies between x and a. Consequently we can always take x so near a that throughout the neighborhood of a we have _/"(£) of the same sign as f'(a) for all values of x in that neighbor- hood. Hence, as x increases through the neighborhood of «, the difference f(x) —f(a) changes sign with x — a; and by definition f(x) is an increasing or decreasing function at a according as /'(a) is positive or negative respectively. (2). Lety'^z) = o and f"(a) 7^ o. Then /(.v) -/(a) = ( - V ~ U) /"(g). Throughout the neighborhood of #, f'\&>) has the same sign as its limit /"{a) ^ o, and therefore does not change its sign as x increases through a. But, as {x — a) 2 also does not change sign as x passes through a, we have the difference /(•*) -A"), retaining the same sign for all values of x in the neighborhood of a, and having the same sign as /"(a). Consequently, by definition, the function f{x) has a maximum or a minimum value f{a) at a according zsf"{a) is negative ox positive respectively. (3). Let /'(a) = o, f"(a) = o, /'"(a) ^ o. Then As before, in the neighborhood of #, /'"{&>) has the same sign as its limit f'"{a) ^ o. But (.# — «) 8 changes its sign from — to -f- as x increases through a. Therefore the difference i .-,' A*) -/(") must change sign as x increases through a, and f{x) is an increasing or decreasing function at a according as f'"(d) is positive or negative. (4). Let /'(a) =/"(*) = . . . =/» = o, but /*«(«) ^ o. Then, by the law of the mean, /(*) -/(«) = i ^jr> f *w- In the neighborhood of a, / M+1 (£) has the same sign d,sf n+x (a). If « + 1 is odd, then (x — a) n+l and therefore f(x) —/(a) change sign as x increases through a; and _/*(.*:) is an increasing or decreasing 106 PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIII. function at a according as f n+1 (rf) is positive or negaiive. If, how- ever, n -\- i is even, then (x — a) n+1 does not change sign, nor does the difference f{x) —/{a), as x increases through a; consequently f{x) is a maximum or a minimum at # according as f n+l (a) is nega- tive or positive. Hence the following 84. Rule for Maximum and Minimum. — To find the maxima and minima values of a given function f{x), solve the equation f\x) = 0. If a be a root of the equation f'(x) = o, and the first derivative of f{x) which does not vanish at a is of even order, say f 2n {a) ^ o, then /"(a) is a maximum \if %n {a) is negative, or a minimum \i/ 2n {a) is positive. EXAMPLES. 1. Find the max. and min. values, if any exist, of (p(x) = x* — gx 2 -)- 24-r — 7. We have '"(o) = o, IV (°) = 4- 0(o) = 4 is a minimum. Show that o is the only root of 0' (•*")• 3. Investigate x 5 — 5.x 4 4- 5-r 8 — I, at # = I, x = 3. 4. Investigate x 3 — 3-r 2 + 3 X + 7> at x = J « 5. Investigate for max. and min. the functions X 3 _ 3 y2 _|_ 6x 4. 7> ^3 _ gx _|_ I5x _ 3# 3-r 5 — 125.x 3 -|- 2l6ar, x 3 -j- 3^ 2 + 6.r — 15. 6. Show that (I — x + -r 2 )/(i -f- x — x 2 ) is min. at x = ^. 7. If -v;'(j — •*") = 2 # 3 > show that jy has a minimum value when x = a. 8. If 3^ 2 j 2 -f- x>' 3 4- 4^7^ = o, show that when x — 3^/2, then y — — 3a is a maximum. D 2 y being then — 9. If 2x h -f- 3 Art. 85.] ON MAXIMUM AND MINIMUM. 107 which is evidently a maximum when/ = o and a is negative, and a minimum when j' = o and a is positive. (2). Labor is frequently saved by considering the behavior of the first derivative in the neighborhood of its roots, instead of finding the values of the higher derivatives there. For example, see Ex. 6, § 85, and also 0{x) = (x - 4) 5 (x + 2)*. Here \x) = 3(3^ - 2)(x - ^Y(x -f 2) 8 . 0' passes through o, changing from -j- to — as x increases through — 2; there- fore 0( — 2) is a maximum. 0' passes through o, but is always positive as x increases through 4 ; therefore 0(4) is an increasing value of (p{x). Also 0' passes through o, changing from — to -I- as x increases through 2/3, and the function is a minimum there. (3). The work of finding maximum and minimum values is fre- quently simplified by observing that Any value of x which makes fix) a maximum or a minimum also makes Cf{x) a maximum or a minimum when C is a positive constant, and a minimum or a maximum when C is a negative constant. f(x) and C -\-f(x) have max. and min. values for the same values of x. (4). If n is an integer, positive or negative, f{pc) and \/{x) \ n have max. and min. values at the same values of the variable. In particu- lar, a function is a maximum or a minimum when its reciprocal is a minimum or a maximum respectively. (5). The maximum and minimum values of a continuous function must occur alternately. (6). A function J\x) may be continuous throughout an interval (a, /?), and have a maximum or a minimum value at x = a in the interval, while its derivative f\x) is 00 at a, but continuous for all values of {x) on either side of a. In this case, to determine the character of f{x) at a, we can use (1) or (2) as a test. Otherwise we can consider the reciprocal i/y , (x) f which passes through o and must change sign as x passes through a, for a maximum or a minimum o(f(x) at a. EXAMPLES. 1. Consider I 5- ioS PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Cn. VIII 3 3. Consider cos x and a sin 2 x -f- b cos 2 jr. 5. Investigate (x 2 -f 2x — i$)/(jc — 5), and also x l — jx -\- 6 x — 10 ' for maximum and minimum values. 6. The derivative of a certain function is (* - l)(* - ^y 2 (x - 3 )»(* - 4)*; discuss the function at x ■ = I, 2, 3, 4. 7. Find the max. and min. values of (a), (x - i)(x - 2){x -3), (<•). x(i - *)(i - x 2 ). [b). -v* - &*» + 22.r 2 - 24-r, (/)• (-V 2 - !)/(** + 3 ) 3 , (c). (x — a) 2 (x — t>), (g). sin x cos*or, [d). (x -a)\x- !>)\ (//). (log.v)/.v. ART. 85.] ON MAXIMUM AND MINIMUM. 109 8. Show that the shortest distance from a given point to a given straight line is the perpendicular distance from the point to the straight line. 9. Given two sides a and b of a triangle, construct the triangle of greatest area. 10. Construct a triangle of greatest area, given one side and the opposite angle. 11 . If an oval is a plane closed curve such that a straight line can cut it in only- two points, show that if the triangle of greatest area be inscribed in an oval, the tangents at the corners must be parallel to the opposite sides. 12. The sum of two numbers is given; when will their product be greatest? The product of two numbers is given ; when will their sum be least ? 13. Extend 12 by elementary reasoning to show that if 2(x r ) =5 x x + . . . + x H = c, n then IL(x r ) = x x . . . x„ is greatest when x x = x 2 = . . . = x M . 14. Apply 13 to show that if. -\- y -\- z = c, the maximum value of xy 2 z % is ^/43 2 - 15. Show that \i x -\- y -\- z z= c. the maximum value of x l y m z n is l l m m n n c l + m + n (/ 4- m + n)i+™+* 16. Find the area of the greatest rectangle that can be inscribed in the ellipse. (Use the method of Ex. 15.) x 2 y 2 —, + -77, = I. [Ans. 2abA a 1 b l J 17. Find the greatest value of Sxyz, if x % y 2 z 2 I" abc a* + T> + 7> =1 - [_ 3 V3 This is the volume of the greatest rectangular parallelopiped that can be inscribed in the ellipsoid. 18. Show that the greatest length intercepted by two circles on a straight line passing through a point of their intersection is when the line is parallel to their line of centres. 19. From a point C distant c from the centre O of a given circle, a secant is drawn cutting the circle in A and B. Draw the secant when the area of the triangle A OB is the greatest. [With C as a centre and radius equal to the diagonal of the square on c, draw an arc cutting the parallel tangent to OC in D. Then DC is the required secant. Prove it.] 20. A piece of wire is bent into a circular arc. Find the radius when the seg- mental area under the arc is greatest and least. \r = a /ft, r = 00 .] 21. Find when a straight line through a fixed point P makes with two fixed straight lines AC, AB, a triangle of minimum area. [P bisects that side.] 22. The product xy is constant; when is x -\- y least ? 23. An open tank is to be constructed with a square base and vertical sides, and is to contain a given volume ; show that the expense of lining it with sheet lead will be least when the depth is one half the width. 24. Solve 23 when the base is a regular hexagon. no PRINCIPLES OF THE DIFFERENTIAL CALCULUS. [Ch. VIII. 25. From a fixed points on the circumference of a circle of radius a, a perpen- dicular A Y is drawn to the tangent at a point P ; show that the maximum area of the triangle APYU 3 V 3V 2 / 8. 26. Cut four equal squares from the corners of a given rectangle so as to con- struct a box of greatest content. 27. Construct a cylindrical cup with least surface that will hold a given volume. 28. Construct a cylindrical cup with given surface that will hold the greatest volume. 29. Find the circular sector of given perimeter which has the greatest area. 30. Find the sphere which placed in a conical cup full of water will displace the greatest amount of liquid. 31. A rectangle is surmounted by a semicircle. Given the outside perimeter of the whole figure, construct it when the area is greatest. 32. A person in a boat 4 miles from the nearest point of the beach wishes to reach in the shortest time a place 12 miles from that point along the shore; he can ride 10 miles an hour and can sail 6 miles an hour ; show that he should land at a point on the beach 9 miles from the place to be reached. 33. The length of a straight line, passing through the point o, />, included be- tween the axes of rectangular coordinates is /. The axial intercepts of the line are a, ft, and it makes the angle B with Ox. Show that (a). I is least when tan 6 = (b/a) 5 . (b). a + ft is least when tan B = {b/af. (c). aft is least when tan B = b/a. 34. Find what sector must be taken out of a given circle in order that the remainder may form the curved surface of a cone of maximum volume. [Angle of sector = 271(1 — V2/3).] 35. Of all right cones having the same slant height, that one has the great- est volume whose semi-vertical angle is tan~i V2. 36. The intensity of light varies inversely as the square of the distance from the source. Find the point in the line between two lights which receives the least illumination. 37. Find the point on the line of centres between two spheres from which the greatest amount of spherical surface can be seen. 38. Two points are both inside or outside a given sphere. Find the shortest route from one point to the other via the surface of the sphere. 39. Find the nearest point on the parabola j 2 = 4/.V to a given point on the axis. 40. The sum of the perimeters of a circle and a square is /. Show that when the sum of the areas is least, the side of the square is double the radius of the circle. 41 . The sum of the surfaces of a sphere and a cube is given. Show that when the sum of the volumes is least, the diameter of the sphere is equal to the edge of the cube. 42. Show that the right cone of greatest volume that can be inscribed in a given sphere is such that three times its altitude is twice the diameter of the sphere. Also show that this is the cone of greatest convex surface that can be inscribed in the sphere. 43. Find the right cylinder of greatest volume that can be inscribed in a given right cone. ART. 85.] ON MAXIMUM AND MINIMUM. m 44. Show that the right cylinder of given surface and maximum volume has its height equal to the diameter of its base. 45. Show that the right cone of maximum entire surface inscribed in a sphere of radius a has for its altitude (23— Vi7)a/i6 ; while that of the corresponding right cylinder is (2 — 2/ V$)*a. 46. Show that the altitude of the cone of least volume circumscribed about a sphere of radius a is 4^, and its volume is twice that of the sphere. 47. The altitude of the right cylinder of greatest volume inscribed in a given sphere of radius a is la/ V3. 48. The corner of a rectangle whose width is a is folded over to touch the other side. Show that the area of the triangle folded over is least when \a is folded over, and the length of the crease is least when \a is folded over. 49. Show that the altitude of the least isosceles triangle circumscribed about an ellipse whose axes are la and 2d, is 3^. The base of the triangle being parallel to the major axis. 50. Find the least length of the tangent to the ellipse x 2 /a 2 -\-y 2 /b 2 — I, inter- cepted between the axes. [Ans. a -j- ^.] 51. A right prism on a regular hexagonal base is truncated by three planes through the alternate vertices of the upper base and intersecting at a common point on the axis of the prism prolonged. The volume remains unchanged. Show that the inclination of the planes to the axis is sec -1 V3 when the surface is least. [This is the celebrated bee-cell problem.] 52. Show that the piece of square timber of greatest volume that can be cut from a sawmill log L feet long of diameters D and d at the ends has the volume 2 LD* 2J D - d 53. A man in a boat offshore wishes to reach an inland station in the shortest time. He can row u miles per hour and walk v miles per hour. Show that he should land at a point on the straight shore at which cos a : cos fj = ti : v, approaching the shore at an angle a and leaving it at an angle /?. [This is the law of refraction.] 54. From a point O outside a circle of radius r and centre C, and at a distance a from C, a secant is drawn cutting the circumference at R and R'. The line OC cuts the circle in A and B. Show that the inscribed quadrilateral ARR ' B is of maximum area when the projection of RR' on AB is equal to the radius of the circle. 55. Design a sheet-steel cylindrical stand-pipe for a city water-supply which shall hold a given volume, using the least amount of metal. The uniform thickness of the metal to be a. If // is the height and R the radius of the base, then H = R. 56. If a chord cuts off a maximum or minimum area from a simple closed curve when the chord passes through a fixed point, show that the point must bisect the chord. PART II. APPLICATIONS TO GEOMETRY. CHAPTER IX. TANGENT AND NORMAL. 86. The application of the Differential Calculus to geometry is limited mainly to the discussion of properties at a point on the curve. Of chief interest are the contact problems, or the relations of a pro- posed curve to straight lines and other curves touching the proposed curve at a point. The application of the Calculus to curves is best treated after the development of the theory for functions of two variables. 87. The Tangent (Rectangular Coordinates). — Lety =/{x), or cp{x,y) = o, be the equation to any curve. The equation to the secant through the points x, j/ and x\ , y x on the curve is y — y _ y x — y B, y ^Px^ X,Y^ /o' X - x -v, - x' (1 X, F being the coordinates of an arbitrary point on the secant. By definition, the tangent to a curve at P is the straight line which is the limiting position of the secant PP 1 when P r ( = )P. But when = )y. The member on the right of equation (1), being the difference-quotient of y with respect to x, has for its limit the derivative of y with respect to x. At the same time the arbitrary point X, Y on the secant becomes an arbitrary point Therefore we have for the equation to the tangent Fig. 17. P x { — )P we have .r 1 ( = )jf and y x ( on the at P X - x = ¥-= Dy (»> in terms of the coordinates x, y of the point of contact. Art. 88.] TANGENT AND NORMAL. 113 The equation to the tangent (2) can be written r- y={ x-, ) ^, (3 ) or in differentials (F-y) dx - (X - x)dy = o, (4) or in the symmetrical form X - x F-y dx dy (5) EXAMPLES. 1. Find the equation to the tangent to the circle x 2 -\- y 2 = a*. Differentiating, we have 2x -\- 2y Dy = o. .•. Dy — — x/y, and the tangent at x, y is Y-y + (X-x)j = Yy + Xx - (*■ + y 2 ) = o, or Yy -f Xx = a 2 . 2. Find the tangent at x, y to x 2 /a 2 -f- y 2 /b 2 = 1. 3. Find the tangent at x, y to x 2 /a 2 — y 2 /b 2 = I. 4. Find the tangent at x, y to y 2 = \px. 5. Find the tangent at x, y to x 2 -f y 2 -j- 2/y -(- 2£X -{- d = O. 6. Show that the equation to the tangent at x, y to the conic (p{x, y)~ ax 2 -\- by 2 -\- 2/ixy -\- 2fx -\- 2gy -\- d = o is (ax + hy +/)X+ (hx + by + g)Y + ( /* + gy + d) = O. 7. Show that the equation to the tangent at x, y to the curve x™-iX , y**-*Y is h — 7 ■ = I- ^w b m 8. Find the tangent at x, y to x 5 = a 3 j 2 . [5^7-^ — 2 Y/y = 3«] 9. The tangent at x, y to x 3 — $axy -f- j 3 — o is (;/2 _ ^ x )K_j_ (^2 _ ^A^ _ ^. 10. Find the equation to the tangent to the hypocycloid x* -{-y * = J, and prove that the portion of the tangent included between the axes is of constant length. 88. If the equation to a curve is given by * = 0(*)i ^ = #(')« then, since dx = cp\t)dt, dy = ip / (f)df, we have for the equation to the tangent \F-y) = p — , (1) /I — COS X ■ = jQ sin x = o, when x(=)o. Also, since = d -f- ip, we have fjD p d 4- tanfl tan0 = i-ptan^ p ^ _ p- f- tan dD e p ~ D o — p tan 0' ^ ' Art. 92. J TANGENT AND NORMAL. 117 Observe that (2) is the same value as that obtained for D^y in §56. Fig. 20. Draw a straight line through the origin perpendicular to the radius vector, cutting the tangent in T and the normal in TV. We call /Wand PT, the portions of the normal and tangent intercepted between the point of contact, P, and the perpendicular through the origin, O, to the radius vector, OP, the polar normal-length and polar tangent-length respectively ; and their projections, OiVand OT, on this perpendicular are called respectively the polar subnormal and subtangent. We have directly from the figure / = p sec = p 4/1 + p\D P e)\ (3) n = p esc ip =Vp 2 -\- (£>ep) 2 ' (4) S t = p tan ip = p 2 D p 6, S n — p cot ^ = Z> d p. (5) W T hen /?p^ is positive (negative), S t is to be measured from to the right (left) of an observer looking from to P. Putting p' =.D e p, we have for the perpendicular from the origin on the tangent (6) P = P 2 iV + p' 2 since pt = pS t . This can be written 1 = -+(£)' if we put p = i/u, for then dp dS I dfo = ~~7»d6' EXAMPLES. (7) 1. In the spiral of Archimedes p = aQ, show that tan if; = Q, and S M is constant. 2. Show that St is constant in the reciprocal or hyperbolic spiral p9 = a. nS APPLICATIONS TO GEOMETRY. [Ch. IX. 3. In the equiangular spiral p = ae^ cot o-, show that ip = a, S t = p tan a> S n = P cot a. 4. If p = a&, show that tan ip = (log a)— *. 5. Show that the perpendicular from the focus to the tangent in the ellipse (I — e cos 0)p = a{\ — ' 3 = 3 x are parallel to the coordinate axes. (x = o, y = o), (.v = ± i, y = ± \*2). ART. 92.] TANGENT AND NORMAL. 119 9. At what point of a a -f- 4,1' — 9 = o is the tangent parallel to x — y = o? (x = - \,y = 2.) 10. The tangents from the origin to X i _ JJ,3 _[_ yPy _|_ 2x y1 _ Q are y — o, 3.V — y = o, jt -f- y = o. 11. The perpendicular from the origin to the tangent at x, y of the curve jf3 _|_ yi — al is / =; \/axy. 12. Show that the slope of the curve x 2 y 2 = « 3 (jf -(- /) to the x-axis is \tc at o, o. 13. If x,y are rectangular coordinates and p, Q the polar coordinates of a point on a curve, show geometrically that when D x y — o we have D 9 p = p tan 0, and verify from the formulae in the text. 14. Show that the curves x 2 y i X 2 y i *+P= s and ^+ v> =1 cut at right angle if a 2 — b' 1 — a' 2 — b" 1 . iii 15. In the parabola x 2 -J- j 2 = d 1 . show that at x, y the tangent is Xyi + Yx* = (axyf, and that the sum of its intercepts is constant and equal to a. 16. The tangent at x, y to (x/a) 2 -f {yjbf = lis Xx/a 2 +(Y+ 2 y)/ Z b*y* = I. Also find the normal. 17. The tangent and normal to the ellipse x 2 -j- 2y 2 — 2xy — x = o at x = 1 are, at (1, o), 2y = x — 1, _y -j- 2* = 2; at (1, 1), 2y = x -\-i, y -{- 2x = 3. 18. In the curve y(x — i)(x — 2) = .*• — 3, show that the tangent is parallel to the x-axis at x = 3 ± |/2. 19. In the curve (x/a) 3 -j- (y/bf = 1, show that (see Ex. I.) JQ ^ _ ~a^ + I2" ~ X * 20. Show that the tractrix c . c + i / c 2_y2 = Jog \/c 2 -y 2 + V < 2 - y* has a constant tangent-length. 21. In the curve y n = a n ~ l x, find the equation to the tangent; and determine the value of n when the area included between the tangent and axes is constant. 22. In p{ae9 -j- be-«) = ab, show that S t = — ab/(aee _ te-8). 23. If p 2 cos 20 = a 2 , show that sin if) = a 2 /p 2 . 24. If two points be taken, one on the curve and one on the tangent, the points being equidistant from the point of contact, show that the normal to the curve is the limit of the straight line passing through the two points as they converge to the point of contact. 120 APPLICATIONS TO GEOMETRY. [Ch. IX. 25. If Q, f\ P are three points on a curve, /'the mid-point ot die arc QP. and Fthe middle point of the chord QP, show that the normal at P is the limit of the line /'Fas Q(=)P, P( = )P. 26. Prove that the limit of any secant line through any two points P, Q on a curve is the tangent at a point .Pas P( = )P, Q( — )P. 27. Show that as a variable normal converges to a fixed normal, their intersec- tion converges, in general, to a definite point, and find its coordinates. Let ( Y - y)y' -f X - x = o and ( Y - y x )y[ +JT-x 1 =o, where y', y[ represent D x y at x, y and x v y x , be the equations of a fixed normal at x, y and a variable normal at x xi y v Eliminating X, we have Y{/i - /) = y\y\ - yy' + - r i - *> = yi(y[ - y') + /to -y) + ( x i - *)• Y. = >'! + - ~^~ x x — X ~y'x -y' I = y + - y" X = x -y when x 1 ( = )^. Also, _ d 2 y where /' = -— -. y ax 1 This point is the center of curvature of the curve for x, y. CHAPTER X. RECTILINEAR ASYMPTOTES. 93. Definition. — An asymptote to a curve is the limiting position of the tangent as the point of contact moves off to an infinite dis- tance from the origin. Or, an asymptote is the limiting position of a secant which cuts the curve in two infinitely distant points on an infinitely extended branch of the curve. 94. We have the following methods of determining the asymptotes to a curve f{x, y) = o: I. The equations to the tangent at x, y and its axial intercepts are Y- -y = ,{X 'dx' Fi = ^y - A, dx -. X — dx If we determine, for x =y = 00 , then the equation to the asymptote is x y Or, if we determine, for x =y ■— 00 , tjx-- m > and either a or b as above, we have for the asymptote y y = mx 4- b or x — \- a. m This method involves the evaluation of indeterminate forms, which must be evaluated either by purely algebraic principles or by aid of the method of the Calculus prescribed for such forms. The algebraic evaluations are of more or less difficulty, and another method will be given in III for algebraic curves. 121 J22 APPLICATIONS TO GEOMETRY. [Ch. X. EXAMPLES. x 2 y 2 1. Find the asymptotes to the hyperbola — — — = i. a 1 b 2 We have A" t - = a 2 /x, Yi — — b' l /y. These are o when x — y = oo Therefore the asymptotes pass through the origin. Also, dy b 2 x b i = 3== ± / * x = 2a is readily seen to be an asymptote. For the others express Dy in terms of x and make x = 00 ; the result is ± 1. Find the intercept in same way. 8. Find the asymptote of the Folium of Descartes x* -\-y* = $axy. See Fig. 49. The asymptote is x -f- y -f- a =0. Putj = wx in the equation to determine slope and intercept. II. We can sometimes find the asymptotes to curves by expansion in a series of powers. Thus, if j-=v+«.+j+3+---» then y = a x -f a x is an asymptote. For, evaluating as in I, we have m = a Q , V i = a v Observe also, if we have > = *(*) +J+J+..., then when x = oc the difference between the ordinate to this curve and that of the curve y = (p(x) continually decreases as x increases. We say the two curves are asymptotic to each other. Art. 94-] RECTILINEAR ASYMPTOTES. 123 EXAMPLES. 9. In Ex. I, I, we have b I o 2 \¥ b I \ a 2 \ As x increases indefinitely, the point x, y converges to the straight-line asymptote ay — ± bx. 10. Solve Ex. 3, I, by expansion. 11. Solve Ex. 6, I, by expansion. Here we find that the given curve and the hyperbola y 2 = x 2 -|- 2ax -\- 4a 2 have the same asymptotes. III. We pass now to the most convenient method of determining the asymptotes to algebraic curves. If the given curve is a polynomial, /"(.v,;) = o, in x and;', or can be reduced to that form, we can always find its asymptotes as follows: Rule 1. Equate to o the coefficients of the two highest powers of x in /{x, mx -f- b) = o. These two equations solved for m and b furnish the asymptotes oblique to the axes. Rule 2. Equate to o the coefficients of the highest powers of x and of y \vif{x, y) = o. The first furnishes all the asymptotes parallel to the jf-axis, the second those parallel to the_y-axis. Proof: (A). The straight line y — mx -\- b ( 1 ) cuts the curve f(x,y) = o (2) in points whose abscissas are the values of x obtained from the solu- tion of the equation in x, f(x, mx -f b) = o. (3) If (2) is of the nth degree in x and y, then (3) is of the nth degree in x, and will furnish, in general, n values of x (real or imaginary). Let (3), when arranged according to powers of x, be A n x" + A n _ x x^ + . . . + A x x + A = o. (3) If one of the points of section of (1) and (2) moves off to an infinite distance from the origin, then one root of (3) is infinite, and the coefficient, A M , of the highest power of x must be o, or A„ = o. This is readily seen to be true by substituting i/z for x in (3), and arranging according to powers of z. Then when z = o, we have x = 00 , and A n = o. In like manner if a second point of intersection of (1) and (2) moves off to an infinite distance on the curve, a second root of (3) 124 APPLICATIONS TO GEOMETRY. [Ch. X. is infinite and we must have the coefficient of .v"^ 1 equal to o, or A» , = O. When (i) and (2) intersect in two infinitely distant points, then (1) is an asymptote of (2), and we have for determining the asymptotes the two equations A„ = °> A n-t = O. These two equations when solved for m and b give the slopes and intercepts on thej>-axis of the oblique asymptotes of (2). EXAMPLES. 12. Consider x 3 = (x 2 -f- 2> al )}'< see Ex. 3. Here x 3 — x 2 y — 3a 2 y — o becomes (1 — m)x 3 — bx l — $a 2 mx — ^a 2 b = o, when mx -\- b is substituted for y. Hence I — m = O and — b = o give y = x as the oblique asymptote. 13. In x 3 -)- y z = 2>°xy, Ex. 8, put y = mx -j- £. .-. (1 -\- m 3 )*? -f yn{mb — a)jr 2 -j- . . . = o, it being unnecessary to write the other terms. Hence m = — I, /; = — (7. Therefore the oblique asymptote is y ■=. — x — a. 14. Show that j = x -(- £# is an asymptote of y z = ax 2 -f- x 3 . 15. The asymptotes of y* — x 4 -f- 2ax 2 y = b 2 x are y ■=. x — \a and y -j- x -\- \a = o. 16. x 3 + 3x 2 j - x)' 2 - 3/ + x 2 - 2xy + 3/ 2 -f 4r -f 5 = o has for asymptotes y + i* + f = o. 7 = x -f- i, ;+-«• = !• (B). If the term ^4 M _ 1 ^*' -1 is missing in (3), or if the value of m obtained from A n = o makes A n _ x vanish, then (3) has three infinite roots when A n = o and A n _ 2 = o, which equations give the values of m and b which furnish the asymptotes. A n _ 2 will be of the second degree in b, furnishing two b's for each m, and there will be for each m two parallel oblique asymptotes, which we say meet the curve in three points at 00 . If also the term A n _ ? x n ~ 2 is missing, or if A n _ 2 vanishes for the value of /;/ obtained from A n = o, then the equations A n = o, A n _ 3 = o furnish three parallel oblique asymptotes, in general, for each m. EXAMPLES. 17. If (x + r) 2 (.r 2 + y 2 -f xy) = a 2 y 2 -f a*(x - y), then A n _ x — (1 -j- vif{\ -\- m -f m 2 ), A H - t - o, An_ 2 = b 2 - a 2 . .*. m = — 1, b = ± a give asymptotes y = — x ± a. Art. 95.] RECTILINEAR ASYMPTOTES. 125 18. In x*{y + xf + 2ay\x + y) + $a 2 xy + a*y = o, the asymptotes are y -|- x = 2^7, _y -)- x -(- 4^ = o. 19. Find the asymptotes to the curves: (a), xy' 2 — x*y = a 2 (x -j- y) 4 P. x — o, y — o, x — y. (b). y* — X s = a 2 x. y — jr. (r). x* — j' 4 = <7-\rj' -f- b 2 y 2 . x -\- y = O, .r = ;'. (C). For the asymptotes parallel to the coordinate axes, the fol- lowing simple process determines them : Arrange f(x, y) = o according to powers ofy, thus: Ay -f (^ + c)y-> + (^ + Gx + #)y- 2 + . . . = o. ( 4 ) If the highest power ofy is, ?i, the degree of the curve, there will be no asymptote parallel to Oy, since then A ^ o. If, however, the term Ay n is missing, or A =0, then for any assigned x one root in the equation (4) in y will be 00 . If, now, Bx -f- C — o, a second root of (4), in >', is 00 at a: = — C/B, and this will be an asymptote to the curve, since D x y is 00 for the same value of x which makes y = 00 . If the terms involving the two highest powers of y in (4) are missing, then Fx 2 + Gx + H = o makes three roots of (4), in j', infinite, and this is the equation to two asymptotes parallel to Oy, and so on generally. In like manner, arranging f{x, y) according to powers of x, we find the asymptotes parallel to Ox by equating to o the coefficient of the highest power of x. Therefore the coefficients of the highest powers of x and y in the equation to the curve, equated to zero, give all the asymptotes parallel to the axes. Of course, if these coefficients do not involve x or y they cannot be o and there are no asymptotes parallel to the axes. EXAMPLES. 20. Find the asymptotes to the following curves: (a). y 2 x — ay 2 = X s -\- ax 2 -j- P. x = a, y = x -f- a, y -\- x -\- a = o. (b). y(x 2 — %bx -\- 2d 2 ) = x 3 — ^ax 2 -\- a*, x = 6, x = 26, y -f- Z a = x -f 3^- (c). x 2 y 2 z= a\x 2 -j- y 2 ). x = ± a, y = ± a. (d). x 2 y 2 - a\x 2 - y 2 ). y -f a = o, y — a = o. (e) y 2 a = y 2 x -\- x 3 . x = a. (/). (x 2 -y 2 ) 2 - 4y 2 +y = o. (g). x 2 (x - yf - a\x 2 +y 2 ) = o. (h). x 2 (x 2 + a 2 ) = (a 2 - x 2 )y 2 . if). x 2 y 2 - jt 5 4. x 4- y. (/). x 2 y 2 =\a+y)\b 2 -y 2 ). (£). y{x - yf - y( X - y) + 2. 95. Asymptotes to Polar Curves. — If f(p, 6) = o is the equa- tion to a curve in polar coordinates, then, when it has an asymp- 126 APPLICATIONS TO GEOMETRY. [Ch. X. tote, that asymptote must be parallel to the radius vector to the point at co on the curve, if the asymptote passes within a finite distance of the origin. The distance of the asymptote from the origin is the limiting value of the polar subtangent when the point of contact is infinitely distant. To determine the polar asymptotes to /{p, (J) = o, determine the values of which make p = co . These values of give the directions of the asymptotes. If the equation can be written as a polynomial in p, the values of are furnished by equating to o the coefficient of the highest power of p. To construct the asymptote when, — a, the direction has been found; evaluate for 0( = )a and p = co the subtangent »(=)o where pu = i. The perpendicular on the asymptote is to be laid off from the origin to the right or left of an observer at the origin look- ing toward the point of contact, according as / is -\- or — respec- tively. EXAMPLES. 21. Let p = a sec + b tan 0. p = oo when = \Tt; also, 2 dO __ (a + b sin 0) 2 dp ~ a sin -J- b ' the limit of which is a -f b. The asymptote is then perpendicular to the initial line at a distance a -f- b to the right of O. Also, when = §tt, p = oo , and the corresponding value of the subtangent gives a — b and another asymptote. 22. Show that p 2 sin (0 — a) -\- op sin (0 — 2a) -\- a 2 — o has the asymp- totes p sin (0 — a) = + a sin a. 23. Find the asymptotes of p sin = aO. 24. Find the straight asymptotes of p sin 4.B = a sin 3©. 25. Show that p cos = — a is an asymptote of p cos = a cos 20. 26. b = (p — a) sin has p sin = /; for asymptote. 27. Determine the asymptotes of p cos 20 = a. Polar curves may have asymptotic circles or asymptotic points. EXAMPLES. 28. Find the asymptotes of pO = a, for = o, = oo . Fig. 57. 29. Find the circular asymptotes of p{Q -j- a) = bQ. and of rt-0 2 _ gQ 2 _ + cos P - efil 2 ' P ~ 4-sin0' P " + sin 0' CHAPTER XI. CONCAVITY, CONVEXITY, AND INFLEXION. 96. On the Contact of a Curve and a Straight Line. — Let y =f(x) be the equation to a curve. The equation to the tangent, §87, at x = a (Y being the ordinate corresponding to x) is r =/(<*) + (* - «)/"(«). The difference between the ordinates of the curve and tangent at any point (by the theorem of mean value) is /(x)-r=^x-a)Y"{S). If f"{a) 7^ o, this difference will retain its sign unchanged for all values of x in the neighborhood of a. Therefore throughout this neighborhood the curve will lie wholly on one side of the tangent. It will lie below the tangent when f"{a) is — , and above it when /» is +. The curve y — f{x) is said to be co?icave at a when /"(a) is negative, or the curve lies below the tangent there; and is said to be convex at a when f"{a) is positive, or the curve lies above the tan- gent there. EXAMPLES. 1. The curve y ■=. e x is always convex, since D 2 e x = e x is always positive. 2. The curve y = log x is always concave, since D 2 log y = —x~ 2 is always negative. 3. The curve y = x 3 -\- ax is convex when x is positive and concave when x is negative, since D 2 y = 6x. 127 128 APPLICATIONS TO GEOMETRY. Points of Inflexion. [Ch. XL y \ p ^M ' A/ ' a ; a a' Fig. 22. Suppose, at x = a, we have f"{a) = o, but /'"(a) ^ o. Then the difference between the ordinates of the curve and tangent at a is Ax)-r=\(x-a)Y'"(S). Since /'"{a) 9^ o, then throughout the neighborhood of a, f"\B,) keeps the same sign as its limit f'"{d). But (x ) 3 changes from - to -f- as x increases through a. Consequently the corresponding point P on the curve crosses over from one side of the tangent to the other as P passes through A. The curve is convex on one side of A and concave on the other. The curve is said to have a point of inflexion at x,y when at this point we have D x y determinate and D l y = o, Dy ^ o. At a point of inflexion x = a a curve is said to be convexo- concave when it changes from convex to concave as x increases through a, and to be concavo-convex when it changes from concave to convex as x increases through a. See the points A and A' in Fig. 22. EXAMPLES. 4. If y — 2{x — af -f- 4_r — i, y" ■==. \z{x — a) = o, when x — a, and y'" = 12. The curve has a concavo-convex inflexion at x = a. 5. Show that every cubic f(x) = ax* -f dx 2 -f ex + d has an inflexion and classify it. Again, suppose at x = a we have /"(a) = o, /'"(a) = o, Then (x - a) f"(a) 9* o. A*) 4! / ,v (£)- In the neighborhood of a, f lv {£i) keeps its sign unchanged, as also does (x — ay. Consequently the curve lies wholly on one side of the tangent, and is convex or concave according a.sf iv (a) is -J- or — . In general, if /"(a) = . . . . =/'"(") = °»/ m+1 ( a ) ^ °> then Art. 96.] CONCAVITY, CONVEXITY, AND INFLEXION. 129 If /;/ -f~ 1 is even, the curve is concave or convex at a according as f"" l {a) is negative ox positive. If ?n 4- 1 is odd, the curve has an inflexion at x = a, and is concavo- convex if/"" l+1 (a) is -|-, and is convexo-concave \i'f" t+l (a) is — . The tangent at a point of inflexion is sometimes called a station- ary tangent, since D x 6 = o there. For, 6 being the angle which the tangent makes with Ox, we have tan 6 — D x y, etc. The conditions for a point of inflexion given above, fory^v, y) = o, are exactly those which have been previously given for a maximum or a minimum of D^y. For y = f(x) has a convexo-concave inflexion whenever f(x) is a maximum, and a concavo-convex inflexion whenever f'{x) is a minimum. The investigation of y —f(pc) for points of inflexion amounts to the same thing as investigating the maximum and minimum values oiy =■ f'{x). It is not necessary to give many examples of finding points of inflexion, since it would be but repeating the work of finding the maximum and minimum values of functions. EXAMPLES. 6. Show that x 3 = (a* -j- x 2 )y has an inflexion at the origin. What kind of inflexion ? 7. Show that a 3 y = bxy -j- ex 3 -f- 2 is an odd integer. 9. When is the origin an inflexion on y n = kx m ? 10. Find the point of inflexion on x* — T>bx 2 -j- a 2 y ~ o, and classify it. [x — b,y = 2& 3 /a?.] 11. Show that the inflexions on/(p, 6) = o are to be determined from td P y d^p See § 56. If we put p = i/u, this takes the simpler form u -\- u'q = o. The polar curve is concave or convex with respect to the pole according as u 4- u'q is -|- or — . The curve in the neighborhood of the point of contact is con- cave or convex with respect to the pole according as it does or does not lie on the same side of the tangent as the pole. 12. Find the inflexion on p sin 8 = aQ. 13. In pB nt = a there is an inflexion when = \/m(i — m). 14. Find the points of inflexion on the curves: (a), tan ax — y. (d). y = e~ a x • (b). y = sin ax. (e). y = (x — i)(x — 2)(x — 3}. (c). y = cot ax. (/). p(B 2 - 1) = aQ 2 . 15. Show that the curve x(x 2 — ay) = a 3 has an inflexion where it cuts Ox. Find the equation to the tangent there. 16. Show that x 3 -\- y 3 = a 3 has inflexions on O x and O y . 17. The inflexions of x 2 y = a 2 (x — y) are at x = o. x ■=. ± a \/$. 18. x = log (y/x) inflects at x = — 2, y = — 2e~ 2 . 19. PG 2 = a has an inflexion at p — a 4/2. CHAPTER XII. CONTACT AND CURVATURE. 97. In the preceding chapter we have studied the character of the contact of a curve with its straight-line tangent. Now we propose to study the nature of the contact of two curves which have a common tangent at a point. 98. Contact of Two Curves. I. Let^' = (x) for x and y, we find the points of intersection of the curves. (1). If (p(a) = ip{a) and {x) - f( X ), we have (x) - i'(x) = (x - a)["(£) - *"(«)] shows that this difference does not change sign as x increases through #, and therefore the curves do not cross at a. (3). If 0(a) = i/:(a), 0'(,z) = J,' (a), 0" '(a) = 0"(«), but 0'» t* ip"'{a), then the curves have a contact of the second order at a] and we have x + !• Solving the equations, we find that x = 1, y = I is a point common to both curves. Also, their first derivatives, Dy. are equal to 3 there, and their second derivatives, D 2 y, are equal to 6; while their third derivatives, D A y, are not equal to each other. Therefore, at the point 1, 1 the curves have a contact of the second order. 2. Show that the straight line y = x — 1 and the parabola \y = x 2 have a first-order contact. 3. Find the order of contact of gy = x 3 — 3-r 2 -|- 27 and gy -j- t,x = 28. [Second.] 4. Find the orders of contact of the curves: (a), y = log (x — 1) and x 2 — 6x -f- 2y -f- 8 = o. [Second.] (&). ty = x 1 - 4 and x 2 + y 2 - zy = 3. [Third.] (c). xy = a 2 and (x — 2a) 2 -\- (y —2a) 2 = 2xy. [Third.] 5. Find the value of a in order that the hyperbola xy = $x — I and parabola y = x -\- 1 -)- a{x — i) 2 may have contact of the second order. 100. Osculation. — (i). We can always find a straight line which has a contact of the first order with a given curve y = (a) + (x - a)\x), £*F= 0"(*)'. The values of a, f3, R determined from the three equations (.r - af + (y- fif = &, (4) x — a + O - fS)(f>'(x) = o, (5) 1 + O - A)0"(*) + [>"(*)] = o, (6) determine the circle of closest contact, of the second order, at x, y on the curve. Solving these equations and writing y f , y" for r , (p" ', we have for the coordinates of the centre of curvature 1 -I- 1/ 2 1 4- v' 2 P=y + ^-, a = x--y^L, (7) and for the radius of curvature (8) (1 +y*)i y Whenever the coordinates x, y are given, we can substitute in these formulae and compute a, f3, and R, and write out the equation to the circle. Observe that the three equations completely determine the circle, and the circle at a point of ordinary position on the curve can have no closer contact with the curve than that of second order. Observe that this is the same circle obtained in § 79, 111. (3), where we con- sidered the circle which was the limiting position of a circle through three points on the curve when these three points converge to x, y 134 APPLICATIONS TO GEOMETRY. [Ch. XII. as a limit. Having a contact of the second order with the curve, the circle of curvature crosses over the curve at the point of contact. This circle is called the circle of curvature of the curve at the point x, y, and R is called the radius of curvature, the point a, /J is called the centre of curvature of the curve at x,y. (3). In general, when the equation of a curve y = tp[x) con- tains a number, n -f-i, of arbitrary constants, we can determine the values of these constants so that the curve shall have a contact of the ?zth order with a given curve y = (x) have an ?ith contact with y = ', ^ = (* 2 + y*f/ 2 a\ 13. Show that if a variable normal converges to a fixed normal as a limit, their intersection converges to the center of curvature as a limit. The equations to the normals at x v y x and x, y are <"-*>£ + •*-* = * <-y-y)% + x-* = o. The ordinate of their intersection is d )\ dy , d _y_ _ dyy dx dx^ which takes the illusory form 0/0 for x x = x. When evaluated in the usual way, we have, when jr 1 (=)x, which is the ordinate of the center of curvature. Substitution of Y — y in the equation of the normal gives X as the abscissa of the center of curvature. 14. Find the radius of curvature at the origin for 2x 2 -j- $xy — \y 2 -f- x 3 — 6y = o. Using Newton's method, ' Aj iy 2 15. Find the radius of curvature at the maximum ordinate of y = e—^x." 1 . What is the order of contact of the circle of curvature ? 16. If f(p, 6) = o is the polar equation to any curve, show that at any point p, G the radius of curvature is (f- + 2p" - pp'" where for brevity we write p' EE D$p, p" = D^p. This follows immediately from substituting for Dy and D l \\ (1) and (2\ £ 56, in (8), § 100. 17. Show that if pn — I, u' = D e u, 11" = D%h, the value of the radius in Ex 16 becomes R = ( " 2 + U " 1)l \u + u") Art. ioi.] CONTACT AND CURVATURE. [37 18. Since at a point of inflexion r" = o. we have there R = 00 . Therefore the inflexion condition for a polar curve is, as found before, u -(- it" = o. 19. If p — nO, show that R = «(i + 6 2 ) i /(2 + 2 ). 20. If p = a , then R = p[i + (log a) 2 ] 1 . 21. If /o = 29 — II cos 2O, ^ = 00 at cos 20 = T » T . 22. Show that R = lab, for p = a sin bQ, at the origin. 23. Find the radius of curvature for the hyperbola x 2 /a 2 — y 2 /b 2 = 1. 24. Find the radius of curvature of: The circle p = a sin 0; the lemniscate p 2 = a 2 cos 29; the logarithmic spiral p = c a0 ', the trisectrix p = 2a cos — a. 25. If R is the radius of curvature of f(x, y) = o, show that R= (dx 2 + y> a ) = °- If we change a by substituting for it another number a l , we get another curve, A x > y> a i) = °> which will, in general, intersect the first curve. The arbitrary constant a in f(x,y, a) = o is called a parameter. All the curves obtained by assigning different values to a are said to belong to the same family of curves, of which a is the variable parameter. Thus f(x,y, a) =o (i) is the equation of a family of curves when we regard a as a variable, and any curve obtained by assigning a particular value to a is a particular member of that family. Thus, in the figure, let the curves i, 2, 3, ... be the particular curves of the family (1), obtained by assigning to a the particular values a x , a 2 , . . . taken in order. Two curves of this family are said to be consecutive when they correspond to consecutive values of a. The sequence of curves corre- sponding toff 1( ff 2 , . . . , as drawn in the figure, intersect in points A, B, C, . . . 138 Art. 103.] ENVELOPES. 139 Illustrations. The arbitrary constant or parameter being a : ( y> a ) = °> ( 6 ) the equation to a curve which passes through the limit of the inter- section of (1) and (2) as (2) converges to (1). Moreover, (6), being a curve distinct from (1), has in general a definite intersection with (1). If, between the equations f(x,y, a) = 0, (1) f*(x, y, a) = o, (6) the variable parameter a be eliminated, we obtain the locus E(x y y) = o (7) of all points in which the consecutive curves of the family fix, y, a) = o intersect as a varies continuously. The curve (7) is called the envelope* of the family (1). Illustration of the Envelope. As the parameter oc varies continuously, the curve f(x, y, a) = o sweeps over or generates a certain portion of the surface of the plane xOy, and leaves unswept a certain portion. The envelope may be regarded as the line which is the bound- ary between these two portions of the plane xOy. 104. The envelope, E{x t y) — o, is tangent to each member of the family f{x, y, a) = o which it envelops. We are not prepared to give a rigorous proof of this statement now. This prouf requires a knowledge of functions of several variables. We can, however, give a geometrical picture which will illustrate the general truth of the statement. For this proof see § 227. Let (A), (B), (C) be three contiguous curves of the family, (A) * Strictly speaking, the equation of the envelope is the equation gotten by equating to o that factor of E(x, v) which occurs only once in £(x, y). See Chapter XXXIX. Art. 105.] EXVELOrES. 141 and (C) intersecting the fixed curve (B) in points P and Q re- spectively. When (A) and (C) converge to coincidence with (B), (A) (B) (Q) Fig. 25. the points P and Q converge to each other and to two coincident points on the envelope. The straight line PQ converges to a common tangent to (B) and the envelope. EXAMPLES. The variable parameter being a, find the envelopes of the following curve families: 1. x cos (x. -f- y sin a. — p = o = f(x, y, a). f' a — — x sin a -f- j cos a. Square and add. Hence jc* 2 -j- jy 2 = / 2 , a circle with radius /. 2. Envelope the family _/ = y — ax — b/a = o. yX = — x + ^/« 2 - . *. a = \/b/x. Hence j 2 = 4&tr. 3. Envelope the family f = y — ax -|- 2^0: -f £a: 3 . /^ = - x + 2£ + 3<$a 2 . .-. a 2 = (jc - 2^)/3^. Hence 27jk 2 ^ = 4(x — lb f. 4. Find the envelope of (x cos a) fa -(- (_y sin a:)/^ = 1 . 5. Find the envelopes of y = ax -j- |/a 2 a 2 ± £ 2 . Er 2 /a 2 ± jj/ 2 /£ 2 = 1.] 6. Envelope the family ;c 2 -f- _>' 2 — 2a;c = r 2 . 105. Envelopes when there are Two Connected Parameters. Let (p(x, y, a, ft) = o (1) be the equation to a curve, involving two arbitrary parameters a and fi which are related by the condition 4>(a, fi) = o. ( 2 ) I. When we can solve (2) with respect to a or /? and substitute in (1), we reduce that equation to that of a family with one parameter. The envelope is then found as before. 142 APPLICATIONS TO GEOMETRY. [Ch. XIII. EXAMPLE. Find the envelope of the ellipses a 2 ^ /j 2 (i) when a -\- ft = c. We have /? = c — a. Therefore a 2 {c — af Differentiating with respect to or, and solving for a, ex ■j+j and /3= cy< J + f Fig. 26. which substituted in (1) give x* -f y% = A II. Otherwise, when it is inconvenient to solve (2), it is generally simpler to proceed as follows : ~Letx,j> be constant, and differentiate (1) and (2) with respect to any one of the parameters, say /3. Eliminate a, fi and a' = da/d/3, between the four equations. (p(x,y, a, (5) = o, (1) (t>lt(x,y, a, (3) = o, (2) $(a, fS) = o, (3) #(*, p) = o. ( 4 ) The result is the envelope E{x, y) = o. For example, take the same question proposed in I. We have for'(i), (2), (3), (4), f , 7 a* r (S 1 x- —JOC or — o, : 2 a + fizzc, a' -f 1 = o. The elimination gives the same result as before. (2) (3) (4) EXERCISES. 1. Find the envelope of a straight line of given constant length, whose ends move on fixed rectangular axes. [jr 1 -|- _y 3 — ^r 5 .] 2. Find the envelope of the ellipses x? r 2 _ a 2 + & ~ ' when the area is constant. [2^ry = c 2 .] A.RT. 105.] ENVELOPES. 143 3. Find the envelope of a straight line when the sum of its intercepts is con- stant, [x- _[_ yh — A] 4. One angle of a triangle is fixed; find the envelope of the opposite side when the area is given. [Hyperbola. J 5. Find the envelope of x/a -\- y/(3 = 1 when a" 1 -f fi m = c>". r jn_ jn_ rn_ "1 |_X'" + i -^ y/rn+i = fW+i.J 6. Show that the envelope of x/l -\- y/m = I, where l/a -\- m/b = 1 is the parabola (x/a)- -f (y/6) h = 1. 7. From a point Pon the hypothenuse of a right-angled triangle, perpendiculars PM. /Ware drawn to the sides; find the envelope of the line MN. 8. Find the envelope of the circles on the central radii of an ellipse as diameters. 9. Find the envelope oiy = lax -f- t* 4 . [16/ 3 -\- 27a- 4 = o. ] 10. Find the envelope of the parabola y 2 = a(x — a). [4y 2 = x-.] 11. Find the envelope of a series of circles whose centers are on Ox and radii proportional to their distances from O. 12. The envelope of the lines x cos 3a -|- y sin 3a = a(cos ia)% is the lemniscate (x 2 -|- y 2 ) 2 = a\x 2 — y 2 ). 13. Find the envelope of the circles whose diameters are the double ordinates of the parabola j 2 = ^ax. [y 2 = \a{a -j- x).] 14. Find the envelope of the circles passing through the origin, whose centers are on y 2 = \ax. [(x -\- 2a)y 2 -\- x' 6 = o.J 15. Find the envelope of x 2 /a 2 -j- J 2 //? 2 = 1, when a 2 -j- fi l = & 2 - [(x ±yf =&.] 16. Circles through O with centers on x 2 — y 2 = a 1 are enveloped by the lemniscate (x 2 -f- y 2 ) 2 = ^a 2 {x 2 — y 2 ). 17. Show that the envelope of La 2 + zMa -f N= o, in which L, M, N are functions of x and y, and a is a variable parameter, is LN= M 2 . 18. In Ex. 17 show that if Z, M, A 7 " are linear functions of x and y, the envelope is a conic tangent to L = o, N = o and having 7J/ = o for chord of contact. Differentiate LN — M 2 = o with respect to x, ... L'N-{- N'L = 2MM'. At the intersection of L = o and J/ = o we have L' N = o; and since there /V ?£ o, we have Z' = o. The D x y from this is the slope of the tangent to the envelope. Hence U = o is the tangent at the intersection of Z = M = o to the envelope, etc. CHAPTER XIV. INVOLUTE AND EVOLUTE. 106. Definition. — When the point of contact, P, of the circle of curvature of a given curve moves along the curve, the center of curva- ture, C, describes a curve called the evohile of the given curve. The evolute of a given curve is the locus of its center of curva- ture. The given curve is called an involute of the evolute. 107. There are two common methods of finding the evolute of a given curve. I. If -' *« -' aa) 1 -f (/'/3)° 3 = (« a - b 2 )\ 144 Hence (aaY -f (V'/3) 3 = (a 2 - b 2 )\ (Fig. 43.) Art. ioS.] INVOLUTE AND EVOLUTE. 145 II. The evolute of a given curve_/(.v, y) = o is the envelope of the normals to the curve. The equation to the normal to/" = o at x,y is X-x+(V-y)y' = o. (1) But ;■ and y' are functions of x, from the equation f= o to the curve. Therefore x is a parameter in (1), by varying which we get the system or family of normals. Hence the required locus is to be found by differentiating (1) with respect to x, keeping X, incon- stant. Thus _i + (jr-jy-y» = o. (2) Eliminating x between (1) and (2), we have 1 -4- v' 2 1 4- v' 2 Y-y = \f and X-x=-y' \f- , y y in which X and JTa.re the coordinates of the center of curvature, S 100, (2). This proves the statement. EXAMPLES. The equation to the normal is y = ax — 2pa — pa 3 . (1) . •. o = x — ip — spa 2 . \ ip ) • which substituted in (1) gives as before in I, \{x — 2fi) 3 = i-jpy" 1 . 2. Find the evolute of the ellipse x 2 /a 2 -f- y 2 /b 2 = 1. Taking the equation to the normal ax see a — by esc a = d l — b 2 . ax sec a tan a -f- by esc a cot a = o. Hence tan a = — (by /ax) 3 , which leads to the same result as in I, (axf + (byf = (a 2 - b 2 f. 108. The normal to a curve is a tangent to the evolute. Let (x-a)*+(y-fi)* = lP (1) be the equation of the circle of curvature at x, y. Then, letting x f y vary on the circle, R remaining constant, we have, on differentiation with respect to x t x — a -f O — fi\\>' = o, (2) i+y+^-w^o. (3) Now let x, v vary along the curve, R being variable. The num- bers a and /? are also functions of x. Differentiate (2), which is the equation to the normal to the curve at x,y, with respect to x. • •• 1 +y* + ' 5 CX ~ 6a*y ' ' ~ 2tf ' These equations are the equations of the evolute. a and being expressed in terms of y, a third variable. 2. Find the coordinates of the center of curvature of the catenary, Fig. 38, ( ~ --) y = la \e a -\- e a ) . y a — x — — \/y' 2 — a 2 , ft = iv. a 3. Find the center of curvature and the evolute of the hypocycloid, a = x + 3 *V, ft =y + 3* § A (<* + pf + (a - /J) 1 = 2 J. 4. In the equilateral hyperbola ixy = a 2 , (a -f /i) 1 - (a - /J) S = 2a 1 . 5. In the parabola x* -\- y* = a , a -\- /3 = 3(.* -J- ^)« CHAPTER XV. EXAMPLES OF CURVE TRACING. 