Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39.48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990.\ ©He d^elUt 3ubvav\u I PRESENTED TO THE CORNELL UNIVERSITY, 1870, The Hon. William Kelly Of Rhinebeck. li&OiEMATICS/lu ELEMENTS ~~~^ OF THE DIFFERENTIAL CALCULUS; COMPREHENDING THE GENERAL THEORY OF CURVE SURFACES, AND OF CURVES OF DOUBLE CURVATURE. SECOND EDITION, CORRECTED AND ENLARGED. Λ BY J.;iRf YOUNG, *5^ PROFESSOR OF MATHEMATICS IN BELFAST COLLEGE. LONDON: PRINTED FOR JOHN SOUTER, 131, FLEET STREET; By J. and C. Adlard, Bartholomew Close. 1836.' An Edition of this Work is published also in 8vo.^for the use of the Universities, price 12s.PREFACE. The object of the present volume is to teach the principles of the Differential Calculus, and to shew the application of those principles to several interesting and important enquiries, more particularly to the general theory of Curve Surfaces, and of Curves of Double Curvature. Throughout these applications I have endeavoured to preserve the strictest rigour in the various processes employed, so that the student who may have hitherto been accustomed only to the pure reasoning of the ancient geometry, will not, 1 think, find in these higher order of researches any principle adopted, or any assumption made, inconsistent with his previous notions of mathematical accuracy. Logical strictness of deduction is an ingredient which every one has a right to expect who opens a work professedly mathematical; and to admit that either in the exposition of the theory of the Calculus, or in its practical applications to physical enquiries, there is ne- cessarily involved any principle that will not bear the most scrupulous examination, is tantamount to a declaration that the subject has no claim to a place among the mathematical sciences. Yet I suspect that it is the adoption of exceptionable prin- ciples, and even, in some cases, of contradictory theories, into the elenfents of this science, that has been the chief cause why it has hitherto been so little studied in a countryIV. PREFACE. where the ancient geometry has been so extensively and so successfully cultivated. The student who proceeds from the works of Euclid or of Apollonius to study those of our modern analysts, will be naturally enough startled to find that in the theory of the differential calculus he is to consider that as absolutely nothing which^ in the Geometrical appli- cations of that theory, is to be .considered as a quantity infi- nitely small; and he will very properly hesitate to receive a conclusion as general, when he is at the same time tpld that the process which led to that conclusion has failing cases; and yet one or both of these inconsistencies pervade more or less every book on the calculus which I have had an oppor- tunity of examining. The whole theory of what the French mathematicians vaguely call consecutive points and consecutive elements in- volves the first of these objectionable principles; for, if the abscissa of any point be represented by x, then the abscissa of the consecutive point, or that separated from the former by an infinitely small interval, is represented by x + da:, although da;, at the outset of the subject, is said to be 0. Again, the theory of tangents, the radius of curvature, prin- ciples of osculation, &c. are all made to depend upon Taylor’s theorem, and therefore can strictly apply only at those points of the curve where this theorem does not fail: the conclusions, however, are to be received in all their generality.* If the foregoing statement be true, it is not to be wondered * I am anxious not to be misunderstood here, and shall therefore state specifically the nature of my objection. In establishing the theory of contact, &c. by aid of Taylor's theorem, it is assumed that a value may be given to the increment h so small as to render the term into which it enters greater than all the following terms of the series taken together. Now how can a function of absolutely indeterminate quantities be shewn toPREFACE. V, at that students so often abandon the study of this science, less discouraged with its difficulties than disgusted with its apparent inconsistencies. To remove some of these harassing impediments to the student’s progress, has been my object in the present volume; and, although my endeavours may not have entirely succeeded, I have still reason to hope that they have not entirely failed. The following brief outline will convey a notion of the extent and pretensions of the book; a more detailed enumeration of the various topics treated of will be found in the table of contents. I Have taken for the basis of the theory the method of limits first employed by Newton, although designated by foreign writers as the method of Jf Alembert; and I have be greater or less than a series of other functions of the same indeter- minate quantities, without, at least, assuming some determinate relation among them ? If we say that the assertion applies, whatever particular value we substitute for the indeterminate in the proposed functions or differential coefficients, we merely shift the dilemma, for an indefinite number of these particular values may render the functions all infinite; and we shall be equally at a loss to conceive bow one of these infinite quantities can be greater or less than the others. It appears, therefore, that the usual process by which the theory of contact is established, applies rigorously only to those points of curves for which Taylor’s development does not fail, and I cannot help thinking that on these grounds the Analytical Theory of Functions, by Lagrange, in its application to Geometry is defective, although I feel anxious to express my opinion of that celebrated performance with all becoming caution and humility. Indeed Lagrange himself has adverted to this defect and observes, ( Thcorie des Fonctions, p. 181,) “Quoique ces exceptions ne portent aucune atteinte a la theorie generate, il est necessaire, pour ne rien laisser a desirer, de voir comment elle doit etre modifier dans les cas particuliers dont il s’agit.” (Seenote C at the end,) But he has not modified the expression deduced from this exceptionable theory for the radius of curvature, which indeed is always applicable, whether the differential coefficients become infinite or not, although, for reasons already assigned, the process which led to it restricts its application to particular points. A 2VI. PREFACE. done so because I consider this method to be as unexcep- tionable as that of Lagrange, and, on account of its greater simplicity, better adapted to elementary instruction. In the first chapter, which is devoted to an exposition of the fundamental principles of the Calculus, I have endea* voured to convey a clear and distinct notion of the true meaning of the expression the symbol employed by mo- dern analysts to represent the limiting or ultimate ratio of the increment of any function y to that of the variable x, on which it depends. The general expression for this ratio is mcrement_°Ly. and increment ofx the ratio or quotient itself is the result of the division here im- plied, a result which will of course vary with the increment of x. In the Calculus, the increment ofx is supposed to vary continually towards, and at length to merge into, zero; and the corresponding quotient, during these changes, is found continually to approach towards, and at length to merge into, a certain fixed and determinate value; which value is there- fore the limiting, or ultimate, or extreme value of the entire series of values through which the varying quotient passes. It is this extreme value which the Calculus selects for its ope- rations, and which is represented by It is obvious that the expression in2remenl °f oc^ which all 1 increment of y along represents the varying quotient of which I Jjave been speaking, takes the form § when that quotient arrives at its extreme value. But it is known that in other enquiries this is often a symbol of dubious import: it is sometimes the indica- tion of incompatible conditions in the analytical statements which have led to it; sometimes the symbol of multiple values; and sometimes wholly unintelligible, or admittingPREFACE. vii. of altogether an arbitrary interpretation. Although, there- fore, in the Calculus, this symbol is limited to a distinct and definite interpretation, yet it would be most inconvenient to use a form that would be perpetually reminding us of vague- ness and ambiguity; and thus the symbol ~ is employed in its stead, the zeros being replaced by dy, άχ. Some wri- ters, however, are very averse to considering dy, da:, as in themselves of absolutely no value, and prefer to call them “ infinitely small quantities,” meaning by “ an infinitely small quantity” that which continually oscillates between something and nothing without ever attaining to either. But as I can form no conception of such things, all mention of them is necessarily excluded from the present work. The Second chapter is a practical illustration of the first. It commences by shewing that if f(x) represent any function of x9 and x be changed into x + h, the new state f(x + h) of the function may always be developed according to the ascending integral powers of the increment h; and this leads to the important conclusion that the coefficient of the second term in the development of the function f (x + h) is the differential coefficient derived from the function f(x); a tact which Lagrange has made the foundation of his theory of analytical functions. The chapter then goes on to treat of the differentiation of the various kinds of functions, algebraic and transcendental, direct and inverse, and concludes with an article on successive differentiation. The Third chapter is devoted to Maclaurin’s theorem, and its application is shewn in the development of a great variety of functions. Occasion is taken, in the course of this chapter, to introduce to the student's attention some valu- able analytical formulas and expressions from Euler, Demoivre, Cotes, and other celebrated analysts, togetherνηι. PREFACE with those curious properties of the circle discovered by Cotes and Demoivre. The Fourth chapter is on Taylors theorem, which makes known the actual development of the function f (x -f* h) according to the form established in the second chapter. From this theorem are derived commodious expressions for the total differential coefficient when the function is compli- cated, and whether its form be explicit or implicit; the whole being illustrated by a variety of examples. The Fifth chapter contains the complete theory of vanish- ing fractions, and points out the principal sources of error which have led some mathematicians to dispute the legiti- macy of the conclusions arrived at in the ordinary exposi- tions of this theory. The charge of bad logic in the processes in question appears to me to arise from a complete misconception, on the part of the objectors, as to what those processes really are. The Sixth chapter is on the maxima and minima values of functions of a single variable, and will, I think, be found to contain several original remarks and improved modes of solution. Chapter the Seventh is on the differentiation and develop^ ment of functions of two independent variables. The usual method of obtaining the development of a function of two variables according to the powers of the increments is to develop first on the supposition that a; only varies, and thaty is constant; and afterwards to consider y, which is assumed to enter into the coefficients, to be changed into y-f Λ. But y may be so combined with x in the function F (ar, y) that it shall, when considered as a constant, disappear from all the differential coefficients, which circumstance should be pointed out, and be shewn not to affect the truth of the result: I have, however, avoided the necessity of shewing this, byPREFACE. IX. proceeding rather differently. The chapter concludes with Lagrange*s Theorem, concisely demonstrated, and applied to several examples. The Eighth chapter completes the theory of maxima and minima, by applying the principles delivered in Chap. VI. to functions of two independent variables, and it also contains an important article on changing the independent variable. The Ninth chapter is devoted to a matter of considerable importance, viz. to the examination of the cases in which Taylor’s theorem fails; and I have, I think, satisfactorily shewn, that, these failing cases are always indicated by the differential coefficients becoming infinite, and that the theo- rem does not fail when these coefficients become imaginary, as Lacroix, and others after him, have asserted. These nine chapters constitute the First Section of the work, and comprise the pure theory of the subject; the remaining part is devoted to the application of this to geo- metry, and is divided into two parts, the first containing the theory of plane curves, and the second the theory of curve surfaces, and of curves of double curvature. The First chapter in the Second Section explains the method of tangents, and the general differential equation of the tangent to any plane curve is obtained by the same means that the equation is obtained in analytical geometry, and is therefore independent of the failing cases of Taylor’s theorem. The method of tangents naturally leads to the consideration of rectilinear asymptotes, which is, therefore, treated of in this chapter, and several examples are given, as well when the curve is referred to polar as to rectangular coordinates; and a few passing observations are made on the circular asymptotes to spiral curves, the chapter terminating with the differential expression for the arc of any plane curve determined without the aid of Taylor’s theorem.X. PREFACE. The Second chapter contains the theory of osculation, which is shewn to be unaffected by the failing cases of Taylor’s theorem, although this is employed to establish the theory. The expressions for the radius of curvature are afterwards deduced, and several examples of their application given principally to the curves of the second order, and an instance of their utility shewn in determining the ratio of the earth’s diameters. The Third chapter is on involutes, evolutes, and consecu- tive curves, and contains some interesting theorems and prac- tical examples. Of what the French call consecutive curves, I have endeavoured to give a clear and satisfactory explana- tion, unmixed with any vague notions about infinity. The Fourth chapter is on the singular points of curves, and contains easy rules for detecting them, from an exami- nation of the equation of the curve. This chapter also con- tains the general theory of curvilinear asymptotes, and com- pletes the Second Section, or that assigned to the considera- tion of plane curves. The Third Section is devoted to the general theory of curve surfaces, and of curves of double curvature; in the First chapter of which are established the several forms of the equations of the tangent plane and normal line at any point of a curve surface, and of the linear tangent and normal plane at any point of a curve of double curvature. In the Second chapter the theory of conical and cylindrical surfaces is discussed, as also that of surfaces of revolution; and that remarkable case is examined, where the revolution of a straight line produces the same surface as the revolution of the hyperbola, whose asymptote is parallel to that line. Throughout this chapter are interspersed many valuable and interesting applications of the calculus, chiefly from Monge, The Third chapter embraces the theory of the curvature ofPREFACE. XI. surfaces in general, and will be found to form a collection of very beautiful theorems, the results, principally, of the researches of Euler, Monge, and Dupin. Most of these the- orems have, however, usually been established by the aid of the Infinitesimal Calculus, or by the use of some other equally objectionable principle; they are here fairly deduced from the principles of the differential calculus, without, in any in- stance, departing from those principles, as laid down in the preliminary chapter. Those who are familiar with these enquiries, will find that I have obtained some of these theo- rems in a manner much more simple and concise than has hitherto been done. 1 need only mention here, as instances of this simplicity, the theorems of Euler and of Meusnier, at pages 227, 233. The Fourth chapter is on txoisted surfaces, a class of sur- faces which have never been treated of, to any extent, by any English author, although, as has been recently shewn, the English were the first who noticed the peculiarities of certain individual surfaces belonging to this extensive class.* For what is here given, I am indebted to the French mathe- maticians—to Monge principally, and also to the Chevalier he Roy, who has recently published a very neat and com- prehensive treatise on curves and surfaces. The Fifth chapter treats on the developable surfaces, or those which, like the cone and cylinder, may, if flexible, be unrolled upon a plane, without being twisted or torn. The Sixth chapter is on curves of double curvature; and the Seventh, which concludes the volume, contains a few mis- cellaneous propositions intimately connected with the theory of surfaces. • See a paper in Leyboum*s Repository, No. 22, by T. S. Davies, Esq. F.R.S., {x), &c.; and, to denote an implicit function, we write F(x, y) = 0, fix, y) = 0, &c. (2.) Let us now examine the effect produced on the function y, by a change taking place in the variable x; and, for a first example let us take the equation y = mx2. Changing, then, x into x + Λ, and representing the corresponding value of y by y, we have y = m (x + A)2, or, by developing the second member, y = mx2 -f- 2mxA + As a second example, let us take the equation y = χ*; and putting, as before, y' for the value of the function, when x is changed into x -f- A, we have y = (x + A)3 = x3 + 3x2A + 3xh2 + A3. We thus see, in these two examples, the effect produced on the function by changing the value of the variable. On account of this dependence of the value of the function upon that of the variable the former, that is y, is called the dependent variable, and the latter, x, the independent variable.EXPLANATION OF PRINCIPLES. 3 Let us now ascertain the difference of the values of each of the above functions of x9 in the two states y and y'. In the first example, In the second, y' — y = 2mxh -f- mh2. y'--y = 3 afih + 3 xh2 -f h3; so that, in the equation y s mar1, if h be the increment of the variable x, we see that 2mxh mh2 will be the corresponding increment of the function y; and, in the equation y = x3, if x take the increment h, the corresponding increment of the function will be 3x2h + 3xh2 -f h3. We may, therefore, in each of these cases, readily find an expression for the ratio of the increment of the function to that of the variable, that y' —. y is to say, the value of the fraction . In the first case, ~ = 2mx -f mh .... (1). In the second, ϋ1^- = 3*» + 3.τΑ + Α» .... (2). h It is here worthy of remark, that, in both these expressions for the ratio, the first term is independent of h; so that, however we alter the value of h, this first term will remain unchanged. If, therefore, h be supposed to diminish continually, and, at length to become 0, the said first term will then express the value of the ratio in this extreme case. This first term, then, is the limit to which the ratio approaches as k diminishes, but which limit it cannot attain till h becomes absolutely 0 In the first of the foregoing examples, 2mx is the limit of the ratio or it is the value towards which this ratio continually approaches when h is continually diminished, and at which it ultimately arrives when these continual diminutions bring it at length to h = 0. In the second example the limit is 3x2. (3.) We may now understand what is meant by the limit of the ratio of the increment of the function to that of the variable. It is the4 THE DIFFERENTIAL CALCULUS. determination of this limit, in every possible form of the function, that is the principal object of the differential calculus. The limit itself is called the differential coefficient, derived from the function; so that, if the function be mx?, the differential coefficient, as we have seen above, is 2mxy and the differential coefficient, derived from the function #9, is 3.T2. In both these cases, as indeed in every other, the respective diffe- rential coefficients are only so many particular values of the general symbol §, to which y—y h always reduces when h = 0. In the first example above, R = 2mx; in the second, $ qs= 3λ*. It appears, therefore, that whatever conflicting opinions may prevail among mathematicians, as to the meaning of the symbol ^ when it occurs in an analytical result, they do not, in the least, affect the pre- cision of the fundamental principles of the Differential Calculus. For here it has a distinct and definite interpretation: it is the symbolical expression for the limit of the value of -—or> *n ot^er words, it represents, solely and exclusively, what the development of — becomes when h = 0. It must not be forgotten that the expression -—implies an operation to be performed, viz. division; and although, in conformity with ordinary usage, and to avoid circumlocu- tion, we have called this expression a ratio or quotient, yet the thing really meant is the result of that operation; just as we call ^/a u the square root of a” although indicates an unperformed process. In each of the equations (1), (2), given above, the right hand member is the correct interpretation of the left; and although, for h =s 0, this left hand member takes the form -g, which, in an isolated state, is altogether meaningless, or may mean anything, yet, being accompanied by its interpretation on the right, it becomes perfectly intelligible and unambiguous. Instead, however, of the general symbol £, a particular notation is employed to represent the limiting ratio, or differential coefficient, inEXPLANATION OF PRINCIPLES. 5 each particular case; thus, if y is the function, and x the independent cly variable, the differential coefficient is represented thus, Ii z were the function, and y the independent variable, the differential coefficient would be ^; the expressions ^ and ~ have, we see, the advantage over the symbol g , of particularizing the function and the independent variable under consideration, and of thus restricting its signification; and this, it must be remembered, is all that distinguishes ~ or ~ from for dy9 dz, d#, are each absolutely 0. This notation being agreed upon, we have, when y = mx2, dI do? = 2 mx, and, when y = x3, do? = As a third example, let the function y = a 3x* be proposed, then, changing x into x + h, and y into y’, we have y = a + 3*2 + 6xh -f 3Λ2, . iLzJL — ex + u·, h and, making Λ = 0, we have, for the differential coefficient, ^=6*. d# If in this example the function had been 3jt2, instead of a + 3jr2, the differential coefficient would obviously have been the same. As a fourth example, let y = ax2 ± b, .·. y = ax2 ± b + 2axh ah2, B 26 THE DIFFERENTIAL CALCULUS. which would have been the same if the constant b had not entered the function. As a last example, take the function y = (a + bx)2 or y = a2 -f- 2abx + b2x2, which, when x is changed into x + A, becomes yf = a2 + 2ab (x -f A) + F (x + A)2 = fl2-f 2α&* + W + 2 (αδ -f 6%) A + 62A2, .·. = 24 (« + 4»)+ 6%, A .·. ^ = 24 (a+ 4*). da? \ / It should be remarked, that of the two parts dy, da·, of which the symbol ^ consists, the former is called the differential of y, and the latter the differential of x. These differentials, although each = O, have, nevertheless, as we have already seen, a determinate relation to each other; thus, in the last example, this relation is such, that ch/= 2b (a + bx) dj?, and, although this is the same as saying that 0 = 2b (a -f- bx) χ 0, yet, as we can always immediately obtain from this form the true value dy ofgor ^,we do not hesitate occasionally to make use of it. From the expression for the differential of a function, we readily see dy the propriety of calling a coefficient, being, indeed, the coefficient of dx.7 CHAPTER XX. DIFFERENTIATION OF FUNCTIONS OF ONE VARIABLE. (4.) Let fix) represent any function of x whatever; then, if x be changed into x -f- A, the general form of the development of f(x -f- A), arranged according to the powers of h, will be f{x + h) = f(x) + A A + B A2 + CA3 + DA4 + • V or, substituting ~ for w, Au — Ziy~yAZ z2 Hence, to differentiate a fraction, the rule is this : From the product of the denominator, and differential of the numerator, subtract the product of the numerator, and differential of the denominator, and divide the remainder by the square of the denominator. c14 THE DIFFERENTIAL CALCULUS. (9.) If it be required to differentiate an expression consisting of several functions of the same variable, combined by addition or sub- traction, it will be necessary merely to differentiate each separately, and to connect together the results by their respective signs. For let the expression be u = aw + by + cz + in which w, y, z, are functions of x. Then, changing x into x + h and developing, w becomes w + Ah -f BA2 + &c. y y + Mh + B'A2 + (*+**>■ 4. Let y = vV-f 6x2. The differential of the root or function under the radical, is 2 bxdx; hence dy == i (o + 26*4» =-7=^= d,, " v a 4-fa2 16 THE DIFFERENTIAL CALCULUS. 5. Let y = (a + bxm)n* The differential of the root or function within the parenthesis, is mbxm—ldx; hence dy =s n (a -f- fcxm)n_1 mbx1*—1 dx, .·· ~ = bmn (a 4- bx"*)*—1 χ”1—*. da? \ / 6. Let y a;2 (0 + tf3)2* The differential of the numerator of this fraction is 2a?do?, and the differential of a + a?3 is 3x2dx, therefore the differential of the denomi- nator is 2 (a -f x3) 3a?3do?; hence (8), , (a -f- x3)22xdx — 6x4 (a + *3) da? 2ax — 4x4 dy=------------(7+^----------------s(j+5vd*' dy 2a? (a — 2a?3) ’ * da? (a -f x3)3 7. + The differential of the root a + s/(b + —) is f (6 + d — x Xa/ 9 c 2c and d — =------------- dx; hence x2 x3 dy dx -4 {a + ^b + -^)r(b + ±) * 4cl«+V(i+—)1« 8. Let y = J x2 -j- a + #*· The differential of a?2 -f a + x2 is 2xdx + (« + xTy~ i X(\x dy_ ____x_______ x ^ J*a+'^« + ®s 2 λ] (a +transcendental functions. 17 9. Let yzsz· _____ v a2 -f a* — x Multiplying numerator and denominator by ^a2 -f x2 -f x, the expression becomes » = ^ + -Χ“ί + *,> .·. dy = d dx -f d *J a2 -f x\ dy 2x , a2 + x2 ··· = — *f------------i— + ■ dx a2 a2 a2\/a2 + x2 _ 2x a2 + 2x2 --- λ2 “V ' “2 a*vV+*2 dy 10. y = a2 — x2 .·. —1= — 2«r. Π. ^ = 4#3 — 2*2-f 7x-f3 .·. ^ = 12λ2 —4*-f T. 12. ^ = (λ -f bx) λ?3 .·. ^ = 3aa?2 + 4δι3. • 13. y = (a -f 6» -f CJ?a + &c.)m .\ ~ === m (a -f fo? -f ολ2 -f &C.)1"—1 (6 -f 2cx -f &c.) 14. y = (a + W* IS. y —a + .· 6(1 — it2) 3 -f x2 dx (3 -f x2)2 s/x ... c dy b , c 16. y = a -f 6 s/x— — —------- · * τν Λ dx 2 s/x ' x2 IT. y=(az3 + 6)2-f 2v/a2-iti(i-.δ) .·. ~= βα*2(α*3 + £) 2 (a2 — 2d?2-f&c) >>/a2 —x2 c 218 THE DIFFERENTIAL CALCULUS. 18. y = x + s/l d y da? Vl —λλ[ι +we have log a = a — 1 — J (a — l)a + J (a — l)s — &c. as dj, it follows, from the expression (1), that d Log a? ___ 1 dx a? log a Unless the contrary is expressed, the differential is always taken according to the hyperbolic system, because the expression is then simpler, log a being = 1. From the preceding investigation we learn, that the differential of a logarithm of a function is equal to the differential of the function divided by the function itself, (13.) To differentiate an exponential function. 1. Let y = ax, then log y — x log a .·. d log y = dx log a, that is, dv — = dx log a .*. dy=zy log adx = log a . a* dx. Hence, to differentiate an exponential, we must multiply together the hyp. log of the base, the exponential itself \ and the differential of the variable exponent. EXAMPLES. 1. Let y = * (a* + **) ^a2— x2 .·. log y = log x -{- log (a2 -f *2) + jlog (a2 — *2), dy da? f 2xdx xAx aA + a2 a;2 — 4x4 Λ ’’’ 7 + a2 + x* “ a2 — x2 == (a2 + a?2) (α2 — a2) ^ therefore, substituting for y its value, we have20 THE DIFFERENTIAL CALCULUS. da? a4 -f a2 x2 — 4x4 λ/α2 — x2 2. y = log λ/ # + «X -f" ^ « Λ? n/ a -p a? — n/ a — a? Multiplying numerator and denominator by the denominator, the expression becomes 2jt s y = log...........— — log x log (α~ν«2-.ή 2a — 2 v a2 — x2 < dy__ 1 a__________a Va2—x2— d«r a: sja2—a:2{α— *Ja2—λ2} n/«2 — x2 {a— s/ a2— jc2 __ — a . 's/a2 — x2^ 3. y = .·. log y = £ log (x2 + 2ar) — I log Vx3+ *2 — x (x3 -f- a2— x), . dy_ dl 3x2 + 2x — 1 y X2 4" 2rtx 3 (x3 X2 — x) (1 — 3a) x2— (a + 2) x — a 3x (x2 + x—1) (x + 2a) dy___{(1 —3a) x2— (a -f- 2)x — a} ^x d*' 3 J (.t2 + x — 1)* (* + 2a)* 4. y = ®»*^ΐ .·. log y m V — 1 log i .·. —= m >/ — 1 ^ > Λ1 — = m ^ — 1 ==wi ^ — 1 . xm*f—:l, dx x From this example it appears, that the rule at (7) applies when the exponent is imaginary.TRANSCENDENTAL FUNCTIONS. 21 5. y=zax*. In this example the variable exponent is xx; hence, calling it z and taking the logarithms, we have ds x&x log z = x log x .·. — = log . 6** +x (2* 4“ 1 )· U# , 1 log- * log * dy log a. a lo. y = a r- =-----------------* da? a? „ i Oog)nx 17. ,β^Λ· .·. ____________f_______________________ * da? a? log a? (log)2 a?.... (log)"—1 x 18. y as xx* .·. ^ = a?**. a?* {— 4" log a? (1 4" log *)}· da? a? (14.) To differentiate circular functions. In order to establish rules for the differentiation of circular functions, it will be necessary first to ascertain what ratio some two of the trigono- metrical lines, as for instance sine and tangent, continually approach, as the arc to which they belong continually diminishes, and which ratio they actually attain only when the arc ultimately vanishes. For this purpose we may take the well known relation tan____sec sin rad9 and, observing that as the arc diminishes the secant approaches nearer and nearer to the radius, with which it actually coincides when the tan arc vanishes, we may conclude that the value of — continually ap- smTRANSCENDENTAL FUNCTIONS. 23 proaches towards unity as the arc diminishes, and that unity must be the ratio with which the sine and tangent vanish simultaneously with the arc.* As, during the whole diminution of the sine and tangent, * That 1 is the ultimate value of —— is evident; for, as in progres- sin ° sing towards this ultimate state, it approaches nearer and nearer to 1, from which it at length comes to differ by less than any assignable quan- tity, however small, the extreme value, or that which terminates this continuous series of results, can be no other than 1 itself. It is no objection, therefore, to our conclusion to say that the geometri- cal relation --^n = is not comprehensive of the extreme case : it is sm rad sufficient that it comprehends all but this, to enable us to infer the extreme boundary at which they terminate. The student will not fail to observe that the ultimate or extreme state of the ratio here adverted to, is actually attained in the case before us. The terms of the ratio not only approach indefinitely near to evanescence, as the secant approaches the radius, but they actually vanish when the two latter lines coalesce. We direct attention to this circumstance because the terms “ ultimate ratio” and “ ultimately equal” appear to be used by Newton and most succeeding writers in a wider sense. Two quantities, for instance, are said to be ultimately in a ratio of equality when they approach continually towards equality; and, before the end of any finite time, come nearer to one another than by any given difference *, although the quantities may be such as to never actually coalesce. Thus, the circle is the limit of all the circumscribed and inscribed polygons; and the three are therefore said to be ultimately equal, though they never actually coalesce. In such cases the limit, or ultimate state, is considered as never accurately reached, which is agreeably to the doctrine of the ancients; and the “ ultimate equality” spoken of refers to the fixed limits, and not to the varying quantities themselves. When, however, the quan- tities compared really coalesce with their limits, then, of course, the ulti- mate equality, or the ultimate ratio, is that of the quantities themselves. Should these become evanescent, as in the case in the text, it may be said that the thing called the ultimate ratio can have no existence. Newton disposes of this objection as follows: “ Perhaps it may be objected that there is no ultimate proportion of evanescent quantities j because the proportion, before the quantities have24 THE DIFFERENTIAL CALCULUS. the arc itself continues of intermediate length, and as the former have, ultimately, a ratio of equality, the ultimate ratio of the intermediate arc to each must be that of equality; hence the ultimate ratios are as follow: vanished, is not the ultimate, and when they are vanished, is none. But, by the same argument, it may be alleged, that a body arriving at a certain place and there stopping, has no ultimate velocity; because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And, in like manner, by the ultimate ratio of evanescent quantities 13 to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish· There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be.,, See the Scholium at the end of the first section of the Principia. * With this quotation from Newton’s celebrated Scholium we ought perhaps to terminate the present note; but having adverted above to the fixed limits, as distinct, in certain cases, from the varying quantities them- selves, we feel inclined to ask whether in strictness such a distinction does not always exist ? Are not the ultimate limits of varying geometrical quantities always things different in kind from the varying quantities themselves? And is there less impropriety in saying that a varying line ultimately becomes zero or 0, than in saying that a varying cir- cumscribed polygon ultimately becomes, or merges into, the circle? Jf we admit that a line, continually decreasing, ultimately vanishes, we must admit that the side of a circumscribed polygon, continually de- creasing, will ultimately vanish. As the diminution goes on, the entire polygon, of which this is a side, will furnish a series of points of contact, separated by intervals continually diminishing. But when the decreasing side actually vanishes, these intervals also vanish; and the hitherto inter- rupted series of points becomes, at the same instant, continuous; the polygon merges into the circle, and this circle is the last of the continuous series of magnitudes at which the varying polygon arrives. The limit, therefore, is actually reached, although not while the original figure con- tinues a polygon.TRANSCENDENTAL FUNCTIONS, 25 tan arc arc sin , sin , tan and therefore, arc ^ sin tan chord ’ chord , chord J. Let it now be required to find the differential of sin x. Changing x into x h, we have (Young’s Trig. p. 46, by putting x \h for A, and £ h for B, in form 4,) sin (x + ti) = sin x + 2 sin \ h cos (# + £ Λ), when h = 0, sin (# -4- h)— sin x sin \ h . . , ,. —K—h----------= -jf-co. C* + **) 5 sin λ h d sin x ——— = 1, .·. —-— = cos x .·. dsmi = cos x dx. \h dx 2. To differentiate cos x, d cos x = d sin 7Γ — x) =s — cos (£ π — x) dx = — sin x dx. Cor. As d cos = — d ver sin .·. d ver sin x = sin x dx. 3. To differentiate tan x, , , . sin x cos x d sin x — sin x d cos x d tan x = d--------=---------------------------------> COS X COS'4 5 x cos2 x + sin2 x cos2x dx 1 cos2# dx = sec2 x dx. 4. To differentiate cot x, d cot x = d tan (£ π — x) = — sec2 (£ π — x) dx = — cosec2 x dx. 5. To differentiate sec x, d sec x = d------- cos x sinx cos2x dx = tan x sec x dx. 6. To differentiate cosec x, o26 THE DIFFERENTIAL CALCULUS. d cosec x = d - dx = — cot x cosec x dx. These six forms the student should endeavour to preserve in his memory. examples. 1. 3/ = sin2x dy = 2 sin xdsinx =2 sinxcosxdx = sin2xdx, ... dx = sin 2x. 2. y = sin"x .·. dy = w sin"—1 x d sin x = ra sin"—1 x cos xdx, dy . , .·. — = n sin"—1 x cos x. d# 3. y — cos mx . \ dy = — sin mxdmx = — m sin wixdx, dx : — m sin mx. 4. « =y tan x", y being a function of x, .·. du = tan x" dy + y d tan x", now d tan x" = sec2 x" d . x" = wx"—1 sec2x" dx, du dy . , o tan x* ~~ 4- yrax"—1 sec2 o?n. d^ d# 5. w = cot xy .·. dw = — cosec2 d * dy. y .·. — =— cosec2x (--------l· \ogx-r-)x . dcr K x dx'transcendental functions. 27 6. y = xeco** .·. dy ss ecoa*da? -f" ^COBJr d cos x == ^ cos * (1 — a? sin a?) (k?, % __ do? = ecoa* (1 —a: sin a?). 7. y = log(a?eC0**) .·. < €cob* — x ginx)^ dy f= — —-------~, and d(a?eC08A)= .veto** 9 ' ' 1 — x sin i? d x 8. y = cos x + *J — 1 sin a?, ,\ ^ = — sin x + s/ — 1 cos x, ax 9. y == cos a? + cos 2a? + cos 3* + dy .·. — =— (sina? -f 2sin2x + 3 sin3a? -f- wth diff. coef. · din It must be observed here, and carefully borne in mind by the student, that this notation is nothing more than a brief and compact mode of representing, symbolically, the results of certain operations dy performed upon a function, y, of x. The operation implied in ^ has been already explained; a like operation, performed on the result, d2 y . d3 y is expressed by a third such operation by and so on; hence d2, d3, &c. are not powers, but symbols, standing in place of the words second differential of, third differential of, &c. There is no im- propriety in calling the expressions do?2, da:3, &c. powers, not, however, of x, but of da:: to distinguish the differential of a power from the power of a differential, a dot is placed in the former case between d and the power. (18.) The following are a few illustrations of the process of succes- sive differentiation: 1. y = xm. do? (m — 1) xm~2, ^~ — m (m — 1) (m —2) o?