I **iv/. jr.. >^9^V k THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA PRESENTED BY PROF. CHARLES A. KOFOID AND MRS. PRUDENCE W. KOFOID THE ELEMENTS DIFFERENTIAL CALCULUS; COMPREHENDING THE GENERAL THEORY OF CURVE SURFACES, CURVES OF DOUBLE CURVATURE. INTKNDID FOR THB USE Or MATHEMATICAL STUDENTS IN SCHOOLS AND UNIVERSITIES. BY J. R. YOUNG, AUTHOR or THE ELEMENTS OF ANALYTICAL CEOMETRT.' REVISED AND CORRECTED, BT MICHAEL O'SHANNESSY, A.M. CAREY, LEA & BLANCHARD, CHESNUT-STREET. 1833. "Entered according to Act of Congress, the 6th of March, in the year 1833, by G. F. Hopkins & Son, in the office of the Clerk of the Southern District of New- York." PrinWd by U. F. UOPKINa & s6n, Now.york. j^m ADVERTISEMENT. This edition of Young's Differential and Integral Calculus is presented to the American public, with a confidence in its favourable reception, proportionate to that which the original acquired in England. The text has not been materially al- tered, though many errors have been corrected, some of which by Professor Dodd of Princeton College, N. J. These volumes will be found to contain a full elementary course of the subject of which they treat, and well adapted as a text book for Colleges and Universities. The second volume, treating exclusively of the Integral Calculus, is now in press, and will be speedily published. New- York, March, 1833. Digitized by tine Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementsofdifferOOyounrich PREFACE The object of the present volume is to teach the principles of the Differential Calculus, and to show the application of these principles to several interesting and important inquiries, more particularly to the general theory of Curves and Surfaces. Throughout these applica- tions I have endeavoured to preserve the strictest rigour in the varioiis processes employed, so that the student who may have hitherto been accustomed only to the pdre reasoning of the ancient geometry will not, I think, find in these higher order of researches any principle adopted, or any assumption made, inconsistent with his previous no- tions of mathematical accuracy. If I have, indeed, succeeded in accomplishing this very desirable object, and have really shown that the applications of the Calculus do not necessarily involve any principle that will not bear the most scrupulous examination, I may, perhaps, be allowed to think that I have, in this small volume, con- tributed a little towards the perfecting of the most powerful instru- ment which the modern analysis places in the hand of the mathema- tician. It is the adoption of exceptionable principles, and even, in some cases, of contradictory theories, into the elements of this science, that have no doubt been the chief causes why it has hitherto been so little studied in a country where the ancient geometry has been so extensively and so successfully cultivated. The student who pro- ceeds 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 application of that theory, is to be considered a quantity infinitely small. He will naturally enough be startled to find that a conclusion is to be taken as general, when he is at the VI PREFACE. same time told that the process which led to that conclusion has fail- ing cafees ; and yet one or both of these inconsistencies pervade more or less every book on the Calculus which I have had an opportunity of examining. The whole theory of what the French mathematicians vaguely call consecutive points and consecutive elements, involves the first of these objectionable principles ;* for, if the abscissa of any point be repre- sented by X, then the abscissa of the consecutive point, or that sepa- rated from the former by an infinitely small interval, is represented by .c + d.v, although dx, at the outset of the subject, is said to be 0. Again, the theory of tangents, the radius of curvature, principles 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 re- ceived in all their generality.! * It is to be rcigrctted that terms so vague and indofinlte should be introduced into the caraci sciences; and it is more to be regretted that English elementary writers should adopt them merely because they arc used by the French, and that too with- out examining into the import these tenns carry in the works from wliichthey are copied. In a recent production of the University of Cambridge, the autlior, in at- tempting to follow the French mode of solving a certain problem, has confounded consecutive points with consec^itive elements, two very distinct things : although neither very intellisible, the consequence of this mistake is, that the result is not what was intended; so that, after the process is fairly finished, a new counter- balancing error is introduced, and tlius the solution righted ! 