109. Until functions of two variables have been studied we are not in position to consider the general problem of curve tracing in the most effective manner. Nevertheless it will be advantageous to apply the properties of curves which have been developed for func- tions of one variable to rinding the forms of a few simple curves, whose figures will be useful in the sequel, before we study functions of more than one variable. no. Principal Elements of a Curve at a Point. — We collect here for handy reference the principal elements of a curve at a point, as deduced in the preceding pages. The notations are the same as there used. I. Rectangular Coordinates. D x y = y', D x y = v" . 1. Equation of the tangent: {Y-y) = (X-xy. 2. Equation of the normal : { F- y y=-{x-.x). 3. Subtangent and subnormal : s, = \y'~\ s n =, y y. 4. Tangent-length and normal-length : / =yVi -\-y'~ 2 , n =yVi +j>' 2 . 5. Tangent intercepts on the axes: Xi = x -yy f -\ Fi=y- xy' . 6. Perpendicular from origin on the tangent: 7. Radius of curvature: ■i+y»|i R = y" \. Coordinates of center of curvature: y 147 14S APPLICATIONS TO GEOMETRY. . [Ch. XV. II. Polar Coordinates. D e p = p' ', Dip = p" . up = 1. 1. Angle between the tangent and radius vector: tan ib = —p. P 2. Angle between the tangent and the initial line: p -j- p' tan 6 tan = —. 7l . ^ p — p tan 6 3. Subtangent and subnormal : ^ ~ ^ - " 5P 6 * - p - ^' 4. Tangent-length and normal-length: 5. Perpendicular from the origin on the tangent: p 2 1 iV + p' 2 / 6. Radius of curvature = , — = «2 + «' 2 . 7v J (P 2 + P' 2 f- P 2 + 2P^ - pp' ( w 2 _|_ w /2)| in. Explicit One-valued Functions. — If the equation to a curve can be solved with respect to the ordinate or the abscissa so as to give y = con- ventionally taken to be the locus of the equation , a v .t v = i + x + - + -- + . . . = e*. 2. j. The curve y = e* is identically the same as the curve in Ex. 6 if we inter- change x and y. (Fig. 33.) 8. Trace the probability curve y = c—* 1 . The ordinate is always -(-; it has a maximum at o, 1 ; it is o when x is ± 00 . There is a concavo-convex inflexion at - 1/V2. Ox is an Show that the curve x = -f- i/V 2 and a convexo-concave inflexion at x = asymptote in both directions, and Oy an axis of symmetry, is as in the figure. (Fig. 34.) 9. Trace the cissoid of Diodes, (2a — y)x 2 = y z . The curve has Oy as an axis of symmetry, and passes through O, and cuts the axes nowhere else. Since Fig. 35. y* _^_ x 2 y = 2 a convex. There are no asymptotes. The curve crosses Ox between x = — a and x = — 2a. Also, y = ± 00 when x = ± co . (Fig. 37.) Fie. 37. Fig. 38. (- --\ 12. Trace the catenary, y = %a\e a -\- e a) , in which a is a positive constant. The curve is the form of a heavy flexible inextensible chain hung by its ends. The ordinate y is a minimum and equal to a when x = o, and is -f- for all values of x. The curve is convex everywhere, y — -\- 00 when x = ± 00 , and there are no asymptotes. The slope continually increases with x. (Fig. 38.) 13. Trace the cubical parabola x 1 = y 2 (y — a), where a is positive. Since x = ± y \/y — a, the point o, o is on the curve. But no other point in the neighborhood of the origin is on the curve y since for such values of y, x is imaginary. The origin is therefore a remarkable point, it is an iso- lated point of the curve, and such points are called conjugate points. For each value of y greater than a there are two equal and opposite values of x. The curve is symmetrical with respect to Oy. The ordinate y is a minimum at x = o. where the tan- gent is horizontal. y" = O gives inflexions at _ 4 _ 4 3 — ~- " 2 which for x A- is Fig. 39. convexo-concave and for x — y = -\- so for x ± 00. is concavo-convex. (Fig- 39-) 3^3 There is no asymptote, and Art. HI.] EXAMPLES OF CURVE TRACING. 1 5 J 14. Trace y = (x 2 — 1)- The curve lies above Ox and has Oy for an axis y Fig. 40. of symmetry, y has a maximum at x = o, and minima at x = ± 1 . There are inflexions at x = ± 1/ \/ r $. The infinite branches have no asymptote. (Fig. 40.) 15. Trace the curve y -(•+*) The ordinate has the limit e when x = ± & . This is the important limit on which differentiation was founded, y has the limit 1 when x = o and continually increases with x. For — 1 < x < o the curve does not exist. The point o, 1 is what is called a stop point, the branch ending abruptly there. For x < — I, and converging to — 1, y is greater than e and is 00 . As x decreases to — so , the curve decreases continually and becomes asymptotic toj = e. (Fig. 41.) EXERCISES. 1. Trace the curves y = sin x, y =. cos x. 2. Trace y = tan x, y = cot x. 3. Trace y = sec x } y = esc jr. 4. Trace j = vers .r = 1 — cos x. 5. Trace y = e x . 6. Trace the curves xy = 1, (* - i)(j - 2) = 3, y(x - i)(x - 2) = 1. 7. Trace the curve y(x — i)(x — 2) = (x — 3)^ — 4). 8. Trace y(a 2 -f- x 2 ) = a 2 (a — x). 9. Trace x\y — a) = a % — xy 2 . 10. Trace a 2 x = _j/(j: — a) 2 . 11. Trace j 3 = jr 2 (2^ — ^r). 12. Trace (x 2 4- 4) = jv^ 3 . J54 APPLICATIONS TO GEOMETRY. [Ch. XV. Tt(a — x) la a)*(x «f- 13. Trace 3^r(i — x)y = I — $x. 14. Trace the quadratrix y = x tan 15. Trace the curve y = sin {it sin x). 16. Trace y = (2x 112. Implicit Functions. — In general, when the equation to a curve is given in the implicit form f(x, y) = o, and we cannot solve for either variable, the investigation requires more advanced treatment than we are prepared to give here. This subject will be taken up again in Book II. The ordinatesto such curves are, in general, several- valued functions of the variable. We give here simple examples of important curves. The student will do well to study the hints given in tracing such curves. 15. Trace the hypocycloid of four cusps, x * + y — a ^- The curve is symmetrical with respect to O, Ox, and Oy. There are two equal and opposite values of y to each x, and two of x to each y, for either variable Fig. 42. less than a. The curve does not exist for values of x or y greater than a. We have in the first quadrant ' — IS and the curve is tangent to Ox at x = a, and to Or at .v = o, y = a. y" being positive in the first quadrant, the curve is convex at any point on it. The curve is sometimes called the asteroid. It is the locus of a fixed point on the circumference of a circle as that circle rolls inside the circumference of another circle whose radius is four times that of the rolling circle. (Fig. 42.) 16. Trace the evolute of the ellipse (axf + (/>y)* = (a* - &*) in the same way as above. (Fig. 43.) Ar EXAMPLES OF CURVE TRACING. 55 Show by inspection that four normals can be drawn to the ellipse from any point inside the evolute. From what points can I, 2, or 3 normals be drawn? Fig. 43. Fig. 45. 17. Trace the parabola y 2 = \px, and its evolute, \{x — ipf = 27 py*. Show that the curves are as drawn. Find the angle at which they intersect. Show from which points in the plane can be drawn I, 2, or 3 normals to the parabola. (Fig. 44.) 18. Trace the curve ( y — x' 1 ) 2 = x 5 . .-. y = x*{i ± **). There are two branches, y = x 2 (l + x i ), y = x 2 (l _**). The first continually increases as x increases from o. The second increases, attains a maximum, and then descends indefinitely, crossing Ox at x = I. Both branches are tangent to Ox at O since is o when x = o. The curve does not exist in the plane to the left of Oy. Ex- amine for asymptotes. Find the inflexion and the maximum ordinate. The origin is a singular point called a cusp of the second species. (Fig. 45.) 19. Trace in the same way the curve x 4 — lax' 1 } 1 — axy 2 -j- a' 2 / 2 = O. 20. Trace the curve 1/2 TZ (X -f- I)X 2 . y is a two- valued function of y z= ± x \'x + I. Ox is an axis of symmetry. The curve passes through the origin in two branches, y = + x \/x~+~\, y = - x \/x + r. The curve does not exist in the plane to the left of x 156 APPLICATIONS TO GEOMETRY. [Ch. XV. and o the ordinate is finite, having a maximum and a minimum. We have for the slopes of the two branches passing through O at x = o. *W *(=)0 V* + As x increases positively, y increases without limit in absolute value. Are there asymptotes? (Eig. 46.) The point in which two branches of the same curve cross each other, having two distinct tangents there, is called a node. In this curve the origin is a node. 21. Trace the curve (bx — cy) 1 = (x — a) 5 . Clearly, x = a, y = ab/c, is on the curve. But these values make the deriva- tive y' indeterminate. Differentiate the equation twice. .-. (b — cy'f — (bx — cy)cy" — \o(x — af — o, and at the point x — a, y = ab/c, (b - cy'f Fig. 47- b/c. Since y is imaginary when x < en b y = - x ± — \/(x - af gives y and the curve is as in the figure. The point a, ab/c, is a cusp of \hejirst species. (Fig. 47. ) 22. Trace the curve 4y~ = 4.x 3 -\- i2x 2 -(- gx . y Fig. 48. 23. Trace the lemniscate, shows that y cannot be greater than x and only equal to^r when they are both o at the origin. The curve is symmetrical with respect to O, Ox, Oy. Also, i - (;)*' and since y -^ x, we have, when x = O, y = o, £■ ± 1, (See 111. (2), § 79.) which are the slopes of the two branches of the curve passing through the origin. Again, x a* - 2(x*+y*) y 'a 2 -f 2(a- 2 + j' 2 ) ' Art. 113.] EXAMPLES OF CURVE TRACING. 157 In the first quadrant y' is -f from x = o to the point determined by 2(-v- - y 2 ) = a\ 4(-<' 2 - )' 2 ) = a\ where it changes ^ign. giving y a maximum, and y' decreases until y' = 00 at ,r = a, v = o. Being symmetrical with respect to the axes the curve is as in the figure. No part of the curve exists for x > a, since the equation is of the fourth degree and a straight line cannot cut the curve in more than four points. Put j' = mx, and plot points on the curve by assigning different values to m. Thus, in terms of the third variable m, we have * = ± a \ TW -> y=± am iL_, (Fig. 48.) 113. General Considerations in Tracing Algebraic Curves. — The equation of any algebraic curve when rationalized is of the form of a polynomial of the «th degree in x and y. It can always be written o = « + «x + -..+««= V (I) where u is the constant term (independent of x and y), u x , u 2% etc., are homogeneous functions or polynomials in x, y of respective degrees 1, 2, etc. If u = o, the origin is a point on the curve. (1). To find the tangent at the origin when u x 7^ o. When « = o, the line^y = mx intersects the curve at O. Substitute mx for y in the equation to the curve. Then, if u x = px -\- £>', the equation (1) becomes (P + m)x + T 2 + . . . = o, (2) where the terms T 2 , etc., contain higher powers of x than the first. Divide the equation (2) by x, which factor accounts for one o root. Then let x = o, and (2) becomes p -f- qm = o, or m = — p/q. This value of m is the slope of the curve at the origin, since now the line y = mx cuts the curve in two coincident points at the origin, and u x = px + qy = o is the equation of the tangent at the origin. If u = o, u x = o, and u 2 = rx z -\- sxy -f- ty 2 . Then, as before, put mx iovy and the equation becomes (r + sm + tm l )x 2 + T 3 + . . . = o, (3) where the terms T 3 , etc., contain higher powers of x than 2. Divide by x 2 , which accounts for two zero roots of (3); in the result put x = o. . • . tm 2 -\- sm -\- r = o (4) is a quadratic giving two values of m, the two slopes of the curve at 0. The equation to the two tangents at O is u = rx 2 -f- -m^ + ^>' 2 = o. 158 APPLICATIONS TO GEOMETRY. [Ch. XV. These are real and different, real and coincident, or imaginary, according as the roots of the quadratic (4) in ??i are real and unequal, equal, or imaginary. The origin being a double point called a node, cusp, or conjugate point accordingly. In like manner if also u 2 = o, the equation of the three tangents at O is u s — o, and the origin is a triple point. Hence, when the origin is on the curve, the homogeneous part of the equation of lowest degree equated to o is the equation of the tangents at O. Further discussion of singular points and method of tracing the curve at a singular point will be given in Book II. (2). A straight line cannot meet a curve of the ;zth degree in more than n points. For, if we put mx -{- b for y in U = o, we have an equation of the wth degree in x for finding the abscissae of the points of intersection oiy = mx -J- b and U = o. If now u r is the term of lowest degree in U, and we put mx for y in U, then x r is a factor and represents r roots equal to o. The linejy = mx cuts the curve £7= o, r times at the origin, and there- fore cannot cut it in more than n — r other points. This will fre- quently enable us to construct a curve by points, when otherwise the computations would be quite difficult. (3). Singular Points. A point through which two or more branches of a curve pass is called a singular point. Illustrations have been given of nodes, cusps, and conjugate points. At a singular point on a curve D x y is indeterminate. Points at which D x y is determinate and unique are called points of ordinary position, or ordinary points. To find a singular point on a curve cp(x, y) = o, differentiate with respect to x. The result will be M-^-Ny' — o, (1) where vl/and N are functions of x and y. At a singular point y' is indeterminate and Af= o, A r = o. Any pair of values of x, y satis- fying the equations = o, M— o, N — o is a singular point. If (1) be differentiated again, we have P + Qy' + Ry'* + Ny" = o. At the singular point N = o, leaving a quadratic in y' for deter- mining the slopes of the curve, if the point is a double point. If a triple point, another differentiation will give a cubic in y' for deter- mining the slopes, etc. If the curve has a singular point whose coordinates are a, /3, and we transform the origin to the singular point by writing x -j- a, y -\- fi for x and y in the equation to the curve, the construction of the curve will be simplified as in (1), (2). Art. ii EXAMPLES OF CTRYE TRACING. T 59 EXAMPLES. 24. Trace the lemniscate, Ex. 23. {x*+y*)* - a \x* -y*) = o. Here n 2 = x- — j'-' r= o is the equation to the tangents at o, or y = ± •*", as before in Ex. 23. Put r = w.r in the equation and compute a number of points. Clearly m cannot be greater than 1. 25. Trace the folium of Descartes, x* 4- y* Zaxy = o. The equation of the tangents at the origin is $xy = 0, or x = o, y = o. We find that x + J + a = o is the only asymptote. Put j == mx, then "*tfW 3#W 2 I-fw 3 ' -* I + ;// 3 ' jr, jj' are finite for o < m < -(- °° ■ Com- pute a number of points corresponding to assigned values of m. Observe that if we change m kito l/m, x and y interchange values. The curve is symmetrical with respect to the line y = x. In the first quadrant there is a loop, the farthest point from the origin being x =y = fa. Determine the maximum values of x and y for this loop. For negative values of m we construct the infinite branches above the asymptote, since y = mx cuts the curve before it does the asymptote. (Fig. 49.) 26. Trace the curve (y — 2) 2 (x — 2)x = (x — i) 2 (x 2 — 2x — 3). Examining for singular points, we find y _ \ X\X — 2)7 7 — 2 Therefore x = I, y = 2 is a singular point. Transform the origin to this point by writing x -f- I for x, y -f- 2 for/. Then the equation becomes y\o? - i) =-- x\* % - 4). Examining for asymptotes, we find the asymptotes x = ± I, y = ± x. The equation to the tangents at is y 2 = \x 2 , .'. y — ± 2x. When y = O; x = ± 2, ^ = o. The curve is symmetrical with respect to Ox, Oy, and O. We need there- fore trace it only in the first quadrant, in order to draw the whole curve. The line y = mx cuts the curve in points whose coordinates are )7 7-2 \± — m 2 Id. — m' These increase continually as m increases from o to 1, and the branch approaches the asymptote as drawn. The coordinates are imaginary for 1 < m < 2, and when m = 2: FlG. 50. x = O, y = O. As ;;z increases from 2 to -f- 00 , x and jj/ are real and increasing, and m — 00 gives jr = ± 1, y -= 00, the curve approaches the asymptote as drawn. The origin is an inflexional node. (Fig. 50.) 6o APPLICATIONS TO GEOMETRY. [Ch. XV. 27. Trace the curve (x -\- $)y 2 = x(x — i)(x — 2). 28. Trace the curve a*y 2 = bx* -j- x b . 29. Trace the dumb-bell a*f- = a-x* — x 6 . 30. Show that ^-f/ = sax 2 )' 2 has the form given in Fig. 51. Fig. 51. 31. Trace x 4 = {x 2 - y 2 )y. The lowest terms are of third degree. The origin is a triple point. The tangents there being y = o, y = ± x. Oy is an axis of symmetry. There are no asymptotes. The line y = mx cuts the curve in one point, besides the origin, whose coordinates are x = m{\ — m 2 ). y = m 2 (i — m 2 ). This shows that there are two loops, in the first and fourth octants, and infinite branches in the sixth and seventh octants. The curve is a double bow-knot and has no asymptotes. (Fig. 52.) * Fig. 53. Fig. 54. 32. Trace the curves y' A — ax 2 — x*, y z = a 3 — x 3 , y 2 (x — a) = (x — b)x 2 . 33. Trace the conchoid of Nicomedes, (*■ + V-){b - yf = ay*, when b =, <, > a. 34. Trace the curves y — (x — i)(x - 2)(x - 3), a 2 x = y(b 2 + x 2 ), x* - y* + 2axy % = o. 35. Show that x 2 y 2 -\- x* = a 2 (x 2 — y 2 ) consists of two loops and find the form of the curve. 36. Show that the scarabeus 4 ( X 2 + yi + 2 axf(x 2 + y 2 ) = b 2 (x 2 - y 2 ) 2 has the form given in Fig. 53. Art. 114.J EXAMPLES OF CURVE TRACING. 161 37. Show that the devil y* — x* -f~ ay 2 -f- &x 2 = o, where a — — 24, 3 = 25, has the figure given (Fig. 54). 114. Tracing Polar Curves.— As in Cartesian coordinates, no fixed rule can be given for tracing these curves. The general directions are the same as before. The particular points are : (1). Compute values of p corresponding to assigned values of 6, or vice versa, according to convenience. Plot a sufficient number of points to give a fair idea of the general position of the curve. (2). Determine the asymptotes, by finding values of which make p = co for the directions of the asymptotes. Place the asymptote in position by evaluating the limit of p 2 D p 6 = — D U H, for the perpen- dicular distance of the asymptote from the origin, as previously directed. Examine for asymptotic points and circles. (3) . The direction of a polar curve at any computed point is given by tan tp = p/p'. (4). Examine for axes or points of symmetry. (5). Examine for maximum and minimum values of p and for points of inflexion. (6). Examine for periodicity. 115. Inverse Curves.— If /(p, 6) = o is the polar equation to any curve, then /(p _I , 6) = o is the polar equation of the inverse curve.* We have been accustomed to put p _I = u, so that /"(a, 6) = o is the equation of the inverse curve. 1. Show that if x, y are the rectangular coordinates of a point on a curve, the equation to the inverse curve is obtained by substituting x2 ~\~ y 2 ' x 2 -\- y' 2 for x and y in the equation to the given curve. 2. Show that the asymptotes of any curve are the tangents at the origin to the inverse curve. 3. Show that a straight line inverts into a circle and conversely. Note the case when it passes through the origin. 4. Show that the inverse of the hyperbola with respect to its centre is the lemniscate. EXAMPLES. 38. Trace the spiral of Archimedes, p = aS. The distance from the pole is proportional to the angle described by the radius vector, tan if> = Q. The curve is tangent to the initial line at O. The intercept PQ between two consecutive revolutions is constant and equal to 2na. Therefore we need only construct one turn directly. The dotted line shows the curve for negative values of 0, which * More generally two polar curves are the inverses of each other, when for the same their radii vectores are connected by p } p z = k 2 . k = constant. 162 APPLICATIONS TO GEOMETRY. [Ch. XV. is the same as the heavy line revolved about a perpendicular to the initial line through O. (Fig. 55.) Fig. 55. Fig. 56. 39. Trace the equiangular spiral p = J> . We can write the equation = b log p, if we prefer, tan ip = b, or the angle between the radius and tangent is constant. p — a for S = o, and p increases as 6 increases. p( = )o for = — 00 . The pole O is an asymptotic point. (Fig. 56.) 40. Trace the hyperbolic or reciprocal spiral pB = a. The pole O is an asymptotic point.' A line parallel to the initial line at a distance a above it is an asymptote. For negative values of 0, rotate the curve through it about a normal to OA at O. (Fig. 57.) 41. Trace the lemniscate p 1 — 2a' 2 cos 20. Fig. 57. 42. Trace the conchoid p = a sec ± b, or (*»+.7 2 X* - a ) 2 = b2x2 - When a < b, there is a loop; when a = b, a cusp; when a >6, there are two points of inflexion. (Fig. 58.) Fig. 58. Fig. 59. Art. '5-J EXAMPLES OK CURVE TRACING. 163 43. Trace the cardioid p = *(i -f cos 0). The curve is finite and closed, symmetrical with respect to Ox. p = 2a, a, o, for 9 = o, \n, tt, and diminishes continually as increases from o to it. Also, tan rp = —"cot 10. As 9(=)it, ii'(=)it, or the curve is tangent to Ox at the pole, which point is a cusp. The rectangular equation is x* -f /-' _ ax = + a j/* 2 +y*. (Fig. 59.) 44. Trace the three-leaved clover p = a cos 30. 45. Trace the curves : (1). p = a sin 20, p=acos20. (2). p = a sin 30, p = a sin 48, (3). p — a sec 2 |0, p = a sec 0. (4). p = a sin 0, p = a sin 3 ^0. 46. Trace the curve p(0 2 — 1) = ^0'-'. 47. Trace p — a vers and p = «(i — tan 0). 48. Trace the evolutes of y = sin x and y = tan x. 49. The Cycloid. The path described by a point on the circumference of a circle which rolls, without sliding, on a fixed straight line is called the cycloid. y N L V R G X ? j A I D A Fig. 60. (1). Let the radius of the rolling circle MPL be «, the point Pthe generating point, M the point of contact with the fixed straight line Ox which is called the base. Take MO equal to the arc MP; then O is the position of the generating point when in contact with the base. Let O be the origin and x, y the coordinates of P, Z PCM = 0. Then we have x = OM - NM - a(B - sin 0), y = PN = a(i - cos 0). The coordinates are then given in terms of the angle through which the rolling circle has turned. OA = 2ita is called the base of one arch of the cycloid. The highest point V is called the vertex. Eliminating 0, we have the rectangular equation x = a cos -1 — ty'zay — v 2 . (Fig. 60.) (2). To find the equations to the cycloid when the vertex is the origin, the tangent and normal there are the axes of x and y, we have directly from the figure x = , and P the tracing point. Then with the notations as figured, we have arc AM = arc PM, or aQ = bcp. Hence x = ON = OL - NL, = (a 4. b) cos B — b cos (8 -f 0), * + b a. V M ^91 C a^ v p/ K y *^£^-— — ■ A\ N L (a -j- &) cos Q — b cos b CK — (a + b) sin Q - b sin (6 + 0), Fig. 63. y = PN = CL — (a -f- b) sin B — b sin for the coordinates of the epicycloid. For the hypocycloid change the sign of b. In this book convexity or concavity of a curve at a point is fixed by the sign of the second derivative of the ordinate representing the function. D%y = + or DyX = -f- means convexity with respect to O x or O y respectively. This is the equivalent of viewing the curve from the end of the ordinate at — 00 , instead of from the foot of the ordinate as is sometimes done. PART III. PRINCIPLES OF THE INTEGRAL CALCULUS. CHAPTER XVI. ON THE INTEGRAL OF A FUNCTION. 116. Definition, — The product of a difference of the variable x 2 — x 1 into the value of the function f{x) taken anywhere in the interval (x^ , x 2 ) is called an element. In symbols, if z is either of the numbers x x or x 2 , or any assigned number between x x and x 2 , the product 2 - x i)A z ) is the element corresponding to the interval (x lt x 2 ). Geometrical Illustration. If y = f(x) is represented by the curve AB in any interval (a, b), and x 19 x 2 are any two values of x in [a, b), then the element corresponding to (x x , x 2 ) is represented by the area of any rectangle x 1 M l Mx 2 , whose base is the interval x 2 — x\ , and alti- tude is the ordinate zZ to any point on the curve segment P x P r 117. Definition. — The integral of a function f(x) corresponding to an assigned interval (a, b) of the variable is defined as follows: Divide (a, b) into n partial or sub-intervals (a, xj, (x x , x 2 ), . . • , (^«_ 2 , Xn-i), { X n-i > <^)> by interpolating between a and b the numbers x H _ l taken in order from a to b. And for con- tinuity of expression let x = a, x n = b. The integral of a function is the limit of the sum of the elements corresponding to the n sub-intervals, when the number of these sub- intervals is increased indefinitely and at the same time each sub- interval converges to zero. 165 i66 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cn. XVI. In symbols, we have for the integral oi/(x) corresponding to the interval (a, b), x r { = ).v r -l r = n In which z r is either x r , x r _ 1 or some number between x r and x r _ lt or as we say, briefly, some number o/*the interval (x r _ T , x r ). At the same time that n = co we must have x r — x r _ x (-=)o. Geometrical Illustration. If y = f{x ) is represented by a continuous and one-valued ordinate to a curve, then the integral of f(x) for the interval («, b) is represented by the area of the surface bounded by the curve, the x-axis, and the ordinates at a and b. ind For, any elementary area, such as (x 3 — x 2 )/[z s ), lies between the areas of the rectangles x 2 P 2 M s x 3 and x 2 N' s P s x a constructed on the subinterval (x 2 , x^), or is equal to one of them, according as z z = x 2 , z s =^ 3 . Also, the corresponding area i ,P 2 P s x s bounded by the curve P 2 P 3 , Ox, and the ordinates at x 2 and x 3 lies between the areas of the same rectangles, in virtue of the continuity of f(x), when x :i — x 2 is made sufficiently small. Hence the sum of the integral elements and the fixed area of the curve lie between the sum of the rectangular areas Afo + N 2 x 2 + . . . + Nnb (i) M x a + M. r v x + . . . + M n x n _ v (2) 1 .' ! RQ be not greater than the greatest of the subintervals into which (<7, />) is ■ In ided. The difference between the areas (1) and (2) is not greater than the area of the rectangle BDQR, whose base is RQ and whose altitude BR is equal to the difference f{b) —/la) and to the sum of the altitudes of A", J/, , N 2 M 2 , .... N H M H . This rectangle BQ has the limit o, since each subinterval has the limit o; and so also lias RQ) while its altitude is finite and constant, or does not change with ;/. Consi quently the areas (1) and (2) converge to the constant area of the curve which lies between them, and so also must the area represented by the sum of the cli me nts oi the integral. Hence the integral oif(x) for (a, b) is equal to the ana of the curve, as enun- i iated. Art. nS.] ON THE [NTEGRAL OF A FUNCTION. 167 118. Evaluation of the Integral of a Function.* — In order thai a function shall admit of the limit which we call the integral for a given interval, the function must, in general, be finite and continuous throughout the interval. Should the function be finite and continuous everywhere in the interval (a } b) except at certain isolated values of the variable, at which singular points it is discontinuous, either infinite or indeter- minately finite, then special investigation is necessary for such singular values, and we omit the consideration of them. We shall assume that the functions considered are uniform, finite, and continuous throughout the interval, unless specially mentioned otherwise. The process of evaluating the limit defined as the integral, in § 117, is called inlegratio?i. In evaluating the limit £ 2(x r - *,_,)/(*,), x r - ^_ 1 (=)o. we are said to integrate the function /"from a = x to b = x n . The numbers a and b are called the boundaries or limits of the integration or integral. The lesser of the numbers a and b is called the inferior, the greater the superior, limit of the integration, f In the differentiation of the elementary functions x a , a x , log x, sin x, and like functions of them and their finite algebraic combinations, we have seen that the derivative could always be evaluated in terms of these same functions. Not so, however, is the case in evaluating the integrals of these functions. The integral cannot be always expressed in terms of these same functions, and when this is the case, the integral itself is a new function in analysis which takes us beyond the range of the elementary functions such as we have defined them to be. We shall be interested, in this book, directly with only those functions whose integrals can be evaluated in terms of the elementary functions. It can be stated in the beginning that there is no regular and systematic law known by which the integral of a given function can be determined as a function of its limits in general. The process of integration is therefore a tentative one, dependent on special artifices. * For Riemann's Theorem : A one-valued arid continuous function in a given interval is always integrable in that interval; see Appendix. Note 9. t Tli e word limit as here employed does not in any sense have the technical meaning limit of a variable as heretofore defined. It is an unfortunate use of th'- word, retained out of respect for ancient custom. It is contrary to the spirit of mathematical language to use the same word with different meanings, or in fact to use two words which have the same meaning. 1 68 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI. The systematizing of the artifices of integration is the object of this part of the text. 119. Primitive and Derivative. — If we have two functions F(x) a.nd/(x), so related that/(.v) is the derivative of F(x), then F(x) is called a primitive of f\x). The indefinite article is used and F(x) is called a primitive oi/\x), because if DF{x) =f(x), then also we have D[F(x) + C] = /(.v), where C is any assigned constant. Any one of the functions f\x) + C, obtained by assigning the constant C, is a primitive of J\x). The primitive of f(x) is the family of functions containing the arbitrary parameter C. Geometrical Illustration. The two curves y=iF{x) + C 1% (i) y _ j? {x) + C 2 . (2) are so related that at any point x their tangents at P x and P n are parallel, and each curve has for the same abscissa the same slope. Their ordinate* differ by a c< >n Fig. 66. stant. Each curve represents a primitive of f(x). Any particular primitive is determined when we know or assign any point through which the curve must pass. 120. A General Theorem on Integration. — If a primitive of a given function can be found, then the integral of the given function from a to X can always be evaluated. The given function being continuous in (a, X). Let_/"(.v) be a continuous function in (a, X), and let F(x) be a primitive <>f/(.\ ). Let x Q = a, x M = X. Interpolate the numbers x x , .... .v„_, between a and X in the interval ( ; provided each subinterval x r - x r _ x (=)o when n = 00 . 17° PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI. Therefore, when n r= oo , and at the same time each subinterval x r — x r _ l [ = )o, we have £ 2(*„ - x r .,V(S r ) = £ 2 (x, - -v r _,y( 2r ), (2) w = oo r = i n - x r=i 2 r being any number of the interval (.r r , ^ r _ x ) ; that is, z r may be x r , or ^ r _ t , or any number we choose to assign between x r and The member on the right in (2) is, by definition, the integral of /(x) from a to A", and we therefore have for that integral £ 2(x r - x^,V(z r ) = F(X) - F(a), m = oo r = 1 which is evaluated whenever we know a primitive of f[x), and can calculate its values at a and X. Observe that it is not necessary that we should know the values of the primitive anywhere except at the limits a and X. The integral is therefore a function of its limits. 121. In the preceding articles of this chapter we have fixed no law by which the values x v . . . , x n _ J were interpolated between a and X. The integral has been denned and evaluated for any distri- bution of these numbers whatever, subject to the sole condition that the intervals between the consecutive numbers must converge to o at the same time that the number of the subintervals becomes indefinitely great. Since it makes no difference how we subdivide the interval of integration, we shall generally in the future subdivide the interval of integration into n equal parts, so that x r — x r _ l = Ax r = h = (X — a)/n, and we shall take the value of the function to be integrated at x r _ t , the lower end of each subinterval. The integral olf(x) from a to X is then F(X) - F(a) = £*2f(x,.)Jx. But observe that f{x r )Ax= F'(x r )Jx = dF(x r ). Hence the integral of /[x) from x = a to x = X is the limit of the sum of the differentials of the primitive function. 122. Leibnitz's Notation. — The notation previously used to represent the integral, while valuable as indicative of the operation ad initio performed in evaluating this limit, is cumbersome, and when once clearly assimilated it can be replaced by a more convenient and abbreviated symbolism. We replace the limit-sum symbol by a Art. 123.] ON THE INTEGRAL OF A FUNCTION. 171 more compact and serviceable symbol designed by Leibnitz. Thus, in future we shall write in the suggestive symbolism (A*)** m £ Zf{x r )Jx, ^ j w = » r = as the symbol for the integral oif(x) from a to X. The characteristic symbol / is a modification of the letter 6", the initial of sum, and is taken to mean limit-sum, or / = f2. The symbol f(x) dx represents the type of the elements whose sum is taken. If F{x) is a primitive oifyx), then F{X) - F(a) = JAx) dx, — C F'{x) dx, = fjF(x). This, then, is the final reduction of the integral; and whenever the expression to be integrated, f(x) dx, can be reduced to the differen- tial dF(x), then F(x) is recognized as a primitive of f{x) and the integral can be evaluated when the limits are known. 123. Observations on the Integral.— Differentiation was founded on the exceptional case in the theorems in limits, wherein we sought the limit of the quotient of two variables when each converged to o. We found that the theorem stating: the limit of the quotient is equal to the quotient of the limits, did not hold, § 15, V (foot- note) in the case when the limit of the numerator and of the denomi- nator was o, but that the limit sought or defined was the limit of the quotient of the variables. Integration is founded on another exceptional case in the theorems in limits. Here we seek the limit of the sum of a number of terms when the number of terms increases indefinitely and also each term diminishes indefinitely. The limit we seek is the li?nit of the sum. The theorem which states: the limit of the sum of a number of variables is equal to the sum of their limits, was only enunciated and proved for a finite number of variables, and does not necessarily hold when that number is infinite. The sum of the limits of an infinite number of variables, each having the limit o, is o and nothing else. The important point in the definition of the integral which makes it a matter of indifference where in the subinterval of the integral element we take the value of the function, is an example of an important general theorem in summation, which can be stated thus: 172 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVL Lemma. If the sum of ?i variables u x , . . . , u n has a determinate limit A when each converges to o for n — oo , so that £( Uj + ...+«„) = A, and there be any other n variables v lf . . . , z>„, such that each con- verges to o for ;/ — oo , and at the same time = I, then also £(V X + . . . + Vn ) = ^ For, whatever be r, *- = « + e„ u r where e r ( = )o, when ?i = oo . Also, ;£^r = ;£^(«r + e r u r ) = £2u r + £2e r u r . If e is the greatest absolute value of e 1? . . . , e M , then ^e r # r | ^ | e2u r — eA, the limit of which is o, and, § 15, III, £Sv r = £2u r = A. This principle is of far-reaching importance in integration, and will be frequently illustrated and applied in the applications of the Calculus. Geometrical Illustration. Let y = F(x) be represented by a curve, and let F'(x) = f(x). Then _f(x) is the slope of the curve or of its tangent at x. We have SQ equal to s F(X) - F { a) = M V P X + M. 2 P 4. yM n B, (i) = 3 JF. Also, the sum of the differentials of F at a, x-y, . . . , is Q 2dF= M x T x -f M 2 T 2 + . . . +M n T n . (2) The difference between this sum and that in (1) is 2dF- 24F=r x T l -{-P 1 T t + . . . +£T n . But we know that the limit of AF _ M r Pr dF ~ M r T r i- 1 when n = ao and Ax( = )o. Hence, by the lemma above, we have F(X) - F(a) =£2JF = £2 dF, = £2Ft(x)dx, wln'i li is another illustration of the integral. Art. 124.] ON THE INTEGRAL OF A FUNCTION. 173 124. The Indefinite Integral. — When we know a primitive of a given function we can integrate that function for given limits. It is therefore customary to call a primitive of a given function the indefinite integral of that function. Indefinite integration is therefore a process by which we find a primitive of a given function. A primitive F(x) of a given function f{x) is called the indefinite integral of f(x), and we write conven- tionally, omitting the limits, f/(x) dx = F(x). This, of course, becomes the definite integral j/( X )dx = F(X) -F(a) when the limits of integration a and X are assigned. The indefinite symbol fA*) dx proposes the question : Find a function which differentiated results mf(x); or, find a primitive oif(x). Before we can solve questions in the applications of the integral calculus, we must be able, when possible, to find the primitive of a proposed function. The next few chapters will be devoted to this object. 125. The Fundamental Integrals. — The two integrals x x (' e x dx and j sin x dx J a v a are called the fundamental integrals. They can be determined directly by the ab initio process, and all other functions that can be integrated in terms of the elementary functions can be reduced to the standard form / du = u by means of these fundamental integrals. 1. We have, where (X — a)/n = h, I x e*dx = £ h[e a + e a+h +..... + *»+«•-«>*], h(=)o p*h T h v 174 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVI. 2. Also, x Jf sin xdx = ^"/^[sin a -J- sin (6? -f- ^) -|- . . . -|- sin (a-\-n—ih)], a A( = )o sin [a -(- J(« — i)h] sin \nh ~^ "sin ±h ~' by a well-known trigonometrical summation.* But the expression under the limit sign is equal to {cos {a — ±h) — cos [a -f- £( 2 » — i)A] j sin \h = (cos (a - iA) - cos (A" - i*)\J^, which, when ^( = )o, has the limit cos a — cos X. x C sin x dx = — cos X -\- cos a. * See Loney's Trigonometry, Part I, § 241, p. 283. CHAPTER XVII. THE STANDARD INTEGRALS. METHODS OF INTEGRATION. 126. As stated in the preceding chapter: ii/{x) is the derivative of F(x) i then F(x) is a primitive of f{x) y or an indefinite integral of f{x). This and the next chapter will be devoted to finding primi- tives of given functions.* This process is nothing more than the inverse operation of differentiation. The word integrate, when used unqualified, for the present means " find a primitive." If we choose to work in derivatives, then in the same sense that Df{x) means, find the derivative oi/(x)\ the symbol D~ l /[x) means, find a primitive oif(x). It is usually preferable to work with differentials and employ the symbol lf(x) dx to mean, find a primitive of f{x), or simply, integrate /%*:). If u is any function of x, then u = I du and is the solution of the integral, The solution of f/{x) dx invariably consists in transforming f(x) dx into the differential du of some function u of x, and when this is done the integral or primi- tive u is recognized. But, inasmuch as every function that has been differentiated in the differential calculus furnishes a formula, which when inverted by integration gives the corresponding integral of a function, we do not consider it necessary that we should always reduce an integral com- pletely to the irreducible form / du. There are certain standard functions, such as those in the Derivative Catechism, which we select as the standard forms whose integrals we can recognize at once, and thus save the unnecessary labor of further and ultimate reduction to du. I * This is the starting-point of the theory of differential equations, an extensive branch of the Calculus. 75 176 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. The Integral Catechism. 1. / cu dx — c I u dx. 2. f (u + v)dx = fu dx + fv dx. 3. / it dv = uv — / v da. A f J Ua + l H-. / u a du = . J a + I 5./^ = log*. 6. I e u du == *•". 7. / a" du = — — . J log a o C ■ j cos au f sin 8. I sin #« du — — . I cos + /<<=?). = sin x -\ cos ax. a 1/8 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 128. Methods of Integration. — The first and simplest method of integrating a given function is, when possible, to Complete the Differential. This means, to transform the integral into / du by inspection, and thus recognize u. Except for the simplest functions this cannot be done directly, and we have recourse to the following. The methods employed by which we reduce a proposed function to be integrated to the irreducible fundamental form / du, or to the recognized form of one of the standard tabulated functions in the Catechism, are I. Substitution. (1) Transformation. (2) Pationa/ization. II. Decomposition. (3) Parts. (4) Partial Fractions. 129. While nearly all the standard integrals in the catechism are immediately obvious by the inversion of corresponding familiar formulae in the derivative catechism, we shall deduce them by aid of the principles of § 127 and the methods of § 128, and the two fundamental integrals / e* dx = e x , / sin x ~ cos x, established in § 125, in order to illustrate the methods of integration laid down in § 128, and to fix the standard integrals in the memory. 130. Transformation [Substitution). — This is a method by which we transform the proposed integral into a new one by the substitu- tion of a new variable for the old one. The object in view being to so choose the new variable that the new integral shall be of simpler form than the old one. Thus, if the proposed integral is j/(x) dx, and we put x = cf)(z), then dx = (*)] (.v) dx. Art. 130.] METHODS OF INTEGRATION. 179 1. Use a substitution to find EXAMPLES. du /du u ' Put u = f v , then du = e° dv, >du 2. Make use of / e 11 du — e u to find / 1 U a du. Put u 1 = £•"'. .*. au a ~ l du = e°dv. Hence a— 1 «<* <&< = — e v u dv — — e a dv a a a e a u a + 1 a-\- I a-\- ^ a ,/ ( ! j, \ M \ « / / « a dx [dx _ Pd(x -f z) _ '"' J \tx 2 ± a 2 = J z ~ J x + z~ _ ° g + = log (* + |/** ± a 2 )- • 131. Rationalization (Substitution). — The object of this process is to rationalize an irrational function proposed for integration, by the substitution of a new variable. Rationalization by substitution is but a particular case of trans- formation by substitution. But, since the direct object in view in rationalization is not generally to reduce the function directly to a standard integral, but to first transform it into a rational function which can be subsequently integrated by decomposition into partial fractions, the process demands separate and distinct recognition. Only a few simple examples will be given here in illustration. The subject will be considered more generally in the next chapter. EXAMPLES. 1. Integrate f(a -f bx*fx* dx. Put a -f- bx z = z 3 . . \ bx 2 dx = z 2 dz. On substitution the integral becomes ?/<* -«*>*=£(? -4)' (a -f- b^)\ S bx* - 30). 2. Integrate / 40b 2 dx (a + bx 2 f Put a + bx 2 = z 3 . . \ x dx = 3c 2 dz/2b. 3 The integral is — ty a + bx 2 . 3. Put a -f- bx — c 3 , and show that x dx f {a + 6 x )i \ b = ~r 2 (bx - la){a + bx)\ 4. Put a 2 — x 2 = c 3 , and show that / _ l( 3 a 2 -\-2x 2 )(a 2 - xrf' (a 2 - x 2 f 2 ° 5. To integrate / dx x* \/i -f x 3 ART. 132. J METHODS OF [NTEGRATION. 183 Put 1 -f 1/.1-2 = z\ .-. dx = - **» dz. The integral becomes /I 2X 1 — I 6- / , . • Put l/x* - I = z*. .: dx = - x* z dz. »' X 2 f/I — x' 1 After substitution the integral becomes 7 A 1 + •*"*) <&:. Put ^ = z*. . .-. J (1 + cos 20) J x/xi+d* But f <&++* = f-^Ldx = f *** + (-**?- Adding, we have 2 /* i/x^d 2 dx = x \/x* + « 2 -f a 2 /* — ^), by Ex. 22, § 130, or Ex. 31, § 137. Art. 133. J METHODS OF INTEGRATION. 185 6. Show, in like manner, that I tfx* — a* dx = U- \/x 2 - d l - \a* log (x + \/x 2 -a*). 7. We can frequently determine the value of an integral by repeating the process of integrating by parts. Thus, integrate /• e ax sin bx dx. Put u — sin bx, dv — e ax dx. du = b cos bx dx, v = —e ax . a /e"* sin bx dx = — e ax sin bx / e ax cos bx dx. a a J But, in the same way, we have / e ax cos bx dx — — e ax cos bx + - / e ax sin bx dx. Substituting and solving, we get the integrals e ax sin bx dx — — (a sin bx — b cos bx), e ax e ax cos bx dx — (a cos bx 4- b sin &r). a 1 -\- b' 1 v Put £/« = tan a, then these integrals can be written sin (for — a) and — === cos {bx — a) \/d 2 -f b 2 \Za 2 + ^ respectively. 8. Use Exs. 5, 6 to integrate /x 2 dx p x 2 dx — - and / . tfx* _|_ a 2 J \/x 2 - a 2 9. Show that / sin—** dx = x sin- 1 .* -\- \/i — x 2 by putting u = sin- 1 *, dv = dx. 10. Use the method of Ex. 5 to show that / \/a 2 — x 2 dx = \x\/d 2 — x 2 -f- \a 2 sin-*—. 11. Use the work of Ex. 10 to get / x 2 dx 4/« 2 — — \x \/a 2 - x 2 -f- la 2 sin-i— . 133. Rational Fractions (Decomposition). — Whenever the func- tion to be integrated is a rational algebraic function, we know from algebra (see C. Smith's Algebra, § 297) that it can always be decom- posed into the sum of a number of partial fractions, each of which is simpler than the proposed function. (See Chapter XVIII.) We do not propose to consider here the general process of inte- grating rational fractions, but merely consider a few elementary examples illustrating the process. If the function to be integrated is the rational fraction (t>(x) 1 86 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. and the degree of is higher than that of ip, we can always divide by ip, so as to get in which the quotient f(x) is a polynomial inland can be integrated immediately. The remainder F(x)/tp{x) is a rational function in which F(x) is a polynomial of one lower degree than tp(x), the general integration of which will be considered later. EXAMPLES. — — X 3 X 2 -(- x — log ( I -f- X). 3 2 e/ * 2 + 4 ^ l0g ^f-2-2- l0g( " 2 - 4) ' dx. 3* + *- x _?? r lJ ==x + i * 2 + 4 * 2 + 4 -*" 2 + 4 * 2 + 4 * * J * 2 + 4 ax -J xax +j3* + 4 2 J * 2 + 4 ' = — .r 2 + — tan-i — — — log (x l + 4). 2 2 2 2 & V ' ^' 4. To integrate / T . & J (x — a)(x — b) We can always write « = -1-1-1 L_\ (x — a)(x — b) a — b\x — a x — b) by inspection. Therefore /dx I . x — a = log . (x — a)(x — b) a — b x — b 134. Observations on Integration. — The processes of Substitution and Decomposition, in their four subdivisions: 1 . Substitution, 2. Rationalization, 3. Parts, 4 . Partial Fractions, constitute the methods of finding a primitive of a given function by reduction to a recognized or tabular form. These may be regarded Art. 134.] METHODS OF INTEGRATION. [87 as the rules of integration in general form corresponding to the rules of differentiation. With this difference, however, that in integra- tion there are no regular methods of applying these rules to all functions as is the case in differentiation. The successful treatment of a given function depends on practice and familiarity with the processes of the operation. Sometimes different processes of reduction lead to apparently different results. It must be remembered, in this connection, that the indefinite integral found is but a primitive of the function pro- posed, and both results may be correct. They must, however, differ only by a constant. Frequently, in reducing an integral to a standard form, we shall have to use all four of the methods of reduction. Experience soon teaches the best methods of attack. In the next chapter we shall consider the subject more generally and make more systematic the methods of reduction to the standard forms. EXERCISES. Integrate Exs. 1 to 10 by the primary method of completiag the differential by inspection. 1. I x* dx, / ax— 3 dx, i 2x~*dx. 2. f(x 2 + I)* xdx = \{x* -f 1 f. m x i _ a r )dx x 4. f(iot* - t-*)dt = 6t* + ±t~ z . 5. f(x~i -f x~i)dx, f(s 2 - i)ds/s, fv dv/(v 2 — 1). 6. / r , * du — lo g V u * + 2U - J tl 2 -j- 2U 7. A/2 _ 2ft-* dt = 2;-* - 6/- 2 + \t 2 - log / 6 . C{a 2 - x 2 f \/xdx, f{Va- ijxfdx, f (x + ifdx. 8. /» 2ax + b j f e* — <^ x j f 2ax -\- b 9 " J ax 2 + bx + c ' J e* + e~* ' J {ax 2 + bx + cf *' //» e x p seclr , J tan-Jjr J sin- 1 * J log x x 10. Write immediately the integrals of I x x x 1 x n ~ l x+i' x + l' x 2 + 1 ' *? + 1 * w + «* cos 2 -^jr, cos - ^ sin .r, tan M ^r sec 2 *. :SS *•/ cos mx cos nx dx = sin [m -\- n)x sin (m — n)x Put x -- — ~2 # Put e x = Z. Put x n = Z. Put ] Og X = z. Put X 2 ■=. z. Put X* = z. PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. , /' cos \^X . . ,— 1. / = — dx — 2 sin y x . J \?x 2. / e* cos e x dx = ? 3. / nx"- 1 cos x M dx = 7 .. f 2X 5 -./r+^ = ? 6. /"_3^_dS r = ? e/ I + JC b - f dx f dx Pu dv -\- v dn J 4/1 — 4jf 2 ' ./ |/L— 2x 2 ' «J |/i — #V ' J I + 4^ 2 ' «/ 9^' + 4' ./ ^r-j/fe 2 — 1' 9. / sin 3.r . — - — =— = — log (a 4- &*«). a -j- bx» nb 5 v ~ y J a + bx*~ |/^ an \ \|a/■ /'(I4-x 2 ) 2 = ? 25. / sin 20 )" l 5 or m 4- >l is a positive integer greater than i. Put x — a = (x — b)z, then a — bz (a — b)z a — b . a — b , x — — • ••• x — a = — , x — b = , dx — - -dz; i — z i — z i — z (i — zy ' and the expression transforms into (I _ z )m+n-2dz (a — b)» l + n - l z m Expand the numerator by the binomial formula and integrate directly. 46. Integrate / sin/* cos q xdx, whenever p -f- q is an even negative integer. Let p -f q = — in. Then sinAr cos?* = sinAr cos-/>- 2 «* = tanAr sec 2 "*, = tan/*(i -j- tan 2 *)"- 1 sec 2 *. Put tan x = t. Then / sin/* cos q xdx = /V(i + t-) n ~^dt. Expand by the binomial formula and integrate directly. 47. Integrate sinAr cos?* dx, whenever / or q is an odd positive integer. Let p — ir -f- I, then I sin 2r + I * cos?* dx = — f (sin 2 *)'' cos?* ^( cos *), = — / (i — cos 2 *) r cos?*{a 2 -b 2 ) dx I . lb 66. /^ 5 X , 2 - 2 = A tan- 1 (- tan *) J a 2 cos 2 * -j- b l %\Vl 1 x ab \a J Divide the numerator and denominator by cos 2 *. /dx — : . Divide the numerator and denominator by l/a 2 4- b 2 , a sin * -f- b cos x ' ' and put tan a = a/b. Then we have , f ~ x = _i_ log tan (i* - $a -f lit). |/ fl 2 + 32,/ cos (* - a) j^a 2 -\-b 2 ' dx -j- £ cos *' # -f- 3 cos * = a (sin 2 £* -|- cos 2 ^x) -\- b (cos 2 -|* — sin 2 -J-*) = (a + £) cos 2 -£* -f- (« — £) sin 2 £*, which reduces the integral to the form of Ex. 66. Divide the numerator and denominator by cos 2 \x, and put z = tan ■£*. Then the integral becomes 68. f — - /dz (« + b) + ( which is standardized. Hence / b; a -\- b cos x \/d 2 -b 2 (\ a + d 2 ) log r ' ' — =— , a < b. i^b 2 — a 1 \'b^-a— \/b — a tan \x I 92 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVII. 69. /' *^_ = 1 tan-i 4 + 5 tan ** e/ 5 + 4snu 3 3 •7/1 t . ^ ! + COS X , I I 70. Integrate / ±- dx = : J (x + sin *)<» 2 (jr 4- sin at) 2 " 71. I x sin x dx = sin x — x cos .*. 72. f l —?dx = log (1 + *) 2 - *. 73. /• **** =-1 ^— . ^ (* 3 + x*y 3 (^3 _j_ x rf /dx . = log (tan-i*). (i -f- x l ) tan-^ & ' __ p dx . \x 4- 1 2 — x 75. / — = 2 sin— 1^ — 2 — = cos-i . J VS + 4* - x 2 \ 6 3 76. f C ° S -V^±dx = sin (log *;. Put * = log z. mm p dx 1 . tan \x — 2 77. / - — : = - log J 4 — 5 sin x 3 2 tan |x — I 78. /" = - tan-i (3 tan x). J 5 - 4 COS 2X 3 CHAPTER XVIII. GENERAL INTEGRALS. General Forms Directly Integrable. 135. The Binomial Differentials. — Expressions of the type x*{a + bxP)i dx, (A) where a, /3, y are any rational numbers, are called binomial differen- tials. This expression is directly integrable in two cases. _ _ TT1 a -\- 1 . I. When — — — is a positive integer. The substitution is a -f- bx$ = z. Then M dx = i?^—r hence (z — a) & zy x a (a -j- 5xfi)v dx = ^ ^ dz. fib~P~ Consequently, when — ~^ — • is a positive integer, the transformed expression can be expanded by the binomial formula and immediately integrated. £f -I- 1 II. When — - 1- y is a negative integer. The substitution is a -f- ^ r/3 = zxfi. For, if we substitute x = i/y in the differential x a (a -(- 6xP) y dx, it becomes — v -?Y-"--2( / 3- /; + X 2 )(I _ X 2 )* x / 2 i / l _ x , d. Ans. - tan- (3 + 4x») (4 - 3x2). 