™-3, ^Lz=m(m — 1) (m — 2) (m — 3) xm~4, cfec. Eiii=2-8D •■■“—π i-St-i &c. (fee. d)> Hence, by substitution, equation (1) becomes y = W + [|j*+| r &y L d.r2 ]*2 + 1 2 · 3 r#y_ ^ d «a?3 ] Λ?3 4* &c.------(2), which is Maclaurin's theorem for the development of a function, according to the ascending powers of the variable. From this important theorem a rule may be deduced, by the aid of which we may obtain the development of a function very readily, from merely knowing the development of the first differential coefficient. Thus, by the theorem just established, the development of the functionmaclaurin’s theorem. 37 Comparing this with the development of the function y above, we see that y must be equal to [y] plus the several terms in the development dy x x of multiplied respectively by x, —, —, &c. Hence we have qx 2 3 this rule, viz. dy e Develop ~ m a series of ascending powers of x9 increase the expo- nent of x in each term by unity, and divide the term by the exponent thus increased. In many cases where the successive differentiation would be laborious, dy the development of ^ may be easily obtained by common algebra, and thus the labour adverted to entirely avoided. An example of this may be seen in the development of tan—1 x at page 50.* EXAMPLES. (20.) 1. Let it be required to develop (a -f x)n, the exponent n being any number whatever, either positive or negative, whole or fractional, rational or irrational. Put y = («+ x)n . . therefore . [y] = an .·. = n (a + a?)"-1................[“^~] = nan-1 ax ax ^- = n(n-lXa + ir)”-2 . . . = n (η — 1) a»-s = —1)(m —2)(a+*)"-», [^-]== n(n— 1) (n — 2) a"-3 J [ d-^. ] = 0 Lda>4 J [i5L] = i L dx5 J &c. ···«*= 1 + A* + Τ^ΐ +1^3+ &C. which is the Exponential Theorem. Since A = log a, we may give to the development the form a*=l + x log a + γ (xlog of + (x log a)3 -f y or, if in the same formula mx be put x, we have 2 cos mx = ym -f- > and from these two equations we deduce the following, viz. y2— 2y cos*-]-1 = 0 .... (1) y2w — 2ym cos mx —1 = 0 . . . . (2).44 THE DIFFERENTIAL CALCULUS. Since these equations exist simultaneously, the latter must have two of its roots or values of y equal to the two roots of the former, and therefore the first member of (2) must be divisible by the first member of (1), or, putting Θ for mx, we have ySm_2y«cos0+ 1 . . . . (3), divisible by — cos “ + 1 · · · · (<)· m But cos Θ = cos (0 -j- 2w?r), n being any whole number, and π = 180°; hence, making successively n = 0, =1, =2, &c. to w = m — 1, we have, since the first equation continues to be divisible by the second in all these cases, Q y2m — 2ym cos 0 -f- 1 = ( y2 — 2y cos ——-1) , „ _ Θ+2ΤΓ , X (y2—2ycos—!----------hi) m X ( y2 — 2y cos - — — -f 1) m Θ “4” βίτ X (y2 — 2y cos---------hi) &c· 1° m factors. m The truth of this equation is obvious, for, while the substitution of Θ ■+· 2ηπ for Θ causes no alteration in the expression (3), the same substitution in (4) gives to that expression a new value, for every value Θ Θ -f 2tt of n, from n = 0 to n=:m — 1; for the arcs —, -----, &c. are all mm different. As, therefore, the expression (3) is divisible by (4) under all these m changes of value, it is plain that these are its m quadratic factors. In this way may any trinomial of the form — 2kym -f 1 be decomposed into its quadratic factors, provided k does not exceed unity, for then k may always be replaced by the cosine of an arc.maclaurin’s theorem. 45 (24.) The geometrical interpretation of the foregoing equation, pre- sents a curious property of the circle, first discovered by De Moivre. To exhibit this property, let P be any point either within ^ & 3 or without the circle whose centre is O, and let the A / circumference be divided into any number, my of equal parts, commencing at any point A. Join the points of division, A, B, C, &c. to P, then, since in the foregoing analytical expression the radius OA is expressed by unity, we shall have, by introducing the radius itself, so as to render the terms homogeneous, the following geometrical values of the above factors, where it is to be observed that Z. POA = — and OP = y. m cos 0 + 1 = OP2» — 20Ρ» X ΟΑ» cos m (AOP) + AO2» y’ — 2y cos—--|- 1 = ΟΡ* —20P χ OA cos AOP + AO* « PA** y* — gycoee + 2^ + 1 = OP* —20P x OA cos BOP + BO* = PB* m y* — 2ycos®±if + χ =OP* —20P χ OA cos COP + CO* = PC* &c. &c. /— 1)^—1 we see is nothing more than a brief and compact expression for certain operations which terminate in the series on the right hand; and we may therefore always substitute for such an expression this its interpretation. In like manner, the 2 /---------------------- form j—- log v — l above, must be regarded as the symbolical expression for certain processes which, when performed, furnish the length of a semicircular arc of radius 1. To render this expression more readily developable, let us put ι-νΛΓϊ for its equal v — 1,48 THE DIFFERENTIAL CALCULUS. in log v — i, and the symbol for π will then be * = ~J= {log (] + - log (1 — V=T)} v — 1 = 7= (^~r+T (^>5- i ■ A* 1*2*3 1·2‘3 A, 1*2 1·2 * 3 · 4 + 16 1 * 2 · 3 · 4 * 5 )*............................... [yl =o· 1-1-1=' dy _ 3 2-4»* r dy,_Q d®3 (1 +*s)s't' (1 + ®»)3 ........1 dr>J — d*y 23x 24® 24 · 3** d®4" ~ (1 + tf2)3 + (1 + x3)3 _ (1 +ϊ3)4 L d*4 J dV ___ 23 · 3 25 · 33 x3 27 · 3x4 d»y Λ, ite3 — (i+i3)3 (i+»4)4 + (ΐ + ι3)» · ’ 2 &C. «fee. .·. y = tan y — J tan3y + £ taney — } tan7y -f «fee. This development is obtainable with far less trouble by means of the rule at page 37. Thus, since by division y~5 = 1 — Xs + *4 — »« + &C., we have, by the rule referred to, y == tan y — § tan3y -f- $ tansy — \ tan7y -f- «fee. tan y being put for x, as before. * The arc whose tangent is 0 may be either 0 or any positive or negative multiple of 7r. We shall, however, suppose here that the arc to be developed does not exceed a quadrant; it must, therefore, become 0 when x becomes 0.MACLAURINS THEOREM. 51 If y = 45°, then tany = 1; .·. arc 45° = 1— ! + $ — * + &c. (27.) From this series an approximation may be made to the cir- cumference of a circle, but, from its very slow convergency, it is not eligible for this purpose.· Euler has obtained from the above general development a series much more suitable, by help of the known formula, ( Young*$ Trig. page 48,) tan (a + b) = tan a + tan b 1 —tana tan 6 for, when a -f- b s=t 45°, tan (a + b) = 1; therefore, tan a -f- tan 6=1 — tan a tan 6. If either tan a or tan b were given, the other would be determinable from this equation. Thus, if we suppose, . 1 .. 1 . , . , tan b η — 1 tan a — —, then-----l· tan b = 1-------, .·. tan 6=----—- · n n η η + 1 Now the value of n is Arbitrary, and our object is to assume it so that the sum of the series, expressing the arcs a, b, in terms of their tangents, may be the most convergent. This value appears to be n 2 or n == 3; therefore, taking n = 2, we have tan α = I, tan b = J. Hence, substituting in the general development a for y, and \ for tan y, and then again b for y, and | for tan b, the sum of the resulting series will express the length of the arc a -f- b = 45°; that is, arc 45°= --- MM., - «L, - - -- .Li 45°. Let now 4a = A, 45° = B, A — B = b = excess of 4a above 45°; then we have 45° = A — b. But ( Trig. p. 48,) ,. x , tan A — tan B 1 tan (A B) tan b j + tan A tan β ~ 239 ' Consequently, if in the general development we replace y by a, and tan y by J, and then multiply by 4, we shall have the length of the arc 4a; and, since this arc exceeds 45° by the arc b, if we subtract the development of this latter, which is given by substituting ^ for tan y, the remainder will be the true development of 45°. Thus ( 4*T~"3752’*’ 45°= J i i 11 ^ ^ 239 3 · 2393 5 · 239* This series is very convergent, and, by taking about 8 terms in the first row and 3 in the second, we find, for the length of the semicircle, the following value, viz. 7Γ = 3 · 141592653589793. If we take but three terms of the first and only one of the second, we shall have π = 3* 1416, the approximation usually employed in practice.MACLAURIN S THEOREM. 53 (29.) The following examples are subjoined for the exercise of the student: 8. To develop » = sin—1 x. . sin3» . 1-3 sin8» , 1*3*5 sin7» , y = sin » 4------- 4----------- A-------------- 4- ’ U— di2 2* C— dx3 2-3’ & Hence the required development is Y_„ . iff A dV Aa dV A3 Y-y+s T + _d? + m+Ac· If A is negative, the signs of the alternate terms will be negative. When we wish for the development of the function Y = F (j- + A) in any particular state; that is, when x takes a given value, we have only to substitute this value for x in the general expressions for the coefficients previously determined, and we shall have the development according to the above form; that is, provided, of course, that the development in such form is possible. But if the value chosen for x render the develop- ment impossible, the impossibility will be intimated to us from the cir- cumstance of some of the terms becoming infinite, as explained in art. (4). It may, however, be proper here to remark, that even in these cases of impossibility, the leading terms of the development, as given by Taylor's theorem, are still true as far as the first term that becomes infinite. But as we propose to devote hereafter an entire chapter to the examination of the failing cases of Taylor’s theorem, we shall not enter into the enquiry here. (31.) The theorem of Maclaurin may be easily deduced from that of Taylor thus: Let x take the particular value x=.0, then56 THE DIFFERENTIAL CALCULUS. [Y] = F(A) = [F(*)] + [ dF(*X d»F(f) d® J 1 't*L d*s >rs+ -d3 F(x) L dx3 J 1 A3 2*3 -f- &c. Now each of these coefficients is constant, and therefore independent of the value of A; hence h may take any value whatever, without affect- ing these coefficients; we may therefore call it x9 it being observed that although x appears in the notation of the coefficients, it does not appear in the coefficients themselves. It follows, therefore, that FW = [F(r)] + [ dFQr) do? ]*+[ d8F(x) da8 J , rdaF(*) 1·2^1 da?3 J 1 ·2·3 -f- &c. which is Maclaurin’s theorem, before investigated. EXAMPLES. (32.) 1. To develop sin (* 4- A) in a series of powers of the arc A. “ da?8 hence, by Taylor’s theorem, u„ d8y . d3y Let ^ = sin a? .·. -^- = cosx, —j-^- =—sin a, = — cos *, &c; _ , A8 A3 sin (x 4- A) = sin a? 4- cos * A — sin x — cos a — 4" &c. A8 A4 = sinI(l-r72+i.2.3.4 • &c.) A3 A5 + COS*(A JT2T3 + 1* 2 · 3 · 4 · 5 “ &C,) The series within the parentheses are respectively equal to cos A and sin A (p. 39,) so that the preceding development merely expresses the known property sin (a? 4- A) = sin a? cos A 4- cos x sin A. 2. To develop cos (a? 4* A).taylor’s theorem. 57 — cos x} d3y . —s sin or, &q. Hence cos (a + A) = cos x — sin xh — cos a? -—- + sin x · -- + p)—^(p> ? + + +-jjr + **····· W . dw d — and thus by continuing the differentiation may all the coefficients in (1)taylor’s theorem. 61 be developed according to the powers of k; but this first will be suf- ficient for our purpose. Substitute in the first two terms of (1) their developments (2), (3) ; and we have Au dl£ V(q + k,p + k) = U + Tk + &C. +-r-* + &C-------------- (4). But k being the increment of the function p, arising from x taking the increment h; and k' being the increment which the function q takes from the same cause, it follows that dV dx2 A2 1 · 2 “b &c*f da? d9q dx2 A2 1 · 2 + &c. Hence, by substitution in (4), we have finally F(y + ^,i> + A)=« + {^ J + ~ £)* + *«· d u d u d y du dx dq dx dp dx (36.) Again: let there be three functions of x, viz. t? = F (p, q, r); then, when x becomes x + Λ, let p, q> r become p + A, q + A', r + k', respectively; and put r' for the latter; then, in the function F (p -f- A, q -|- A', /), r' enters as a constant; hence, as above, where u = F (p, 9, τ') .·. putting r + k" for r', di? _ a dw d* . “=0+d;v+&c·’ a?=d7 + a i dp dp dp dr But A" = -jj A + &c., consequently, dv li· d, *" + dec. G62 THE DIFFERENTIAL CALCULUS. FO + *,y + A')»-+A") = «, + {J dr dv d,r ' dq djr d» d£ Ax ’ dp Ax* -j- &c. dt>____dt> dr dw d^ dp dp dx dr da? ‘ dq da? ' dp da? and so on for any number of functions. Hence the rule is to differen- tiate the expression with regard to each of its constituent functions severally, as if all the others were constants; their sum will he the required differential. Cor. If p is simply x} then, in the function u = F (x} q)7 du dw | du dq da? da? ' dj dx 9 and in the function u = F (a?, q, r), d u da?' du du dq du dr dx'dq da? dr da? (37.) We must not confound here the on the left, with that on the right, in these equations; for the former denotes the total differential coefficient, of which the latter forms but a part, and is therefore called a partial differential coefficient. It is to be regretted, however, that analysts are not agreed as to the best means of distinguishing total, from partial, differential coefficients; and accordingly, in most works on the calculus, the same symbol is applied indiscriminately to both; a circum- stance likely to prove a frequent source of perplexity to the learner; and to avoid which we shall, throughout this volume, always distinguish the total differential coefficient by enclosing it in brackets; so that the two equations above will be written thus:taylor's theorem. 63 (38.) We shall now add a few examples, showing the application of the rule deduced in last article. EXAMPLES. Now _ _ . . fdu. Au . dw dy l. Letw = cotxy .·. {—} = -—\- — —· da? da? dy da? — = — cosec2a?y-----------=— cosec2xy. yxv-x do? da? du 0 dxy 0 . — = — cosec2xy —-— = — cosec2xy. a v log x d y dy .*. 1^} =s — ay cosec2ay (— + log a? da? x ax S. Let u = ■ ΧΛ + (* + v* + V1 + i/ xf Put X 4- t/x 4- l/x 4- $/a?^= q .·. u = F (x, q)f and dw aqx d u ax2 dy 1 11 d?-(lS + ?s)l’ “ <>*- +W* + 3J+4X hence d« _ ar> . 1 1 1 a?x IT-} —-------------j 0 H------£ H-----i H—;)----------------. (a?2 4“ 92)8 2 a?* 3 a?* 4ar (x2 4" 92)5 3. Let w = log tan —, y being a function of x. V d u x x 1 d tan — sec2 — · — _______y _ y y . tan — dx . x tan — y y sm — cos — y y64 THE DIFFERENTIAL CALCULUS. d u d tan — l_ y o x x sec4* — · —- y y2 X dy tan — du y y X X tan — y x x * y* sin — cos — y y hence W & _ y—«Λ*. y2 sin ~ cos — y y 4. Let u = log (x — a -f- x2 — 2a#) .·. { —} = — .1 \/ x* — 2 ax 5, Let u = (cos a?)ein x .da. / / i sin2#. .·. ? —} = (cos #)8in* (cos x log cos #-------------). d# cos# Implicit Functions. (39.) Hitherto we have treated of explicit functions only, or those whose forms are supposed to be given. We shall now consider implicit functions, or those in which the relation between the indepen- dent variable #, and function y, is implied in an equation between the two, and which may be generally expressed by F(*, y) = 0. The deductions in article (36) will enable us very readily to find the dy coefficient ~ from such equations, without being under the necessity of solving them, a thing indeed often impossible. If we turn to the corollary in the article just referred to, and substi- tute y for q, we findIMPLICIT FUNCTIONS. 65 But here u = F (x, y) *= 0, therefore = 0, for u' — u being always 0, - j^U is always 0; hence dw d* __________0 da? dy da? This equation shows that if we differentiate the several terms con- tained in the function u with respect to x only, and then with respect to y regarded as a function of x, we may equate the sum of the results du to zero, and thus determine ^; or we may employ for the same purpose the following formula immediately deduced from that above, viz. dy du m du' da? da? * dy ’ which shows that after having transposed the terms all to one side of the equation, we must differentiate the expression as if y were a constant, and then divide the resulting coefficient, taken with a contrary sign, by that derived from the same expression, on the supposition that a? is a constant. EXAMPLES. 1. Let y2 — 2mxy + a2 — a = 0. Differentiating the several terms according to the first formula, we have 2yg_2^_2OTxg + 2* = ° .·. 2(y — mac) ^ = —i) dy___my — x ' * dx y — mx g 2 By the second formula we have66 THE DIFFERENTIAL CALCULUS. ----p = Zmy — 2x, -^ = 2y— 2?wx.·. ~ as , Jf do? di/ 2. Let x3 + 3axy -f- 3/3 = 0* da? y — z — ■!- =— 3is —3ey, i«3» + V...ie_e+fgr do? dy ^ * dr ^—-* P + y* If the second differential coefficient be required,* we have dy day C°*+ys)(2* + «3g) —(*2+ay)(a+2yg) dr2 dr7 (ax -f- y*y dy u. or, substituting for ~ its value just found, c%__ Qx-fj/2) (2ax2+2xy2 —ax2—a2y)—(x2-fay) (a2x+a^2—■2xay—2at/g^ dr2~ ~~~ (ax +1/2)3 ' ______2xy4 4- (toy 4- 2x4y — 2g3xt/ (a# + y2)3 ______2xy (y3 + 3axy -f" x3) — 2a3ay ““ (ax + y*)8 that is, since x3 + 3axy + y3 = 0, d2y ___ 2a3xy dr2 (ax -f- y2)3 3. Let my3 — a?y = iw, to develop y, according to the ascending powers of x, dw dw dy -r- = 3m,* —*.·*-. da? dy * dr %my2—x' therefore, calling the successive differential coefficients by /?, y, r, , z 3 3V 3y8-f 2>3ayp , * 3ay8— w3^"^ W ^ * (3*y8 —»i3)2 ^ ’ i— 2*32 »3w3(2yp-f xjp2) _n ·’· L8*]— 3x;?/2__m3 (3xy8 —m3)8 * ■ 2*3y3 2*3*3 m3y8 2*3*4 ··· 3afya— m3 (3ay2—w3)8 m3 /j. (fee. (fee. Hence, by Maclaurin’s theorem, ___ ___________ 3#7 £c y ««3 we 5. Let y8 + 2xy + x2 sss a8, to find dr68 THE DIFFERENTIAL CALCULUS. v2 V oo — b dv 6. Let -r- ■ ., = 23/----------# + $, to find / ( tf^x* 5 which each becomes $ when x = a, and which each obviously contain the factor x — a in both numerator and denominator. In these cases, therefore, we at once see that the values of the fractions when x = a are, severally, ■y> “““ ® > 0) &C. In certain other cases the value, although not so easily seen as in the foregoing instances, may, nevertheless, be soon ascertained, by perform- ing a few obvious transformations on the proposed fractions. Take the following example: -f Fw Q)A3 + Ac. /<*)+/'(*)* + +f"(*)h3 + &c. where F'(,r), ¥\x)}hc.fXx),f"(x), &c. are put for the successive diffe- rential coefficients divided by as many of the factors 1*2*3, &c. as there are accents. If in this we substitute a for x, then, since both F(jr) and f(x) vanish, the fraction becomes, after dividing numerator and denominator by A, F'Q) + F"(«)A + F'"(a)A» -f &c. f\a)+f'(a)h +f"Xa)h* + bc. and this fraction, when A = 0, must obviously be equal to ; that is F(«) Ha) /(«) /(«)* If, however, both F'(e) and f'(a) are also 0, then, expunging these terms from the fraction (2), and dividing numerator and denominator again by A, we have, when A = 0, F(«) F\a)_ /(«) “ /'(«) ’ F (a) and so on, till we at length obtain for a fraction of which the JW numerator and denominator do not both vanish, and such a fraction we eventually shall obtain in virtue of the preceding lemma. Hence the following rule to determine the value of a fraction whose II74 THE DIFFERENTIAL CALCULUS. numerator and denominator both vanish when x = a, viz. For the numerator and denominator substitute their first differential coefficients, their second differential coefficients, and so on till we obtain a fraction in which numerator and denominator do not both vanish, for x = q; this will be the true value of the vanishing fraction. EXAMPLES. Oje____fox 1. Required the value of------when x = 0. x — log a. a* — log 6.0* .·. = log a — log b = log A · a3___3a _i_ 2 2. Required the value of —— 2- g--------- when xs= 1· ϊ’ —— OX “p OX o F'Q) __ 3x*-3 # F7(q) = . f'(x) 4x3—12x + 8 ** f'(a) differentiating again, F77 (x) ___ 6x F" (a)_____ ___^ 3. Required the value of -i—s“?.f C0SJ when x = 90®. sin a? + cos x — 1 F7 (x) ____ cosx-|-sinx F7(q) _________ /' (x) cos x — sin x * * /7 (q) 4. Required the value of x-f i2 — (n 1 )a xw+* -f~ (2ro2 -f- 2rc — 1) xn+9 — na (1 — x)3 when x = 1. F'(«) _ /'<*). 1 4» 2x — (w 4“ 1 )V* + + 2) (2n2 -f- 2w — 1) χ«+» — wa (λ + 3)x«+* 3(1 x)a F'(q) _ 0 ’’’ /'(«) ~ °vanishing fractions. 75 F*(«) _ /"(*> ” 2-tt(tt+l)1»"-l+(n+])(2tt» + 6ra*-f3«-2)a’"-(»+2)(>3+3”i)1',+ l 6(1—*) F"(a) _ 0 Λ /"(<*) " 0 F"'(*) _ /"(')“ -(»MtXn+l),*^+(n»+tt)(2nH6»!l+3n-2)i’,-l--(»+1)(?H-2)(»3+3n»)*» V" (a) »(» + l)(2»+ Q_F(e) ’ ' /"'(«) _ 6 /(<*)’ 5. Required the value of a — >/ a2 — φ3 a?2 when x=r 0. F"(a) _ 1 /"(a) 2a * 6. Required the value of X S-—*-22- when a? — 90°. cos x F'(a) _ , 7w’" 1__x 7. Required the value of-when * = 1. cot x — F'(a) _ 2 /'(«) ~ it ' 8. Required the value of ----------when * = «. Λ log a — log x F'(«) /'(«) »a*.16 THE DIFFERENTIAL CALCULUS. (42.) If, in the application of the foregoing rule, we happen to arrive at a differential coefficient, which becomes infinite for the proposed value x = ay we must conclude that the development, in a series of ascending integral powers of A, is impossible for that particular value of the variable; and that, therefore, the rule, which is founded on the possibility of this development, becomes inapplicable. The process, however, to be adopted in such cases, is still analogous to that above; depending upon the development of the numerator and denominator of the proposed fraction; but here this development must be sought for by the common algebraical methods. F C As before, let be a fraction which becomes £ when we change /w x into a. Substitute a + A for x, and let the terms of the fraction be developed according to the increasing powers of A, either by involution, the extraction of roots, or some other algebraical process; then we shall have F(a + A) AA* +BA^ + tt ; 2° * = *'; 3° λ < of. In the first case, by dividing the two terms of the fraction by h*, and then supposing A = 0, there results F (a)_____0_ /(«) ~ A' 0. In the second case the result of the same process is FQ) _A /(a) A' In the third case, by dividing the two terms of the fraction by A*, and then supposing A = 0, the result isVANISHING FRACTIONS. 77 F (a) A ’ /(«) “Ο “*· It appears from these results that the development of the numerator and denominator need not be carried beyond the first term, or that involving* the lowest exponent of h ;* and according as the exponent in the numerator is greater than, equal to, or less than, that in the denomi- nator, will the true value of the fraction be 0, finite, or infinite. We have, therefore, the following rule : Substitute a -|- A for x, in the proposed fraction. Find the term con- taining the lowest exponent of A, in the development of the numerator; and that containing the lowest exponent of A, in the development of the denominator. If the former exponent be greater than this latter, the true value of the fraction will be 0; if less, it will be infinite. But if these exponents are equal, divide the coefficient of the tenn in the numerator, by the coefficient of that in the denominator; and the true result will be obtained. This method, which is applicable in all cases, may frequently be employed advantageously, even where the preceding rule applies. EXAMPLES. n W . , , r (a? — 3aa? + 2a2)^ 9. Required the value of --------'—----- when x = «. (X3_a3)i Substituting a + A for x, we have F(a-f-A) A^(A— a)3 f ____________ (—ah)^ -f- &c. /(“+*) Ai(3ai _|_ 3aA _J_ (3a*h)i + &c. • The first term which actually appears in the development is of course meant here: those which may vanish in consequence of the coefficients vanishing not being considered. t To develop this according to the ascending powers of A, we must write it thus: (—a -j-h)^y and apply the binomial theorem, when we shall Ϊ2 _ i + - a 3 A + &c. 378 THE DIFFERENTIAL CALCULUS. Since the exponent of A in the numerator exceeds that in the denomi- nator, we have F(a) f — a2 (see p. 69.) Substituting a A for x, when } = a, F(« 4" Λ)_(e + A)5 — eft 4* A^_ A^ 4r VANISHING FRACTIONS. 81 The following example belongs to this case: 15. Required the true value of the difference x tan x — \ π sec xf when * = 90°. _l_______1_ /(*) FQ) 1 F(*) + /(*) 1____________1____ . ^ 7Γ sec x x tan x x sin x — £ tt 1 cos a? a? tan x x £ 7Γ sec x by substituting cos x for —i—, and then multiplying numerator and deno- sec x minator by £ πχ tan x, F'(x)-/'(*) x cos x 4- sin x — sin x ... F(a)—/(a) 1. It should be remarked that in this, as well, indeed, as in the preceding cases, the transformation requisite to reduce the expression to the form $, in many instances at once presents itself to the mind; when of course it will be unnecessary to recur to the preceding formulas. The example . sin x just given is one of these instances, for since tan x ==--------, and J e ' 7 cos x sec x as the proposed expression at once reduces to x sin x —27Γ cos x which is the required form. (46.) We shall terminate this chapter with a few miscellaneous examples for the exercise of the student. xn —— 1 16. Required the value of----r-, when x = 1. x —· 1 Ans. n,82 THE DIFFERENTIAL CALCULUS. 17. Required the value of ax2 -f- ac* — 2 acx ba? — 2bcx -j- be8 when x = 2___a2\§ 24. Required the value of ---------when x= a. (x — ay Ans. (2a)^. 25. Required the value of-- — ,-, when x = 1. x—1 log x Ans.$. ax — x 26, Required the value of —:————— ----------when x = a. ^ ar — 2 Ip (B + Chy~P + DAi_/3 + &c.) Putting S for the sum of the series within the parentheses, it is obvious β—d that h may be taken so small that Sh may be less than any pro- posed quantity, A; and that therefore, if ti be such a value, we must have A// > S β—Λ which establishes the proposition. As SIt is less than A, for h = h', the expression continues less than A for every value of h less than h\ (49.) Let us now enquire by what means we may determine those values of x which render any proposed function, y =F (j), a maximum or a minimum. In order to do this, let x be changed into x db h, then, by Taylor’s theorem,ΜΑΧΙΜΑ AND MINIMA. 85 F(4?±A)=F(af)± A + -^· A* dV hs 1 · 2 * dx3 1 ·2·3 d*y dr4 rA4 1 · 2 * 3 · 4 ± F(a±A), and, consequently, for every such value that can occur among those for which Taylor’s theorem does not fail, we must have the condition Γ&] Ld*J • d2y d#8 A8 1 * 2 : [—] L d*3 J A3 1*2*3 + &c. < 0 . . . (1). But if the supposed value render the function a minimum, then, for all the intermediate values of A between A = A' and A = 0, we must have and, consequently, F(a)° · · · <*>· It has, however, been proved above, that a value may be given to A small enough to render the first term in each of the series (1) and (2) greater than the sum of all the other terms; and that this first term will continue greater for all other values of A between this small value and 0; so that, for each of these values of A, the sign belonging to the sum of the whole series is the same as that of the first term; it is impossible, therefore, that either of the conditions (1) or (2) can exist for both -f- [^] ^ an<* — 1-dlP ^ un^ess [jgp = 0; we conclude, therefore, that of the values of x, for which Taylor’s theorem holds, those only can render the function a maximum or minimum which fulfil the condition,86 THE DIFFERENTIAL CALCULUS. and even these must be tested as follows: Expunge the first term from each of the series, (1), (2), then in virtue of the condition just stated, we must have, in the case of a maximum, d2y A2 d3y A3 2±^Γ2^ + &ε·<°···(^ and in the case of a minimum, Λ _A*_ d Sf h? L dx2 J 1*2 L dx3 J 1-2*3 + &c. > 0 . . . (4). But the former of these conditions cannot exist for any of the values of A between h = K and A = 0, by virtue of the foregoing principle, d2y . . . d unless is negative; nor can the latter condition exist unless [-^—] is positive; that is, supposing that these coefficients do not vanish from the series (3) and (4). We may infer, therefore, that of the values of x which satisfy the condition dx = 0, those only among them that also satisfy the condition d 2y -j— < 0 belong to maximum values of the function, while those fulfill- d2y ing the condition > 0 belong to minimum values of the function. It is possible, however, that some of the values derived from the equation dy d2y t~ = 0 may, when substituted for x in cause this coefficient to dx * dx3 vanish, in which case the conditions (1), (2), become r d3y ·, -l· Γ ——1 — - L da3 J l A3 2 · 3 + [iV] Ldx4 J A4 1-2-3.4 ± &c. < 0 (5), + [jQL] —iL “ L dx3 J 1-2-3 A4 -f [-^-] T L dx4 J 1 · 2 · 3 · 4 ± &c. > 0 . . . (6), andΜΑΧΙΜΑ AND MINIMA. 87 d \ which are both impossible unless [~pf] = 0, for reasons similar to dV those assigned above, and, unless, also [-pj-] < 0 in the case of a maximum, and > 0 in the case of a minimum; that is, on the supposition that this coefficient does not vanish from the series (5) and (6.) If, however, this coefficient does vanish, then, for reasons -similar d®y to those assigned in the preceding cases, the following coefficient — d®y must also vanish: the condition of maximum will then be Γ1 L d*6J d6y < 0, and the condition of minimum [— 6 ■] > 0, and so on. It hence appears, that to determine what values of x correspond to the maxima and minima values of the function y = F( &c. stopping at the first, which does not vanish. If this is of an odd order, the root that we have employed is not one of those values of x that renders the function either a maximum or a minimum; but if it is of an even order, then, according as it is negative or positive, will the root employed correspond to a maximum or to a minimum value of the function. (50.) It must, however, be remarked, that should any of the roots of dy the equation ^ = 0 cause the first of the following coefficients, which does not vanish, to become infinite, we cannot apply to such roots the foregoing tests for detecting the admissible values, and for distinguishing the maxima from the minima, because the true development of the88 THE DIFFERENTIAL CALCULUS. function for any such value of x begins to differ in form from Taylor’s development, at that term which is thus rendered infinite (4); so that we cannot infer, from Taylor’s series, whether the power of A, which ought to enter this term, is odd or even. In a case of this kind, therefore, we must find, by actual involution, extraction, &c. the true term that ought to supply the place of that rendered infinite in Taylor’s series for r = a. If this term take an odd power of A, or, rather, if its sign change with the sign of A, then x^za does not render the function either a maximum or a minimum; but if the sign does not change with that of A, then the value of x renders the function a maximum or a minimum* according as the sign of this term is negative or positive. To illustrate this case, suppose the function were y = A + (* — a$ da? 3 (a? — a)§ d** 9 '■* ' Now the equation ~ =0 gives r = a, so that if any value of x could render the proposed function a maximum or a minimum, this probably would be it. We cannot, however, without examination, be confident on this point, because, although suitable values of x may exist, yet they may be such as to cause Taylor’s theorem to fail even at its commencement, and consequently cannot be among the roots of the equation In order however to test the root just found, sub- stitute its value in 4-^-: the result we see is infinite, and we cannot dr* therefore infer the state of the function from this coefficient; substi- tuting, then, a ± A for x in the proposed, we have F (a ± A) s= b ± A^;maxima and minima. 89 and, as /J obviously changes its sign when h does, we conclude that the function proposed becomes neither a maximum nor a minimum for the proposed value of x. Again, let Λ y = b ■+· (x — a)3 d# 3 v =T («-««)- dv The equation ~-=0 gives = a value which causes d,r2 become infinite: therefore, substituting a h for x in the proposed, we have F (α ±Λ) = έ +A3; and, as the sign of Λ3 is positive whatever be the sign of h} we conclude that the value x = a renders the function a minimum. (51.) There remains to be considered one more case to which the general rule is not applicable, and which, like the preceding, arises from the failure of Taylor's theorem. We have hitherto examined only those values of x for which Taylor’s development is possible, as far at least as the first power of h, but, as before remarked, we cannot say that among those values of x, which would render the coefficient of this first power infinite, there may not be some which cause the function to fulfil the conditions of maxima or minima; therefore, before we can conclude in any case that the values of x, deduced from thg condition ~ = 0, comprise among them all those which can render the function ax a maximum or minimum, we must examine those values arising from dy the condition = oo, by substituting each of these ± h for x in the proposed equation, and observing which of the results agree with the conditions of maxima and minima in (47.) It will be frequently i 290 THE DIFFERENTIAL CALCULUS. found, however, that this new condition can only be satisfied by x s=s oo, which is of course not admissible inasmuch as it does not admit of preceding and succeeding values. In the two examples above, for instance, this new condition furnishes no admissible value for x, so that, in the first example, no suitable value exists, and in the second only one, and that renders the function a minimum. (52.) If the function that y is of x be implicitly given, that is, if u = F (x, y) = 0 $ then, by (39), we have, for the differential coefficient, __ du^du da? da? * dy * * * * ' '* d y d u and therefore, when = 0, we must have -5- = 0; hence, the values ’ dx ’ d.r corresponding to maxima and minima are determinable from the two equations* u = 0 -x £«•3 (2). Having found from these the values of x that may render y a maximum or a minimum,f as also the corresponding values of y itself, we must d2y substitute them for x and y in , when those values of y will be * Other values may be implied in the condition ~ = , which leads d u to — = 0, but to ascertain which of these are applicable would require us tofSolve the equation for y. f Gamier, at p. 271 of his Calcul Differentiel, says, that, by means of the equations (2), “on obtient les valeurs de x et y par lesquelles F(a?, y) devient ou peut devenir maximum ou minimum j” but this is evidently a mistake, since, by hypothesis, F («*?, y) is always = 0.ΜΑΧΙΜΑ AND MINIMA. 91 maxima that render this coefficient negative, and those will be minima that render it positive. But those values that cause it to vanish, belong d3v neither to maxima nor to minima, unless the same values cause also -~ dx3 to vanish, and so on. The second differential coefficient may be readily derived from (1); for, putting for brevity dy M da? N we have d2y da?2 dM dM da1 dy % da?' M ( dN da’ dN dVs da?' N2 which, because or —, = 0, becomes for the particular values of x resulting from this condition, r d 2y d 2 u . rd n_ ^ dx2 1 — ~ ~da^~ ~ ^ * · * ('3·)' By differentiating the above expression for Vy_ da?2 we should find [-*] =. 1 da?3 J . d 3u rd«. and so on. ·■·«>, (53.) Before we proceed to apply the foregoing theory to examples, we shall state a few particulars that may, in many instances, be serviceable in abridging the process of finding maxima and minima. 1. If the proposed function appears with a constant factor, such factor may be omitted. Thus, calling the function Ay, the first differential dy d y d y 1 coefficient will be A~, and A~ = 0 leads to = 0, also — = 0 ax ax dx dy Adi leads to — = 0, so that A may be expunged from the function. dy92 THE DIFFERENTIAL CALCULUS. 2. Whatever value of x renders a function a maximum or minimum, the same value must obviously render its square, cube, and every other power, a maximum or minimum; so that when a proposed function is under a radical, this may be removed. The rational function may, however, become a maximum or a minimum for more values of x than the original root; indeed, all values of x which render the rational function negative will render every even root of it imaginary; such values, therefore, do not belong to that root; moreover, if the rational function be = 0, when a maximum, the corresponding value of the variable will be inadmissible in any even root, because the contiguous values of the functioi? must be negative. 3. Whatever value of x renders a function a maximum, the same value will obviously render its reciprocal a minimum, and vice versa. Hence if it be found more convenient to take the reciprocal of any pro- posed function, instead of the function itself, we are at liberty to do so, recollecting, however, that the maxima and minima values of the changed function correspond to the minima and maxima of the original one. examples. (54.) 1. To determine for what values of x the function y = a4 -f- b3x — Λ2 becomes a maximum or minimum, % = i3 — 2A, = — 2Λ da? dar From the second equation it appears that, whatever be the values of x, dy given by the condition = 0, they must all belong to maxima. ax b3 From b3 — 2c2x = 0 we get x = ; hence when x = —— .·. y == a* 4- , a maximum. 2 cl * 4 c*ΜΑΧΙΜΑ AND MINIMA. 93 du The equation ~ = would give, in the present case, a? = «, a value which is inadmissible (61). 2. To determine the maxima and minima values of the function y = 3a2x® — b4x c5 & = 9Λ*-ί·,^· = 18Λ da? dx2 putting 9— ό4 =s 0 .·. x = ± · 3a d2w Substituting each of these values in we infer from the results that . b2 , 26« when x = —- . . . . y s=s c5-----—, a minimum, 3a 9a 62 t x = — — . . . .y=c5-f* t:—> » maximum. 3 (t ad 3. To determine the maxima and minima values of the function y = >/2ax. Omitting the Tadical w = 2ax — = 2a, dx as this can never become 0 or «, we infer that the function has no max- imum or minimum value. 4. To determine the maximum and minimum values of the function 2/ = λ/ 4a2x2— 2αχ®. Omitting the radical and the constant factor 2a (53), u = 2 ax2 — a·3, .·. ^ = 4ax — 3x2, = 4a — 6x, dx dx294 THE DIFFERENTIAL CALCULUS. .·. x (4a — 3x) = 0 .·. x = 0, or x = — · d2w Substituting each of these values in — —, the results are 4a and — 4a; hence when x = 0 . . . y = 0, a minimum, 4a 8^3 . x = — . . y = —a2, a maximum. If, instead of the preceding, the example had been y = >/2ax3 — 4a2x2> we should have had da 0 4 d2a — = 3x2 — 4a#, ——- = 6x — 4a; do? cur 4a .·. x (3x — 4a) = 0, .