1 1 am anxious not to be misunderstood here, and shall therefore state specifi- cally the nature of my objection. In establisliing the theory of contact, &c., bff 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 inde- terminate quantities be shown to be greater or less than a series of other functions of the same indeterminate 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 particu- lar values may render the functions all infinite; and we shall be equally at a loss to conceive how one of these infinite quantities can be greater or less tlian the others. It appears, therefore, that the usual process by which the theory of con- tact is established, applies rigorously only to those points of cur\xs for which Taylor^s development does not fail, and T cannot help thinking that on these grounds the .^nalylical Tliem-y of Functions, by Lagrange, in its application to Ge- PREFACE. Vll If this statement be true, it is not to be wondered at that students so often abandon the study of this science, less discouraged with its difficulties than disgusted with its inconsistencies. To remove these inconsistencies, which so often harass and impede the student's pro- gress, has been my object in the present volume ; and, aUhough 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 JVeivton, although designated by foreign writers as the method of d' Member t. I consider this method to be as unexceptiona- ble as that o( Lao-range, and on account of its greater simplicity, better adapted to elementary instruction. The First Chapter is devoted to the exposition of the fundamental principles ; and in explaining the notation I have been careful to im- press upon the student's mind that tlie differentials dx, dy, &c. are in themselves absolutely of no value, and that their ratios only are sig- nificant : this is tlie foundation of the whole theory, and it has been adhered to throughout the volume, without any shifting of the hypo- thesis. In the Second Chapter it is shown, that i£fx represent any function of X, and x be changed into x + /t, the new state y (x + h) of the function may always be developed according to the ascending inte- gral powers of the increment h ; and this leads to the important con- clusion that the coefficient of the second term in the development of the function f{x-\- h) is the diflferential coefficient derived from the function /x; a fact which Lagrange has made the foundation of his ometry is defective, altliough I feel anxious to express my opinion of tliat celebra- ted performance witii all becoming caution and humility. Indeed Lagrange him- self has admitted this defect, and observes, {Thiorie des FoncHons, p. 181,) " GLuoi- que ccs exceptions nc portent aucune atteinte ci la thfeoric g^nferale, il est n6ces- saire, pour ne rien laisscr k desirer, de voir comment die doit etre modifier dans les cas particuliers dont il s'agit" (See note C at the end.) But he has not modified the expression deduced from this exceptionable thebry for the radius of curvatiue, which indeed is always applicable whether the differential coefficients become infinite or not, although, for reasons already assigned, (he process which led to it restricts its application to particular points. ^111 PREFACE. 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 M.aclaurin' s theorem, and its ap- pHcation is shown 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 valuable analytical formulas and expressions from Euler, Demoivre, Cotes, and other celebrated analysts, together with those curious properties of the circle discovered by Co fes and Demoivre. The Fourth Chapter is on Taylor's theorem, which m.akes known the actual development of the function /(a; + h) according to the form established in the second chapter. From this theorem are de- rived commodious expressions for the total differential coefficient when the function is compUcated, 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 vanishing frac- tions. The Sixth 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 processes. Chapter the Seventh is on the differentiation and development of functions of two independent variables. The usual method of obtain- ing the development of a function of two variables according to the powers of the increments, is to develop first on the supposition that x only varies and that y is constant, and afterwards to consider y, which is assumed to enter into the coefficients, to be changed into y -\- h. But y may be so combined with x in the function F {x, y) that it shall, when considered as a constant, disappear from all the differential co- efficients, which