5 4/3 1 / I2 _ gx 2 dx A 1 , 2(3 + 4*2)4 -f ex ~^7 — ; — ^T Ans - — lo g , -« ^ (4 - 3*-)(3 + 4**)' 20 s 2(3 _|_ 4x 2 } i _ ^ 137. Integration of — / ; qX — *dx. (C) a -f- 2 ox -\- ex 2 This is a particular and simple case of the rational fraction which will be treated generally in § 148. On account of its special impor- tance we give it separate treatment here. Let L represent the linear function p -f- ax. Let Q represent the quadratic function a -\- zbx -f- ex 2 . I. Consider / — . Completing the square in Q, we have f d ± = f J a -j- 2b x -f- ex 1 J ( cdx (ex _|_ bf - (b 2 - ac) Put ex -\- b ■=. z. Then the integral becomes dz 3 /. (b 2 - ac)' This is standardized, and depends on whether b 2 — ac is positive or negative. If negative, the roots of the denominator are imaginary and the integral is an angle, the standard 13. If positive, the roots of the denominator are real and the integral is a logarithm, the standard 14 (§126). If ac > b 2 , f dx 1 ex -4- b IT = / z3 tan ' / a ' (l) If ac < 3 2 , / ^ _ 1 ex -\- b — \/b 2 — ac Q 2\/b 2 — ac ex -j- 3 -J- 4/0^ — ac 196 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. II. Consider fi* PL _ pc-qb fdx_ q_ A " J Q ' c J Q + 2cJ - Since the derivative, Q', of Q is a linear function, we can always determine two constants A and B, such that L = A + BQ\ or p -J- ?•* = -4 -f- 2^i? 4- icBx. Equating the constant terms and coefficients of x, B — q/2c, A =p — bq/c. 'dQ Q The first integral has been reduced in (1), (2), and the second is log Q. In working examples, carry out the process and do not substitute in the general formula. EXAMPLES. 1 f xdx (\ ~ 2 1 I 2X + 4 Iv '■ J X* + 4 * + 5 J { (* + 2) 2 + I "^ 2 X* + 4* + 5 f **' = - 2 tan-'(^ + 2) + 1 log (x 2 + \x -f 5). 2. f^-^L = _ _I_ tan-x^ti _j_ 1 log (*a 4. 2 x + 3). J^ + 2X+3 ^ |/2 2 gV ^ ^^ 3. /* — = — ? f- log (* -f 1). J x 2 4- 2x + I .* + 1 ^ 5V ^ ; 4 - / , - fe f "/ ) f r e = 7 lo S ^ + 4-r + 5) - tan-*(* + 2). «/ •*" t" 4-*" ~r 5 2 5 - / ; iT — 2 = - lo ? (3 - ■*)■ J 3 + 2X — X 2 6. f ^fr 1 "^,^^ ^ = 2jt - log (* 2 + 6x + io)* + 11 tan-i(x 4- 3). 7 - f- *7* y l\ = x - lo § (^ + 2^ + 2)2 4- 3 tan-(x 4- 1). j/ JV —J— 2^* *-j— 2 /i'T.r) — - dx, where F{x) is any polynomial in x, divide J\x) by Q until the remainder is of the form L/Q, and integrate. 138. Integration of (/ + fl* j ^ ^ |/tf -f- 2&V -|- C.V~ Let, as in § 137, L and ^ represent the linear and quadratic functions respectively. I. Consider rdx J Q T square f—= r f Complete the square in the quadratic, and then dx Art. 138.] GENERAL INTEGRALS. 197 which is the standard 11 or 12 according as b 2 is greater or less than ac. If a and c are both negative and ac > b 2 , the function is imaginary. We have, according as the roots of Q are real or imaginary, —7=- log [ex — I — 3 — I — \/c{a 4- 2bx -f- C.* 2 )], 1 . ex -f- ^ — — sin * — — , \/c \/ac -j- £ 2 as the corresponding values of the integral. II. Consider Write, as in II, § 137, L = A -\- BQ\ and determine A and B. Then , § 137, L al on th< r dx J LQ? The first integral on the right was reduced in I, the second is 2(2*. III. Consider J ^0} a dx dz 1 — pz Put p -f ox = i/z. . ■. — — t = , x = — . r ' * 7 p + -2X+ 3+i /2 J (x - l)*/x*- - 2x + 3 |/ 2 X - I 12. r (* + 3)^ = tfj + 2x+ - + log (x + j + ^ + 2 ^ + 3)2. «/ |/x 2 -j- ix -|- 3 Reduction by Parts. 139. Integration of Powers of Sine and Cosine. / sin n xdx = 1 sin" -1 * sin * dx. Put & = sin" -1 *, dv = sin x dx; . t. du — {n — 1) sin" -2 * cos x dx, v = — cos x. Hence, applying the formula for parts, isin n xdx= — sin* -1 .* cos x -\- (n — 1) / sin" -2 * cos 2 xdx, = — sin" -1 * cos x -f- (n — 1) / sin"~\*(i — sin 2 x) dx, — — sin" -1 * cos *•-[-(>/— 1) / sin n ~ 2 xdx— (n— 1) / sin M * for each pair of conjugate imaginary roots there is a fraction C + Z?-v for each pair of conjugate multiple imaginary roots of order s there are s fractions of the types ^, + ^1 , ^, + ^, , 1 E, + xF s x* _j_ „.* _|_ ^ "T (^2 _j_ ^ _j_ ^s T • • • -r ( v > + a v + py In these partial fractions the numbers A, B, C, D, F, F, etc., are constants. Since there are exactly as many of these constants as there are roots oif(x), they are n in number. If now we equate F(x)/f(x) to the sum of the partial fractions and multiply the equation through by /(.v), we shall have F(x) equal to a polynomial in x of degree n — 1. When we equate the constant terms and the coefficients of like powers of .1* on each side Art. 141.J GENERAL [NTEGRALS. 103 of this equation, we have // linear equations in the constants A, />, C, etc., which serve to determine their valui The integral of the rational function then depends on r ix and r { E+ Fx)dx J (■*-«)' J (**+& + & The first of these can be integrated immediately, the second is always of the type f {E + xF)dx f dz /• zdz J u* -v +■*]'" { + 'J W+W + J W+W' wherein x = a-\-z. The last integral on the right is r *** = * r A z 3 = _±_ -1 J (« 2 +'« , ) r *J (rf+jy 2 (r-i)(^+^- 1 - To integrate the first integral on the right, j put 2 = b tan 6. .-. dz = dsec 2 6d6. Then 7wTn == ^f C0S ' r " e ' ls ' which can always be integrated by parts, § 139. Hence the rational function can always be integrated. EXAMPLES. J x A — 4jc We have here single real roots; hence x i + 6x - 8 x 2 + 6x - 8 ^ ^ C .* a — 4* x(x — 2)(x + 2) # ' JC — 2 ' X + 2 ' Clearing of fractions, x t + 6x - 8 = A{x - 2)(x + 2) -f- ^(x + 2) + C(* - 2)*, (1) = (A -f j9 -f- C)x 2 -f 2(^5 - C)x - \A. Equating coefficients, A + B -|- C = 1, 2(^ - C) = 6, - 4^ = - 8. .-. A = 2, j5 = 1, C = — 2. Hence the integral is -2 _j_ 6x - E ~~$x~ . X\X - 2) = l08 r . g (X + 2) 2 If we assign particular values to a - in (1), we can find A, B, C more easily. Thus put x = o, then — 4^ = — 8; put x = 2, then * Provided these « equations are independent, which they are. -j- See also Ex. 88, at the end of the chapter. / x 2 _|_ d x _ £ ^3 _ ^ ^ = 2 log x + log (x - 2) - 2 log (x + 2), 204 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. SB = 8; put x = — 2, then SC = — 16, which give the constants at once. The general principle involved in this abbreviated process is: when there are only single roots, put x equal to each root in turn, and the constants are immediately determined. 2 - f i iX Tu X) ^ o\ = t log (* - 3) + I log (x + 2). J (x - 3){* -T 2 ) 3- f . *i*. , = ! log (* + 3) + i log (x - i). 4 - Ix^Vx^ix ^ = rJogx + l log (x - 2) + i log (, + 3 ). 6 - f x * 3 lx-6 d * = log [( * + 3)2 (X ~ 2)L 7. T X)* 3 + (C - 3^ - 2i?)^ 2 -{- (sA -\- B - C + D)x - A. .: I = A + D, o = C - $A - 2D, o = 3A + B-C + £>, i = - A. Whence A = - i, .5 = 2, C = I, Z> = 2. •T 3 + I _ _ I _J 2 , * , 2_ x(x — I) 3 x "*" (# — I) 3 "^ (Jf - I) 2 "" jr - I ' /Jtr 3 -(- I II — — dx = — log x — ■ s (- 2 log (x — I), *(.*• — i) 3 (x — i) 2 x — i [ (x - i) 2 = log X (x - I) 2 — 7-t— = . Here there are a pair of imaginary roots. (x + i)(x z + I) x A Lx + J / (7+~l)(** + I) - ! + , + I -t «■ ' Art. 141.] GENERAL INTEGRALS. 205 Clearing of fractions, x = A(l + x 2 ) + [Lx + M)(l + *), = (A + J/) + (Z + J7> + (^ + ^)-v 2 . Equating coefficients, Z -f A = o, Z -f M — I, ^ -f J/ = o. Z = {, J/ = i , ^ = _ \, r x dx 1, I + x 2 1 ••• J (T+ ix,- +1) = 4 ,og (TT^F + 2 tan " 1 " 13 - /rq^ We have * + ■** = (I + r)(1 - * + x ' l) - 1 ^ Zx + M ■"'- I +X 3 "~ T+^r + I - a- + x 2 ' Clear the fractions and put * = — 1. Then A ~ \. Substituting this, we get 2,{Lx -f M) = 2 — x. /dx _ 1 /» <&■ 1 /* (2 — *)<&■ I + jt 3 ~ 3«/l + A:~'3 t /l— jr-f-x 2 ' II I 2x I log (i + x) — -j: log (I - x + x 2 ) 4 ^tan-i 3 6 V3 y 3 14. f * = I log - I + " + " 2 + —tan- 2 " + ' e/ I - X 3 6 g I - 2X -f- X 2 ^ |/7 15 f x2dx 1/3 v 3 - 2 x 2 + (^4-^ + ^ + 2?. .-. ^ = — i 5 B = 3, C = 2. Z> = o. 2X 3 _|_ x _j_ 3 -x+3 , 2x 1 + (x 2 -f- i) 2 (x 2 -f i) 2 ' x 2 4- i /a* 2 — j j P ut x — tan Q> tnen * ne integral becomes /cos 2 a?9 = 10 4- i sin 20 = -1 tan-*# -I . 1 2(jH + I) - /^^-=^^)+t^- T -Io g( , 2+r) . 21. [J*Lt*L* m = ,.; 3 "- 24 _, + -^ tan- 2 ^- 3 3 X +3V 3(* 2 - 3-*" + 3) 3 |/ 3 4/3" 22. ffi + J-ZJ c/x = ■ 2 ~ * ; 4- log (^ + 2)* ,/ (x 2 + 2) 2 4 (x 2 + 2) ^ & v ' ; 4 y 2 4/2 142. Trigonometric Transformations. — On account of the simple character of the reduction formulae in §§ 139, 140, it is often advan- tageous to transform many algebraic integrals to these forms, and con- versely many trigonometrical formulae can be transformed into useful algebraic forms. * EXAMPLES. 1. Put x = a tan 6, then /x vi dx r _ a m-n+i / s i n w0 cos w_ w - 2 fa (a 2 4- x' 2 ) in J 1 0, then /x™ dx , C («2 _ x 2f« J 2. Put x = a sin 0, then x'" rt'.r * sin cos"-^ 3. Put x = a sec 0, then x»' dlr r cos M - w - 2 /x'» rfx /* ^. — a m - M +i I , dQ. ( A 2 — a 2 f- n J sin"-'0 4. Put x = 2a sin 2 0, then /X™ dx /•sin 2 "'-"+ ] ,„ (2ax — x 2 ) in J cos*-'0 5. Make the same transformations in the above integrals when m or n is negative. * The reduction formulae for the binomial differentials are given in the Ap- pendix, Note 10. Art 144. J GENERAL INTEGRALS. 207 The general integral A x m dx can always be transformed to the trigonometric integral when the signs of a and c are known, whatever be the signs of m and n. EXAMPLES. 1 . Integrate by trigonometrical transformations f \/a* — x 2 dx, f \/x l — a 2 dx, f \/x 2 + d l dx, /dx r dx /* dx \Zd 2 — x 2 ' J \/x 2 - a 2 ' J \/x 2 -f a 2 ' Rationalization. 143. Integration of Monomials. — If an algebraic function con- tains fractional powers of the variable x, it can be made rational by the substitution x = z n , where n is the least common multiple of the denominators of the several fractional powers. J i-\-x i For example, + Put x = s 4 . The transformed integral is : 3 (i -f z) dz * I + z 2 Consequently the integral is |jc* — 2x1 — ^xi -\- 4 tan- 1 ^ — 2 log (1 + x*). Again, any algebraic function containing integral powers of x along with fractional powers of a linear function a 4- dx can be ration- alized by the transformation a -f bx = z n , in the same way as above. EXAMPLES. /x^ dx 2 —=== tt (5* 3 + t* 2 + 8x + 16) s/x - 1. \/x — 1 35 P x dx 2 2a -\- bx , 2 2. / f3 = T .._T._.. , by a + £x = 2 2 . J {a + bx> P ^ a + ^ Complete the differential, integrate and compare results. dx , / . n 2 2 yx — 1 -\- 1 = \og(x -f |/x - 1) = tan— + V* ~ * v 3 v 3 r (x 2 ± i)dx _ r _ J x \/x* -f- ax 2 -f~i ~ ^ 4/z 2 + Put x T x-i ± 2 144. Observations on Integration. — As we have remarked be- fore, comparatively few functions have primitives which can be expressed in a finite form of the elementary functions. For example, 2o8 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. / \/y dx, when y is a polynomial in x of degree higher than the second, is not, in general, an elementary function and cannot be expressed in finite form in terms of the elementary functions. \iy is of the third or fourth degree, the integral defines a new class of functions called elliptic functions. Functions that are non-integrable in terms of the elementary functions can frequently be expanded by Taylor's series and the integral evaluated by means of the infinite series. Any rational algebraic function of x and \/ax 2 -\- bx -j- c can be rationalized and integrated as follows : Factor out the coefficient of x 2 and \ety = \/±x' i -\- px -j- q. The rational function F{x,y) is rationalized in x: I. When the coefficient of x 2 in y is positive, by the substitution \/x 2 4- px -\- g = z — x. Then , = Z -Z± , s _ x = t±*±l, dx= ^±J*+fi liz . p -f- 2Z p -f- 2Z {p -j- 2Zy J ^ ' J \P+2Z' P+2Z )(pJf2Zf II. When the coefficient of x 2 is negative and the roots of the quadratic a, (5 are real, then — x 2 -j- px -j- g = {x — a)(fi — *)■ The function F(x,y) is rationalized by either of the substitutions \/ — x 2 4- p x -\-q = \/{x — a)((3 — x) = (x — a)z or (/? — x)z. ]dz, (*_«), = !£= -f- (3 (ft — a)z\ z dz T hen x = - — ~- , dx = — - v — — — ' dz, (x — a)z = y - — - ; . i 4- z l (i 4- z 2 ) 2 v i -j- z 2 .-. fF( X ,y ¥x = 2{ a-P)fF(^- + 1 + ^ /(M When the roots of — x 2 -\~ px -\- q are imaginary the radical is imaginary. 145. Integration by Infinite Series. — We know that if a function A x ) = a o + a i x + V* + • • • in an interval ) — H, -\- H {, then also its primitive is equal to the primitive of the series for this same interval (§ 72). Hence J/(x)dx = a Q x + \a x x 2 -f- \a 2 x* + . . . EXAMPLES. f dx - - 4- i - 4- 111 x " 4- * 3 5 ^- 4- ' «/ "4/! _ x 5 _ I 2 6 2.4 11 "^ 2.4.6 16 ' 2. / = 2 4/sin -r 1 -f • + . . • ) J |/sin.r V 2 5 ^2.4 9 / Art. 145.] GENERAL INTEGRALS. 209 Put sin x = z. .-. i/x — dz/cos x, and the integral is r * J \w q m 4- n ' 2 ! J X s — Jx -\- 6 20 (.r — i) 5 ra f x ' idx _ _ I r (3^ 2 - 7 4-7)^ Jx 5 - 7jc 4- 6 " _ 3 J x* - 7x 4- 6 ' = l 1 „ g( .^ 7 , + 6) + | )1( , s (--^ + 3) Art. 145.] GENERAL INTEGRALS. 213 67. / , - F *L_ = _i + I 1 lo B '_*_. J jc*(0 — x) ax a i a — x 68 - J ( — - ov - 2) = J3i + 2 lo s jri- / , ' ta D --* +I 7, -/i^+- 3 = 67^T) + l0g (X - I)* 18 |/2 ^2 Notice x* — 4x-(-3 = (jc— i) 2 (x 2 -f 2.r -f 3). 72 - / (*-.)»(£-»,+ ,) =^1 + '° g j.-Zr+« + ' tan "' ( - r - "' 3 ' J (x 2 + a 2 ) 2 ~ 2T 3 an ' a + 2a 2 x 2 + a 2 ' 74 /* ^ Jw ^. l 1 ! , J x(i + x 3 ) 2 3 g i + ^ r 3i + x 3 Put X 3 = 2. 75. / = — log — — 5T — tan- 1 .*-. J x- K i + x + x 2 -f- x 3 ) 4 g '* ' i/x _ 1 x* I X 2 -f- X 3 ) ~ 4 ° g (I + X)\l + X 2 ) ~ 2 76. / ; , , „ -- ; — = - tan- dx _ 1 _ x 3 + 3* X 2 d£t I 2X 3 = -7- tan- " 77. /* X ' ldx - a f x>dx 1 x 2 « 79. If/(jc) = (* - atj) . . '. (x - a M ), and T^x) is a polynomial of degree less than n, show that / --}-{ dx = 2 7T - L . log (x - a r ). 80. Show that any algebraic function involving integral powers of x and frac- tional powers of a -4- bx y = p+qx can be rationalized by putting y = z m , where m is the least common multiple of the denominators of the fractional powers. Apply to Exs. 81, 82. a/x — b -f- \/x — a g V — i / — =•• |/x — b — yx — a fJ^ZT^ = |/(* - a)(x -b)- ^(a - b) log 214 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XVIII. 83. If f(x) is a rational function of sin x, cos x, then f(x) dx is rationalized by the substitution tan \x = z. -,, . 2z i — z 2 2.dz 1 hen sin x = , cos x — — , , ax = -. I -j- Z 2 ' I + Z 2 ' I -f z 2 In particular, when m — I, n — I, or m ~\- n is even, say 2.r, we get for these respective cases s\n m x cos n xdx = — c w (i — c l ) r dc, = -f- s™(i — s 2 ) r ds, Pn dt where j = sin x, r = cos .*•, / = tan x. /dx -, where Q x , Q 2 are any quadratic functions of x. Q1Q2 Write out Q x ~ 1 in partial fractions. This reduces the integral to § 136, (B), or to § 138, (D). 85. In general, if /(x, y) is any rational function of x and y, where y 2 — a -f- 2bx -f- ex 2 = c(x — a)(x — ft), then any one of the following substitutions will rationalize f(x, y)dx: y = a$ 4- xz, =-. z + xc\ = z(x — a)\/c. f dx _ C x ~ %dx _ 2m+3 Put u — x~* n +~, J (a 2 4- x 2 ) n ~~ J (1 4- a 2 x—' i ) n ' X * ' dv ^ or * ne other factor. r dx _ 1 _ \ ^ x _ J* dx 1 J (a 2 4- .r 2 )« — 20 V - 1) J (tf 2 4- * 2 )»-i + ^" ~ 3 V ( f b /(x) dx = F{a) - F(b), •*■ C A x ) dx- - C f{x) dx. That is, interchange of the limits is equivalent to a change of sign of the definite integral. This is also at once obvious from the original definition of an integral. For dx has opposite signs in the two limit-sums j a f{x) dx and f h A x ) dx > while they are equal in absolute value. 215 216 PRINCIPLES OF THE INTEGRAL CALCULUS. [Cii. XIX. 148. New Limits for Change of Variable. — If we transform the integral $*f(x)dx by the substitution of a new variable for x, then we have to find the corresponding new limits. Let the substitution be x = -<** sin wjc tf.* = -5— -. 20. / c - ax cos mx dx _ a 2 + w 2 J a 2 + 0* / += ° dx * u — ; 7 ; 9 = ■ — , when ac > "-* dx. Use this to show that the value of the above integral is Jo PKP + i) .-.(/ + * - when ^ is a positive integer, and therefore whenever / or ^ is a positive integer the integral can be evaluated. 23W. f**li — x$dx = . 23( 2 ). [x*(i-x)l* + 3 01 / — 1 — , „ = — ■ . Where a > b. 31. / fl 4- b cos Va 2 _ p 32 f x s ^ n x dx = *• Jo 150. A Theorem of Mean Value. — Since in t/.r <£c keeps the same sign throughout the summation, XX X mf dx < { f{x) dx < M f dx, where m and J/ are the least and greatest values respectively of the func- tion f{x) in {x , X). Therefore the integral lies in value between m(X — x ) and M(X — x Q ). Sincey^Jt) is continuous in the interval, there must be a value of or, say £, in (x Q , X), for which ( X f(x)J X = (X-x )f{$), f{£) being a value of the function between m and M, its least and greatest values. Art. 150. J ON DEFINITE INTEGRATION. The value 219 <& — a j Xo is called the mean value of the function in (a , A"). If F{x) is a primitive off[x), then F{x)-n-\) = (x--\)A$), since F'(x) =/(x). This is the familiar form of the Law of the Mean as established in the Differential Calculus. The theorem of mean value for the Integral Calculus can be estab- lished directly from the definition of a mean value. For, if Ax == (X - x )/n, then f/[x)dx=£ 2 Jxf(x r ), = '(X-x ) f A*o)+A*\) + ..-+A**) m « =00 If the limit of the arithmetical mean of the n values of the function at the points of equal division of (x , X) be indicated by /([£), the result is the same as above indicated. If y=/(x) ^y dx ■ x Geometrical Illustration. is represented by the curve AB, then : area (x Q AZBX). This area lies between the rectangles x ATX and x SBX, constructed with x Q X as base and the least and greatest ordinates to the curve respectively as altitudes. There is evidently a point c, between x Q and X at which the ordinate £Z = /(q) is the altitude of a rectangle x Q RQX, inter- mediate in area between the greatest and least rectangles, whose area is equal to that bounded by the curve. EXAMPLES. 1. Find the mean value of the ordinate of a semi-circle, supposing the ordinates taken at equidistant intervals along the diameter. Let x 2 -f- y 2 = a 2 be the circle. Then 1 /.+« _ 2a.f_„ r x 2 dx = \rta, viz., the length of an arc of 45' 220 PRINCIPLES OF THE INTEGRAL CALCULUS. [Ch. XIX. 2. In the same case, suppose the ordinates drawn through equidistant points measured along the circumference. Then the arc length is the variable, and the mean ordinate is 2 sin dQ = —a. 7tJ n it 7tJ We shall see later that this is the ordinate of the centroid of the semi- circumference. 3. A number n is divided at random into two parts; find the mean value of their product. n Jo % x)dx = — n 2 4. Find the mean value of cos x between — n and -J- tt- 5. If Af r *(y) is the mean value of y = f{x) in (x v x 2 ), show that: (a). M*{2x*+zx- i) =8i. (b). M]{2 _3*+5*i-*i)=Y. (c). M*(x+ i)(* + a) = i2i. (d). M Q in (sin B) — 2/tt. 6. Find the mean distance of the points on the semi-circumference of a circle of radius r, from one end of the semi-circumference, with respect to the angle. Wo M^-- 2r cos dO = ¥-; Tt By the mean value of n numbers is meant the »th part of their sum. To estimate the mean value of a continuous variable between assigned values, we take the mean of n values corresponding to equi- distant values of some independent variable and find the limit of this average when the number of values is increased indefinitely. The mean value depends on the variable selected. See Exs. i and 2 above. liy is a function of /, then the mean value of y with respect to / for the interval {f lt Q is 1 /»'a dt. 151. An Extension of the Law of the Mean. — If is any integer, is due to Schlomilch and Roche. 153, The Definite Integral Calculated by Series.— If f{z) can be expressed in powers of (z — a) by Taylor's series, for all values of z in (a, x), then also can the primitive of f{z), and the definite integral of the function is equal to that of the series, taken term by term, between a and x. Hence, integrating between a and x, A*) =A") + ( 2 - ")/'(") + ^r^-V"(«) + • • • , we have £/{z) dz = (x-a)/{a) + £=#/"(«) + ^=^/"(«) + • • . (■) In particular, put at = o, then we have £/(z) dz = a -f(a) - a l/'(a) + a l/"(a) -..., (,) a formula due to Bernoulli. Knowledge of the derivatives at a serve therefore to compute the integral. When a = o in (1), then pW dz = a/(o) + £/'(o) + ~f\o) + ..., (3) which is Maclaurin's form, and is more convenient, in general, for computation than (2). EXAMPLES. 1. Deduce Bernoulli's formula (2), § 153, by using the formula for parts, ffix) dx = xf(x) -fx/'(x) dx. J-x \ 3 5 2! 7 3! / 3. / *-* a dx — x ■ - -f . .. Jo 3 5 2! 7 3! 4. ^log (tan ' = VJ- * Io g( 2 + VD< 39. Tutting x = a sin 6, I x 1 \/a z — x 2 dx = a* I sin 2 G cos-0 > dx. EXAMPLES. 1. Area of the circle. Taking x 2 -f- )' 2 = d 2 as the equation of the circle, .-. y — ± \/a 2 — x 2 . y -, 3 . P Pi / a '0 £ c a \A Fig. 70. Taking the positive value of the radical, we have for the area x P P x x x , If x x = a, we get the area of the semi-segment x P A. If x = o, and x x = a, we have the area of the quadrant OB A equal to i> x' 2 dx = \ita 2 . If is the angle POA, then y = a sin 0, x — a cos 0. . •. - %xy. But xy is the area of the rectangle ONPM. The area of the segment POP' of the parabola cut off by a chord perpendicular to the diameter is two thirds the rectangle MPPM'. 4. Area of the hyperbola. Let x 2 /a 2 — y 2 /b 2 = 1 be the equation to the curve. Then the area of A PN is A = J y dx, a 2 dx. log (x + \/x' y P/ \^Y i>^~. A T = \xy — \ab log 5. Area of the catenary. The equation to the curve is y =\a\e« + e « ) . . O VPN is x C x l~ -~\ A = J \a \e a + e a ) dx, The area O VPN is Fig. 73. = U 2 \e a — e a ) — a Vy' 1 If NL is perpendicular to the tangent at P, show that the above area is twice that of the triangle PLN Observe that tan LNP = Dy, LN = y cos LNP, etc. 6. Show that the area of a sector of the equilateral hyperbola x 2 — y 2 = a 2 included between the .r-axis and a diameter through the point x, y of the curve is 7. Find the entire area between the witch of Apiesi and its asymptote. The equation is (x 2 -\- 4a 2 ]y = 8tf 3 . Ans. \-itd 1 . Art. 155.] ON THE AREAS OF PLANE CURVES. 229 8. Find the area between the curve y = log x and the .v-axis, bounded by the ordinates at x = I and x. Ans. x(log x — I) -f- I. 9. Find the area bounded by the coordinate axes and the parabola x^ -\- y^ = at. Ans. \a 2 . Ans. \nab. 10. Find the entire area within the curve ©'+ ©*- 11. Find the entire area within the hypocycloidx\ + 7* = a--. Hint. Put x = a sin :< 0, y = a cos :t 0. Ans. |7Ttf 2 . 12. Find the entire area between the cissoid (2a — x)y 2 = X s , and its asymp- tote x Ans. 13. Find the area included between the parabola x 2 = 4.av and the witch y(x* + 4* 2 ) = 8a 3 . Ans. a\2it - f ). The origin and the point of intersection of the curve give the limits of the integral. 14. Find the area of the loop of the curve /2 — (r _ n\lr _ h\1 Hint. Let x z 2 . Ans. 1 \V- a ? "Q *5\ < 15. Find the whole area of the curve a 2 y 2 — x 3 (2a — x). Fig. 74. Ans. Fig. 75. 16. Find the area of the loop c_ Ihe curve ay - x\b -f x). The area of the loop is . 2 /»° 32/^ A = S I X 2 A/b + X dx - ■ 5 W_* T ^ 3-5-7^ Put 3 -f x = z 2 . 17. Show that iiy =f(x) is the equation of a curve referred to oblique coordi- nate axes inclined at an angle go, then the area bounded by the curve, the x-axis, and two ordinates at x , x x is A = sin go I y dx. 18. The equation to a parabola referred to a tangent and the diameter through the point of contact isy 2 = kx. Show that the area cut off by any chord parallel to the tangent is equal to two thirds the area of the parallelogram whose sides are the chord, tangent, and lines through the ends of the arc parallel to the diameter. 19. The equation to the hyperbola referred to its asymptotes as coordinate axes is xy = c 2 . If (» is the angle between the asymptotes, show that the area between the curve, x-axis, and two ordinates at x , x l is c 2 sin go log ( — ) • 20. If y = ax n is the equation to a curve in rectangular coordinates, show that the area from x = o to x is 2 30 APPLICATIONS OF INTEGRATION. [Ch. XX. 156. If the area bounded by a curve, the axis of y s and two abscis- sae x , x v corresponding to the ordinatesj' , y v is required, then that area is pyx A = J x dy. EXAMPLES. 1. Find the area of the curve/ 2 = px between the curve and the j-axis from y ■= o to y = y. 2. Find the area of the curve y = e x between the curve, the j-axis, and ab- scissae at y = 1, y = a. Check the result by finding the area between the curve and the jf-axis for corresponding limits. Also find the area bounded by the curve, the /-axis, and the negative part of. the ^r-axis. 157. Observe that in the examples thus far given the portion of the curve whose area was required has been such that the curve was wholly on one side of the axis of coordinates. It is evident that if the curve crosses the axis # between the limits of in- tegration, then, y being positive above the ^f-axis FlG ' 76, and negative below it, those portions of the area above Ox are positive, those below are negative. The integral ~ ri ydx £ is then the algebraic sum of these areas, or the difference of the area on one side of Ox from that on the other side. EXAMPLE. Find the area oiy = sin x from x = O to x We have Jo f I! 'T- Jo : I ' sin x dx = — cos x h _io But / sin x dx = 2, '2rr sin x dx = — I. Fig. 77. ... A*" = A" + A*" = 2 158. It is evident that the area considered can be regarded as the area generated or swept over by the ordinate moving parallel to a fixed direction, Or. Art. 160.J UN THE AREAS OF PLANE CURVES. If we have to find the area between two curves 2/ y x = 0(.v), r 2 = ip(x), and two ordinates at a and b t such as the area LMXR in the figure, that area can be computed by finding the area of each curve separately. But if it is more convenient, the (J area is Fig. 78. j\y, ~J\) dx = j\,/:(x) - V -x^IQ = y r Let a and b be the abscissae of the extreme tangents aA and bB. Then the area of the curve is A = jf (> ' 2 "^ dX% This result also holds if the curve cuts the axis of x. EXAMPLE. Find the whole area of the curve (y — tnxf - Here y = mx ± \/a 2 yx A mx mx (y 2 -}'i) dx > x 2 dx = na 2 . 160. The area of any portion of the curve K'T (') Fig. 80. 232 APPLICATIONS OF INTEGRATION. [Ch. XX. is ab times the area of the corresponding portion of the curve f{ X , y) = c. (2) For (1) is transformed into (2) by puttings = ax' ', y = by' in (1); and hence y dx, from (1), becomes ab y' dx' , and we have Cy **=*£'*• EXAMPLES. 1. The entire area of the circle x 2 -j- y 2 = 1 is n. Hence that of the ellipse x 2 /a 2 + y 2 /b 2 = I is abit. 2. Find the whole area of the curve ( — J -|- l^-\ = In Ex. 11, § 155, it is shown that the area of X i _J_ y\ _ J is f 71. Hence that of the proposed curve is \itab. 3. Check the result in Ex. 2 by putting x = a sin 3 0, y — b cos'0. Then ydx = ^ab sin 2 cos 4 d(p. I sin* 2 cos 4 d(p = %rtab. 161. Sometimes the quadrature of a curve is to be obtained when the coordinates are given in terms of a third variable, or is facilitated by expressing the coordinates in terms of a third variable. Thus if x = 0(/), y = tit), the element of area is ydx— ip(t)cp'{t)dt. EXAMPLES. 1. Find the area of the loop of the folium of Descartes, x z ._j_yj _ ^axy. Put y — tx\ then - _3fL_ v - 2afi I - 2 ' 3 •'• ^=or+^p 3 ^ and fydx = 9 a 2 f-^- i / 2 (i - 2/- 3 )<# 6a 2 ga 2 Fig. 81. y ~* J (i + z 3 ) 3 1 4- *» 2(1 + *»)*• The limits for / are o and 00 . Hence A = |a 2 . 2. In the cycloid, x — a{t — sin /), y = a(i — cos /), . •. f y dx = a 2 /Vers 2 / dt = q& J sin* \t dt. Taking t between o and 7T, we get T,na 2 for the entire area between one arch of the cycloid and its base. Art. 162. J ON THE AREAS OF PLANE CURVES. 233 3. Eind the area of the ellipse using x' 2 /a' 2 -\- y' 2 /b' 2 = 1, where x = a cos 2 ^. When the law of change of /, the length of the tangent, is given as a function of 6, the area can be evaluated. If t — f(6) be this relation, the curve t =_/"(#), considering / as a radius vector and # the vectorial angle, is called the directing or director curve of the generating line. EXAMPLES. 1. Show that the area swept over by a line of constant length a laid off on the tangent from the point of contact is 7ta' 2 , when the point of contact moves entirely around the boundary of a closed plane curve. 2. The tractrix is a curve whose tangent -length is constant. Find the entire area bounded by the curve. (Fig. 84.) The area in the first quadrant is generated by the constant length PT = a turning through the angle \it as the point P moves from J along the curve J PS asymptotic to Ox. Therefore the area in the first quadrant is Irta 2 , and the whole area bounded by the four infinite branches is ita 2 . 236 APPLICATIONS OF INTEGRATION. [Ch. XX. 3. Check the above result by Cartesian coordinates and find the equation to the tractrix. We have directly from the figure dy —-— - tan PTN dx yd* : . . ' . y dx = — ya 2 — y 2 dy. Fig. 84. Hence the element of area of the tractrix is the same as that of a circle of radius a. It follows directly that the whole area of the tractrix is rta 2 . This gives an example of finding the area of a curve without knowing its equation. To find the equation of the tractrix, we have dx \/a 2 dy Integrating, we get x = — ya 2 — y 2 -j- a log \/a 2 - y 2 since x = o wheny = a. This is said to be the first curve whose area was found by integration. 4. Show that the area bounded by a curve, its e volute, and two normals to the curve is *o where p is the radius of curvature of the curve, and Q the angle which the normal makes with a fixed direction. 164. Elliotts Theorem. — Two points P x and P on a straight line describe closed curves of areas (P^ and (P 2 ). The segment P X P 2 moves in such a manner as to be always parallel and equal to the radius vector of a known curve p =/"(#) called the director curve. It is required to find the area of the closed curve described by a point P on the line P X P 2 which divides the segment P X P 2 in constant ratio. Art. 164.] ON THE AREAS OF PLANE CURVES. 237 Let (P), (P x ), (P 2 ), (A) be the areas of the closed curves described by the corresponding points as shown in the figure. Let Fig. 85. P X P 2 and P X P 2 , Fig. 86, be two positions of the segment. Produce them to meet in C. .--^ Let p = P X P 2 , P,P/PP 2 = mjm 2 . PJ> = m l -\-m 2 *—P x P t = kj>, PP. ^p x p 2 = k 2 p, (1) 2 m l -f m 2 where k x -f- k % = 1 . The element of area P X P 2 P 2 'P X is, § 163, if CP X — r, d(P 2 ) - d(P x ) = |(p + rf dd - JH dd, = pr dd + \tf dd. In like manner the element of area P X PP'P X ' is d(P) - d(P x ) = l(k lP + rf dd - p dd, = k x pr de + %k x Y de. Multiply (1) by k x and eliminate k x pr dd between (1) and (2), remembering that k x -f- k 2 = 1. Then d(P) = WJ + w\) - *AHA)- Integrating for a complete circuit of the points P x and P 2 about the boundaries of the curves, we have {P) = K(Pd + K( p .) r KK( A ), (3) where the area of the director curve is given by (A)=ffffid0, the limits of the integral being determined by the angle through which the line has turned. 238 APPLICATIONS OF INTEGRATION. [Cn. XX. In particular, if P X P 2 = p is constant and equal to a, we have Ho/ditch's theorem, {P) = kjft + K{P 2 ) - RV/' /ft If a chord of constant length a moves with its ends on a closed curve of area (C), the area of the closed curve traced by the point P which divides the chord in constant ratio m : n is (P) = (C) (in -f n y ' = ( C ) ~ c x c 2 7t, if P is distant c and c 2 from the ends of the chord. EXAMPLES. 1. A straight line of constant length moves with its ends on two fixed intersect- ing straight lines; show that the area of the ellipse described by a point on the line at distances a and b from its ends is nab. 2. A chord of constant length c moves about within a parabola, and tangents are drawn at the ends of the chord; find the total area between the parabola and the locus of the intersection of the tangents. Am. \itc l . The area between the parabola and the curve described by the middle point of the chord is the same. 3. It can be shown that the locus of the intersection of the tangents in Ex. 2 to the parabola y 2 = \ax is {y 2 - 4ax){y* -f 4rt' 2 ) = a 2 c 2 . Check the result in Ex. 2 by the direct integration / z dy = \c 2 it fromjj' = - 00 to/ = -f 00. z being half the distance from the intersection of the tangents to the mid-point of the chord. 4. Tangents to a closed oval curve intersect at right angles in a point P\ show that the whole area between the locus of P and the given curve is equal to half the area of the curve formed by drawing through a fixed point a radius vector parallel to either tangent and equal to the chord of contact. 5. If p v B x and p 2 , B 2 are the polar coordinates of points P x and P 2 on a straight line, then the radius vector p of a point on this straight line which divides the segment P X P 2 — X so that PP X = k x X, PP 2 = k 2 \, is determined by P 2 = k -iP\ + V2 2 - k A Xi ' (0 This is Stewart's theorem in elementary geometry. If (p is the angle which p makes with P 1 P 2 , then Px > = p* + k*W - 2k x Xp cos 4>, p 2 2 = p 2 + k*X* + ik 2 \p cos 4>. The elimination of cos

and Elliott's theorem follows immediately on integration. = -K> + !*? & - \ r 2 dQ, = ;-/, dO -f U? dO. = lr' dO - \{r - /, Y do, = r/ 1 dO - \l* dO. = /. r dO + m - l*)d Art. 1O4.J OX Till-; AREAS OF PLANE CURVES. 239 The geometrical interpretation of (2) is as follows : Let A. = P x l\ be constant Construct the instantaneous center of rotation / of X as 1\1\, turns through JO. Then /',/','. PP\ J'JV (Fig. SO) subtend the angle M at /. The center / being considered as origin or pole. (2) follows at once. The extension to the case when A. is variable is immediately evident. 6. Theory of the Polar Planimeter. In Fig. 86, let P X P 2 = / be constant. At P let there be a graduated wheel attached to the bar P X P 2 in such a manner that the ;ixk- of the wheel is rigidly parallel to P x P r This wheel can record only the distance passed over by the bur at right angles to the bar. Let P 1 P=l 1 , PP 2 -l v Let CP = r. Then with the symbolism of § 164 we have d(P 2 ) - d(P } d(P) - d(P,) Adding these two equations, d(P 2 ) - d{P,) But r dO = dR is the wheel record for a shift of the bar. Integrating, we have for the area bounded by the curves traced by P 2 and P x and the initial and terminal position of the bar W - (*\) = K R 2 - *i) + Wi - A 2 )(9 2 - 9i), r 2 being the initial and terminal angles which the bar makes with a fixed direction, and P l} P 2 the initial and terminal records of the wheel. Notice that when the wheel is attached to the middle of the bar (P 2 ) - (/\) = 1{R 2 - P x ). The path of P x is a circle in Amsler's instrument. EXERCISES. 1. Find the area of the limacon p = a cos -\- b, when b > a. Ans. (b* -f W)rc. 2. Show that the area of a segment of a parabola cut off by any focal chord in terms of/, the chord length, and/, the parameter, is i/ 2 /2. 3. Show that the area of the curve x 2 y 2 = (a — x)(x — b) is 7l{a* — b^) 2 - 4. Show that the whole area between the curve y(a 2 -\- x 2 ) = ma 3 and the .r-axis is m-jta 2 . 5. Show that the whole area between the curve y 2 (a 2 — x 2 ) = ft and its asymptotes is 2itb 2 . 6. Show that the area between the curve and the axes in the first quadrant for (x/a)i -f- ( y/bf = 1 is ab/20. 7. Show that the area of a loop of the curve y 4 — 2c 2 y 2 -f- a 1 * 1 = o is 2f r 3 n a ^ 8. The locus of the foot of the perpendicular drawn from the origin to the tan- gent of a given curve is called the pedal of the given curve. (1). The pedal of the ellipse (x/a) 2 -f (y/b) 2 = 1 is p2 = a 2 cos 2 + b 2 sin 2 0. 240 APPLICATIONS OF INTEGRATION. [Ch. XX. Show that its area is \-n{a 2 -f- b 2 ). (2). The pedal of the "hyperbola {x/af — (y/b) 2 = I is pr = a 2 cos 2 Q — b- sin 2 9. Show that its area is ab -\- (a 2 — b 2 ) tan— *(a/b). 9. If y x , y 2 , y s be three ordinates, y 2 being midway between y x and_y 8 , of the curve y = ax 3 -\- bx 2 -\- ex -j- d, show that the area bounded by the curve, the .r-axis, and the ordinates y x and y % is If we transfer the origin to x 2 , o, and put x x = — h, x 2 = + /*, the equation of the curve can be written y = ax 3 + fix 2 4- yx + 8. We have for the area ,+k J ydx = 2h(\ph 2 + d), and \h(y x +>' 3 + 4y 2 ) nas this same value. This is called Newton's rule. 10. Show that the area of any parabola y — ax 2 -+- bx -f- c, from x — — h, to x — -f- /i, can be expressed in terms of the coordinates x l , y x and x 2 , y 2 of any two points on the curve, whose abscissae satisfy x x x 2 — — £// 2 . Ans. A = Zh * 1 ** ~~ ** y \ x x — x 2 The mean ordinate in the interval is Let/ and q be two undetermined numbers. Then P>\ + qy ( 2 > P + 9 = -ii (3> give determinate values of / and ^, provided •*"i 2 , * 2 2 > W = °» x 1 , * 8 , o i,i,i or jr x jr 2 = — \h 2 . Theri ;'« = p)\ + qy 2 » and the values of / and q from (2), (3) give the result. 11. In Elliott's theorem, § 164, (3), show that the mean of the areas of the curves described by all points on the segment P X P 2 is ^[(^i) + (^)J — \{^)- 12. A given arc of a plane curve turns, without changing its form, around a fixed point ;n its plane; what is the area swept over by the arc? Art. 164.] ON THE AREAS OF PLANE CURVES. 241 13. If a curve is expressed in terms of its radius vector r and the perpendicular from the origin on the tangent/, prove that its area is given by I /* pr dr 2 J tf-przrp' 14. Lagrange's Interpolation Formula. We have seen, in the decomposition of rational fractions, that when tp(x) = (x - a t )(x - a 2 ) . . . (.r - q„), and F\x) is a polynomial in x of degree less than n, F(x) __ « I F(a r ) See § 133, and Ex. 79, Chapter XVIII. If F\x) is any differentiable function of x, then, since vanishes at x = ^ x , . . . , # M , and the second term is a polynomial of degree n — I, we have, § 98, II, lemma, where r is some number between the greatest and least of the numbers x, a x , . . . , a n . The formula is called Lagrange's interpolation formula. The member on the left computes the value at x of an unknown function when its values at a x , . . . , a n are known, with an error which is represented by (x - a x ) . . . (x>-a») ^ n\ v 15. Gauss' and Jacobi's theorem on areas. If F\x) is any polynomial of degree 211 — 1, then the exact area of the curve y = Fix) between x = p, x = q can be computed in terms of n properly assigned ordinates. Let ^ X) ~ ~ x-a r f'(a r )> where, as in Ex. 14, ip(x) = (x — a x ) . . . (x — a n ). Then J{x) = F(x) — L(x) is a polynomial of degree 2n — I, in which F[x) is of degree 2« — 1, L(x) of degree n — 1. Also, J{x) vanishes when x = a v . . . , a n . Hence F(x) - L{x) = A 0(x) iP(x), where A is some constant and (p{x) some polynomial of degree n — I, since ip(x) is of degree n. Integrating between p and q, r Fx) dx - r L(x)dx = a r ^{x) ^) dx. Jp Jp Jp 242 APPLICATIONS OF INTEGRATION. [Ch. XX. Jacobi has shown as follows that we can always assign a x , . . . , a n , so that I

(x) f\x)dx= f*2p )* or the proposition is established.* If the degree of F(x) is 2«, then the area can be expressed in terms of n -f- I ordinates taken at the roots of / d \ »+* \dxj [{X ~^ X ~ ^ M+I = °- The area of y = 7^x) can be expressed in a singly infinite number of ways if one more than the required number of ordinates be used, in a doubly infinite number of ways if two more than the required number be used, and so on. 16. Show that the area of y = a o + a \* + a -2 x ' 2 + a z^ > from — h to -f h, is equal to where y x and y 2 are the ordinates at .v = ± A/4/3, Give a rule and compass construction for placing these ordinates. *See Boole's Finite Differences, p. 52. CHAPTER XXI. ON THE LENGTHS OF CURVES. Rectangular Coordinates. 165. Definition of the Length of a Curve. — A mechanical con- ception of the length of a curve between two points on it can be obtained by regarding the curve as a flexible and inextensible string without thickness, which when straightened out can be applied to a straight line and its length measured. The curvilinear segment is then said to be rectified. The rigorous analytical definition of a curve and of its length is a more difficult matter. If y is a function of x such that y, Dy, D 2 y, are uniform and continuous functions in an interval x = a, x = /?, then the assem- blage of points representing y =/(x) in (a, f3) is called a curve. We can demonstrate * that if P and P are any two points on this curve, we can always take P and P x so near together that the curve between P and P lies wholly within the triangle whose sides are the tangents at P and P x and the chord PP V And also, if Q, R be any other two points on the curve between P and P lt then, however near together are Q and R, the same property is true for Q and R. Fig. 87. If we divide the interval (a, b) into n subintervals and at the points of division erect ordinates to the points A, L, . . . , B, etc., on the curve, then draw the chords through these points, and the tangents to the curve there, we shall have two polygonal broken lines ALMNB inscribed, and ATRSVB circumscribed, to the curve AB. Appendix, Note II. 243 244 APPLICATIONS OF INTEGRATION. [Ch. XXL Let c r represent the length of the rth chord, and t r that of the rth side of the circumscribed line. Clearly, whatever be the manner in which (a, b) is subdivided or to what extent that subdivision be carried, we shall always have 2c r < 2f r and £ 2c r — £ 2t = o. I I «=oo I n = x> i If we interpolate more points of division in (a, b), then 2/ decreases while 2c increases. Consequently 2t and 2c converge to a common limit. This limit we define to be the length of the curve between A and B. 166. Let P be a point x, y on a curve, the length of which between A and P is s. Let P be a point on the curve having coordinates x -f- Ax, y -f- Ay, and let the length of the curve between P and P x be As, the length of the chord PP X be zfc. Draw the tangents at P and P v Let the angle which PT makes with Ox be 0, and the angle between TP and 77^ be J0. / 1? then, by § 165, Ac < As < / -f / x . But, from the triangle ^77^, (Jc) 2 = / 2 -f /j 2 -f 2// x cos A6, — (/4-/J 2 - 4#, sin 2 \A6. Ac V 4^i ' + 4 (' + «' //#. 4// x /(/ -j- Z^ 2 can never be greater than 1, and when A6( — )o, Ac\ — )o, t -f- / x (=)o, also sin 2 \A6{ — )o. Therefore when Ax( = )o > we have Ac \ A*( = )o Since, by definition, As lies between Ac and / -f t x 'Ac £ we also have Now, or £Ac _ A.r( = )o {Acf = (Ax)* - j'Acy _ (Ac\ /As \Ax) ~~~ \Js) \Ax (4>') 2 > )'~+® Art. 166.J ON THE LENGTHS OF CURVES. 245 Therefore, in the limit, for Av( = )o, or * = V I + ($)'*■ (I) Hence the length of the arc of the curve from A to P is In like manner, using 4>' instead of Ax, we obtain -f'A^m*- » In differentials EXAMPLES. 1. Rectify the semi-cubical parabola ay 2 = :*3. Here y x% dy a i' ' ' dx =16)' ds dx -(■ +5 s -rhz) '*-ll h gx\i ~4a) -i the arc being measured from the vertex. This was the first curve whose length was determined. The result was obtained by William Neil in 1660. 2. Rectify the ordinary parabola y % = 2ax. We have DyX = y/a. 1 /•y , .-. s=- Ya*+y*dy, a t/0 1 1-7. 1 a , y -\- Vy 2 4- <* 2 the arc being measured from the vertex. I - -*) 3. Rectify the catenary y — \a \e <* -f- ^ a / . We have Dy — \\e a — e a ) . .-. s = \ ( X \e~" -f *~ «/ <£r = ^a V s - * a / • 246 APPLICATIONS OF INTEGRATION. [C11. XXL Show that s = PL Fig. 73. Also, NL = constant. The catenary is therefore the evolute of the tractrix represented by the dotted line in the figure. 4= Rectify the evolute of the ellipse (ax)* -f- (by)* - (a 2 - b 2 )*. Write the equation in the form ©'+» put x = a sin 3 0, y = (5 cos 3 0. . •. s = 2 3 f (a 2 sin 2 -f ft 2 cos 2 + -•")'+ ' "* <£ + y*-\ 9. Find the length of the tractrix (see § 163, Fig. 84) dy fdy — a I — = — a log y -f- const. Measured from the vertex, x = o, y = a, S = a log (a/y). x 2 v 2 10. Length of an arc of the ellipse —-)- — = 1. Put .r = a sin 0, y =.- b cos 0. Then 5 = /% 2 cos 2 -f <* 2 sin 2 0)^/0 r = a J (1 — e 2 sin 2 0) , — r = Jj Ac _ zfc A'e~As~~A~d' *' d6~dd ... <&2 = ^ p 2 _|_ p 2 ^ ds dc . ds nss = -^ , since — = 1, § 166. dc -r>+^) , -r^+^)'* Otherwise we can deduce the formula for the length of an arc in polar coordinates directly from the corresponding formula in rect- angular coordinates. - — . For x = p cos 6, y = p sin 0. . •. <£r = dp cos — p sin 6 dd, dy = dp sin -{- p cos i). 4. Show that the length of the arc of log p = aB, from the origin to (p, 0), is 5. Find the length of p -f 2 = a 2 , 6. Find the length of p — a sin 0, from o to \it. 7. Find the length of p = a sec 6, from o to ±7t. 8. Show that the entire length of p = a sin 3 \B is f #£. 9. Show that the entire length of the epicycloid 4(^2 _ fl 2)3 _ 27a i p 2 sin 2 0, which is traced by a point on a circle of radius \a rolling on a fixed circle of radius a, is \2a. 10. Find the entire length of the curve p = a sin 20. 11. Show that the length of the hyperbolic spiral pB = a is [ yd 1 -\- p l — a log a + \/a* + p* P _JP 2 12. Show that the length of the parabola p = a sec 2 iB, from 8 = — \it to = 4- i7r, is 2rt(sec \it 4- log tan \n). 168. Geometrical Interpretations of the Differential Equations ds' 2 = dx 2 + ^/ ! and /y^ M /B dx Art. iC8.] ON THE LENGTHS OF CURVES. -49 II. We can in the case of polar coordinate lengths of certain circular arcs as follows: Let OA be the initial line and P a point on the curve /(p, 0) = o, PT the tangent at P. Draw OC perpendicular to p = OP, cutting the normal at P / in C. Then n = PC is the normal length, and S, t = OC is the subnormal. Let 9 be the independent variable, then d r j — AQ is an arbitrarily chosen angle. We have the differentials ds, dp, p dQ proportional to the sides of the triangle POC, or to n, S„, p, respectively. For we have ixhibit ds, dp, and p dO as the @°= & + P* But dp = S n dQ, by §92, (5). Also, S n 2 + p 2 = n 2 . Hence ds = tide. Draw .PC 'parallel and equal to OC. Strike the arcs PN, PL, and PM with centers C radius S n , C radius n, O radius p, having the common central angle AQ — dQ. Then ds = PL, dp = PN, pdQ = PM. It is interesting to notice that the rectilinear triangle PLNis a right-angled tri- angle similar to PCO; the sides of which, PL, PN, NL, are equal to the chords jr subtending the arcs PL, PN, and PM respectively. Therefore, in the triangular figure PLN whose sides are the circular arcs PL, PN, and an arc LN with, radius p equal ^ to PM, ;we have the sides {PN) = dp, (PL) = ds, (LN) — p dQ. Also, the angles between the circular arcs are (Z) =&- if>, (P) = if>, (N) = i7t+ dd. and ds 2 = dp 2 -+- p 1 dQ 2 , dp = ds cos ip, p dQ = ds sin zl: In order to prove these statements, it is only necessary to show that the rectilinear segment LN is equal to the chord subtending PM. Let x, y be the chords subtending PL, PN. Then from the rectilinear triangle PNL we have LN 2 = x 2 -\-y 2 - 2xy cos LPX. But / LPN = iff -f \AQ - \AB = if). Also, x = 2n sin \AQ, y = iS n sin \AQ. .-. LN 2 — 4(S n 2 + ri 2 - 2nS n cos rj>) sin 2 \AQ, = 4p 2 sin 2 \AQ = (chord MP) 2 . The remainder follows easily. Observe that if we draw PM perpendicular to OP, as in Fig. 93, and put PM = 8p, MP X = dp, then we have, for AQ(=)o, jTa^ _ r sp _ t rsp _ X*- z \ JL^~ l'^- 1 ' Therefore the difference equation Ac 2 = dp 2 -\- dp 2 leads at once to the differential equation ds 2 = dp 2 + p 2 dQ 2 . 250 APPLICATIONS OF INTEGRATION. [Ch. XXI. 169. Radius of Curvature and Length of Evolute. If/" (Xj r) = o is the equation of a curve, then sec 2 6> dy f a d\y Hence if R is the radius of curvature at dd dx' R (1 +y 2 r a dx sec v — - n = ds dd 1 y dd since ds = sec 6 dx. Therefore ds = R dd. The angle A6 = dS is the angle between the tangents at P and P v and is equal to the angle between the normals at P and P x . 170. The length of the arc of the evo- lute of a given curve is equal to the differ- ence of the corresponding radii of curvature of the involute. Let x, y be a point on the involute cor- responding to the point a, ft on the evo- lute. Then we have for the radius of curvature R 2 = («-*)* + (ft -J') 2 - Fig. 94. Differentiating, we get R dR = (a - x)(da - dx) -f- (ft -J')(dft - dy), = (a-x)da + (ft-j>)dft, (1) since (a — x)dx -\- (ft — y)dy — o, this being the equation of R, the normal to the involute. If 6 is the angle which the tangent to the involute at x, y makes with Ox, then, since R is tangent to the evolute, R makes with Ox the angle