·. x = 0, or x = 3 the same values as before; but the first corresponds here to a maximum, since it makes negative; this value, therefore, must, by (53), be rejected. If, indeed, we substitute 0 ± h for #, in the proposed function, it becomes _________ y = *J—4α2Α2=£2αΛ3, where h may be taken so small as to cause the expression under the radical to be negative for all values of h between this and 0. 5. To determine the maxima and minima values of the function y = a + \^a3 — 2 a2x + ax2· If y is a maximum or minimum, y — a will be so; therefore, transposing the a, and omitting the radical (53), u = a3 — 2a2x -j- ax2 da . d2w _ = -2«’ + 2«*, ^r = 2«,ΜΑΧΙΜΑ AND MINIMA. 95 .·. —2α2 -f 2αχ = 0 .·. x = a, .·. when x = a . . . y = «, a minimum. 6. To determine the maxima and minima values of the function y (a — x)2 In solving this example we shall employ the third principle at p. 68, which is often found useful, when the proposed function is a fraction with a de- nominator more complex than the numerator. Instead of the function itself, we shall, therefore, take its reciprocal, which will give us a more simple form, and it is plain that the maxima and minima values of the re- ciprocal of a function correspond respectively to the minima and maxima values of the function itself. Omitting, then, the constant a2, and, taking the reciprocal, we have u a2 — 2«x -f- x1 x a2 x 2 a -f- x. du dx =-^ + 1> d hi 2a2 dx2 x9 !±a··· hence x = a makes u a minimum, and x = — a makes it a maximum; therefore, when x = a . . . y = ac , a maximum, x = — a . . . y — — £a, a minimum. T. To determine the maxima and minima values of the function y=.b-\- V(d? — a)5. Omitting b and the radical, we have « = (x — a)5 ; dx = 5 (x — a)4, dx2 = 4 . 6 (x — a)3 ;96 THE DIFFERENTIAL CALCULUS. A .·. 5(x —a)4 = 0, .·.*=«, .·. [^]=0- As this coefficient vanishes, we must proceed to the following, which dft« however all contain x — a, and therefore vanish, till we come to —. CHr = 2 · 3 · 4 · 5;* as therefore the first coefficient which does not vanish is of an odd order, the function does not admit of a maximum or a mi- nimum. 8. To determine the maxima and minima values of the function y = xx ^ = **(l+logx),^-=I*{i- + (l+logx)»}. · The factor xx can never become 0, therefore, (1 + log *) = 0, .·. log x = — 1; • * = e-i=-i r d *y _ ,1,« ··· ^ = (τ)-ϊ .·. when x = -i-, x* = (—V, a minimum. e v e 7 9. To determine the maxima and minima values of y in the function u = a*3 — 3 axy -J- y3 = 0 ~ = 3αλ — 3ay, .·. (52) d«a? * All the coefficients preceding the nth will always obviously vanish from the function P (x ± a)n 5 so that if n be odd, i = + a renders the function neither a maximum nor a minimum. But when n is even, then a maximum or minimum obviously exists.ΜΑΧΙΜΑ AND MINIMA. 97 x* — 3axy > .·. y = — .·. xe — 2a3x3 = 0, 3x8—3«y = 0 * « .·. x = 0, or Λ? = α 3/2 .·. (52) ~ φ—>*' * <· S- -1— 2 2 — or----- a a .·. when a? = 0 .... y = 0, a minimum, ι = β V 2 . . . . j| = e V 4 , a maximum. 10. To divide a given number a into two parts, such that the pro- duct of the with power of the one and the wth power of the other shall be the greatest possible. Let x be one part, then a — x is the other, and y = xm (a — x)n = maximum, dy . .*. -r- = mxm~l (a — x)n — nxm (a — x)n—1 = *" which give (a — op)*1-'1 {ma — (rn + n) x] = 0, .·. # = 0, or a — x = 0, or ma — (m -j- n) x = 0, x = 0, x = a, ma m + w The first and second of these values are inadmissible, because the num- ber is not divided when x = 0, or when x = a. Substituting the third value in98 THE DIFFERENTIAL CALCULUS. [1^-] =-[*]—·[·- #]*-· {«I [a — »7 + · [*]’} which is negative because each factor is positive; hence the two required parts are ftvfA/ nu, 9 —:— ana ---------;—, being to each other as m to n. m n m + n Cor. If m == n the parts must be equal. An easier solution to this problem may be obtained as follows: Put — = p and determine x, so that we may have u = ocP (a — x) = a maximum, du . „ , ··· ^ =piP~x (a a?) — xP — xp—1 {pa — 1) #} = 0, .·. x = 0, or pa — (p + 1) x = 0 .·. x = —r · P -r 1 This last value substituted in d2w -— = xF-2 {/>« — (/? + 1)*} — (p + 1) Λ*-* causes the first term to vanish; the result is therefore negative, so that x = ——— = ——— corresponds to a maximum value of u, and there - p -f- 1 m -j- n fore (53) to a maximum value of un = xm (a — XX * There is obviously no necessity to recur to the second differential co- efficient to ascertain whether this value render the function a maximum or a minimum, since it is plain that there is no minimum unless one of the parts may be 0.100 THE DIFFERENTIAL CALCULUS. ab xs/ a2 -f- b2 —” Omitting the constant ab, inverting the function (Ex. 6,) and squaring» we have u = a2x2 + b2x2—a?4, .·. ^ = 2a2x + 2b2x — lx3 = 0, da? . ·. χ = 09 a2 -\-b2 — 2x2 = 0 , a2+62 »24-*'2 The first of these values is inadmissible; from the second we find that r2 = r'2 hence the conjugates are equal. For the second differential coefficient we have Λ 2,, 45-=a^ + w-ito·, \a2 4- b2 This being negative, shows that x = ■ 1 2 corresponds to a max- imum value of u, or to a minimum value of y; so that the conjugates here determined, include an angle whose sine is the least possible; and this happens when the angle itself is the greatest possible (being obtuse), as well as when it is the least possible. 13. To divide an angle 0 into two parts, such that the product of the wth power of the sine of one part, by the with power of the sine of the other part, may be the greatest possible. Let x be one part, then 0 — x is the other, and sinnar . sin”1 (0 — x) = maximum, .·. n log sin x + m log sin (0—x) = maximum, n cos x m cos (0 — x)___^ sin x sin (0 —x) ,maxima and minima. 101 .·. n tan (0—x) = m tan x, ,\n: m :: tan x : tan (0 — x), λ + w : n — *» :: tan x -|- tan (0 — x) : tan λ? — tan (0 — x), :: sin 0 : sin (2* — 0),# .·. sin (2x— 0) = —-— sm 0 n + m _ Λ . ,tn —m . .\2^φ = ». ,dw. du i du dz ^d® d<® dz dd? 9 * dw._____du du dz____________ *■ dy dy ' dz dy ' . .du du dz , du . dwds, . du == (—— -f- — -—) d® -f- (-——— ——) dy = 0, dd? dz d® dy dz dy' * «*+$**■ * The brackets are employed here for the same purpose as at (37), viz. to imply the total differential coefficient derived from u, considered as a function of a single variable. This form it will be necessary to adopt whenever u contains, besides ®, other variables that are functions of ®, provided we wish to express the total coefficient with respect to ®. No ambiguity can arise from our calling these same coefficients partial in one sense, and total in another. They are partial coefficients in relation to the whole variation of u, but they are total coefficients as far as that variable is concerned whose differential forms the denominator; and it may be remarked here, once for all, that when we enclose a differential coefficient in brackets, we mean the total differential coefficient to be understood, arising from considering the function, whose differential is the numerator, as simply a function of the variables whose differentials form the deno- minator.108 tHE DIFFERENTIAL CALCULUS. Thus: let Ax2 -f- By2 -f- C z2 —1=0, ··· Φ=2α*+^!=ο’ idL“l = 2B, + 2C,| = 0; du = (Ax + Cz dx + (By + Cz dy = 0. (58.) If u = F (z), z being a function of x and y, the two differential coefficients are (33) d u____d u dz du d u ds dx dz dx dy dz dy and the total differential is therefore . du dz Ά . du dz Λ du=z— — dx -f — — dy. dz dx· dz dy * Now it is worthy of notice, that the ratio of the ttuo partial differen- tial coefficients, is independent of F, so that this may be any function whatever. Thus ^ du dx du du dz dz dx· du dz dz dy' dz ’ dx dz dy 9 which is an important property, since it enables us to eliminate any, arbitrary function F of a determinate function f{x, y) of two variables. We shall often have occasion to employ it in discussing the theory of curve surfaces. By means of this property too we may readily ascertain whether an expression containing two variables is a function of any proposed combination of those variables. For, calling this combination z and the function u, we shall merely have to ascertain whether or not the above condition exists; or, which is the same thing, whether or not the condition du dz du dzFUNCTIONS OF TWO INDEPENDENT VARIABLES. 109 exists. For instance, suppose we wished to know whether u = xA -f 2x*y2 +y4 is a function of ζ = χ2 Here £=4i3+4^> S=4^+4^ %=*’ ·■· £=(4χ3+4V) 2y _ (4^+V) 2i=0i consequently, since the proposed condition exists, we infer that u is a function of z. We shall now proceed to apply Taylor’s theorem to functions of two independent variables. Development of Functions of two Independent Variables. (59.) In tlie function s = F (r, y) suppose x takes the increment h. the function will become F (r -|-h}y), y remaining unchanged, since it is independent of x; then, by Taylor’s theorem, „ , , , - n , ds j- d*z h· d*z A3 (*+ ,, y) _ z + ^A,4- da?a γΤ2* + "d^T i .2*3 + &c---------(1). But if y also take an increment A, then z will become , dz d% ** . dh A3 , t * + dy*+ dy* l-2 + dy» 1·2·3 + &ε' . d z so that in the expression (1) we must for substitute d# for ddr d** d*z j *-+_&* + _J£. d»T da: dor , V 1-2 dx daz dj^ ; 1-2-3 *}· F(a? H- hy y + £) = 1 d2z d2s d2z + 4- (44 A2 + A:2) * 2 dx2 1 dx dy dy* 1 d3z + 2—(ϋτΑ3 + 3 **«+■- 4- &c. where the partial differential coefficients in each term are identical with those which appear in the differential of the preceding term, as the actual differentiation shows, thus : d*=£de+S^----<* i)· d z dzr the coefficients ^, being functions of x and y, we have , dz d22 d2z ddx — dr* ^^dxiy^ and, consequently, dz d2z d2z άψ= *&**+-&**’ dSss='d^'dxa+2d^;da'dy+l?'d3'“ · · · ·(2)· * This can be regarded as the second differential of the function z only on the supposition that the variations of s and y are both constant and independent; so that we cannot deduce from this expression, that for the second differential of z, on the supposition that y is a function of x. For l 2114 THE DIFFERENTIAL CALCULUS. In like manner, these coefficients being functions of x and y, we have d A - d3z dx -f- - d3z Ay d dx2 dx3 dx2 Ay d2s d3z dx d3z Ay d AxAy dx2dy dx Ay2 d2z d3z dx + Ch . Ay; df Ay1 d+*/· '(«) d#2 dy dy &c. Hence dy2 &c. «fee, + 3 dy dy dy2 &c. , d(/(y»2 . , &(/(y)Y ** T dy 1 · 2 1 dy2 !·■«· (1). ; + Now, instead of this development, we should obviously have obtained that of F (*) = F(y + xf(z)), if in place of z and its differential coefficients we had employed F(*) and its differential coefficients. We should then have had w= F(y + ^Λ») du- du dz dx dz dr therefore [ u ] = F (y) in mvif{) Lda?J Ay JKV) d2tt __ A2u d£v2 , dff _d^*_ A»2 ~~ As2 dz dx2 ,dF(y) dFQr) d(/(y))» _ dy (/(y)) dy dy dy Hence /«=rs---/^=i(V d(/(y»a_l ^(/O))3, 1 dV_9 dy· _ dy J'2 dy q1 1 dy2 q3 dy2 q3 dy 8* 9 -yy7>&c- Hence, by the formula f3), we have, by putting for y its value —, 9 r 1 r9 z= — + — -r + 6 r* , 8*9 r7, e — I· L —m —.1 * «1» . r, we have, by putting 3/ for α (1 — e2) and x for — e cos ώ, F (r) = F(y + xf(r)) = F(y + i-r) = (y + i· r)». Hence, by the formula (2), d-^y» , = + + -----fl y ~ dy y 1 dy 1 d’-^ly , dy * x9 dy2 1*2*3 -f- «fee.Lagrange’s theorem. 119 = y* -Myn-^- + w(w + i)^JL- + η(Λ + 1)(Λ + 2)2/»ρ^ + &c. n. /, «ecosw . n(n+J) , = αη (1 — e2)n (1 — — --j- —7—7;—- e* cos* ω — η(^+_1) (» + 2) , . 1 * 1*2 e3 cos3 ω 4- &c. 1*2*3 3. It is required to revert the series a -f" /3z 4" yz2 4- £z3 4" &c· = that is, to express the value of z in terms of the coefficients. Here * = — ψ — ψ (r + & + &c-)=y+/(*)» therefore, by the formula (3), (y +^+ *«·)* , “=y — (ύ + fy+ &«·) + ,y* dy 1 · 2 d» —(y + It/ -f &c.)» t dy8 1 · 2 · 3 + ,+*c· = «_ X4-y4 —&c· y β* βν β120 THE DIFFERENTIAL CALCULUS. +ι£»·+-^*·+*'· + &c. where y = — —, consequently β β* ^β* 0* ^ 4. Given 1 — « -)· e* = 0 to develop log &, according to the powers of a. log % = a + i a2 -f- i a3 + i aA + F [x±h,y± A], and, consequently, (60), _ dz . dz [± r-A ± —i da: d y d2z dh dh Λ:] + * [^h*±Z^hk+-—> / J a L dr* da: dy dy2 If, therefore, of the small values which we suppose h and k to take, h be the smaller, a part of k may be taken so small as to be less than h; or, which is the same thing, equal to one of the values of h between the proposed value and 0, so that we have h' = k'; therefore, the above condition is [±£±&A' + il5-±2d^ + ^A'S+&c-<0· This condition being similar to (1) art. (49), we infer, by the same reasoning, that dz dz ±ώ±^=0, which cannot be, for both the signs ±, unless ····<')· By continuing to imitate the reasoning in (49), we find that these same conditions must exist for all the values of the variables that render the function a minimum. Hence (49), we have, in the case of a maximum, the condition i [^L+2 — 1 dx2 - dxdy + d2z dy2 ] A'2 -f &c. < 0, M122 THE DIFFERENTIAL CALCULUS. and in the case of a minimum, hi dh dh , -------l· 2---------\- da?2 da: dy . dh dy2 ] A'2 *f &c. > 0, so that, supposing these first terms do not vanish for the values of x and y given by (1), the condition of maximum is d2z _dh_ , L d*2 ± 2 d* dy + d2z dy* ] <0, and the condition of minimum, [ d2z da?2 ± 2 dl Ay ^ d»z Ay1 ] >0, In either case, therefore, the expression within the brackets must have the same sign independently of the sign of the middle term. To deter- mine upon what other condition this depends, let us represent the expression by A ± 2B -f- C or A (1 ± 2 5- + A A B8 B2 Adding — — — == o to the quantity within the parentheses, its form is A{(i±Jr+J-gj. Now this expression will always have the same sign as A, provided C B2 C has, and that — > that is, AC > B2 or AC — B2 > 0, because then the factor of A will be necessarily positive. Hence, besides (1), the condition that a maximum or a minimum may exist is AH Ah AH ‘dr1 dy» Mi dy ^ ••••(2), and we are to distinguish the maximum from the minimum by ascertaining whether the proposed values of x and y renderFUNCTIONS OF TWO VARIABLES, ETC. 123 dh dx2 < 0 or > 0, or, which amounts to the same, whether d2g dy2 < Oor > 0, d2* d2* λ since and have the same sign. Should any of the values determined from (1) cause the coefficient of h'2 to vanish, there will be no maximum or minimum for those values, unless the coefficient of the following term vanishes also. EXAMPLES· (65.) 1. To determine the shortest distance between two straight line» situated in space. Let the equations of the two lines be i = a-) fx' = a'z' + a' > and-J .... (1) yr=.bz + $y = + /3' then the expression for the distance between any two points (z, y, s), (a/ y' 2r) is (Anal. Geom. p. 226), ~D*=u = (x—x'f + (y —y')* + (z — *')* = (« - a· + as - «V)5 + (β-β' + bz- b'z'y + (i - *')’ and this expression, containing the two independent variables z, z' is to be a minimum. Hence by the conditions (1), p. 121, ^ = 2 (*—z')-f 2a^*—aV)+25(/3—/3'-j-5z—6Y)==0 > . (2), ^ = —2 (z — z/)+2a'(a—a'+0z— α'ζ')+2Ρ(β — β'+bz — &V)=0 ) dz and from these equations the proper values of z, z' may be readily deter- mined, which substituted in the expression for u, render it the least pos- sible. That these values really belong to a minimum is evident, because,124 THE DIFFERENTIAL CALCULUS. ^-=2 ()+<·’ d*2 v d*'2 : and this proves that d 2u d 2u ~^2 are both positive, and that d2w d 2u d2w x ds2 "d?*· “ (did?) > °- Since the equations of a straight line passing through two points (x, y> z), (x'f y'> O are x—x’=2a'{z— %') ^ y —-y'=(* — s') j we have, by substitution, when these points are on the lines (1), a — a' -az — a'z' = a" (? _ z') β—β' 4- Λ* — 5V = 5" (*—»') hence, if this line be that in question, we have, by combining the equations (2) with these, the conditions 1 -f- aa" + bb" = 0, 1 4* ®V' 4- b'b" = 0, which conditions show, that this minimum straight line is perpendicular to both the lines (1). (See Anal. Geom. p. 228.) From these conditions we get „ V— b ___________ d’—a a alb—ab'* alb—ab,} by means of which, and the equations (3), the expression for D becomes D = ^5^0 - «Τ + (b - V? + (ab - air, in which, if we substitute the value of 2 —σ' deduced from (2), we obtain, finally, Ό (b — b')(a — a') —(a — a!) (β—β') ~ s/(a— o')2 + (6 —by + (a'b—ab')2 If the numerator of this expression vanish, we shall have D == 0; so that, in this case, the lines will intersect. Indeed, the condition of inter» section of the two lines, (1), we know {Anal. Geom. p. 225,) to heFUNCTIONS OF TWO VARIABLES, ETC. 125 (* - V) (a - a') =0- a') (β - /?). 2. Among all rectangular prisms to determine that which, having a given volume, shall have the least possible surface. Representing the three contiguous edges of the prism by a, y, :, and the volume by a3, we have but since u = 2xy ·+· 2xz 2yz = a minimum, jcyz = a3 .·. % = 2xy -J- 2a3 2a3 . . -----------= a minimum· y * Therefore we must have the conditions that is, d u______ d u d«a? * dy 2a3 ' Λ 2α3 2*—-r o, from which we obtain If these values really correspond to a minimum they must fulfil the conditions r Λ r i w Λ r d2u / NOT ^ „ Ι-Ίϋ3^ > ’ > ’ "d»’" 1 > ’ and these conditions are fulfilled, since [ϋϋΐ-4 ίΐϋΐ-4 ri^l-2 1 it2 J —4’ [ iyi J ~ 4’ Μ* V ~ ’ Hence the required prism must be a cube. In the preceding example we might have concluded, without recurring to these conditions, that the results obtained belong to the required mini- mum, there being obviously no other maximum or minimum, except that which belongs to x = 0, y = 0, z = a>, these being the values which cause the differential coefficients to become infinite, (see art. 50.) M 2126 THE DIFFERENTIAL CALCULUS. 3. To divide a given number, a, into three parts, such that the continued product of the wth power of the first part, the nth power of the second part, and the p\h power of the third, may be the greatest possible. ma na pa The three parts are - , so that the m+ »-(-/? ηι-\-η-\-ρτη-\-η-\-ρ three parts are to each other as the exponents of the proposed powers. 4. To determine the greatest triangle that can be enclosed by a given perimeter. The triangle must be equilateral.* On changing the Independent Variable. (66.) It is frequently requisite to employ the differential coefficients d y d2w dr’ ^C** *nw^ck x *s considered as the principal variable under a change of hypothesis, x, and consequently y, being assumed as a function of some new variable t. It is therefore of consequence to ascertain what changes take place in the expressions for these coefficients in such cases. This we may do as follows: Since according to the new hypothesis y = F(x) and a? =/(*), therefore, (33), dy dy da? dy dy e do? (dy) dt do? dt * * cU dt * dt (cto?)* dy do? where, for brevity, (dy) is put for ~ and (dx) for -j^. d2y d2y dx2 dy d2x dt2 dx2 dt* da? d£2 * For more examples the student may refer to Jephson’s Fluxional Calculus, to Gamier s Calcul Differentiel, or to Puissant’s Problemes de Gtom&rie.CHANGING THE INDEPENDENT VARIABLE. 127 d *y dx2 _____(Jv) (Αχγ _ (dV> (to) - (d^) (Ay)' di* (da?){ ^ ' (d*)3, In a similar manner we might, if necessary, find the expression for d3w d?"· It appears, therefore, that dy __(dy) do? (dx) d'ty _ (dV) (d*) — ( the brackets denoting the values when x = a. * For brevity, Mn will be here used to denote the «th differential coeffi- cient derived from MAn.131 H+ --1 P FAILURE OF TAYLOR S THEOREM. Hence, by substitution, in the foregoing equations, [dF(ld*A-] = B + 2CA + 3DA* H---------M,A“-> + N,A + Ac.........(2). daF(a- + i) c + 2.M +................MjA*-11 Ν,Α + p dx2 * &c. + ] βΝ.+,θϊ ‘+icc.«£=±I &c. &c. all the succeeding differential coefficients being obviously infinite, because the exponent ~ — 1, which is already negative, continually diminishes by unity. (70.) It follows, therefore, that if the true development of F(jt + A), arranged according to the increasing exponents of A, contain for x = a132 THE DIFFERENTIAL CALCULUS. a fractional power of A, comprised between the powers Anand h "+1, then the several terms of this development will be correctly determined by Taylor’s theorem, as far as the term containing A" inclusively; but the terms beyond this become infinite, and therefore do not belong to the true development. If a term, in the true development of F(a -f A), contain a negative power of A, this should be the leading term, as the arrangement is according to the increasing exponents; therefore, this first term, when Λ· = α and A = 0, must be infinite, and consequently all the differential coefficients, (2), (3), &c. must be infinite. (71.) The converse of these inferences are true, viz. 1°. dn+l Y(x) general development of F(jt + A), the coefficient —--— If, in the is the first which becomes infinite for a particular value of x, then, in the true development, arranged according to the increasing exponents of A, the term immediately succeeding that which contains An, will contain a fractional power of A, the exponent being between n and n -f- 1.* For it is obvious, that in order that the n -f- 1th may be the first of the »+L coefficients (2), (3), &c. which contain a negative power of A, A p must be the first fractional power of A which enters the development (1), n -f ~ being between n and w+ 1. 2°. If, for a particular value of x, the function F(zr) become infinite, then will all the differen- tial coefficients become also infinite; and the true development will * This is of course on the supposition that the development is possible according to some powers or roots of the increment A, with finite coeffi- cients ; but when no such development is possible, then the appearance of rd«+rF(*)_ ^ dxn+l * will indicate that the coefficient of hn+x will itself involve some transcen- dental function of A not developable according to any powers or roots of A with finite coefficients.failure οί taylor’s theorem. 133 contain a negative power of A, or else a log h, a cot A, &c. For, if the true development of F(a -f- A) did not contain a negative power of A, nor a log A, a cot A, &c. F(a), which this becomes when A = 0, could not be infinite; hence, such a function of A must enter, and therefore, as shown above, all the differential coefficients become infinite, for x =a. It is, therefore, necessary to examine the function F(a), before we deduce the coefficients from F(.r). (72.) To obtain the true development of the function for those par- ticular values of the variable, which cause Taylor’s theorem to fail, the usual course is to recur to the ordinary processes of common algebra, after having substituted a -f A for x in F(#). Suppose, for example, the function were and that we required the development of F(.r + A), for x = a. Taking the differential coefficient, we have As, therefore, the first differential coefficient becomes infinite for the proposed value of x9 we conclude that the true development of the function for that value, when arranged according to the increasing exponents of A, has a fractional power of A in the second term, the exponent of this power being between 0 and 1; for it is plain that no transcendental function of A can enter. Substituting, then, a -f h for x in F(ar), we have F(:r) = 2ax — x2 -f a x1 — a2, >/x2 — a8 l0g(l~a) + I~“’ which, for x = a, becomes - —-; so that the true development of F(x -f- A) agrees with Taylor’s form, as far, at least, as the first two terms. Differentiating again, we have d2F(V) dx2 1 4x s/x -f 2 log (x — a) +3, which, for x = a9 evidently becomes infinite, inasmuch as log 0=— oo . Hence we infer that the true development differs from Taylor’s form atFAILURE OF TAYLORS THEOREM. 135 the third term. The first three terms are as follows: (a + A)i + AslogA = ai + y a-1A +(logA-j-^ a-*)As + *c. The third term involves log h, which we know is not capable of development according to the powers of A, (see p. 38). SCHOLIUM. (73.) In the preceding remarks on the development of functions for particular values of the variable, we have said nothing about the values of A, the increment of that variable, having indeed considered that in- crement as indeterminate, or rather of arbitrary value. It must, how- ever, be observed that, although the particular value which we give to x does not, in any case, fix the value of A, it may nevertheless fix the limit between which and 0 all the values given to A must be comprised, in order that for particular values of a, Taylor’s development may not fail.* This fact is very plain, for, if the development holds for all values of x from x = a up to x == b9 but fails for x = 6, in consequence of a differential coefficient becoming infinite, then will the development hold when a -f- A is substituted for x in F(a), provided A be taken between the limits A = 0 and A = b — a, or, more strictly, provided it does not exceed these limits, since throughout this interval [F(a)] and [F(,r + h)] necessarily involve the same radicals. In like manner, if the development hold for all values of x from x = b down to x = a, but fail for x = «, then will the development hold when b — A is substi- tuted for x for all values of b from A== 0, to A = b — a, but it will not hold for the value of A immediately succeeding this last; and it is obvious that A will always be subject to such restrictions unless the development holds, not merely for x = a, but universally. When, therefore, we find that for xz=a the differential coefficients do not any * It must be remembered that for Taylor’s theorem to hold for a = a, F(a) and F(a -f- A) must contain the same radicals.136 THE DIFFERENTIAL CALCULUS. of them become infinite, all that we can conclude is that the develop- ment of F(a ± h) is according to Taylor’s theorem for all values of h between some certain finite value h', which may indeed be indefinitely small, and 0, and it is only when this is not the case that the theorem is said by analysts to fail. We have thought it necessary to point out these circumstances to the student, seeing that some authors, from not attending to them, have fallen into very important errors, and have laid down erroneous doctrines with respect to the foiling cases of Taylor’s theorem. Thus Mr. Jephson, at page 191, vol. i. of his Fluxional Calculus,* a work containing much valuable information, says, ((It may further be observed that Taylor’s theorem always fails when the assigned value of x causes any of the terms to become imaginary, and that this may take place without causing the function itself to be imagi- nary ; thus take fx = c + x2 x — a if we suppose x = 0,fxz=c, f'x = 0, but f''x, f"x .... all contain —1.” From this it would appear that Taylor’s theorem may foil to give the true develop- ment in other cases besides those which cause the differential coefficients to become infinite, which, however, is not true. Whenever, for any particular value of x, Taylor’s coefficients become imaginary, we must infer, agreeably to the statement in (4), that the function F(# + h) becomes imaginary for that value of x; h being of course limited, as above explained. In the example just quoted, where ,_______ JL a2 f'x=.2* v x — a + ■. ' —» V x — a /#x = 2 n/ x — a -f * ■--------------——j > v x — a (λ· — a)* &c. / — a A2 -|- A3 -f- Ac., 2 V —a and this is the true development, for A must not exceed the limits A = 0, and A = e, since x — a causes the differential coefficient to become infinite, and therefore the development to fail. With regard to the failing cases of Maclaurin’s theorem, it may be observed that they are very different from the failing cases of Taylor’s. Whatever be the form of the proposed function, its general develop- ment, according to Taylor’s theorem, never fails; but the failure of Maclaurin’s theorem always arises from the form of the proposed function, and it is the general development that fails, and consequently all the particular cases. For it is obvious that every function will fail to be developable by Maclaurin’s theorem, whose form is such that, for x = 0, it, or any of the differential coefficients derived from it, becomes infinite. Before terminating these remarks it may be proper to observe that the student is not to attribute what analysts have been pleased to term the failing cases of these theorems to any defect in the theorems them- selves; on the contrary they would be very defective if they did not exhibit such cases. All that is meant is, that the function in particular states may fail to be developable according to Taylor’s series, and under particular forms it may fail to be developable according to Maclaurin’s series; so that, in fact, these theorems fail to give the true development only when that development is impossible. (74.) Let us now examine implicit functions, and let us suppose that x = a causes a radical to vanish from F(.r) in consequence of a factor of it vanishing; we have seen (68) that such radical will reappear in some of the differential coefficients; suppose it appears in the first, which requires that the factor spoken of be a: — c, then for x = a this coeffi- cient will have more values than the proposed function, as it contains a radical more. But if the function that F(.r) ory is of x be only implicitly given, that is by means of an equation without radicals, we know (52) dy Ithat the expression for will be also without radicals; and from138 THE DIFFERENTIAL CALCULUS. such an expression it does not at first seem clear how we are to deduce the multiple values alluded to, and which might be obtained by solving the equation for y, and thus introducing the radical. But since the d y expression for ^ appears under the form of a fraction, viz. (52), dy _ dx d u dx du * • · (O, dy we readily perceive that one case is possible, and only one, in which this fraction may take a multiplicity of values besides those implied in y, viz. the case in which it becomes $; that is, when the following conditions exist simultaneously, viz. du dx • · (2); so that these conditions are those which must exist for every value of x dy which destroys a radical in F(tf) but not in (75.) Whenever, therefore, any particular value of x destroys a radical d u in y, but not in then the expression (1) must take the form to admit of the necessary multiple values. The rules laid down in (41) enable us to determine the true value of the fraction (1) in the proposed circumstances, that is, when it takes the form for there is but one independent variable, viz. x. By diffe- rentiating numerator and denominator separately, we have, by the article referred to, d2w ( d2w dyFAILURE OF TAYLOR^ THEOREM. 139 d*w d2w_ dy d8a ^ da* dx dt/ da? dy8 This being a quadratic equation furnishes two values for [j“]> timi/in dy are all that, belong to the fraction (1) or to [^], unless, indeed, both numerator and denominator of (3) also vanish for a value of x given by the conditions (2), in which case we must differentiate again as in art. d« (41), when the value of [-jy will be given by an equation of the third degree which will furnish three values, and so on; and in general, if the fraction, or [^], admit of n values, the equation which determines them can be obtained only by differentiating n times, which will lead to an equation of the rath degree, and it is plain that the radical destroyed in F(x) must be of the -i-th degree, seeing that it gives to n additional values. Suppose, for example, we had the function y = x -j- (x — a) n/x — b, ,.4-ι + Λ=ϊ+-^, d,r n/ x—b and for x = a, [y] = a> [^]= 1 ±^a-b·, d y so that the radical which disappears in y appears in this, therefore, has two values. Now let the same function be given in an implicit form and without radicals, viz.140 THE DIFFERENTIAL CALCULUS. u = (y — x)2 — (x — a)9 (x — &) = 0, du du .·. ^ = —2 (y —!) — (* — «) (3*— 24 + a), — = 2fy — x). Since for x = a, y = a, therefore both these expressions become 0; dy 0 hence [g^] = —. Taking, then, the differentials of both numerator and denominator of the fraction, 2 (y — s) + (x—a) (3*— 26 "h «)___ 0 2(y — x) 0 , we find when j = a that it becomes ,4, = ,+ —‘ therefore, Φ-1 T^l. ,d!t. ··· φ=' as before. (76.) Suppose now that the radical which disappears from F(.r), by dy reason of a factor, disappears also from —, and that it appears in d9y which is the same as supposing that the vanishing factor is (jt — af. d x* In this case ^ will have the same number of values that F(x) has for d2y x = a, but additional values will belong to g-^-, so that we must have da·8 0 ~QFAILURE OF TAYLOR^ THEOREM. 141 and the true values, when the function is implicit, will all be determined by the principles already employed. For example: the explicit function y = jc -f (* — «)a gives when x = a, [y] = «,[|] = i,[S-] = ±W«, and we shall see that the same values are equally given by the implicit function (y — x)1 2 = (x — a)*x, d2y by applying the foregoing process to . For by successively differentiating, and representing, for brevity, the several coefficients derived from y by p', p", &c. we have 2(p'—-1) (y — x) = (x — ay (5x — a) O'] = 1 ( p' — 1)* + p" (y — x) = 2 (x — a)2 (5x — 2a) . r„//i r2 (* — α)2($Λ? — 2a)— (p' — l)2., 0 ••Cp] = C-----------------------------------j==T — Γβ(*—«)(&*—3a)—2p*(p'- 1) ___ 0 p' —1 ]“ 0" _ _60x — 48a — 2pm (y — 1) — 2/'% 12a —2 [ p"2] ~L p" ΟΊ .•.3[p"2]=12a [p"] = ± 2 v/a, as before. The two examples following will suffice to exercise the student in this doctrine, which is merely an extension of the principles treated in Chapter V. to implicit functions. 1. Given y3 = (x—- α)3 (λ? —6),142 THE DIFFERENTIAL CALCULUS. dv to determine the values of when x = a, Q&’ 2. Given (y — χ'Ϋ — (r — a)4 (x — b) = 0 dw d2v to determine the values of and ——, when x = a. or d#·* We here terminate the First Section, having fully considered the various particulars relating to the differentiation of functions in general.SECTION II. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE THEORY OF PLANE CURVES. CHAPTER» X. ON THE METHOD OF TANGENTS. (77.) We now proceed to apply the Calculus to Geometry, and shall first explain the method of drawing tangents to curves. The general equation of a secant passing through two points (x', y), (x", y'), *n any plane curve, is {Anal. Geom. p. 197), y—y y-y" x' — x" (x — x'), y’ — yn, being the increment of the ordinate or proposed function corresponding to x' — x"> the increment of the abscissa or independent variable. The limit of the ratio of these increments, by the principles of the calculus, is dy. d^’ that is to say, such is the representation of the ratio or quotient when x' — x" = 0, and, consequently, y' — y" = 0. But when this is the case the secant merges into the tangent. Hence the equation of the tangent, through any point (#', y) of the plane curve, is »—»'=% (χ-χ,> — 0)· dy It appears, therefore, that the differential coefficient ~ > for any pro-144 THE DIFFERENTIAL CALCULUS. posed point in the curve, has for its value the trigonometrical tangent of the angle included by the linear tangent and the axis of x, that is, provided the axes are rectangular. If the axes are oblique, the same coefficient represents the ratio of the sines of the inclinations of the linear tangent to these axes. (See Anal. Geom. p. 19.) By means of the general equation (1) we can always readily deter- mine the equation of the tangent to any proposed plane curve wheii the equation of the curve is given, nothing more being required than to determine from that equation the differential coefficient. Suppose, for example, it were required to find the particular form dy for the ellipse. We here have to determine from the equation and which is dy______BW # dx' AV * therefore the equation of the tangent is BV (*', y) being any point in the curve, and (x, y) any point in the tangent. Again let it be required to determine the general equation of the tangent to a line of the second order. By differentiating the general equation Ay* + B*y + c*'* + By + Ex'+f=o, we have 2A^'-S'+ ^-^ + + 2C*'+ D-g-+ E = 0, dy ^ 2 Cxf + By -{- E d*' 2Ay' 4- Bx' D ’ so that the general equation isMETHOD OF TANGENTS. 145 2(V + By 4-E (78.) As the normal is always perpendicular to the tangent, its general equation must be, from (1), It is easy now to deduce the expressions for the subtangent and sub- normal. For if, in the equation of the tangent, we put y = 0, the resulting expression for x — x' will be the analytical value of that part of the axis of x intercepted between the tangent and the ordinate y of the point of contact, that is to say, it will be the value of the subtangent Tή (ΑηαΙ. Georn. p. 84), If, instead of the equation of the tangent, we put y = 0 in that of the normal, then the resulting expression for x ·— x' will be the value of the intercept between the normal and the ordinate y% that is, it will belong to the subnormal Ny, As to the length of the tangent T, since T = >/y2 + T/*, we have, in virtue of (3), Also, since the length of the normal, N, is N = ^y,% + N/2, we have, by (4), ------(6). Ο146 THE DIFFERENTIAL CALCULUS. The foregoing expression^ evidently apply to any plane curve what- ever ; that is, to any curve that may be represented by an equation between two variables, whatever be its degree, or however complicated its form. We shall now give a few examples principally illustrative of the method of drawing tangents to the higher curves, for which purpose we shall obviously require only the formula (3), for it is plain that to any point in a curve we may at once draw a tangent, when the length and position of the subtangent is determined. Or, knowing the point ( r > y% we may, by putting successively x = 0 and y = 0, in the equa- tion (1), determine the two points in which the required tangent ought d»/ to cut the axes, and then draw it through them. If — is 0 at the proposed point, the tangent will be parallel to the axis of x, because, dV as remarked above, is the value of the trigonometrical tangent of the inclination of the linear tangent to the axis of x, and for this reason also the tangent will be parallel to the axis of y when — is infinite at the proposed point. EXAMPLES. (79.) 