circumstances should be pointed out and be shown not to affect the truth of the result : I have, however, avoided the ne- cessity of showing this, by proceeding rather differently. The chap- ter 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 chapter VI. to functions of two independent variables, and it also contains an important article on PREFACE. IX changing the independent variable, a subject very improperly omitted in all the English books. The Ninth Chapter is devoted to a matter of considerable import- ance, viz. to the examination of the cases in which Taylor's theorem fails ; and I have, I thuik, satisfactorily shown, that these failing cases are always indicated by the differential coefficients becoming infinite, and that the theorem does not fail when these coefficients become imaginary, as Lacroix, and others after him, have asserted. Besides the correction of this erroneous doctrine, which has been sanctioned by names of the highest reputation, another very remarkable over- sight, though of far less importance, is detected in the Calcul dcs Fonctions of Lagrange, and is pointed out in the present chapter : it has been unsuspectingly copied by other writers ; and thus an entirely wronflf solution to a very simple problem has been printed, and re- printed, without any examination into the principles employed in it ; and which, I suppose, the high reputation of Lagrange was consider- ed to render unnecessary. These nine chapters constitute the First Section of the work, and comprise the pure theory of the subject ; the remaining part is devot- ed to the application of this to geometry, and is divided into two parts, the fii-st 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 tan- gents, 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 con- siderationof 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 ob- servations made on the circular asymptotes to spiral curves, the chap- ter terminating with the differential expression for the arc of any plane curve determined without the aid of Taylor's theorem. The Second Chapter contauis the tlieory of osculation, which is shown to be unaffected by the failing cases of Taylor's theorem, al- though this is employed to estabhsh the theory. The expressions for the radius of curvature are afterwards deduced, and several examples B X PREFACE. of their application given principally to the curves of the second order, and an instance of their utility shown in determining the ratio of the earth's diameters. The Third Chapter is on involutes, evolutes, and consecutive curves, and contains some interesting theorems and practical examples. Of what the French call consecutive curves, I have endeavoured to give a clear and satisfactory explanation, unmixed with any vague notions about infinity. The Fourth Chapter is on the singular points of curves, and con- tains easy rules for detecting them, from an examination of the equa- tion of the curve. This chapter also contains the general theory of curvilinear asymptotes, and completes the Second Section, or that assigned to the consideration of plane curves. The Third Section is devoted to the general theory of curve sur- faces, and of curves of double curvature ; in the First Chapter of which are established the several forms of the equations of the tan- gent 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 cur- vature. In the Second Chapter the theory of conical and cylindrical surfa- ces is discussed, as also that of surfaces of revolution ; and that re- markable case is examined, where the revolution of a straight line produces the same surface as the revolution of the hyperbola, to which this line is an asymptote. Throughout this chapter are interspersed many valuable and mteresting appUcations of the calculus, chiefly from Monge. The Third Chapter embraces the theory of the curva- ture of 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 Bupin. Most of these theorems 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 instance, departing from those principles, as laid dovvn in the preliminary chapter. Those who are famiUar with these inquiries will find that I have obtained some of these theorems in a manner much more simple and concise than has hitherto been done. I need only mention here, as instances of this simphcity, the theorems of Euler and of JVfewsmer, at pages 182 and 186. PREFACE. XI The Fourth Chapter is on twisted mrfaces, a class of surfaces which have never been treated of, to any extent, by any English author, al- though, as has been recently shown, 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 mathematicians, to JVLonge principally, and also to the Che- valier Le Roy, who