T r cos - y 2 f The volume of the entire pseudo-sphere is %rta z , or one half that of a sphere with radius a. 9. Find the volume generated by the revolution of the catenary I - -A y = %a\e a -\- e a J about Ox from O to x. Ans. \ita{ys -J- ax). 10. The volume generated by revolving the witch (x 2 -f- \a 2 )y asymptote is 4^r 2 « 3 . 8a 3 about its Fig. 99. 175. To find the volume of the revolute generated by a closed curve revolving about an axis in its plane, but external to the curve. We take the difference between the volumes of the revolutes generated by MABCN and MADCN. Hence the volume of the solid ring generated by ABCD revolving about Ov is where x x = RD, x 2 = RB, and the limits of the integral are y = OM, Y = ON. A corresponding integral gives the volume about Ox* Art. 176.J OX THE VOLUMES AND SURFACES OE REVOLUTES. 259 EXAMPLES. 1. The solid ring generated by the revolution of a circle about an axis external to it is called a torus. Show that the volume of the torus generated by the circle (x - fl)»+y = r 2 (a £^ r) about Oy is 2X 2 r 2 a. We have x. 2 = a -\- |/r 2 — y 2 , x x = a — |/r 2 — y*. V y = 7t \a 4/r 2 — y 2 dy — 2it t r l a. Observe that the volume is equal to the product of the area of the generating circle into the circumference described by its centre. 2. Show that the volume of the elliptic torus generated by (f ~ ff , y _ _ {c > a) about Oy is 27t 2 abc. 176. The Area of the Surface of a Re volute. — We know, from elementary geometry, that the curved surface of a cone of revolution is equal to half the product of the slant height into the circumference of the base. The area of the curved surface of the frustum included between the parallel planes AD and BC is i therefore / 7i{VD.AD- VC-BC). / Since BC/AD = VC/VD, we deduce for the ^= surface generated by the revolution of CD about Fig. 100. BA the area 2 7TMN.CD, where MN joins the middle points of AB and CD. In the figure of § 174, Fig. 98, subdivide, as before, the interval (a, b) into n parts; erect ordinates to the curve AB at the points of division. Join the points of division on the curve by drawing the chords of the corresponding arcs, thus inscribing in the curve AB a polygonal line AB with n sides. Let PP' be one of the sides of this polygonal line. The curved surface of the frustum of a cone generated by the chord Ac = PP r revolving about Ox has for its area 27r yJry Ac = 27t{y + \Ay)Ac. We define the surface generated by the revolution of the arc of the curve AB about Ox to be the limit to which converges the sur- face generated by the revolution about Ox of the inscribed polygonal line, when the number of the sides of the polygonal line increases indefinitely and at the same time each side diminishes indefinitely. 260 APPLICATIONS OF INTEGRATION. [Cn. XXII. To evaluate this limit, we have for the area of the surface gen- erated by the curve AB n = » i &*(=)o n — £ ^27iyds. Since for each pair of corresponding elements of these two sums we have 27i(y + \Ay) Ac £ 2 7ty ds Hence we have, by definition of an integral = i. >x = b /*x = o S x = 27t yds, •' x = a In like manner, if AB revolves about Oy, we have for the area of the surface generated /*x = b S y = 2 7T I Xds, EXAMPLES. 1. Find the surface of the sphere generated by the revolution of the circle y 2 = a 2 — x 2 about Ox. dy x ds a We have — = , — = — . dx y dx y .-. S x = 27t I y ds — 27t{r 2 — x x )a. Hence the area of the zone included between the two parallel planes is equal to the circumference of a great circle into the altitude of the zone. If x 2 = -f- a, x x — — a, we have the whole surface of the sphere \na 2 . 2. Show that the curved surface of the cone generated by the revolution of y = x tan a about Ox, from x = o to x — //, is nli 2 tan a sec a. Verify the for- mula deduced for the surface of a frustum in § 176. 3. Surface area of the paraboloid of revolution. Let j 2 = 2tnx revolve about Ox. Then m j 3/// i 1 4. Let 2a be the major axis of a 2 y 2 -f b 2 x 2 = a 2 b 2 , and e its eccentricity. Then wd have for the surface of the prolate spheroid _ 27tbe >+« \a 2 . . / , — , sin- 1 A s ' = - L \? - x dx = 2nah V 1 ~ ' + — ) ■ Art. 177.] ON THE VOLUMES AND SURFACES OF REVOLUTES. 261 5. Show that the surface of the pseudo-sphere is S x = 27ta I dy — 2iza* = I rotating about the j-axis generates a solid whose volume is |7r. 5. The volumes generated by y = e* about Ox and Oy are respectively * f° e**dx = *jr, 7T J\^zy? d >' = 27r - 6. The curve JV = sin * rotating about Ox and 0/, respectively, for x = o, at = "#, generates the volumes at Am** 4r = \it\ it Jf{j - 2 ^) cos * <** = 27r2 - 7 The volume generated by one arch of the cycloid x = a(B- sin 6), y = *(i - cos 9), rotating about Or, is 327T and each side of the polygon converges to o, the area of the polygon converges to A, the area of the curve, and the prism and cylinder have the same altitude H. The volume of the cylinder is the limit of the volume of the prism and is therefore AH. 179. Volume of a Solid. — Consider any solid bounded by a surface. Select a point and draw a straight line Ox in a fixed direction. Cut the solid by two planes perpendicular to Ox at points X y , X 2 distant X x and A' 2 from 0. Whenever the area A of the section PM oi the solid by any plane PM perpendicular to Ox, distant x from 0, is a continuous function 264 Fig. 102. Art. 179.] ON THE VOLUMES OF SOLIDS. 265 of a, then the volume of the solid included between the parallel planes at X. and X 2 is V= f* % Adx. To prove this, divide the interval between A r x and X 2 into a large number of parts, n. Draw planes through the points of division perpendicular to Ox, thus dividing the solid into n thin slices, of W Fig. 103. which MPP X M X is a type. Let A be the area of the section PM, and A x that of section P X M X at a distance x x from 0. Let A V be the volume of the element of the solid included between the sections at x and x x , and x x — x = Ax the perpendicular distance between the sections. We can always take Ax so small that we can move a straight line, always parallel to Ox, around the inside of the ring cut out of the surface by the planes at x and x\ in such a manner as to always touch this part of the surface and not cut it, and thus cut out of the element of the solid a cylinder whose volume is less than A V. Let the area of the curve traced by this line on the plane PM be A'. Then the volume of this cylinder is 6 V' = A' Ax. In like manner, we can move a straight line parallel to Ox around the ring externally, always touching and not cutting it. Thus cutting out between the planes of the sections at x and x x a cylinder of which the element of volume of the solid is a part. Let this straight line trace in the plane PM a curve whose area is A" . The volume of this external cylinder is A" Ax. Hence we have A' Ax < AV < A" Ax, or A' < ^ a 2 . Also, for x' 1 -f- y 2 = a 2 the function is o, while for any arbitrarily assigned values of x and/ whatever, such that x 2 ~\-y 2 < a 2 , the func- tion has a unique determinate positive value. Geo- metrically speaking, the function exists for any point on or inside the circumference of the circle x 2 -\-y 2 = a 2 in the plane xOy, and the point representing the function for any such assigned pair of values of x, y, is a point on the surface of the hemisphere x 2 +y 2 + z 2 = a 2 which lies above xOy. The circle x 2 -\- y 2 = a 2 is called the boundary of the region of the variables for which the function z = + \/a 2 - x 2 - y 2 is defined, or exists in real numbers. In general, a function z = f(x, y) of two independent variables is represented by the ordinate to a surface of which z = f(x, y) is the equation in Cartesian orthogonal coordinates. The study of a function of two independent variables corresponds, therefore, to the study of surfaces in geometry, in the same sense that the study of a function of one variable corresponds to the study of plane curves as exhibited in Book I. 183. Function of Dependent Variables. — Let z = f(x,y) be a function of two independent variables x and y. Since x and y are independent of each other, we can assign to them any values we choose in the region for which z is a denned function of .v and_>>. Art. 183.] THE FUNCTION OF TWO VARIABLES. I. In particular, we can hold y fixed an which case z is a function of the single variable x. For example, let y r= b be constant, then Z =/(.V, i) (I) is a function of the single variable x. If z =zf[x,y) be represented by a sur- face, then equation (1), which is nothing more than the two simultaneous equa- tions z=/(x,y), y=b, is represented by a curve AB in a plane x'O'z' , parallel to and at a distance b from the coordinate plane xOz. Or, is the curve of inter- section of the surface z =zf(x,y) and the vertical plane y — b, as exhibited by the simultaneous equations (2). The equation z =f(x, b) of this curve is referred to axes O'x' ', O'z' of x and z respectively, in its plane x'Oz\ (*) II. In like manner, if we make x remain constant, say x = a, and let_>> vary, then z = f(x 9 y) becomes *=A*,y)> (3) a function of y only, and is represented by a curve AB in a plane y'O'z', Fig. 112, parallel to and at a distance a from the coordinate plane yOz. Or, it is the curve of intersection of the surface z =f(x,y) and the plane x = a, whose equations are z=/(x,y), J x = a. \ (4) Fig. 112. III. Again, since x and y are inde- pendent, we can assign any relation we choose between them. For example, instead of making, as in I, II, x and y take the values of coordinates of points on the line x = a or y = b in xOy, we can make them take the values of coordinates of points on the straight line x — a y — b = -■ = r > (5) cos a sin a which is a straight line through the point a, b in xOy and making an angle a with the axis Ox. Substituting x = a -f- r cos a, y = b -{- r sin a 276 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIV. in z =/[x,y) for x and y respectively, and observing that r is the distance of x, y from a, b measured • on the line (5), we have =/(< r cos i\ sin a). (6) Fig. 113. If a is constant, (6) is a function of the single variable r, and is the equa- tion of a curve APB cut out of the surface z ■=/{x i y) by a vertical plane through (5), and the curve has for its equations a y — b (7) cos a sin a The curve (6) is referred in its own plane, rO'z\ to O'r, O'z' as coordinate axis. The coordinates of any point P on the curve being IV. In general, x and _y being independent, we can assume any relation between them we choose. For example, we may require the point x, y in xOy to lie on the curve ). \ = y) = c - Geometrically, this is nothing more than the equation to curve LMN, Fig. 115, cut out of the surface z = /[x, y) by the horizontal plane z = c, at a distance c from xOy. Its equations are *=A*>J>)> \ Z = C. ) The lines cut on a surface by a series of horizontal planes are called the contour lines of the surface. In particular, if z = o, then f{x, y) = o is the equation in the xOy plane of the horizontal trace of the surface 2 = f(x, y), or the curve ABC cut in the horizontal plane by the surface. In the same way that the implicit equation in two variables defines either variable as a function of the other, the implicit function /(jv, y, z) = o is an equation defining either of the three variables as a function of the other two as independent variables, and can be represented by a surface in space having x, y, z as the coordinates of its points. 185. Observations on Functions of Several Variables, — The general method of investigating a function of two independent variables is to make one of the variables constant and then study the (-) Fig. 5- 278 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cll. XXIV. function as a function of one variable. Geometrically, this amounts to studying the surface represented by investigating the curve cut from the surface by a vertical plane parallel to one of the coordinate planes. Or, more generally, to impose a linear relation between the variables x and y, and thus reduce the function to a function of one variable, as in § 183, III, which can be investigated by the methods of Book I. Geometrically, this amounts to cutting the surface by any vertical plane and studying the curve of section. As we have seen in § 184, and as we shall see further presently, the study of functions of two variables is facilitated by reducing them to functions of one variable, and reciprocally we shall find that the study of functions of two or more variables throws much light on the study of functions of one variable. 186. Continuity of a Function of Two Independent Variables. Definition. — The function z = f{x, y) is said to be continuous at any pair of values x, y of the variables when corresponding to x, y we have/~(..r, y) determinate and for x x (=)x, y x (^=.)y J independent of the manner in which x x and y x are made to converge to their respective limits x andy. The definition also asserts that £[f(x 1 >y 1 )—A*>j>)]^.°> tor x x (=)x, y x (=)v. In words: The function z =f(x,y) is continuous at x,y when- ever the number z x = f(x\,y l ) converges to z as a limit, when the variables x x , y x converge simultaneously to the respective limits x,yin an arbitrary manner. Geometrically interpreted, the point P xi representing x\ , y x , z x , must converge to P, representing x, y, z, as a limit, at the same time that the point N, representing x x , j\ , con- verges to M, representing x,y; whatever be the path which N is made to trace in xOy as it converges to its limit M. A function _/"(.#, y) is said to be continuous in a certain region A in the plane xOy when it is continuous at every point x, y in the region A. An important corollary to the definition of continuity oi/(x y y) at x, y is this: Whatever be the value of f(x, y) different from o, we can' always take .v , y so near their respective limits x, y that we shall have/i^Vj , y x ) of the same sign a.sf(x, y). 187. The Functional Neighborhood. — A consequence of the definition of continuity of z =zf(x,y) is as follows: Art. 187.] THE FUNCTION OF TWO VARIABLES. 2 79 If /(.i\ y) is continuous in a certain region containing r c = o gives r — a 2 /(a 2 -f- ^ 2 ) 4/2", and Z>|r = — . Hence the values of x, y, z as before. The first method, in which we substitute for/ in terms of x, is only possible when we can solve the condition to which the variables are subject, with respect to one of them. The second method, in which we express x and y in terms of a third variable, is always possible, although perhaps cumbersome. The class of problems such as the one proposed and solved here should be care- fully considered, for we propose to develop more powerful methods for attacking them. But it should not be forgotten that those methods themselves are developed in the same way as is the solution of this particular problem. The student should accustom himself to seeing curves referred to coordinate systems in other planes than the coordinate planes, for in this way a visual intuition of the meaning of the change of variables, and a concrete conception of the corresponding analytical changes which the functions undergo, is acquired. CHAPTER XXV. PARTIAL DIFFERENTIATION OF A FUNCTION OF TWO VARIABLES. 1 88. On the Differentiation of a Function of Two Variables. — A function of two independent variables has no determinate deriva- tive. It is only when the variables are dependent on each other that we can speak of the derivative of a function of two variables. The derivative of a function of two variables is indeterminate unless the variable is mentioned with respect to which the differentiation is performed and the law of connectivity of the variables given. 189. The Partial Derivatives of a Function of Two Independ- ent Variables. — Among all the derivatives a function of two variables can have, the simplest and most important are the partial derivatives. Let z = f(x,y) be a function of the two independent variables x and y. The simplest relation we can impose between x and y is to make one of them remain constant while the other varies. We then reduce the function 2to a function of one variable, to which we can apply all the methods of Book I for functions of one variable. For example, \ety be constant and x variable. Then z =f(x, y) is a function of x only, and it can be differentiated with respect to x by the ordinary method, and we have D ±Ax x >y) -f(x,y) 1- This is called the partial derivative of the function z or f with respect to x. To obtain the partial derivative of f(x, y) with respect to x, makejy constant and differentiate with respect to .v. Correspondingly, the partial differential of /[x, y) with respect to x is the product of the partial derivative with respect to x, D x f, and the differential of x or x\ — x = Ax. If we represent the partial differential of/* with respect to x by d x f, then we have d x f = f x (x,y)dx, and the corresponding partial differential quotient is It is customary to employ the peculiar symbolism designed by 282 Art. 190.] THE FUNCTION OF TWO VARIABLES. 2»3 Jacobi for representing the partial differential quotient or derivative of /{x, y) with respect to x. Thus the above will hereafter be written (the svmbol 3 is called the round d) df = d f f dx ~ dx ' The symbol d being used instead of d to indicate the partial differential as distinguished from what will presently be defined as the total differential, which will be represented as formerly by d. In the same way, if we make x cons/an/, then f[x, y) becomes a function of one variable y, and has a determinate derivative with respect toy. This derivative we call the partial derivative oi/(x,y) with respect toy, which is written and defined to be dy >i( = )> A x >yi) -A x >y) y x -y 190. Geometrical Illustration of Partial Derivatives. — If z =f{x,y) is represented by the ordinate to a surface, then at any point P(x, y, z) on the surface draw two planes PMQ and PMR parallel respectively to the coor- dinate planes xOz and yOz. These planes cut out of the surface the two curves, PK und PJ respectively, passing through P. z=f(x,y) (y constant) is the equation of the curve PK in the plane PMQ. z=f(x,y) (x constant) is the equation of the curve P/in the plane PMR. Draw the tangents PT 'and PS to the curves PK and P/in their respective planes, and let them make angles and ip with their horizontal axes, as in plane geometry. Then we have — = tan 0, ox dz 3y tan xp. Therefore the partial derivatives of f(x, y) with respect to x and y are represented by the slopes of the tangent lines to the surface z =f(x, y), at the point x,y, z, to the horizontal plane xOy. These tangents being drawn respectively parallel to the vertical coordinate planes xOz, yOz. Also, draw PV parallel to MQ, and PU parallel to MR. Then we have VT = (x l - x) tan 0, US = (y f -y) tan fa 284 PRINCIPLES AND THEORY OF DIFFERENTIATION. [C11. XXV- if Q is x lt y, and A' is x, y' . Or represent the corresponding partial differentials of f with respect to x and^' at /*(.*, y, z). Thus the partial derivatives and differentials of f(x i y) are interpreted directly through the corresponding interpretations as given for a function of one variable. 191. Successive Partial Derivatives. — If z = /[x, y) is a func- tion of two independent variables x and y, then, in general, its partial derivative with respect to x, is also a function of x and y as independent variables. This deriva- tive can also be differentiated partially with respect to either x or y, as was /(or, y). Thus, differentiating again with respect to x, y being constant, we have the second partial derivative of f with respect to x. In symbols d 2 /(x,y) _ dx* -/"(*' J')- In like manner/*^r, y) can be differentiated partially with respect tor, a- being constant. Thus we have for the second partial differen- tial quotient of_/~with respect first to x and then toy dydx S^X'Sh Similarly, differentiating fy(x, y) partially with respect to_>', we have dy* ~ dy -Jyy\ x >y)> and with respect to x we have dY(x,y)_ d/;(x,y) =z dxdy dx Jy*y~>J)- Thus we see that the function z — f{y\ y) has two first partial derivatives, dz dz a? a7' and four second partial derivatives, d*z dh dh &z dx*' df' dydx 9 dxdy Each of these give rise to two partial derivatives of the third Art. 192.] THE FUNCTION OF TWO VARIABLES. 28S order, and generally the function has 2* partial derivatives of the «th order, of the forms d n z b n z dx'dy*' dy>dx*' where p and q are any positive integers satisfying/ -(- q = n. These «th derivatives, however, are not all different, for we shall demon- strate presently that dx p and dy in the denominators are interchange- able when the partial derivatives are continuous functions, and that b n z d n z dxt dyi ~ dydx*' or the order of effecting the partial differentiations is indifferent The number of partial derivatives otf(x,y) of order n is then n -\- 1 EXAMPLES. 1. If z = x 1 -f- axy -\- cos x sin y, dz .'. ~ = 2x -\- ay — sin x sin/, ox dz by- = ax -f- cos x cos y. 2. if f(*,y) = - 2 + i>- 1, dx IX ~a* 3. In Ex. I, d 2 z dy~ dh -— — = a — sin xcosj/=-— — , q> ojc ox qj/ = 2 dx 2 4. In Ex. 2, show that cos x smy, ay dH = — cos x smy. ey dy dx dx dy 192. Theorem. — The partial derivatives are independent of the order in which the operations are effected with respect to x and y. In symbols, if z — f(x,y), we have d 2 z d 2 z dx dy dy dx' Consider the rectangle of the four points M, (x,y); M v (x v y x ); Q, (x x ,y); R, (x,yj. The theorem of mean value applied to a function of one variable x gives =A'{S,y), (1) where B> is some number between x x and x. (See Book I, § 62.) 286 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXV. Form the difference quotient of (i) with respect to_>', a_ 4/ /(. Wl ) -Aw) -A*,*) +A*.y) Ay Ax O-i-J-K^-*) = mS ' y ]Zf S ' y) =fW,v), (-) where rj is some number between j^ andj>. The value (2) is therefore equal to the second partial derivative of/*, taken first with respect to x, then with respect to y, at a pair of values £;, rj ofx,y. Geomet- rically, at a point B,, 77 in the rectangle MQM X R. In like manner, taking the difference-quotient of/" first with respect to y, we have %- *"'}-?■* -r*.«. (3) where rf is some number between y x and y. Now taking the difference-quotient of (3) with respect to x, we have A Af = /(. Wl ) -f{x,y x ) -f(x v y) +f(x,y) Ax Ay " (x 1 -x)(y l -y) — x —x ~~ - /r?/ ' * ^ ' 7 '' ^ where £,' lies between x x and ~r, 7;' between y Y and y. The value of (4) is then equal to the second partial derivative of/", taken first with respect to y and then with respect to x at some point B,' ', rf , also inside the rectangle MQM X R. But (2) and (4) are identically equal. Hence we have ayis, y) __ dy(Z>, V ') This relation is true whatever be the values x v y v If now the functions *S and */ dy dx dx dy are continuous functions of x and_y in the neighborhood of x,y, then since £,' ', rf and £, 7; converge to the respective limits x, y when x l ( = )x,y l ( = ]}>, the two members of (5) converge to a common limit at the same time, and therefore ay = &/ (6) dy dx dx dy' * ' Art. 192.] THE FUNCTION OF TWO VARIABLES. 287 Incidentally, equations (2) and (4) show that the difference- quotients A Af _ AY A A/ _ AY Ay Ax = Ay Ax ' Ax Ay ~ Zix Ay converge to a common limit whatever be the manner in which Ax( = )o, Ay( = )o, and that common limit is by dy or dy dx dx dy ' Observe that in the symbols dydx J ~ Jxy the operations are performed in the order of the proximity of the vari- able to the function. In like manner, making use of the result in (6), we have dy\_ d d d/ _ d d df d dy dy) dx dy dx dy dx dx ~~ dy dx 1 ' dy dy dx 2 dy dy dx 2 ' and similarly for other cases. Hence, in general, tf+Qf & + , then dc dc 8. If z — e x sin y -\- ey sin x, show that \ 2 /S^X 2 7 + W = '" + ^ + 2eX+y Sin {X +y) ' /dz' 9. From z = xvy*. show that 9^ dz (■* + y + lo g *)* 10. Show that if ^ is the angle between the plane xOy and the tangent plane to the surface z = f(x, y) at x. y, then — (©■+©•. Let /*. (x, / , z) be on the surface, and PRS the tangent plane. Draw MN — / perpendicular to RS. Then ^ = iM". 9/ _ .P/I/ 9/ _ PM dJc ~ RM' dy ~ SM' Since RS. NM - RM- MS, and RS 2 = RM 2 + JfS*, .-. (MN)~* = (RM)~* + (MS)**, and therefore tan 2 ;' = MAT* I PM\* \~RM Also, )'+ (m) - ff ) + (I) (I) 2 - (!) '■ 11. In Fig. 121, let A 7 " be any point in the trace of the tangent plane with xOy. Let NM make an angle 9 with (9x, and the tangent line NP to the surface make an angle with the horizontal plane xOy. Then the triangle RSM is the sum of the triangles RMJV, NMS, or RM-SM = SM-NM cos 6 + RM-NM sin 0, PM PM „ , iW . _ •A^- = -^7 cosG + -^7 sm0 - Therefore the slope of a tangent line to the surface at x, y, z, whose vertical plane makes the angle with zOx, is ^_ C os + -/ - 9x ~ 9) tan (0 12. Find the tangent line to a surface at /'which has the steepest slope. From Ex. II we have — tan = — — sin -f i- cos G = o. ,/Q ^ dx dy Art. 192.] THE FUNCTION OF TWO VARIABLES. 89 The values of sin S, cos from this equation put in (1), Ex. II, give for the tangent line of steepest slope -*-© + ©■ Observe that this is the slope of the tangent plane in Ex. 10. 13. If (x, y) = o. Let P, (x,y) and P x , (x„ y x ) be two '2/ points on the curve ■> z on tne surface. The derivative of y with respect to x in (p{x, y) = o is the limit of the difference-quotient y x _ y _ MP X _ MQ cot MP X Q __ tan MPQ x x ~^x ~ PM ~ MQ cot MPQ ~ tan MP X Q ^ being between x and ^ 1? 77 between j and / x (by the theorem of the mean). Therefore, when x x ( = )x, y x (=)y, d(p d y_ _ £ y x - y _ _ ^^ dx T x x — x d(p' ~¥ This usually saves much labor in computing the derivatives of implicit functions in x and y. The important results of Exs. 10, II, 12, and 13 are deduced here geometrically to serve as illustrations of the usefulness of partial differentiation. They will be given rigorous analytical treatment later. 14. Employ the methods of Book I, and also that of Ex. 13, to find D x y in the following curves: x^/a 1 — y 2 /b 2 —1=0, x sin y — y sin x = o, ax % y -j- by 2 x — \xy = o, e x sin y — log y cos x = o. 15. Show that the slope of the tangent at x, y on the conic ax 2 -f- by 2 -f- 2hxy -)- 2ttx -\- 2vy -f- d = o dy ax -f- hy -f- u lS 1x ~ ~~ hx -\- by + v CHAPTER XXVI. TOTAL DIFFERENTIATION. 193. In the partial differentiation of f{x, y) we made x or y remain constant during the operation, and differentiated the function of the one remaining variable by the ordinary methods of Book I. We now come to consider the differentiation ofy^, y) when both x and y vary during the operation of evaluating the derivative. Such derivatives are called total derivatives. In order to make clear the nature of the total derivative of a function z=f{x,y), consider the simple case when there is a linear relation between x \nd i', x — x' y — y' _ 7 = ~lnr " r ' where / = cos 6, m = sin 6, and the differentiation is performed with respect to r. Let x', y'\ I, m ; be constant. Then r varies with x and y, and r*=(x- x'f -f- (y - y')\ Also, x andjy are linear functions of r, and x = x' -J- Zr, y =y f -|- mr. Substituting these values of x and y in f(x, y), we reduce that function to a function of the one variable r, and it becomes f( x > + lr,y' + mr). (1) The derivatives of this function with respect to r can now be formed by the methods of Book I. Thus we get by the ordinary process of differentiation W ?L W etc dr' dr^ dr*' " for the successive derivatives of f with respect to r. These are called the total derivatives of/* with respect to r. Both variables x and r vary with r. We can give a geometrical interpretation to this total derivative as follows: The equation x — x> y —y' r , v — j — — — r ( 2 ) l m v ' 290 Art. 194.] T< )TAL PIFFEREXTIATK )N. 291 is the equation of a straight line through x', y' in the horizontal plane xOv, making an angle 6 with Ox. r being the distance between the points x', y' and x, y on the line. Let 0' be the point j/f.y. Draw O'z' vertical. The vertical plane rO'z' through the Fig. 123. line (2) cuts the surface representing z=f(x,y) in a curve PP X , whose equation in its plane, referred to O'r and O'z' as axes of coordinates r and z, is z=f{x' + lr,y' + mr). (3) Let P 1 be a point on this curve whose coordinates in space are x x , y\ , z x and in rO'z' are r x , z v Let r x — r = Ar. Then, by definition, the derivative of z with respect to r at x, y is the limit of the difference- quotient, when r 1 ( = )r, *, - *_A x i>yi) -A x >y) r i~ - r r , ~ - r Hence we have dz _ ~dr ' ~ = tan GO, where go is the angle which the tangent PM" to the curve PP X at P, and therefore to the surface, makes with O'r, or the horizontal plane xOy. Observe that as a? , y converge to x, y, the point M converges to M along the line M X M. By assigning different values to 6 we can get the slope of any tangent line to the surface, at P, with the horizontal plane. In particular, when the line (2) is parallel to Ox or Oy, or, what is the same thing, when 6 = n or \n, the total derivative becomes a partial derivative, as considered in the preceding chapter. 194. The Total Derivative in Terms of Partial Derivatives. — It is in general tedious to obtain the total derivative, after the manner indicated in § 193, by reducing the function directly to a 292 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI. function of one variable, and generally it is impracticable. We now develop a method of determining the total derivative in terms of the partial derivatives. Let z =f(x, y), where x and y are connected by any relation cf>(x, y) — o. To find the derivative of z with respect to /, where / is any differentiate function of x and y. Let z take the value z lt and / become / 1? when x, y become Let v be constant and x 1 be a variable. Then the law of the mean is applicable to the function f(x y , y) of the one variable x\, and we have A*, y) = (•*, - *)^AZ>y). (1) where £ is some number between x l and x. In like manner, let x be constant and_>> vary, then, by the law of the mean, /(. Wl ) -A^.y) = 0\ -yy^A*i> n)> ( 2 ) where ?; is some number between y x and y. Adding (1) and (2), we have A*> , >\) -/(•*, >') = K - x )d$A5>y) + O'i -y) rrj A*i >v)- (3) Therefore the difference-quotient with respect to / is z j^r t =A{S,y)^+A{^v)% (4) Ax, Ay, Az, At converge to o together, and at the same time x x (=)x, ^(=)^,j^ 1 (=)y, V( = ]>'- Also > ¥(Z>j>) and *A*i>v) dg drj have the respective limits y&y) and sa^ ox oy if these latter functions are continuous in the neighborhood of x, y. Passing to the limit in (4), we have for the total derivative oif(x,y) with respect to /, at x, y, df^dfdx dfdy dt dx dt "*" dy dt' ^ 5) The geometrical interpretation of (1) is this: In Fig. 123 we have M t (x,y); M %i [x lf yfr (J, (-x\,y); R, (x, yj. Also, A*x>y) -A x >y) = Q' K = p Q' tan qtk. Art. 195.] TOTAL DIFFERENTIATION. 293 But, since on the curve PK there must be a point A", (4f, y, z) at which the tangent is parallel to the chord, In like manner for equation (2), AWi) -A*vJ) =£P X = - ^A'tan LKP V But, since there is a point F, (x x , t], z) on the curve KP X at which the tangent is parallel to the chord, we have - tan ZA^ 1 = 1/(^,7,). 195. The Linear Derivative. — An important particular total derivative is the case considered in § 193. Suppose there is a linear relation between x and^', such as x — a y — b I m = r. Then x = a -f- Ir, y = b -J- mr. To find the total derivative of f(x, y) with respect to the variable r, we have dx , dy — = /, ~~ =m. dr dr I = cos 6, m = sin #, being constant. Therefore This is a much simpler way of evaluating this derivative than that proposed in § 193. As before (see Ex. 11, § 192, § 193), tan go = ~ = -£~ cos B 4- -£- sin 8 (2) dr dx ay v ' is the slope to the horizontal plane of a tangent line to the surface, in a vertical plane making an angle 6 with xOz. Again, suppose, as in § 194, that x and y are related by 0(jr, y) = o, and we wish the derivative of f with respect to s, the length of the curve (p(x, y) = o, measured from a fixed point to x, y. Then, putting / = s in (5) § 194, df_^d/dx dfdy ds dx ds "^ dy ds W dx dy But -r = cos 6, — = sin #, where is the angle which the tan- ds ds gent to - M dx=dj, J-dy = d ) f Observe that dx y dy are the partial differentials of/! Hence d/=S x /+B y /; (2) or, the total differential of/ at x, y is equal to the sum of the partial differentials there. The value of the differential at a fixed point depends on the values of dx and dy, which are quite arbitrary. The geometrical interpretation of the differential is as follows: In Fig. 123, let dx = MQ and dy = MR. Draw PR', Q'M' y Q" S parallel to MR. Then 3*/ = Q'Q" = M S and d y f=R'R" = SM". .-. df- M'S + SM" = M'M"; or, the differential of the function is represented by the distance from a point in the tangent plane to the surface at P from a hori- zontal plane through P. 197. The Total Derivatives with respect to x and^. — If, in the total derivative df _ df dx^ ¥^ ~di ~~d^~dt"^~d^~dt 1 we take / = x, then the total derivative of/ with respect to x is dx ~ dx~T~ dy dx' \ ' If we take / = y, then TAL 1 HFFERENTIATION. 95 x and v varying as the coordinates of a point on some curve MH in the horizontal plane. Fig. 124. If, P x is x 19 y x9 z 19 then, in Fig. 124, z x - z = /P 1 = J'P; 9 x x - x = N'M> = J'P'. dz Therefore — = tan a is the total derivative of z with respect to x. That is, the total derivative of/" with respect to x is repre- sented by the slope to Ox of the projection P'T' of the tangent PT to the surface on the vertical plane xOz. The tangent PT being in a vertical plane through P which makes with xOz the angle dy determined by — = tan 6, as determined from cf){x, y) = o. That dx 4y , is, ~ is the slope to Ox of the horizontal projection MN oi the tangent PT. In like manner the total derivative of/" with respect toy is equal to tan /3, this being the slope to Oy of the projection of the same tangent PT on the perpendicular plane^6te. Equations (1) and (2) are immediately determined from the total differential ¥ 4T= 9 £* dy dy by dividing through first by dx and then by dy. In Fig. 124 we have df =JT = J'T' = J"T" 9 and equations (1) and (2) can be verified by the differential quotients taken from the figure directly. 296 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI. 198. Differentiation of the Implicit Function f(x, y) — o. — An important and valuable corollary to the total differentiation of the function z = /"(.v, r) is that which results in giving the derivative of y with respect to x in the implicit function f(x, y) = o. Since z — o in z =/[x, y) gives f(x, y) = o, and in f(x, y) = o are admissible only those values of x andjy which make z constantly zero, the derivative of z with respect to any variable must be o. Therefore, from (5), §194, or (1), § 196, § 197, K dy dx dx ~~ ~ ~df' ¥ This has been geometrically interpreted in Ex. 13, Chap. XXV. In general, the plane z = c, c being any constant, cuts the surface z =y(-v, y) in a contour line, or curve in a horizontal plane, at dis- tance c from the horizontal plane xOy. The equation of this curve in its plane \sf(x, y) = c. In the same way as above, o. ¥ dx dx ~dt d/dy ~*~ dy dt ~ dt ' dc ~ It dy ¥ •'■ dy_ dx _ dt ~~dx ~di = - dx ' ¥ dy which corresponds to the slope of the tangent to the contour (at the point x y y, c) to the vertical plane xOz. EXERCISES. 1. if jc- 3 + y 3 — z ax y = c i ^ n< ^ Dx}'- Here *L = S{x * - ay), ¥= S (y* - ax) dy _ x 1 — ay dx ~ ax — y 1 ' dy y log y* - y 2. Find D x y in x m /a** -f y**/6** = 1 3/ _ tnx™-* df _ wj""- 1 dx ~ a m ' dy ~ h™ 3. If x log y — y log x — o, then ax x log xy — x 4. Let x = f> cos 6. Find the total differential of x. -— = cos Q, — - = — ft sin 9, dp OQ ' ' .*. dx = cos 6 dft — ft sin dQ. Art. 19S.J TOTAL DIFFERENTIATION. 297 5. Find the slope to the horizontal plane of the curve 1 =x+y. J 6. Find the slope to xOy (the steepness) of the curve cut from the hyperbolic paraboloid % — x 2 /d i — y 2 /b 2 by the parabolic cylinder y 2 = 4/>x. We have dz dz dx , dz dy tan co = — = - ds dx ds ' dy ds' s being the length of the parabola y 2 = \px. Here dz 2x dz iy dx ~ a 2 ' dy ~ b 2 ' *-[■+»"] tan go VP + which is the declivity of the curve in space at x, y, z. Find the points at which the tangent to this curve is horizontal. 7. If u = tan-'(//x), du = (x dy - y dx)/(x* -\- y 2 ). 8. If z = xy, dz = yxy-i dy -f- xy log x dy. 9. Find the locus of all the tangent lines to a surface z — f(x, y) at a point (a, b, c), P. Through /'draw a vertical plane, Fig. 123, rMP, whose equation is x — a y — b ■ — 7— = = r. (1) I m ' Then the equation to the tangent line, PM", to the surface at P, in the plane rMP in terms of its slope at a, b, c, is z- c _ df r ~ dr' z and r being the coordinates of any point on the tangent line. But at a, 6, c df _ d/O, b) da df{a, b) db dr ~ da dr db dr Therefore the equation to the tangent line to the surface at a, i>, c, whose hori- zontal projection makes / with Ox (where / = cos 6, m = sin 6), is Z - C = rl ^ + rm db- ^ Eliminating rl and rm between (1) and (2), we have "-' = e-4t£ + Cr-»)fr (3) an equation of the first degree in x, y, z, which is the locus in space of the tangent lines at a, b, c on the surface. This locus is a plane, Exercise 1, Chap. XXIV, touching the surface at a, b, c, and is defined to be the tangent plane to the surface at a, b, c. 298 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVI. 10. Show that the equation to the tangent plane to the surface z = ax 2 + by 2 at any point x', y', s' on the surface is z -\- z' = 2{axx' -\- byy'). 11. Use the equation to the tangent plane to verify Ex. 12, § 192. The direction cosines of the plane are proportional to — , — , — I. Hence if /, m, n are these cosines, / _ m n 1 V ~ ¥ = da db 1 V I+ © ,+ ® Also, sec 2 ^ = i/« 2 , giving the same result as Ex. 12. 12. Show that when 3/ a/ d~x = °' fy = °' (1) the tangent plane to the surface is horizontal at values of x,y satisfying z =f(x,y) and (i). 13. Show that the curve on the surface z =/(x, y) at all points of which the tangent plane to the surface makes the angle 45 ° with xOy is the curve cut on the surface by the cylinder ©"+ (I)'- 14. Apply Ex. 13 to show that the cylinder x 2 -f- y 2 — \a 2 cuts the sphere x 2 -\- y 2 -\- z 2 ■= a 2 in a line at every point of which the tangent plane to the sphere is sloped 45° to the horizontal plane. Draw a figure and verify geometrically. 15. The equation x 2 -j- y 2 = a 2 represents a vertical cylinder of revolution whose axis is Oz and radius is a. Find the equations of the path of a point which starts at x = a, y = o, z = o and ascends the cylinder on a line of constant grade k. This curve is the helix, a spiral on the cylinder, having for its equations z . z x = a cos -j- , y = a sin j— ka' ' ka> CHAPTER XXVII. SUCCESSIVE TOTAL DIFFERENTIATION. 199. Second Total Derivative and Differential of z = /(x t y). It has been shown in § 194, (5), that dt dx dt T dy dt ' {I) where x and y are any differentiate functions of/. If we differentiate again with respect to /, then d 2 / _ d /df dx\ d /df dy ~di 2 ~ df\dx~ ' dt)^"di [dy "dt ~T7t[dx)^~dx dfi '^~dtdt\ey)^~dydt 2 (2) Since — , ~ are functions of x andj' to which (1) is applicable, in the same way we have d /df\_ 9 /df\ dx H(¥\ 4y Jt [dx) ~ dx \dx) ' ~df ~*~ dy \dx) ' ~di ' ~ dx 2 dt + dy dx dt ' W d L (d/\ _ a 2 / dx , a 2 / dy (4) Also, dt \dy J ~ dx dy dt ' dy 2 dt Substituting in (2) and remembering that a 2 / _ d*f dx dy dy dx* we have finally for the second total derivative of f(x,y) with respect to/ /«> J xy> J yy 1 *»*»'• j and the first and second derivatives of y with respect to x by y' , y" . Putting/" = z = o in (7), § 199, we have o =/£ + 2/;;y +/;;y 2 +/;/'. Buty = —f'x/fy. Substituting this and solving fory, >py _ '/a/i/i -m/jf -/;;(/:r In like manner we get, by interchanging x and y, the second derivative D'~x. Otherwise deduced from (8), § 199. 201. Higher Total Derivatives. — We shall not have occasion to use the higher total derivatives of z = f{x, y) above the second. They, however, are deduced in the same way as has been the second, by repeated applications of the formula for forming the first derivative. For the third total derivative of/" with respect to / see Exercise 35 at the end of this chapter. Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 301 The higher total derivatives of/"(*v, y) with respect to an arbitrary function / of x and y become very complicated and are seldom employed in elementary analysis. There is, however, an important particular case in which the higher derivatives of/[x f jr) require to be worked out completely — that is, when x and y are connected by a linear relation. This case we now consider and call it linear dif- ferentiation. 202. Successive Linear Total Derivatives. — To find the «th derivative of/(x,y) with respect tor, when x and y are linearly related by x — a y — b a, b, /, m being constants. The first derivative is, as found before, £--='!+"£ <■> Differentiating again with respect to r, we have Otherwise this follows immediately from (5), § 199, wherein dx dy d z x d 2 y ' = '' IF = 1 ' Ir= m < lr>=dF = °- Differentiating (2) again with respect to r, and rearranging the terms, we have 2-'S +»* 3i + «•*&+-?■ « We observe that (1), (2), (3) are formed according to a definite law. The powers of /, m. and their coefficients follow the law of the binomial formula. , , 9 a ^ J If we consider the symbols — , — as operators on/, and write conventionally dx> dy = \dx) \dy)' then we can write dy / d d y = \ l *Z + m -x- dr 2 \ dx dv dV (.9 dr< \ d.r )/, . (5) 302 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVII, in which the parentheses are to be expanded by the binomial formula d 3 and the indices of the powers of—— and — — taken to mean the num- dx dy ber of times these operations are performed. We can demonstrate that this law is general and that we shall have d V _ // a , - d dx" as follows. First, observe that (7) For Also, d df _ d df d df _ 3 df dr dx dx dr ' dr dy dy dr d_df _d 2 f dx df dy dr~ dx ~~ dx^ dr + dy dx~dr' d df J d / df 3/\ dx~~dr~~dx\ dx~ Jr?n ly) i .ay . dy — * 5—5 + m (8) dx 2 dx dy ' which proves the first equality in (8), and the second is proved in the same way. Now assume (7) to be true. Differentiating again with respect to r, we have d H +f dr H+ f d /d d \« - = d-r[ / dx+ m -W) / > / a dydf The memoria technica (7) being true for n = 3, it is true for 4, and so on generally. Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 303 EXERCISES. 1. Given x 2 -f y* = a\ find Dy. 2. If -x- 3 + xf - ay* = o. / = (3^ + y*)/2y(a - x). 3. If (* + „*)' = a^/ 3 - «=). *- **- ~" 2 ) 4. If z" = dt u(a 2 + t 2 -f u 2 )' X + Z dz 22 2 ^ — s' ' _ d Y / d f _ 2 f 2 a.r 2 / 9y a ' 12. If two curves 0(.r, y) = o, ^(.r, j) = o intersect at a point x, y, and if go be their angle of intersection, prove that tana» = *«#-#»« . 0* #y + 0y 4>x 13. Show that two curves

') ^ _ — 7 2 — 2xy* -f- 3^, show that a s , dz *dx+>'dy = SZ ' , 3»« , a 2 2 ., a 2 s * a a^ + ^a^ + 'V = 2 °*- 21. If *= 22. If . = iZJ! . d -l - Z. a 2 c _ 2 a 2 c aT 2 ~~ a a> T x - y ' 3* (.r - ;>) 2 Art. 202.] SUCCESSIVE TOTAL DIFFERENTIATION. 305 23. If v — nz —fix — mz\ then m -"- -}- n— = 1. 24. If «=;'*, prove 1^ =>*-i(i + log j*) = u 'j x . 25. If z = \/x^fJ 2 , Prove ^|- + y ~\ \ = o. If 8 = j/x 3 -j-;< 3 , prove Lr_ -f/_j 3 = 26. The curve x 3 -\~ y 3 — 3.1 = o has a maximum ordinate at the point I, \/2, and a minimum ordinate at — 1, — 4/2. 27. The curve p(sin 3 -|- cos 3 0) = a sin 2O has a maximum radius vector at the point a \^2, \it. 28. The cur maximum ordinate at the point — 4-, 2. 29. The curve x K -\-y i — $xy % — 2 = has neither maximum nor minimum ordinate. 30. Show that (o, 2) gives y a maximum, and ± \ 4/3, — £ a minimum, while I 4/3, I makes x a maximum, and — | 4/3, | gives x a minimum in the cardioid (x 2 -f y 2 f — 2y(x 2 + j 2 ) — x 2 = o. 31. In x x -j- 2ax 2 y — ay 3 = o, y is a minimum at j*r — ± a. 32. In 3a 2 / 2 -f- xy 3 -\- ^ax 3 — o, y is a maximum for x = 3a /2. 33. Investigate the conic ax' 2 -\- 2/ixy -f- by 2 = I, for maximum and minimum coordinates. 34. If R is the radius of curvature of/(jr, y) = o, and 6 the angle which the tan- gent makes with a fixed line, show from ds = R dB and 6 = tan— 1 dy/dx, that R = (i+Z 2 ) 1 = (^ 2 + ^»)i _y" *f 2 _y dx — dy d 2 x The first when x is the independent variable, the second when the independent variable is not specified and dx, dy are variables. 35. The third total derivative oif(x, y) with respect to any variable t is dy _ (dx 8^ dy d \ 3 , d/ d 3 x df d 3 y dfi ~ \di dx"^"dt dy] J + dx~ ~dF ~*~ ~ty ~dt 3 Fdy d 2 x dx d 2 / ld 2 x dy d 2 y dx\ dydty dy~\ + 3 [fx 2 ~dT 2 It + dx~dy \dt J dt + di 2 ~dt)^~ dy 2 dt 2 ~dt J CHAPTER XXVIII. DIFFERENTIATION OF A FUNCTION OF THREE VARIABLES. 203. We are particularly interested here in the differentiation of a function w =/(x, y, z) of three independent variables, for the reason that when w — o we have f(x, y, z) = o, the implicit function of three variables, which can be represented by a surface in space, and also because the treatment of the function of three variables assists in the discussion of the implicit function of three variables. We do not attempt to represent geometrically a function w of three independent variables. However, corresponding to any triplet x = a, y = b, z = c, there is a point in space which represents the three variables x, y, z for those particular values. When, corresponding to any triplet x, y, z, the function /(.v, r, z) has a determinate value or values it is denned as a function of A\ V, Z. The function /"is a continuous function of x } y, z at x, v, z when for all values of x lt j\, z, in the neighborhood of x, y, z we have the number f(x x ,y xi z t ) in the neighborhood oi/[x, y, z). Differentiation of w =/[x,y, z). — Let x, y, z and x 1 ,y 1 , z x be represented by two points P, P^ in space. Complete the parallelopiped PRQP X with diagonal PP X , by drawing parallels to the axes through P and P v Then in the figure we have the coordi- nates of P, (-Vj, r, z), and of Q, (.\\, r, , z). Let PP X = Ar, and let /, m, oc n be the direction cosines of the angles which PP X makes with the axes Ox, FlG - I2 5- Or, Oz, respectively. Then we have 306 204. x l — X = Mr, y\ — V = ?nJr, z. — z — nAr. Art. 205.] FUNCTIONS OF THREE VARIABLES. 307 Applying the theorem of mean value for one variable, letting sr, r. x in succession alone vary, we have Av?J -Aw) = (*, - «1/K*^i0. /(w) -A* j*) = 0\-J')A(- x \vz), A x iJ' z ) - A*y z ) = (*i - *1/1(£j«)i where £, j', z; x\, ?/, 0; ^, y 1} C, are points such as Z, M, A r , respectively, on the segments PR, RQ, QP V By addition, we have Now let / be any differentiate function of x, y, z, such that /— f x , when x, y, z become x 1} y x , z v Then for the difference- quotient of w with respect to /, -f-zrr =ft(&*)-4i +f^ x ^-Jt +S&*iS&T/' , . . , . . dw dw dw If now the partial derivatives — -, — — , — — are continuous func- dx dy dz tions throughout the neighborhood of x, y, z, we have, on passing to limits in the above equation, the total derivative of f with respect to /, dt dx dt ~^ dy dt^ dz dt' { } The process is obviously general for a function of any number of variables, and if F is a function of n independent variables i\ , . . . , v n , then the derivative of F with respect to /, a function of these variables, is dF _ V^dF dv r ~df ~ / , dzy ~df m 1 Second Total Derivative of w =/(x, y, z). — We can differ- entiate (1) with respect to / and obtain in the same way d v _ l dx d d y d dz d \ V d f d2x 1 d f d2 y \ d l_ d2z ~d¥~ \Yt^ Jr dtdy'^didz)^~^dx~dt^ + tydf>^~dzdT r ^ 205. Successive Linear Differentiation. — Of chief importance are the successive linear total derivatives oif(x, y, z) with respect to r when x — a y — b z — c _ I m n where a, b, c, I, m, n are constants. Then x = a -j- Ir, y = b -f- mr, z = c -f- «r, and dx ■ , dy dz dr dr dr are constants, their higher derivatives are o. 308 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXVIII. Equation (i), § 204, becomes ¥ = l f + •¥• + » d f « dr ox ay dz We can differentiate (1) again with respect to r and get d*f / t d d d \2 or obtain the result directly from the equation (2) in § 204. We can show, as for two variables, that the «th linear total derivative can be expressed by 3M£+ 4 +■£)>. o) where the parenthesis is to be expanded by the multinomial theorem and the exponents of the operative symbols indicate the number of times the operation is to be performed on/". EXERCISES. 2. If xey 4- log z — yz — o, -- = - v ' . dy 1 -yz 3. If u = log(x 3 + y z + 2 3 - 3-ri's), «L 4- «; + «i = 3(* + >' + s) -1 - 4. If 7t/ — log (tan x + tan/ -f tan. 2), w ' x sin 2x -\- w' y sin 2y -f- w\ sin 22 = 2. 5. If w = (x 2 -\-y 2 + s 2 ) - *, show that 3 2 «/ 9' 2 w 3 2 w _ a* 2 " + ~dy z "*" ai 5 6. If w = e*v, dx °'™ d2 = (1 + 3^ + *y*V- 7. If re; = .r^ 4 + ^ 2 2 3 +jcy 2 2 2 , 7^; y2 = 6^'S 2 -4- 8j0. az 8. Show that — = 00 at the point (3, 4, 2) on the surface * 2 + 3 s2 + xy - V' z ~ 3 X - 4= = o. d 2 z 9. Show that — = o at the point (— 2, — 1, o) on the surface 4^ 2 + 2 2 - 5*3 + 4>'- +y - 2« — 15 = O. 10. - \ = — at the point (1 2, — 1) of the surface dxd J' 343 x 2 — y 2 -\- 2z 2 -\- 2xy — \xz -\- x — y -\- z — 5 =0. 11. Show that the second total derivatives of w = f(x,y, z) with respect to x, y, z are respectively z\ dt>f{x,v) . , when x — a, y = 0, dr*h> drf = ('as. + "»)*'• *>■ since / = (jtr — a)/r, m = ( y — b)/r, from (1). Hence 3°9 310 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIX. In like manner let x = B,. y when r = ct; Z, r/ being num- bers respectively between a and x, b and_^. Then Substituting the values of (4) and (5) in (3), we have the Law of the Mean Value extended to functions of two variables, or b - n + nrihy.U x - a ^+ "-<} /(*». (6) 207. The geometrical interpretation of § 206 is as follows : Given the ordinate to a surface at a particular point a, b, and the partial derivatives of the ordi- nate at that point. To find the ordinate at an arbitrary point x,y. Let z =f(x,y) be the equa- tion to a surface on which A, (a, b, c) is the point at which the coordinates and partial de- rivatives of z are known. Let P be the point on the surface at which x, y are given and z or f(x, y) is required. Pass a vertical plane through A and P, cutting the surface in the curve AP and the horizontal plane in the straight line BM, whose equation is Fig. 126. r x — a = r. The equation of the curve AP cut out of the surface by this verti- cal plane is z = f(a -j- Ir, b -f mr) , referred to axes Br, Bz' and coordinates r, z, in its plane rBz' The law of the mean is applied to this function of the variable r, resulting in (3). Then, since these derivatives are linear, they can be ex- pressed in terms of the partial derivatives of z at a, b, and (3) is trans- formed into (6). 208. Expansion of Functions of Two Variables. — Whenever the function (2), § 206, of the one variable r can be expanded in Ari. 2io] EXTENSION OF THE LAW OE THE MEAN. 311 powers ofr by Maclaurin's series as given in Book I, then we can make n = 00 in (6), and we have A *' y) = Zk \ (x ~ a) ^ + °' - 6) is } A "' b) ' and the function /\x t y) can be computed in terms off{a, b) and the partial derivatives at a, b. 209. Functions of Three Variables. — Following exactly the same process as in § 206, for we have the law of the mean for three variables, where £, r/, C are the coordinates of some point on the straight-line segment joining the points in space whose coordinates are x,y, z and a, 5, c. Whenever the function of one variable r, f[a -j- Ir, b -\- mr, c -j- nr), can be expanded in an infinite series of powers of r by Maclaurin's series, Book I, then we can make n = 00 in (i), and have A*,y,')=£fy {(— 4a+^- b ^-4c} ^'^ (2) p = o 210. Implicit Functions. — The law of the mean enables us to express the equation of any curve or surface in terms of positive powers of the variables, and permits the study of the curve or surface as though its equation were a polynomial in the variables. Thus if z =f(x,jy) is constant and o, then f(x,y) = o is the equation of a curve in the plane xOy. The equation of any such curve can, by (6), § 206, be written in the form c fi = o In like manner, by (1), §209, the equation to any surface f(x, y, z) = o can be written n ° =Zh \ ( x -4a+^-4i+(°-4c } Aa > b > c)+R - (2) 312 FRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXIX. jR n being (5), § 206, for equation (1) above, and the correspond- ing value in (1), § 209, for equation (2). 211. The law of the mean as expressed in this chapter is funda- mental in the theory of curves and surfaces. It permits the treatment of implicit equations in symmetrical forms, which is a far-reaching advantage in dealing with general problems whose complexity would otherwise render them almost unintelligible. A most useful form of the equations for two and three variables is obtained by putting x — a = h, y — b = k, z — c = /, and in the result changing a, b, c into x, y, z. Thus for two variables fix + h,y + k) = £^(* £ + *J) A*,**- (1°) For three variables ST 1 1 / A*+h>y+hz+i)^2^ 7]\ h dx+ ^ +/ a^ EXERCISES. 1. Show that the equation of any algebraic curve of degree n can be written as either or n I / d d \ r 0= r?M**> +,1 >*) /{0 ' 0) ' ^ 2. Show that any algebraic surface of »th degree can be written in either of the equations = £ij j(» -.)! + (, -*) 1 + (—')£ \ r A«. *.0. (i) r = o v a )• (2) (x 1- y — } fix, y) is called a concomitant of/(x, y). dx dy] 3. The functioi Find the concomitants of a homogeneous function /(x, y) of degree n. In (10), § 211, put h = gx, k — gy, then .A* + **. y + £>') = 2^- (* s + ;' ^j /(-*-, ;')• Since/ is homogeneous in jc and r of degree », /(* + gx, y + *r) =/ {(1 + *)*. ( J + *)* } = (* + *)Vl*. jO. pr / 9 9 \ r Art. 211.] EXTENSION OF THE LAW OF THE MEAN. 313 This equation is true for all values o£g including o. Therefore, equating like powers of g, we have „ a 2 / 32/ 0-7" (*E +'£)">=•" In the same way, if/(.r, y, z) is homogeneous of degree n, we find, by putting h = gx, k = gp, I = gz in (11), § 211, as above, the concomitants of/(x, ;>', 2), / 9 a a \ r (* dJ: + > a? + * lb) 7 = *" -!)•••(«--+ 0/, for r = I, 2, . . . , ». The concomitant functions are important in the theory of curves and surfaces. They are invariant under any transformation of rectangular axes, the origin remaining the same. CHAPTER XXX. .MAXIMUM AND MINIMUM. FUNCTIONS OF SEVERAL VARIABLES. 212. Maxima and Minima Values of a Function of Two Inde- pendent Variables. Definition. — The function z = f(x, y) will be a maximum at x = a, y = b, when f (a, b) is greater than_/(jt-, y) for all values of x and y in the neighborhood of a, b. In like manner/^, b) will be a minimum value of/"(.r, y) when /{a, b) is less than f(x , j/) for #// values of .v, v in the neighborhood of tf, b. In symbols, we havey^tf, b) a maximum or a minimum value of the function f{x y y) when is negative or positive, respectively, for all values of x, y in the neighborhood of a, b. Geometrically interpreted, the point/*, Fig. 115, on the surface representing z =f{x i y) is a maximum point when it is higher than all other points on the surface in its neighborhood. Also, P is a minimum point on the surface when it is lower than all other points in its neighborhood. This means that all vertical planes through P cut the surface in curves, each of which has a maximum or a minimum ordinate z at P accordingly. Also, when P is a maximum point, then any contour line LMN, Fig. 1 1 5, cut out of the surface by a horizontal plane passing through the neighborhood of P, below P, must be a small closed curve; and the tangent plane at P is horizontal, having only one point in common with the surface in the neighborhood. Similar remarks apply when P is a minimum point. When the converse of these conditions holds, the point P will be a maximum or minimum point accordingly. 