1. To draw a tangent to a given point P in the common or conical parabola. By the equation of the curve __ P_. dx 2y hence, at the point (xf, y')y we have B. A M X Hence, having drawn from P, the perpendicular ordinate PM, if we setMETHOD OF TANGENTS, 147 off the length, MR, on the axis of x, equal to twice AM, and then draw the line RP, it will be the tangent required. 2. To determine the subtangent and subnormal at a given point (a/, y) in the parabola of the nth order, represented by the equation 3. To determine the subtangent at a given point in the logarithmic, curve. The equation of this curve related to rectangular coordinates is which shows that if the abscissas x be taken in arithmetical progression, the corresponding ordinates y will be in geometrical progression; so that the ordinates of this curve will represent the numbers, the logarithms of which are represented by the corresponding abscissas, a being the assumed base of the system. Hence, calling the modulus of this base m, we have, by differentiating (13), Hence the remarkable property that the subtangent is constantly equal to the modulus of the system, whose base is a. y = ax11, dv ~ = naxn—{. d® dy _ i da? 4. To determine the subtangent at a given point in the curve whose equation is a3 — 3axy -j~ y3 = 0. Ar=3x» — 3ay —3 da? x tan Z P = y_ x d# » + y_ djj X Ax therefore r times this expression is the value of the polar subtangent, dy But the differential coefficient , which implies that x is the principal variable, ought to become, when the variable is changed to ω, (66), dy dy # Ax d JL· da> Ax d + y Ay Αω Now, from the above formulas of transformation. Ax dr Ay dr do; ’ Αω - cos ω — r sm ω, —~~ — —— Αω da sin ω + r cos ω, da? dr .... (1). y Αω —. r — sin ω cos ω — r* sin-4 ω Αω X dy dd> dr “"r“di7 - sin ω cos ω -j- r2 cos2 ω (2). da? dr (3). X · da> = r — - cos2 ω — r2 sin ω cos ω y- Ay Αω *\4 II - sin2 ω-fr3 sin ω cos ω (4), Subtracting (2) from (1), we get the numerator of the foregoing expres- sion for tan £ P equal to — r2; and adding (3) and (4), we get the o 2150 THE DIFFERENTIAL CALCULUS. denominator equal to r — ; consequently, — r r2 tan Z. P = —— ·*· RF = subtangent, dr dr da> da> Also FP2 r2 dr FN = —— = r2 -r- —· = — = subnormal. FR dr dw dw It may be proper to observe here that in our figure the tangent from P falls to the left of PF, thus making /. Ρ = ω — a. If, on the con- trary, the tangent fall on the other side, then we should have L P =*—ω, in which case ta„zr = -^· dw (81.) We shall apply these formulas to spirals, a class of curves always best represented by polar equations. Hence, if a represent the base of the Napierian system, since the modulus will be 1, the subtangent will be equal to the radius vector, and therefore the angle P will at every point of the curve be equal to 45°, because tan L P = i. Since, by the equation of this curve, log r = ω log a, it follows that, if a denote the base of any system, the various values of the angle, or cir- cular arc ω, will denote the logarithms of the numbers represented by the corresponding values of r. Hence, the propriety of the name logarithmic spiral.METHOD OF TANGENTS. 151 In this curve tan f_ P = r -j- = αω -f- ~ = m; d(t> 771 hence the curve cuts all its radii vectores under the same angle; and it is therefore sometimes called the equiangular spiral. 9. To determine the subtangent at any point in the Spiral of Archimedes, its equation being r = αω dr da> —- = αω2 = τω =T., άω so that FR is equal to the length of the circular arc to radius r, compre- hended between FR, FAj when, therefore, ω = 2ττ, the subtangent equals the length of the whole circumference. The spiral of Archimedes belongs to the class of spirals represented by the general equation r = αωη. When n — — 1, we have νω = a, and the spiral represented is called the hyperbolic spiral, on account of the analogy between this equation and xy = a. It is also called the reciprocal spiral. 10. To determine the polar subtangent at any point in the hyper- bolic spiral. T/ = «· 11. To determine the polar subtangent at any point in the spiral whose equation is r = ααΓ"^. τί=2βωέ = ^. ' r 12. To determine the polar subtangent at any point in the parabolic spiral, its equation being r = αω^. 13· To determine the polar subtangent at any point in a spiral whose equation is r ss aw*. «+i T,= r »152 THE DIFFERENTIAL CALCULUS. (82.) In these examples the expression for the polar subtangent only is employed; but that for the polar subnormal, deduced immediately from the former at page 150, is also of use. It serves to furnish us very readily with the length of the perpendicular drawn from the pole of the curve to the tangent through any point (r, ω), and, as this is an element frequently introduced into enquiries connected with Physical Astronomy, we shall here show how it may be obtained. Calling this perpendicular p, it is obvious, from similar triangles, that PN : PF :: PF : p, PF2 '''P~ pn“ But PN = n/pF2 -|- FN2, hence, introducing the expression for FN, V r2 or p3 r4 »-s + (i:)s Ndto> In the three curves of the second order, when the focus is the pole, we have, {Anal. Geom. p. 137), 2m 1 -}- e cos ω dr from which ~, being obtained and substituted in the expression for p2, will give Λ 4m2r »2 =----------------—, F 4m — r (1 — e3) which, when e = 1, that is, when the curve is a parabola, becomes p% = mr, where m is the distance, FA, of the focus from the vertex. Hence we have, in the parabola, p2=FP.FA.RECTILINEAR ASYMPTOTES. 153 When e < 1, that is, when the curve is an ellipse, in which case 2Ba , . B2 m = ——, and 1 — e2 = ——, we have A A* . BV BCa. FP F 2A —r FP When e > 1, that is when the curve is an hyperbola, and consequently B2 1 — e» = — then a_ BV _BCS. FP P — 2A+»·- F'P Rectilinear Asymptotes. (83.) -4 rectilinear asymptote to a curve may be regarded as a tangent of which the point of contact is infinitely distant, so that the determi- nation of the asymptote reduces to the determination of the tangent on the hypothesis that either or both y' = co, x' = co, the portions of the axes between the origin and this tangent being, at the same time, one or both finite. The equation of the tangent being , or y2 = ~ — 2> it is plain that x = + a renders y= co, we infer, therefore, that the curve represented by this equation has two asymptotes, each parallel to the axis of y, and at the distance a from it. If both expressions are infinite, there will be no asymptote corres- ppnding to x = 00 . If both expressions are 0, the asymptote will pass through the origin, dy and its inclination 0, to the axis of x will be determined by —,=tan 0. If for y' = 00 one or both of the above general expressions are finite, there will be an asymptote, and its position may be determined as in the foregoing cases. EXAMPLES. (84.) 1. Let the curve be the common hyperbola, of which the equa- tion is y — — a2— a2, a d y bx 00 a ^x2 — a2 hence the general expressions (83) are x2— a2 o2 x —--------= — x XRECTILINEAR ASYMPTOTES. 155 and ba “ «Λ.2_α2 I „»■ 1 — -T both of which are 0, when x = « ; hence an asymptote passes through the origin. Also dy _ , b 1 dx - “ a ’ J-S when x = ao, therefore; a the inclination of the asymptotes to the axis of x, they are both repre- sented by the equation b y = H-----X, y a 2. To prove that the hyperbola is the only curve of the second order that has asymptotes. The general equation of a line of the second order, when referred to the principal diameter and tangent through the vertex as axes, is y2 = mx -f- nx2, ____ 2y2 mx + 2 nx2 — 2 y2 mx dy m + 2nx m + 2wx m -f- 2nx dx y dy____mx-\-2nx2 2y2—mx — 2 nx2 dx ^ 2y 2 y mx_____ 2 s/mx -f- nx2 Dividing numerator and denominator of each of these expressions by x, they reduce to and ----h 2 n - + Λ and these, when x = <®, or indeed when y = ao, become156 THE DIFFERENTIAL CALCULUS. Hence the curve will have asymptotes, provided n be neither 0 nor negative, that is to say, provided the curve be neither a parabola nor an ellipse, but if it be either of these, there can exist no asymptote ; therefore the hyper- bola is the only line of the second order which has asymptotes. (85.) When the curve is referred to polar coordinates, then, since the radius vector of the point of contact is infinite when the tangent becomes an asymptote, it follows that if for r = co the subtangent is finite; this subtangent may be determined by (80) in terms of o>, and ω may be found from the equation of the curve, so that there will thus be deter- mined a point in the asymptote and its direction, which is all that is necessary to fix its position. There will always be an asymptote if ω is finite, for r = co. If, for r = co, ω is also co, there exists no asymptote. 3. Let the curve be the hyperbolic spiral. By ex. 10, art. 81, the subtangent at any point is constant, and equal to a, therefore there must be an asymptote; also by the a equation of the curve ω == — =0, when r = oo , T therefore the asymptote is perpendicular to the fixed axis, at the distance a from the pole. Neither the logarithmic spiral, nor the spiral of ΛπϊΜη^Ιββ have an asymptote. 4. Let the spiral whose equation is ___ αω2 a ω2 — 1 1—ω~2 be proposed, which admits of a rectilinear asymptote, because ω = 1 renders r = c®. The direction, therefore, of the asymptote is ascertained, and consequently the direction of the infinite radius vector, since they must be parallel. It remains, therefore, to determine the subtangent, or distance of the asymptote from the pole dr 2αω—3 ______ 2r3 d<«> (1 — ω—2)2 αω3 9 because ω = 1 when r =s oo .RECTILINEAR ASYMPTOTES. 157 (86.) Although we do not propose to treat fully in this place of curvilinear asymptotes, yet we may remark in passing, that if r should be finite although ω be infinite, it will prove that the spiral must be continued for an infinite number of revolutions round its pole, before it can meet the circumference of a circle whose radius is this finite value. In such a case, therefore, the spiral has a circular asymptote. If, more- over, the value of r for ω = co be greater than the value of r for every other value of ω, the spiral will be included within its circular asymptote, but, otherwise, it will be without this circle. 5. Thus in the spiral whose equation is (r2 — ar) ω2 — 1=0, or ω = * » \/r2 — ar ω is infinite when r = ay and for all less values of r, ω is imaginary; hence the spiral can never approach so near to the pole as r = a, till after an infinite number of revolutions; so that the circumference whose radius is a is within the spiral, and is asymptotic. If, on the contrary, the equation had been 1 ω = -/ '■ ....> V ar—r2 then also r s=a gives ω = oc, but for all other real values of ω, r is less than a, so that this spiral is enclosed by its asymptote circle, the radius of which is a. 6. To determine the rectilinear asymptote to the logarithmic curve. The axis of x. 7. To determine the equation of the asymptote to the curve whose equation is y3 = ax2 -f- x3. The equation is y = x + i a. 8. To determine the rectilinear asymptote to the spiral whose equation is r = ai*T The fixed axis is the asymptote. p158 THE DIFFERENTIAL CALCULUS. 9. To determine whether the spiral shown to have a rectilinear asymptote in ex. 4 has also a circular asymptote. The circle whose centre is the pole and radius = a is an asymptote. (87.) Before terminating the present chapter, it will be necessary to exhibit the expression for the differential of the arc of any plane curve, as we shall have occasion to employ this expression in the next chapter. Let us call the arc AB of any plane curve s, and the coordinates of B, x, y; let also BD be a tangent at B, and BC any chord, then if BE, ED are parallel to the rectangular axes, BC will be increment of the arc $, corresponding to BE = h9 the increment of the abscissa x. Now, putting tan DBE = a, we have ED = ha .·. BD = n/a2 + A2a2 and BD + DC s/A2(l + -f CE BC “ n/a2 4- CE2 CE This ratio continually approaches to or to unity, as h diminishes. and this it actually becomes when h = 0. Consequently, since the arc BC is always, when of any definite length, longer than the chord BC, and shorter than BD + DC,* it follows that, when h = 0, the ratio of the arc to either of these must be unity; therefore in the limit arc BC chord BC arc BC chord BC but chord BC *J^2. J££?=· The parameters which enter (2) being arbitrary, they may be deter- mined so as to fulfil as many of the conditions y = Y, dy_dY (Py _daY da? cb?> dx2 dx2* (3), as there are parameters, but obviously not more conditions. We shall thus have the values of A, B, C, &c. in terms of x, and of the fixed parameters a, b, c, &c.; which values, substituted in (2), will cause so many of the leading terms in both series to become identical, whatever be the value of x. Other corresponding terms of the two series may, indeed, be rendered also identical, but this can take place only accidentally, not necessarily. Hence, whatever particular value we now give to x, the resulting values of the corresponding coefficients will necessarily agree to the extent mentioned, that is, as far as the n first terms, if there are n constants originally in (2); and this is true, even if such particular value of x render any of the coefficients infinite, inasmuch as they are always identical as far as these terms, but no further. We know, however, that in those cases where any of the coefficients become infinite, (1) and (2) will fail to represent the true developments of the ordinates y, Y' at the proposed points. Nevertheless, as the two series have been rendered identical, as far as n terms, should they both fail within this extent, the terms which supply these in the true develop- ment, must necessarily be identical. (See note C at the end.) Now the greater number of leading terms in the two developments, which are identical, the nearer will the developments themselves ap- proach to identity, provided, at least, A may be taken as small as we please; for if n — 1 terms in each are identical, we may represent the difference of the two developments byOSCULATION, AND RADIUS OF CURVATURE. 161 A.Aa + S — (A'„A“'+ S'), or AnA“— A\ha' + S — S' . . . . (4), where S, S' represent the sums of the remaining terms in each series after the nth. Hence, Λα being the lowest power of h which enters this expression, for the difference it follows from (47), that a value may be given to h small enough to cause the term Anha to become greater than all the other terms in (4), and consequently, for this small value, A„Art — A'nh*' > S —S', and, therefore, the whole difference (4) is greater than twice S — S', but when the nth term is the same in both developments, as well as the preceding terms, then the difference (4) is reduced simply to S — S', which we have just seen to be less than (4). Consequently the de- velopments approach nearer to identity, for all values of h between some certain finite value hi and 0, as the number of identical leading terms become greater. When the first of the conditions (3) exist, the curves have a common point; when the second also exists they have a common tangent at that point, and are consequently in contact there, and the contact will be the more intimate, or the curves will be the closer in the vicinity of the point, as the number of following conditions become greater; so that of all curves of a given species, that will touch any fixed curve at a proposed point with the closest contact whose parameters are all deter- mined agreeably to the conditions (3). No other curve of the same species can, from what is proved above, approach so nearly to coinci- dence with the proposed, in the immediate vicinity of the point of con- tact, as this; so that no other of that species can pass between this and the proposed. A curve, thus determined, is said to be, in reference to the proposed curve, its osculating curve of the given species. (89.) It appears, from what has now been said, that there may be different orders of contact at any proposed point. The first two of the conditions (3) must exist for there to be contact at all; therefore, when only these exist, the contact is called simple contact, or contact of the p 2162 THE DIFFERENTIAL CALCULUS. first order; if the next condition also exist, the contact is of the second order, and so on; and it is obvious, that of any given species, the oscu- lating curve will have the highest order of contact, at any proposed point, in a given curve. Tf the curve, given in species, has n parame- ters, the highest order of contact will be the n — 1th, unless, indeed, the same values of these parameters that fulfil the n conditions (3)> should happen also to fulfil the n + 1th, the n + 2th, &c.; but this, as observed before, can take place only accidentally, and cannot be pre- dicted of any proposed point, although we see it is possible for such points to exist. (90.) At those points in the proposed curve, for which Taylor’s de- velopment does not fail, contact of an even order is both contact and intersection, and contact of an odd order is without intersection; before proving this, however, we may hint to the student that contact is not opposed to intersection, for two curves are said to be in contact at a point, when they have a common tangent at that point; and yet, as we are about to show, one of these curves may pass between the tangent and the other, and so intersect where they are admitted to be in contact. To prove the proposition, let us take the difference (4), which, when Taylor’s theorem holds, is (A„ — A'„)Aa + S — S' .... (5), An A'„ being here the n — 1th differential coefficients. If these are odd, the contact is of an even order; also & being odd, hA will have con- trary signs for A= +A' and — ti, and, therefore, since for these small values of h, the sign of the whole expression (5) is the same as that of the first term, the difference of the ordinates corresponding to x h, and to x — h, will be the one positive and the other negative, so that the two curves must neces- sarily cross at the point whose abscissa is x. But if a is even, the contact is of an odd order, and the difference (5) between the ordinates of the two curves corresponding to the same abscissa, x + h, will, for a small value of h, have the same sign, whether h be positive or negative; so that, in this case, the curves do not cross each other at the point of contact.OSCULATION, AND RADIUS OF CURVATURE. 163 (91.) The student must not foil to bear in remembrance, that the proposition just established comprehends only those points of the pro- posed curve, at which none of the differential coefficients become infinite, from the first to that immediately beyond the coefficient which fixes the order of the contact. For it is only upon the supposition that the true development, within these limits, proceeds according to the ascending integral and positive powers of h, that the foregoing conclusions respect- ing the signs of the difference (5) can be fairly drawn. (See note C.) (92.) From the principles of osculation now established, it is evident that any plane curve being given, and any point in it chosen, we may always find what particular curve, of any proposed species, shall touch at that point with the closest contact, or which shall most nearly coin- cide with the given curve in the immediate vicinity of the proposed point. Thus, an ellipse or a parabola being given, and a point in it proposed, we may determine the circle that shall approach more nearly to coincidence with that ellipse or parabola in the vicinity of the pro- posed point, than any other circle, and which will therefore better represent the curvature of the given curve at the proposed point than any other. On account of its simplicity and uniformity, the circle is the curve employed to estimate, in this way, the curvature of other curves at proposed points; that is, the curvature is estimated by the curvature of the osculating circle, or rather as the curvature of a circle increases as the radius diminishes, and vice versa, it is usual to adopt, as a fit representation of the curvature, the reciprocal of the radius. The osculating circle is called the circle of curvature, and its radius the radius of curvature; and, from what has been said above, it follows that the determination of the curvature at any point in a proposed curve, reduces itself to the determination of this radius: to this, therefore, we shall now proceed.164 THE DIFFERENTIAL CALCULUS. Radius of Curvature. PROBLEM I. (93.) To determine the radius of curvature at any proposed point of a given curve. The general equation of a circle being (*-*)2+(ί-/3)2 = Λ it becomes determined as soon as we fix the values of the parameters ar β, r, and these may be determined, so as to fulfil any three indepen- dent conditions, but not more. In the case before us, the conditions to be fulfilled are those of (3), art. (87) : that is to say, putting p', p% &c. for the successive differential coefficients derived from Y = F(x), the equation of the given curve, the conditions to be fulfilled are *=Y’ d£=f dy _ d' (y—/3) = ° .... (2), yCy-^^-r2---------(3). From the second equation (* — *)2 = y'2(y — β)Κ Substituting this in the first,RADIUS OF CURVATURE. 165 (P'*+ l)(y-i8)2=r* Adding this last to the third, there results P'* + 1 y-P which, substituted in (2), gives x _a= £&* +!) p" Consequently, — p'(p* + 1> (/’ + l)3 β=2/ + £1+1, p" (/2 + i)l ds3 dx3 N3 (See p. 159). These equations completely determine the osculating circle, whenever the coordinates x, y of the proposed point are given. Should this point be such as to render p = 0, then the expression, for the radius of curvature at that point, becomes r = — and the tangent at the proposed point must be parallel to the axis of x (78). The sign of p" will always make known the direction of curvature; for, if a linear tangent be drawn at the proposed point (x, y), we shall have for the ordinate of the tangent, due to the abscissa x ·+· A, the expression y +p'h; and for that of the curve, due to the same abscissa, y + p h + ip"A2 + &c.; and it is obvious that if p" is plus, this latter ordinate, for a small value of h, must exceed the other; and thus the tangent will proceed from the point in a direction between the curve and axis of x9 that is, the curve will, at that point, be convex towards the axis of x. If, on the contrary, p" is minus, then the ordi- nate of the curve will be less than that of the tangent; so that the curve proceeds from the point between the tangent and axis of x, and is thus concave towards that axis. Should p" = 0 at the proposed point, r will be infinite, whether j/= 0 or not, so that the osculating circle then becomes a straight line; as, therefore, this straight — —' line has contact of the second order, the parts of the ( curve in the vicinity of the point will lie on contrary sides of it, as in166 THE DIFFERENTIAL CALCULUS. the annexed diagram (90); that is, supposing pe" is neither 0 nor λ . If pf"=z[o, and the next following differential coefficient neither 0 nor oo, the contact will be of an order which is unaccompanied by intersection. The curve will be convex at the point if the first finite differential coeffi- cient be of an even order and -f-; and it will be concave when the sign of this coefficient is —. A point at which the tangent intersects the curve, or at which the curve changes from convex to concave, is called a point of inflexion, or, a point of contrary flexure. The analytical indications of such points will be more fully enquired into when we come to speak of the singular points of curves. (94.) By referring to equation (2) above, which has place even when the contact is but of the first order, we learn that the centre (λ, β) of every touching circle, is always on the normal at the point of contact; for that equation is the same as <*-»’— dx We shall now apply the general expression, for the radius of curva- ture, to a few particular cases. EXAMPLES. (95.) 1. Required the radius of curvature, at any point in a parabola» Differentiating the equation of the curve, fx=z4mxt we have 2yp' = 4m .·. o' =----- y Vyp" + 2pf* = 0 .·. —1-, y _ N3 _ N3 r— y* pn 4to* As the expression for the normal diminishes with x, the vertex is the point of greatest curvature, r being there equal to 2m, half the parameter; moreover, p" being negative, the curve is always concave to the axis of x.RADIUS OF CURVATURE. 167 2. To determine the radius of curvature at any point in an ellipse. By differentiating the equation ay + &V = a262, we have a?ypf b2x = 0 .·. p' = — b*x ~a?y b* 4- «V2 b4 +«y2+*=o .·. f=- __(pn + i)^ («v2+ft4*2)^ <«y (<*4j2 + r— f/' α6,3 44 e4i« .(1). or, using the expression for r in terms of the normal, we hare Jil=2lN3 j^p" b4 (2).* Since (Anal· Geom., p. 87), bfi aN = bit .·. r = —— .... (3). ab y ba At the vertex of the transverse axis r == — = semiparameter = the a normal, (Anal· Geom.y p. 69). ft2 At the vertex of the conjugate r=—· b From equations (2) and (3) it follows that, in the ellipse, the radius of curvature varies as the cube of the normal, or as the cube of the diameter parallel to the tangent through the proposed point. It is often desirable to obtain rasa function of X, the angle included between the normal and the transverse axis. For this purpose we have, since * This expression will become identical with (1) if we substitute for N3 its value given in the Analytical Geometry, p. 84, viz.168 THE DIFFERENTIAL CALCULUS. a:2 = α2 (1—η^), y2 = N4sin2X, and (Anal. Geom. p. 84) Ns = + N2 sin2 λ b2 ) + N2sin2X, ... N»{l-(1-A’)sin=\} = ^; but, (Anal. Geom., p. 68^) a (1—e9sin9X)* ... r=a42N3:= * _ = ^ a (1 — e2 sin4 X)^ (1 — e2 sin9 X)^ The curvature — is greatest at the vertex of the transverse, and least b2 at the vertex of the conjugate axis. At the former point r = —, and at a9 the latter r = ~, as shown in the preceding page. 6 b2 1--=*>, b2 (96.) The present is a very important problem, being intimately connected with enquiries relative to the figure of the earth. By means of the last expression for r, the ratio of the polar and equatorial diameters of the earth, may be readily deduced, when we know the lengths of a degree of the meridian in two known latitudes, L, l, for these lengths may, without error, be considered to coincide with the osculating circles through their middle points; and, since similar arcs of circles are as their radii, we have, by putting M, m for the measured degrees, and R, r for the corresponding radii, R : r :: M : m,RADIUS OF CURVATURE. 169 but R = -?.(1-ga) and, = -a(1-ea) , (1 — e2 sin2 L)2 (1 — If, therefore, we differentiate on the supposition that ω is the indepen- dent and γ the dependent variable, and denote the first and second differential coefficients by p, and p„, we shall have (y cos ω—&) (prosit)—y sinw)4-(y sin ω—β)(ρ/8Ϊηω+γ cosa>)=0...(2) (p,cos ω —y sin o>)24-(y cos ω — Λ) (pe cos ω— 2j0, sin ω—y cos ω) 4* (p, sin ω-\-γ cos ω)24"(y si*1 ω—j3) sin <*>4-2p, cos ω —y sin <*>)==0...(3) If from the two latter equations we determine the values of y sin ω — β and y cos ω — a, and substitute them in (1), we shall obtain the follow- ing expression for r in functions of y and its differential coefficients, viz. Q 2174 THE DIFFERENTIAL CALCULUS. r (r3+;>,*)* γ* + 2ρ*—γρ„ but we shall arrive at this expression more readily by first deducing from the equations y = y sin ω, x = γ COS ω the differential coefficients = y cos ω +p, sin ω = (dy) ^ = -y8in/“)*, we may put the above expression for r under the form . N3 ' ·* + ty? — ΊΡ» ------(5), 5. To determine the radius of curvature at any point in the loga- rithmic spiral ω y =a λ ω αγ ___a ____ y ___ dw m d*r _ y _ dw ~mi~p,r HenceRADIUS OF CURVATURE. 175 r — (y* + pf)i . j I „jvi _ y J i _i_ J r“r* + V-m,_Cr +Λ) -r>l +»J r Λ 1 + tonyp («»·<· 80) = y cosec P. It appears, therefore, that the radius of curvature is always equal to the normal. 6. To determine the radius of curvature at any point in the curve whose equation is y = 2 cos ω ± 1 (5 ± 4 cos ω)* Γ SI —— ....— · 9 ± 6 cos ω (101.) The general expression for the radius of curvature at any point (γ, ω) of a curve referred to polar coordinates, may be concisely exhi- bited in terms of y and p, the radius vector and perpendicular from the pole upon the tangent. For differentiating the expression for p at (82), and substituting p, dr for — , and y for r, we have dp ^ 2yp,pf + y3p, — y*p,p„ dw (r2+/>,’)* which is obviously equal to the reciprocal of the expression for r in equation (4), multiplied by yp,. Consequently, r = a^-= —; dp dp do> dw that is, (p. 127,) If from the centre of curvature a perpendicular were drawn to the radius vector, it is plain that the angle included between this perpen-176 THE DIFFERENTIAL CALCULUS. dicular and the radius of curvature would be equal to P, the angle included between the radius vector and tangent.* The perpendicular here supposed to be drawn, would obviously bisect that chord of the osculating circle which passes through the pole, and which, for dis- tinction, is called the chord of curvature. Hence, from the similar right-angled triangles, we should have the following equality, viz. P dy λ chord of curvature = r — = p ~r~> Ύ r dp an expression which is of use in Astronomy .f CHAPTER XXX. ON INVOLUTES, EVOLUTES, AND CONSECUTIVE CURVES. (102.) If osculating circles be applied to every point in a curve, the locus of their centres is called the evolute of the proposed curve, this latter being called the involute. The equation of the evolute may be determined by combining the equation of the proposed curve with the equations (2), (3), p. 129, con- taining the variable coordinates α, β of the centre. As these three equations must exist simultaneously for every point of contact (x, y), the two quantities x> y may be eliminated, and, therefore, a resulting equation obtained, containing only a and β, which equation, therefore will express the general relation between a and β for every point (j, y); in other words, it will represent the locus of the centres of the osculating circles. * Although neither this perpendicular, nor that from the pole to the tangent, called p, are actually exhibited in the figure at page 148, yet the student, bearing in mind that the centre of curvature is always on the normal, will find no difficulty in supplying these two lines. t See Young’s “ Treatise on Mechanics,” p. 212.ON EVOLUTES. 177 Or, representing the equation of the proposed curve by y = F(jt), we shall have to eliminate x and y from the equations (p. 130) F'0^+1) r ' P = y + P" y = F<>), when the resulting equation in α, β will be that of the evolute. examples. (103.) To determine the evolute of the common parabola »* = 4m* .·. P' = ~~ ···?"=— -pr> 1 + f,*=*l±±2l = 1 + 5L, £ = -£, y* x p 2 m w2 a — 2wi .\ a = a? -f- --f- 2m =3x-f- 2m .·. x =---> 1 2m r 3 /j = y_ -iL _y = ^£.3 = ...X = tl'E), H * 4m2 * y y* £ k 4 ' m — m* •·ρ=^α-2ι,ν> which is the equation of the evolute. If the origin be removed to that point in the axis of x whose abscissa is 2m, then the equation becomes ]32 = 4 27m the locus of which is called the semicubicalparabola. It passes through the origin because /3=0 when a—0 j therefore the focus of the proposed involute is in the middle, between its vertex and the vertex of the evolute. (Anal. Geom. art. 100.) The curve con-178 THE DIFFERENTIAL CALCULUS. sists of two branches symmetrically situated with respect to the axis of x, or of a, and lies entirely to the right of the origin, for every positive value of a gives two equal and opposite values of β, and for negative values of a and β is impossible. It is easy to see, therefore, that the form of the curve is that represented in the margin. 2. To determine the evolute of the ellipse. By example 2, page 16?, we have ' — _ —_____hl_ ^ a2y * ^ a2y3 * . i , _ «V + p' xy2 • * 1 T V —-----Γ1----’ “77 — aAy2 p b2 . Λ_____ 1/(ey+*v> ··*“* -----—,ts~y-----------------7Φ------- Now, since, by the equation of the curve, a2y2 = aH»2 — b2x2 or b2x2 = a%2 — a2y2, .·. a*y2 -f- 64i2 = b2 (a4 — c2x2) or = a2 (b4 -f- c2y2), c3 being put for a2 — 62. Hence, by substitution, c2x3 _ c2u3 * == ~π*~’ β ~—h Substituting these values in the equation of the involute, we have ..^ι+„φΐ=^. a2b2 or, finally, dividing all the terms by----------, we obtain for the evolute the 3 equation (6/3)* + (aa)$ = J = (a* —ON EVOLUTES. 179 If λ = 0, then β = ± —, so that the curve meets b the axis of y in two points, c, d, equidistant from the origin O. If β = 0, then a = ± —, so that it a also meets the axis of x in two points, b, a, equi- distant from 0. If a is numerically greater than c2 — the ordinates become imaginary, and if β is numerically greater than — the abscissas become imaginary; therefore the curve is limited by the b four points a, b, c, d, and touches the axes at those points. It consists, therefore, of four branches symmetrically situated as in the figure. 3. To determine the evolute of the rectangular hyperbola, its equa- tion between the asymptotes being xy = a2. The equation of the evolute is (4α)1. THEOREM. (104.) Normals to the curve are tangents to the evolute. Let the equations of the curve and of its evolute be y=zV(x) and /3 = Φ(λ). The normal to the curve is y—y = — “t (*—*')· The tangent to the evolute is These lines both pass through the point (*, β) since the centre of cur- vature is always on the normal. We have therefore to prove that180 THE DIFFERENTIAL CALCULUS. d/3 _______1^ da p' p'* 4-1 In order to this let us put for shortness ■■■ ——f, then, (page 165,) a=x—pff, β = \i+f, Ψ ••■S"1 -p'i d/3 , , 4/ · dx p dx * * da pf> which was to be proved. THEOREM. (105.) The difference of any two radii of curvature is equal to the arc of the evolute comprehended between them. Differentiating the equation (,V — β)2 + 0* — »)2 = r2, on the hypothesis that a is the independent variable, we have da dr da But equation (2), page 164, <)£ββί· * It will be remembered thatON EVOLUTES. 181 hence but by last article, / , N clr y_/3 = (*-a)g, and Cx _ «)’ (ig- + 1} = r* . . . (1). -<*—> ··(*>· Dividing (2) by the square root of (1), we have that is (87), dr t , . ----— = — .·. —« = r±a constant, da da for otherwise there could not be — ^ (1 a da Hence if r, r' be the radii of curvature of any two points, and s, s' the corresponding arcs of the evolute, then r ± const = — s ' r' ± const = — s' so that the difference of the two radii is equal to the arc of the evolute comprehended between them; therefore, if a string fastened to one extremity of this arc be wrapped round it and continued in the direction of the tangent at the other extremity as far as the involute curve, the portion of the string thus coinciding with the tangent will by R182 THE DIFFERENTIAL CALCULUS. (104) be the radius of curvature at that point P of the involute curve which it meets, and, consequently, by the above property, if the string be now unwound, P will trace out the involute. On Consecutive Lines and Curves. (106.) Every equation between two variables may always be con- sidered as the analytical representation of some plane curve, given in species by the degree of the equation, and determinable both in form and position by the constants which enter it, provided, that is, these constants are fixed and determinate. If, however, the equation contains an arbitrary or indeterminate constant a, then, by assuming different values for λ, the equation will represent so many different curves varying in form and position, but all belonging to the same family of curves. Now if we consider the form and position of one of these curves to be fixed by the condition a = λ', another, intersecting this in some point, (r', y), may be determined from a new condition a = + h; and if h be continually diminished, this latter curve will approach more and more closely to the fixed curve, and will at length coincide with it. During this approach, the point of intersection (f, y') necessarily varies, and becomes fixed in position only when the varying curve becomes coincident with the fixed curve. In this position the point is said to be the intersection of consecutive curves, so that what mathematicians call consecutive curves, are, in reality, coincident curves, and the point which has been denominated their point of intersection may be deter- mined as follows: (107.) Let F (x,y,x') = 0 .... (1) represent any plane curve, xf being a parameter, and for any intersecting curve of the same family let xf become x' + h, then, since however numerous these intersecting curves may be, the x, y of the intersections belong also to the equation (1); it follows that as far as these points are concerned, the only quantity in equation (1) which varies is x', therefore, considering x, y as constants in reference to these points, weCONSECUTIVE CURVES. 183 have, by Taylor’s theorem, FO, y.x' + h) = F(*,y, τ') + **<·£>?> A + d»F(*,y,y> A* , d*'s 1 . 2 + but F(a, y, a·7) = 0, therefore F(a, y, af -f A) _ dF(a, y, a/) , d2F(j?, y, x') A , e A dr7 + da7* 1 . 2 + but F(a, 3/, a' 4- h) = 0, therefore Λ dF (a, y, *') d2F(a,y,a') A , ^ _ 0 = —d?—+—S?