has recently published a very neat and compre- hensive little 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 miscellaneous propositions intimately connect- ed with the theory of surfaces. From the foregoing brief analysis, it will appear evident to those familiar with the present state of mathe- matical instruction in this country, that I have introduced, into a little duodecimo volume, a more comprehensive view of the theory and applications of the differential calculus than has yet appeared in the English language. But I have aimed at more than this ; 1 have en- deavoured to simplify and improve much that I have adopted from foreign sources ; and, above all, to estabhsh every thing here taught, upon principles free from inconsistency and logical objections ; and if it be found, upon examination, that 1 have entirely failed in this en- deavour, I shall certainly feel a proportionate disappointment. I am not, however, so sanguine as to look for much public en- couragement of my labours, however successfully they may have been devoted : it is not customary to place much value, in this coun- try, upon any mathematical production, of whatever merit, that does not emanate from Cambridge. The hereditary reputation enjoyed by this University, and bequeathed to it by the genius of Barrow, of jyeivton, and of Cotes, seems to have endowed it with such strong claims on the public attention and respect, that every thing it puts forth is always received as the best of its kind. If this be the case with Cambridge books, of course it is also the case with Cambridge men, and accordingly we find almost all our public mathematical situations filled by members of this University. It is true that now * See Leybourn^s Repository, No. 22. Xll PREFACE. and then, in the course of half a century, we find an exception to this ; one or two instances on record have undoubtedly occurred, where it has been, by some means or other, discovered that men who had ne- ver seen Cambridge knew a little of mathematics, as in the case of Thomas Simpson, and of Dr. Hutton ; but such instances are rare. It is not for me to inquire into the justice of this exclusive system ; but, while such a system prevails, there need be little wonder at the decline of science in England : while all inducement to cultivate sci- ence is thus confined to a particular set of men, no wonder that its votaries are few. It is to be hoped, however, that in the present " liberal and enlightened age," such a state of things will not long continue, and that even the poor and unfriended student may be cheer- ed up, amidst all the obstacles that surround him, in the laborious and difficult, but subUme and elevating career on which he has entered, by a well-founded assurance that his exertions, if successful, will not be the less appreciated because they were solitary and unassisted. May 12, 1831. J. R. YOUNG. CONTENTS SECTION I. On the Differentiation of Functions in general. •Article Page 1. A Function defined - - - - - - -1 2. Effect produced on the function by a change in the variable - 2 3. Differential coefficient determined - - - - - 3 4. General form of the development of/ (a; + A) - - - 5 5. The coefficient of the second term in the general development is the dif- ferential coefficient derived from the function /a; - - - 8 6. To differentiate the product of two or more functions of the same variable 9 7. To differentiate a fraction - - - - - -10 8. To differentiate any power of a fimction - - - - ib. 9. To differentiate an expression consisting of several functions of the same variable - - - - - - - -12 10. Application of the preceding rules to examples - - - ib. 11. Transcendental functions - - - - - -15 12. To find the differential of a logarithm - - - - ib. 13. To differentiate an exponential function - - - - 16 Examples on transcendental functions - - - - ib. 14. To differentiate circular functions - - - - - 19 15. Differentiation of inverse functions - - - - - 21 16. Forms of the differentials when the radius is arbitrary - - - 24 17. Successive differentiation explained - - - - - 25 18. Illustrationsof the process - - - - - - 26 19. Investigation of Maclaurin's Theorem - - - - - 28 20. Application of Maclaurin's theorem to the development of functions - 29 21. Deduction of -EuZer's expressions for the sine and cosine of an arc, by means of imaginary exponentials - - - - - 31 22. Demoivre^s formula, and series for the sine and cosine of a multiple arc - 32 23. Decomposition of the expression!