213. Conditions for Maxima and Minima Values of /(x, 1). — Let z =/{x, y), x and r being independent. To find the conditions that z shall be a maximum or a minimum at x, y. I. Any pair of values x\ y' in the neighborhood of x, y can be expressed by x' = x -\- lr, y' =y ~\- mr, 3M Art. 2I3-] MAXIMA AND MINIMA VALUES. 315 where / = cos 8, m = sin 8. Then z -=-J\x -\- Ir, y -\- ??ir) is a function of the one variable r, if 8 is constant. If z is a maximum or a minimum, we must have, by Book I, dz d 2 z . . . — = o, -— ^ negative or positive, respectively, for all values of 8. That is, — = cos 6 ~~ 4- sin -^— = o. dr dx dy This must be true for all values of 8. But when 8 = and 8 = J.7T, we have }&A x >y) = ° and jjA*>y) = ° (0 respectively. Equations (1) are necessary conditions in order that #, _y which satisfy them may give z a maximum or a minimum. But they are not sufficient, for we must in addition have d 2 z „d 2 f , d' 2 f 3f — = l 2 /~ + a^^-4- + »'ft. (2) dr* ox 2 ax ay ay 1 v ' different from o and of the same sign for all values of 8. When (2) is negative for all values of 8, then z at x,y is a maximum; and when (2) is positive for all values of 8, then z is a minimum. Pot %=A, ^L=H, % = S. dx 2 ' dxdy dy 2 The quadratic function in /, m (see Ex. 19, § 25), Al 2 + 2Hlm -f Bm 2 , (3) will keep its sign unchanged for all values of the variables /, m, pro- vided AB - H 2 is positive. Then the function (3) has the same sign as A. (a). Therefore the function f(x, y) is a maximum or a minimum at x, y when df _ df _ d 2 fdy /dvy_. (A . , 9 2 / a 2 / . and is a maximum or a minimum according as either -^ or -^, is negative or positive respectively. (b). If AB — Zf 2 = — , then will (2) have opposite signs when m = o and m/l = — A/H\ also when / = o and m/l — — i//i?. The function cannot then be either a maximum or a minimum (see Ex. 19, § 25). 316 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. (c). If AB — ZT 2 = o, and A, B, H are not all o, then the right member of (2) becomes (/AJ-jnBy _ {rnB + lHf A = B and has the same sign as A or B for all values of 6, except when vi J I = — A/H. Then (2) is o. This case requires further examina- tion, involving higher derivatives than the second; as also does the case when A, B, Zfare all o. To sum up the conditions, we have _/*(.*•, y) a maximum or a mini- mum at x, y when /i = o, /; = Or f'yy = T max. min., max. min.. J xx Jxy Jxy Jyy + • If the determinant is negative, there is neither maximum nor minimum; if zero, the case is uncertain.* To find the maximum and minimum values of z = f(x, y), we solve f' x = o, f' y = o, to find the values of x, y at which the maxi- mum or minimum values may occur, then substitute x, y in the conditions to determine the- character of the function there. The value of the function is obtained by either substituting x, y inf(x, y), or by eliminating x, y between the three equations ^ =A x >J)> /'* = °> f'y = ° for the maximum or minimum value z. This method employed for finding the conditions for a maximum or a minimum value of z =f(x, y) has been that which corresponds geometrically to cutting the surface at x, y by vertical planes and determining whether or not all these sections have a maximum or a minimum ordinate at x, y. II. Another way of determining these conditions is directly by the law of mean value. We have A *.S) -A*,y) = (*' - -) a ^ + (/ --r)^- For all values of x', y' in the neighborhood of x, y we have £,, rj also in the neighborhood of x, y. If the values/"',., f' y are different from o, then the values/"^, f'^ are in the neighborhoods of their limits and have the same signs as those numbers for all values of x', y' in the neighborhood of x, y. Therefore the difference on the left of the equation changes sign when x' — .v, as y' passes through;-, iif y 9* o. In like manner this difference changes sign whenj'' =y, as x' passes * For examples of the uncertain cast.- in which the function may be a maximum, a minimum, or neither, see Exercises 22, 25. at the end of this chapter. Art. 213.] MAXIMA AND MINIMA VALUES. 317 through x, \if' x ^ o. Hence it is impossible f or fix, y) to be a maximum or a minimum unless f' x = o and f y = o. When f' x — o, fy — o, we have A v'.y)- A .v ) ,.) = (.v'-.v)^ +2 (.v-.v)C,'-,) fl ^ /+(l .-,,^. If the member on the right of this equation retains its sign unchanged for all values of x' , y' in the neighborhood of x, y, the function will be a maximum or a minimum at x, y. But in this neighborhood the sign of the member on the right is the same as that of its limit, (.v - *)»g + >(-' - *)(? -y)^ + (/ -y)%. This gives the same conditions as in I, and leads to the same results. EXAMPLES. 1. Find the maximum value of z ~ 2> ax }' — x * ~ }' z - This is a surface which cuts the horizontal plane in the folium of Descartes. Here dz dz te = 3<*y-3x 2 , ~= 3 ax- 3 y 2 , (I) d 2 z d 2 z c d 2 z The equations (1) furnish lay - 3 x 2 - o, 3 ax - 3 y 2 = o. (3) for finding the values of x, y at which a maximum or a minimum may occur. Solving (3), we have x = o, y = o, and x = a, y = a. For x = o, y = o, d 2 z b 2 z I d 2 z \ 2_ 2 dx 2 dy~ 2 ~ \dx~dy) ~ ~~ 9a ' and there can be neither maximum nor minimum at o, o. Fur x = a, y = a, d 2 2 d 2 z 1 Mr. dx* dy 2 \ dx dy) ~ + 2 7 a ' d 2 z and since — == — 6a, we have the conditions for a maximum value of z at a, a dx 2 fulfilled. Hence at £, a the function has a maximum value a 3 . 2. Show that a z /2y is a maximum value of (« - x )[ a -y)(x+y -*)• 3. Find the maximum value of x 2 -f- .27 -\-y 2 — ax — by. Ans. \{ab — a 2 — b 2 ). 4. Show that sin x -f- sinj -f- cos (x -*- y) is a minimum when x =y = \n, a maximum when x =y = ±7t. 5. Show that the maximum value of (ax + by + c) 2 /(x 2 + y 2 + 1) is « 2 + b 2 + r 2 . 318 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. 6. Find the greatest rectangular parallelopiped that can be inscribed in the ellipsoid. That is, find the maximum value of Sxyz subject to the condition x 2/ a * +y*/P + z 2 /c 2 = i. (I) Let u = xyz. Substituting the value of z in this from (i), we reduce «toa function of two variables, *"-£-? a kg hbf gf c an 2 du 2 From — = o, — — = o, we find the only values which satisfy the con- dx ay ditions x = a/ 4/3, y = b/ 4/3. These give z = c/ 4/3, and the volume required is 8abc/$ 4/3. 7. Show that the maximum value of x 2 y^z i , when 2x -f- $y -\- \z — a, is (a/qj 9 . 8. Show that the surface of a rectangular parallelopiped of given volume is least when the solid is a cube. 9. Design a steel cylindrical standpipe of uniform thickness to hold a given volume, which shall require the least amount of material in the construction. [Ra- dius of base = depth.] 10. Design a rectangular tank under the same conditions as Ex. 9. [Base square, depth = -J side ol base.] 11. The function z = x 2 -f- xy -\- y 2 — 5^ — 4^+1 has a minimum for x — 2, y= 1. 12. Show that the maximum or minimum value of z - ax 2 -f by 2 + 2hxy + 2gx -f zfy -f c (1) is I a h i h b gf c \\ e have -^ =^ + ^ + ^ = o, -— =A* + *y+/=a (2) Multiply the first by x, the second by y, subtract their sum from (i), and we get «=£*+.# + «• (3) Eliminating x and j between (2). (3). the result follows. The condition shows that when ab — Ii 1 is positive, the above value of z is a maximum or minimum according as the sign of a is negative or positive. If ab — k 1 = — , then z is neither maximum nor minimum. We recognize the surface as a paraboloid, elliptic for ab — Ji 1 positive, and hyperbolic when ab ~ h 2 = -. 1 3. Investigate z = x 2 -4- 2>. v ~ — x )' ~\~ 3- v — 7. 1 ' ~\~ 1 f° r maximum and min- imum values of z. 14. Investigate max. and min. of x* -\- ) A — x 2 -J- .at —J' 2 . x = o, j- = o, max. ; x = y = ± 4, min. ; x = — ^ = ± \ 4/3, min. 15. The function (jr — v)- — ty{x — 8) has neither maximum nor minimum. 16. The surface x 2 -f- 2r 2 — 4.x- -4- 4.'' + 3 C ~h J 5 = ° has a maximum s-ordinate at the point (2, — 1, —3). 17. The function x 4 -\- ) A — 2x 2 -4- \xy — 2r- has neither maximum nor minimum for x — O, y = o; but is minimum at (-(- 4/2, — 4/2 ), (—4/2, -f- 4/2 ). Art. 214.J MAXIMA AND MINIMA VALUES. 319 18. Show that cos x cos a -f- sin x sin a cos (y — /j) is a maximum when X — ex, y = /J. 19. Show that x' 2 — 6xy 2 -\~ cy* at o, o is minimum if c > 9. and is neither maximum nor minimum for other values of c. Hint. Complete the square in x. 20. Show that (I -f- x 2 -\- y 2 )/(i —ax — by) has a maximum and a minimum respectively at £ _ j _ 1 ± 4/1 + a 2 -j- />* a ~ b a 2 + b' 1 " 21. Show that 3, 2 make x*y 2 (6 — x — y) a maximum. 22. Show that a, b make (2ax — x 2 )(2by — y 2 ) a maximum. 23. Show that 3+44/2 is a maximum, — 6 — 4 4/2 a minimum, value of y* - 8j-3 + i8r* - Sy + ** - 3 x 2 - 3 x. 214. Maxima and Minima Values of a Function of Three Independent Variables. Let u = /(x,y, z), x 1 — x = lr = h, y x —y = mr = k, z x — z = ?ir = g. As before, if u is a maximum or a minimum at x,y } z, we must have u = y r (~v -+- /r, y -f- «r, -f- nr )> a maximum or a minimum for all values of /, ?n, ?i, or du du du du , C=fi', F=f» f G=f'J zi H=f£) negative (positive) for all values of h, k, g, then will u be a maximum (minimum). The condition that (2) shall keep its sign unchanged for all values 2,20 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Cn. XXX. of h, k, g has been determined in Ex. 20, § 25, where it is shown that when A H and A H B\ A H G H B F G F C are both positive (2) has the same sign as A for all values of h, k, g. Therefore /{x y y, z) is a maximum or a minimum at x,y, z, determined from f x =°> fy' = °> fz = °> when we have ftl _ ^ max. J xx \ mm. fxxjyx f" f" J xy J yy = + , f f f J xx J yx J zx J xy J yy J zy f" f" f> J xz J yz J zz The conditions for maximum or minimum can be frequently- inferred from the geometrical conditions of a geometrical problem, without having to resort to the complicated tests involving the second derivatives. EXAMPLES. / = X 2 + y-> + Z 2 fx = 2x—y+i=o, f y = ■ *=—h y=—h - xy. O, f z ' = 2Z — 2 = O. = I. give / = - f. Also, /;; = 2, /;; = 2, f z ' z = 2, /j = - 1, /£ = , /;; = o. 2 1 1 2 2 I o 120 002 Therefore — 4/3 is a minimum value of/". 2. Find the maximum and minimum values of ax 2 -f- by' 1 -f- cz 1 -\- 2fyz -f- 2gxz -j- 2/z.rj' -|- 2«-r -f 2WJ/ -(- 2Ws -|- d. Here /*' = 2(<;x -f- >#y -|- 5a -f- u) = o, ' // = 2(//a- + ^+> + z')=o, fz = 2(gx +fy + «r + a/) = O. Multiply the first by .*, the second byj, the third by subtract the result from the function/. ... f=ux + vy + wz + d. Eliminating x, y, z between (1) and (2), we have (1) Add together and (2) / a h g u h b f v g f c w u v w d a h g , h b f gf c which is a maximum or a minimum according as a — T a h I z= -f, h b a h % h b^f if c = =F , the upper and lower signs going together. Art. 215.] MAXIMA AND MINIMA VALUES. 321 3. Find a point such that the sum of the squares of its distances from three given points is a minimum. Let x x% y v »!,... x 3 , y. A , z v be the given points. Then /= ^ [(* -x r f -(- (y -y r f + (4 - *,.)»], /; = 22(X - X r ) =z o = 3 x - 2x r , fy = 2 2(y -y r ) = o = zy - 2y r , f z = 22{z - 2,) = o = 32 - ^ 2r . • •• * = !(*! + *« + * 8 ), ;' = K.i'i + ^ 2 + M ^ = 1^ 4- z 2 + z 3 ). The point is therefore the centroid of the three given points. fxx —fyy = f'z'z = °\ /^ = /** ~ f'y'z = °- Show that the solution is a min- imum. Extend the problem to the case of n given points. 4. If w = ax 2 -j- byx -\- dz 2 -f- Ixy -f- wyz, show that x — y = 2 = o gives neither a maximum nor a minimum. 215. Maximum and Minimum for an Implicit Function of Three Variables. — To find the maximum or minimum values of z in f(x, y, z) = o. Since the total differentials of/" are o, we have d y=( dx l+ d ^+ d 4)^+% d ' x +P^ dh =°- < 2 > Also, at a maximum or a minimum value of z we must have ¥ , ¥ , k dx + iy d y dz = ^ = dz for all values of dy and dx. It is therefore necessary that ¥ ¥ ¥ dx=°> 8y-=°> dz*°' (3) Substituting these values in (2), we have at the values of x,y which satisfy (3), and make dz = o, ft - , dx ^ dy S) %/ dz- ^ , In order that this shall retain its sign for all values of dy and dx, we must have /£/£' - (/^') 2 = +• (4) 322 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. Then the sign of d 2 z is that offj/x*. (See Ex. 19, p. 31.) Hence z will be a maximum (minimum) at x, y, z, determined from fx = °> fy' = °> /= °> when/" 2 y^ is positive (negative), provided (4) is true. EXAMPLES. 1. Find the maximum and minimum of z in 2x 2 -f- 5/ 2 -f- z 1 — $xy — 2 X — 4}> — I = o. fx = 4* - 4;' -2 = 0, /; = 10,' - 4* - 4 = o, give x = f, J = 1, 2f = ± 2. /» = 22 = ± 4, /i; = 4, /^/;; - (/gy> = 24. s is therefore a maximum and a minimum at |, 1. 2. Show that z in z 3 -f- 3-r 2 — 4*7 -(- jj's = o has neither a maximum nor a minimum at x = — -fa, y = — T | F , z = — ^. 216. Conditional Maximum and Minimum. — Consider the determination of the maximum or minimum value of z =. f{x i y) i when x andjy are subject to the condition 0(x,y) = o. Geometrically illustrated, z — /(-v,y) and (p(x,y) = o are the equations of the line of intersection of the surface z =y~and the ver- tical cylinder = o. We seek the highest and lowest points of this curve. Since, at a maximum or minimum value of z, * = j£dx +?L& = o, (1) also ^dx-]-—dy=o, (2) we have, eliminating dy, dx, the equation fx '* = ° (3) to be satisfied by x, y at which a maximum or minimum occurs. Equation (3) together with = o determines x and y for which a maximum or minimum may occur. I fsually the conditions of the problem serve to discriminate be- tween a maximum, minimum, or inflexion at the critical values of x, v. The test of the second derivative, however, can be applied as follows: We have &z = f xx dx* + 2/%; dx dy + /;; df + f x d\x + /; dy, ( 4 ) which must keep its sign unchanged for all values of x, y satisfying = o in the neighborhood of the x,y also satisfying (3). But we also have 0^ dx* + 20.;; dxdy + 0;; df + 0.; d*x + 0; J\v = o. (5) Art. 217.] MAXIMA AND MINIMA VALUES. 323 To eliminate the differentials from (4). (5), multiply (4) by 0, , (5) by /J, and subtract, having regard for (3). In the result substitute for dy/dx from (2). When this is negative (positive) we have a maximum (minimum) value of »2. The form of the test (6) is too complicated to be very use- ful, and it is usually omitted. EXAMPLES. 1. Find the minimum value of x 2 -\- y 2 when j and y are subject to the condi- tion ax -\- by -\- d = o. Condition (3) gives bx = ay. Therefore, at ad bd y - <# + &' J ~ a* + b 2 ' we have X ^ y a 2 + P ' which can be shown to be a minimum by (6). Otherwise we see at once from the geometrical interpretations that this value of .r 2 -f- y 2 must be a minimum. First. \/x 2 -\- y 2 is the distance from the origin, of the point x, y which is on the straight line ax -\- by -f- d = o, and this is least when it is the perpendicular from the origin to the straight line, which was found above. Second, z = x 2 -\- y 2 is the paraboloid of revolution. The vertical plane ax -\- by 4- d = o cuts it in a parabola, whose vertex we have found above, and which is the lowest point on the curve. 2. Determine the axes of the conic ax 2 -\- by 2 -\- ihxy — 1. Here the origin is in the center, and the semi-axes are the greatest and least distances of a point on the ciu've from the origin. We have to find the maximum and minimum values of x 2 -\-y 2 , subject to the above condition of x,y being on the conic. Let u = x 2 4- y 2 and = ax 2 -f- by 2 -f- ihxy — 1 = o. Condition (3) gives x ax -\- hy y ~~ by -\- hx' Multiply both sides by x/y and compound the proportion, and we get (a — u~ l )x 4- hy — o, hx + (b — u-*)y = o. Eliminating x and/, there results for determining the maximum and minimum values of «. 217. The whole question of conditional maximum and minimum is most satisfactorily treated by the method of undetermined multinliprc of Lagrange. The process is best illustrated by taking an example sufficiently general to include all cases that are likely to occur and at the same time to point out the general treatment for any case that can occur. 324 PRINCIPLES AND THEORY OE DIFFERENTIATION. [Ch. XXX- To find the maximum and minimum values of u =f(x,j>, z, w) (rj when the variables x, y, z, w are subject to the conditions x dx + x dx + tp' y dy -f ip> x dz -f i/>£ afc, = o. ) Multiply the second of these by A, the third by //, A and jjl being arbitrary numbers. Add the three equations. (A + A0i + tf>$dx + (/; + \K = o, 1 A + x & + M>i = °i /; + A0; + / ART. 217.J MAXIMA AND MINIMA VALUES. 325 Multiply by 0, b, c and add. Also, transpose and square. Then 2{ax -f by + cz) + (a 2 4- b 2 + c 2 )X = - id 4- (a 2 4- 4 2 4- c l )X - o, 4(* 2 +}' 2 + * 8 ) - (« 2 + * 2 + ^)A- 2 = 4" - (« 2 -f * 2 + < 2 W = o. /- d |/ fl 2 _j_ £2 + f 2 • The problem is to find the perpendicular distance from the origin to a plane. 2. Find when u = .r- -J- y 2 4- s 2 is a maximum or minimum, .r, jr, z being sub- ject to the two conditions x 2/a 2 4- y 2 /6 2 4- 2 2 /^ 2 = 1, & 4- my 4- »« = o. Geometrically interpreted: Find the axes of a central plane section of an ellip- soid. Equations (6) give ix 2x 4- A — 4- }xl =0. 22 2z + \ — -\- un = o. Multiply by jf, _y, z and add. We get A = — «. Therefore ua 2 l 2 ub 2 m 2 uc 2 n 2 lx = — — — , my - 2(« - a 2 ) ' ' 2(« - 3 2 ) ' 2(z< - <: 2 ) Hence the required values of u are the roots of the quadratic a 2 / 2 b 2 m 2 + C- it'- ll — a 2 u — b 2 11 — c 2 3. Find the maximum and minimum values of u — a 2 x 2 4- b 2 y 2 4- c 2 z 2 , x, y, z being subject to the conditions x 2 4- y 2 4- z 2 — I, /.r 4- my 4- «z = O. The required values are the roots of the quadratic /z/(« _ a 2 ) 4- ;« 2 /(w — b 2 ) 4- « 2 /(« — c 2 ) = O. 4. Find the maximum and minimum values of u = x 2 4- y 2 4- z 2 when .*, y, z are subject to the condition ax 2 4- £y 2 4- cz 2 4- 2/V2 4- 2gxz 4- 2/foy s= 1. Geometrically interpreted: Find the axes of a central conicoid. The conditions (6) give x 4- {ax 4- hy 4- £z)A = o, ; + (^ + ^+/zU = 0, ^ + (^4-^4- «W = o- Multiply by .*, _y, 2 and add. . •. A = — #. Eliminating .r, jy, z from the above equations, a — it—*, h , g h , b - u-x, f g , f , c-u-i The three real roots of this cubic, see Ex. 17, § 25, furnish the squares of the semi-axes of the conicoid. 326 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXX. 5. Show how to determine the maxima and minima values of x 2 -j- y 2 -\- z 2 subject to the conditions (*« 4- yl -f S 2)2 _ a 1 x 1 _^ p y 2 _j_ C 2 2 3 f Ix -j- my -\- nz = o. EXERCISES. 1. Show that the area of a quadrilateral of four given sides is greatest when it is inscribable in a circle. 2. Also, show that the area of a quadrilateral with three given sides and the fourth side arbitrary is greatest when the figure is inscribable in a circle. 3. Given the vertical angle of a triangle and its area, find when its base is least. 4. Divide a number a into three parts x, y, z such that x m y n zP may be a maximum. x y z a Ans. —————— p m+n+p 5. Find the maximum value xy subject to the condition x 2 /a 2 -\-y 2 /b 2 = i. This finds the greatest rectangle that can be inscribed in a given ellipse. 6. Find a maximum value of xy subject to ax -J- by = c, and interpret the result geometrically. 7. Divide a into three parts x,y, z, such that xy/2 -\-xz/$ -j-yz/4 shall be a maximum. Ans. x/21 = y/20 = z/6 = a/47. 8. Find the maximum value of xyz subject to the condition x i/a 2 -\-y 2 /b 2 -f z l /c 2 = 1 by the method of § 216. 9. Show that x -j- y -\- z subject to a/x -f b/y -j- c/z = l is a minimum when x/ \/a ' = yj \/b~= zj \/c '= \/~a "-f \Zb~-\- \/7. 10. Find a point such that the sum of the squares of its distances from the corners of a tetrahedron shall be least. 11. If each angle of a triangle is less than 120 , find a point such that the sum of its distances from the vertices shall be least. [The sides must subtend 120 at the point.] 12. Determine a point in the plane of a triangle such that the sum of the squares of its distances from the sides a, b, c is least. A being the area of the triangle. xyz 2A a b c a 2 4- b 2 -f c 2 13. Circular sectors are taken off the corners of a triangle. Show how to leave the greatest area with a given perimeter. [The radii of the sectors are equal ] 14. In a given sphere inscribe a rectangular parallelopiped whose surface is greatest; also whose volume is greatest. [Cube.] 15. Find the shortest distance from the origin to the straight line. Z x x -f m x y +n l z=p l , l 2 X + m% y _|_ n%Z — p 2 . The equations of the planes being in the normal form. Art. 217.] MAXIMA AM) MINIMA VALUES. We have, if u 2 = x 2 + y 2 -\- z 2 , 2x + /jA + /> = O, 2J -f- Wj/l -f- w 2 z/ := O, 2z -(- «,A -f- «.,/* = o. Multiply these by x, y, z in order and add. Multiply by l x , m x , n x in order and add. Multiply by / 2 , m 2 , n 2 in order and add. Whence the equations 2«2 + p x \ -f p^ - O, 2 P\ + ^ + cos 9 /* = °, 2 Pi + cos ^ + V- = o. Since / x 2 -+- w x 2 -f »j 2 = / 2 2 + w 2 2 + « 2 2 = 1, /^ -f m x m 2 -f »j» 2 = cos 0, where is the angle between the normals to the planes. Eliminating X and /i, we have A u 2 sin 2 I cos I COS I »1 2 +A 2 - 2^ 2 COS which result is easily verified geometrically as being the perpendicular from the origin to the straight line. 16. A given volume of metal, v, is to be made into a rectangular box; the sides and bottom are to be of a given thickness a, and there is no top. Find the shape of the vessel so that it may have a maximum capacity. If x, y, z are the external length, breadth, depth, * + V 3« = i*. 17. Find a point such that the sum of the squares of its distances from the faces of a tetrahedron shall be least. If Fis the volume of the solid, x, y, z, w the per- pendicular distances of the point from the faces whose areas are A, B, C, D, then x _ y _ z __™ _ iv A ~ B = C ~ ~ A 2 + B 2 + C 2 + D 2 cz)e~ is given by c \ I a 2 b 2 c 2 \ ^z-\j 2 U 2+ ^ + rV 18. Of all the triangular pyramids having a given triangle for base and a given altitude above that base, find that one which has the least surface. The surface is %(a -f- b -f- c) 4/r 2 -j- k 2 , where a, b, c are the sides of the base, r the radius of the circle inscribed in the base, h the given altitude. 19. Show that the maximum of (ax + by -i- r*\*-«*** - Py 2 ~ a b a 2 x fj 2 y y 20. Show that the highest and lowest points on a curve whose equations are W then two roots of (4) are 0, and the line (3) cuts the curve in two coincident points at x,y, and is by definition a tangent to the curve at x, y. Eliminating /, m between the condition of tangency (5) and the equation to the straight line (3), we have the equation to the tangent atx,y, (*—>£+<'->>£-* w the current coordinates being X, Y. The corresponding equation to the normal at x> y is A' - x r-y dF dF ' V* dx dy Art. 219.] APPLICATION TO PLANE CURVES. 331 EXAMPLES. 1. Use Ex. 3, § 211. to show that if F\x, y) = c is the equation to a curve, in which F(x, y) is homogeneous of degree n, then the length of the perpendicular from the origin on the tangent is _ nc 2. If F[x, y) = u n -\- zt»_ l -f- . . . -f- u x -\- u Q = o is the equation of a curve of ;/th degree, in which u r is the homogeneous part of degree r, show that the equation of the tangent at x, y is dF dF X dx + Y dy~ + U ' 1 - 1 + 2Un ~ 2 + ■ ' ' + mt 9 = °- If X, Y is a fixed point, this is a curve of the {n — i)th degree in x, y which intersects F(x, y) = o in n(n — 1) points, real or imaginary. These points of intersection are the points of contact of the n(n — 1) tangents which can be drawn from any point X, Y to a curve F = o of the «th degree. 3. If A', Y be a fixed point, the equation of the normal through X, Yto F — o at x, y is This is of the «th degree in x, y, which intersects F= o in « 2 points, real or imaginary, the normals at which to F = o all pass through X, Y. There can then, in general, be drawn n 2 normals to a given curve of the nth. degree from any given point. 4. Show that the points on the ellipse x 2 /a 2 -j- y 2 /b 2 =1 at which the normals pass through a given point a, (5 are determined by the intersection of the hyperbola xy(a 2 - d 2 ) = aa 2 y - (3b 2 x with the ellipse. 5. If F(x, y) = o is a conic, show that its equation can always be written » = *.,»>+ |t»— )^0'-*)s}'+*{(*-«>5+Cr-*)oj , «lO (a). Show that the straight line whose equation is x — a y — b — 7- = ~ = r* ( 2 ) where / = cos 6, m = sin 0, cuts the curve in two points whose distances from a, b are the roots of the quadratic o = m.a, t>+r(l± + «|) F+ V (/i. + mff* (3) (b). Show that dF OF is the equation of a secant of which a. b is the middle point of the chord. (c). Show that the equations dF dF — - = O, r— = Q, dx dy solved simultaneously, give the coordinates of the center of the conic. (d). Show that x — a y — b . .dF , dF — = < and / \- m — = o / m dx dy 33 2 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. are the equations of a pair of conjugate diameters of the conic F = o. whose center is a, b. 6. If k 2 < i, show that the tangent to x 2 /a 2 -j- y 2 /6 3 = k 2 cuts off a constant area from x l /a 2 -f- y 2 /b 2 = I. 7. In Ex. 5, show how to determine the axes and their directions in the conic F — o. by finding the maximum and minimum values of r in the quadratic (3). as a function of 6, the center of the conic being a, b, 220. The Inflexional Tangent. — Kx an ordinary point x, y on the curve F(x, y) = o, the straight line X - x Y - v - — = ■ =0 (1) I VI cuts the curve in points whose distances from x, y are the roots of the equation in r, o = If we have l 8x- +m -W = °> ^ the line (1) cuts the curve in two coincident points at x,y, and is tan- gent to the curve there. If, in addition to (3), / and m satisfy ^+ w !) v =°' (4) then the line cuts the curve F = o in three coincident points at x, y, provided ' a a \ 3 ldx- + m Jy) F *°- W In this case the line (1) has a contact of the second order with F = o at x,y, and this point is an ordinary point of inflexion. This means that the value of l/m — tan 6 in (3) must be one of the roots of the quadratic in l/m (4). Eliminating / and m between (3) and (4), we have a condition that x, y may be a point of inflexion, 'F» Fp - 2F» Fi F; + F>> F* = o. (6) To find an ordinary point of inflexion on F — o, solve (6) and F — o for x and y. If the values of x, y thus determined do not make both F' x and F' vanish, and do satisfy (5), the point is an ordinary point of inflexion. The solution of equations (6) and F = o is generally difficult. In .general, if x, y is an ordinary point satisfying F = o, and /dF a ?£j?Y>- \dy dx~. dx dy ' '" °' Art. 221.] APPLICATION TO PLANE CURVES. 333 r = 2, 3, ...,» — i, and /dF d OF a \3)- 3a- cLv dy) * °' then when n is odd we have a point of inflexion at which the tangent cuts the curve in n coincident points at x, y. When n is even x t 1 is called a point of undulation and the curve there does not cross the tangent but is concave or convex at the contact. The conditions for concavity, convexity, or inflexion at an ordinary point on F— o can be determined as in Book I. For, differentiating F = o with respect to x as independent variable, ° ~~ dx + Jydx' JL. ^±\* F < d JL d Jy dx "f dx dy J ~t~ dy dx* At an ordinary point d x F ^ o or d y F ^ o. Hence the curve is convex, concave, or inflects at x, y according as (± + ±±Yjr ( d Z±_ d ZAY F dy _ \dx ^ dx dy J \ dy dx dx dy J dx z ~ dF " /dF\s ~dy \dy) is positive, negative, or zero. EXAMPLES. 1. Show that the origin is a point of inflexion on a 3 y = bxy -\- ex* -\- dx 4 . 2. Show that x — b, y = 2b 3 /a 2 is an inflexion on x 3 — 3&r 2 -f- a 2 y = o. 3. Show that the cubical parabola y 2 = (x — a) 2 (x — b) has points of inflexion determined by 3.* -f- a = 4A Hint. Solve the conditional equation for (x — a)/{y — b). 4. If y 2 = f(x) be the equation to a curve, prove that the abscissae of its points of inflexion satisfy 2f(x)f"{x)= \f\x)\ 2 . II. Singular Points. 221. If at any point x, y on a curve F(x,y) — o dF , dF — - — o and — - = o, dx dy the point x, y is called a singular point. dv dF /dF Since -f- = — -r— / -=— > the direction of a curve at a singular point dx ox I dy is indeterminate. 334 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. 222. Double Point If at a singular point the second partial derivatives of F are not all o, we shall have d 2 F d 2 F d 2 F o = (A' - *)• w + 2 (-V - X )(V-y) wj;i + { r-jf ^ . Divide through by (A" — x) 2 and let X(=)x. Then d 2 F d 2 F ( d\> *F d 2 F / dy\ d 2 F/dy\2 x 2 + 2 dx dy \dx ) + Of \dx ) ' This quadratic furnishes, in general, two directions to the curve at x, y. Such a point is called a double point. The two straight lines d 2 F d 2 F d 2 F pass through the point x, y and have the same directions there as the curve, and are therefore the two tangents to the curve at the double point. The coordinates of a double point on F(x, y) = o must satisfy the equations F=o, F_; = o, F;=o. (i) The slopes of the tangents there are the roots t x and / 2 of the quadratic fiF^+2tF£ + F£=o. (2) (A). Node. If the roots of the quadratic (2) are real and different, then F" F" — F" 2 — — ( A the curve has two distinct tangents at x, y, and the point is called a node. The curve cuts and crosses itself at a node. (B). Conjugate. If the roots of the quadratic in / (2) are imaginary, or F"x F yy — F xy 2 = +, (4) the point is a conjugate, or isolated point of the curve. The direction of the curve there is wholly indeterminate. There are no other points in the neighborhood of a conjugate point that are on the curve. For the equation to the curve can be written • = {(*-* \^ + ^-y)r y Y F + i{(*-*> 8 T + ('-■»£}* For arbitrarily small values of X — x and Y — y the sign of the second member is that of the first term, and (4) is the condition that Art. 222.] APPLICATION TO PLANE CURVES. this term shall keep its sign unchanged. Therefore the equation can- not be satisfied for X, Y in the neighborhood of x, r. (C). Cusp-Conjugate. If the roots of (2) are equal, or 7" F" — F" 2 — XX ■*■ yy ■*■ X y the point maybe either a conjugate point or a cusp. one determinate direction there and a double tangent, assumes that F xy F ' xx , F y ' y ' are not independently o. sideration of the cusp -conjugate class is postponed. (5) The curve has Equation (5) Further con- Illustrations. 1. The following example, taken from Lacroix, serves to illustrate the distinc tion and connection between the different kinds of double points. (a). Let y 2 = (x — a)(x — b){x — c), (1) where a. b, c are positive numbers, and a < b < c. The curve is real, finite, two-valued, and sym- metrical with respect to Ox for a < x < b. It does not exist for x < a or b < x < c\ it is finite and sym- metrical with respect to Ox for all finite values of x > c. The ordinate is 00 when x = co . The curve consists of a closed loop from a to b, and an infinite branch from c on. The curve is shown in Fig. 127. V (b). Let c converge to b. Then the loop and open branch tend to come together, and in the limit unite in /' = (*- a)(x - b)\ (2) giving at b a node. (See Fig. 128.) (c). Let b converge to a in ( I). The closed oval continually diminishes, shrinking to the point a. *e^ Fig. 128. In the limit we have y j/2 = ( X - af(x - c), (3) which consists of a single isolated or conjugate point x = a, and an open branch for x > c. (Fig. 129.) (d). Let c and b both converge to a. The oval shrinks to a, and the open branch elongates to a also, resulting in y" 1 = (x - af % (4) which has a cusp at a. (Fig. 130.) Fig. 129. 2. A clear idea of the meaning of singular points on a curve is obtained when we consider the surface z = I\x, y), which for any constant value of z is a curve cut out of the surface by a horizontal plane. For example, using (1), Ex. 1, we have the surface z = (x - a)(x - b)(x -c)- y\ is symmetrical with respect to the xOz plane, Fig. 130. which and cuts the xOz plane in the cubic parabola z = (x — a)(x — b)(x — c), and the horizontal plane in the curve j2 - (x - a)(x - b)(x - c). 336 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. A moving horizontal plane cuts the surface in curves of the same family. For example, DD is an open branched curve; BB is a curve with a node as in Fig. 128; A A is a curve with a closed oval and one open branch as in Fig. 127; so also is CC. As the horizontal cutting plane rises until it reaches a maximum point Z"on the Fig. ni. surface the closed oval shrinks until it becomes the point of contact of the horizon- tal tangent plane, which plane cuts the surface again in the open branch T. The point of touch T is a conjugate of the curve TT and part of the intersection of the surface by the plane. If the cutting plane be raised higher, to a position J, the oval and conjugate point disappear altogether and the section is only the open branch /. Observe that the tangent plane at the node of BB is also horizontal, but the ordinate to the surface is there neither a maximum nor a minimum. The node of BB is a saddle point on the surface. To illustrate the cusp, consider the surface z — (x — a) 3 — y 2 . This cuts xOz in z = (x — a) z , and the horizontal plane in y 2 = (x — a) 3 . All planes parallel to yOz cut the surface in ordinary parabolse. All sections of the surface by horizontal planes are open branched curves, none having cusps except that one m xOy. All horizontal sections for _/£ z negative have inflexions in the plane x = a, and their tangents there are parallel to Ox. The horizontal sections above xOy have no inflexions. As the plane of the horizontal section below xOy rises, the inflexional tan- gents unite in the tmique double tangent at the cusp in the plane xOy. 3. The above considerations will always enable us to discriminate between a conjugate point and a cusp of the first species,* when the singular point is of the cusp-con jugate class under condition (5). For, let F[x,y) = o have a point of this class, and let F(x, y) = o be the equation of the curve referred to the singular point as origin and the tangent there as x-axis. The point is a cusp of the first species if F\\\ o) changes sign as x passes through o. If F(x, o) does not change sign as x passes through o, the point is either a conjugate or a cusp of the second spi cies. If in the neighborhood of such a point no real values of .r, y satisfy the equation, the conjugate point is identified. Also, the conjugate points on F = o are the values of x, y which make z — F -x maximum or a minimum. * A cusp is of the first species when the branches of the curve lie on opposite sides of the tangent there. If both branches lie on the same side of the tangent, the cusp is of the second species. Art. 222,.] APPLICATION TO PLANE CURVES. 337 The only forms that double points on an algebraic curve can have, besides the conjugate point, are nodes and cusps. (See Fig. 133.) Node. Cusp, first species. Cusp, second species. Fig. 133. In fact, all other singular points of algebraic curves are but combi- nations of these, together with inflexions. EXAMPLES. 1. Show that the origin is a node of y 2 (a 2 -j- x 2 ) = x 2 (a' 2 — x 2 ), and that the tangents bisect the angles between the axes. 2. Show that the origin is a cusp in ay 2 = x 3 . 3. Find the singular point on y 3 = x 2 (x -j- a). [Cusp.] 4. Investigate b(x 2 -\- y 2 ) = x 3 at the origin. 5. Investigate x 3 — Z ax y -)- JK 3 = o at the origin. 6. Find the double point of (bx — cy) 2 ■=. (x — af, and draw the curve there. [x = a, y = ab/c. Cusp.] 7. The curve ( y — c) 2 = (x — a) k {x — b) has a cusp at a, c, if a ^ b\ conju- gate if a < b. 8. Investigate y 2 = x(x -j- a) 2 and x& -\- y%~ = a* for singular points. 9. Investigate at the origin the curve F= ay 2 — 2xy 2 -f lyx 2 — ax 3 -j- by 3 -f x* -f ;/» = o. Here F x = O, F y = o, F'J 2 — F' x ' x F y ' y — o, at the origin, and the third partial derivatives are not all o. The origin is a point of the cusp-conjugate class, and j 2 = o is the double tangent. Since F K x, o) = — ax 3 -f- x 4 - changes sign as x passes through o, the origin is a cusp of the first kind. 223. Triple Point. — If x, y satisfy the equations F=Fi = f; = f>> = f;; = f;; = o, (i) and do not make all the third partial derivatives of F vanish. Then we have at any point X, Y on the curve Divide by {X — x) 3 and make X( = )x. We have the cubic in / for finding the three directions of the curve at x, y, o = F- x + z tF> x 'J y + Z PF% + fij%. (2) The solution of this gives, in general, three values of / = dy/dx, furnishing the three directions in which the curve passes through x, y, 33 8 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. which is a triple point on the curve. The equation of the three tan gents at x, y is 9 3)3 dx ' ' dy Some forms of triple points are shown in Fig. 134. Fig. 134. EXAMPLES. 1. Show that x* = (x 2 — y 2 )y has a triple point at the origin. 2. Investigate at O the curve x 4 — $axy l -f- 2a }' i = °- 224. Higher Singularities. — In general, if d r F dx» dy* ~ for all values of p -f- q = r, and r = o, 1, 2, . . . , » — 1, then the curve ^ = o has an //-pie point at x,y, and in general passes through the point n times. The equation of the n tangents there is ( 9 3 ) n Their slopes are the roots of {r x + 'sy) F =°- Examples of multiple points are shown in Fig. 135. Fig. 135. EXAMPLES. 1. Investigate x*> -\- y b = $eufiy % , at o, o, 2. Investigate (y — x 2 ) 2 = x b . at o, o. 3. In x 5 -f- bx* — a*y 2 = o, the origin is a double cusp. 4. Determine the tangents at the origin to y 2 = x 2 (i - x-). [ x ± y - o.] Art. 225. J APPLICATION TO PLANE CURVES. 339 5. Show that x* — $axy -\- y* = o touches the axes at the origin. 6. Investigate x* — ox-v -\- by* = o at o, o. 7. Show thato, o is a conjugate point on ay 2 — .r ! -f- bx 2 = o if a and b are like signed, and a node when not. 8. Show that the origin is a conjugate point on y\x 2 — a-) = x*, and a cusp on (y — x 2 ) 2 = x 3 . 9. Investigate ( r — x 2 ) 2 = x H at o, o, for ;/ < 4. 10. Investigate {x/a)* -\- {)'//>)* = 1, where it cuts the axes. 11. Find the double points on x* — 4ax :i -f- 4<2 2 x 2 — b 2 y 2 -f- 2/>'[v — a 4 — 5* = o. 12. Also on x* — 2ax' 2 y — ax/ 1 -j- a 2 y 2 = o. 13. Find and classify the singular points on x* — 2ax 2 y — axy' 1 -j- a 2 y 2 = o when a = I, a >' I, «, 2, it is the equation of a C0«£ with vertex at the origin, and which cuts the horizontal plane z = 1 in the curve F = o, which curve is the subject of investigation. Consequently any investigation of F x = o carried on for a homo- geneous function in a - , j/, 2 is applicable to the curve F — o when in the results of that investigation we make z = 1. III. Curve Tracing. 226. In the tracing of algebraic curves, the following remarks are important. (I). If the origin be taken on a curve of the nth degree, at an or- dinary point, the straight line^ = mxca.11 meet the curve in only n — 1 other points. If a curve has a singular point of multiplicity m, and this be taken as origin, the Ymey = mx can meet the curve in only n — m other points. Therefore, if any curve of the »th degree has at the origin a sin- gularity of multiplicity n — 2, the line y = mx can meet it in only two other points besides the origin, and by assigning different values to m we can plot the curve by points conveniently. (II). If any curve has a rectilinear asymptote, and we take the jy-axis parallel to this asymptote, we lower the degree of the equation in y by 1. If there be m parallel asymptotes, and we take the y-axis parallel to them, we lower the degree of the equation in y by m. If the degree of the equation in y can thus be made quadratic or linear in ;', then by assigning different values to x, the curve can be plotted by points conveniently. (III). In any algebraic equation of a curve F= o, when the origin is on the curve, the coefficients of the terms in x,y are the re- spective partial derivatives of the function Fat o, o. Therefore the homogeneous part of the equation of lowest degree equated to o is the equation of the tangents at the origin. The origin is a singular point whose multiplicity is that of the degree of the lowest terms ; it is an ordinary point if this be I. (IV). The Analytical Polygon. — Newton designed the follow- ing method of separating the branches of an algebraic curve at a singular point, and tracing the curve in the neighborhood of that Art. 220.J APPLICATION TO PLANK CURVES. 34i point. The method also determines the manner in which the curve passes off to 00 . Let F(x t y) beany polynomial in x andjr which contains no con- stant term. Then F(x t y) = 2 C^y = o is the equation of a curve passing through the origin. Corresponding to each term Cr-x-y 1 , plot a point with reference to axes Op, Oq, having as abscissa and ordinate the exponents p and q of .v and y respectively. Thus lo- A lf . . . , A 10 , draw Fig. 136. the simple polygon A l A z A h A^A % A l(i A l in such a manner that no point shall lie outside the polygon. Such a polygon is determined by sticking pins in the points and stretching a string around the sys- tem of pins so as to include them all. The properties of the polygon are :* (1). Any part of the equation F = o, corresponding to terms which are on a side of the polygon cutting the positive parts of the axes Op, Oq, and such that no point of the polygon lies between that side and the origin, when equated to o is a curve passing through the origin in the same way as does F — o. Thus, if we strike out of F = o all terms except those correspond- ing to terms on the side A Z A 5 , we have left a simple curve which passes through the origin in the same way as i^= o. In like manner, if we strike out all terms save those corresponding to points on the side A X A % , we have another simple curve passing through the origin in the same way as does F = o, and so on. (2). Any part of the equation F=o corresponding to points which lie on a side of the polygon cutting the positive parts of the axes Op, Oq, and such that no point of the polygon lies on the opposite side of this line from the origin, when equated to o gives a simple curve which passes off to infinity in the same way as does F= o. Thus the part of F = o corresponding to the side A^A % gives such a curve. Again, the part corresponding to A 8 A 10 gives another such curve. (3). Any side of the polygon which cuts the positive part of one axis and the negative part of the other merely gives one of the axes Ox or Oy as the direction of an asymptote to F = o, and these are more simply determined by equating to o the coefficients of the highest powers of x and of jy in F= o. Such a side is A^A^ (4). Any side of the polygon which is coincident with one of the *For a demonstration of these properties see Appendix, Note 12. 34 2 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. axes Oq, Op, as A X A^> merely gives the points of intersection of F = o with Ox or Or accordingly. (5). Any side of the polygon which is parallel to one of the axes Op, Oq gives rectilinear asymptotes parallel to an axis, or the axis as a tangent to the curve according as the side falls under conditions (2) or (1). EXAMPLES. 1. Trace jr 6 -\- 2a 2 x s y — Py 3 = o. Numbering the terms in the order in which they occur, we have A x , A 2 , A 3 , in the polygon corresponding to the terms of the equation. The curve passes through O in the same way as does the curve x 6 -\-2a 2 x 3 y = x 3 {x z +2a 2 y)=:0. corresponding to A X A 2 , or as shown in Fig. 138. Also, the curve passes through O in the same way as does 2a 2 x'y — b % y* = y(2a 2 x 3 — by 2 ) =r o, corresponding to A 2 A Z , as shown in Fig. 139. The curve passes off to 00 in the same way as does the curve 6 — ^3 V 3 — n . or x 2 = by, Fig. 140. Fig. 138. x° — oy A = o corresponding to A X A % The form of the curve is therefore as in Fig. 141, Fig. 140. Trace the following curves: 2. x 4 — 2ax 2 y — axy 1 -\- d l y 4. ay 2 — xy 2 — 2yx 2 -\- ax 4 — x 2 = O 5. x* - o 2 xy _|_ b 2 y 2 — o. 7. x* — laxy 2 -f- 2ay z - O. 9. x* -f a 2 xy — y 4 — o. 11. a 2 (x 2 +y 2 )-2a(x-y) 3 +x 4 +y 4 =0. 13. a(y- x) 2 (y + x ) =y 4 + x 4 . 15. ox(y • 17. Trace ^2 _ ^. — ax 3 y — axy 3 -\- a 2 y axy -f axy 2 + a 2 y 2 : a < I. 8. x h -2a 3 x 2 -^^a 3 xy — 2a 3 y 2 -\-y 10. a\x 2 - y 2 ) + x*+y* = 0. 12. a(y 2 — x 2 ){y - 2x) =y*. 14. x* — axy 2 -\- y 4 = °- 16. x* - a 2 xy + b 2 y 2 = o. — o, near the origin. 18. x* — a 2 xy 2 = af. 20. x* -f ax 2 y = ay 3 . 22. .r :> +;- 5 = $ax 9 y. 24. (x - 2\v 2 = 4_r. 26. {y — x)(y-4x)(y + 2x): 19. ,rxy Sax 2 av°. 21. x(y -xY = b 2 y. 23. (x - 3lr 3 = (y - i)x\ 25. (x - i)( x - 2)y 2 = x 2 . 27. (j, -*)*(;, + *)(, + 2*) 1 6a 4 . Art. 228.] APPLICATION TO PLANE CURVES. 343 IV. Envelopes. 227. Differentiation of functions of several variables affords a method of treating the envelopes of curves, which in general simplifies that problem considerably and gives a new geometrical interpretation of the envelope. For example, we can supply the missing proof, in § 104, that the envelope is tangent to each member of the curve family. When x, y moves along a curve of the family F(x,y,a) = o, (1) a is constant, and we have on differentiation But if x, y moves along the envelope, a is variable, and on dif- ferentiation of (1) dF f dF , dF , — dx -f- — - dy -f —~da— o. (?) dx ' dy ' da vo/ Also, on the envelope - — = o. Therefore -3-, from (2) and (3), are the same at a point x, y common to the curve and its envelope. 228. Again, let a, /3, y be variable parameters in the equation F(x,y, a, ft, y) = o, (1) where a, f3, y are connected by the two relations cp(a, /?, y) = o, (2) dip ^-da da dip "^ d/3 *+? dy = 0. Multiply the second of these by A, the third by //, and add. Deter- mine X and pt so that the coefficients of da and d/3 are zero. Then 344 PRINCIPLES AND THEORY OF DIFFERENTIATION. [Ch. XXXI. if we take dot as the independent variable parameter, the differentials d/3 f dy are arbitrary and we can assign them so that the remainder of the equation shall be zero. Then BF 30 dif; dF 30 30 ^ + A a/i + / ^ = ' (5) dy+ X d r + M d r = ' < 6 > The envelope is the result obtained by eliminating a, f3, y, X, /j. between the six numbered equations. If we have only two parameters and one equation of condition, the particular treatment is obvious; as is also the treatment of the gen- eral case when we have n variable parameters connected by n — i equations of condition. 229. We can get a concrete geometrical intuition of the relation of curves of a family and their envelope, by letting z be a variable parameter and considering F(x,y, z) = o as the equation of a surface in space. Then the curves of the family are the projections on the horizontal plane xOy of horizontal plane sections of the surface, obtained by varying z = a. EXAMPLES. 1. Find the envelope of a line of given length, /, whose ends move on two fixed rectangular axes. We have to find the envelope of x/a -f- )'/b = 1 when a 1 -\- b' 2 = I*. .-. x/a* = Xa, y/b* = Xb. Hence A = a-*, and a = (Z 2 *)*, b = {l*y% and the envelope is x* + y 3 = / 3 . 2. Find the envelope of concentric and coaxial ellipses of constant area. Here x'-Jd 1 + y 2 /b°- = 1 and ab = c. .-. x */a* = Ab, y*/b* = Xa. .-. 2cX = 1. The required envelope is the equilateral hyperbola 2xy — c. 3. Find the envelope of the normals to the ellipse. Here a*x/a - b*y/'/S = a* — b* and a* /a* -f fi*/b* = 1. .-. a*x/a* = Xa/a*, b*y//3* = - X/S/b*. .-. X = a* - b*. I [en< e i;ivr the required envelop a _ I ax \ * /? _ I by \ * a - \a* - b* ) ' /; "" \«, is x m ~ l -j-_y«+« — r'"+i. 5. Show that the envelope of x/l-\-y/m = 1, where the variable parameters /, w are connected by the linear relation l/a -f- /////; = 1, is the parabola ©'+»'■" 6. Show that if a straight line always cuts off a constant area from two fixed intersecting straight lines, it envelops an hyperbola. 7. Show that the envelope of a line which moves in such a manner that the sum of the squares of its distances from n fixed points x r , y r is a constant k 2 , is the locus ^Xr — k 2 , ^X r )' r r ^>*r, X 2x r y r , 'Sy;. — Z- 2 , 2y r , y 2x r , 22y r , n , 1 x , y ,1,0 Let the line be Ix -f my -j- p — o. Then k 2 = l 2 2xl + ™^y*r + np 2 + 2m/ 2y r + 2// 2x r + 2/m 2* r ;>v, = ^Z 2 -|- bm 2 + r/ 2 4- 2/w/ -f 2«-// -f- 2/^/w. Also, P -\- m 2 — I, / and wz being direction cosines of the line. Hence we have al -f- /;/// -f gp -f A/ -f 4j^ = o, /,/ _|_ bm +fp + Am + I//;/ = o, g l 4. //« -\- cp -\- o + i^ =0. Multiply by /, m, p in order and add. . •. A = — k 2 . Eliminating /, m, p, ju between the equations (a _ k 2 )l + hm 4- gp 4- ijux = o, ^/ 4- (^ _ £2) w 4-/^4- 4//J - o, gl 4. / w 4. c p + ^ _ o, */ 4- ^^ 4- / 4" ° = °> we have the result. 8. Show that the envelope of a straight line which moves in such a manner that the sum of its distances from n points x r , y r is equal to a constant k, is a circle whose center is the centroid of the fixed points and whose radius is one «th the distance k. Let Ix 4- my -\- p = o be the line. Then P -\- m 2 — 1, and k = P2x r 4- n£Ey r -\- np, — al -\- bm 4- cp. Here we have a 4- Ax 4- 2/// = o, b 4- Ay 4- 2/z-w = o, f 4- A 4- o =0. . \ A = — c = — n. Multiply these three equations by /, m, p in order and add. Hence k -\- 2/1= o. 346 PRINCIPLES AND THEORY OF DIFFERENTIATION. |Ch. XXXI. The equations a — nx = i/, b — ny = km, squared and added, give the envelope (*-^)'+(-^y=er- 9. Find the envelope of a right line when the sum of the squares of its distances from two fixed points is constant, and also when the product of these distances is constant. 10. A point on a right line moves uniformly along a fixed right line, while the moving line revolves with a uniform angular velocity. Show that the envelope is a cycloid. 11. Show that the envelope of the ellipses x 2 /a 2 -\-y 2 /b 2 =. I, when a 2 -f- b' 1 = k\ is a square whose side is k 4/2. 12. Show that the envelope of line xa™ -\- yb»<- = c™+i, when a n -\-b n = d n , is +y -m 13. Find the envelope of the family of paraboloe which pass through the origin, have their axes parallel to Oy and their vertices on the ellipse x 2 /a 2 -\- y 2 /b 2 = x. [A parabola.] 14. The ends of a straight line of constant length a describe respectively the circles (x ± c) 2 -j- y 2 = a 2 . Show that the envelope of the curve described by the mid-point of the line, c being a variable parameter, is \{x 2 + y 2 — \a 2 )x 2 -f- a 2 y 2 = o. 15. Find the envelope of a family of circles having as diameters the chords of a given circle drawn through a fixed point on its circumference. [A cardioid. ] 16. In Ex. 14 show that the area of each curve of the family is %ita 2 when c > \a. Also, show that the entire area of the envelope is \a\^Tt — Vz\ PART VI. APPLICATION TO SURFACES. CHAPTER XXXII. STUDY OF THE FORM OF A SURFACE AT A POINT. 