*~ ΓΤ2 + *°· hence, when the curves are consecutive, that is, when A = 0, we have the following conditions, viz. to determine a and y. Suppose, for example, it were required to determine the point of intersection of consecutive normals in any plane curve. Representing the equation of the curve by and any point in the normal by (a, y)9 we have for the equation of the normal, This corresponds to the first of equations (2), a' being the parameter; hence, differentiating with respect to a', of which y' is a function given by the equation of the curve, we have184 THE DIFFERENTIAL CALCULUS. (y —/)/—p'2 —1==0· , , £±I •••y=y' + p V hence (93) consecutive normals intersect at the centre of curvature. (108.) If we eliminate the variable parameter x' by means of the equations (2), the resulting equation will belong to every point of inter- section given by every curve of the family w=F(*,y,a/) = 0 .... (1), and its consecutive curve; for whatever value we suppose x? to take in the equations (2), the result of the elimination will obviously be always the same. Hence this resulting equation represents the locus of ail the intersections, and we may show that at these same intersections this locus touches every individual curve in the family. The equation (1), where x' represents a function of x, y, determined by the second of the conditions (2) in last article, is obviously the equation of the locus of which we are speaking, and the same equation, when xf takes all possible values from 0 to ± oo, furnishes the family of curves, which we are now to show are all touched by this locus. Taking any one of this family, and differentiating its equation (1), / being constant, we have , du , , du , du = — d# 4- —- du = 0. dx dy Differentiating also the equation (1) of the locus, x' being given by the second condition of (2) in last article, we have , du , , du . , du . . du s= — dx + — dy + — dx = 0, dx n dy * 1 dx' but by the condition just referred to -7-7- = 0, at the point where the dj· curves whose equations we have just differentiated meet in this conse-consecutive curves. 185 cutive position; hence, since at those points each of these equations dy give the same value for it follows that they have contact of the first order; we infer, therefore, that the equation (1), when x' is determined from the second of the conditions (2), last article, represents a curve which touches and envelopes, or is enveloped by, the entire family oj curves represented by the equation (l), x being any arbitrary constant. Thus, as we already know, the locus of the intersections of normals with their consecutive normals is a curve which touches them all at their points of intersection, being the evolute of the curve to which the normals belong. The following examples will further illustrate this theory. EXAMPLES. (109.) 1. To determine the curve which touches an infinite series of equal circles, whose centres are all situated on the same circumference. Let the equation of the fixed circle be a* + /3*=Ra .... (1,) the centre of that circle being the origin of the coordinates. The equation of the moveable circle, (*, β) being the varying centre, * will be (*—«)* + (y—β)1—** = o. But as (λ, β) is always on the fixed circumference (1), we must always have between a and β the relation a* + /3* = R» .·. β = Hence by substitution, (i — ay + (y — Vr*—<*2)2 — i-J = 0=!« .... (2). Here a is the variable parameter, and, consequently, the equation repre- sents the entire series of moveable circles j if, therefore, we differentiate R 2186 THE DIFFERENTIAL CALCULUS. this equation with respect to a, and equate the result, —, to zero, this dec new equation, combined with (2), will enable us to eliminate *, and the result of this elimination will, by the preceding theory, be the analytical representation of the curve which touches all the moveable circles at those points where each is intersected by its consecutive circle. Now ^ = - (X - a) + Or - (Ri _ ,«)"*» — (*—«) + — »=o, vR3—a —x R*—a* -|-ey=0, .·. RV = «»(*’+ v a2 + ya this substituted in (2), or in its development, x2 — 2ax + y2 — 2y VR2—a2 + R2 = r*, givee 2Rx* R2i2 ** + ? —7=- W {**- π-}) + Rs=>·’> ·/*»+! *® + ys whence and, consequently. *’»+ y* — + r» _ rit Vi2-l-y2 X* + y2 — 2R \/x2 -f- y2 + R* = r2, VdT+y2 —R==±r, x2+i/2=(R±r)2, an equation representing two circles, whose radii are respectively R -f rCONSECUTIVE CURVES. 187 and R — r. Hence the series of circles are touched and enveloped by two circular arcs, having these radii, and the same centre as the fixed circle. 2. Between the sides of a given angle are drawn an infinite number of straight lines, so that the triangles formed may all have the same surface; required the curve to which every one of these lines is a tangent. Let the given angle be 0, and, taking its sides for axes, we have, for the equation of every variable line, y=zctx + /3 .... (1), and, putting successively, y = 0 and x = 0, the resulting expressions for x and y denote the sides of the variable triangle, including the given angle, so that these sides are — — and β: hence, calling the constant surface s, we have β3 s = — ■— sin 0 .·. , 2a β2 sin0 2s ’ hence the equation (1) is the same as y β2 sin 0 2s *+β · · · · (i), where β is considered as an arbitrary constant. But if for this arbitrary constant we substitute the function of x, arising from the condition ~ = 0, then (2) will represent the locus of the intersections of each variable line, with its consecutive line, which locus touches them all. Differen- tiating then with regard to β, we have β sin Θ s x + ] = 0 .·. β s x sin θ’ this substituted in (2), gives, for the equation of the sought curve, s y- 2x2 sin Θ x + x sin Θ xy: 2 sin 0 ’ or rather188 THE DIFFERENTIAL CALCULUS. hence the carve is an hyperbola, having the sides of the given angle for asymptotes. 3. The centres of an infinite number of equal circles are all situated on the same straight line: required the line which touches them all. Ans. They are touched by two parallels to the line of centres. 4. From every point in a parabola lines are drawn, making the same angle with the tangent that the tangent makes with the diameter at the point; required the line touching them all? Ans. They are touched by a point, viz. the focus, in which therefore they all meet. CHAPTER XV. ON THE SINGULAR POINTS OF CURVES, AND ON CURVILINEAR ASYMPTOTES. Multiple Points. (110.) If several branches of a curve meet in one point, whether by intersecting or touching each other, that point is called a multiple point. In the former case the point is said to be of the first species, and in the latter of the second species, and we propose here to enquire how, by means of the equation of a curve, these points, if any, may be de- tected. Multiple points of the first species. When the curve has multiple points of the first species, we readily arrive at the means of determining their position from the consideration that at such points there must be as many rectilinear tangents as there are touching branches, and, con- sequently, as many values for dy d? the tangent of the inclination ofSINGULAR POINTS OF CURVES. 189 any tangent through the point {x, y) to the axis of x\ so that the equa- tion of the curve being freed from radicals and put under the form F(x, y) == 0, its multiple points of the first species will all be given analytically by the equation , du % du 0 ^ dx 9 dy 0 * so that no systems of values for x and y can belong to multiple points of the first species, but such as satisfy the conditions du dx 0, as well as the equation of the curve. Having, therefore, determined all such systems of values by solving the two last equations, the true values of p' for each system will be ascertained by proceeding as in (41), and those systems only will belong to multiple points of the first species that give multiple values to p\ Let us apply this to an example or two. EXAMPLES. (111.) 1. To determine whether the curve represented by the equation ay* _ X*y _ fa-3 = 0, has any intersecting branches, g = _3x*C+*), J = 3 At the points where branches intersect we must have 3a2 (y b) = 0, Say2 — x3 = 0, or .·. x = 0, y=z 0, = V Sab2, y = — b 5190 THE DIFFERENTIAL CALCULUS. this second system of coordinates do not satisfy tbe proposed equation, and therefore do not mark any point in the curve; the first system, which is admissible, shows that if there exist any multiple point it must be at the origin. Hence, to ascertain the true value of pf at this point, we have, by differentiating both numerator and denominator in the expression for p', [/] = [ 3^(y + 6), _ JO 3 —a3 J 0 . [6x(y + b)+3xV £ 'L 6ayp' — 3x2 J 0 6 (y +b) + 12p'x + 3ay L 6aypn -j- dap'2 — 6x J 6b 6a [p'f [/]=V-5 therefore, as this has but one real value, the curve has no intersecting branches. 2. To determine whether the curve represented by the equation at* -f" 2ax?y — ay3 = 0 has intersecting branches rf?/ Am — = 4i(*s + ay)=0, — = a (2*® — 3,a) = 0. There is but one system of values that can satisfy these three equations, viz. x = 0, y=z 0; so that if there are intersecting branches they must intersect at To determine, therefore, whether at this point p' has multiple have [/] = 4x (a? + ay) JO la(3,s —2x4) J 0 the origin, values, we __ 6a2 4“ %ay + ____ *■ 3aypf— 2ax 0 _ 4«[p'] ~ 3a [/Js — -2aSINGULAR POINTS OP CURVES. 191 .·. 3a [/>']*— 6a [//] = 0, .·.[/] =0, or[/]=±>/2; hence three branches of the curve intersect at the origin; the tangent to one of them at that point is parallel to the axis of a?, and the tangents to the other two are symmetrically situated with respect to the axis of y, since they are inclined to the axis of x, at angles whose tangents are + sj2 and — a/®· (112.) Should the values of;/, corresponding to any values of* and y, which satisfy the equation of the curve, be all imaginary, we must infer that, although such a system of values belong to a point of the locus, yet that point must be detached from the other points of the locus; for since, if the abscissa of this point be increased by A, the de- velopment of the ordinate will agree with Taylor’s development, as far, at least, as the second term for all values of A, between some finite value and 0, it follows that all the corresponding ordinates between these limits must be imaginary, so that the proposed point is isolated, having no geometrical connexion with the curve, although its coordi- nates satisfy the equation. Such a point is called a conjugate point. (113.) From what has now been said, it appears that, by having the equation of a plane curve given, those points in it where branches intersect, as also those which are entirely detached from the curve, although belonging to its equation, may always be determined by the application of the differential calculus, and independently of all con- siderations about the failing cases of Taylor’s theorem, except, indeed, those connected with the theory of vanishing fractions. We shall now seek the analytical indications of Multiple Points of the Second Species. (114.) The several species of multiple points, or those where branches of the curve touch each other, the differential calculus does not furnish the means of readily determining from the implicit equation of the192 THE DIFFERENTIAL CALCULUS. curve. We know that at such a point, p' cannot admit of different values, since the branches have one common tangent, and we know, moreover, that if Taylor’s theorem does not fail at that point, we shall, by successively differentiating, at length arrive at a coefficient which, being put under the form o 9 the different values will indicate so many different touching branches; for if no coefficient gave multiple values for the proposed coordinates x', y', then the ordinates corresponding to the abscissas between the limits x' and x ± h, h being some finite value, would each have but one value, and, therefore, different branches could not proceed from the point (\x', y'). But we have no means of ascer- taining ά priori which of the coefficients furnishes the multiple value. When, however, the equation of the curve is explicit, then the multiple points of either species are very easily determined. Thus, if the equa- tion of the curve be y =z (x — a)2 s/ oc — b -f- c, ex we at once see that x = a destroys the radical in y and p', that reappears in p"; therefore, at the point corresponding to this abscissa, there will be but one tangent, and yet two branches of the curve proceed from it on account of the double value of p". Hence the point is a double point of the second species, the branches have contact of the first order, and, because p' = 0, the common tangent is parallel to the axis; if the radical had been of the third degree, the point corresponding to the same abscissa would have been a triple point, &c. It appears, therefore, that when the equation of the curve is solved for y, there will exist a multiple point, if in the expression for * a radical is multiplied by the factor (x— a)m. If m = 1, the branches of the curve intersect at the point whose abscissa is λ* = cr, because then p' at that point takes the same values as the radical, but if m > 1, then the branches touch, because then the radical is destroyed in p' for x = a; in both cases the index of the radical will denote the number of branches which meet in the point. Such, therefore, are the geometrical significations of the cases discussed in (76) and (76).SINGULAR POINTS OP CURVES. 193 Cusps, or Points of Regression. (115.) A cusp, ox point of regression, is that particular kind of double point of the second species in which the two touching branches terminate, and through which they do not pass, so that on one side of such a point, viz. on that where the branches lie, the ordinate has a double value, and on the other side the contiguous ordinate has an imaginary value. The cusp represented in the first figure, where the branches are one on each side the common tangent, is called a cusp of the first kind, and that in the second figure, where the branches lie both on one side, a cusp of the second kind. (116.) It is obvious that cusps can exist only at those points, the particular coordinates of which cause Taylor’s theorem to fail; for if Taylor’s theorem did not fail at such a point, then the ordinates in the vicinity, corresponding both to x -f- h and to x — A, would be both possible or impossible at the same time. We are not, however, to infer that when the adjacent ordinates are on the one side of any point real, and on the other side imaginary, that a cusp necessarily exists at that point; for it is plain that the same analytical indications are fur- nished by the point which limits any curve in the direction of the axis of x, or at which the tangent is perpendicular to that axis, as in the third figure. It becomes important, therefore, in seeking particular points of curves, to be able to distinguish the point which limits the curve in the direction of the axes from cusps. Is (117.) Now, at the limits, the tangents to the curve are parallel to dy the axes, the limits are therefore determined by the equations — = <» d y and -r- = 0; and they fulfil, moreover, the following additional con- s194 THE DIFFERENTIAL CALCULUS. ditions, viz. 1°, the ordinate, or abscissa, whichever it may be, that is parallel to the tangent, immediately beyond the limit, must be imaginary; but if it be ascer- tained that this is not the case, the point is not a limit but a cusp of the first kind, posited as in the annexed figures, or else a point of inflexion; the latter when the contiguous ordinates are the one greater, and the other less than that at the point. 2°, Besides the first condition, there must exist also this, viz. that immediately within the limit the double ordinate or abscissa, whichever may be parallel to the tangent, must have one of its values greater and the other less than at the point; but if both are greater or both less, the point is not a limit but a cusp of the second kind, posited as in the annexed figures. Hence, when the branches forming the cusp touch the abscissa or the ordinate of the point, they may be dis- dv covered by seeking among the values which satisfy the equations ^ = 0 dy and = co, those which do not fulfil both the foregoing conditions. Let us illustrate this by examples. EXAMPLES. (118.) 1. To determine whether th£ curve whose equation is (y — i)3 = (* — «)* has a cusp at the point where the tangent is parallel to the axis of y. By differentiating dy 2 x — a dJ —T * (y — by* this becomes infinite for y = b, therefore the point to be examined is (a, b). In order to this, substitute a ± A for * in the proposed equation, and we have, for the contiguous ordinates,SINGULAR POINTS OP CURVES. 195 y = b + J$, which is not imaginary either for -f- h or — h; the point (a, b) is therefore a cusp of the first kind, and posited as in the figure, since the contiguous values of y are both greater than b. 2. To determine whether the curve whose equation is y — a = (x — b)^ + (x — b)% has a cusp at the point where the tangent is parallel to the axis of y · Here the coefficient ^ becomes infinite for x = b, therefore the point da? to be examined is (6, a). Substituting b -f h for x, we have t/ = + + For negative values of h this is imaginary, therefore the curve lies entirely to the right of the ordinate y = a, so that the condition 1° pertaining to a limit is fulfilled. To the right of this ordinate the two values of y, cor* responding to a value of h, ever so small, are both greater than y ■= a, so that the condition 2° is not fulfilled; the point (Jb, a) is therefore a cusp of the second kind, and posited as in the cut. 3. To determine the point of the curve whose equation is (y — a — $y = (x—by, at which the tangent is parallel to the axis of y. The differential coefficient becomes infinite for x = b, therefore the point to be examined is (b, a + b). Substituting b -{- h for x, y == 4* "I" -j* ht negative values of h render this imaginary, therefore the condition 1° is fulfilled; positive values give two values for y, and as h may be taken so small that h$ may exceed Λ, and since, moreover, the two values of h^ are the one positive and the other negative, it follows that the real ordinate contiguous to the point has one value greater, and196 THE DIFFERENTIAL CALCULUS. the other less, than that at the point of contact; hence the condition 2° is also fulfilled, and thus the point marks the limit of the curve, which, there- fore, lies to the right of the ordinate, through x = b (119.) Having thus seen how to determine those cusps where the branches touch an ordinate or abscissa, we shall now seek how to discover those at which the tangent is oblique to the axes. The true development of the ordinate contiguous to such a cusp must be of the fonn dx' y' + ■ h Ah + β*β + &c. and the corresponding ordinate of the tangent will be hence, subtracting this from the former, and calling the difference Δ> we have (120.) Now, in order that the point (x' y') may be a cusp, this diffe- rence, for a small value of h, must have two values; and to be a cusp of the first kind, these two values must obviously have opposite signs; but since A may be so small that Aha may exceed the sum of all the follow- ing terms, ha must have two opposite values; hence, a must be a fraction with an even denominator, and, conversely, if λ be a fraction with an even denominator, the point (#', yf) will be a cusp of the first kind. d2w Hence, at such a point, is either 0 or oo : 0 if β > 2, and co if β < 2. (121.) In order that the cusp may be of the second kind, both values of δ must have the same sign, therefore h* cannot admit of opposite values for the same value of A, consequently a must in this case be either a whole number, or else a fraction with an odd denominator;SINGULAR POINTS OF CURVES. 197 and conversely, if a be either a whole number or a fraction with an odd denominator, the point (a/, y) will be a cusp of the second kind, provided, of course, that a has two values. The position of the branches will depend on the sign of A. We shall now give an example or two. (122.) 4. To determine whether the curve whose equation is yz=Lx±& has a cusp. Here y is possible for positive values of x, and imaginary for all nega- tive values; hence there may be a cusp at the origin. To ascertain this, put h for x, in the equation, and we have, for the contiguous ordinate, the value V = h ± A2. The coefficient of h being 1 s= -χτ*> we see that the tangent da? to the curve at the origin is inclined at 45° to the axes, and, since § has an even denominator, the origin is a cusp of the fii 5. To determine whether the curve whose equation is y — a = x -f- bx2 + cx% has a cusp. Here y is imaginary for all negative values of x, therefore the point (0, a) may be a cusp. Substituting h for i, we have y = a + A + i*s + c*i kind. As before, the tangent is inclined at 45° to the axes, and, » ✓ since the exponent of the third term is a whole number, and r the whole expression admits of two values, in consequence of the even root it follows that the proposed point is a cusp of the second kind. The branches are situated to the right of the axis of y, because h must be positive, and they are above the tangent because bh2 is positive. 6. To determine whether the curve whose equation is (2y + x+ 1)2=2(1 — x)5 s 2 has a cusp.198 THE DIFFERENTIAL CALCULUS. Here values of x greater than 1 are obviously inadmissible, and to the value x = 1 corresponds y — — 1; hence the point having these coordinates may be a cusp. Substituting 1 -j- h for x, we have y = — i + JA + A*, therefore the tangent to the curve at the proposed point has the trigono· metrical tangent of its inclination to the axis of x equal to £, and since the fraction j has an even denominator, the point is a cusp of the first kind. Because h is negative, the branches are to the left of the ordinate to the point which is below the axis of ocy because this ordinate is negative. Points of Inflexion. (123.) Points of inflexion have been defined at (93), and we have there shown that a point of this kind always exists when its abscissa causes all the differential coefficients to vanish between the first and the wth, provided the wth be odd and become neither 0 nor co . The simplest _ d2y d 3y indication therefore of a point of inflexion is [—j^r] = 0, and ] neither 0 nor co ; such indications, however, cannot be furnished by any point at which the tangent is parallel to the axis of y9 since in this case [~] and all the following coefficients become infinite. Neither can these indications take place at any point, for which Taylor’s theorem fails after the third term. It becomes, therefore, of consequence, in examining particular points of a curve, to be able to detect the existence of points of inflexion by some general method, independently of the differential coefficients beyond the first. The only general method of doing this is that which we have already employed for the discovery of cusps, and which consists simply in examining the course of the curve in the immediate vicinity, and on each side the point in question. Points of inflexion are somewhat similar to cusps, each having some of the analytical peculiarities common to both, and to the limiting points of curves, as already hinted at in (116.) But the characteristic propertySINGULAR POINTS OF CURVES. 199 of a point of inflexion is, that the adjacent ordinates on each side are the one greater and the other less than the ordinate at the point. This peculiarity distinguishes a point of inflexion from a limit, inasmuch as at a limit the ordinate imme- diately beyond is imaginary; and it distinguishes it from a cusp of the first kind, inasmuch as at such a cusp the adjacent ordinates are either both greater or both less than at the point, or else, as is the case when the tangent at the point is oblique to the axes, one of these ordinates is imaginary, the other double. We have then first to ascertain at what points of the curve inflexions may exist, or to find what points are given by the conditions d 2y_____P d,#2 Q = 0 or , or, which is the same thing, what points are given by the separate conditions P = o, Q = o, we are then, by examining the course of the curve in the vicinity of each point, to determine to which of them really belong the characteristic of an inflexion. Thus the means of distinguishing points of inflexion being sufficiently clear, we shall proceed to a few examples. EXAMPLES. ]. To determine whether the curve whose equation is y = 6-f (x—a)3 has a point of inflexion where the tangent is parallel to the axis of x. Here p' = 3 (x — a)2, and when the tangent is parallel to the axis of x, p* ~ 0, .·. x = a and y =s b9 at the proposed point. In the vicinity, x = a -|- h, .·. y = A3,200 THE DIFFERENTIAL CALCULUS. which is greater than b, the ordinate of the point when h is positive, and less when h is negative; the point (a, b) is therefore a point of inflexion. 2. To determine whether the curve whose equation is y3 = xs or y ss has an inflexion at any point. this becomes oo for x = 0, therefore a point of inflexion may exist at the origin. Putting h for x, we have which is greater than 0, the ordinate of the point, when h is positive, and less when h is negative; hence there is an inflexion at the origin. Also the equation of the tangent being y = the ordinates corresponding to x = ± h are both less than those given by the above equation; hence the curve lies above the tangent to the right of the origin, and below it to the left, as in the figure. 3. To determine whether the curve whose equation is y — x = (x — a)3 has a point of inflexion / = l + |(x —α)ί,/'==|. $(* — a)_i, this becomes infinite for x = a, therefore a point of inflexion may exist at the point (a, a). In the vicinity of this point x = a + A, .·. y = a + h + h$, which is greater than a when h is positive, and less when h is negative; hence (a, a) is a point of in- flexion. As the corresponding ordinates of the tangent y = a ± A, one, viz. y = a + A, is less than that of the curve, and the other greater, the curve bends, as in the figure.SINGULAR POINTS OF CURVES. 201 On Curvilinear Asymptotes. (124.) Two plane curves, having infinite branches, are said to be asymptotes to each other, when they approach the closer to each other as the branches are prolonged, but meet only at an infinite distance.* Hence, since the expression for the difference of the ordinates cor- responding to the same abscissa in two such curves becomes less and less, as the abscissa becomes greater and greater, and finally becomes 0, when the abscissa becomes oo, it follows that that expression can contain none but negative powers of x, without the addition of any constant quantity. For, if a positive power of x entered the expression for the difference, that expression would become not 0 but oo, when x = oo, and, if there were a quantity independent of «r, the difference would be reduced to this quantity, and not to 0, for x = 0. Hence two curves are asymptotes to each other, when the general expression for the difference of the ordinates corresponding to the same abscissa is Δ = A'x~ “ + Β'χ-β + C'x~y + &c...........(1), or when the general expression for the difference of the abscissas cor- responding to the same ordinate is Δ = + B'y~P + C'y~V -f &c.........(2), and conversely, when the curves are asymptotes to each other; one or both these forms must have place. If for one of the curves whose corresponding ordinates are supposed to give the difference (1) there be substituted another, which would reduce that difference to Β'χ~β + νχ~Ί + &c. this new curve would be an asymptote to both, and would obviously, * Spirals meet their asymptotic circles only after an infinite number of revolutions; these we do not consider here, having adverted to them at (86).202 THE DIFFERENTIAL CALCULUS. throughout its course, continually approach nearer to that which it has been compared to, than the one for which we have substituted it does. In like manner, if a third curve would further reduce the difference (1) to cy-r + &c. this third curve would approach the first still nearer, and all the four would be asymptotes to each other. It appears, therefore, that every curve of which the ordinate may be expanded into an expression of the form y = At“ + Bi‘+ . . . . A'iT* + B'*-0+&c............(3). admits of an infinite number of asymptotes. Since the general expression for the ordinate of a straight line is y ss Ax + B, for the difference between this ordinate and that of a curve at the point whose abscissa is x, to have the form (1), the equa- tion of the curve must be ι, = Α* + Β + Α'*-“ + Β'*-0 +&C. .... (4), this equation, therefore, comprehends all the curves that have a rec- tilinear asymptote, and among them the common hyperbola, whose equation is y = ± (**—A*)* = T 4 * T i ΑΒλτ-1 + &c. A A The curves included in the equation (4) are therefore called hyper- bolic curves. The other curves comprised in the more general equation (3), not admitting of a rectilinear asymptote, are called parabolic curves. The common hyperbola, we see by the above equation, admits of the g two rectilinear asymptotes y=±L—-x, and of an infinite number of A hyperbolic asymptotes. As an example of this method of discovering rectilinear and curvili- near asymptotes, let the equation my3 — xy3 = mx3SINGULAR POINTS OF CURVES. 203 be proposed. The development of y in a series of descending powers of x, is (Ex. 9, p. 68,) mA y=z-m-— &c. therefore the curve has one rectilinear asymptote, parallel to the axis of x, its equation being y= — m; the hyperbolic asymptote next to this, and which lies closer to the curve, is of the fourth order, its equation being yx3 -f- mx3 -f- mA = 0. Again, let the equation of the proposed curve be b y=---------. (x2 — a*'f =bxr-l-\-& c....(1); also, since x2 — a2 sss ~ .·. x = α + i · “ 3Γ”2 + &c..........(2). y λ From (1) it appears that the curve has a rectilinear asymptote, coinci- dent with the axis of x, its equation being y = 0; the hyperbola whose asymptotes coincide with the axes is also an asymptote, its equation being xy = b. From (2) it appears that the curve has another rectili- near asymptote, parallel to the axis ofy, its equation being x = a; the hyperbola next to this is of the third order. If we consider the radi- cal, in the proposed equation, to admit of either a positive or a nega- tive value, then there will be two rectilinear asymptotes, parallel to the axis of y and equidistant from it, as also two hyperbolic asymptotes, symmetrically situated between the axes.SECTION III. ON THE GENERAL THEORY OF CURVE SURFACES AND OF CURVES OF DOUBLE CURVATURE. CBAPTSR X. ON TANGENT AND NORMAL PLANES. PROBLEM I. (125.) To determine the equation of the tangent plane at any point on a curve surface. Let (τ', y, z') represent any point on a curve surface of which the equation is * = F(x,y), then the tangent plane will obviously be determined, when two linear tangents through this point are determined. Let us then consider, for greater simplicity, the two linear tangents respectively parallel to the planes of xz, zy ; their equations are and z — z' =ss a (x — x') y=y' z' = b(y — y') J--------(2)· But since these are tangents to the plane curves, which are the sections through (τ', y, s'), parallel to the planes of xz, zy, therefore (77)TANGENT AND NORMAL PLANES. 205 a dz' dr , d/ P, »=*>=✓· Moreover, the traces of the plane through the lines (1), (2), upon the planes of xz, zy9 being parallel to the lines themselves, a and b must be the same in the traces as in these lines; and, since they are the same in the plane as in its traces, it follows that the equation of this plane must be z — x'==p'(x(2/--2/') .... (3), in which the partial differential coefficients p', q', express the trigono- metrical tangents of the inclinations of the vertical traces to the axes of x and y respectively. For the angle which the horizontal trace makes with the axis of x, we have, by putting z = 0, in (3), pr tangent of inclination = —t · (126.) If the equation of the surface is given under the form w = F(x,y, z) = 0 . . . . (4), then the expressions for the total differential coefficients derived from u considered as a function, first of the single variable x, and then of the single variable y, are (57) .dw, dw , dn . U = ;s+d^=0 Au. du du dy ^ dy dz ^ ^ from which we get the values du du 2s „/________iJy. ^ du’ * dw ’ ds dz τ206 THE DIFFERENTIAL CALCULUS. hence, by substituting these expressions in (3), the equation for the tangent plane becomes <‘-0S + <—«o£ + Gr-JOg = o ·..· W. PROBLEM II. (127.) To determine the equation of the normal line at any point of a curve surface. We have here merely to express the equation of a straight line, per- pendicular to the plane (3), and passing through the point of contact (y, y‘, s'). Now the projections of this line must be perpendicular to the traces of the tangent plane, or to the lines (1), (2); hence the equations of these projections must be x — x' + p (z — z') = 0 % y—/ + »'(* — *0=° which together, therefore, represent the normal. (128.) If we represent by λ, β, γ, the inclinations of this line to the axes of depending entirely on the nature of the directrix, and a, b being given by the generatrix. (133.) Now, by differentiation, this function may be eliminated (58), hence, — bp' 1 — up' 1 — bq — a(f ’ «/ + i/=lora^+ i|=l-(4), which is the general differential equation of cylindrical surfaces. (134.) The same equation may be immediately deduced from the general equation of a tangent plane, to the cylindrical surface. Thus, the equation of any tangent plane, through a point (x*9 y, z) being 2 — s' =y (x — *') + / (y—y')t the condition necessary for it to be always tangent to the cylinder on which this point is situated, is merely that it may be always parallel to its generatrix (1), and this condition, expressed analytically, is {Anal. Geom. p. 232,) ap' + V— 1 = 0 .... (4); this is, therefore, the relation which must have place between the partial differential coefficients derived from the equation of the surface, in order that that surface may be cylindrical, and it agrees with the relation before established. If, in this equation, we write for p', q', their values deduced from the equation u = 0 of any cylindrical surface, as exhibited in (126), it becomes d u d u d uCYLINDRICAL AND CONICAL SURFACES. 211 PROBLEM II. (135.) Given the equation of the generatrix, to determine the cylin- drical surface which envelopes a given curve surface. Since the cylinder envelopes the given surface, the curve of contact is common to both, therefore every tangent plane to the cylinder touches the enveloped surface in that curve. The equation of any of these tangent planes is z—z'=p! (x — x') + y'Cy — y)> whether p’ and q' be derived from the equation of the surface, and take those particular values which restrict them to the curve of contact, or whether p and q' be derived from the equation of the cylinder, and preserve their general values; because in the one case the contact of each tangent is confined to a point in the curve of contact, and in the other case the contact extends along the whole length of the cylinder. Hence, for the curve of contact, the condition (5) must have place, as well as for the entire surface of the cylinder. The mode of solution is, therefore, obvious; we must deducep’ and q' from the equation of the given surface, and substitute them in (5); the result, combined with the equation of the given surface, will obviously represent the curve for which/?' and q' are common to both surfaces; that is to say, we shall thus have the equations of the directrix, and that of the generatrix being also given, the particular cylindrical surface becomes determined. (136.) If the proposed curve surface be of the second order, then the equation (5) will necessarily be of the first degree in x, y, z, and will, therefore, represent a plane; so that the combination of this, with the equation of any surface, must necessarily represent a plane section of that surface; we infer, therefore, that if any cylindrical surface circum- scribe a surface of the second order, the curve of contact will always be a plane curve, and consequently of the second order, and therefore the cylinder itself must be of the second order. As an example, let it be required to determine the cylindrical sur- face having the generatrix (1) in Problem I., and enveloping the ellipsoid, whose equation is212 THE DIFFERENTIAL CALCULUS. U= Ax2 + Bi,2+Cs2 — 1=0 .... (1). The directrix being on this surface, is included in the equation (1), and the same directrix being on the enveloping cylinder, is included in the equation (5) of last problem; hence the combination of these two equations represents the directrix itself to the exclusion of all other points; that is, the equations of the directrix are Ax2 + By2 -f- Cs2— 1 = 0 j 4 · · · · (2)> Aax + Bby + Cs =0 which equations correspond with those marked (2) in Problem I.; and to get the cylindrical surface sought, it remains for us to proceed as directed in Problem I. Eliminating x, y, z, from the two equations above, combined with those of the generatrix in Problem I., there results the following relation between a and β, viz. (A*2 + B/32 — 1) (Aa + Bb + C) = (Aa* + Bbβyi; in which equation we are to substitute, agreeably to Problem I., the values of a and β, given by the generatrix, in terms of x, y, z. The result of this substitution is {Α(λ? — as)2 + B(y — δε)2— ]}(Aa2 + B62 + C) = {Aa (x — az) + (y —bz)}2 which is therefore the equation of the cylindric surface sought. In order to reduce it to a more commodious form introduce Cz — C# within the brackets of the second member, which will then be {(Ααχ + Bby + Cz) — (Aa2 -f B62 + C>}2 and by actually squaring and transposing, the equation ultimately takes the following form, viz. (Ax2+ By2+ Cs2 — 1) (Aa2-f-B£2+ C) = (Aax + Bby + Cz)2, showing that the cylinder which it represents touches the proposed ellipsoid along the curve situated in the plane (2) above, viz. in Ααχ + Bby -f Cs = 0.CYLINDRICAL AND CONICAL SURFACES. 