/*" — 2j/ cos. 0+1 into its quadratic factors - - - - - - - -33 24. Demotfre's property of the circle - - - - - 34 25. Cotes'* properties of the circle - - - - - - 35 XIV CONTENTS. ^rtklt Pag* 26. JbAn BamouHi's development of ( -y/ — 1) - * " - ib. Developments of tan. x and tan. ~^x - • • - 36 27. Evler^s series for approximating to the circumference of a circle - 38 28. Bertran(Ps more convergent series - - * - - - ib, 29. Examples for exercise - - - - - - -39 30. Investigation of Taylor's Theorem - - - - - 40 31. JV/ocZawrin's theorem deduced from Taylor's - - - - 42 32. Application of Taylor's theorem to the development of fvmctiona - ib- 33. Of a function of a function of a single variable - - - - 44 34. Examples of the application of this form - - - - 45 35. Form of the differential coefficient derived from the function u = Y{p,q,) where ji and q are functions of the same variable - - - ib. 36. Form of the coefficient when the function is v = F ( p, g, r,) - - 46 37. Distinction between partial and total differential coefficients - - 47 38. Examples -..-..--48 39. Differentiation and development of implicit functions - - - 49 40. On vanishing fractions - - - - -.- -52 41. Application of the calculus to determine the true value of a vanishing fraction - - - - - - - -53 42. Determination of the value when Taylor's theorem fails - - 56 43. Determination of the value of a fraction, of which both numerator and de- nominator are infinite - - - - - - 59 44. Determination of the value of the product of two factors, when one be- comes and the other ao- - - - - -60 45. Determination of the value of the difference of two functions, when they both become infinite - - - - - * ib_ 46. Examples on the preceding theory - - - - - 61 47. On the maxima and minima values of functions of a single variable - 63 48. If the function F (a -j- A) be developed according to the ascending pow- ers of h, a value so small may be given to h that any proposed term in the series shall exceed the sum of all that follow - - 64 49. Determination of the maxima and minima values in those cases where Taylor's theorem is applicable - - - - - ib. 50. Determination of the values when Taylor's theorem is not appUcable - 66 . . dy 51. Maxima and minima values which satisfy the condition — = od - 68 ax 52. Conditions of maxima and minima, when the function is impHcitly given ib. 53. Precepts to abridge the process of finding maxima and minima values 69 54. Examples - - - - - - - -70 55. On the cautions to be observed in applying the analytical theory of maxi- ma and minima to Geometry - - - - - 79 56. Differentiation of functions of two independent variables - - 81 57. Form of the differential when the function is implicit - - - 82 CONTENTS. X? JkUele Page SS. The ratio of the two partial differential coefficients derived from u = Fz, z being a function of a; and t/, is independent of F - - - 83 59. Development of functions of two independent variables - - 84 60. The partial differential coefficients composing the coefficient of any term in the general development are identical with those arising from dif- ferentiating the preceding term - - - - - 87 61. Maclaurin's theorem extended to functions of two independent variables 88 62. Lagrange's Theorem - - - - - - -89 63. Applications of Lagrange's theorem - - - - - 91 64. Maxima and minima values of functions of two variables - - 94 65. Examples - - - - - - - -96 66. On changing the independent variable - - - - 99 67. On the failing cases of Taylor's Theorem . - - . 100 68. Explanation of the cause and extent of these failing cases - - ib, 69. Particular examination of them . - - . - 102 70. Inferences from this examination ----- 103 71. The converse of these inferences true ----- 104 72. To obtain the true development when Taylor's theorem fails - - ib. 73. Correction of the errors of some analysts with respect to the failing cases of Taylor's theorem ..---- 106 74. On the multiple values of — in implicit functions - - - 108 dx 75. Determination of these multiple values . - - . 109 / x -\- a, when we replace a: by a: + h, is, by the above mentioned theorem, f\{x-\- h) -{- a\ = V{x-\-a) + h = {x + «)^ + i{x + a)-^ h— :^{x + af^ h^ + &c. where, in the case x =^ — a, all the coefficients become infinite, and the develqiment, according to the positive integral powers of ^-, be- comes in this case impossible ; for the function then becomes merely \/ h or h\, in which the exponent of ^ is fractional. The impossibility of the proposed form of development in such particular case is always intimated, as in the example just adduced, by the circumstance of infinite coefficients entering it, for imaginary coefficients would imply merely that the function f{x + h) for the assumed value of a; becomes imaginary, and not that the development failed. A particu- ar examination of the cases in which the