230. We shall in the present chapter use f{x,y) and F(x, y, z), when abbreviated intrj/"and F, to mean a function of two and three vari- ables respectively. The functions immediately under consideration are z =f(x,y) and F(x,y, z) = o. The first expresses z explicitly as a function of x andjr, and is to be regarded as the solution of the implicit function F— o with respect to z. It is to be observed generally that since results obtained from the investigation of F — o are translated into those for z =fby d>+«F &+ z — m -j- n m 4- n m -\- n is a j>oint on the straight line through the points x l9 y lt z l and x 2 ,y 2 , z 2 , which divides the segment between these points in the ratio of m to n. By varying /;/ and n we can make x' ,y' , z' the coordinates of any point whatever on this straight line. But the point x, y' , z' must be on the surface (i), since, on substitution, these values satisfy (i). Therefore, whatever be the two points whose coordinates satisfy (i), the straight line through these points must lie wholly in the locus representing (i). This is Euclid's definition of a plane surface. The intercept of the plane on the axis Oz is —D/C. Therefore, when C = o, the intercept is oo , or the plane is parallel to Oz. Hence (i) becomes Ax + By + D — o, the equation of a plane parallel to Oz, cutting the plane xOy in the straight line whose equations are z — o, Ax -\- By -\- I) = o. We use orthogonal coordinates unless otherwise specially men- tioned. If /, m, n are the direction cosines of the perpendicular from the origin on the plane and p is the length of that perpendicular, the equation of the plane can be written in the useful form Ix 4- my 4- nz — p = o, (2) where Z 2 4" nj2 + n% — J • II. The Straight Line. Since the intersection of any two planes is a straight line, the equations of a straight line are the simultaneous equations V + ^'+0 + A = o. ) K6) The equations (3) of a straight line can always be transformed into the symmetrical form x-a __y-b _ z- c -r - -5- - ~r ~ *' (4) where a, b, c is a point on the line ; /, m, n, the direction cosines of the line; and A is the distance between the points x, y, z and a, b, c. III. The Cylinder. A cylinder is any surface which is generated by a straight line moving always parallel to a fixed straight line and intersecting a given curve. The moving straight line is called the clement or generator, and the fixed curve the directrix of the cylinder. With reference to space of three dimensions and rectangular coor- dinates, any equation /(•v-.r) = o (5) is the equation of a cylinder generated by a straight line moving Art. 233.] STUDY OF THE FORM OF A SURFACE AT A POINT. 349 parallel to Oz and intersecting the plane xOy in the curve /"(.v, r) = o. Yox f{x,y) = o is nothing more than the equation /(•*, j; *) = o in three variables, in which the coefficients of z are zero, and which is therefore satisfied by any x, y on the curve fix, y) =0 in xOy and any finite value of z whatever. In like manner f(y, z) = o, f[x, z) = o are cylinders parallel to the Ox, Oy axes respectively. IV. The Cone. A cone is a surface generated by a straight line passing through a fixed point, called the vertex, and moving according to any given law, such as intersecting a given curve called the directrix or base of the cone. Any homogeneous equation of the «th degree in x, y, z, such as F(x,y, z) = o, (6) is the equation of a cone having the origin as vertex. Let a, ft, y be any values of x, y, z satisfying (6). Then, since (6) is homogeneous, ka, kft, ky will also satisfy (6), and we shall have F(kx, ky, kz) = k n F(x,y, z) = o whatever be the assigned number k. The coordinates of any point whatever on the straight line through the origin and a, ft, y can be represented hy ka, k(i, ky. Therefore all points of this straight line satisfy (6). When the point a, ft, y describes any curve, the straight line through O and a, ft, y generates a surface whose equation is (6), and this is by definition a cone. If we translate the axes to the new origin — a, — b, — c, by writ- ing x — a, y — b, z — c, for x, y, z in (6), we have F{x — a,y — b, z — c) = o, (7) a homogeneous equation in x — a, y — b, z — c, which is the equa- tion of a cone whose vertex is a, b, c. 232. General Definition of a Surface. — If F(x,y, z) is a continu- ous function of the independent variables x,y, z, and is partially dif- ferentiate with respect to these variables, we shall define the assemblage of points whose coordinates x, y, z satisfy F(x,y, z) = o (1) as a surface, and call (1) the equation of the surface. 233. The General Equation of a Surface. — Let F(x,y,z) =z o be the equation of any surface. Then, by the law of the mean, we can write F(x,y, z) = F{x',y, z') 35© APPLICATION TO SURFACES. [Cir. XXXII- in which the summation can be stopped at any term we choose, provided we write £, ?/, Z instead of x f , y', z' in the last term, where £, 7/, Z is a point on the straight line between x, y, sand x^y', z' . We can therefore always write the equation to any surface in the standard form + Zt\ \ ( ' v - x "> h + o -->'') 1+ (« ~ *') i } '*= - (0 This enables us to study the function as a rational integral function of x, y, z. If the equation of the surface be given in the explicit form z = f(x,y), then in like manner, by the law of the mean, we have for the equation to the surface . =A*',y) +^p { (* - *o £ + (.-, -/) ±, } >, in which the summation stops at any term we choose, provided in the last term we write B, for x' and 77 for y'; £, ?/ being a point on the line joining x,y to x\ y'. 234. Tangent Line to a Surface. — A tangent straight line to a surface at a point A on the surface is defined to be the limiting posi- tion of a secant straight line AB passing through a second point B on the surface, when B converges to A as a limit along a curve on the surface passing through A in a definite way. To find the condition that the straight line x — x' _y —y' _ z — z' _ I ~ m ~ 71 shall be tangent to the surface F(x, y, z) = o. The equation of a surface in implicit form is, § 233, (1), F( x ',y, 0') + (.v - *') g + (y -jOgJ + (• - O a7\+* = °- ( 2 > Substitute IX, mX, nX for x — x' , y —J' r , 2 — z , from (1) in (2). We have the equation in A, o = F { ,;y, 0') + (/ g + m % + n g) A + R, (3) for determining distances from x',y', z' to the points in which (1) intersects the surface (2). \{ F{x' , y' , z') = o, or x',y', z' is on the surface, one root of (3) is o. If in addition 7 dF dF dF Art. 235.] STUDY OF THE FORM OF A SURFACE AT A POINT. 351 two values of A are o, or two points in which the secant (1) cuts the surface (2) coincide in x', y' , z' , and the line will be tangent to the surface at x' , y' , z ', and have the direction determined by /, w, ;/. Observe that in conditions (4) and P -+- m 2 -f- n 2 = 1 we have only two relations to be satisfied by the three numbers /, m, n, and therefore there are an indefinite number of tangent lines to a surface at a point .1', y', z' '. If the equation to the surface be in the explicit form z = f{x,y), or . = *+<*-*)& + ,-y)V + J l, (5 ) then, as before, the straight line (1) meets the surface (5) in x',y', z' when z = z' and other points whose distances from x',y', z' are the roots of the equation in A, ox oy J The condition of tangency is that a second point of intersection shall coincide with x',y' , z' , or , of df 235. Tangent Plane to a Surface. — When the locus of the tan- gent lines at a point on a surface is a plane, that plane is called the tangent plane to the surface at that point. The point is called the point of contact. Tangent plane to F[x,y, z) = o at x',y', z f . The straight line x — x' _ y — y' _ z — z' I m n is tangent to the surface F= o at x',y', z' when F(x',y', z') — o and 7 dF dF BF ,\ / a7 + ra a j 7 + w a? = °- (2) If now at x', y', z' the derivatives dF dF dF dx" dy" b\z' are not all o, we obtain the locus of the tangent lines to F= o at x',y,z f by eliminating /, m, n between (1) and (2). Therefore this locus is (*-*og+u-y)f + (— 0^ = 0. (3) Equation (3) is of the first degree in x,y, z, and therefore is a plane tangent to F = o at x',y', z'. 352 APPLICATION TO SURFACES. [Ch. XXXII. Tangent plane to z -=.f(x,y). Eliminating /, m, n between (i) and l & +m W~ n = °' (4) we have as the tangent plane to z — f^x',y'. 236. Definition of an Ordinary Point on a Surface. — We have just seen that when at any point on a surface F — o the first partial derivatives, dF dF dF dx' gp dP are not all zero, the surface has at that point a unique determinate tangent plane. Such a point is called a point of ordinary position, or simply an ordinary point. On the contrary, if at x,y, z we have d x F = o, d y F = o, d z F = o, the point is called a singular point on the surface. We shall see presently that the surface does not have a unique determinate tangent plane at a singular point. EXAMPLES. 1. Find the conditions that the tangent plane to z =/[x, y) shall be parallel to xOy. Ans. d x f = d y /= o. 2. Find the conditions that the tangent plane to F(x, y, z) = o shall be hori- zontal. Ans. d x F = d y F = o, d z F ^ o. 3. Show that the tangent plane at x', y', z to the sphere x 2 -f- y 1 -|- z 2 = a 1 is XX> + yy' 4- zz> - a 2 . x* , y c-2 4. Find the tangent plane to the central conicoid 1 1 = 1. a b c xx' yy' zz' Ans. V ~-\ = I- a b c 5. Show that the tangent plane to the paraboloid ax 2 -[- by % = iz at x' . y', z' is nxx' -j- byy' = z -f- z'. 6. Show that the tangent plane to the cone Fx,y, z) = o, having the origin as vertex, is xd x 'F -f- yd y >F -\- zd z >F — o. This follows directly from the fact that F is homogeneous, and therefore the tangent plane is dF t dF t dF dF , dF dF x d X ' + y w + Z ^ = x te+ y w + * d~T = n ^ x '>y'> ■o = °- where n is the degree- of the cone. Art. 236.] STUDY OF THE FORM OF A SURFACE AT A POINT. 353 7. Find the equation to the tangent plane at any point of the surface jr* 4- y* -f- z* = a*, and show that the sum of the squares of the intercepts on the axes made by the tangent plane is constant. 8. Prove that the tetrahedron formed by the coordinate planes and any tan- gent plane to the surface xyz = a 3 is of constant volume. 9. Show that the equation of the tangent plane to the conicoid ax' 1 -\- by- -f- cz 2 -f- 2fyz -j- 2gxz -f 2/ixy -\- 2nx -f- zvy + zwz -f d = o, at x J , y', z', is (ax' + hy' -f gz' 4 u)x + (Ax' -f by' + fz' + v)y + (^ +jy + «* + «0* + «y + z// -f wz' + fl? = o. 10. Show that »[ -*>gl+< K -'>| +<*-*>£' F = o is the general equation of any conicoid, and that dF _ dF _ dF _ dx dy dz are the equations of the center of the surface. 11. Show that the plane dF n dF dF ( ,_ a) _ +0 ,_ / Q_ + ( ,_ r) _ =o cuts the conicoid F = o in a conic whose center is a, /?, y. and therefore this is the tangent plane when a, ft, y is on the surface. 12. Show that the locus of the points of contact of all tangent planes to the surface F = o, which pass through a fixed point a, (5. y, is the intersection of F = o with the surface dF dF dF 13. This surface is of degree n — 1 when F — o is of degree n. For, let F— U n -f- ... + U x -\- U . where U r is the homogeneous part of degree r. Then, as in two variables, we have the concomitant dU r , dU r dU r X Hx- + y ~dy- + Z -dz- = rU - Therefore the tangent plane at x, y, z may be written dF dF dF X dx~ JrY -dy- +Z -dz- = nUn + ( " ~ l)Un - 1 + ' * ' + U " OT dF dF dF x a7 + Y T + z a* + ^" r + 2Un ~ 2 + ■•• + (*- ^ + W ^o = o. since £4 -f~ . . . -f- U Q = o. 14. Find the condition that the plane lx -f- 09/ -f »2 = o shall be tangent to the cone F = ax 2 -f ^/ 2 4- r2 2 4- 2/J/3 4- 2gxz 4- 2hxy = o at *', /, z'. The equation to the tangent plane at x', y', z' is, Ex. 6, dF dF dF X dS+Sd/^ Z d7=°' 354 APPLICATION TO SURFACES. [Ch. XXXII. In order that this shall be identical with ix -f- my -f- «* = o, the coefficients of x, J', z must be proportional. 9^ / dF OF I . dF I dF I Hence ax' + hy' -j- #2/ = /A, //jf' -f- £/ -f-/z' = wA, ^ + />'' + c* = nX, lx' -j- m )'' ~\- nz' = o. In order that these equations shall be consistent we have g I — o, * / / < the required condition. 15. Generalize Ex. 14 by finding the conditions that the plane may cut, be tan- gent to, or not cut the cone except in the vertex. Eliminating z, the horizontal projection of the intersection of the plane and cone is two straight lines (an 2 -j- d 2 — 2ghi)x 2 -f- 2(hn 2 -j- dm — gmn —fhi)xy -j- (lui 1 -j- cm 2 — 2fmn)y 2 = o. These will be real and different, coincident or imaginary, according as (an 2 -f- cP — 2ghi)(bri 2 -\- cm 1 — 2fmn) — (Jin 2 -j- elm — g??m — f/n) 2 , which can be written as the determinant A = a h g I h b f m 8 f c n I m 11 O is negative, zero, or positive, respectively. 16. Show that the projections of the two lines in 15 can be written dj dzl dA , W X --0~h X} ' + Ta y -=°' with similar equations for the projections on the other two coordinate planes. 17. Show that A in Ex. 15 can be written da do dc where , 3D d g D 3D l ~zr-~ -J- ¥ lm M = ~ A > a k g h b f g f c 237. Conventional Abbreviations for the Partial Derivatives. — The elementary study of a surface is usually confined to those properties which depend only on the first and second derivatives, that is, on the quadratic part of the equation to the surface when the equation is expressed by the law of the mean. This being the case, it is of great convenience in printing and writing to have compact symbols for the first and second partial deriv- atives. These derivatives being the coefficients of the first and second powers of x,y, z in the equation, it is customary to represent them by Art. 238.] STUDY OF THE FORM OF A SURFACE AT A POINT. 355 the same letters as are conventionally employed as the coefficients of the terms in the general equation of the second degree in three vari- ables. We shall hereafter frequently write : When F(x, v, z) = o, "£ '•%• »■'£■ d*F = d*F d 2 F Fm ™ '■Q*™ ff m * F dy dz dx dz dx dy When z = /{x,y), * = d Z = d Z = d V = d Y , = d V P dx' q dy ' T ~ dx*' * ~ dxdy' df' 238. Inflexional Tangents at an Ordinary Point. — We have seen, §§ 234, 235, that there are an indefinite number of tangent lines to a surface at an ordinary point, lying in the tangent plane and passing through the point of contact. If the second partial derivatives of F = o are not all o, there are two of these tangent lines that are of particular interest. (1). Let z =f(x,jy) be the equation of a surface. The straight line I m n ' cuts the surface z = _/"in points whose distances from the point x, y, z on the surface are the roots of the equation in A -(^H-g-.) + j(/i + ^)/ + >=.i, w we have seen that (1) is tangent to z = / 'at x, y, z. If in addition we have /, m, n satisfying the condition two roots of (2) are o, and the line (1) cuts the surface in three coin- cident points at x, y, z. The conditions pi -\- qm — n = o, rP -j- 2slm -f- lm 2 = o, 35 6 APPLICATION TO SURFACES. [Ch. XXXII. determine two straight lines, in the tangent plane, tangent to the surfaces =/"at the point of contact. Each cuts the surface in three coincident points there. These are called the inflexional tangents at x, y, z. They are real and distinct, coincident, or imaginary, according as the quadratic condition rl 2 -{- 2s/m -f- trri 1 = o, in l/m, has real and different, double, or imaginary roots, or accord- ing as rt — s 2 = _ &f &f dx 2 dy \dxdy) is negative, zero, or positive. Since any straight line, such as (i), cuts any surface of the »th degree in n points, the straight lines in any plane cut the curve of sec- tion of a surface of degree n in n points. Therefore a plane cuts a sur- face of the nth degree in a plane curve of degree n. The tangent plane to a surface of degree n cuts the surface in a curve of degree n passing through the point of contact. But each of the inflexional tangents to the surface cuts this curve in three coincident points at the point of contact. Each is therefore tangent to the curve of section at the point of contact of the tangent plane, which is there- fore a singular point on the curve of section. This point is a node, conjugate point, or cusp according to the value of condition (5). Com- pare singular points, plane curves. Eliminating l,m,n between (1), (3), (4), we have for the equa- tions of the inflexional tangents at x, y, z Z-z=(X-x)p + (V-y)q, {X - xfr + 2 (X - x){F-y)s + {V-yft = o. The second is the equation of two vertical planes cutting the first, the tangent plane, in the inflexional tangents. (2). If the equation of the surface is F— o, then the straight line (j) cuts the surface in points whose distances fvomx,y, z are the roots of the equation in A, ^ ,/dF dF dF\ X 2 / d d a \a^ , „ If x,y, z is on the surface, or F(x,y, z) = o, and LI + Mm +Nn == o, the line (1) is tangent at x,y, z. If in addition /, m, n satisfy the condition { / 3x+ m 6y+ n dz) F =°> Art. 239.] STUDY OF THE FORM OF A SURFACE AT A POINT. 357 the line (1) cuts the surface in three coincident points at x,y, z. The conditions P + m 2 _|_ f? s- lf (6) LI -f- Mm -f Kn = o, (7) ^Z 2 + Bm 2 -f C* 2 + 2*>*» -f 2G/» + 2^/w = o, (8) determine the directions of the two inflexional tangents. Eliminating /, m, n between (1), (7), (8), we have the equations of the inflexional tangents a.tx,y, z, {<*-* )-& + (*-*#+ (*—)£ } /■= o, ( 9 ) {(^- J; )^+(i'-^)| + (Z- e )l}V=o. (xo) The first is the tangent plane, which cuts the second, a cone of the second degree with vertex x,y, z, in the two inflexional tangents. These tangents will be real and different, coincident, or imaginary, according as the plane (9) cuts the cone (10), is tangent to it, or passes through the vertex without cutting it elsewhere. That is, ac- cording as the determinant (see Ex. 15, § 236) A H G L H B F M (11) G F C N L M No is negative, zero, or positive. 239. Should the second partial derivatives also be separately o at x, v, z, and r the order of the first partial derivatives thereafter which do not all vanish at x, y, z, then there will be at x, y, z on the sur- face r inflexional tangents, which are the r straight lines in which the tangent plane at x, y, z cuts the r planes or the cone of the rth degree, {(X- x) l x + ( r- y) d - + ( z- Z ) £}> = <». These r inflexional tangents to the surface are the r tangents to the curve cut out of the surface by the tangent plane at the point of contact, which point is an r-ple singular point on the curve of section. EXAMPLES. 1. Show that the inflexional tangents at any point x' , /, 2' on the hyperboloid x^/a 1 4. y 2 /b 2 — z 2 /c* = 1, lie wholly on the surface and are therefore the two right-line generators passing through the point. Show that their equations are y - y bx'z' ± acy' ay' z' q= bcx 35 8 APPLICATION TO SURFACES. [Ch. XXXII. 2. Show that the inflexional tangents at a point x, y, z on the "hyperbolic parab- oloid x 2 /a 2 — y 2 /b 2 = 2s lie wholly on the surface, and that their equations are X - x _ Y-y _ Z -z a ~ ± b ~ x y a T I the upper signs going together and the lower together. 3. Show that the inflexional tangents to the cone ax 2 -f by 2 -\- cz 2 -j- ifyz -J- 2gxz -f 2/kr_y = O are coincident with the generator through the point of contact. 4. Show that at a point on a surface at which any one of the coordinates is a maximum or a minimum the inflexional tangents are imaginary. 240. The Normal to a Surface at an Ordinary Point. — The straight line perpendicular to the tangent plane at the point of contact is called the normal to the surface at that point. Since the equation to the tangent plane at x, y, z is the coefficients of X, Y, Z are proportional to the direction cosines of the normal, and we have for the equation to the normal at x, y, z or EXAMPLES. 1. Show that the normal at x, y, z to xyz = a z is Xx - x 2 = Yy - y 2 = Zz - z 2 . 2. Find the equations of the normal to the central conicoid ax 2 -f- by 1 + &* = *• X - x _ Y-y _ Z- z ax ~ by ~ cz 3. Show that the normal to the paraboloid ax' 1 -f- by 2 — 22 has for its equations X - x Y - v by Z. 241. Study of the Form of a Surface at an Ordinary Point. — We may study the form of a surface at an ordinary point by examining it (1) with respect to the tangent plane, (2) with respect to the conicoid of curvature, (3) with respect to the plane sections parallel Art. 242.] STUDY OF THE FORM OF A SURFACE AT A POINT, 359 to the tangent plane, (4) with respect to the plane sections through the normal. 242. With respect to the tangent plane: (1). Let z —/(x,v). Then the equation of the surface is Let A", J^, ^ be a point in the tangent plane in the neighborhood of the point of contact x, y, z. Then the difference between the ordinate to the surface and the ordinate to the tangent plane is z-z I= i{(A--.v ) i + (^-, ) ;;;>. This difference is positive for all values of X, Y in the neighbor- hood of x, y when dyay /a 2 / y dy S? aF ~~ \dx~dy) an a? are positive (Ex. 19, §25). Then in the neighborhood of the point of contact the surface lies wholly above the tangent plane, and is said to be convex there. In like manner Z — Z is negative throughout the neighborhood when rt — s 2 is positive and r is negative at the point of contact. Then the surface in the neighborhood of the point of contact lies wholly below the tangent plane and is said to be concave there. (2). Let F(x, y, z) = o. In the same way we have the equa- tion of the surface, (X-x)F' x +(Y-y)F' y +(Z-z)F', = and for that of the tangent plane at x, y, z, (X - x)F' x + {V-j)F;+ {Z 1 - z)F' z = o. On subtraction, (z^F'-l j ( *_^ +( r-^+(ir-*)± } V. Therefore, at x,y, z, by Ex. 20, § 25, when \A H\ and A \H B\ A H G H B F G F C are positive, the surface is convex when A and F' z are unlike signed, concave when A and F' are like signed. 360 APPLICATION TO SURFACES. [Ch. XXXI.. Observe that a surface is concave or convex at a point when tie inflexional tangents there are imaginary, and conversely. Wheu a surface is either concave or convex at a point, its form is said to «e synclastic there. When the inflexional tangents are real and different the surface does not lie wholly on one side of the tangent plane in the neighborhood of the point of contact, but cuts the tangent plane in a curve having a node at the point of contact and tangent to the inflex- ional tangents. At such a point the form of the surface is said to be anticlastic, and the surface lies partly on one side and partly on the other side of the tangent plane in the neighborhood of the contact. The conditions that a surface may be synclastic or anticlastic at a point are, (n), § 238, A H G L H B F M G F C N L M N o -}- synclastic y — anticlastic. The hyperboloid of one sheet and the hyperbolic paraboloid are the simplest examples of anticlastic surfaces, these being anticlastic at every point of the surfaces. The surface generated by the revolution of a circle about an external axis in its plane generates a torus. This surface is anticlastic or synclastic at a point according as the point is nearer or further from the axis of revolution than the center of the circle. 243. With Respect to the Conicoid of Curvature. (1). The explicit equation z = /(x,y), or t = z+(x - x) V +{ r- y) !+I { { x- x) l s+( r- y) £ J 7, shows that in the neighborhood of x,jy, z the surface differs arbitra- rily little from the paraboloid z = z+{ x- x) d J- +( y- y) ^ + 1 { { x- x) ±- +{ r- y) I } '/ This is called the paraboloid of curvature of the surface at x,jy, z. It has the same first and second derivatives at x, y, z as has the surface z —f, and therefore, at that point, has, in common with the surface, all those properties which are dependent on these derivatives. Obviously, the surface is synclastic or anticlastic according as the paraboloid is elliptic or hyperbolic. From analytical geometry, the discriminating quadratic of the paraboloid rx 2 -f- ty 2 -j- 2sxy -f- 2px -\- 2ay — 20 -\- k = o is X 2 - (r + t)X + (rt — s 2 ) = o. Art. 245.] STUDY OF THE FORM OF A SURFACE AT A POINT. 361 This gives the elliptic or hyperbolic form according as rt — s 2 is positive or negative. (2). In the same way, the implicit equation F(x,y, z) — o, or dx + &-*)&+ l r -')l + '*-')sV*=*' shows that in the neighborhood of x,y, z the surface differs arbitrarily little from the conicoid of curvature whose equation is the same as the left member of the equation above when equated to o. The form of the surface at x, y, z is the same as that of the conicoid of curvature there, and they have the same properties there as far as these proper- ties are dependent on the first and second derivatives of F. The discrimination of the conicoid can be made through the discriminating cubic (see Ex. 17, p. 30) A — A, H , G | = o, H , B -A, F G , F , C-X\ and the four determinants A H G L\ H B F Ml G F C N\ as in analytical geometry.* 244. The Indicatrix of a Surface. — At an ordinary point x,y, z on a surface, at which the second derivatives are not all o, a section of the surface by a plane parallel to and arbitrarily near the tangent plane differs arbitrarily little from the section of the conicoid of curvature made by this plane. Such a plane section of the conicoid of curvature is called the indicatrix of the surface at x, y, z. Points on a surface are said to be circular (umbilic), elliptic, para- bolic, or hyperbolic according as the indicatrix is a circle, ellipse, parabola (two parallel lines), or hyperbola (two cutting lines). 245. Equation to Surface when the Tangent Plane and Normal are the 2-plane and 2-axis. — If the equation is z =f(x, y), then since z — o, p = o, q = o at the origin, the equation is 2Z = rx 2 -f- 2sxy -\- ly 2 -f- 2R. The equation of the indicatrix at the origin is z = rx 2 -j- 2sxy -\- ly 2 , * See Frost's, Charles Smith's, or Salmon's Analytical Geometry. 3 62 APPLICATION TO SURFACES. [Ch. XXXII. z being an arbitrarily small constant. This is an ellipse or hyperbola according as rt — s 2 is positive or negative, giving the synclastic or anticlastic form of the surface there accordingly. 246. Singular Points on Surfaces. — If, at a point x, y, z on a surface F ■=. o, we have independently dF OF dF a — = °> a - — °> a~~ ox ay az the point is said to be a singular point. If the second derivatives are not all zero, then all the straight lines whose direction cosines /, ??i, ?i satisfy the relation d 3 \2 (1) 4+ dy dz F = (2) will cut the surface in three coincident points at x,y, z, and are called Eliminating /, m, n by means of the equation to the tangent lines. line and (2), we obtain the locus of the tangent lines at x,y, { { x-x)± + { r S)j^ + ( Z -z) % a ~dz F= o. (3) This is the equation of a cone of the second degree, with vertex x, y } z, which is tangent to the surface /=o at the point x, y, z. The form of the surface at x, y, z is therefore the same as that of this cone. Such a point is called a conical point on the surface. When this cone degenerates into two planes, then all the tangent lines to the surface at x, y, z lie in one or the other of two planes. The point is then called a fiodal point. The condition for a nodal point is that equation (3) shall break up into two linear factors, or A H G H B F G\-. F C (4) A line on the surface i^^oat all points of which (4) is satisfied is called a ?wdal line on the surface. Such a line is geometrically defined by the surface folding over and cutting itself in a nodal line, in the same way that a curve cuts itself in a nodal point. If r is the order of the first partial derivatives which are not all zero, then the surface has a conical point at x,y, z of order r, and a tangent cone there of the rth degree whose equation is (X 6 dx C->)|r + ^— )w} >=0 - (5) 247. A singular tangent plane is a plane which is tangent to a surface all along a line on the surface. For example, a torus laid on a plane is tangent to it all along a circle. The torus has two singular Art. 247-] STUDY OF THE FORM OF A SURFACE AT A POINT. 363 tangent planes. All planes tangent to a cylinder or cone are singular. EXERCISES. 1. The tangent plane to yx 2 = a 2 z at x lt y v z x is zxx x y x -J- y x\ — a 2 Z = 2(/'': r Find the equation to the normal there. 2. The tangent plane to z(x 2 -\- y 2 ) = 2kxy at x lt y x1 z t is 2x(x l z l - ky x ) + 2y(y 1 z 1 -kx 1 ) -f z(x\ -f y\) - 2kx x y x = o. The tangent plane meets the surface in a straight line, and an ellipse whose projection on the xOy plane is the circle (x 2 + y 2 ){x\ - y\) + {x\ + yi){yy x - xx x ) = o. Show that the s-axis is a nodal line. 3. The tangent plane to a 2 y 2 — x 2 (c 2 — z 2 ) at x v y lt z x is XXl (f - z \) - a 2 yy x - zz x x\ + x\z\ = O. At any point on Oz, F x ' = F y ' =± F z ' = o, show that at any such point there are two tangent planes z_ ± \t=£ 4. Show that the tangent plane at x x , y x , z x to x $ _j_ jj/3 _j_ 2 3 — $xyz = a z is x{x\ - y x z x ) -f y{y\ - x x z x ) + z{z\ - x x y x ) = a\ 5. The tangent plane at x x , y x , z x to x m y n zP = a is ;;/ n p — x -\ y -f- — r z = m -\- n -\- p. x x y x z 6. Show that (2a, 2a, 2a) is a conical point on xyz — a(x 2 -\- y 2 -f- z 2 ) -\- 4a 3 = o, and find the tangent cone at the point. Aits, x 2 -\- y 2 -\- z 2 — 2yz — 2zx — 2xy — o 7. Show that the surface \a 2 ^ b 2 + c 2 ) 6 \a 2 ^ b 2 ) c 2 ^ 4 has two conical points. The tangent cone at o, o, o is $x 2 /a 2 -\- 3y 2 /b 2 -f- z 2 /c 2 = o. 8. Determine the nature of the surface ay* + bz 2 + x(x 2 + y 2 + z 2 ) = O at the origin. The origin is a singular point, the tangent cone there is ay 2 -j- bz 2 = o. If a and b are like signed, the origin is a cuspal point around the x-axis. 9. A surface is generated by the revolution of a parabola z 2 = = Z 2 + M 2 + .V 2 . 144. 3 68 APPLICATION TO SURFACES. [Ch. XXXIII. Also, Q being a point on the surface, (X- x)L + (V-y)M+ (Z- z)N ■ i=I ^L+v-A^+iz dx dz\ F+ 2T : AP + BnP -f Cn % -f 2 Fmn + 2 Gln + zHlm, since £T = o for Q( = )J>, and X - x _ Y-y _ Z-z _ (i) is the equation of the tangent PR. The derivatives Z, A, etc. , of course being taken at P. 252. If the equation of the surface be f(x,y) — z = F — o, then since L ~p, M = q, N= — 1, C = F — G = o, (1), § 251, becomes 1 rP 4- 2slm 4- ^ 2 (1) K 1/i +/ 2 + ^ 253. To Find the Principal Radii at Any Point on a Surface. — We have only to find the maximum and minimum values of R in (!)> § 2 5 J > § 2 52. I. In (1), § 251, let /, m, n vary subject to the two conditions IL + mM+nN =0, P + w 2 -f ?z 2 = 1. Then, by the method of § 217, Al + Zfa + £« + AZ + fjil = o, #7 -f- Bm + Fn + AJ/-f jura = o, GI + Zw + Cn +\N -f yuw = o. Multiply by /, m, n, respectively and add. . *. jx = — k/R. ... (A - k/R)1 + #« + G« + AZ = o, HI + (B — K/R)m -f Z« -f \M— o, G7 + Zrc +(C-/c/Z)«+ AiV^= o, Z/ -f j&/>w + Nn = o. Eliminating /, m, n, A, we get the quadratic A -k/R, H , G , Z , B - k/R, F H G L , M , M C - k/R, N N ' , o Art. 254.] CURVATURE OF SURFACES. 369 the roots of which are the principal radii of curvature at the point at which the derivatives are taken. II. If z = f(x,y) be the equation to the surface, then in (1), §252, we have /, m, n subject to the two conditions pi -\- qm — 11 — o, and 1 2 -\- m 2 -f- ?i 2 = 1, which reduce to the single condition (1 +/»)/» + ipqlm -f (1 + f)m 2 = 1. Applying the general method for finding the maximum and minimum values to (1), § 252, rl + sm + A[(i + f)l + 'pqm\ = o, si + Im + \\_pql + (1 + q-)m\ r= o. Multiply respectively by / and m and add. Whence A = — k/R. Eliminating / and m from \rR - (1 + p 2 )*]l + (sR - pq K )m = o, (sR - p qK )l -f \iR — (1 -f- f)K\m = o, there results the quadratic [rR - (1 +J)k] \iR - (1 + ?>] - 0* - ^/c) 2 = o, or (rt - s 2 )R 2 - [r(i + ? 2 ) + /(i + f) - 2pqs-] K R + /c 4 = o, for finding the radii of principal curvature. In this equation *•=!+/» + ,». The problem of finding the directions of the principal sections and the magnitude of the principal radii of curvature is the same as that of finding the direction and magnitude of the principal axes of a section of the conicoid Ax 2 + By 2 + Cz 2 -f 2Fyz -f 2 G x z + *Hxy = 1, made by the plane Lx -j- i?/y -f- A% = o. 254. To Determine the Umbilics on a Surface. — At an umbilic the radius of normal curvature is the same for all normal sections. Consequently equation (1), § 251, for any three particular tangent lines will furnish the conditions which must exist at an umbilic. Through any umbilic pass three planes parallel to the coordinate planes cutting the tangent plane there in three tangent lines whose direction cosines are l x , m lf o; / 2 , o, « 2 ; o, m z , ;/ 3 , respectively. Then equating the corresponding values of k/R in (1), § 251, AQ+Bm* + iBfa = Al? + Cn? + aGfo = Bmf+Cnf+zFm^. Also, since these three tangent lines are parallel to the tangent plane, the equations Z/j + Mm x — ZI 2 -f Nn 2 = Mm 3 + Nn z = o give 1 ~~ Z 2 4- M» x L 2 + M 2 ' 37° APPLICATION TO SURFACES. [Ch. XXXIII. and I lf m l have opposite signs. The same equations give like values for / 2 , n 2 , etc. On substitution we obtain the conditions which must exist at an umbilic, AM 2 + BL 2 - 2HLM _ AN 2 + CL 2 - iGLN L 2 4- M 2 I 2 + N 2 BN 2 -f CM* - 2FMN W M » + A' 2 These two equations in x,jy, z, together with the equation to the surface, give the points at which umbilics occur. If the equation of the surface isf(x,y) — z = o, results are cor- respondingly simplified and the conditions which must exist at an umbilic are immediately obtained from the fact that k/R is constant for all values of /, m, n, satisfying the identical equations — = rl 2 -j- 2slm -f- tm 2 , R I = (i +p 2 )P + 2pqlm + (i + q*)m\ Whence results, from proportionality of the constants, K r s t R i+p 2 pq i-\-q 2* (0 255. Equations (2), § 254, are very simply obtained by seeking the point on the surface z = f(x,j>) at which the sphere % we get x — a-f (z — y)p— o, y - P + (* - r) ? = °> 1 +? + <* — y)r = °> 1 + ? 2 + (* — r) / = °> pq + - r) s = °- , v /? I + P 2 I + ^ Also, ^ = — (2 — x)|/i + /~ + q 2 , since the direction secant of the normal with the £-axis is — (1 -\- p 2 -f- q 2 )*' 256. Measure of Curvature of a Surface. — The measure of cur- vature of a surface is an extension of the measure of curvature of a curve in a plane, as follows: Art. 256. J CURVATURE OF SURFACES. 371 The measure of entire curvature of a curve in a plane is the amount of bending. Let P x and P % be two points on a curve whose distances, measured along: the curve, from a fixed point are s x and s 2 . Let cp x and c\ be the angles which the tangents at P lt P 2 make with a fixed line in the plane of the curve. Then the whole change of direction of the curve between P and P 2 is the angle 2 — X . This angle is also the angle through which the normal has turned as a point P passes from P x to P 2 along the curve. This angle between the normals is called the entire curvature of the curve for the portion P x P r It can also be measured on a standard circle of radius r, as the angle between two radii parallel to the nor- mals to the curve at P lt P r If P X P 2 be the subtended arc in the standard circle (Fig. 145), the whole curvature of PJP 2 is proportional to P X P;, or A- $1 = ^-^. If the standard circle be taken with unit radius, the entire curvature of P X P 2 is measured by the arc s % ' — s x on the unit circle. The mean curvature, or average curvature, of P X P 2 is the entire cur- vature divided by the length of the curve P x P 2i 02 - 0i _. V ~ S l *t — *i " '. -V or, is the quotient of the corresponding arc on the unit circle divided by the length of curve P X P 2 . The specific curvature of a curve, or the measure of curvature of a curve at a point P, is the limit of the mean curvature, as the length of the arc converges to zero. It is therefore the derivative of (ft with respect to s. But since ds = Rdcp, where R is the radius of cur- vature of the curve at a point, we have for the specific curvature d

= &• The unit solid angle, called the steradian, is that solid angle which 37 2 APPLICATION TO SURFACES. [Ch. XXXIII. cuts out an area A equal to the square on the radius. In particular, if we take as a standard sphere one of unit radius, then gj = A, or, the area subtended is the measure of the solid angle. Definition. — The entire curvature of any given portion of a curved surface is measured by the area enclosed on a sphere surface, of unit radius, by a cone whose vertex is the center of the sphere and whose generating lines are parallel to the normals to the surface at every point of the boundary of the given portion of the surface. Horograph. — The curve traced on the surface of a sphere of unit radius by a line through the center moving so as to be always parallel to a normal to a surface at the boundary of a given portion of the sur- face is called the horograph of the given portion of the surface. Mean or average curvature of any surface. The mean or average curvature of any portion of a surface is the entire curvature (area of the horograph), divided by the area of the given portion of the surface. If £ be the area of the given portion and od the entire curvature, the mean curvature is ^ Specific Curvature of a surface, or curvature of a surface at a point. The specific curvature of a surface at any point, or, as we briefly say, the curvature of a surface at a point on the surface, is the limit of the average curvature of a portion of the surface containing the point, as the area of that portion converges to o. In symbols, the curvature at a point is doo dS' Gauss* Theorem. The curvature of a surface at any point is equal to the reciprocal of the product of the principal radii of curvature of the surface at the point, or *A 146. Let 6" be any portion of a sur- face containing a point P. Draw the principal normal sec- tions PM — As 2 , PN = As v .-.js=/is 1 -js. z =p f 1 /i(p 1 -p:^(p 2 Ago = A Acr 2 = A(p x >A(p 2 . A 2 . Am. f s , - ^. 2. A surface is formed by the revolution of a parabola about its directrix; show that the principal radii of curvature at any point are in. the constant ratio 1 : 2. 3. Find the principal radii of curvature, at x, y, z, of the surface z . z A x 2 4- y 2 4- a % y cos x sin — = o. Ans. ± — — — . a a a 4. Show that at all points on the curve in which the planes z = ± — cut 2ab the hyperbolic paraboloid 22 = ax 2 — by 2 the radii of principal curvature of the latter surface are equal and opposite. This curve is also the locus of points at which the right-line generators are at right angles. 5. Show from (6), § 248, that the mean curvature of all the normal sections of a surface at a point is 2 U,+ Rj' 6. Show that at every point on the revolute generated by a catenary revolving about its axis, the principal radii of curvature are equal and opposite. 7. Show that at every point on a sphere the specific curvature is constant and positive. 8. Show that at every point of a plane the specific curvature is constant and o. 9. Show that at every point on the revolute generated by the tractrix revolving about its asymptote, the specific curvature is constant and negative. This surface is called the pseudo-sphere. 10. If the plane curve given by the equations x/a = cos S -f log tan -£■ Q, y/a = sin 0, revolves about Ox, the surface generated has its specific curvature constant. 11. If R v R 2 are the principal radii of curvature at any point of the ellipsoid on the line of intersection with a given concentric sphere, prove that *T+V const . 12. Prove that the specific curvature at any point of the elliptic or hyperbolic paraboloid y 2 /b -\- z 2 /c = x varies as {p/z^, p being the perpendicular from the origin on the tangent plane. 13. In the helicoid/ .= x tan (z/a) show that the principal radii of curvature, at every point at the intersection of the helicoid with a coaxial cylinder, are con- stant and equal in magnitude, opposite in sign. 14. Prove that the specific curvature at every point of the elliptic paraboloid 2z = x 2 /a -\- y 2 /b, where it is cut by the cylinder x 2 /a 2 -j- y 2 /b 2 = 1, is (^ab)-i. 15. Prove that the principal curvatures are equal and opposite in the surface x 2 (y — z) -\- ayz = o where it is met by the cone (x 2 + dyz)yz — (y — z) 4 -. 16. The principal radii of curvature at the points of the surface a 2 x 2 = z 2 K x 2 + y 2 ), ■ where x = y = z, are given by 2R 2 -\-'2 4/3 aR — ga 2 = o. 374 APPLICATION TO SURFACES. [Ch. XXXIII. 17. Prove that the radius ot curvature of the surface x™ -\- y m -j- z™ = a m at m — 2 an umbilic is 3 2m aj{m — 1). x y z 18. Show that — = — = — is an umbilic on the surface a b c jfi/a + y 3 /b -f z z /c = k 2 . 19. Show that x = y = z = (a^)^ is an umbilic on the surface .xyz = a&r and the curvature there is \{abc)~^. x 2 y 2 z 2 20. Find the umbilici on the ellipsoid — -\- —4- — = 1. a* 0* c 1 a 2 (a 2 — b 2 ) c 2 (b 2 c 2 ) Ans. The four real umbilics are x 2 = —- , z 2 = —^ . a 2 — c- a 2 — c' 1 21. At an ordinary point on a surface the locus of the centers of curvature of all plane sections is a fixed surface, whose equation referred to the tangent plane as 2-plane and the principal planes as the x- and ^-planes, is (^ +y* + z 2 ) ^L + >l^ = 2{x 2 + yi) . 22. Show that an umbilicus on the surface (x/a)i + (y /6)* + (z /cf = I 23. If F = o is the equation of a conicoid, show that the tangent cone to the surface drawn from the vertex or, ft, y touches a surface along a plane curve which is the intersection of F = o and the plane <* " a> ^ + ( -" " /S) ^ + (2 -r)^+1* A x) = o. 24. Find the quadratic equation for determining the principal radii of curva- ture at any point of the surface *(■*) + H¥\ + *(*) = o, and find the condition that the principal curvatures may be equal and opposite. 25. Show that the cylinder (a 2 4. c 2 )b 2 x 2 + (<5 2 + c 2 )a 2 y 2 = (a 2 + b 2 )a 2 b 2 cuts the hyperboloid x 2 /a 2 -j- y 2 /b 2 — z 2 /c 2 — 1 in a curve at each point of which the principal curvatures of the hyperboloid are equal and opposite. 26. Show that the principal radii of curvature are equal and opposite at every point in which the plane x = a cuts the surface x{x 2 + y> + z 2 ) = 2a{x 2 + y 2 ). 27. In the surface in Ex. 24 show that the point which satisfies 4>»{x) = rP"(y) = X "(*) is an umbilic. 28. Find the umbilici on the surface 2z = x 2 /a -\- y 2 /b. Ans. x = o, y = — \/{ab — b 2 ), z = ±(a — b), if a > b. 29. Show that z — f(x, y) is generated by a straight line if at all points a 2 / a 2 / 8x 2 ~dy f — l— '-A V 2 ~ \dx by) This is also the condition that the inflexional tangents at each point of the sur- face shall be coincident. Such a surface is called a torse or developable surface. CHAPTER XXXIV. CURVES IN SPACE. 257. General Equations. — A curve in space is generally denned as the intersection of two surfaces. A curve will in general have for its equations x {x>y, z ) = °, 0,( x >y> z ) = °- C 1 ) If between these two equations we eliminate successively x,y, z, we obtain the projecting cylinders of the curve on the coordinate planes, respectively, $i(y> z ) = °> &(■*> z ) = °> t\( x >y) = o. Any two of these can be taken as the equations of the curve. 258. A curve in space is also determined when the coordinates of any point on the curve are given as functions of some fourth variable, such as /, x= 0(/), y = f{t), *=*(/). (2) The elimination of / between these equations two and two give the projecting cylinders of the curve. 259. Equations of the Tangent to a Curve at a Point. — If the equations of the line are (1), the equations of the tangent line to (1) at x,y, z are the equations to the tangent planes to cf) l = o, 2 = o, taken simultaneously, or Since the tangent line is perpendicular to the normals to these planes, the direction cosines /, m, n of the tangent line are given by / m n 1 vF^raqsTx ~ N ^ - N A ~ L > M 2 - l,m= h - where ** = (Mjr t - M^f + (A\£, - nay + (A*; - L w- L,, M v N x being the first partial derivatives of X at x,y, z, and similarly Z 2 , M %i N 2 are those of 2 . 375 (2) (3) (4) 376 APPLICATION TO SURFACES. [Gil XXXIV. 260. If s is the length of a curve measured from a fixed point tox,y, z, then the direction cosines of the tangent to the curve at x, y, z are dx dy dz /= *> '"=Js< *=&' and the equations of the tangent are X -x _ Y-y Z-z dx dy dz ds ds ds If the equations to the curve be given by (2), § 258, then — = (p'(t) — , etc., and the equations (2) become X — x _Y — y _Z — z In general the equations to the tangent are X-x _ Y-y _ Z-z dx dy dz ' without specifying the independent variable. 261. The Equation to the Normal Plane to a Curve at x,y, z is . __ .dx t/T - . dy dz . , <*-*> *+>*+ i z -»>.s=°' (i) the normal plane being defined as the plane which is perpendicular to the tangent at the point of contact. Regardless of the independent variable, (1) becomes (X - x)dx + (Y-y)dy + (Z - z)dz = o. (2) EXAMPLES. 1. Find the tangent line to the central plane section of an ellipsoid. The equations of the curve are The equations of the tangent at x, y, z are X-x Y — y Z - z b 1 c l c l a z a 2 t> 2 2. Trace the curve (the helix) x = a cos /, y = a sin /, z = bt. Show that the tangent makes a constant angle with the x, y plane, and that the curve is a line drawn on a circular cylinder of revolution cutting all the elements at a constant angle. Art. 262.] CURVES IN SPACE. 377 3. Find the highest and lowest points on the curve of intersection of the surfaces 2z = a x 2 -f by 2 , Ax -f By -f- Cz + D = o, from the fact that at these points the tangent to the curve must be horizontal. 4. Show that at every point of a line of steepest slope on any surface F = o we must have dF , OF J ■—-. dy — — dx — o. dx dy 5. Show that the lines of steepest slope on the right conoid x = y/(z) are cut out by the cylinders x 2 -\- y 2 = r 2 , r being an arbitrary radius. 262. Osculating Plane. — UP, Q, R be three points on a curve, these three points determine a plane. The limiting position of this plane when P, Q, R converge to one point as a limit is called the osculating plane of the curve at that point. The coordinates x,y, z of any point on a curve are functions of the length, s, of the curve measured from some fixed point to x, y, z. Therefore, if s x be the length to x lt y lt % lf dx - s' -j- l/£ 2 oV (1) The equation to the plane through P can be written A(X - x) + B{T -y) + C(Z - 2) = o. If this passes through <2 and i?, then A(x l -x) + B{y x -y) + Cfo - 2) = o, ) A(x 2 -x) + B(y 2 -y) + C(* 2 - 0) = o. [ Substitute the values of the coordinates from (1) and (2) in (4) Divide by ds, 6s 2 , and let o\r(=)o. . \ Ax' + .#/ 4- Cz' — o, ,4.*" + By" -f C*" = o. (3) (4) 4). (5) Eliminating -4, i?, C between (3) and (5), we have the equation to the osculating plane at x,y, z, (6) X- x, x' x" r-y, y y Z -z z' z" I 378 APPLICATION TO SURFACES. [Ch. XXXIV. Or, regardless of the independent variable, X — x, Y — y, Z — z — o. (7) dx dy dz d 2 x d 2 y dh 263. To Find the Condition that a Curve may be a Plane Curve. — If a curve is a plane curve, the coordinates of any point must satisfy a linear relation Ax -f By + Cz -f D = o, where A, B, C, D are constants. Differentiating, Adx -\-Bdy + Cdz = o, Ad 2 x -f BcPy + Cdh = o, Ad*x + j&/"> + Cdh = o. Eliminating ^4,-5, C, we have the condition dx, dy, dz d 2 x, d 2 y, d 2 z d 3 x, d 3 y, d 3 z which must be satisfied at all points on the curve. 264. Equations of the Principal Normal. — The principal normal to a curve at a point is the intersection of the osculating plane and the normal plane at the point. Let /, m, n be the direction cosines of the principal normal at x,y, z. Then, since this line lies in the normal and osculating planes, y z z' x' + m \z"x" + n lx' -f- my' -(- nz' : x' y x"y> = o, These conditions are satisfied by / = x". m since y z' +y +*' x y x»y x"y" z" x f y' z' x"y" z" = o. or Also differentiating x' 2 -\-y' 2 + z' 2 = 1, .'. x f x"+y'y" +z'z" = o. Therefore the equations of the principal normal are X - x _ Y -y __ Z-z x" y" z" X-x Y-v Z-z d'x d 2 y d 2 z (1) (2) 265. The Binormal. — The binomial to a curve at a point is the straight line perpendicular to the osculating plane at the point. Art. 267.] CURVES IN SPACE. 379 Its equations are therefore, from (6), § 262, A' - x F- v Z -z y - z " _ y " z ' ~ Yx" -z" x' ~ x'y" - x"y'' (4) Dividing through by ds 3 , the equations can be written without specifying the independent variable. 266. The Circle of Curvature. — The circle of curvature at a given point of a space curve is the limiting position of the circle passing through three points on the curve when the three points converge to the given point. Clearly, the circle of curvature lies in the osculating plane and is the osculating circle of the curve. To find the radius of curvature. Let a, /?, y be the coordinates of the center, and p the radius of the circle of curvature at x, y, z. Then (* _ af + (y - (5f + {z - yf = p*. Let x,y, z vary on the circle. Differentiate twice with respect to s. Then (*- «)*" + (y- PV + (z - y) z" + x'* + y' > + z f * = o. But x' 2 4- y' 2 -[- z' 2 = 1. Also, the line through x,y, z and a, ft, y is the principal normal, whose direction cosines, by (1), § 264, are <*■" / = with similar values for m and n. Since x — a = /p, v — fi = mp, z — y = tip, The center of the circle is a = x — ip, etc. 267. The direction cosines of the binormal are / = p[y'z" - z'y"\ m = p{z'x" - x'z"\ n = p{x'y" -/*"). For, by (4), §265, I _ m _ n . v > z " _ z ' y " - z ' x " _ x 'z" ~ x'y" -y'x" * ^ Also differentiating x* 2 -\-y f 2 + 2' 2 = 1, . '. *'*" + J>>" + *V = O. The sum of the squares of the denominators in (1) is (*' 2 + y 2 + 2 / 2)( ^/ 2 + y/ 2 + z » 2) _(*'.*" +>/' + s'*") = l/p 2 . Hence the results stated. 380 APPLICATION TO SURFACES. [Ch. XXXIV. 268. Tortuosity. Measure of Twist. Definition. — The measure of torsion or twist of a space curve is the rate per unit length of curve at which the osculating plane turns around the tangent to the curve, as the point of contact moves along the curve. If the osculating plane turns through the angle Jr as the point of contact P moves to Q through the are As, the measure of torsion at P is dr_ _ fAr its ~ £ As ' when As( = )o. The number _|_ m y 4. nz > = o, (3) fa" -\- my" -f nz" — o. (4) sin 2 At sin 2 Ax ( JtN | 2 As 2 Ar 2 \ As j 1 * Let' As( = ) Then, in the limit, ')*= / dn \ m Ts- dm\* **/ + / dl t \ n Ts~ l - ds) H dm ~ds~- — m diy ds) ' Since P + m 2 + n*= 1, ■■■ 4 ds 4- dm ds dn + n d7-~ - 0. Art. 270.] CURVES IN SPACE. 381 Differentiating (3) and using (4), /v + »y + »v = oi (5) Differentiating, P -f- m 2 -j- «- = 1. (6) . '. //' -f mm! -\- ?iri = o. (7) From (5) and (7) we get /' m' n' mz' — ny' "~ nx' — lz' ~~ I'y — mx' 9 and each of these is equal to I'x" m'y" n'z" (8) mz'x — ny x nx'y" — Iz'y" ly'z" ' v." — nv'/x" n.Tc'v" — /c'v" ' h>'v" mx'z" (9) _ I'x" -f m'y" -f n'z" lL-\- mM + nN ' Differentiating (4), l' x " _|_ m 'y" _j_ n'z" -f- &"' -f my'" -f »*'" - . Therefore (8) is equal to l x "> -j. my'" + nz"' _ x'"L+y'"M+z'"N IL + mM+nN ~ L 2 -j- M 2 -f N 2 * Remembering that /, m, n; x',y', z' are the direction cosines of two lines at right angles, {mz' — ny') 2 -j- (nx' — lz') 2 -j- {ly' — mx') 2 Therefore, by (2), §268, and (8), ( dT V- ( x'"L+y"M+z'"N \ 2 \ds) ~ \ L 2 + M 2 -f- N 2 ) ' or 1 dt - — — = p 2 >- V)y'" + (z- y)z'" = o. 3*2 APPLICATION TO SURFACES. [Ch. XXXIV. Eliminating between the last three equations, (* \y z = '" - y'x"'f -f (/*'" - z'y"') 2 + (z'x" f - x'z'"f\. Clearly the circle of curvature lies on the sphere of curvature. Let P, Q, R ', J be four points on a curve and in the same neighbor- hood, R and p the radii of spherical and circular curvature. Then, C being the center of the circle through P, Q, R, and £ S that of the sphere through P, Q, R, J, we have directly from the figure, Ap Ax SO R 2 (4ti W£)' R 2 '+£)' \ v v v "^^^^i ti\ \ J J c r2 = P' + dx Fig. 147. 271. The expressions for the value of the radius of curvature and measure of torsion in § 266, and (10), §269, have been worked out with respect to s, the curve length, as the independent variable. These can be written in differentials, regardless of whatever variable be taken as the independent variable. Represent by dx dv dz 2 \, z, is dF , dF 7 dF J — dx + - Jr dy + ^-dz= o. (1) dx dy dz v y But in F(x,y, z, a) = o, as x,y, z vary along the envelope, a also varies, and the equation to the tangent to the envelope is dF 7 dF J dF ' dF 7 — - dx -f- — - dy -f — - dz -f —- da — o. (2) dx dy dz da v ' Since at any point x,y, z common to the surface F= o and the envelope, that is all along the characteristic, we have F* = o, the planes (1) and (2) coincide. EXAMPLES. 1. Show that the envelope of a family of planes having a single parameter is a torse (developable surface). Let z = x(P(a) +)'ip(a) -f *(<*). dz dz lit .-. — = 4>(a), -- = tp(a); :• x