213 PROBLEM III. (137.) To determine the general equation of conical surfaces. Let y\ z') be the vertex of the conical surface, then, since the generatrix always passes through this point, its equations, in any po- sition, will be x — xf = a (z — z') y — y' = b (z — 2') Also, let the equations of the directrix be 2)=0,/(* *,y,2)=0 .... (2), then, since for every point in this line, the equations (1) and (2) exist together, we may eliminate the variables x, y, z; the result will be an equation, containing the fixed constants x\ y', and the indeterminates a, 6; therefore, solving this equation for b, we shall have b = , by differentiating each member of this equation with respect to x and y9 and dividing the results as in (58), we have (y-y')p' _ z — z' — jx — ^p * —z' —(y—y)/ (x — a/)g' which reduces to 2 — z =/ (s — a*') + q' (y — y'), the differential equation of conical surfaces in general.214 THE DIFFERENTIAL CALCULUS. (139.) This same equation, like that of cylindrical surfaces, may be obtained more readily by the consideration of the tangent plane, which, as it always passes through the vertex (j/, y, z') is, in every position, represented by the equation * — 2'=//(χ —χ')+/ (y—y>, this relation, therefore, must exist between the partial differential co- efficients p\ q, for every point of the surface, in order that it may be conical. As in Problem I. if for p'y q\ we substitute their values derived from the implicit equation of any conical surface, the differential equa- tion becomes d u dl 0. PROBLEM IV. (140.) Given the position of the vertex, to determine the equation of the conical surface that envelopes a given curve surface. Since the cone envelopes the proposed surface, the curve of contact is common to both, so that the tangent planes to the cone touch also the given surface, according to this curve. The.equation, therefore, of the tangent plane * -s' = y(x — x7) +5/(2/— y> .... (1), holds equally for any point on the conical surface, and for any point in the curve of contact. Hence, if the values of p, q', be derived from the equation of the given surface, and substituted in (1), this, combined with the equation of the given surface, must represent the curve com- mon to both surfaces, that is, the directrix of the cone. Therefore, the vertex and directrix being known, the equation of the required conical surface becomes determinable. (141.) If the given curve surface be of the second order, the equation (1) will be also of the second order; but, nevertheless, the combinationCYLINDRICAL AND CONICAL SURFACES. 215 of these two equations will be that of a plane, for a surface of the second order may be generally represented by the equation Ax2 + By2 + Cz2 ^ + 2Dyz + 2Exz 4“ 2Fxy > = K . . . . (2), + 2Gx «4* 2Hy -J- 2Jz ) which gives dz ___ Ax ~f- Fy 4- Ez + G dx Ex + D^ Cz + J dz ___ Fx 4~ By -f Dz 4- H dy Ex 4- By 4- Cz 4- J * substituting these values for p' and q', in the equation (1), and sub- tracting from the result the equation (2), we have (Ay4-F/4-Ez'4-G)^ 4-(F*'4-B/4-Dz'4-H)y 4-(Ex'4-D/4- Cz'4-J)z = 0 . 4- Gx* 4- H/ 4- Jz' 4“ K „ (3), which is the equation of a plane; therefore, the conical surface which circumscribes a surface of the second order, must itself be also of the second order. (142.) The above proof is from Monge (Application de l· Analyse a la Geometrie), but it may be rendered much more concise, by assuming the axes of reference so as to give the general equation of the surface a simpler form. Thus, let the axis of x pass through the centre, if the surface have a centre, or be parallel to its diameters if it have not, and let the other two axes be parallel to the conjugates to this, the form of the equation will then be Az24-Bya4-Cx*4-2Fx = G .... (4),216 THE DIFFERENTIAL CALCULUS. These values, substituted for p' and q' in (1), convert that equation into , (F + Caf) («? —a/) , By(y — y) __ A *”* + ------------—-----------+ Tz -0, or As2 + By2 4 Ca;2 4 Fa? — A z'z — By'y — Cx'x — FV = 0. The difference between this and (4) is A z'z 4 B tfy 4 Cx'oc 4 Fa? 4 Fa?' = G, the equation of a plane. (143.) Referring again to Monge’s process, we may remark, that if we accent the constants in the general equation (2), it may be taken as the representative of another surface of the second order, for which the plane of contact with a circumscribing cone, whose summit coincides with that of the former cone, will be represented by the equation (AV 4- Fy 4- EV 4- G') x > 4 + 4 (FV4-BV 4-DV4-Hf)y (EV 4* r>V 4- CV 4- J') * ► =0 . . . . (5). GV 4- Hy 4- JV 4* K' J Now, although equation (3) be multiplied by an indeterminate con- stant, p, the result will still represent the same plane, and this plane will obviously be identical to that represented by (5), provided the co- efficients of the variables sc, y} z, are the same in both equations, that is to say, provided we have the conditions p (Ax' 4- Fy 4- Es' 4- G) = AV 4- F'/ 4- EV 4- G' p (FV 4- By 4- Ds'4- H) = FV 4- BY + DV 4- H' p (EV 4- By' 4- Cz 4- J) = EV 4 D'/ 4- CV 4 J' P (GV 4- Hy* 4- Jz' 4- K) = GV 4- Hy 4- JV 4 K'. As therefore the four quantities, V,y, /, p, are arbitrary, they may beCYLINDRICAL AND CONICAL SURFACES. 217 determined so that these conditions shall be fulfilled, the four equations being just sufficient to fix the values of these four quantities, and as each of them enters only in the first degree, they will each have but one value. It follows, therefore, that there is a certain point, and only one, from which, as a vertex, if tangent cones be drawn to two given surfaces of the second order, their planes of contact shall coincide. The com- mon vertex will be at the intersection of those diameters to each of which the plane of contact is conjugate; since it has been shown above that the vertex of the tangent cone is always situated on that diameter of the surface to which the plane of contact is conjugate.* (144.) We may here observe, that as we have not fixed the origin of the axes to any particular point on the diameter which has been taken for the axis of x9 nor indeed the diameter itself, we may consider the diameter to be that passing through the vertex {x'9 y9 z') of the cone, and this point to be the origin; in which case x'9 y9 z' will each be 0, and the equation of the plane through the curve of contact will then be simply hence, the plane through the curve of contact is conjugate to the dia- meter through the vertex of the cone. If this vertex be supposed in- finitely distant, the same result will belong to the circumscribing cylin- der, viz. that the plane of the curve of contact is conjugate to the diameter parallel to the generatrix of the cylinder. Surfaces of Revolution. (145.) The surfaces of revolution, considered in the Analytical Geo- metry, comprise those only in which the revolving curve is always situated in the plane of the fixed axis. We shall here treat of surfaces * For these and other kindred properties, the student is referred to Mr. Davies’s paper on Geometry of Three Dimensions, in Ley bourn’s Repository, vol. 5. u218 THE DIFFERENTIAL CALCULUS, of revolution in general, the revolving curve being any how situated with respect to the axes. Sections of the surface, in the plane of the axis, are called meridians. problem v. (146.) To determine the equation of surfaces of revolution in general. Let the equations of the generating curve be F(tf, yy z) = 0, /(*, y, z) = 0 . . . . (1), and those of the fixed axis x = az -f* et y = bz+fi then, since the characteristic property of surfaces of revolution is, that every section perpendicular to the fixed axis is a circle, we shall have first to determine a plane perpendicular to the line (2), and then to ex- press the condition that this plane, combined with the surface, always represents a circle whose centre is on (2). Now, the equation of the required plane is (Anal. Geom. p. 235,) z + ax + ty = c . . . . (3), and the condition is, that it must give the same section as if it were to cut a sphere, whose centre we may fix at pleasure, but whose radius will vary with the section, that is, it will depend upon c in equa. (3). Assuming the centre of this sphere at the point where the line (2) pierces the plane of xy, its equation will be (Anal. Geom. p. 241,) (* — *)2 + (y ~ β)2 + £a = r2 .... (4). Hence, supposing r to be the proper function of c, the equations (1), (3), (4), must all have place together; hence we may eliminate x9 y9 z, and thus determine what the relation between r and c must necessarily be, to render these equations coexistent. The result of the elimination will obviously lead to c = (r2), hence, substituting for c and r theirSURFACES OF REVOLUTION. 219 values in terms of the variables, we have, finally, * + ax + by = {(x — «)* + (i/ — /3)2 + z2} .... (5), for the relation which must always exist among the co-ordinates of every point, in every circular section. This, therefore, is the equation of surfaces of revolution in general. (147.) If the fixed axis be taken for the axis of z9 then a, a; b, β, are each 0, therefore, in this case, the general equation becomes *=* (*» + *» + *«)---------(6), which, solved for z9 gives the form 2 = ψ(^ + y*) .... (T). (148.) There is one case of this general problem, viz. that where the generatrix is a straight line, revolving round the axis of z9 but not in the same plane with it, that deserves particular notice. Let us take, for axis of x9 the shortest distance between the axis of z and the generating line; then this axis will be perpendicular to both (65); the equations, therefore, of the line will be * = ± y = ± K also, for any variable section perpendicular to the axis of z, z = c, a;2-j-t/2-|-s2 = r2. Eliminating x9y,z9 we have for c and r2 putting their values above, we have λ2 + bh2 = x2 -f- y1. By putting successively x = 0, y = 0, in this equation, the resulting forms belong to hyperbolas; hence the surface is the hyperboloid of re-220 THE DIFFERENTIAL CALCULUS. volution of a single sheet. The equation of the hyperbola correspond- ing to x = 0 is so that y = ± hz is the equation of the asymptotes (see Anal. Geom. p. 99,) hence the generating straight line, in its first position, is in a plane with, and parallel to, one or other of the asymptotes of that hyperbola in its first position, which would generate by revolving round the axis of z, the same surface as the line; these two lines, therefore, continue parallel during the revolution of both; the one, viz. the asymptote, generating the conical surface asymptotic to the hyperboloid generated by the other line, viz. the line x = ± a, y = hz, or x—± a, y = — bz, and it therefore follows that these four lines will be the sections made on the surface by two tangent planes to the asymptotic cone, drawn through any diametrically opposite points in its surface; these will cut each other on the surface, two and two, and include an angle equal to that between the asymptotes; so that the surface may be generated by the revolution of either of these intersecting lines. We shall shortly see that hyperboloids of one sheet, in general, admit of two distinct modes of generation by the motion of a straight line. (149.) Eliminating the indeterminate function 0, in (5), which de- pends on the nature of the generating curve (1) by differentiation, as, in the preceding problems, we find p' -j- a _x — a p'z 9' + t> ~~ y — 0 + jV from which results the partial differential equation (y — β—bz)tf— (a?—* — az)q—b(x — a) — a (y — /3) = 0 . . (1), and when the axis of z coincides with that of revolution, this becomesSURFACES OF REVOLUTION. ‘221 yp' — xtf = 0. The differential equation of surfaces of revolution may also be ob- tained from the consideration of the normal, which must always cut the axis of revolution, being situated in the meridian plane. Thus, the equations to the normal are (127) and as these must exist simultaneously with the equations (2), we may eliminate x, y, z9 and the result will necessarily be the required relation between p, q% and the variable co-ordinates x\ y‘, s', of any point on the surface. (150.) A given curve surface revolves round a given axis, to deter- mine the surface which touches and envelopes the moveable surface in every position. The enveloping surface touches the moveable one in every position; if, therefore, we take any particular position of the latter, their combi- nation will give the curve of contact; this curve being common to both surfaces, the tangent planes, at all its points, are common to both sur- faces; hence, the values ofp', q(, which vary only with the tangent plane, are the same for both surfaces, as far as this common curve is concerned, and it is evidently by the revolution of this curve round the fixed axis that the enveloping surface is generated. Hence, to determine this curve, we must deducep', q, from the given equation, substitute them in the general equation (1) of surfaces of revolution, since there is a line on some such surface to which they belong, as well as to the given surface; and then, to determine what this line really is, it will be ne- cessary merely to combine this last result with the equation of the given surface: we shall thus obtain the equations of the generating curve, and the position of the fixed axis being previously known, the enveloping surface is determinable by Prob. V. PROBLEM VI.222 THE DIFFERENTIAL CALCULUS. (151.) As an illustration of this, let us suppose a spheroid to revolve about any diameter, to find the equation of the surface enveloping it in every position. Let the surface be referred to the principal diameters of the spheroid, then the equations of any other diameter will be from which we derive substituting these values in the general equation, for all surfaces of re- volution round the proposed axis (1), that is in the equation (y — bz) jf— (x — az)tf + ay — bx = 0, and we have hence, combining this with the given equation, we have, for the gene- rating curve of the envelope, the equations hence, the envelope itself is to be determined thus. We must elimi- nate x, y, z, by means of (2), and the equations x = az, y = bz ... . (1), and the spheroid itself may be represented by the equation x2 4* y2 + ***** — m*> ay=xbxf χ2 Η” 2/* + λ!h* = m2 ay = bx |---------(8). z 4- ax -{- by = c x2 -f- y2 4- z2 = r2 oi any circular section, the result will beCURVATURE OF SURFACES. 223 (r2n2 — m2) (a2 -f- b2) = { putting for r and c their values in terms of r, y9 z, we have, finally, {n2(x2 + y2 + s2) — m2} (a2 + δ2), s= {(2 + a® + fy) ^n2— ^ ^171,2—*2—, which is the equation of the enveloping surface.* CHAPTER XXX. ON THE CURVATURE OF SURFACES IN GENERAL. (152.) The simplest method of contemplating surfaces is by consi- dering them as produced by the motion of a line straight or curved, which, in all its positions, is subject to a fixed law. Viewed under this aspect, surfaces seem to divide themselves into two distinct and very comprehensive classes, viz. those whose generatrices must necessarily be a curve, and those whose generatrices may be a straight line. If, in this latter class of surfaces, the law which regulates the generating straight line be such that through any two of its positions, however close a plane may always be drawn, then it is obvious, that in every such surface, if a plane through the generatrix in any position, but not through any other points of the surface, that is, if a tangent plane, be drawn, this plane, if supposed perfectly flexible, might be wrapped round the surface, without being twisted or tom; or, on the contrary, the surface itself might be unrolled, and would then coincide in all its points with the plane. Surfaces of this kind are, therefore, very properly distin- This solution is from Hymers’s Geometry of Three Dimensions, p. 145.224 THE DIFFERENTIAL CALCULUS. guished by the name Developable Surfaces; the simplest of these are the cone and cylinder. (153.) We see, therefore, that these surfaces are such that a plane may be drawn through any two positions of the generatrix, and which, if turned round one position, supposed fixed, will pass through all the intermediate positions of the other. But if the law of generation is such that this cannot have place for any two positions, however close, then the tangent plane, through one position, could plainly never be brought to pass also through another position, however near, without being twisted. Such surfaces, therefore, are properly designated by the name Twisted Surfaces. These two kinds of surfaces will be separately discussed hereafter; the particulars in the present chapter relate to curve surfaces in general. Osculation of Curve Surfaces. (154.) Let the equations of two curve surfaces be * =f(x> &)> z = y)> when referred to the same axes of coordinates. The first of these sur- faces we shall suppose fixed, both in magnitude and position, by the constants a, 5, c, &c., which enter its equation, being fixed. The second surface we shall suppose fixed only in form, by the form of its equation being given, but indeterminate as to magnitude and position, on account of the arbitrary constants A, B, C, &c. which enter its equation. Let now the variables x and y take the increments h and k; then, for the first surface, we have (60) dz dz d2z d2z A2 + 2 - - hk + dx dy and for the second, d2zOSCULATION OP CURVE SURFACES. 225 Z' = Z + ^*+®* + iKJ« +*«.** + da; 1 d?j T d*dy ^ or, more briefly, dy d2Z dy2 k2) + _______? ^ R cos2 φ — r sin2 φ which becomes infinite when R r sin2 φ = R cos2 φ, or when tan φ s= ± λ/-— > but for all positive and negative values of φ between this and 0, p will be positive, while beyond these limits p will be negative.231 CURVATURE 0Γ DIFFERENT SECTIONS. It appears, therefore, that if from the origin two straight lines are drawn in the tangent plane, inclined to the axis of x at the angles tan = R R + s/~ and tan 0 = — , these will coincide with the surface; all the sections between the sides of the two opposite angles thus formed will be convex, all the sections between the sides of the other two opposite supplementary angles will be concave; so that the two straight lines which we have seen may be drawn from the proposed point to coincide with the surface, separate, the convexity from the concavity at that point. (163.) In order to determine whether the principal radii at any point are both of the same sign or not, we may observe that the expressions (8) for these radii at art. (158) may be put under the forms r' + s/\rf -f t')2 — 4 irU — s'2) _________________Γ_________________ / _j_ ^ V -f i')« — 4 (r't' — s'* (10), from which forms we immediately see that the radii will have the same sign, viz. positive if r*f — s'2 > 0, and contrary signs if r’t* — s'2 < 0; this last condition, therefore, exists in the case just considered. (164.) We shall terminate these remarks by showing that a para- boloid of the second order may always be found, such that its vertex being applied to any point in any curve surface, the normal sections through that point shall have the same curvature for both surfaces. For, take the planes of the principal sections for those of xz, yz, then the radii of these sections being R, r we know that a paraboloid, whose vertex is at the origin, will in reference to the same axes be represented by the equation {Anal. Geom. p. 263,) 2r 31 2R 9 r and R being the semi-parameters of the sections of the paraboloid on232 THE DIFFERENTIAL CALCULUS. the planes of xz, yz. Now the equation of a normal section of this paraboloid, by a plane whose inclination to that of xz is φ, will be obtained by substituting in this equation / cos φ for x, af sin for y, z remaining the same for all normal sections (Anal. Georn. p. 281); hence, the equation of the section in question is . cos2 ώ sin2 0 ,, 2Rr v 2r 2R ' * * R cos2 0 ± r sin2 0 so that the semi-parameter, and, consequently, the radius of curvature (94) of this parabolic or hyperbolic section, is Rr R cos2 0 ± r Β\η2φ> the very same as the radius of curvature of the corresponding section of the proposed surface, be this what it may (159). Hence, this paraboloid has the same curvature in every direction that the proposed surface has at the origin of the coordinates. As a tangent plane at the point of osculation touches both surfaces, and as both have the same curvature there, it follows that if a secant plane, parallel to and at a small distance from the tangent, continually move towards it, the varying section will approach nearer and nearer in form to a curve of the second order, in which it will finally vanish when the secant comes to coincide with the tangent; and the direction of the axes of this evanescent curve will be the same as those of greatest and least curvature through the point. Dupin, who first noticed this curious circumstance, calls the curve to which the section tends the indicatrix of the surface at that point. A surface may thus be regarded as covered with elliptic, parabolic, and hyperbolic points; the first class including circular points which are the “ombilics” or umbilici of the surface. It is demonstrable that on every surface there exists a curve of double curvature along which the points are all parabolic; and which curve divides the surface into two regions, in one of which the points are all hyperbolic, and in the other all elliptic. Moreover, in the hyperbojic region there exists a curve of double curvature, along which the hyper- bolic points are equilateral; and this curve subdivides the hyperbolic region into two, such that the angle of the asymptotes is acute through-CURVATURE OF DIFFERENT SECTIONS. 233 out the one, and obtuse throughout the other. For the investigation of these curious particulars we must refer the student to a paper on the “Courbure des Surfaces,” by M. Gergonne, in the “Annales de Ma- th£matiques,” 1831. PROBLEM II. (165.) To determine the radius of curvature at any point in an oblique section. Take the tangent to the section through the point as axis of x, the point itself for the origin, and the axis of z' in the plane of the section; then, calling the normal the axis of z, the normal section through the axis of x, s, and the oblique section s', we have, at the proposed point (87), (ds) =(d$'). Now at the same point y = 7577., (98), but if (prz) the axis of z' be transferred to the axis of z, then s = z' cos Θ .·. (d2z) = (dV) cos 9; hence, by substitution, (ds}2 Ύ == (d^) cose = P coa Θ · · · · (J)> where γ is the radius of the oblique section, and p the radius of the normal section through the tangent to the former; so that y is the pro- jection of p on the plane of the oblique section, which remarkable property is the theorem of Meusnier. It immediately follows from this theorem, that, if with the radius of any normal section of a curve surface a sphere be described, and through the tangent to that section at the normal point, planes be drawn, cutting both the sphere and the proposed surface, every section of the sphere will be an osculating circle to the corresponding section of the surface; because, if the normal radius of the sphere be projected on any of these sections, the projection will obviously be the radius of that section, and the same projection is, by the above theorem, the radius of curvature of the corresponding section of the proposed surface. x 2234 THE DIFFERENTIAL CALCULUS Lines of Curvature and Radii of Spherical Curvature. (166.) In speaking of plane curves we have already explained (106) what is to be understood by consecutive normals and consecutive curves. We propose, in the present article, to consider the intersections of any normal at a point of a curve surface, with its consecutive normal; but here it must be remarked that consecutive normals to curve surfaces do not necessarily intersect, as in plane curves; for, before coinciding, these normals, although ever so close, need not be both in the same plane; and, in such a case, when they become* consecutive, or coincide, they coincide throughout at once, having even then no point in common that before coinciding was a point of intersection. Hence such conse- cutive normals have no point of intersection. If, however, upon any curve surface there can be traced a line, such that the noYmal to the surface at every point of it is intersected by the consecutive normal, that line will have peculiar properties. Such a line is called, by Monge, a line of curvature. (167.) To determine the lines of curvature through any point on a curve surface. Let the surface be referred to any rectangular axes whatever, then (x', y, z',) being any point on it, we have, for the equations of the normal, Let now the independent variables x', y', take any increments Λ, k; the equations of the normal to the corresponding point will be PROBLEM III, (A) (B)LINES OF CURVATURE. 235 Now, if the normals (1), (2) intersect, their equations must exist simul- taneously ; therefore, since A = 0, B = 0, dA , _dA_ k_ da/ + dy' h dB dB k_ dx + dy h . . . (3) The coordinates (#, y, z,) of the intersection of the proposed normals will be obtained by the combination of the four equations (1) and (3) in terms of x', y, z'9 which are fixed, and of the increments k, h. But from four equations three unknowns may be always eliminated, and the result of this elimination will be an equation between the other quanti- ties ; hence then there exists a constant relation between the increments k, A, when the normals intersect; these increments are therefore depen- dent; consequently the y, x', of which these are the increments, must be dependent;* therefore, when the normals are consecutive, that is, when h = 0, the equations (3) become dA dA d/ da/ + dj/ da/ dB dB_ jy_ dar d/ da?' or, by substituting for A and B their values (1), l + + + — *) (^+i'-^-)=° · · · · (4)> % + ?'(?' + ?' + + = ° · · · · (s)> * If this should appear doubtful to the student, its truth may be shown by removing the axes of x, y, to the proposed point, in which position h, k9 will be the variable coordinates of the line of curvature, and these will merely take a constant when the axes are replaced in their first position.236 THE DIFFERENTIAL CALCULUS. from which, eliminating z' — z, we have the following equation for determining -~r ax' {(, + W-rtf) ij£ + {(1 + r'-(l + j")<'}-J£-- (i+^>s'+y?'/=o) or, which is the same thing, (: i + f m ) H ay* da?'8 + ( i + y 1 + P‘\ ¥( dy' ¥ ' to! (I±il -) ¥¥ as 0 • · (·)· d y This being a quadratic equation furnishes two values for —the tan- gent of the inclination of the projection of the line of curvature, through (xf, y, z)j on the plane of xyf to the axis of x. Hence, there are two directions in which lines of curvature can be drawn through any pro- posed point; and if in (6) we substitute for p', qf, &c. their general values in functions of x, y7 that equation will then be the differential equation which belongs to the projections of every pair of lines of curvature; so that every line on a curve surface which at all its points satisfies this equation, will be a line of curvature. In the foregoing equation if we assume i + p'2 __ yy __ i + y* / ” e ’ the tangent will not be restricted by that equation to parti- cular values, inasmuch as the equation then becomes identical, and dy will therefore be satisfied for any value of Hence, from the points at which the foregoing conditions have place, there issue lines of cur-LINES OF CURVATURE. 237 vature in every direction. These points are i( ombilics ” and the condi- tions which they must fulfil become the same as those already noticed at (160), when, as is the case there, p' = 0, q' = 0. (168.) Between every pair of lines of curvature there exists a very remarkable relation: it is that they are always at right angles to each other. To prove this it will only be necessary to place the coordinate planes, which have hitherto been arbitrary, so that the plane of xy may coincide with, or at least be parallel to, the tangent plane at the point to be considered, in which case p' and q' are both 0; and, consequently, the equation (6) becomes . d*'a “r s' &»' — 1=0 dy' therefore, calling the two roots or values of , tan Θ and tan 8', we have, by the theory of equations, tan Θ tan θ' =— 1, which proves that the projections of the two lines of curvature through the origin, are perpendicular to each other, and consequently the lines themselves are perpendicular to each other. d m'2 Moreover, the equation (7), if divided by =tan2#, becomes iden- tical to equation (6), page 228, which determines the inclinations of the principal sections; hence, the lines of curvature through any point, always touch the sections of greatest and least curvature at that point. Also, in the same hypothesis, with respect to the disposition of the co- ordinate planes, z' = 0, as well as p\ q ; therefore the equation (4) or (5) gives ____ 1 __ tan Θ 2 r* -f- s' tan Θ9 °Γ s' + H tan Θ* but if the plane of xz coincide with a plane of principal section, it will, as we have just seen, touch the line of curvature, and then 8 = 0, so that238 THE DIFFERENTIAL CALCULUS. and these are precisely the expressions found at (159), for the two radii of curvature of the principal sections at the proposed point, in reference to the same axes; hence we infer, that the consecutive normals to the surface at any point, intersect at the same points as the consecutive normals to the principal sections. These points of intersection are no other than the centres of curvature of the surface at the proposed point, for if spheres be described from these centres to pass through the pro- posed point, they will touch there, since both have the same normal, and therefore the same tangent plane; and these two spheres have the same curvature as the surface in the two directions of the lines of cur- vature, since consecutive normals to the surface in these directions, cut that through the point at the centres of these spheres; also the plane sections, tangential to these directions, have the corresponding sections of the spheres for their osculating circles, since the consecutive normals, at their point of contact, also intersect at these centres; therefore, the radii of curvature of the surface at any point, coincides entirely with the radii of curvature of the principal sections through that point, so that (160) if the radii are both equal at any point, the curvature of the sur- face is uniform all round that point. (169.) The annexed figure is intended to give an idea of the disposition of the lines of curvature on the surface (S,) drawn through points P,P', &c. PT, PT, &c. are the normals to the surface at those points, and as each is intersected by its consecutive normal, the locus TP ... of these intersections is a curve. The locus too of the normals PT, PT, &c. themselves, form a surface, throughout perpen- dicular to the proposed; this surface, thus generated by the motion of a straight line PT along the curve P P'. . . and each position intersecting its con- secutive position, is obviously a developable surface; one of whose edges is the line of curvature PP'.. . and the other the line of centres TP . . . which latter is called the edge of regression of the developableRADII OF SPHERICAL CURVATURE. 239 surface. Proceeding, in like manner, along the other line of curvature through P, we have another developable normal surface, whose edge of regression is the locus of the centres of curvature belonging to this second line of curvature. Applying similar considerations to every point on the surface (S), we shall thus have an infinite number of de- velopable normal surfaces at right angles to each other, and which will obviously form together two continuous volumes; and the edges of regression will, in like manner, form two continuous surfaces, or sheets, being the locus of all the centres of curvature. These surfaces, there- fore, bear the same relation to the original surface, as that which in plane curves we have called the evolute bears to the involute. It would be quite incompatible with the pretensions of the present volume to extend any further our enquiries into the properties of lines of curvature. For more detailed information respecting these remark- able lines, the student must study the illustrious author by whom they were first considered, Monge, in his Application dc VAnalyse a la Giometrie, a work abounding with the most profound and beautiful speculations on the subject of curve surfaces and curves of double cur- vature; and which, together with the Developpements de Geometric of Dupin, constitute a complete body of information on a very attractive and important branch of mathematical study, the cultivation of which, however, has been almost entirely neglected hitherto in this country.* Radii of Spherical Curvature. (170.) We have already seen that the radii of spherical curvature or simply the radii of curvature at any point of a surface, are identical to the radii of the principal sections through that point; and have given tolerably commodious formulas for the calculation of these radii, when the axes to which the surface is referred originate at the proposed point, the plane of xy being coincident with the tangent plane, and the axis of • The only English Mathematician, I believe, who has produced public proof of his having given much attention to these enquiries, is Mr. Davies, of Bath, whose papers on surfaces, ) t A4=l +/2 + ?'s,RADII OF SPHERICAL CURVATURE. 241 the equation for determining z — z' becomes (s_2'). + ±(t_l') + J = 0-----------(1), and the roots of this substituted in the equation R = (z — z')k, give R = ^ (A ± •Jh'—igK)----------(2) =-----------------------(3). h ± va*—5gk* (172.) Thus the radii of curvature are determined; and the directions of the lines of curvature, and therefore also of the principal sections, are determined by Problem III.; consequently, the radius of curvature of an oblique section, any how inclined to coordinate planes, any how situated with respect to the surface, may now be determined by help of the formulas (9) and (1) at pages 229 and 233. It appears from (3) that the surface will be convex or concave in the direction of a line of curvature in the immediate vicinity of the point, according as g > 0 or g < 0. If g = 0 the equation (3) shows that one of the radii will be infinite. When the functions of /, sf9 represented by p'9 qf, r', s', ?, are complicated, the expressions just deduced for the radii of curvature will obviously be complicated in the extreme. They are, however, easily manageable when the proposed surface is of the second order, as Dupin has shown in his Developpements for both classes of these surfaces. We shall here give the solution for surfaces which have not a centre, that is, for paraboloids; the process for the other class, or for central surfaces, being exactly the same, but the reductions rather longer. γ242 THE DIFFERENTIAL CALCULUS. PROBLEM V. (173.) To determine the radii of curvature at any point in a paraboloid. The general equation of paraboloids being we have ® * y a+¥=2*’ p'=T’ ■■■1 +/’+ S +1 +1 = *’ .-.A — (I -f — +(1 +|L) ,A + B+2s B2/ A AB Hence, generally, whatever be the paraboloid, we have, for the coeffi- cients in equation (1) pa. 241, the values 7 = A + B + 2,, -. = ΑΒ(χ,+ |3 + 1>, and consequently for R we have Aa + + 1 x iA-fB+2« ,A+B4-2* 2 *2 i/2 t---2^- t^)-ab(^ + ^2 + i)}. The sum of the two radii are, therefore, R + ,, = J X» + b»+ 1 * (A + B + 2*), but (128) the first of these factors is the reciprocal of the cosine of the inclination <* of the normal at the point {x, yy s,) to the axis of z, .·. (R + r) cos « as A + B + 2s,TWISTED SURFACES. 243 which is the expression for the sum of the projections of the radii of curvature on the axis of z; A, B being the semi-parameters of the sections on the planes of xz, yz (Anal. Geom. p. 263). If the point be at the vertex, then x = 0, y = 0, z = 0, and the values of R then become __ A + B R=—3T-± J < A + B )2 — AB = A + B A—B R = A, r = B, and these are also the radii of curvature of the two parabolic sections on the planes of xz, yz (95), so that these sections which we have already called the principal sections in the Analytical Geometry, are really the principal sections, or those of greatest and least curvature. A similar process leads to similar inferences for central surfaces of the second order. CHAPTER IV. ON TWISTED SURFACES.* (174.) We have already stated (152) a twisted surface to be one whose generatrix is a straight line moving in such a manner along its directrices that it continually changes the plane of its motion. The present chapter will be devoted to the consideration of this class of surfaces. Proceeding from the simpler kinds to the more general, we shall first examine the surfaces whose directrices are straight lines as * This is the class of surfaces called by the French Surfaces Gauches, and which, together with the class of developable surfaces, they include under the general name of Surfaces R£glees, expressive of their mode of generation by straight lined generatrices. There has recently appeared, in244 THE DIFFERENTIAL CALCULUS. well as the generatrices; then those having one of its directrices a curve, afterwards those having two curvilinear directrices, and lastly those having three directrices of any kind. Twisted Surfaces having Rectilinear Directrices only. PROBLEM i. (175.) To determine the surfaces generated by a straight line moving parallel to a fixed plane, and along two rectilinear directrices not situated in one plane. Let the fixed plane, called the directing plane, be taken for that of xy, and the plane parallel to the two directrices for that of xz; then the equations of these directrices will be x = as + * ϊ c x 5=5 a'z + * (1)------- and ------(2), y==/3 > Cy = /3 and the generatrix, being parallel to the plane of xy, will be represented by the equations * z = b, y = mcc + n . . . · (3). As this line has always a point in common with (1), the four equations (1), (3) exist together, therefore, eliminating x, y, z, we have, among the variable parameters, the relation β = m (ab + λ) + n . . . . (4), the parameters a, a, β, being fixed by the position of the directrices, but the others being variable. In like manner, since the line (3) has also always a point in common with (2), the four equations (2), (3) exist together therefore, eliminating Ley bourn’s Repository, No. 22, a very masterly enquiry into the history of these surfaces, from the pen of Mr. Davies, wherein the claims of the English to the first consideration of “ rule surfaces” is fully established.TWISTED SURFACES. 