a + Xa = °- Also, &z . t da &z da dH da ... da w=--^ a) ^' dy = 1j){a) dj' -d7dj =!£+ (ft- *>?£ = «»• (4) This is the equation of a surface passing through the intersection of (1) and (2). But for any fixed values x,y, z, a, fi satisfying (1) and (2) there are an indefinite number of surfaces (4) obtained by varying a lt fi x , all of which cut (1) in lines passing through x, y, z. Consequently there are of these surfaces (4) two particular surfaces, dF _ dF _ da'~~ °' dfi 7 ~~ °' which cut (1) in lines passing through x, y, z. If now the point x,y, z has a determinate limit when a x (-=)a, fi x ( = )fi, then the three surfaces F(a, fi) = o, F' a (a, fi) = o, F&a, fi) = o, pass through and determine that point. These surfaces (5) intersect, in general, in a discrete set of points. If, however, we eliminate between them a and fi, we obtain the equa- tion to the locus of intersections. This locus is a surface called the envelope of the family (1). 275. The Envelope of the Family F(x,y, z, a, fi) = o is Tan- gent to Each Member of the Family. — The tangent plane to any member of the family is dF 7 dF 7 dF , , x _■**+ - dy + aT ^ = o. (1) 3^8 APPLICATION TO SURFACES. [Ch. XXXV. As the point x, y, z moves on the envelope, a and /3 vary. The plane tangent to the envelope is dF dF dF dF dF ^ x + T/» + aT* + a^ da + ^ dP = °- (2) At a point x,y, z common to the envelope and one of the surfaces, we have F' a = o, Fp = o, and therefore the planes (i) and (2) coincide. Since this point is the intersection of the line whose equations are F' a = o, Fp = o with the surface F = o, the envelope is tangent to the surface at a point, and not along a line. 276. Use of Arbitrary Multipliers. — K F(x,y, z, a, /3) = o, where a, (i are two arbitrary parameters connected by the relation (p(a, /?) = o, then F = o is a family of surfaces depending on a single variable parameter. The equation of the envelope is found by the elimination of a, ft, da, d/3 between F = o, = o, F' a da-\- F'pd/3 — o, cp' a da + tpfrd/3 — o. This is best effected, as in the corresponding problems of maximum and minimum, by the use of arbitrary multipliers. Thus the equation of the envelope is the result obtained by eliminating a, [3, X from F= o, = o, F' a + \0' a = o, F'p + A0£ = o. The family of surfaces, represented by F = o, containing n parameters which are connected by n — 1 or n — 2 equations is equivalent to a family containing one or two independent parameters respectively. Such a family, in general, has an envelope. The problem of finding the envelope is generally best solved by intro- ducing arbitrary multipliers to assist the eliminations. If more than two independent variable parameters are involved, there can be no envelope. Foi in this case we obtain more than three equations for determining the limiting position of the intersec- tion of one surface with a neighboring surface. From these three equations x,y, z could be eliminated, and a relation between the parameters obtained, which is contrary to the hypothesis that they are independent. In general, if F— o contains n arbitrary parameters a x , . . . , a n connected by the n — 1 equations of condition X = o, . . . , M _j = o, the equation of the envelope is found by eliminating the 27i — 1 numbers a x , . . . , a n , X lt . . . , A M _,, between the 2ti equations F = O, 0, = O, . . . , 0„_, = o, *"K) + WK) + • • • +A*_i + fl 2 _ dl ' y + 4) -/(-v.^)]->' + 4) - (A. >0^ 9« _ r> £ & Ay J a Ay Art. 279.] DIFFERENTIATION OF INTEGRALS. 393 Hence, when 4>'( = )o, we have du and, generally, du /** df ~1 i dx ' d n u _ r b ay dy»~Ja dr ^ dx . EXAMPLES. 1. K r-e-axdx = I Jo a be differentiated n times with respect to a, we get 1 n \ x n e~* x dx — — 1 1: no or 2. From r dx = -?- «/o U 2 + «) " 2ai Z* 00 <**" _ 1-3-5 . . • (2» — i) ?r X (* 2 + a ) n+t ~ 2.4.6 ... 2« 2 a w+ ±' The value of a definite integral can frequently be found by this method. Thus: 3. Let u = f* ( f" ~ l ) -dx. Jo lo S * Then * = f l ^ **£* = Z* 1 W. = _L_. «fa J log x J a 4- I 5^-j = '<>? (« + '). constant being added since u = o when a = o. 4. Find /""log (1 4- a cos 0)^9. ^/w. ar log (1 + ^1 - a 2 ). 279. Integration under the Integral Sign. I, Indefinite Integral. Let F(x,y)=f f(x,y)dx. Then will f[fA*>s)**] & = f[ff( x > y¥y\ dx - Let v = / y(jtr,j^)^j/. Then ~ =f(x,y). Also, ty$ v dx = / V ^ r = fA x >y) dx = F i x >y)- .'. fvdx =JF(x,y)dy y f I JA*>yyy \ dx =f J jf(x>y) d * \ d y- 394 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVI. Hence the order in which the integration is performed is indif- ferent. This shows that in indefinite integration when we integrate a function of two independent variables, first with respect to one vari- able and then with respect to the other, the result is the same when the order of integration is reversed. This being the case, we can represent the result of the two integrations by either of the compact symbols f $/*<* = f }/***■ As in differentiation, the operation is to be performed first with respect to the variable whose differential is written nearest the func- tion, or integral sign. II. The same theorem is true for the definite double integral of a function of two independent variables when the limits are constants. Let ff(x, y)dx = X(x, y), ff(x, y)dy = F(x, y) ; fff(x,y)dxdy =fff(x,y)dydx = F(x,y). Then p/dx = X{ x% ,y) - X{x v y), pfdy = T(x, ,.,) - F(x, A ); i/yi •C 1 !n fdy \ dX = F{X "- h) - r(x >' J > ] - F(X *' y > ] + *l*»-ti' The last two values are the same. Hence C\r;y dx \ dy =i:;\s:y d A^ or P pA*, yY* *» = P P A*.y).r dx = -5 7- . Jo °- + b C^ P a , r*\ b da / / l *-«* sin 6x da dx = / , Jo J„ Ao « 2 + * 2 ' J/tco^-,, or If a =r o, a x = 00 , then 3. Evaluate Put — sin ^x sin /^ f-^*- k= r e~* % dx. r e -aHt+x*) a dx _ k - a \ r Fe-«^+**)adadx = kfe-< Jo Jo Jo *V« = k\ fe-^+^ada^ 1 * 2 Jo 2 1 -j- x l Vo i-h^ = ^ tan ^J - i7r = &. He-^dx = \ \/n, Jo 2a and Also, and and This gives the area of the probability curve. 280. If F(x, y, z) is a function of three independent variables, the same rules as for a function of two independent variables govern the triple integral j j JFdxdydz. Examples of double and triple integrals will be given in the next chapter. CHAPTER XXXVII. APPLICATIONS OF DOUBLE AND TRIPLE INTEGRALS. Plane Areas. Double Integration. 281. Rectangular Formulae. — If x, y are the rectangular coor- dinates of a point in a plane xOy, then doj = Ax Ay = dx dy is an element of area, being the area of the rectangle whose sides are Ax and Ay. Let the entire plane xOy be divided into rectangular spaces by parallels to Ox and Oy, of which Ax Ay is a type. The area of any closed boundary drawn in the plane is the limit of the sum of all the entire rectan- gular elements of type Ay Ax included in the boundary, when for each rectangle Ax( — )o, Ay{ — )o. For the area within the closed boundary A is equal to A = 2 Ay Ax plus the sum of the fractional rectangles which are cut by the Fig. 148. boundary. This latter sum can be shown to be less than the length of the boundary multiplied by the diagonal of the greatest elementary rectangle, and therefore has the limit zero. Hence A =£2 Ay Ax, taken throughout the enclosed region, when Ax( = )Ay( = )o. The summation is effected by summing first the rectangles in a vertical strip PQ and then summing all the vertical strips from R to T; or, first sum the elements in a horizontal strip PL, then sum all the horizontal strips in the boundary from Sto U. These summations are clearly represented by the double integrals y t u u ^ ^d s N / / \ 7 \'t 7 Y • 1 M 1 -? 7 Xz : 7 L\ *=^== : lllTz V ■e^ w , X r M i r' r* M A6 _ £Ap jAiAO - I+ U p' = *i when Ap( = )A£)( = )o, the area within any closed boundary is equal to A = £2pApAd when Ap( = )A6(=)o. 398 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cm XXXVII. This summation can be effected in two ways : (i). We can sum the checks along a radius vector R S, keeping A 6 constant, then sum the tier of checks thus obtained from one value of 6 to another. (2). We can sum the checks along the ring UV, keeping p and Ap constant, then sum the rings from one value of p to another. These operations clearly give the double integrals «=m(p) JfOi /»P=«PW , , a /•Pa /»»=mp; JQ . ' / pdpdd, / / ddpdp. EXAMPLES. 1. Find the area between the two circles p = a cos 0, p = b cos 0, b > a. IT /» 2 /»£ COS (I). ^ = / / P ^P <#, Jo Ja cos _ r 2 ^(£2 _ fl 2) COS 20 dQ =j (b 2 - c 2 ). -IP -»-£ ncos -7 /»« /»COS A * dB pdp+ / P <# P *P, Jo Jcos -1 ^ which gives the same result as (1). The double integration is not necessary for finding the areas of curves; it is given here as an illustration of a process which admits of generalization. Volumes of Solids. Double and Triple Integration. 283. Rectangular Coordinates. — Let x,y, z be the coordinates of a point in space referred to orthogonal coordinate axes. Divide space into a system of rectan- gular parallelopipeds by planes parallel to the coordinate planes. Let Ax, Ay, Az be the edges of a typical elementary parallelopiped. Then the volume x Ax Ay Az is the elementary space volume. The volume of any closed surface is the limit of the sum of the entire elemen- tary parallelopipeds included by the surface when Ax( = )Ay( = )Az( = )o. V = £2 Ax Ay Az, taken throughout the enclosed space. (1). Let x, y, Ay, Ax be constant. Sum the elementary volumes Art. 283.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 399 between the two values of z, obtaining the volume of a column MS of the solid. The result expresses z as a function of x and r given by the equation or equations of the boundary. (2). Let x, Ax be constant. Sum the columns between two values of y for Ay( — )o. The result is the slice of the solid on the cross-section x = constant, having thickness Ax. (3). Sum the slices between two values of x for Ax(=z)o. The result is the total sum of the elements, expressed by the integral Jxi Jy=(x) Jz — K{x,y) f x * r*z dy £y x dx. dx. Clearly, if more convenient we may change the order of integra- tion, making the proper changes in the limits ot integration. EXAMPLES. 1. Find the volume of one eighth the ellipsoid ^ + ^- + --1. Jo Jo x y j j n II \~ ^/ dx = * 7tabc ' \ be 4 2. Find the volume bounded by the hyperbolic paraboloid xy = az, the xOy plane, and the four planes x = x x , x = x 2 , y = y x , y = y 2 . xy V= [**[** fdzdydx, Jx 1 Jy x Jo Jx x Jy x a J Xl 2a 2 x dx, I \a = — (x 2 — x x )(y 2 - v l )(x 1 y l + x 2 y 2 -f x x y 2 -f x 2 y x \ t\a = i{x 2 - X X ){X 2 -y x \Z x + 2 2 +^ 3 + **)• 4oo INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVII. The volume is therefore equal to the area of the rectangular base multiplied by the average of the elevations of the corners. This is the engineer's r.ule for calcu- lating earthwork volumes. Polar Coordinates. — The polar coordinates of a point P in space are p, the distance of the point from the origin; 0, the angle which this radius vector OP makes with the vertical Oz; and 0, the angle which the vertical plane POz makes with the fixed plane xOz. Through P draw a vertical circle PM with radius p. Prolong OP to R, PR = A p. ■x Draw the circle R Q in the plane POM with radius p -f dp. If A A is the area PRQS, then / AA _ AppA6~ l ' We may therefore take dA = p dp dd. This area revolving around Oz generates a ring of volume 2 7T p sin 6 dA. Therefore the volume generated by dA revolving through the arc ds — p sin 6 dcp is in the same proportion to the volume of the ring as is the arc to the whole circumference, or the element of volume is p 2 sin 6 d(p dp dd. We divide space into elementary volumes by a series of concen- tric spheres having the origin as center, and a series of cones of revolution having Oz for axis, and a series of planes through Oz The volume of any closed surface is the limit of the sum of the entire elementary solids included in the surface when Ap{ = )A{ = )A0( = )o. Or, the volume is equal to the triple integral V = f ffp 2 sin 6 d dp dd, taken with the proper limits as determined by the boundaries of the surface. EXAMPLES. 1. Find the volume of one eighth the sphere p = a n it V- f 2 f 2 ( P a p 1 dp • sin QJO.Jfa = - 2 / * smBdB'd 3 J=o Je=o IT Art. 285.] APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 401 The first integration gives a pyramid with vertex O and spherical base a 2 sin i/0 - x 2 - y 2 P \x=a f'y- yax-X 1 dy dx J x =o Jy=° \/a 2 -x*-y 2 ' Integrate directly, or put sin 2 = x/(a -f- x) and integrate Hence S = 2a\it — 2). (2). Again, \S = / pQ sec y dp. «'o p — a cos b = a sin y. .-. y = \ic — Q $S = — a 2 fo cos dQ = — a 2 [Q sin -f cos 6]*" = - a\\ir — I) Lengths of Curves in Space. 287. As in plane curves, the length of a curve in space is defined to be the limit to which converges the sum of the lengths of the sides of a polygonal line inscribed in the curve. Art. 287. j APPLICATION OF DOUBLE AND TRIPLE INTEGRALS. 405 Since \~£) A ** = Aa * + Ay2 + **» with similar values for the derivatives —, — , './N'-*.®"*®'* with corresponding values for s when >■ or 2 is taken as the indepen- dent variable. If the coordinates of a point on the curve are given in terms of a variable /, then and (D:= p* |j+ (if- EXAMPLES. 1. Find the length of the helix x = a cos - , y = a sin - , measured from z = o. Take z as the independent variable. Then dx a . 2 dy a z - = -7-sin-, — =—cos — ; dz b b 1 dz b b' 2. Find the length, measured from the origin, of the curve 2ay — x 2 , bd l z = x z . '-/(«-^i)^=/('+=h=- +^-*+* 3. Show that the length, measured from the origin, of y = a sin #, 42 = a 2 (.*r + cos x sin x), is x -\- z. 4. Find the length of _ 2 !x 3 y — 2 |/a* — *, ^ = # — — a(~» measured from the origin. ^»^- J = •* -\-y — *• 4©6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVII. 5. Find the length, measured from the horizontal plane, of the curve x 2 y 2 /- --' 288. Observations on Multiple Integrals. — The problem of integration always reduces ultimately to the irreducible integral dE being the element of the subject to be integrated. Or this may be taken as the starting-point and considered as the simplest element- ary statement of the problem for solution. This, in simple cases, may be evaluated directly, otherwise it may be necessary to integrate par- tially two or more times with respect to the different variables which enter the problem. There may be several different ways in which the elements can be summed. A careful study of the problem in each particular case should be made in order to determine the best way of effecting the partial summations, with respect to the limits at eacli stage of the process. One is at perfect liberty to take the elements of integration in geometrical problems in any way and of any shape one chooses, as the limit of the sum is independent of the manner in which the subdivision is made (see Appendix). This should be verified by working the same problem in several different ways. The applications of multiple integration in mechanics are numerous and extensive. Further application beyond the elementary geometrical ones given here is outside the scope of the present work. EXERCISES. In these exercises the results should be obtained by double and triple integra- tion, and also by single integration whenever it is possible. 1. Find the volume bounded by the surfaces x 2 -f- y 2 = a 2 , z = O, z = x tan a. pa p Ya* - x* px tan a Ans. 2 / / / dz dy ax = $a z tan a. Jo «/o Jo 2. Find the volume bounded by the plane 2 — 0, the cylinder (x- a) 2 + (y -bf = A' 2 , and the hyperbolic paraboloid xy = cz. Ans. it — R 2 . 3. Find the volume bounded by the sphere and cylinders x i + y i ± z i - a 2 , x 2 + y 2 = b\ p 2 = a 2 cos 2 + b 2 sin 2 0. Ans. $(16 - 37t)(a 2 - b 2 )l Art. 2SS.J APPLICATION OF DOUBLE AND TRIPLE ENTEGRALS. 407 4. A sphere is cut by a right cylinder whose surface passes through the center of the sphere ; the radius of the cylinder is one half that of the sphere rt. Find the volume common to both surfaces. Ans. \{7t — 4 )a 3 . 5. Show that the volume included within the surface ^(-v'.-) = °- \a b c J is abc times the volume of the surface F(x, y, z) = o. 6. Show that the volume of the solid bounded by the surfaces z = o, x 2 -\-y 2 = ^az, ^-fjj/ 2 = 2nr, is \nc^/a. 7. Find the entire volume bounded by the positive sides of the three coordinate planes and the surface 8. Find the volume bounded by the surface **+}>*+** = a*. Ans 9. Find the volume of the surface abc 90' QMOMt)*- ** *-* 10. Show that the volume included between the surface of the hyperboloid of one sheet, its asymptotic cone, and two planes parallel to that of the real axes is proportional to the distance between those planes. 11. Find the whole volume of the solid x i/a* + y 2 /b 2 + z'/c* = 1. Ans. \itabc. 12. Find the whole volume of the solid bounded by {x 2 _j_y _j_ ^2)3 _ 2ja s xyz. Ans. | *> £ + {* £ -')■ = °» are of the second order and first degree. 294. Solution of a Differential Equation. — To solve a given differential equation *\x,y>y') = °> dy wherejy = — , is to find the values x and^ which satisfy the equa- tion. Thus, if the values of x and y which satisfy the equation 0O~, y) = © satisfy a differential equation F = 0, then = o is a solution of ^= o. The solution of a given differential equation may be a particular solution or it may be the general solution. The general solution in- cludes all the particular solutions. Or the solution may be a singular solution, which is not included in the general solution. The complete solution of a differential equation includes the general solution and the singular solution. The meaning of these solutions will be developed in what follows. The solution of a differential equation is considered as having been effected when it has been reduced to an equation in integrals, whether the actual integrations can be effected in finite terms or not. Equations of the First Degree and First Order. 295. The simplest type of an ordinary differential equation of the first order and degree is dy=/(x)dx. (1) Integrating, we obtain the solution y = F( x ) + c, (2) where F(x) is a primitive of/(x) and c is an arbitrary constant. For a particular assigned value of c, (2) is a particular solution of (1), Art. 295.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 411 and is the equation of a particular curve in a definite position. At each point of the curve (2), is the slope, or direction of the curve (2). For different values of ewe have different curves. The ordinates of any two such curves differ by a constant. Equation (2) is then the equation of a family of curves having the arbitrary parameter c. This singly infinite sys- tem of curves, or family of curves with a single parameter, is the general solution of the differential equation (1). 296. Every equation of the first order and first degree can be written Mdx -f- N dy = o, (1) where, as has been said before, MandJVare either constants, functions of a: or r, or functions of x andj'. 297. Solution by Separation of the Variables.— This solution consists in arranging the equation Mdx + Ndy = o, (1) so that it takes the form (p(x)dx + ip(y)dy = 0. (2) The process by which this is effected is called separation of the variables. When the variables have been thus separated the solution is obtained by direct integration. Thus, integrating (2), fcP(x) dx + fip(y) dy = c, where c is an arbitrary constant, and is the parameter of the family of curves representing the solution. I. Variables Separated by Inspection. — A considerable number of simple equations can be solved directly by an obvious separation of the variables. The process is best illustrated by examples which follow. EXAMPLES. 1. Find the curve whose slope to the jr-axis is — x/y, and which passes through the point 2, 3. The geometrical conditions give rise to the differential equation & x 1 1 j - — = — — , or y ay -f- x dx = o. ax y The solution of which, obtained by integration, is the family of circles x 2 + y = c 2 . The particular curve of the family through 2, 3 is x 2 + y 2 = 13. 412 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. 2. Find the line whose slope is constant. - = m gives the family of parallel straight lines y = mx -f- c. 3. Find the curves whose differential equation is x dy -f- y dx = o. The variables when separated give dx . dy \. 4~ = o. x y .'. log* -f- log y = c, or xy = k. Otherwise we may write the solution xy — e c . This is a family of hyperbolae having for asymptotes the coordinate axes. If we observe that x dy -\- y dx is nothing more than d(xy), the solution xy = c is obvious. 4. Find the curve whose slope at any point is equal to the ordinate at the point. tt d y d ? j Here -y- = y. .-. -- = dx. dx y Hence \ogy = x -{- c , or y = e x + c = e c e x = ae x , which is the exponential family of curves. 5. Find the curve whose slope is proportional to the abscissa. Ans. The family of parabolae jk = ax 2 -f- c, in which c. the constant of integra- tion, is the parameter. 6. Find the curve whose slope at x, y is equal to xy. Ans. y = c e ^ x • 7. Find the curve whose subtangent is proportional to the abscissa of the point of contact. dx dx dv Here y — - = ax. . • . — = .a — gives ' dy x y & log x = a log y -(- c, or y a == kx. 8. Find the curve whose subnormal is constant. y - — a gives y- = 2.ax -\- c, the parabola. X 9. Find the curve whose subtangent is constant. Ans. y — ce a . 10. Find the curve whose subnormal is proportional to the «th power of the ordinate. What is the curve when n is 2 ? 11. Find the curve whose normal. length is constant. Here the geometrical conditions give the differential equation ■J- ♦■*)■- y-x lJ r ;;- --*■ ••• <** = \/a--y* Integrating, x — c = — (tf 2 — y 2 )^, or the family of circles {x-c) 2 +y 2 = a\ with radius a. having their centers on the .r-axis. 12. Find the curve in which the perpendicular on the tangent drawn from the foot of the ordinate of the point of contact is constant and equal to a. Art. 297.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 413 The differential equation of condition is ++®' °t= = dx. v/> The solution is therefore the family of curves c + x - a log (y + \f y 2 - a 2 ). When c = o this is the catenary with Oy as axis. 13. Find the curve in which the subtangent is proportional to the subnormal. 14. Determine the curve in which the length of the arc measured from a fixed point to any point Pis proportional to (1) the abscissa, (2) the square of the abscissa, (3) the square root of the abscissa of the point P. (i). A straight line. (2). The condition is ds 2 = dx 2 + dy 2 = ^ dx* a dy = |/r 2 — a 2 dx. The solution of this is c -f- ay = \x \/x 2 - a 2 - \a 2 log \x + y ' x 2 - a 2 ] , (3). The geometrical condition can be written s = 2 \/ax. ds — ^ — dx. dx 2 -f dy 2 = ds 2 = —dx gives Ax x dy •j ! Put x = z 1 and integrate. The result is the cycloid c -j- y = ^x{a — x) + a sin- 1 \a Ex. 14, really leads to a differential equation of the first order and second degree, which furnishes two solutions which are the same. 15. Find the curve in which the polar subnormal is proportional to (1) the radius vector, (2) to the sine of the vectorial angle, (i). p — ce a $. (2). ft = c — a cos 0. 16. Find the curve in which the polar subtangent is proportional to the length of the radius vector, and also that curve in which the polar subtangent and polar sub-normal are in constant ratio. Ans. ft = cent. 17. Determine the curve in which the angle between the radius vector and the tangent is one half the vectorial angle. Ans. p = c(i — cos 0). 18. Determine the curve such that the area bounded by the axes, the curve, and any ordinate is proportional to that ordinate. X If fl is the area, £1 - ay. . •. d£l — y dx — a dy. . : y = «"! 19. Determine the curve such that the area bounded by the x axis, the curve, and two ordinates is proportional to the arc between two ordinates. dy £1 = as. .•. y dx = a ds, dx = a , . 414 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. This gives, on integration, the catenary c + x = a log (y + \/) ri ~ <)• 20. Find the curve in which the square of the slope of the tangent is equal to the slope of the radius vector to the point of contact. The parabola x* -\- y^ — £*, or (x — yf — 2c{x -f- y) -f c 2 = O. 21. Solve M dx -f- N dy, when Mx ± Ny = o. (i). 7l/r -f- A> = o gives M/N = — y/x. Substituting in the equation, — - = — . . •. x = cy. (2). Mx — Ny — o gives M/N — y/x. dx dy Substituting in the equation, (- — = o. . •. xy = c. II. Solution when the Equation is homogeneous in x and y. — When the equation Mix -f N dy = o is such that M = (p(x, y), N = ip(x, y) are homogeneous functions of x and^ and of the same degree, the solution can be obtained by the substitution^ = zx. We have N - >Hx,y) ~ w> Divide the numerator and denominator by x n , n being the degree of

x , assign to h, k the values which satisfy to a x h + bjz + c, = o, [ , ^ a 2 A + ^ + c 2 = o. j" Then (2) becomes dy' _ ^7 r f / -f- ^y dx' a 2 x' -f 3^' ' This is homogeneous and can be solved by § 297. \{f{x',y') = o is the solution of (4), then/(^ — h,y — k) is the solution of (1). II. Uab = ah, let - 2 = ^ = fl*. (4) Then (1) becomes a x b x dy _ a x x + b x y + e x ^ dx m(a x x -\- b x y) -f- £ 2 Put = 0^ -\- b x y. Then. (5) becomes dz , . z -\- c. 0, dx 1 x ??iz -\- c 2 in which the variables can be readily separated. 41 6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cn. XXXVIII. EXAMPLES. 1. Solve (37 - 7x +7)dx + {jy - 3* + 3)dy = o. Ans. (y -x+ i)\y + x - if = c. 2. Solve (2x -\-y + i)dx -f- (4* + *y — t)dy — o. Ans. x -\- 2y -\- log(2x -\-.y — i) = c. 3. Solve. ( 7 j + x + 2)a& - (3* + 5/ + 6)4> = o. Ans. x-)-5;-f 2=4x-;| 2) 4 . 299. The Exact Differential Equation. — The differential equation Mdx-\-Ndy = o is said to be an exact differential equation when it is the immediate result of differentiating an implicit function /"(or, _>') = o. In fact, if u =f(x,y) = o, then ', such that du — M dx -\- N dy. (1) Since M=. — , il/ contains the derivatives of only those terms in u which contain a. Integrating (1) with respect to x (y being con- stant), we have u=fMdx+> 9y yw (3) As was said, is independent of x and so also is ^— , as is verified ay by differentiating (3) with respect to x; JL I # - — /Vy 2 dy = r Therefore the solution is x 3 — 2x 2 y — 2xy 2 -f- y 3 =. c. 2. Solve (x 2 + y 2 )(x dx + y dy) -f x dy - y dx = o. x i _L y2 y Ans. L ^- + tan-i — — c. 2 ' x 3. Solve (a 2 + $xy - 2y 2 )dx 4. {ix—yfdy - o. Ans. a' 2 x -)- $y A — 2xy 2 -\- ^x 2 y = c. 4. Solve (2ax -f- by 4- g)dx 4- (2cy -|- bx + e)dy — o. Ans. ax 2 -\- bxy -f- 9' 2 4" <£"■* + 9' — &' 5. Solve (w dx 4- «<^) sin (oti 4~ n )') — ( n l ^ x "f m d)') cos («x 4- my). Ans. cos (w,r 4- «»') 4" s i n ( w -*" + m )') = f - 6. Solve 2x(jc + 2j)^x -j- (2x 2 — y 2 )dy = o. Ans. x' A 4" 3 a " 2 J' — V s = c - 303. Non-Exact Equations of the First Order and Degree. — We have seen that when a primitive equation f[x,y) = o is differentiated there results the exact differential equation = o we eliminate any constant occurring in /and 0, we get another equation, */'(.v, r, r') = o, which is a differential equation satisfied at every point on /•=. o. Therefore /"= o is a primitive of ?/' — o. But //' = o will not be an exact Art. 303.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 419 differential of the primitive/" = o, although/" = o is a solution of the differential equation tp = o. To fix the ideas, consider the equation ax -f- by -\- cxy -j- k = o. (1) The exact differential equation of (1) is (a -j- cy)dx -f- (b -J- cx)dy = o. (2) When (2) is integrated the constant of integration restores the parameter k of the family (1) and (1) is the solution of (2). That is to say, the family of curves (1) obtained by varying the parameter /£ gives the solution of the exact differential equation (2). The constant k was eliminated from (2) by the operation of differ- entiation and restored by the process of integration. Eliminate a between (1) and (2) by substituting a -J- cy — — = — X from (1) in (2). There results the differential equation ■ dy dx by + k x{b -U ex) ' 1 dx c dx (3) or b x bb -\- ex bdy dx c dx by -|- k x ex + b' Integrating and adding the arbitrary constant — log c', log (by -|- k) -\- \og(cx -f b) — log x — log c' — o. {by -f- ^)(<^^ + ^) — c/jc > or (^c — c')jc -\- b 2 y -{- be xy -\- kb — o. Putting the arbitrary parameter in the form kc — c' = ab, this equation becomes the original primitive ax -j- by -\- cxy -\- k = o. This equation with the variable parameter a is the solution ot the differential equation (3). The differential equation (3), or (by -f- ^)^jc — x(b -\~ cx)dy — o (4) is not an exact equation, for A (by + k) = 5, A ( - bx -ex") = - b - 2C*. But (1) is the primitive of (3) as well as of (2). Again, if we eliminate first b and then c between (1) and (2), we 420 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. shall get two other differential equations, neither of which is exact, but each of which has (i) for solution with variable parameters b and c respectively. Observe particularly that if (4) be multiplied by i/x 2 , it becomes an exact differential, =^r- <* — & =°> (5) since 1 l b y + k \ - d I b + cx \_b dy \ x 2 j ~~ dx \ x J x r Integrating this exact equation (5) under the rule §302, the solution is qx -J- by -f- cxy + k = o, the same equation as (1) with q for parameter. 304. Integrating Factors. — In the preceding article we have seen that the same group of primitives can have a. number of different differential equations of the first order and degree. The form of any particular differential equation depending on the manner in which an arbitrary constant has been eliminated between the primitive and its exact differential equation. In the example above, when the differential equation was not exact, it was made exact by multiplying by 1/x 2 . Such a factor is called an integrating factor of the differential equation which it renders exact. The number of integrating factors for any equation Mdx + Ndy=o (1) is infinite. For, let /u be an integrating factor of (1). Then ju(Mdx -f- N dy) is an exact differential, say dn, and fi(M dx + N dy) — du. Multiply both sides of this equation by any integrable function of u, say /(a), fi/{u){Mdx -f- Ndy) =/(u)du. (2) The second member of (2) is an exact differential, and therefore also is the first. Hence, when /u is an integrating factor of (1), so also is /Jf(u), where f{u) is any arbitrary integrable function of u. In illustration consider the equation y dx — x dy = o. This is not exact, but when multiplied by either—,, — , or -- 1 y 2 xv x~ it becomes exact and has for solution x — = constant. y Art. 305. J INTEGRATION OF DIFFERENTIAL EQUATIONS. 421 The general solution of the differential equation M dx -\- N dy — o consists in finding an integrating factor jj. such that }A(Mdx + Ndy) = o is an exact differential, then integrating by the method given as the solution of the exact equation. The integrating factor always exists, but there is no known method by which it can be determined generally. The rules for determining an integrating factor for a few important equations will now be given. 305. Rules for Integrating Factors. I. By Inspection. — While the process of finding an integrating factor by inspection does not, strictly speaking, constitute a rule, in the absence of a general law for finding the integrating factor it is an important method of procedure. An equation should always be ex- amined first with the view of being able to recognize a factor of inte- gration. The process is best illustrated by examples. EXAMPLES. 1. Solve y dx — x dy + f(x)dx = o. The last term is exact; its product by any function of x is exact. Therefore any function of x that will make y dx — x dy exact is an integrating factor. Such a factor is obviously 1/x 2 . - ydx-xdy , Ax) dxt=2 or '( x 2 ' ~X~ T 3 +4?*-* gives the solution y_ X +f>=<- 2. Solve y dx -\- log x dx = : x dy. Ans. ex -f y -f- log x -f I = O, 3. Solve (1 + xy)y dx + (1 — xy)x dy — o. (Factor i/x 2 y 2 ). Ans. ex = ye x y. 4. Integrate x a y&(ay dx -f bx dy)= o. Obviously x ka-i-ykb-i j s an integrating factor, where k is any number. On multiplying by the factor we get axka-\ykb dx -\- bxkaykb-\ dy = jd{xka-ykb s ) = o, the solution of which is evident. 5. Integrate xay£{ay dx + bx dy) -f- x*\yt\(a x y dx -f b x x dy) = O. 422 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXVIII. The factors x ka — »— ^ykb— 1-£, x i l a l -i-a 1 ^k l i> 1 -i-^ i make the expressions x*yP(ay dx -f bx dy) and x a tyPi(a } y dx + b x x dy) exact differentials respectively, whatever be the values of the arbitrary numbers k and k v Therefore, if k and k x be determined so as to satisfy ka — i — a = k 1 a 1 — i — a v kb - i - p = k x b x - i - /5 V the factors are identical and these values of k and k x furnish the integrating factor of the equation proposed. 6. Solve (y z — 2yx 2 )dx -f (2xy 2 — x' s )dy — o. Ans. x 2 y 2 (y 2 — x' 2 ) = c. 7. Solve the equation [y + xf(x* + y 2 )]dx = [x - y/(x 2 + y 2 )]dy. (l) This is the differential equation of the group or family of rotations. Put x 1 _|_ y% _ ,,2. Rearranging (i), ydx-x dy -f /(r2) .{xdx + y dy) = o, 2(7 djc — Jf dx - x dy) - (x dy - y dx) + f(r 2 )dr 2 = o, or y*d(j) -*d U) + f K r 2 )dr 2 = o. An integrating factor is obviously ^ 2 . Whence x- y 1 r 2 Integrating, tan-.*-ta„-i£ + /*-*S** = ,. 7 x J r l II. Whenever an integrating factor exists which is a function of x only or ofy only, it can be found. Making use of the fact that e z is always a factor of its derivative: (a). Let 2 be a function of x. In e 2 (Mdx -f iV^) = o, put -flf ' = *W, -AT' = *W. Then __-^_ -j =**A +^ dy 9>/ ox ax dx The condition that dM _dN dy dx or dz = — dx. N d y - Art. 305.] INTEGRATION OF DIFFERENTIAL EQUATIONS. 423 If, therefore, dM _ dN cP{x) N is a function of x only, then z = / »\fJr"Q* + *\. (3) This is the solution of the linear equation (1). 424 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch, XXXVIII. 2. Bernoulli's Equation. — The equation known as Bernoulli's I + * = <*"• in which P, Q are functions of x or are constants, reduces to Leibnitz's linear equa- tion. For, multiply by ( — n -f- i)/y n , and put v = y~ n + l . Equation (i) becomes *L + (I _ n)Pv = (I - n)Q, which is linear in v. 3. Solve f\y) -^ +Pf(}') = Q, where P, Q are functions of x. Put v = f(y)- The equation becomes *£ + Pv = Q > which is linear in v. III. When Mx ± Ny ^ o, //^re #r£ two cases in which the inte- grating factor of M dx -\- N dy = o c#« <$ + *■ (log ^.OgQ, where tt = log (xy), v — log (x/y). Art. 305. J INTEGRATION OF DIFFERENTIAL EQUATIONS. 425 This case is otherwise solved by the substitution y = zx. see §297,11. (2). Divide by Mx - Ny M dx 4- N dy 1 Mx -f Ny Mx-Nv' = ^ Mx^fy °* ^ + ^ l0 ^ (*/»■ If M=y(xy), then 7J/ air 4- A T > + J V'j^)^- — (.r -f .r i / ^> / )'^ / == o. Ans. y = ex. 9. (x 2 + j/ 2 -f- 2*)<&r 4- 2y ^' = O. ^«J. .r 2 4- y 2 = ce- x . 10. (3-r 2 - y 2 )dy = 2xy dx. Ans. x 2 - y 2 - cy\ 11. 2xy dy = (x 2 4- y 2 )dx. Ans. x 2 _ j 2 - ^ x y' s 12. (x 2 / — 2xy 2 )dx — (x 3 — 3x 2 /)^v. ^«i. h log^ = c. y x 13. (3^ 2 J 4 4- 2xy)dx = (.r 2 — 2x 3 y 3 )dy. Ans. x*y* -\- x 2 — ' = {in -f- ;ri')^r. ^//.r. y — mx -|- <; |/i 4- x 2 . ^ ' tan v , . _, .x 3 — -zx -\- c 19. 4-H — = (•* — J ) sec y> Ans. sin i' = ^ ! . dx ^ x 4- I V 3(* + I) 20. - ; ■ — x — y. Ans. y = x — i 4- a?-*. 21. '-'- + J = ^J' 3 - -<4«J. — = x 4- £ + c* 2 *. /v 1' 22. -— = //— 4- e*x*, Ans. y = x n (e x 4- c). dx x 23. *+l=i£, = I . A,.4 = i + «=-. 306. Solution by Differentiation. — A number of equations can be solved, by means of differentiation as equations of the first order and degree. EXAMPLES. 1. Let i> = -f- . Let the differential equation be dx x=f(p). (1) Differentiating with respect to/, dx=f\p)dp. Since dy = pdx, this gives the equation dy=f\P)pdp. • •• y =jf\P)pdpj r c. (2) The elimination of/ between (1) and (2) gives the solution. 2. In like manner, if the differential equation is y=AP)* (*) on differentiation we have dy =/'(/>) dp. ... pdx=f\p)dp, d*=nfdp. ... x= f£fdp+c. w The elimination of/ between (1) and (2) is the general solution of (1). 3. x = jp + log /. ^;w. x + I = ± |/2j 4- r 4- log ( - 1 ± \/2y 4- c). 4. x 2 / 2 = 1 4- p\ Ans. e 2 y 4- 2cxey -\- c 2 = o. 5. ;/ = «/ + ty 2 . /*»j. x± \fa 2 4- 4' - e-*)dx = o. ^'"- ye* = x 4- c 7. cos 2 x = (*" + «■)(•*■ + 0" b sin x 4" cos •* 9. ' 2 - 2a 2 * 2 - 2/, 2 ;- 2 = c. 20. dy = (xY - l)*y dx. Ans. y*{x> + 1 4" «**) = "• 21. 2x7 ^ 4- (j 2 - * 2 ) d y = °- j^_ s - f + x2 = cy ' 11. (x 4- y)dy +{*- y¥* = o- ^ w - lo s ^ 2 + y ' + tan_I ^" = £ ' 23. (xy 4- x 2 / 2 4- *y + J)/ ^ + (^ 3 " • r2 >' 2 ~ ^ + I)x ^ J ' = °* Ans. x 2 ;' 2 — 2jtj/ log cy — I. OA ^. -L ; i- v - - ^ ;/J - xn y - axJ r c - '*' dx ^ x y x n ' 25 ' ^ - * 'dx' \ I+ dx) K l + ** CHAPTER XXXIX. EXAMPLES OF EQUATIONS OF THE FIRST ORDER AND SECOND DEGREE. 307. The equation of the First order and Second degree is a dy quadratic equation in — of the form where A, Bare, in general, functions of xa.ndy. dy We shall represent-^- by p. Equation (1) can be written symboli- (*x cally A*,y,p) = o. (2) 308. There are three general methods which should be made use of in solving (1): (1). Solve for y ; (2). Solve for x ; (3). Solve for p. 309. Equations Solvable fory. — If (2) can be solved for y, the equation becomes y = F(x,p). (1) Differentiate with respect to x. 8F dF dp •*• P = Zx-+WTx ( 2) dp This equation (2) is of the first order in — . The elimination oi p between (1) and the solution of (2) furnishes the solution of (1). The elimination of p is frequently inconvenient or impracticable. When this is the case, the expression of x and^y in terms of the third variable/ is regarded as the solution. EXAMPLES. 1. Solve/ + 2xy = x 2 +y 2 . (i) .-. y = x+ \/p. Differentiating, , J dp 428 Art. 309.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 429 or dx = — — * — . / 2 \ . M - 1 •'• * = £log y - +', /*+ ' , I _L <.2*-2«: or /* = ~—L (3) Eliminating /, we have for the solution k + e 2x 2. Solve x — yp = ap 2 . (!) Differentiate y = , with respect to x, and put the result in the form dx 1 ap dp P(i-p') i-f Solving this linear equation, p x = • (c + a sin-ip). (2) 4/1 -^ Substituting in (1), y — _ a p -f (<: + sin-*/). (3) Vi - P 1 The values of x, y expressed in terms of the third variable/ in (2), (3) furnish the solution of (1). 3. Clairaut's Equation. — The important equation, known as Clairaut's, y=px+f(P), (1) can be solved in this manner. Differentiate with respect to x. ... P=t + x f x+np) % or, [*+/'(/)] %. = «. (2) The equation (2) is satisfied by either dp x + f\P) = 0, or f x = o. The solution of (1) is obtained by eliminating p between either of these equa- tions and (1). dp ~- = o gives p = c, constant. Therefore one solution is y = cx+f(c), (3) which is the family of straight lines with parameter c. The second solution is the result of eliminating p between y =px+f { p), and o = x +f'{p). S ^ The second of these equations is the derivative of the first with respect to/; x and y being regarded as constants,/ as a variable parameter. This result is 43° INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX. clearly the envelope of the family of straight lines representing the first solution (3). This envelope is called the singular solution of(l). Thus the general solution of Clairaut's equation (1) is effected by substituting an arbitrary constant for/ in the equation. The singular solution is the envelope of the family of straight lines representing the general solution. 4. Lagrange's Equation. — To integrate y = xf{p) + i\p\ (1) Differentiating with respect to x and rearranging, dp^ AP)-P ^ AP)-P This is a linear equation in x and can be solved by § 305. II, Ex. I. Eliminating/ between (1) and the solution of (2), the solution of (1) is obtained. Otherwise x and v are obtained in terms of the third variable/. 5. Solve y - (1 -\- p)x + /'-'• dx Differentiating, 1_ ^ — _ 2/. dp Solving this linear equation, x = 2(1 — p) -\- ce~P\ .-. y = 2 -/ 2 + (i +p)ce-P. 6. Solve x 2 (y —px) = yp 2 . dv which is Clairaut's form. .-. v — cu -j- c 2 . Hence y 2 — ex 2 -\- c 1 . 310. Equations Solvable for x. — When this is the case A*> y> p) - ° becomes * = F(y,PY (*) Differentiate with respect to y. i___bF bF dp . ■'■ j-Jt + lp4>' This is of the first order in -f . The elimination of p between (1) dy and the integral of (2), or the expression of x and y in terms of/, furnishes the solution of (1). EXAMPLES. 1. Solve x = = ;'+/ 2 - I dp — = I 4- 2/ 4- » "r / T * dy dy = - . 2 pr,ip . p - 1 y = c — L/ 2 + 2/ + : 2log(/ - I)], x= - c — [2/ + 2l >g(i>- 0]- 2. X = J ' ~f" log/ a . A/is. y = c - a log(/ - - 0. X = C + a log- / / — I 3. Solve p 2 y 4- ^px = y. Ans. J 2 = 2CX + <- 2 Art. 313.] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 431 311. Equations Solvable for />. — The equation /'(.\\ 1, />) = o is a quadratic in />. If this can be solved in a suitable form for integration tor/, it becomes \P- {*>y)\\P -*(x,y)\ = o. Each of the equations p = 0(-v, r) and p = ip(x, y) is of the first order and degree in ~, and their solutions are solu- dx tions of (1). Such solutions have already been discussed. EXAMPLES. 1. Solve p 2 — (x -\- y)p -f- xy = o. (/ -*)(P -J>) = dy gives Then the equation becomes y = x/(p). (2) Differentiate (2) with respect to x and rearrange. . dx = f\p) dp * p-(fP)' EXAMPLES. 1. Solve xp 2 — 2yp -f- ax — °- dns. 2cy = c 2 x 2 -f- #• 2. Solve j = yp 2 + 2/x. ^4«j. y 2 = 20: -j- ^ 2 - 3. x 2 p 2 — 2xj'^> — ly 2 = o. ^//j\ ry = x 3 , xy = c. Orthogonal Trajectories. 313. A curve which cuts a family of curves at a constant angle is called a trajectory of the family. We shall be concerned here only with orthogonal trajectories. If each member of a family of curves 43 2 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX. cuts each member of a second family of curves at right angles, then each family is said to be the orthogonal trajectories of the other family. At any point x, y where two curves cross at right angles, the rela- tion pp' = — i exists between their slopes/, p' . 314. To Find the Orthogonal Trajectories of a given Family of Curves. Let a ) = o ( 1 ) be the equation of a family of curves having for arbitrary parameter a. Let f(x,y,p) = o (2) be the differential equation of the family (1), obtained by the elimina- tion of the parameter a. The differential equation f(*,y, -))=°, or . f[f>y> - -§) =0 > < 3) is the differential equation of a family of curves, each member of which cuts each member of (i)at right angles. Therefore the general integral of (3), ip(x,y, b) = o, (4) is the equation of the family of orthogonal trajectories of (1). EXAMPLES. 1. Find the orthogonal trajectories of the family of parabolze y 2 — /\.ax. Differentiating and eliminating a, the differential equation of the family is dy _ y dx 2x The differential equation of the orthogonal trajectories is dx y dy ~ 2x' The integral of which is x 2 -f- \y % = c 2 , a family of ellipses. 2. Find the orthogonal trajectories of the hyperbolae xy = a 2 . The differential equation is y + xp — o. The differential equation of the orthogonal trajectories is dx y -' Ty= °' giving the hyperbolae x 2 — y 2 — c 2 for trajectories. 3. Find the orthogonal trajectories of y = mx. 4. Show that x 2 -\- y 2 — 2cy =0 is orthogonal to the family Art. 316 ] EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 433 5. Find the orthogonal system of — +<- — 1, in which 6 is the parameter. Aus. x- -f /-' = a 2 log x 2 +c. 6. Find the system of curves cutting x 2 -f Py* — b 2 a~ at right angles, a being the parameter of the family. Ans. yc = x^. The Singular Solution. 314. We have seen in the rase of Clairaut's equation, § 309, Ex. 3, that there may exist a solution of a differential equation which is not included in the general solution. Such a solution, called the singular solution, we now propose to notice more generally. 315. Singular Solution from the General Solution. Let (x,y, c) = o (1) be the general solution of the differential equation f(x,y,p) = o. (2) A solution of the differential equation (2) has been denned to be an equation (1) in x, y such that at any point x, y satisfying the dy equation (1) the x,y, and p = —derived from this relation satisfies (2). The general solution (1) being the integral of (2) satisfies the con- dition for a solution. Also, however, the envelope of the system of curves (1) is a curve such that at any point on it the x, y, p of the envelope is the same as the x,y, p of a point on some one of the sys- tem of curves (1), and must therefore satisfy (2). Consequently the envelope of the family (1) is a solution of (2). This is a singular solution. It is not included in the general solution, and cannot be derived from it by assigning a particular value to the parameter c. We may then find the singular solution of a differential equation (2) by finding the envelope of the family (1) representing the general solution of (2). Thus the singular solution of (2) is contained in tp(x,y) ■= o, which results from the elimination of c between (p(x, y, c) = o and y>P) = ° and fi( x > y> p) = °- dv Since at any x,y satisfying (2) the x,y, — of (2) is the same as dy the x,y, -—-of a point on (1), the equation (2) must contain a solu- dx tion of (1). EXAMPLES. 1. Find the general and singular solutions of p 2 -j- xp = y. This is Clairaut's form, and the general solution can be written immediately by- putting/ = const. However, independently, we have on differentiation dp dp -v- = O gives p — c, and y = ex -j- c 2 for the general solution. Differentiating with respect to c and eliminating c, we find the singular solution 4)' -)- x 2 = o. Integrating the other factor, x -\- ip = o, or eliminating/ between this and the differential equation, the same singular solution is found. 2. Find the general and singular solutions of the equation y = px -j- a 4/1 -\~ p 2 . Am. x 2 -f- y 2 = a 2 . 3. Find the singular solution of x 2 p 2 — 2> X )'P + 2 J>' 2 -)- ^r 3 — o. ^;zj. jr 2 (j' 2 - 4T 3 ) == o. 317. The Discriminant Equation. — The discriminant of a func- tion F[x) is the simplest equation between the coefficients or constants in i^) which expresses the condition that /"has a double root. If F has two equal roots, equal to a, then F(.v) = (x - ay(x), where is some function which does not vanish when x = a. Hence, differentiating and putting x = a, we have the conditions for a double root at a, F(a) = o, F'(a) = o, F'\a) ^ o. Eliminating a between /iV) = o, F\a) = o. or, what is the same thing, eliminating x between F(x) = o, F'(x) = o, we obtain the discriminant relation between the coefficients, the condition that F(x) shall have a double root. 318. ^-discriminant and /-discriminant. Let cp(x, y, c) = o be the general solution of the differential equation y(jt, y, p) = o. Art. 320.J EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 435 (1). The equation t/)(x, y) = o which results from the elimination of c between the equations (p(x, y, c) — o and (fi' c (x, y, c) = o is called the c-discriminant, and expresses the condition that the equa- tion y) = ° which results from the elimination of/ between the equations f(x, y, p) = o and /;(x, y, p) = o is called the /-discriminant. It expresses the condition that the equa- tion/" = o, in/, shall have equal roots. 319. c-discriminant contains Envelope, Node-locus, Cusp- locus. — The c-discriminant is the locus of the ultimate intersections of consecutive curves of the family cf)(x, y, c) = o. It has been previously shown that the envelope of the family is part of this locus, and also that the envelope is tangent to each member of the family. Suppose the curves of the family have a double point, node, or cusp. Then, in case of a node, two neighboring curves of the family Fig. 155. intersect in two points in the neighborhood of the node, which con- verge to the node-locus as the curves converge together. In the neigh- borhood of the envelope two neighboring curves intersect in general in but one point. In the case of a cusp, two neighboring curves intersect, in general, in three points in the neighborhood of the cusp-locus. Two of these points may be imaginary. We may expect to find the envelope occurring once, the node-locus twice, the cusp-locus three times as factors in the c-discriminant 320. /-discriminant contains Envelope, Cusp-Locus, Tac-Locus. ■ — If the curve family f(x,y, p) ~ o has a cusp, then for points along the cusp-locus the equation vanishes for two equal values of/, as it does also for points along the envelope. But, in general, the — of the cusp-locus is not the same as the/ of the curve family and therefore does not satisfy the differential equation. 43 6 INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XXXIX. Again, at a point at which non-consecutive members of the curve family = o are tangent the x, y, p of the point satisfies the equa- tion f = o. The locus of such points is called the lac-locus. The dy — of the tac-locus is not the same as that of the curve family = o, and the tac-locus therefore is not a solution of/" = o. 321. It has been shown by Professor Hill (Proc. Lond. Math. Soc, Vol. XIX, pp. 561) that, in general, the f the envelope once, c-discriminant contains -J the node-locus tw ice, (the cusp-locus three times, C the envelope once, /-discriminant contains 1 the cusp-locus once, (the tac-locus twice, as a factor. This serves to distinguish these loci. Of these, in general, the envelope alone is a solution of the differential equation. It may be that the node- or cusp-locus coincides with the envelope, and thus appears as a singular solution.* The subject is altogether too abstruse for analytical treatment here. EXAMPLES. 1. xp 2 — (x — a) 2 = o has the general solution y + c = i-*- 3 - 2ax K 90' + c ? = 4*(* - 2 a Y- The /-discriminant condition is x(x — a) 2 = o, the ^-discriminant condition is x(x — 3a) 2 =0. x = o occurs once in each, it also satisfies the differential equation and is the singular solution or envelope, x = a occurs twice in the /-discriminant and does not occur in the c dis- therefore the tac-locus. x = 3^ the c- and does not occur in the/dis- : T> a is therefore a node locus. Show that [y -j- c) 2 = x 3 is the general solution of o is a cusp-locus. There cnminant. x = occurs twice in criminant. x = 2. \p 2 =: gx, and x 3 = singular solution. 3. Solve and investigate the discriminants in p 2 + 2xp = y. General solution (2x* -]- 3x1' -f- c) 2 = 4(x 2 -(- j) s . No singular solution. Cusp-locus x 2 -\-y =0. 4. In %ap s = 27J', show that the general solution is ay 2 = (x — cf, singular solution y — o, cusp-locus j 3 = o. 5. Find the general and singular solution of y = xp — p 2 . Ans. y = ex — c 2 , x 2 = 41'. Fig. 156. *Proc. Lond. Math. Soc, Vol. XXII. p. node- and cusp-loci which arc also envelopes.''' 216. Prof. M. J. M. Hill, On Art. 321. J EQUATIONS OF FIRST ORDER AND SECOND DEGREE. 437 EXERCISES. Find the general solutions of the following equations. 1. p 2 = ax 3 . Ans. %${y 4 cf = 4^. 2. p 3 = '/ + 4/ 2 - x 2 = o. Put .r a — 3/ 2 = z^ 2 . Ans. 3(x 2 -j- j 2 ) ± \cx 4 r 2 = o. 16. (x 2 + j 2 )(i +/) 2 - 2(x + y)(i + /)(* +#) + (x +ypf = o. ^«j. x 2 +y 2 — 24* 4 y) 4 ^ 2 = o. 17. x H — = a. ^«f. (y + f, 2 4- (* - a) 2 = 1. ^/i+/ 2 18. y = px -\-p — / 3 . Ans. y = ex 4 c — r 3 . 19. j 2 - 2/A7 — 1 = p 2 (i — x 2 ). Ans. (y - ex) 2 = 1 -\- c 2 . 20. y = 2px 4 y 2 p 3 . Put y 2 = z. Ans. y z = ex -\- ^c 3 . 21. x 2 (y - px) = yp 2 . Ans. y 2 = ex 2 4 c 2 . eh 2 22. {px -y){py 4 *) = /fc 2 /. ^»j. / 2 - or 2 = - — -. 23. y = xp 4 -t/£ 2 4 a 2 / 2 . ^»j. v = ex 4- |/^ 2 + « 2 ' 2 i singular solution x 2 /« 2 -\-y 2 /b 2 = 1. 24. ^ = p(x — b) 4 a//, singular solution, j 2 = 4a(x — b). 25. (^ — xp)(mp — n) — mnp. Ans. (y — ex)(me — n) = nine, singular solution, (x/m)^ ± {y/nf- = 1. 26. y 2 - 2xyp + (1 + x 2 )p 2 = I. Ans. (y - ex) 2 = 1 - e 2 , singular solution, y 2 — x 2 = 1. Ans. y CX' t" X 2 - } ,2 C 2 = I. a? - c 2 438 INTEGRATION FOR MORE THAN ONE VARIABLE. {Ch. XXXIX. 27. / 3 - 4-r;'/ -f 8i< 2 = o. Ans. y = c{x - c)\ singular solution, 2jy = 4^. 28. Find the orthogonal trajectories, A being the variable parameter, of the following curve families: (1). ~ f jz = 1. Ans. x 2 + y 2 = a 2 log x* 4- c. (2). * 2 + ni l y 2 = m*\\ 29. Find the orthogonal trajectories of the circles which pass through two fixed points. Ans. A system of circles. 30. Find the orthogonal trajectories of the parabolse of the «th degree a n-iy — x n . Ans. ny' 1 -f- x 2 = c 2 . 31. Find the orthogonal trajectories of the confocal and coaxial parabolae y 2 = \\{x -j- A). Ans. Self-orthogonal. 32. Find the ortho-trajectories of the ellipses x 2 /a 2 -f- y 2 /b 2 = A 2 . Ans. y b% = ex"*. 33. Show that if is the differential equation of the family of polar curves cp(p, 6, c) = o, then (aw|)=o is the differential equation of the orthogonal system. 34. Find the orthogonal trajectories of/? = a(i — cos 0). Ans. p = c(i -\- cos 6). 35. Also the ortho-trajectories of — (1). p n sin nQ = a n . Ans. p n cos nB = c H . (2). p =z log tan 6 -f- a. Ans. 2/p = sin 2 6 -{- c. CHAPTER XL. EXAMPLES OF EQUATIONS OF THE SECOND ORDER AND FIRST DEGREE. 322. The differential equation of the second order and first degree is an equation in x y y, p, q, A x > y> P> 9) = °> dy dp d 2 y where p = — , q = — = — 2 , and in the equation q occurs only in the first degree. We shall attempt the solution of the equation for only a few of the simplest cases. We have seen that the general solution of the equation of the first order and degree gave rise to a singly infinite number of solutions, represented by a family of curves having a single arbitrary parameter, this parameter being the constant of integration. In like manner, the general solution of the equation of the second order and first degree, involving two successive integrations, requires at each integration the introduction of an arbitrary constant. The general solution, therefore, contains two arbitrary parameters, and is correspondingly represented by a doubly infinite system of curves, or two families, each having its variable parameter. The process by which a differential equation of the second order is derived from its primitive is as follows. Let y< a 9) = o. 439 44° INTEGRATION FOR MORE THAN ONE VARIABLE. [Ch. XL. EXAMPLE. The simplest equation of the second order is d * y - h Here the integrations are immediately effected. <■(£)-** ... £_*,+ * c l being the first constant of integration. Integrating again, the general solution is y = %&x 2 -f- c x x -f- = F ^- The differentials involved are exact, and it is only a question of integrating twice. The solution is . •. y = / dx I F(x)dx -\- c x x -f- c r Ex. q = xe x . Ans. y = (x — 2)e* -\- c x x -f- c y 325. Form/(7, q) — o. — Here PutfU/. Then £=*=**=,* dx dx z dx dx dy dy The equation becomes pdp = F{y)dy. Art. 326.] EQUATIONS OF SECOND ORDER AND FIRST DEGREE. 441 Integrating, dy dx = The integral of this gives the solution. 1. Solve -— = a 2 y. dx 2 EXAMPLES. 2 / I\y)dx = a 2 y 2 . Put c x — a 2 c. dy .-. dx = a ^y 2 + c Hence ax = log(y + ^y 2 -\- c) -f- r 2 . Show that this can be transformed into y = c{e ax -f cj = sin -1 — , 2 c or y = c sin (ax 4- ^ 2 )> Multiply the differential equation by 2p and obtain the first integral directly as in Ex. 1. Examples 1 and 2 are important in Mechanics. 3. Solve q \/ay = I. Ans. 3* = 2a\y^ — 2c 1 )(y* -{- cj* -\- c Y 326. Form/(/>, q) = o. ), or -«" %=*(£)■ « l=^>' ... „/ •n/) ' 44 2 INTEGRATION FOR MORE THAN ONE VARIABLE.. [Ch. XL. This is an equation of the first order, the solution of which is that of the required equation. EXAMPLES. It dy 1. Solve dx 1 Integrating p—*dp -\- adx, we have for the first integral dx ax -J- c .-. y = log (ax + c) + d, or ey = c x x -+- c % . d*y dy — 2. Solve a -±- = -f. Ans. y = c x e a -f- c 2 , dx 2 dx 3. q = p' 1 + I. Ans. e~y = c 2 cos (x -f- f i). 4. <7 + / 2 + i = o. Ans. y — log cos (x — c x ) -\- r 2 . 327. Form f(x, p, q) = o. — Such equations are reduced to the first order in x and/ by the substitution q = -J-. "- A*>P>!) =/(^A ^) = EXAMPLES. d*y dy ^ m (I -\- x 2 ) -^- 2 -{- x •— -\- ax = o is equivalent to ± 4- x p 4- -^- The first integral is The second integration gives ^ = * 2 — «"**■ (2) I. ^tftfAf of the Auxiliary Equation Real and Unequal. — The func- tion (2) vanishes if ?n be one of the roots of the auxiliary equation m 2 -f- Am -j- B = (m — m x )(m — m„) = o. (3) Hence y = e^i^is a solution. Also, jy = ^'"1* is a solution for any arbitrary constant c v In like manner y = c 2 e m * x is a solution. The sum of these two, ^ = ^i* -f- c 2 e M **, (4) is also a solution, and is the general solution of (1) since it contains two independent arbitrary constants, c x and c r II. Roots of the Auxiliary Equation Real and Equal. — If m l = m 2 , the solution (4) fails to give the general solution, since then and c x -f- c 3 = c' is only one arbitrary parameter. The solution in this case is immediately discovered on differentiat- ing (2) with respect to m. For then d 2 xe" tx dxe mx -f A -f Bxe mx = (2m -f- ^Jg"^ -f (w 2 -|- Am + B)xe mx . It m = /a is the double root of (3), then (3) and its derivative vanish when m = ju. Consequently y = .are' 1 -* is a solution, and also is y = £#£'*•*. Hence the sum of the two solutions cV*and £*#"' is the general solution of (1) when /a. is a double root of (3), or ,, = <^( c ' _|_ cx). (5) III. 7?oo/j- of the Auxiliary Equation Imaginary. — When the roots of (3) are imaginary and of the forms m l = a-\- id, m 2 = a — id, 444 INTEGRATION FOR MORE THAN ONE VARIABLE. . [Ch. XL. where i = \/ — i, these roots may be used to find the solution. For (4) becomes y — Cl e {a + ib ^ + c/ a ~ ib)x , = e ax {c x e ibx -f c 2 e~ ibx ). We have by Demoivre's formula e ibx — cos j) X _|_ i s j n i x ^ e -ibx __ cos fa _ 1 s [ n l Xt Therefore the solution is y = e ax { (c 1 + c 2 ) cos bx -f- (^ — c 2 ) z' sin &r}, = e ax (k l cos for -f- k 2 sin for), (6) where ^ = ^ -f- c 2 , k 2 = (c l — c 2 )i. If the arbitrary constants c x and c 2 be assumed conjugate imaginaries, the constants k and k 2 are real. By writing tan a = £j/£ 2 , or cot/? = ^/^ 2 , the solution (6) may be written respectively y — c'e ax sin (for -f- a), = c"*"* cos(for - /?). (7) EXAMPLES. 1. Solve q — p = 2y. The auxiliary equation is m 2 — m — 2 = (m -j- l)(^ — 2) = O. The general solution is therefore jr = c x e~ x -j- ^ 2 *. 2. If ? - 2/ 4-/ = 0, (m— I) 2 = o, .-. y = e*(c x + ^). 3. Solve q -\-2>p = SW- m i _j_ yn — 54 = (m — 6)(m -f 9). .-. y = c l e 6x -j- c 2 e— 9 X . 4. Solve q -j- 8/ -f 257 = O. /w 2 -f- 8/« -f- 25 = o gives m = — 4 ± 3 |/— 1. . •. j — ^— 4^(^ cos 3 jt -4" <£ 2 sin 3^). 330. Solution of the Equation y4, i? being constants. Put jc — £ z , then z = log #. Also, r/r ^dfe _ 1 ' e — y2ev -f- c 2 -j- c x H — ■ or 2ef = c 2 sec 2 ^* -f- c 2 ), according as the first constant of integration is + c 2 , o, or -c 2 . 4. xq -\-p = O. Ans. y = c x log x -f c r e % dx -j- c r 6. x 2 q — 2y. Put z = 2y/x 2 . .'. xy = qx 3 -f- c v 7. q + 127 = Jp. Ans. y = c x es* + c 2 e**. 8. 3(q + y) ' — 10/. Ans. y = c x e& + c^ x . 9. q + \p - y. Ans. ye** =C x e* VI + c 2 e-x Vs . ax bx 10. ab(y -f q) = (a 2 -f b 2 )p. Ans. y = c x e b + c 2 e*. 11. ^L = if. Ans. y = c x &* + c+-** + c y dx s dx 446 INTEGRATION FOR MORE THAN ONE VARIABLE. [Cir. XL. 12. q — 6p -J- i$y = o. Ans. y — [c x sin 2x -\- c 2 cos 2xyv c . 13. q — lap -)- b 2 y = o. Ans. According as a > or < b, y — cax(c x gxV a *-i>* _|_ c 2 e- x ^*' 1 -P i ), or ^(q sin .r \/b' 2 — a 2 + x )- cPy 21. .r -7-3- = 2. ^4«j. 7 = c x -f- f 2 jf -)- c z x 2 -|- x 2 log .r. 22. -^ — sin 3 .r. A;is. y = c x -\- c 2 x -j- c % x 2 -f- |cos .r — ^cos 3 x. 23. £V 3 = a. Ans. (c x x + c,) 2 = ^j 2 - a. JC X 24. a 2 ^ 2 = 1 -f- P"' Ans. 2y/a = c x e<* -(- ^-'^ rt -j- r 2 . 25. «V 2 = ( J +/ 2 ) 3 . >*»*• (•* + 1 -f- f 7T. Whatever assigned value x may have, we can always assign an integer j.i corresponding to an arbitrarily chosen integer ??i, for which - i < a m x - /* ^ + -J, Put jf ;w+i = tf' w Jtr — yu, and let ... x >- x= _l±3n±l } x "-x-= 1 ~ Xm+l , a" 1 a m ' and .#' < .r < #". The integer m can be chosen so great that x' and x" shall differ from .v by as small a number as we choose. We have 00 f{x') — J\x) V~^ cos {a n 7tx') — cos (tf M 7r.r) ;// — 1 (ai)' COS (i7 M ^'^ / ) — COS (a n 7tx) a n {x' — x) cos {a m+n 7rx') - cos (« W+M 7r^) \ bm+n COS jfl 7TX J - COS (fl- ■ -g^ n=o * Taken from Harkness and Morley, Theory of Functions. 45 45 2 APPENDIX. Since cos ( a" 7t x) — cos (a n 7tx) . x -\- x \ \ 2 I a\x' - x) \ 2 I x' - x 2 and since^the absolute value of the last factor on the right is less than 1, then the absolute value of the first part of (i) is less than in — 1 o and therefore less than — = — — , if ab > 1. ab — 1 Also, since a is an odd integer, cos (a m+n 7tx') — cos [a M (ju — i)n~\ — — ( — i)* 4 , cos (a w+n 7tx) = cos (a"jj7r -j- ^Wi^) = ( — i) M cos («"*,+,»). Therefore CO E cos (fl'-'^o') — COS (a'" +w ^Jt') .%- — j; = ( _ i^oty-V* ;+^Kv. t ), All the terms under the 2 on the right are positive, and the first is not less than §, since cos (x m+1 7t) is not negative and 1 -j- x m+J lies between 4- and j. Consequently /(*') -/(X) _ . ., „ m z(2 , 7T1J (- D-H-* g+a^), (n) Jt" — „V where £ is an absolute number > 1, and 1} lies between — 1 and -f- 1. In like manner where £,' is a positive number > 1, and rf' lies between — 1 and -j- 1. U ab be so chosen as to make ab > 1 + -|tt, 2 7T that is, - > -7 , 3 ab - 1 the two difference-quotients have always opposite signs, and both are infinitely great when m increases without limit. Hence _/"(.\) has neither a determinite finite nor determinate infinite derivative. SUPPLEMENTARY NOTES. 453 Every point on such a line, if line it could be called, is a singular point. Some idea of the character of the geometrical assemblage of points representing such a function can be obtained by selecting two particular fixed points A, B of the as- semblage. Between A and />, in progressive order, select points P v P.,. . . . representing the function corresponding to x v j„, . . . Consider the polygonal line AI\ P., . . . B. Increase the number of interpolated points in- definitely, and at the same time let the dif- ference between each consecutive pair con- verge to o. Then, since the function J\x) is continuous, each side, P,P, „ of the broken line converges to o. But, instead of each angle between consecutive pairs of sides of this polygonal line converging to two right angles, it, as their lengths diminish indefi- nitely, as was the case when we defined a curve with definite direction at each point; let now these angles converge alternately to o and in. The polygonal line folds up in a zigzag. The point P converging to the neighborhood of a true curve AB. But the difference-quotient at any. point of the zigzag assemblage has no limit, it becomes wholly indeterminate as the two values of the variable converge together. It is also possible that the length representing the sum of the sides of the polygonal between any two points of the assemblage at a finite distance apart (however small) is infinite in the limit. Such functions are but little understood and have been but little studied. It is possible that they may have in the future far-reaching importance in the study of molecular physics, wherein it becomes necessary to study vibrations of great velocity and small oscillation. NOTE 2. Supplementary to § 42. Geometrical Picture of a Function of a Function. 00*; can represent the function z If z =f(y), where y geometrically as follows: Draw through any fixed point in space three straight lines Ox. Oy, Oz mutually at right angles, so that Ox, Oy are horizontal and Oz is vertical. These lines fix three planes at right angles to each other. xOy is horizontal, xOz and yOz are vertical. The relation^ = (x)\; (P'"Q'"), *=Aj>)- The derivative D x y is represented by the slope of P'Q' at P' to Ox. The derivative D y z is represented by the slope of the tangent to P'"Q'" at P'" to Oy; the derivative D x z by the slope to the axis Ox of the tangent at P" to P"Q" . The function of a function is represented by a curve in space. NOTE 3. Supplementary to § 56. The nth Derivative of the Quotient of Two Functions. Let y = u/v. Then this product, we have vy. Applying Leibnitz's formula to u = vv u' = I ? -V V' u" 2\ — v' ~2 y + v' y> 1! 1! u" V n 1 v n-i ?i\ n\ {71 — 1)! 1 ! + v n ~ 2 y" (;/ — 2)! 2T . . . +v y To find y n , the wth derivative of u/v, in terms of the derivatives of from the 11 -J- 1 equa- te u and v. Eliminate y, — tions. We get ' (« - 1): U V o u' v' — — V o 1! 1! u" v" v'_ iT 2T 1 ! {?i -\- 1) rows SUPPLEMENTARY NOTES. Also, in particular, if u = i, we have 455 l -I)' V v o o . V IT v o . • • 3- V" 7'' ~2\ l\ v . • • n rows NOTE 4. Supplementary to § 56. To Find an Expression for the «th Derivative of a Function of a Function. Lets =y"(r), where j' = 0(-v). To find the «th derivative of z with respect to x. We have, by actual differentiation, A =fi?*, A" =f y "y* s -if,wi +/;/;'■ The law of formation of these first three derivatives of /"with respect to x shows that the wth derivative must be of the form A = AA + AA' + ■ ■ ■ + 4JS, (1) where the coefficients, A r , contain only derivatives of >' with respect to x and are therefore independent of the form of the function/". Conse- quently, if we determine A r for any particular function f we have determined the coefficients whatever be the function f Let then r\ Then in (1) we have - D%y - by = (y - i)~ A + y C— 0' Hence, when b = j>, we have 1! A r _ x + A r . (2) ^r = -A D&y - b) A which means that (y — 3) r is to be differentiated n times with respect to x and in the result y substituted for b. 45 6 APPENDIX. This gives the nth derivative of y with respect to x in terms of the derivatives of/" with respect to y and those of y with respect to x, and is the generalization of the formula dx JKJ) dy dx We can give another form to (3), as follows. L,ety = b when x — a. Then y — b = 0(jv) - 0(a) = (* - a)z>, (4) where v stands for the difference-quotient

) r = D n x {pc - ayv, But, Dt{x - a) r rr r(r -°i) . . . (r -/ + i)(* - a)-* = o when p > r, = o when p < r and .v = a. = r! when p = r and # = a. Therefore (3) becomes d\ n „, . Y^ dx, )A,)=jc,,/,,„(a"(*a^);.. « Notes 3 and 4 give some idea of the complicated forms which the higher derivatives of functions assume. NOTE 5. § 64. Footnote. If a function /"(.r) and its derivatives are continuous for all values of a- in (nr, ft) except for a particular value a of x at which J\a) = 00 , then all the derivatives o{/(x) are infinite at a. Let ATj < x 2 < a. Then /(■g-/(-v,) = (.v 2 -.v I )/'(5), where B, lies between ^ 2 and x v Let a — x x be a small but finite number, and let x 2 ( = )a. Theny^.v.,) is infinite, andy^v,) is finite. .-. (*-*,}/"(£) = 00. Since a — x y is finite, f\£>) = °o ; and since/"'(c?) is finite if a — £ is finite, we must have a — £( = )o and /» = CC . SUPPLEMENTARY NOTES. 457 In like manner we show that /"(a) — co , and so on. Corollary. If /[a) — co , then / '(a) = co , and also ./i.v, becomes co when x = a. For, considering absolute values, if f \a) — co , then also l°gf( a ) — °° • By tne theorem established above, if \og/[x) is co when x = a, then also becomes co when x = a. NOTE G. Supplementary to Chapter VI On the Expansion of Functions by Taylor's Series. 1. This subject cannot be satisfactorily treated except by the Theory of Functions of a Complex Variable. The present note is an effort to present in an elementary manner by the methods of the Differential Calculus a fundamental theorem regarding the elementary functions. An elementary function may be defined to be one which does not become o or co an infinite number of times in any finite interval, however small. Such functions are also called rational. A function _/*(.%■) is said to be unlimitedly differentiable at x when all the derivatives f r {x) of finite order are finite and determinate at x, We consider only those functions which are such that neither the function nor any of its derivatives become o or oo an unlimited number of times in the neighborhood of any value x considered. 2. In the same way (hat a function of the real variable x may be o for an imaginary number p -f- iq t such a function may be oo for a complex number p -\- iq, where i = V—i. For example, the func- tion * (X) ~ llf /4/w 5. It follows, therefore, that if / Remembering that the modulus or absolute value of any number x -}- iy is l / 2 4- Vi that c \p + iq -x\ then of two numbers / x -(- ^ and/ 2 -f- ^2 that one is nearest jt for which we have the difference least. SUPPLEMENTARY X* >! 1 5. 459 has a finite limit different from o, it is necessary that £-■ \ = V " + 1 /"(•>) 6. Since at all points of absolute convergence of S and at all points of infinite divergence of S I the boundary between absolute convergence and infinite divergence of 6" is marked by the values of x, y which satisfy £t\™ = *• 7. The locus £/"(*) = x, for an arbitrary and great value of n, will be a close approximation to the boundary line we seek. Differentiating, this locus has the differ- ential equation dy _ y f n +\x) dx n f\ X) --(■ ] +; 1 y f n ^{x\ w tiich for n arbitrarily great gives , in the limit > dy dx ' 1*. in virtue of r y f H+ \x) It Jjn+i f\x) ' ' on the boundary. 8. Therefore the absolute value o(y is equal to the absolute value of a linear function of x, of the form y 1 = J k — x 1 2 , for all values of x andj^on the boundary. This is the equation of the family of boundary lines having the parameter k. These lines are fixed by the fact that whenever /"(.v) or f r (x), for any finite r, is 00 we have^ — o. 9. If, therefore, f{x) — co when x — p, the corresponding bound- ary lines for a real pole p are the two straight lines f = (P ~ *)\ or y — x — p and y = — x -\-p. 460 APPENDIX. If /(-v) = co when x = p -j- *<7, then the corresponding boundary lines for a complex pole/ -f- ? !7 are tne two branches of the rectangu- lar hyperbola f = \P + *g - x\ 2 , or (-v - />)* +>=*• having for asymptotes jr = x — p and j' = — x -f- />. 10. Therefore for any function having real and complex poles the boundary lines consist of pairs of straight lines crossing Ox at 45 at the real poles and of right hyperbolae having as asymptotes similar straight lines crossing Ox at the real part of the complex pole. The vertices of the hyperbola corresponding to the pole/ -[- iq are/, ± q. 11. The region of absolute convergence of 6' is that portion of the plane (shaded) such that from any point in it a perpendicular can be drawn to Ox without crossing a boundary line. The nearest boundary lines to Ox make up the boundary of the region of converg- ence of S. It consists of straight lines and hyperbolic arcs. Fig. 159. The boundary line of the region is symmetrical with respect to Ox. The ordinate at any point of this boundary line of converg- ence is the radius of convergence for the corresponding abscissa, and is equal to the distance of its foot from the nearest pole point. For any point on the boundary / n _^ 1 f n {x) ' ' is less than 1 for any point inside, and greater than outside, the region. for any point SUPPLEMENTARY NOTES. 461 12. If a function has two real poles », /)', and no pole between at, ft, the region of absolute convergence consists of a square between a and (3. If between a and (j their is an imaginary pole p -J- /r/ such that/ lies between ^ and /j, the imaginary pole has no inllu- ence on the region of convergence if [>--K« + /S)] 2 + ? 2 > \{a-P)\ If, however, the hyperbola^' 2 = (p — -*) 2 -{- q 2 cuts off a portion of the square of convergence. 13. Theorem II. If f(x) is a one-valued, determinate, unlim- itedly differentiate function (having only a finite number of roots or poles in any finite interval), then Zi o for all values of x and y for which the series is absolutely convergent. That is, for all values oiy less in absolute value than the radius where/ -j- iq is the nearest pole of f(x) to x. Equation (1) is not true for any value oiy such that \y\ >R. Proof: The construction of the region of absolute convergence shows that from any point P in this region can be drawn two straight lines making angles of 45 with Ox to meet Ox without crossing or touching the boundary of absolute convergence. At any point x, y in the region of absolute convergence the series 00 -3 /-+«(*) is absolutely convergent. But _dS _dS ~ dx ~~ dy ' Hence = £(* + *> = (dx + dy), = o, if x -\-y is constant. Therefore all along the line x -\- y — c, in the region of absolute convergence, £ must be cons/an/. This line passing through any 462 APPENDIX. point P in this region meets Ox without touching the boundary. At the point where x -\-y = c meets Ox we have_y = o, x = c, and S =/(c) =/(.v +.,■). Consequently all along any such line passing through the region of absolute convergence, and therefore at any point whatever in this region, we have o 14. What is the same thing, 00 o for all values of xa.ndjy which make the series absolutely convergent.* If we make the investigation in the form (1), the regions consist of parallelograms on the line v = x as diagonal, and having for sides the straight lines X =p, X = 2V — p, corresponding to a real pole/, and hyperbolae or x 1 — 2xy -f 2pv = p l -f- f, corresponding to a complex pole/ -J- * at the zeros oiffx). There the curve has vertical asymptotes, but closes up on the asymptote as n increases. Also, f n {x) cannot have the same zero point for an indefinite number of consecutive integers n unless the function is a polynomial. Again, if at any assigned point x the derivatives are alternately o, the radius of convergence is fixed by / '+i)j^7 } \ = \ Rl ' since for absolute convergence we must have ' / '•{" + /"(•«) < 1. * It being understood that y is at a finite distance from any value of the variable it which the function is 00 . SUPPLEMENTARY NOTES. 463 This simply means that there are two poles that arc equidistant from the value x. If the poles of a function are all real, if is im- possible for more than alternate derivatives to be zero continually. If there are more than two poles equidistant from x, then at least one must be complex. If there be three equidistant poles from x, then one must be real and two imaginary, /> ± iq, and conjugate. Then the derivatives at .v are o alternately in pairs and the radius of convergence there is R*\ = \£n(n +!)(»+ 2)^L- y and so on. Points x, equally distant from several poles, are the singular points on the boundary. Elsewhere, for three poles, we can always write f"( v) f n (x\ f n+J (x\ /~ M+2 ( y) -C+xX-+»)/^ - nf±±. ( . + l) ^.(. +8 )£_U , the limit of which is A 3 , and converges to the value R % at the singular point as a- converges to the x of such a singular point. The generaliza- tion of this is obvious. EXAMPLES. 1. The region of absolute convergence and of equivalence of the Taylor's series of the functions tan x, cot x, sec x, esc x, consists of the squares whose diagonals are the intervals between the roots of sin x, cos x, respectively. 2. In particular tan x is equivalent to its Maclaurin's series for all values of x in ) -\n. + \7t{._ Also for sec x in the same interval. cot x, esc x are equal to their Taylor's series in the interval )o, it{ , the base of the expansion being \%. x 3. Expand by Maclaurin's series. v e* — I J Put y equal to the function. Then ye x — y — x. Apply Leibnitz's formula, and put x = o in the result. We have for deter- mining the derivatives of y at o, Making n = I, 2, 3, . . . , we find these derivatives in succession, and there- fore x x . B, „ B 9 , , B % = i-l + ^L X * -^L x *+^x* -5 ' -7 I A I 'fit e* — I 2 2! 4 wherein B x = *, B 2 = fa B z = fa B x = fa B, = fa . are called Bernoulli's numbers. They are of importance in connection with the expansion of a number of functions. Since e ±in = — 1, the poles ± ire are the nearest values of x to o at which the function becomes 00 . The series is therefore convergent and equal to the function for x hi ) — 7t, -\- 7t. 4^4 APPENDIX. 4. Show that forx in ) — In, + £#(, ?+! =3-- ttC* " I) + ~4V ( 24 - x > - iy ( 26 - ■>+ • either directly or from x x 7.X C x + I _ = a /y(n + l) (* + *) jk^t |r= i^' („2 _ w -2) _> _ / (« -f I) -f . (« -J- 2) ^ r I = I R 2 . we have (1 - x 2 ) - 2x R — R 2 = o, or R I = 1 x ± I. Therefore if jr is the base of a Taylor's series for y, the function is equal to the series in )x — 1, x -J- i(. If jr = o. the Maclaurin's series is equal to the function in )- 1. + i(. When x = o, the differential equation gives y(n+2) — („2 _ m 2 )y(g\ which gives the coefficients in the series. 7. Treat in the same way cos {jh sin— *x). 8. For what values of x is the Maclaurin's series corresponding to the function y in (1 - x 2 )y" - xy' — a 2 y = o equal to the function? W< >rk as in 6. The function is e a sin -1 *. 9. In general, any function y satisfying a differential equation (I _j_ ax 2 )y("+^ -\-px(n + b)yV l + *) + q{n — c)(n — d)yi») = o, where , k being an arbitrary constant. But if/ is a pole of/js), then 2 = o when £ =p. Hence o = k — £ 7/3 />, or /£ = e^p. Therefore, corresponding to any assigned £, the boundary corre- sponding to the pole/ is fixed by which is a circle about the origin in the z-plane with radius £=\P-Z\, since /? is arbitrary. 3. lip is the nearest pole oif(z) to C, then for all values of z for which |»|<^ = |/-C| the series is absolutely convergent, and is infinite for any value of z \i\z\>R. 4. Put £ = z + C. Then z = B, - Q. The series o is absolutely convergent at all points £, inside the circle Cdescribed about x C as a center with radius R=\Z-P\, FlG - l6 °- p being the nearest pole of f{z) to C. For any assigned value of B, in this circle the series S is con- stant with respect to C, since r ! « I and this is o when ?i = 00 . Now we can always move C np to (r) A and a being positive constants, a > 1. This function is one- valued, finite, continuous, and unlimitedly differentiate for all finite values of the real variable x. It has, however, infinitely many com- plex poles ±—#r-> r=i, 2, 3, ... an infinite number of which are in the neighborhood of x = o, which is therefore an essentially singular point. For the nth. derivative oif(x) we find (i = -f- V ' — 1) o At x = o, f(o) = e-\ f am + *(o) = o, / 2 "'(o) = (— i)" ! (2m)\ e~^ 2m . Therefore the Maclaurin's series is 00 X - ^ 7 x 2r ■ s =2 ( - ir ^' (2) o This series is absolutely convergent for all finite real values of x. *This problem was first solved by Cauchy, by means of singular integrals. See any text on the theory of functions of a complex variable. 4 68 APPENDIX. Now let A < It |.*| < • •• A*)> I -f- X 2 I -J- a 2 *'" and 6" < *-*. In particular, let x = a - *. > i -f~ -*' 2 i + # 2 -* 2 Fig. i6i. /(*"*) > > i -(- i/fl i -\- a e~ l > .S when x = a~± y when — ■ — > e -i or a > . a -\- i = ■> = e — i The function /*(*) and the series £ are different. In the figure the solid line is the curve y =f(x), the dotted line the curve -a; y — S, constructed with exaggerated ordinates, for the values X = log 2, a = 2.* NOTE 9. Supplementary to § 118. Riemann's Existence Theorem. Any function f{x) that is one-valued and continuous throughout an interval (a, b) is integrable for that interval. Let the numbers x xt x 2 , . . . , x n _ 1 be interpolated in the inter- val {a, b) taken in order from a = x Q to b =. x n . We have to prove that the sum of the elements S H = 2/{Zr)(Xr ~ *r-i) (I) converges to a unique determinate limit, when each subinterval con- verges to zero, whatever be the manner in which the numbers x r are interpolated in (a, b). I. The sum S n must remain finite for all values of n. For f(x) is finite, and if M and m are the greatest and least values oif(x) in m{b — a) < £„ < Af(£ — a). Also, since /*(*) is continuous, there exists a value B, in (a, <$) at which S. = (* - «)/(*), (2) /"( (3) where B,' r is some number in the subinterval [x Ti x r _ 1 ). Form similar sums of elements for each of the n subintervals of (iV Let/ = n[ -f- . . . -f- »*. Add the « sums of elements such as (3). Hence s f = s„ i +... + s„, n , I = i (x r - *,-,)/(«). (4) I This is a new element sum containing/ > « elements, which is to be regarded as a continuation of (1) by the interpolation of new numbers in each subinterval of (1). Subtracting (4) from (1), we have s. - s, = i[/&o -/<«)](*, - -v,-,)- Let 6 be the greatest absolute value of the difference between the greatest and least values of f{x) in the subinterval (x r — x r _ 1 ), r — 1, . . . , n. Then, since f{z r ) and f{£' r ) are values of f(x) in {x r , x r _ I ) ) S n -.S,\<\62(x r -x„), I x i > • • • > X m—\ i occurring in (1) and (6), thus dividing (a, b) into m -f- n intervals. 47° APPENDIX. Interpolate in each of these m -\- n intervals new numbers, thus dividing (#, b) into m -f- n -\- p subintervals. Form the element- sum S m + n + p corresponding to these subintervals. Then, by II, S n and S m + n + * converge to the same limit. In like manner S m and S m + M + P converge to the same limit. Therefore S n and S m converge to a common limit. The uniqueness of the limit of (i) under any subdivision whatever of (a, b) is demonstrated. This theorem gives the means of denning analytically the area and length of a curve, and the volume and surface area of a solid. NOTE 10. Supplementary to § 135. Formulae for the Reduction of Binomial Differentials of the form x a (a 4. bx n ) y dx. Put y — a + bx n . Then Dxyv = ax a -y y -f- nybxf^- M - i yr- 1 i — aax a -i\n-T- -j- (a a -j- ny)bx aJ r^- x y*- x > (1) = (a -f- ny)x a —[yy — anyx a —]yy— 1 . (2) In (1), put a = m— n+i, y=p + i, then Bx m - n+i y +l =a(m — n+ i)x m - n y -f- (np + m + i)bx m yt (A) In (2), put a = m-\-i, y = p, then Dx m+ y = (np + m+i )x m y p — anpx m y*-\ (B) In (1), put or = wz -I- 1, y=^-fi, then D x ™+yfi+i — a{m + i)* w y -f {np -f- w + « + i)^ w+ V- (C) In (2), put « = w -f 1, y=p+i, then jr^w+yn = ( H p + m + n + i)x" ! yt +l — an(p + i)x m yK (D) Integrating the formulae (A), . . . , (D), we have the formulae of reduction, where y = a -{- bx n : ^=t£^ (a) f xmyPdx = * W ^ +I _ (»/ + » + »+ i)* />y& (C) J *y ax (m+i)a (m + i)a J K > ^'W* = h , , — r — / x m yt +1 dx. (D) SUPPLEMENTARY NOTES. 47 NOTE 11. Supplementary to § 165. H> =/(- r ) be represented by a curve, and y, By, B\y are uniform and continuous, then we can always take two points P and P x on the curve so near together that the curve lies wholly between the chord and the tangents at P and P y Let x, y be the coordinates ofP, and X, J^those of P', any point on the curve between P and P v The tangent at P has for its equation r t =/(.v) + (A' - x)/\x). At any point x, y of ordinary posi- tion, not an inflexion, the difference between the ordinate to the curve and the tangent is AX)-r l = {X - x) ' i /'\S), (1) where £, is some number between x and A^. We can always take X so near to x that /"{£) keeps its sign the same as that oif"\x) for all values of B, in (x, X). Therefore the difference (1) keeps its sign unchanged in (x, X) or the curve is on one side of the tangent, for this interval. The equation to the chord PP 1 is r. =/0) + (A - x)/'{s x ), where/X^i) is the slope of the chord PP V The difference between the ordinates of the curve and chord is /(A) - ¥ c = (A - x)[f\8) -/'(£,)]. (2) Let x l be so near x that/"^^), f'(£i^ have the same sign as/\x). Then this difference (2) keeps its sign unchanged for all values of A in (x, x x ). It can now be easily shown that (2) and (1) have opposite signs, and there can always be assigned a numbers* so near x that the curve PP X lies wholly in the triangle formed by the tangents at P, P x and the chord PP V NOTE 12. Supplementary to § 226, IV. Proof of the Properties of Newton^ Analytical Polygon. I. Let there be any polynomial in x and y, such as / = w + ?< + A,„x*'"f (0 wherein the exponents a, ft of each term satisfy the linear relation a a + bfi — c, (2) c being taken a positive number. 47 2 APPENDIX. Let f be arranged according to ascending powers of y, so that fi x < ft % < . . . Then = x**[a i + A m (yx~y + . . J, (3) r^V'U*"'-- V • • • G'*~"- fa-ftM- (4) where k x , . . . , kp _p x are the roots of the equation in / = yx a , A i + A /*-K' + . . . +Aj--* ==o. Therefore the locus of f = o consists of x = o, y = o, and the parabolic curves r = K x >. (r = i, . . . . , § m - /»,). b_ 2. In (3), let_y = &* a , /£ being constant. Then b b f=A 1 x ai Z? i x 8l « + A 2 x a *A fi *x fi >«-{- . . . , = Ajf l x a * +fii 7+AJl! it x at+fi *T+ . . . , = x T {a i #*+A/*+ . . .) = ^T.r a , A' being constant. 3. Let /' be a function A'xr-'yfi' , or the sum of a finite number of such functions, such that the exponents a', ff of each term satisfy the linear equation aa' + bfi' = <;'. _*_ Then, as in 2, \ety = Ar a , and we have in the same way c' f = K / x a i K' being a constant. 4. Let a, b and c, c' be positive numbers. Then /' - K 'x^ f ~ K > where je andj> satisfy y 1 = kx 5 . (1). If c' > c, then //' — — = o, when .r( = )o, j/( = )o. SUPPLEMENTARY NOTES. 473 £>. (2). If c' < c, then /' 1, ~7T = o, when x = 00 , _^ = cc . 5. We are now prepared to prove § 226, IV, (1), (2). Let *\x,y) = 2C r x>jft = o. (1). Let/ represent that part of /"which corresponds to a side of the polygon as prescribed in § 226, IV, (1), and F' represent the remainder of F. Then F=/+F', F F' 7=1 + 7* Through each point corresponding to terms in F' draw a line parallel to the side corresponding to/* Then by 3, (1), we have /F . FF' ? =I ' smce iz =0 * when x{ — ) o, y( = )o. Therefore in the neighborhood of the origin F = o and /= o are the same. But the form of/" = o in the neighborhood of the origin is that of a parabola y* = kx*. Hence F = o goes through the origin in the same way as does f = o, whose form is that of a parabola of type y* = &e*. (2). Let /"represent that part of F corresponding to a side of the polygon as prescribed in § 226, IV, (2), and F' the remainder of F. F F' Then = T + Draw parallels to the side corresponding to/*, through all points corresponding to terms in F' '. Then by 3, (2), we have /F . FF' 7 =I ' since i/ =o ' when x = 00 , y = 00 . Therefore F = o and/*= o pass off to 00 in the same way. Also, /" = o passes off to 00 , as does a parabola of type,/* = kx b . Note. — The same process can be extended to surfaces, using a polyhedron in space. The part of the equation corresponding to a plane face such that there are no points between that face and the origin gives the form of a sheet of the surface at the origin. Likewise the part corresponding to a plane face such that no point lies on the side opposite to the origin gives the form of a sheet at 00 . The plane faces in each case cutting the positive parts of the axes. INDEX. [The numbers refer to the pages, ,] Absolute number, 2 Anticlastic surface, 360 Appendix, 451 Archimedes, spiral of, 117, 161 area of spiral, 234 length of spiral, 248 Areas of Plane Curves, rectangular coordinates, 226, 396 polar coordinates, 233, 397 Asymptotes, rectilinear, 121 to polar curves, 125 Auxiliary equation, 443 Axes, of a conic, 323, 325 of a central plane section of a coni- coid, 328 Base of Expansion, 88 Bernoulli. definite integral by series, 222 differential equation, 424 Binomial differentials, 193, 470 Binomial formula, 67 Binormal, 378, 379 Bonnet, 131 Boundary of region of convergence, 460 Cantor, definition of number, 5 Cardioid, 118, 163 area, 234; length, 248 surface of revolute, 261 volume of revolute, 401 orthogonal trajectory of, 438 Catenary. normal-length, 1 16 radius of curvature, 134 center of curvature. 146 curve traced, 152 area, 228 ; length, 245 volume of revolute, 258 surface of revolute, 261 differential equation, 446 Cauchy, theorem of mean value, 79, 87, 222 theorem on undetermined forms, 93 on expansion of functions, 467 Caustic by reflexion, 390 Circle, area, 227, 234 length of perimeter, 246 Circle of curvature, for plane curves, 100, 134 for space curves, 379 Cissoid, tangent and normal, 115 subtangent, 116 curve traced, 151 area, 229 Clairaut's equation, 429 Coordinates of center of curvature, 133 Computation of, e, 84; logarithms, 86; it, 89 Concavity and Convexity, 127 Concavo-convex, 128 Conchoid of Nicomedes, 160 Concomitant, 312 Convexo-concave, 128 475 476 INDEX. Cone, volume of, 257, 266 equation of, 349 Conic, center of, 331 Conicoid of curvature, 360 Conjugate points. 334 Connectivity, law of, 19 Conoid, volume of, 266 Consecutive numbers, 7 Constant, 4 Contact, of a curve and straight line, 127 of two curves, 130 Continuity, theorem of, 23 of functions, 278 Continuum, 3 Convergency quotient, 14 Cubical parabola, 135, 150 Curvature, 130 radius of, 133 circle of, 134 of surfaces, 365 measure of, 370 spherical, 381 Curve tracing, 147, 340 Curves in space, 375 Cusp, 151, 155, 336, 337 Cusp-conjugate point, 335 Cusp-locus, 435 Cycloid, tangent to, 114 curve traced, 163 area, 232 length, 252 surface of revolute, 261, 263 volume of revolute, 262, 263 Cylinder, equation of, 348 Decreasing function, 74 Definite integration, 215 Degenerate forms of differential equa- tion of second order, 440 Degree of differential equation, 409 Descartes, 26 Develop ible surfaces, 374 Devil, 161 Difference, of the variable, 35 of the function, 35 quotient, 36 Differential, 55, 63 quotient, 55, 64 coefficient, 55 relation to differences, 57 total, 294 Differentiation of, logarithm, 41 power, 42 sum, 43 product, 44, 69 quotient, 45, 454 inverse function, 46 trigonometrical functions, 44, 45, 46 circular functions, 48 exponentials, 49 function of a function, 49, 70, 453, 455 implicit function, 66, 296 function of independent variables, 282, 306 under the integral sign, 391 Differential equations, first order, 409 second order, 439 Discriminant equations, 434 Double points, 334 Double integration, 396 Dumb-bell, 160 Edge of envelope, 389 Elements of curve at point, 147 Elliott's theorem, 236 Ellipse, tangent and normal, 113 subnormal, 1 15 radius of curvature, 125 evolute, 144, 154 area, 228, 233 arc length, 246 length of evolute, 246, 251 normal to, 331 orthogonal trajectory, 433 [NDEX. 477 Ellipsoid, volume of, 268, 399 Elliptic functions, 208 Elliptic paraboloid, 268, 280 Envelopes, of curves, 138, 343, 435 of surfaces, 385 Epicycloid, 164, 248 Equiangular spiral, 118, 162, 234 Equilateral hyperbola, evolute of, 146 area of sector, 228 Euler's theorem on curvature, 366 Eulerian integral, 217 Evolute of a curve, 144 length of, 250 Exact differential, 416 Exponential curve, 150 Family of curves, 138 Finite difference-quotient /(*) -A*) x — a «th derivative of, 138 Folium of Descartes, tangent, 1 13; asymptote, 122 traced, 159; area, 232, 234 Fort, 18 Fresnel's wave surface, 390 Function, definition, 19, 273, 274 explicit, implicit, 19, 277 transcendental, 20 rational, irrational, 20 symbolism for, 21 uniform, one- valued, 21 continuity of, 22, 278 difference of, 35 derivative of, 36 difference-quotient of, 36 increasing, decreasing, 74 Function of a Function, geometrical picture, 453 «th derivative of, 455 Gamma functions, 217 Gauss, theorem on areas, 241 theorem on curvature, 372 Geodesic line, 384 Groin, volume of, 267 surface of, 408 Harkness, 451 Helix, 376, 384, 405 Holditch's theorem, 238 Homogeneous, coordinates, 339 differential equation, 414, 431 Horograph, 372 Hyperbola, tangent, 113; asymptotes, 122 radius of curvature, 137 area, 228, 233 orthogonal trajectory, 432 Hyperbolatoid, volume of, 269 Hyperbolic sine, cosine, 29 Hyperbolic spiral, traced, 162; subtangent, 117 area, 234 ; length, 248 Hyperbolic paraboloid, volume, 399 Hyperboloid of revolution, volume, 258 Hypocycloid, tangent, 113; evolute, 146 traced, 154, 164 ; area, 229 length, 246 volume of revolute, 258 surface of revolute, 261 Increasing function, 74 Indicatrix of surface, 361 Infinite, infinitesimal, 2, 7 Inflexion, 128 Inflexional tangent, 332, 355 Integer, definition, I Integral, definition, 165 indefinite, 173; definite, 215 fundamental, 173 Integration, definition, 167 by transformation, 178 by rationalization, 182 by parts, 183 478 INDEX. Integration by partial fractions, 185 under the /'sign, 393 Integrating factor, 420 Interval of a variable, 4 Intrinsic equation of curve, 251 of the catenary, 252 of the involute of circle, 252 of the cycloid, 253 Illusory forms, 95 Inverse curves, 161 Involute of a curve, 144 Jacobi's theorem on areas, 241 Lacroix, 335 Lagrange, theorem of mean value, 78 differential equation of, 430 interpolation formula, 241 Leibnitz, «th derivative of product, 69 symbol of integral, 170 linear differential equation, 423 Lemniscate, 99 traced, 156, 159 ; area, 234 ; length, 253; area revolute, 263 Lengths of curves, plane curves, 243, 247 curves in space, 404 L'Hopital's theorem, 94 Limit, of a variable, 7 principles of, 7 theorems on, 8 of (I + 1/2) 2 , 16 of integration, 167 Li ma 9011, area of, 239 Linear differentiation, 293, 301, 307 Linear differential equation, 443 Line of curvature, 384 Logarithmic curve, traced, 150 ; length, 246 Logarithmic spiral, area, 234 ; length, 248 Lpxodrone, 384 M u l.iunn 5 series, 83, 467 Maximum and Minimum, 103 independent variables, 314 implicit functions, 321 conditional, 322 McMahon, 416 Mean Value, theorem of, 76, 218 formula by integration, 220 for two variables, 309 Mean Curvature, 371 Meunier's theorem, 366 Modulus of a number, 3, 458 Morley, 451 Neil, 245 Neighborhood, 7, 278 Newton, binomial formula, 67 radius of curvature, 135 rule for areas, 240 analytical polygon, 340, 471 Nodal point and line, 362 Node, 156, 337 Node-locus, 435 Non-exact differential equation, 418 Normal, to a curve, 114, 115, 330 to a surface, 358 Normal plane, 376 Oblate spheroid, volume, 258; surface, 262 Omega, 2 Order of differential equation, 409 Ordinary point, on curve, 329; on surface, 352 Orthogonal trajectories, 432 Osculation, 132 Osculating plane, 377 Parabola, tangent, 113; subnormal, 116 radius of curvature, 134 evolute, 144; area, 228, 234 arc length. 245. 248 length of evolute, 251 orthogonal trajectory, 432 INDEX. 479 Paraboloid of revolution, volume, 258; surface. 260 Parameter, 138 Partial derivatives, 282 Pedal curve, 239 Plane, equation to, 347 Planimeter, 239 Pole of a function, 457 Primitive of a function, 168 Principal, sections of surface, 365 radii of curvature, 365, 368 normal, 378 Pringsheim, 87, 467 Probability curve, traced, 151; area, 395 Prolate spheroid, volume, 257 Pseudo-sphere, volume, 258; surface, 261 Pursuit, curve of, 446 Quotient of functions, nih derivative, 454 Radius of convergence, 88 Radius of curvature, plane curves, 100, 133, 250 at point of inflexion, 135 for surfaces, 365 for space curves, 379, 389 Real number, 3 Reciprocal spiral, 1 17. 126 Revolute, definition, 255 volume, 256 ; surface, 259 Riemann's existence theorem, 468 Roche, 222 Rolle's theorem, 75 Root of a function, 457 Saddle point, 336 Scarabeus, 160 Schlomilch, 222 Semi-cubical parabola, 245 Sequence, 14 Singular points, on a curve, 158. 333 Singular points on a surface, 352, 362 Singular solutions of differential equa- tions, 433 Singular tangent plane, 362 Singularity, essential and non-essential, 457 Solution of differential equations, gen- eral, particular, complete, 410 by separation of variables, 410 when M and N are of first degree, 415 by differentiation, 426 when solvable for y, 428 when solvable for x, 430 when solvable for/, 431 Specific curvature, 371, 372 Sphere, volume, 257, 400; surface, 260, 403 Spherical curvature, 381 Steradian, 371 Stewart, 238 Stirling, 83 Straight line, equations, 348 Subtangent, subnormal. 115, 116 Successive differentiation, 62 Surface, definition, 349 general equation, 349, 361 of solids, 255, 402 Synclastic surface, 360 Table of derivatives, 52 Table of integrals, 176 Tac-iocus, 435 Tangent, to plane curves, 112, 116, 330 length, 115 to space curves, 375 Tangent line to surface. 350 Tangent plane, 35 1 Taylor's series, 82, 86, 87. 221. 457. 467 Tortuosity, 380, 382 Torse (developable surface), 374 Torus, volume, 259; surface. 261 tangent plane to, 364 480 Total, derivative, 291, 294 differentiation, 290 differential, 294 Tractrix, tangent-length, 119; area, 235 arc length, 246 volume of revolute, 258 Trajectory, 431 Transcendental function, 83 Triple, point, 337; integration, 398 Trochoids, 164 Umbilic, 361, 369 Undetermined forms, 92 INDEX. Undetermined multipliers, applied to maxima and minima, 323 applied to envelopes, 343, 388 Undulation, point of, 333 Variable, definition, 4 difference of, 35 Volumes of solids, 255, 398, 400, 401 Weierstrass, 5 example of derivativeless function, 45 1 Witch of Agnesi, tangent, normal, 115 traced, 152; area, 228 volume of revolute, 258 Zero, 2 MATHEMATICS Evans's Algebra for Schools. By George W. Evans, Instructor in Mathematics in the English High School, Boston, Mass. 433 pp. i2mo. $1.12. Aside from a number of novelties, the book is distinguished by two notable features : (1) Practical problems form the point of departure at each new turn of the subject. From the first page the pupil is put to work on familiar material and on operations within his powers. Difficulties and novelties arise in a natural way and in concrete form and are met one at a time, and he is led to see the need for each operation and preserved from regarding algebraic processes as a species of legerdemain. (2) The book contains nearly 3,500 examples, none of which are repeated from other books. The exercises are graduated according to difficulty and are adapted in number to what ex- perience has shown to be average class needs. Problems are carefully classified with reference to the several types of equations arising from them, and the pupil is specially drilled upon typical forms (as, for example, " the clock problem," " the cistern problem," "day's work problem," etc.) and upon generalized forms. Paul H. Hanus, Professor in Harvard University : — The author has certainly been successful in presenting the essentials of ele- mentary algebra in a thoroughly sensible way as to sequence of topics and method of treat- ment. C. H. Pettee, Professor in the N. H. College of Agrictilture : — I have actually become tired look- ing over algebras, geometries, and trigonometries that have no ex- cuse for existence. Hence it is with real pleasure that I have examined Evans's Algebra for Schools. The author evidently knows what a student needs and how to teach it to him. E. S. Loomis, Cleveland {Ohio) West High School : — To pass gradually from arithmetic to all gebra. to bridge that intellectua- chasm in the minds of many, is no little thing to do. Evans has done it more nearly than any other author I have read. I like his scheme of models, but above all I like his coor- dinating algebra and the other sciences. I wish I could teach the book, it is so full of good things. Jas. E. Morrow, Principal Al- legheny {Pa.) High School:— I find more to commend in this algebra than in any book on the subject since the publication of 's in 1869. 32 Mathematics Gillet's Elementary Algebra. By J. A. Gillet, Professor in the New York Normal College, xiv + 412 pp. i2mo. Half leather, $1.10. With Part II, xvi 4-512 pp. 121110. $i-35- Distinguished from the other American text-books covering substantially the same ground, (1) in the early introduction of the equation and its constant employment in the solution of problems ; (2) in the attention given to negative quantities and to the formal laws of algebra, thus gaining in scientific rigor without loss in simplicity ; (3) in the fuller development of factoring, and in its use in the solution of equations. James L. Love, Professor in Harvard University : — It is un- usually good in its arrangement and choice of material, as well as in clearness of definition and ex- planation. J. B. Coit, Professor in Boston University : — I am pleased to see that the author has had the pur- pose to introduce the student to the reason for the methods of al- gebra, and to avoid teaching that which must be unlearned. F. F. Thwing, Manual Train- ing High School, Louisville, Ky. : — Two features strike me as being very excellent and desirable in a text-book, the prominence given to the concrete problems and the application of factoring to the so- lution of quadratic equations. J. G. Estill, Hotchkiss School, Lakeville, Conn.: — The order in which the subjects are taken up is the most rational of any algebra with which I am familiar. Gillet's Euclidean Geometry. By J. A. Gillet, Professor in the New York Normal College. i2mo. Half leather, $1.25. 436 pp. This book is " Euclidean " in that it reverts to purely geo- metrical methods of proof, though it attempts no literal repro- ductions of Euclid's demonstrations or propositions. Metrical applications and illustrations of geometrical truths are inter- spersed with unusual freedom. " Originals " are made an integral part of the logical development of the subject. Percy F. 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The early propositions, and a few difficult and funda- mental propositions later, are proved at length to furnish models of demonstration. 2. The details of proof are gradually omitted, and a large part of the work is developed from hints, diagrams, etc. 3. The problems of construction are introduced early, and generally where they may soon be used in related propositions. Oren Root, Professor in Hamil- ton College, N. Y. : — I like the book, especially in that it gives " inventional geometry" while giving the fundamental propo- sitions. Geometry is taught very largely as if each proposition were an independent ultimate end. Pupils do not grasp the interlock- ing relations which run on and on and on unendingly." Mr. Keig- win's book, compelling pupils to use what they have learned of re- lations, must help to prevent this. C. L. Gruber,/^?. 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Revised, x -+- 399 pp. i2mo. $1.20. Newcomb's Elements of Plane and Spherical Trigo- nometry. (With Five-place Tables.) With Logarithmic and other Mathematical Tables and Examples of their Use and Hints on the Art of Computation. By Simon Newcomb, Pro- fessor of Mathematics in the Johns Hopkins University. Revised, vi r 168 + vi + 80 + 104 pp. 8vo. $1.60. Elements 0/ Trigonometry separate, vi + 168 pp. Si. 20. Mathematical Tables, with Examples of their Use and Hints on the Art of Computation, vi + 80 + 104 pp. $1.10. The Tables, which are to five places of decimals, are regu- larly supplied to the United States Military Academy and to Princeton University and Yale University for the entrance examinations. Newcomb's Essentials of Trigonometry. Plane and Spherical. With Three- and Four-place Logarithmic and Trigonometric Tables. By Simon Nkwcomb, Professor of Mathematics in the Johns Hopkins University, vi + 187 pp. i2mo. $1.00. Much more elementary in treatment than the foregoing. Newcomb's Elements of Analytic Geometry. By Simon Newcomb, Professor of Mathematics in the Johns Hopkins University, viii + 357 pp. i2mo. $1.20. Corresponds closely to the usual college course in plane analytic geometry, but is so arranged that a practical course may be made up by omitting certain sections and adding Part II, which treats of geometry of three dimensions. The sec- tions omitted in the practical course, together with Part III, form an introduction to modern projective geometry. 3 6 Mathematics Newcombs Elements of the Differential and Integral Calculus. By Simon Newcomb, Professor of Mathematics in the Johns Hopkins University, xii + 307 pp. i2mo. $1.50. A complete outline of the first principles of the subject without going into developments and applications further than is necessary to illustrate the principles. Nipher's Introduction to Graphic Algebra. For the use of High Schools. By Francis E. Nipher, Professor in Wash- ington University. i2mo. 66 pp. 60 cents. Eighteen of the most elementary graphs illustrating the solution of equations. It is thought that none of these graphs is beyond the capacities of high-school pupils. By injecting some such material here and there into the ordinary instruction in algebra, new meaning can be given to mathematical opera- tions and new interest to the whole subject. Phillips and Beebe's Graphic Algebra. Or Geometrical Interpretations of the Theory of Equations of One Un- known Quantity. By A. W. Phillips and W. Beebe, Professors of Mathematics in Yale College. Revised Edition, 156 pp. 8vo. $1.60. ^ „f.'fi*4f