245 x, y, z9 we get for a second relation among the three arbitrary parameters the equation β'=*» (a'b -f- «') -f* n . . . . (5). By means of the two relations (4) and (5) among the parameters which enter (3), we may eliminate them and thus obtain the sought equation in jt, y, z. Subtracting each from (3), we have y — β =ra(x — az — a) y — β' — m{x — a'z — *'), eliminating m we obtain, finally, (a-a')yz + )y + (α'β—αβ') z + (β'—β)χ = αβ'—α'β........(0) for the equation of the surface, which is therefore of the second order. Let us now enquire what particular kind of surfaces of the second order this equation includes. By applying the criteria (3), p. 292, Anal. Geom., we find that the surfaces are not central, they must, therefore, be paraboloids. By putting x — k we find in the resulting equation for any section parallel to the plane of yz, that the squares of the variables are absent, therefore, Anal. Geom., p. 166, these sections are all hyper- bolas. We infer, therefore, that the surface (6) is always a hyperbolic paraboloid. If in the equation (6) we make z equal to any constant quantity, the equation will always be that of a straight line, being indeed necessarily one of the positions of the generatrix; also, if we put y equal to any constant quantity, we find that every section parallel to the plane of xz is a straight line, so that through every point on the surface of a hyperbolic paraboloid there may be drawn two straight lines, their assem- blage constituting two distinct series situated in two distinct series of parallel planes, and hence there are two distinct ways in which the surface may be generated by the motion of a straight line, but not more than two ways, since the equation (6) represents a straight line only on the two hypotheses assumed above: and as no two of the positions of the same generatrix, however close, can be in the same plane, the hy- perbolic paraboloid is a twisted surface. (176.) We may show at once, by setting out with the equation of the hyperbolic paraboloid, that two straight lines pass through every y 2246 THE DIFFERENTIAL CALCULUS. point on its surface, and, moreover, that these lines are both in the tan- gent plane at that point. Thus the equation of the surface is (Anal. Geom. p. 266,) px* -pfy*=:pp'z .... (1), and that of the tangent plane through (*', y, z'9) 2pxx/ — 2p'yy' = pp' (z+ z') .... (2), the relation among the coordinates x'9 y, z\ of the point of contact being of course px,a—ρψτ=ιρρ·ζ .... (3). Adding together equations (1) and (3), and subtracting (2) from the sum, there results p(jB — a/)* —p (y — y)2 = 0, which is the condition necessary to be satisfied for every projected point (<τ5 y,) common to the surface (1) and the plane (2), seeing that it has resulted from the combination of their equations. Such condition being satisfied by every point in the lines represented by the equation y— y'=±(x— s') */£-, it follows that the lines of which these are the projections are common to both surface and tangent plane, so that the tangent plane cuts the surface according to two straight lines passing through the point of contact. PROBLEM II. (177.) To determine the surface generated by the motion of a straight line along three others fixed in position, so that no two of them are in the same plane. Let us first consider the case in which the three directrices are all parallel to the same plane. Assume the axes of x and y in the plane passing through one of theTWISTED SURFACES. 247 directrices (B), and parallel to the other two (B'), (B'). Let the axis of x coincide with (B), and the axis of y be parallel to (B')> and let the axis of z be drawn to pass through both (B') and (B'); then the equa- tions of the directrices will be (B) y=o. 2 = 0 and another in common with (B"), we have the additional conditions α' = ?»γ + ρ, /3' = rc/-f q .... (S). Eliminating now the arbitrary parameters m, n, p, q, by means of (1), and these equations of condition, we shall arrive at the equation of the surface. The equations (3) give, in conjunction with (1), n P = *—y x — y which values, substituted in (2), give * — r Ϋ 2 = β'—Ύ (y—y') *y + Φ’—β) χ* + (λ—«') yz ^ + Φύ— β'7) * + («'y#—ay) y + (αβ—βα) s / = 0 + */3'y-*73y' } for the equation of the surface. By applying the usual criteria, {Anal. Geom. pages 292, 294,) we find that the surface must be a hyperboloid, and as the squares of the variables are all absent from the equation, no intersection {Anal Geom. p. 281) can possibly be an imaginary curve; hence the surface must be a hyperboloid of a single sheet, and it is obviously twisted, since the generatrix constantly changes the plane of its motion. (179.) We may, as in the preceding problem, by commencing with the equation of this surface, show that through every point on it two straight lines may be drawn, and that they will both be in the tangent plane through the point. Thus the equation of the surface isTWISTED SURFACES. 249 and that of the tangent plane through o'> o a»+lϊ e·-·1----(2)’ the relation among .τ', y, 2' being fixed by the equation (3). α2_Γ 6* c2 * * Adding together equations (1) and (3), and subtracting twice equation (2) from the result, we have (*-*Ύ , (y-y')2 («-Ο» m a2. + fr2 c2 ° ’ ’ * * ( h a relation which must have place for every point common to both the surface and the tangent plane. Also, subtracting (3) from (2), x') y (y —y) (z — O _ Aa 0 . . . . (5). Now, in order to ascertain whether the points fulfilling these conditions can lie in a straight line, let us combine them with the equations of a straight line through (τ',y, z), viz. a? — #'=«'(2 — z'), y — y' = fr' (z—z') .... (0). Substituting in the equations (4) and (5) these expressions for x — x', y—yf>we have , a'2 , fr'2 1 x fr2 c2> , aV fry ■ΊΓ + &2 c2' a2 + fr2 c2 ’ fry fr2 250 THE DIFFERENTIAL CALCULUS. these relations, therefore, must exist among the constants in (6), for it to be possible for that line to belong to the surface. From the second of these we readily deduce a rational value of a'9 which, substituted in the first, b' will be given by the solution of the quadratic, which will furnish two values, so that two lines passing through the point of contact may be drawn, that shall be common to both the surface and the tangent plane. Twisted Surfaces having but one Curvilinear Directrix. (180.) In surfaces of this kind the generatrix moves along a straight line and a curve, remaining constantly parallel to a fixed plane, called the directing plane. Such surfaces are called conoids; and that they are twisted surfaces is plain, because a plane to pass through two positions of the generatrix must pass through the rectilinear directrix, and become, therefore, fixed, so that it cannot be moved round one position without ceasing to pass through two. The directing plane is usually taken for that of xy9 the origin being at the point where the straight directrix pierces it. PROBLEM III. (181.) To determine the general equation of conoidal surfaces: Let the equations of the straight directrix be # = ?wz2, yzsznz . . . . (1), and those of the curvilinear directrix, F(x,y,z) = 0, f(x,y, z) = 0 .... (2). The equation of the generatrix, being in every position parallel to the plane of xy, must always be of the form z = «, y = /3i + r .... (3), a and β being variable parameters. As this line has always a point in common with the line (1), theirTWISTED SURFACES. 251 equations exist together; hence, eliminating x, y, g, by means of these four equations, we have the condition na = βma -f γ or y = na — βτηΛ; so that the equations (3) of the generatrix become z = a, y — na = β(χ — ma) .... (4)J but this same line has also a point in common with the curve (2); hence, eliminating x, y, z, by means of the four equations (2), (4), we have an equation containing only constants and the variable parameters *, β, which equation, solved for «, gives *=k/3) · · · · (5)· But, by equations (4), a ■ Zy β y — e x —mz ’ hence, by substitution in (5), Z=0( y— x — mz,y which expresses the general relation among the coordinates of any point of the generatrix in any position; therefore this is the general equation of a conoidal surface. (182.) If the straight directrix coincide with the axis of z, then m = 0, n = 0, and the conoid is represented by the general equation Ζ = φ(£), whether the axis of z, or the straight directrix, be perpendicular to the directing plane or not; if it is perpendicular, the conoid is called a right conoid. In these cases the equations of the generatrix are simply Z = a, y = βx. (183.) As an example, let it be required to find the equation of the252 THE DIFFERENTIAL CALCULUS. inferior surface of a winding staircase, the aperture or column round which it winds being cylindrical. To conceive the generation of this surface, let us suppose a rect- angle to be rolled round a vertical column, which it just embraces, the line which was the diagonal of the rectangle will then become a winding curve called a helix, and it will make just one turn round the column, its horizontal projection being a circle; if immediately above this another equal rectangle be applied to the column, the vertical edges when brought together being in a line with those of the first, the diagonal of this will form a continuation of the helix, and in this way will be exhibited the trace of the edge of the surface in question on the vertical column, or the curvilinear directrix; the other directrix is the axis of the cylinder, the directing plane being horizontal. Now for every point in the diagonal of a rectangle the abscissa has a constant ratio to the ordinate, the axes being the sides including the diagonal; so that, reckoning from the foot of the helix, the circular abscissas and vertical ordinates corresponding are in a constant ratio. Hence, taking the centre of the cylindrical base for the origin, and drawing the axis of y through the foot of the helix, calling h the height and 2 nr the base of one of the rectangles, or of the cylinder, we shall have, for each point of the helix, these relations, viz. ninn , i 2 h ar *4“ V* —» , x = r sin —y — = —— . . . . (1), * r s 2πν and for the generating line, the equations 2 = *, ν = βχ .... (2). Tf from the two last of (1) we eliminate the arc $ we shall have the following equations of the projections of the curve 2τΓ2 ® = rsin(-^-) .... (3), eliminating x, y, z, from the equations (2), (3) we haveTWISTED SURFACES. 253 y in which equation if we substitute for * and β the values z and — given by (2), we shall obtain, finally, — X = sin (27Γ 4-)» or — = tan (2π 4~)> s/lf+tf K h y K h which is the equation of the surface, that is, of the twisted helixoid. (184.) It remains to determine the differential equation of conoidal surfaces. In order to this, we must eliminate the arbitrary function Φ in the equation by differentiation, as in the several similar cases in Chapter II., we thus obtain the equation pf p’ (my — nx) — (y — nz) y' y' (my — nx') + (x — mz) * which reduces to pf (x — mz) + 4 (y — nz) = 0, or, when the conoid is right, simply to p'x + q'y = 0, because then m =0, n =0. (185.) The same results may be at once obtained from the conside- ration of the tangent plane; for (x’, y', z’) being any point on the surface, the equation of the tangent plane is z — z?—p'(x — x) + y')> which touches the surface along the generatrix through (xy, z'), and this being every where at the same distance z' from the horizontal plane, it follows that if in the above equation we put z = z\ the result z254 THE DIFFERENTIAL CALCULUS. //(* — *') + q' (y — ^) = 0 will express the relation between the x, y of every point in this gene- ratrix. But at that point where it cuts the straight directrix, the x, y have the relation x = mz'y yz=znz') so that, by substitution, we have p' (mz' — x') -j- q' (nz' — y) = 0, for the relation among the coordinates of every point (x'y y', a') on the surface, which agrees with that deduced above. Twisted Surfaces having Curvilinear Directrices only. (186.) We now proceed to consider those surfaces which cannot have a rectilinear directrix, or rather those whose directrices may be any lines whatever. We shall first suppose two directrices. PROBLEM IV. (187.) To determine the general equation of surfaces generated by a straight line which moves along any two directrices (D) (D') whatever, and continues at the same time parallel to a fixed plane. Taking, as before, the directing plane for that of xy, the equation of the generatrix in any position will be * = «> 9 = βχ + γ .... (1), the parameters all varying with the varying positions of the generatrix. Let now the equations of the two fixed directrices be (D) F(x, y, z) = 0, /(x, y,z) = 0-------(2), (D') F,(*,y,*)=0, . (3).TWISTED SURFACES. 2 55 Then the condition is, first that the generatrix meets (D), or that their equations (1), (2) exist together ; hence, by eliminating the coordinates of the common point from these four equations, we shall obviously obtain an equation containing only constants and the variable para- meters α, β, y; that is to say, we shall obtain among these parameters a relation Φ(α, β, γ,)=0. Proceeding in the same manner with the equations (1), (3) which also exist together for a certain point, we obtain a second relation *(*> β> y>) = 0· By means of these two equations we may eliminate any one of the parameters; therefore, eliminating first γ and then β, we have β=:φ(α), γ = ψ(α); hence, substituting for these variable parameters their values in functions of the variable coordinates as furnished by equation (1), we have, for the general relation among these coordinates, the equation y = jΓ0(«) + Ψ(Ο · . · · (4). This then is the general equation of all surfaces generated as announced, whatever be the form of the directrices; when these forms are given, the forms of φ and ψ become determinable by the above process, and then the general equation (4) takes the particular for belonging to the individual surface. (188.) Let us now determine the general equation of these surfaces in terms of the partial differential coefficients. Putting the equation (4) in the form y — αψ(ζ) = ψ(ζ), the ratio of the partial coefficients of each side, taken relatively to x and y, will, by the principle in (58), be — 0(0 Ki that is, 4 = 9 9 1 — *(*)>256 THE DIFFERENTIAL CALCULUS. an equation from which the arbitrary function φ(ζ) is eliminated. Applying the same principle to this last equation, we have ‘JZ dx d Φ i _ v_. ' iy q' ’ that is, putting according to the usual notation ¥ _ .j ¥ _ ¥ _j ¥_ — « dx ~ ’ dx ~ dy ~ ’ dy ~ ’ Τ' (f*—p'sf , tfs' — $1 __ p’ — - ψ whence an equation from which both the arbitrary functions are eliminated, and which must be fulfilled for every point in every surface generated as in the problem, whatever be the directrices. We see that as two arbitrary functions were to be eliminated, the process has led to a partial differen- tial equation of the second order. problem v. (189.) To determine the general equation of surfaces generated by the motion of a straight line along three curvilinear directrices (D), (D'), (D*). We shall first remark that the motion of the generatrix is entirely governed by these conditions, for if we take any point on the first directrix (D) and conceive two cones whose bases are (D'), (D*) to have this point for their common vertex, these cones will obviously intersect each other in all the straight lines that can be drawn from the point to the curves (D'), (D#), the positions of these lines are therefore fixed by these intersecting cones, and these are fixed by their bases; hence, all the lines that can be drawn from the point to the lines (D'), (D*) areTWISTED SURFACES. 257 determinate both in number and position, this being true for every point in (D), it follows that the surface generated by all these lines is determinate, and it is now required to find its equation. As there is here no directing plane, the equation of the generatrix in any position will take the form # = αζ + γ, y = · · · · (1)> and, since it always has a point in common with (D), we may eliminate by means of the equation of (D) combined with these, the coordinates of that point: the result will furnish a condition among the variable parameters. In like manner, employing the equation of (D'), we shall arrive at another equation of condition, and, lastly, the equation of (D#) will furnish a third equation. By means of these three equations any two of the parameters a, β, γ, δ, may be eliminated, and we shall obtain three equations of the form 0 = γ = ψ(α), δζ=ζ τγ(λ). Substituting these expressions for β, γ, δ, in the equations (1), we shall have x = as -f- ψ(α), y = ζψ(α,) -f π(α), two equations which have place for every surface generated as proposed, the functions which fix the directrices being quite arbitrary. If these functions are known, or the directrices fixed, we may then eliminate the parameter & by means of these equations, and thus deduce the equation of the individual surface; but the general relations among the coordinates for all the surfaces of this family can be exhibited only by means of two equations as above. The general relation among the partial differential coefficients belonging to all this family of surfaces, may, however, be ascertained in a single equation by eliminating, as in last problem, all the arbitrary functions by successive differentiation; this will lead to a partial differential equation of the third order, for which see Monges Application de VAnalyse a la G'tomitrie, p 195. (190.) We shall terminate the present chapter with the following example: z 2258 THE DIFFERENTIAL CALCULUS. On the opposite sides of the hori- zontal parallelogram ABDC are de- scribed two vertical semicircles, and perpendicular to their planes is drawn the straight line OY through the centre of the parallelogram; taking this straight line and the two semi- circles as directrices, it is required to find the equation of the surface gene- rated by a straight line moving along them. Let the axes of coordinates be the perpendicular horizontal lines OX, OY, and the vertical OZ, then the equations of the three directrices will be x =s 0, 2=0... y = — b, (a? — a)2 + z2 = x2 . . . . (2), y = + b, (x 4. a)2 4- z* = r2 . . . . (3). The equations of the generatrix, since it always passes through a point (j3, 0, 0) in the axis of y, will take the forms *=»(y — β)> z = y(j)—P) · · · · (4)» and the condition to be fulfilled by this line is, that it rests on each of the semicircles; or that at certain points, x, yf z, are the same in the equations (2), (4) and (3), (4) ; hence, eliminating these first from (2), (4), and then from (3), (4), we have these relations among the variable parameters, viz. {*(*> +/3) + a}2+ y2(*+/3)2 = r*--------($), {αψ-β) + α}2+γ2ν-β)2=**-----------------(6), which, by subtraction, give β (ba2 *-J- dot, -f- by2) = 0. This condition is satisfied by the value β = 0; but this is not admissible, since it would restrict the generatrix to pass always through the origin,TWISTED SURFACES. 259 and have no motion along OY; hence, dividing bywe have the re- lation net *2 + r2 + T=0 · * · ·(7)> between the parameters λ, y. Substituting the value of y2, given by this equation in (5), it becomes (b2 — jS2) aet s= (r2 — a2) (8), and by means of these equations, together with those of the generatrix, we may readily eliminate the parameters; thus the values of λ and y, given by (4), are x z > Ύ : iy—β and these, substituted in (7), give y —β *(*’ + **) and finally, these substituted in (8), give for the surface the equation ■ {y + + ((Λ _aS) <++*), t which is the same as {axy + b (x2 + z2)}2 = b2 r2 x2 -f δ2 (r2 — a2) z2.260 CBAPTSXt V. ON DEVELOPABLE SURFACES AND ENVELOPES. (191.) When in an equation between three variables u = F(a’, y, zy a) = 0 there enters an arbitrary constant a, that equation, by giving different values to «, will represent so many different surfaces all belonging to the same family. If we fix one of these by any determinate value of a, another, intersecting this, will be represented by changing a into a + A, h being some finite value. If h be now continually diminished, the intersection will continually vary, and will become fixed only when the varying surface becomes coincident with the fixed surface. In this position the intersection is said to belong to consecutive surfaces, and it may be determined both in form and position by a process similar to that employed at (106). Thus a being the only variable concerned in the intersections, we have, by Taylor’s theorem, , du Έ , d2u A2 , . F(i,j,i,a + A) = K + — A + — +&c.=0, but, since u = 0, therefore, d u d2w da, da2 -j- Vv *,) = «> . . . . (2), then the equations to tangent planes to each will have the forms 2 — S,=J>, (* — !,)+ ?,(«» — Vl) , . . . (3) and ε— z2=j?2(a; — x2) —ya) .... (4), and for these planes to belong to both surfaces, their equations must be identical; that is, we must have the conditions Pi—Pv 9\—92 · · · · (5)> S1— Pi*x — ?i2/i = z2—Pi*»— Mi · · · · (6)· By means of the six equations marked, five of the coordinates may be determined in terms of the sixth, x; hence, if these functions of x, be now substituted for their values in the remaining equation, we shall obtain a result containing only the variable parameter xv and which will conse- quently represent the family of planes which generate the developable surface sought. Calling this result P = 0, the generatrix of the surface will be given by the equations from which, eliminating xlt we have the equation of the surface sought; and this equation, combined with that of each surface separately, will give the two curves of contact.264 THE DIFFERENTIAL CALCULUS. PROBLEM. (195.) To determine the differential equation of developable surfaces in general. The general equation of the generating plane, arranged according to the variable coordinates x, y, z, of &ny point in it, is z =p'x + g'y + z—ft*—· · * · (1)» and this plane remains the same for every point in the generatrix, as well as for the point (/, /, *'), so that the quantities y · · · · (2), remain constant, although x, y', z' all vary, provided this variation is confined to the rectilinear generatrix, for which y is always a function of xf but not else; hence, the conditions which restrict the point xfyy'yz' to the generatrix on which it is first assumed, is, -that the differential coefficients derived from (2), y being considered as a function of x, are all 0; and it is plain that if any two be 0, the third will be 0 also; hence, differentiating the two first, we have / -f- s' = 0, and s' -f-1' ^ = 0, 1 dx dx where ^ fixes the position of the rectilinear directrix for which the ex- dy pressions (2) remain constant. Eliminating, then, we obtain the following equation, which must hold for every directrix, viz. rY — s'2 = 0 . . . (3). This, therefore, is the equation which the differential of the equation of every developable surface must accord with; or, in usual terms, it is the differential equation of developable surfaces in general.DEVELOPABLE SURFACES AND ENVELOPES. 265 (196.) We shall exhibit another method of obtaining the equation (3), from the general equations (2), art. (192). Differentiating the first of these, in which a is a function of x and y implied in the second, we have the two partial differential equations P1 ?'=/(*) -jp + **'(«) ;jp + ΨΟΟ + ί'Ψ' (*) jp’ but, in virtue of the second of the equations (2), these become p'=0(«), 0' = ψ(Λ), consequently, p' must be a function of /, and may therefore be repre- sented by V = «·(/)· Eliminating now the arbitrary function π by differentiation, as in (58), we have τ' sf 7 = T as before. (197.) We have as yet considered only the simplest class of surfaces, whose intersections, with their consecutive surfaces, are given by the general equations (1), art. (191), viz. plane surfaces, the intersections being straight lines. It is obvious, however, that whatever be the sur- faces, the intersections are still given by the equations (1), and the en- velope of these surfaces, found by eliminating from them the arbitrary parameter «. This parameter, however, may enter the equation of any particular family of surfaces in an infinite variety of different forms and ways; it may enter into only one of its terms, or be combined with several; a simple power only of it may enter, or a complicated function; and still, entering only as a parameter, the general equation, under all these changes, will still preserve the same character, and represent but one family of surfaces. With the envelope, however, it will be different; this depends as well on the arbitrary parameter, as on the variables which enter the general equation, since the value of this parameter must a a265 THE DIFFERENTIAL CALCULUS. be found from one of the equations (1), in terms of x, y, z, and this value substituted in the other for the equation of the envelope. Never- theless, since, as just observed, the individual surfaces represented by the two equations (1), for every particular value of the parameter in whatever form it may enter, is always of the same degree, it follows that each individual intersection (1), will uniformly be a curve of the same order, and which will change its order only when the order of the surface changes. This curve of intersection, or of contact, common to all the envelopes of the same family of surfaces, is called, by Monge, the Cha· r act eristic. Considering any of the characteristics (1) separately, we may enquire what are the points in which it is intersected by the consecutive charac- teristic ; and the method of determining these intersections is analogous to that already explained in (107) and (186), that is, we must combine with the equations (1) of this curve their differentials taken relatively to a; hence, the consecutive intersections for any particular position of the characteristic, will be determined by the equations F(x, y, ε, a) = 0 dF(g,y, 2>*)_0 da d*F(x, y, *, *) _ Λ da2 * each of these separately represents a surface, any two together a line common to both, and all three the point or points common to their intersection, a being considered constant in the results of the operations indicated. By solving these three equations for x, y, and z, we shall obviously obtain known values for the coordinates of the points of inter- section required, which of course are all situated on the envelope. Now, if from the three equations above, we eliminate a, we shall have two equations in x, y, z, existing together; which, being the same for the intersections of every pair of consecutive characteristics, must represent the locus of these intersections, and be situated on the enve- lope. It will therefore be a line which touches and encompasses all the characteristics, in the same manner as the envelope touches and embraces all the enveloped surfaces. It must then form an edge of theDEVELOPABLE SURFACES AND ENVELOPES. 267 envelope, or the line in which its sheets terminate, and it is therefore called, by Monge, the edge of regression of the envelope, In the developable surfaces, we have seen that the characteristic is a straight line, and the consecutive intersections of the characteristic, in every position, obviously form the edge which limits the locus of the charac- teristics, that is, the developable surface. The consideration of envelopes, characteristics, and edges of regres- sion, have been successfully employed by Monge, and succeeding writers, to remove several difficulties in the higher departments of the integral calculus, that do not appear to be otherwise clearly explicable; but it would be out of place here to do more than to hint at the im- portance of these researches; to pursue them to their fullest extent the advanced student must have recourse to the profound work of Monge, before referred to, viz. Application de VAnalyse d la Geomttrie. We shall conclude the present chapter with one example on the determina- tion of the envelope. (198.) The centre of a sphere of given radius moves along a given plane curve, it is required to determine the surface which envelopes the sphere in every position. Let the equation of the given curve, along which the centre moves, be £ = *(*>---------------------------0)> so that for every abscissa a of this curve, the ordinate corresponding will be . . . (3). * + φ(λ)) 0'(*) = o 3 The last equation is that of a plane, passing through the point (a 0(<*))> or centre of the sphere; it is, moreover, perpendicular to the tangent to the curve (1) at this point, for the equation of this tangent is268 THE DIFFERENTIAL CALCULUS. <Ρ-β) = *'(«) (*<«')-*(“))> and that above is y-0(.) = -J-5 (*-«), so that whatever be the form of φ, the characteristic is always a great circle of the moveable sphere, of which the plane is normal to the curve. The species of the curve which is the characteristic being, however, constant, as observed in art. (197,) however φ and a may vary, the species may be at once determined by assuming α = 0, φ(α) = 0, which reduces the equa- tions of the characteristic to sa + y8 + = ** | or y* + sa = ra, which belongs to a circle; the species, therefore, is a. curve of the second order. To determine the equation of the envelope, we must eliminate a from (3), and the resulting equation in x, y, z, will belong to the envelope; thus, if the curve (1) be a circle of radius a, then φ(α) ss */a* —a* ·*· 0'(a) = Vo* substituting these values in the equations (3), they become (*—*)’ + (y — >/a*—«*)* + Js = r\ ay — x ,l/aa — a,f and determining, from this last equation, the expression for a, and substi- tuting it in the preceding, we shall obtain, finally, (a ± n/i2 + y2)2 = r2— za, for the equation of the envelope.269 CHAPTER. VX. ON CURVES OF DOUBLE CURVATURE. (199.) In the preliminary chapter to the present Section, we investi- gated the expressions for the tangent lines and normal planes to these curves; we shall now discuss their general theory. As, however, in the course of this discussion, we Shall sometimes have occasion to employ the differential expression for the arc of a curve of double curvature, we shall commence by seeking the form of this expression. (200.) We know that the projecting surface of every curve of double curvature, is a cylindrical surface, (see Anal. Geom., p. 311;) if, there- fore, this cylindrical surface be developed, the curve will become plane, its length will be unaltered, and the curvilinear base of the project- ing cylinder, which we shall here suppose to be vertical, will become a straight line on the plane of xy\ hence, for the plane curve referred to this straight line t, and the axis of z, we shall have (87) the expression (ds)* = (d2)*+(d02, but t being itself, in reality, the arc of a plane curve, we have (d02=(d*)2+(dy)*, hence, by substitution, (ds)* == (d*)2-Kdy)2-Kdz)2, which is the differential expression required. a a 2270 THE DIFFERENTIAL CALCULUS. Osculation of Curves of Double Curvature. (201.) Let V =/(*), * = F(*)------(1), and Υ=ψ(*), Ζ = Ϋ(χ)--------(2), be the equations of two curves of double curvature, or rather of the projections of these curves on the planes of xy, xz. Then, if we con- sider the constants, a, b, c, &c. which enter the first pair of equations as known, and the constants A, B, C, &c. belonging to the second pair as arbitrary, these latter may be determined so that the curve to which they belong may touch the proposed or fixed curve (1), in any given point, more intimately than any other curve of the family (2). For, giving to x any increment, h, we have, by Taylor’s theorem, d *y dr3 A* + dy___ dY dx dx 9 i*-I™. &c dis — dx2 ’ (3), the projection of the curve (2), on the plane of xy, shall touch more intimately the projection of (1) on that plane, than the projection of any other curve of the family (2). In like manner, if the constants which enter the second of the equa- tions (2), be all of them determined from the conditions dz dZ dh d2Z 9 * 9 cto? dx da? dx ν and, for contact of the second order, we must have the two additional conditions d2y _ d»Y_ d2z __ daZ "diT — d*a » d** — d** ’ and so on. (202.) From these principles, we may very easily deduce the equa- tions of the tangent at any point of a curve of double curvature. Thus the equations of any straight line in space, are i/ = A* + B; z=z A'x+B'---------------------------(1), and these correspond to the equations (2) above; and as four arbitrary constants enter,' the conditions (5) may be fulfilled by them; thus, taking the two last conditions, we have, by accenting the variables of the curve, di' d*' A', and, therefore, the two first require that d*' so that the equations (1) of the tangent, through any point (a1', y\ z') are272 THE DIFFERENTIAL CALCULUS, , dy, PROBLEM I, (203.) To determine the osculating circle, at any point in a curve of double curvature. In finding the osculating circle, at any point of a plane curve, we had, of course, the plane of that curve given, but, in the present case, we have to determine both the plane of the circle and its radius. Now let us suppose that r is the radius of the osculating circle, and that β, β, γ, are the coordinates of its centre; then it is plain, that the circle will be a great circle of the sphere whose equation is and, since the plane of the circle passes through the point (α, β, γ), its equation must be of the form x— a + m(y — β) + η(ζ — γ) = 0 . . . . (2). These two equations, combined with those of the proposed curve, give the values of x, y, z> common to all, and therefore belong to the point where the circle (1), (2) meets the curve. We have, therefore, to diffe- rentiate the equations (1), (2) successively, and to consider, agreeably to the conditions (3), (4), art. (201), that the resulting differential co- efficients belong as well to the proposed curve at the point, as to this circle. For contact of the first order, we have (*—«) + («/—β)ρ' + (χ—y)i'=o · · · · (3)» 1 + mpf + nq‘ = 0 . . . . (4); and for contact of the second order we have, in addition, (Χ_Λ)2-[-(3,_β)2_|_(2_ y)2 = r2 .... (1) · i+/2 + y*+/'(y-/3) + y'(s-r) = o-------(5). mp,f + n ’ 9 — *-*+γψ-ΥΊ' (y-/3)~ 77'-W’ (*~y)=0> x-r=9---^r·9 (,*—*) + ψ,(3—β) · · · · O’)» hence, the three conditions (2), (4), (6), determine the plane of the os- culating circle, and which is called the osculating plane, through the proposed point (x, y, z). Equation (7) then represents this plane. For the coordinates of the centre of the osculating circle we have, from equations (1), (2), (3), (np' — mq')r (n~qf)r M * —ys (m —p')r S M > where M is put for the expression */{(np’ — (λ —+ (m— p')*}. Substituting these values in (5), we have, for the radius of oscula- ting circle, (l+y»+yj)M (n — q’) p* — (m —p') q” Hence, putting for m and n the values already deduced, and restoring the value of M, we have274 THE DIFFERENTIAL CALCULUS. (i+/H-i's)l_______ r— J {ρΤ* + q"* + (p’f — >l' v")*i ’ Π + p'8 + 4*) (?'/' + Ί *?") * — x f%+fi+(/?"—?y')* ’' Λ , <1+*· + «'*) fP" + but not more, the plane of these elements being the osculating plane at the point. The process, then, is to assume the equation of a plane through the point x — x — y) -f n(z — x') = ° .... (1), and to subject it to the condition of passing also through the points (x -f dx', y 4- dy, z' -f dz') and x7 4- 2dx' 4- dV, y' 4- 2dy 4- dy. z' 4- 2dz' 4-‘dV. Such a process the student will at once perceive to be exceedingly exceptionable; for, besides the vague notion attached to the infinitely small consecutive arcs, the expressions x 4- dx, y 4* dy, and the like, mean no more in the language of the differential calculus, than x,y, &c.; for dx, dy, &c. are not infinitely small, but absolutely 0, as we have all along been careful to impress on the mind of the student. The process is, however, susceptible of improvement, thus : suppose the plane (1) passing through one point (x*, y', z') of the curve, passes also through a second point, of which the abscissa is x' Ax', where Δx' means the increment of x; then, substituting x1 Ax for xr, the equation (1) becomes x — X 4- m (y — y) 4“ » ~ *0 — (Δ*' 4- m A y 4- nAsf) = 0. ♦. (2), which, in virtue of (1), is the same as Axf 4* m Ay' 4* nAz =s 0, J 4* m -—7- 4* n Ax Az' 17 0. Suppose now that these two points merge into one; that is, let Ax' s= 0, then276 THE DIFFERENTIAL CALCULUS. d z' i+m- + n-=0 (3); hence the plane becomes determinable by the conditions (1), (3). Again, let this plane pass through a third point, x' -|- Ax', then sub- stituting this for x' in both the equations (1), (3), they will furnish the additional condition dy dz' ^d“?+”Ad?=0; hence, dividing by Ax' and supposing this third point to coincide with the former; that is, supposing Ax' = 0, we have the new condition dy dx'* + n (4). The equations (3) and (4) determine m and n, and thence the plane (1), which is such as to pass through but one point of the curve, and at the same time to be so placed that the most minute variation from this position will cause it to pass through three points of the curve. (207.) By whatever process the osculating plane is determined, the radius of the osculating circle may be easily found from considerations different from those at (203). For, as the linear tangent to the curve, must also be tangent to the osculating circle, it follows that the centre of this circle must be on the normal plane, as well as on the osculating plane; it must, therefore, lie in the line of intersection of this normal plane, with its consecutive normal plane; hence, if this line be deter- mined, the combination of its equation with that of the osculating plane, will give the point sought. Now (188) the line of intersection of con- secutive normal planes is * — — yO + ZC* — 5') = ° } p''(y-2/') + 9''(*-O-p's-9,4-i=0r therefore the centre is to be determined by combining these equations with that of the osculating plane, viz.CURVES OF DOUBLE CURVATURE. 277 z (* — *') + ψ(ν—ΐ/), being precisely the same equations as those employed before, for the same purpose. If the origin be at the point, and the tangent be the axis of x, then a/, y, s', p', q', are each 0; therefore the equations of the line of intersection are * = 0, y P"’ and the equation of the osculating plane pf'z — q"y=z 0; this, therefore, is perpendicular to the line of intersection, (Anal. Gtom., p. 234.) (208.) The expressions in (203) for the coordinates of the centre of the osculating circle, will become very simple by introducing the sub- stitutions furnished by art. (204); the results of these substitutions will be a = #-f-ra dsa? /3 = y -f- r2 d*9 y = z -f- r* d 9* de® the independent variable being s. (See Note D.j PROBLEM ΙΓ. (209.) To determine the centre and radius of spherical curvature at any point in a curve of double curvature. We are here required to determine a sphere in contact with the proposed curve at a given point, such that a line on its surface in the direction of the proposed may, in the vicinity of the point, be closer to the curve than if any other sphere were employed. In the direction of the curve, the z and the y of the sphere must be both functions of x; so that the equation of the sphere is resolvable into two, corresponding b b278 THE DIFFERENTIAL CALCULUS. to the equations (2), art. (202), which two equations belong to the curve which osculates the proposed. The actual resolution of the equation into two is obviously unnecessary; it will be sufficient in that equation to consider x as the only independent variable. The general equation of a sphere is (I-_a)2+(y_/3)2 + (2_y)2 = r2-------------(I), and the particular sphere required will be that whose constants are determined from the following differential equations : χ — α+ρ'(ν — β) + ΐ'(ζ-7) = 0 .... (2), f (y- β) + f <* ■- r) +i + p's + i'5=0----------· (S), P"'(y-/3) + i",(»-y) + 3(p'p" + ?'i") = 0-----------(4)· These four equations fix the values of the parameters *, β, γ, r, and, therefore, determine both the position and magnitude of the osculating sphere. If the origin of coordinates be at the proposed point, and the linear tangent be taken for the axis of ay the determination becomes easy, for x, y, z, being each = 0, as also p\ qr, the foregoing equations (2), (3), (4), become * = 0, p"P + q"y = 1, ρ'''β + f y = 0, r ν'" * · P ” p" q'" — p"f f> y fpm — q’" p" ’ hence, by substitution in (1), s/ p'”2 + (/"* (210.) We already know that if to every point in a curve of double curvature normal planes be drawn, the intersections of these planes with the consecutive normal planes will be the characteristics of the develop- able surface which they generate, and the intersection of any characteristic with the consecutive characteristic will be a point in the edge of regres- sion, corresponding to the given point on the proposed curve. NowCURVES OP DOUBLE CURVATURE. 279 equation (2) above being that of the normal plane, this point is deter- mined by precisely the same equations (2), (3), (4), as determine the centre of spherical curvature; these points, therefore, are one and the same, as might be expected; hence, the locus of the centres of spherical curvature forms the edge of regression of the developable surface gene- rated by the intersections of the consecutive normals. If then, by means of one of the equations of the proposed curve and the three equations of condition mentioned, we eliminate x9 y, z9 and then perform the same elimination by means of the other equation of the curve and the same conditions, we shall obtain two resulting equations in α, β9 γ, which will be the equations of the edge of regression. PROBLEM III. (211.) To determine the points of inflexion in a curve of double curvature. Since a curve of double curvature, as its name implies, has curvature in two directions, if at any point its curvature in one direction changes from concave to convex, the point is called a point of simple inflexion. But if at the same point there is also a like change of curvature in the other direction, the point is then said to be one of double inflexion. in other words, if in but one projection the tangent cross the projected curve, the point is one of simple inflexion; but if the tangent cross the curve in both projections, then the point is one of double inflexion. As in plane curves the tangent line has contact one degree higher at a point of inflexion, so here the contact of the osculating plane is one degree higher. Hence, at such a point, besides the conditions in (203), which fix the osculating plane, we must, at a point of simple inflexion, have the additional condition arising from differentiating (6), viz. mpr,r -j- nq" — 0. Eliminating ~ from this and equation (5), we have which condition renders the expression for the radius of spherical"280 THE DIFFERENTIAL CALCULUS. curvature at the point infinite, as it ought. Unless, therefore, this condition exist, the point cannot be one of inflexion; but the point for which the condition holds may be one of inflexion, yet to determine this the curve must be examined in the vicinity of the point. As to points of double inflexion, it is evident,, from what has been said (123) with respect to plane curves, that such points must fulfil the conditions p" = 0, or oo, /'= 0 or oo, and these render the radius r, of absolute curvature, infinite or 0. Evolutes of Curves of Double Curvature. (212.) In speaking of the evolutes of plane curves we observed (105) that the evolute of any plane curve was such, that if a string were wrapped round it and continued in the direction of its tangent till it reached a point in the involute curve, the unwinding of this string would cause its extremity to describe the involute. But besides the plane evolute hitherto considered, there are numberless curves of double curvature, round which the string might be wound and continued in the direction of a tangent till it reached the involute, which would equally, by unwinding, describe this involute; and generally every curve, whether plane or of double curvature, has an infinite number of evolutes, as we are about now to show. (213.) If through the centre of a circle, and perpendicular to its plane, an indefinite straight line be drawn, and any point whatever be taken in this line, then it is obvious that the point will be equally distant from every point in the circumference of the circle; so that, if a line be drawn from it to the circumference, this line, in revolving round the perpendicular under the same angle, will describe the circumference. Such a point is called a pole of the circle; so that every circle has an infinite number of poles, the locus of which is determined when the positions of any two are given. (214.) Now, as respects curves of double curvature, we have seenCURVES OP DOUBLE CURVATURE, 281 that the centre of the circle of absolute curvature corresponding to any point, is in the line where the normal at this point is intersected by its consecutive normal, the centre itself being that point in this line where it pierces the osculating plane, which (204) is the plane drawn through the tangent line perpendicular to this line of intersection, or characteristic; hence the characteristic corresponding to any point in the curve, is the locus of the poles of curvature at that point; and the intersection of this characteristic, with the perpendicular to it from the corresponding point of the curve, is that particular pole which is the centre of absolute curvature, the perpendicular itself being the radius. As the locus of the poles corresponding to any point is no other than the characteristic, the locus of all the poles corresponding to all the points of the curve must be the locus of all the characteristics, and therefore (192) a developable surface, (215.) Suppose now through any point, P, of the curve, a normal plane is drawn of indefinite extent, the characteristic or line of poles corresponding to the point will be in this plane; let, therefore, any straight line be drawn from P to intersect this line of poles in the point Q, and be continued indefinitely. If this normal plane be conceived to move, so that, while P describes the proposed curve, the plane continues to be normal, the characteristic will undergo a corresponding motion, and will generate the developable surface corresponding to the curve described by P; and this motion of the characteristic will cause a corresponding motion of the point Q, not only in space, but along the arbitrary line from P, which has no motion in the moving plane. As, therefore, Q moves along the characteristic, successive portions QQ' of the line, PQ will apply themselves to the surface which the moveable characteristic generates, and there form a curve to which always the unapplied portion QP is a tangent. Now the nprmal plane being in every position tangent to the surface throughout the whole length of the characteristic, it is obvious that, in the above generation of this surface, nothing more in effect has been done than the bending of the original normal plane, supposed flexible, into a developable surface. If, therefore, we now perform the reverse operation, that is, if we unbend the normal plane, the point P will describe the curve of double b b 2282 THE DIFFERENTIAL CALCULUS. curvature, and the curve QQ' traced on the developable surface will become the straight line PQ; so that the curve of double curvature may be described by the unwinding of a string wrapped about the curve Q' Q, and continued in the direction QP of its tangent, till it reaches the point P in the proposed curve. It follows, therefore, that the curve Q'Q is an evolute of the curve of double curvature proposed; and, moreover, that, as the line PQ originally drawn was quite arbitrary, the proposed curve has an infinite number of evolutes, situated on the developable surface which is the locus of the poles of the proposed; hence the locus of the poles is the locus of the evolutes. If the original line PQ be perpendicular to the corresponding line of poles or characteristic, then, since this characteristic moves in the moving plane while PQ remains fixed, PQ cannot continue to be per- pendicular to the characteristic; but the radius of absolute curvature is always perpendicular to the characteristic, this radius therefore cannot continue to intersect the characteristic in the point Q; so that the locus of the centres of absolute curvature is not one of the evolutes of the proposed curve. (216.) Should the curve which we have all along considered of double curvature be plane, then, indeed, since the characteristics are ail parallel, and perpendicular to the plane of the curve, the line PQ, once perpendicular, will be always perpendicular to the characteristic; so that then Q will coincide with the centre of curvature, PQ being no other than the radius of curvature, the locus of the centres being the plane evolute before considered. But when PQ is not drawn perpen- dicular to the original characteristic, but is inclined to it at an angle a, then it always preserves this inclination during the generation of the cylindrical surface, which is the locus of the poles; therefore every curvilinear evolute of a plane curve is a helix described on the surface of the cylinder, which is the locus of the poles of the plane curve. Every curve traced on the surface of a sphere, has, for the locus of its evolutes, a conical surface whose vertex is at the centre of the sphere; because the normal planes to the curve being also normal planes to the spheric surface, all pass through the centre.CURVES OF DOUBLE CURVATURE. 283 (217.) From what has now been said, it is obvious that if from any point in a curve a line be drawn to touch the developable surface which is the locus of its poles, and its prolongation be wound about the surface without twisting,* it will trace one of th*e evolutes; and, as the string may be drawn to touch the surface in every possible direction, it follows that every developable line on the surface will be an evolute. If the curve be plain, the evolutes are all on the cylindrical surface whose base is the plane evolute. As obviously a developable line is the shortest on the surface that can join its extremities, it follows that the shortest distance between two points of an evolute measured on the surface, is the arc of that evolute between them. PROBLEM IV. (218.) Having given the equation of a curve of double curvature to determine those of any one of its evolutes. All the evolutes of the curve being on the same developable surface, the equation of this surface must be common to them all; and we have already seen (192) how the equation of the surface is to be determined, so that it only remains to find for each evolute a particular equation which distinguishes it from all the others, and determines its course on the developable surface. In order to this, let us consider that each evolute must be such that the prolongation of its tangent at any point always cuts the involute, or, which is the same thing, the tangent to the projection of the evolute at any point passes through the corresponding point in the projeetion of the evolute; therefore, considering the plane of xy as that of projection, we have, for the tangent at any point (x% y') * This is what I understand Monge to mean, when he says (App. de VAnal. de Giom., p. 348,) “ si I’on plie librement sur cette surface le pro- longement de cette tangente.” It seems not improper to call such lines placed on a developable surface developable lines, and those which form curves on the developed surfaces twisted lines. Of these two species of lines all the former are evolutes, but none of the latter are.284 THE DIFFERENTIAL CALCULUS. in the projected evolute, and, since the same line passes through a point (x9 y) in the projected involute, its equation is also hence, combining this equation with that of the developable surface, determined agreeably to the process pointed out in art. (192), and eliminating x, y being a given function of x, we shall have two equations in x\ y, z'9 of which one will contain partial differential coefficients of the first order, and which together will represent all the evolutes. To find that particular one which is fixed by any proposed condition, it will be necessary to discover, by the aid of the integral calculus, the primitive equation from wiiich the differential equation mentioned is deducible; this primitive equation will involve an arbitrary constant, whose value may be fixed by the proposed condition, and thus the equations of the particular evolute will be determined. We shall terminate this Section by subjoining a few miscellaneous propositions. y —y of — x y) · · · · 0)> y=4>(x) .... (2), *=/(*> Φ(χ)) = Ψ(*) · · · · (3):286 THE DIFFERENTIAL CALCULUS. where x*, y'9 z', are the coordinates of the proposed point on the surface. άφ(χ) · Now —is the total differential coefficient derived from the function dx z ==/(#, y)9 in which y is considered as a function of x> given by the equation (2), that is d^(*> _ _„/ , . d0(*) . Ax dar ^ ^ Ax ’ hence, by substitution, the equations of the tangent in space become (*') λ ■ ■ (*)■ y—/=· *0^ s —s' = (p' + ?'-M^-)(i— Now, to obtain the locus of the tangents whatever be the curve through the point (x'9 y’9 sf)9 we must eliminate the function if a centre exists, or will show, by their incongruity, that the surface has no centre. Thus, suppose the equation of the surface is of an even degree, then we must equate to 0 the coefficients of all the odd powers and combinations of x, y, z, since the terms into which these enter would change signs when the variables change signs; we obtain in this way the equations of condition. If the equation of the surface be of an odd degree, then we must equate to zero the coefficients of all the even powers and combinations of x9y, z; so that only odd powers and combinations may effectively enter the equation, for then whether the variables be all -}- or all — the function f (x9 y, z) will still be 0. Now the differential calculus furnishes us at once with the means of obtaining the several expressions which we must equate to zero without actually substituting x + xi9 y -\-yt9 z + z/9 for x9y, z, in the equation of the surface. For if we conceive these substitutions made in the function f(x, y9 z), we may consider the result as arising from xt9 y,9 zl9 taking the respective increments x,y, z, and we know that every such function may, by Taylor’s theorem, be developed according to the powers and combinations of the increments; and that the several terms of the development consist each of the partial differential coefficients of the preceding term, the first being f(x„y„ *,)· Hence, if the coefficients of the first powers of x, y} z, are to be respectively zero, then we have to equate to zero each of the partial coefficients derived from u, = βχ,, y„ z,) = 0, or, which is the same thing, from u = f(x, y, z) = 0 the proposed equation; if the coefficients of the second powers and combinations of x} y, z, are to be rendered each 0, then we shall have to equate to zero each partial coefficient derived from again differentia- ting, and so on. As an illustration of this, let the general equation of surfaces of the second order‘288 THE DIFFERENTIAL CALCULUS. Aa?2 4* Ay2 4* A V 2Bya + 2B'zx + 2B"xy ^ l =0 = u ....(]) 4- 2Cx 4- 2C'y 4- 2C's + E 5 be proposed, then the degree of the equation being even, the coefficients of the odd powers of the variables in the equation, arising from putting v + V + V» z + for x9 \h * are to be equated to 0, and as the equation is but of the second degree, these odd powers will be of the first; hence we have merely to equate the first partial differential coeffi- cients to 0, that is g = A* +B'S + B'y + C =0 du ~ =S My 4- B* 4- B'* +C'=0 dy ~ = A'2 4 B'z 4- By 4- C# = 0 The values of x, y, z, deduced from these equations are the coordi- nates y,, of the centre. These values may be represented by N Ν' N' Χ·~~Ό* VlO’ %t Ό* where D = AB2 4- A'B'a 4- A"B'2 — AA'A'—2BB'B', so that the surface has a centre if D is not 0; but if D = 0 and the numerators all finite, the surface has no centre; and, lastly, if D = 0 and either of the numerators also 0, then the surface has an infinite number of centres, and is, therefore, cylindrical. The equations of condition (2) are the same as those at page 292 of the Analytical Geometry. PROPOSITION III. (221.) To determine the equation of the diametral plane in a surface of the second order which will be conjugate to a given system of parallel chords.MISCELLANEOUS PROPOSITIONS. 289 Let the inclinations of the chords to the axes be α, β, y, then the equation of any one will be x=zmz +p, y=znz + g ... . (]), where cos a cos β cos y, cos y For the points common to this line and the surface, we must combine this equation with equation (1), last proposition, and we shall have a result of the form Rz2-f Sz + T = 0 .... (2), which equation will furnish the two values of z corresponding to the two extremities of the diameter, and therefore half the sum of these values will be the z of the middle, that is, this z is 2 = .·. 2Rs + S = 0-----(3), which is obviously the differential coefficient derived from (2), or, which is the same thing, the total differential coefficient derived from (1) last proposition, in which x and y are functions of z given by the equations (1). This differential coefficient is, therefore, .dw, ___dw d# du dy du______ *ds it? dz dy ds dz ° d u ( d u , d u = sIy + i = 0'·'®- where p and q, the only quantities which vary with the chord, are eliminated; hence, this last equation represents the locus of the middle points of the chords or the diametral surface, and it is obviously a plane. By actually effecting the differentiations indicated in equation (4) upon the equation (1), last proposition, we have, for the equation of the required diametral plane,290 THE DIFFERENTIAL CALCULUS. m (Ax + B'z + B"y + C) + n (A'y + Bs + B"x + C') + A"* + By + B'x 4-C" = 0, or (Am + B' + B"n) * 4- (A'n 4- B 4- Wm) y 4- (A" 4- Bn 4- B m) z 4- Cm 4- C'n 4- C" = 0. PROPOSITION IV. (222.) A straight line moves so that three given points in it constantly rest on the same three rectangular planes; required the surface which is the locus of any other point in it. Let the proposed planes be taken for those of the coordinates, and let the coordinates of the generating point be «r, y, z, and the invariable distances of this point from the three points resting on the planes of yzy x z, and xy, X, Y, Z. The coordinates of these three points will be Then, since the parts of any straight line are proportional to their projections on any plane, each part having the same inclination to it, it follows that if we project successively each of the parts X, Y, Z, on the three coordinate planes, we shall have the relations In the plane of yz, 0, y, zr x x — x" X Y z y—y x y__ Y Z---Σ' Z—2'' 2 X Y Z J But the part X of the moveable straight line comprised between theMISCELLANEOUS PROPOSITIONS. 291 generating point (y, z) and the point (0, y', zf) resting on the plane of y> z, has for its length the expression X2 (w — w')2 (z— z')2 >=^ + S^ + L^ri-····^ but from the equations (1) y—}/ _ y z — 2? _ 2 X Y9 X z5 hence, by substitution, (2) becomes x* v* z* ____L £_l jL— i X2 “ Y2 * ' consequently, the surface generated is always of the second order. The surface would still be of the second order if the three directing planes were oblique instead of rectangular, as is shown by M. Dupin, in his Oeveloppements, p. 342, whence the above solution is taken. PROPOSITION V. (223.) To determine the line of greatest inclination through any point on a curve surface. The property which distinguishes the line of greatest inclination through any point is this, viz. that at every point of it the linear tangent makes with the horizon a greater angle than any other tangent to the surface drawn through the same point of the curve. Now, as all the linear tangents through any point are in the tangent plane to the surface at that point, that one which is perpendicular to the trace of the tangent plane will necessarily be the shortest, and therefore approach nearest to the perpendicular, that is, it will form a greater angle with the horizon than any of the others. We have, therefore, to determine the curve to which the linear tangent at every point is always perpendicular to the horizontal trace of the tangent plane to the surface through the same292 THE DIFFERENTIAL CALCULUS. point; or, which is the same thing, the projection of the linear tangent on the plane of xy must be perpendicular to the trace of the tangent plane. Now the equation of the projection of the linear tangent at any point is , dy' / y—y ~ΊΰΓ^~ and, by putting z = 0 in the equation of the tangent plane, we have, for the trace in the plane of xy, the equation — Zf — p'(x — x') + q {y —- y'), and, since these two lines are to be always perpendicular to each other, we must have throughout the curve the general condition __ £_ dx p' = 0, p' and q' being derived from the equation of the surface; so that the values of these being obtained in terms of x and y, and substituted in the equation just deduced, the result will be the general differential equation belonging to the projection of every curve of greatest inclina- tion that can be drawn on the proposed surface. To determine that passing through a particular point, or subject to a particular condition, we must, by help of the integral calculus, determine the general pri- mitive equation from which the above is deducible; this primitive will involve an arbitrary constant, which may be fixed by the proposed condition, and thus the particular line be represented. As an example, let us take the right conoid, of which the equation by (182) is Then for the ratio of the two partial differential coefficients p'Tq\ which ratio is independent of the form ψ (art. 58), we have JL x dy — —- = -p-, by the condition above; y293 where c is any constant quantity· The lines of greatest inclination are therefore such that their horizontal projections are concentric circles; the common centre being the point where the vertical axis, that is, the axis of z9 pierces the plane of xy. Again, let the surface be of the second order, and represented by where c' is the number whose logarithm is c, or d = log'-1 f, and c is an arbitrary constant; hence the horizontal projections of the lines of greatest inclination will be parabolic curves when A and B have the same sign, and hyperbolic curves when the signs of A and B are different. (See ex. 6 and 7, page 148.) Suppose the point of departure is in the plane of zx; then, since at that point y = 0, and x = x', it follows that the equation of the line of greatest inclination through the point must be satisfied for these values; so that we must have then JL B _ dy u d y x άχ * * y a which equation is evidently the differential of B log y — A log x = c; and this leads to b , A B /A /1X , y -4-λ* zzcy or y = cx .... (1), (2) .... 0 = c'xA .*. c' = 0. c c 2294 THE DIFFERENTIAL CALCULUS. Such, then, is the value of the arbitrary constant in the general equa- tion (1) of these lines, for that particular line which we have selected from among them, so that this line coincides with the section made by the plane of xz; its projection being a straight line, which is a variety of the parabola. PROPOSITION VI. (224.) The six edges of any irregular tetraedron or triangular pyramid are opposed two by two, and the nearest distance of two opposite edges is called breadth; so that the tetraedron has three breadths and four heights. It is required to demonstrate that in every tetraedron the sum of the reciprocals of the squares of the breadths is equal to the sum of the reciprocals of the squares of the heights. Let the vertex of the tetraedron be taken for the origin of the rectan- gular coordinates, and let also one of the faces coincide with the plane of xz, then the coordinates of the three corners of the base will be • 0, 0, I 0, I y"', s'", and the equations of the three edges terminating in the vertex will be x = 0 y = o Now the perpendicular distance between each of these edges and the opposite edge of the base will evidently be equal to the perpendicular demitted from the origin on a plane drawn through the latter edge and parallel to the former. Hence, denoting the three planes through the edges of the base by Ax + By -f- Cs = 1 | Ex -f- Fy + Gs = 1 | Id? -f* Ky -f- Lz = 1, they must be drawn so as to fulfil the conditions, (See Anal. Geom. p. 232,)MISCELLANEOUS PROPOSITIONS. 295 Cz =1 Ax" -f-C2"=l Ai',//4-By"-f-C2'"=0 Gs' =1 Ex0 -j-Gs" =0 lx‘ +L*" =1 Ia/"+Ky"-}-Ls'"==l Ls' =0 These conditions fix the following values for A, B, C, &c. viz. af" _J_ 1________s^_ a/'V7 ^ s' * x'z i a77y77*7 , x"y" z'" y"'Z' 1 s' 1 ff"7s" ' s'" p ___ _ Γ _________ -p __________ ___I w______________. 2' ’ *V ’ ~ y‘" ^ yy'V y"V ’ :0, 1 = 1, K 1 Hence, calling the breadths B, B', B", we have {Ami, Geom. p. 236,) 1 ™ ™ _(/V~y"s")^(x'V~x"VWV")HfxY'/)2 B* “ A+B+C— («VO* i__ F2 , p2, r2 __ (yvwwv^o»4-w,,)< V2 -r r ■ ^ — /A/v^ B'2 (x'y"zf)* _j____p ■ k2 ,Ls^(^)2+(^-^:)2 B"2 — +* Ί- ^ — (Iyv>2 Hence β2 + β,2 + β//2 — (*' — z")Yf2 4- {(s"— *V" — a'Y"}2 + . H-WV)i...,.(l). { (2' — Ζ"')χ" + Λ"}2 + (yV)i + (X"_X'")V)2 5 Again, the expressions for the heights or perpendiculars demitted from each of the points (0, 0, 0) j (0, 0, s ); (x , 0, z ); {x > y > z ), upon the plane which passes through the other three are, severally, (Anal, Geom, p. 236,)296 THE DIFFERENTIAL CALCULUS. H2 = (Vy'Y)2 (z" — z)*y"*+ {(s'" —s')*"+ (*' — s")*"T + (y"V')a frV'O»_________ H'2 = (sV7 + *,,/«'/)*+ (a7,/3/",)s "(y'"Os + (*"'Oa’ “ (*"z')2 • £[2 · H'2 ' £L"2 * H///2 + (zf — z'')x,u\2 + v i-r(*V,/0,-(2)· 2(x'/3/,//)2+(y/V02+(x/V/,--x,,V')2+(3/,/'2+x,//?+x'/2>/2 ’ which expression is the same as that before deduced, and thus the theorem is established by a process purely analytical. This remarkable property was discovered by M. Brianchon, and formed the subject of the prize question in the Ladies’ Diary for 1830. Elegant solutions upon different principles by Messrs. Woolhouse and Godward may be seen in Professor Leybourn’s Mathematical Repository, No. 25. end of the differential calculus.NOTES Let Note (A), page 51. On the Summation of slowly Converging Infinite Series. a — bx -f* cx2 — c?t3 -f“ &c. = S, .·. — bx -{- cx2 — dx3 -f- &c. = S — a .·. — b 4- cx — dx2 4“ cx3 — &c. = ----- x Consequently, by adding these two last equations together, and represent- ing the numerical differences, b — c, c — d, d — e, Δη+1; and consequently, half the sum of the two limits will be a superior limit still nearer. We may conclude, therefore, that if we mul- tiply the final term in the inferior limit by and add the product to that limit, we shall thus obtain a near superior limit. It is scarcely necessary to observe here, that in what has hitherto been said, the coefficients in the proposed series, S, are supposed continually to diminish, as also the several series of differences, which supposition is con- formable to what usually occurs in practice, S being a convergent series. As an example, let it be required to sum the converging series ■-Τ + Τ-Τ + Τ-Π + ·* It will be advisable to actually sum up first a few of the leading terms, and then to apply the formula (B), or rather (C), to the remainder of the series. The work may be arranged as follows: + 1 — *3 + *2 + *142857 + -111111 — -090909 •744012 = + -076923 10256 — -066667 7844 + -058823 6191 — -052632 5013 + -047619 2412 1653 1178 of first six terms. 759 475 284NOTES. 301 The numbers *076923, *010256, *002412, *000759, and 000284, are the respective values of b, Δ2, Δ3, and Δ4, in (C); and to deduce from these the value of S, it will be necessary merely to add half the last number, 284, to the preceding; half the sum to the next; and so on, to the last, adding half the final sum to the number *744012, previously found. The remainder of the operation is therefore as follows: •076923 10256 2412 759 284 •082767 11687 2862 901 142 •041384 5844 1431 450 •744012 •785396 = Inferior limit. •785400 = Superior limit. .·. *785398 = — very nearly. This value of we may be assured,, from the small interval between the limits, cannot possibly differ from the truth by more than a unit in the final decimal. It is in fact true even in the last figure. We have separated the two parts of the process in this example for the purpose of clearer illustration; but they may be combined, as in the fol- lowing example: To find the value of the converging series 1 39 32.52 32.52.72 ϋ2 ' 22.42 22.42.62 22.42. 6s . 82 which occurs in the expression for the time of vibration of a pendulum in a circular arc: Dd302 NOTES. + 1 — •25 •140625 — 97656 + 74768 — 60562 •807175 50889 7010 43879 1696 5314 548 + 38565 1148 4166 336 34399 812 31045 3354 + 654 2023 8022 327 54900 1012 4011 27450 •807175 834625 = inferior limit •834628 = Superior limit •834626 = S very nearly. A value which cannot differ from the truth by more than a unit in the last decimal. It is in fact true in all its places.* * For further particulars respecting this method of summing infinite series, see a paper by the author of the present work in the Philosophical Magazines for May and July, 1835.NOTES. 303 Note (B), page 117. Demonstration of the Theorems of Laplace and Lagrange. Let it be required to develop the function u = Ψ05), where z = F(y -|- xf (z)). By differentiating the second of these equations, first relatively to x, and then relatively to y, we have ^ = F(y + xf(z)) (/(z) + */'(z) ^ = F'(» + */<*)) 0 + xf (*) fy)· dz dz Multiplying the first by —, and the second by —, dy ax and subtracting, there results ds ,, v dz Λ da , di „ x da? but, since u or Ψ(ζ) depends only on 2, we shall have du , % dz dw Tv di due v da? dy dy therefore, eliminating Ψ(ε), we get du dz du dz da? dy dy da? * At page 88 we put /'(z) to represent the differential coefficient of /(s) relatively to a?$ here the same symbol denotes the coefficient relatively to z.304 NOTES. άζ or putting for its value (1), and making for abridgment/(«) = Z, this ur dz last expression becomes divisible by —, and reduces to du______^ 4u da? d y . (A), d u d u so that we may always substitute for — the quantity Z —■· If we differentiate the preceding equation relatively to x, we shall obtain d 2u dx^~ dx (2), d u dz but the expression Z — being no other than/(z) Ψ'(ζ)—, that is to say a dz function of z multiplied by —we may consider it as the differential co- ily efficient of some new function of x, which we may represent by and we shall then have dZ — d«, _ du . dy d2u, sss Z ——» and —— ■: ~ j ay ay dr dr dy therefore (2), inverting the order of the differentiations in this last ex- pression, d2u d2ux dx2 dy di I Aul dx dy (3), now, it must be observed that the relation (A) exists, whatever be the func- tion u j it therefore exists for the function ux; hence _ z d“i do? dy di/. Substituting then in (3) for - its value here exhibited, and afterwardsNOTES. 305 for - its equal Z there results dy dy d*u dZ dux ~W dZ2 d u d y • (B). dx2 dy dy Differentiating this last equation relatively to x, we shall obtain d3w du d2Z2-F ____dy dx3 dxdy * and considering, as before, the function Z2 — to be the differential coeffi- μy cient of some new function of z, viz. uv we shall have, by inverting the order of the differentiations, d d2—2. dtto du , d3u dx = Z2 —, and — =——— dy d y da3 dy2 • (4), but the equation (A) subsisting for every function u9 must have place for _d%_ = z du2 dx dy 1 therefore (4), d u, d3u _X = Z3 —, ---- dx dy dx3 d y2 (C). The analogy among the expressions (A), (B), (C), is obvious, and we shall now show that this analogy continues uninterrupted ; that is, gene- rally, if then dn—2 Zn~l du dy dx”-1 dyn~2 dnu dn-lZn — dy dx” I II -----(M), • (N). D d 2306 NOTES. d u For considering, as before, the function Zn—1 —- to be the differential dy coefficient of some new function of s, viz. w*_p so that _ , d u dun—, , % z-'3j=^r we have, by differentiating (M), d nu d*»—i dun—j dx dxn dyn—1 but the equation (A) subsisting for every function of z subsists for un-l, therefore dun~x _ z dun-l ' hence (5) dnu ~dxn dx “ dy ’ dn—i Z" — d«-y(s)^ ______dy J dy dyn—1 (Jyn Let now x = 0 in the original function, and in each of the coefficients d n d2u dnu Λ. . ——, —— .... ——; then we have dx dx2 dx« ’ M = ^(F(y)) = 0(y), W = F(y) .·./[*] =/(F(ji)) = +(y), and *■ dxn ^ dy"-"1 Consequently, by Maclaurin’s theorem, . .. ,άφ(ν) * . ’ Jr . «=*(?) +*(y)T +- 1 . 2 l3 dy2 1.2.3 4* C08v=rl?- By employing these expressions M. Cauchy has arrived by rather a novel process at the theorem of Memnier, given at p. 233· Let the equation of any curve surface be u=z F(x, y, *) = 0,NOTES. 311 upon which is traced any curve MG', determined by the equation y=φ(*)> joined to the preceding. If through the tangent MT to this curve, and also through the normal MN of the surface, we draw a plane, we shall be fur- nished with a normal section MG, of which the radius of curvature r, at M, will be some portion of the normal MN. Also the radius of curvature r7 of the assumed curve MG7, at the same point, will be some portion of the line MN7, perpendicular to the tangent MT. Now, considering s to be the independent variable, we have, for the in- clinations of r7 to the axes the expressions above, viz. r' d8z ~A&~’ and the inclinations of r or of MN to the axes are (129) d u d u Au v v —, v — · cU* Ay Az Hence, calling the angle N'MN, between the two radii, ω, we have (Anal. Geom. p. 239) . , Au d8x Au Ay Au Ah v K Ax As* ^ Ay As* ^ Az As* ' But the equation of the surface, considered as one of the equations of the curve MG7, gives after two successive differentiations, still regarding s as the independent variable, Au A*x Au A*y Au Ah ______ Ax As* Ay ds2 Az As* I A*u dx2 A?u Ay2 A*u ds8 da8 ds2 Ay* ds8 Az* ds8 ^ A*u dx dy ^ ^ A*u dx dx A*u Ay ds Ax Ay ds2 dx ds dj2 Ay Az dfa Now whatever be the curve MG7, provided only its tangent MT = x' remains unchanged, the second member of this last equation will remain312 NOTES. unchanged, because the values of which are the same as d* di ds those of -^r, ~^r> remain unchanged. Therefore this second member being substituted in the expression for cos ω leads to a result of the form r' = K cos ω, K being a constant expression for all the curves on the proposed surface which touch MT at the point M. Put now, in this expression, ω = 0, then r' becomes r, therefore r = K, consequently r' = r cos ω, which result comprehends the theorem of Meusnier; since, if the curve MG' is plane, its plane will coincide with N'MT, and the angle ω of the two radii will become the angle formed by the plane N'MT of the oblique section with the plane NMT of the normal section passing through the same tangent MT.—Leroy, Analyse Appliquee a la Giom6trie, p. 268 We may take this opportunity of remarking that, in our investigation of this theorem, at p. 233, it might easily have been shown, without referringto article (86), that da/2 _ = ]+tan’e, because, by the right-angled triangle, a/2 = xa sec2 Θ = ®2 (1 4- tan2 0) da'2 ds* —- = 1 -f tan2 0 = dx2 dac2 Note (E), page 136. The erroneous doctrine adverted to at page 136 is laid down also by Lacroix, in his quarto treatise on the Calculus, vol. I. p. 340, from whom, indeed, Mr. Jephson seems to have adopted it. The principle as stated by Lacroix is (< que la serie de Taylor devient illusoire pour toute valeur qui rend imaginaire Pun quelconque de ces terms; et que cela peut arriver sans que la fonction soit elle-m&me imaginaire.” THE END. Printed by J. and C. Adlard, Bartholomew Close.