general form of the deve- lopment fails to have place, will form the subject of a future chapter ; at present it is sufficient to apprise the student that such failing cases may exist. (6.) By transposing the first term in the general development of f{x + ^.), we have f{x + h) —fx = A^ + Bli" + C/i=' + &c. ,. n^±^-Z±. = A + Bh+ Ch^ + &c. h hence, when h = 0, dx from which result we learn, that the coefficient of the second term, in the development of the function f{x + h), is the differential coefficient derived from the function fx ; so that the finding the difierential coef- ficient from any proposed function, fx, reduces itself to the finding the coefficient of the second term in the general development of f{x + h), or of the first term in the developed difference f{x + h) -fa- Having obtained this general result, we may now proceed to apply it to fimctions of different terms ; but it will be proper previously to observe, that those constants which are connected with the variable in the functiouyx, only by way of addition or subtraction, cannot appear in the coefficient A ; because A, being multiplied by h, can contain THE DIFFEHENTIAL CALCULUS. 9 no quantities which are not among those multiplied by x + ^ in f{x -\- h), or by x infx. (6.) To differentiate the product of two or more functions of the same variable. Let y, 2, be functions of x, in the expression u = ayz. By changing x into x + h, the function y becomes 1/ =y+ Ah+ Bk" + Ch^ + &c. . . . (1), «nd the function z becomes z' = z-\- A7i + B7i2 + CV-{- &c. . . . (2). Hence, when ^ = 0, we have from (1) h da? ' and from (2) z' — z _ dz _ ., h dx ' ' Muhiplying the product of (1) and (2) by a, we have u' = ayz -f a {Az + A'y) h + &c.* = ayz+a{^^z+^y)h + kc. therefore, af-pz4- -r-y) being the coefficient of the second term of the development of «', we have du dy , dz -J- = az -~ -t ay -;- dx dx dx .'. du =■ azdy + aydz . . . (3). Hence, to differentiate the product of two functions of the same va- rictble, we must multiply each by the differential of the other, and add the results. It will be easy now to express the differential of a product of three functions of the same variable. Let. u = wyz be the product of three functions of .r ; then, putting v for wy, the ex- pression is u = vz; hence, by (3), * tt' is that value wliich w attains when the functions y and z have varied by virtue of the variation h of the variable x on which they depend. Ed. 2 10 THE DIFFERENTIAL CALCULUS. du = zdv + i^dz, butt) = toy ; therefore, by (3), dv = ydic + wyd ; consequentiy, by substitution, dn = sydw + ziody + wydz . . . (4), and it is plain that in this way the differential may be found, be the factors ever so many ; so that, generally, to differentiate a product of several functions of the same variable, ice must multiply the differen- tial of each factor by the product of all the other factors, and add the results. If we suppose the factors to be all equal to each other, we shall obtain a rule to differentiate a positive integral power. Thus the differential of the function U = X^ = X'X'X'X.... is du = af*-' dx + af^* dx + af"'^ dx + &c. to m terms, that is du = maf^^ dx .•. —- = m3f^\ ax This form of the differential is preserved whether m be integral or fractional, positive or negative ; but, to prove this, we must first dif- ferentiate a fraction. (7.) To differentiate a fraction. Let u = -,y and z being func- tions of X ; therefore uz = y, and duz = dy, that is, by the last article, zdtc + udz = dy .: du = — , or, substitutmg - for u, ^^^zdy-ydz^ z^ 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. (8.) To differentiate any power of a function. The form of the differential when the power is whole and positive has been already established. Let then THE DIFFERENTIAL CALCULUS. 11 m u = y" be proposed, y being a function of a?, and^ being a positive fraction. Since u" = tj'", .'. nw""' du = my"^^ dy, Now consequently, , . mil — m m (m ~ 1) = 1, n n du = — y dy. Let now the exponent be negative, or u = y-H 1 .*. m" = v~"' = — J yrn .*■. du" = d — but .'. nW^^ du = — jn?/~""~' dy, and dtt = — -— dy, or, substituting for u its equal y~'^, we have Jn" = nM^■'d«, and d -;j = — m -^-^;j- dy = — wj/"*""' dy, m du = y dy. Hence, generally, to differentiate a power, ive must multiply together these three factors, viz. the index of the power, the power itself dimi- nished by unity, and the diffei'ential of the root. This rule might have been deduced with less trouble, by availing ourselves of the binomial theorem, for, supposing inu = y^ that the increment of the function y becomes k when the increment of a? be- comes h, we have u' = {y + ky and, by the binomial theorem, the 12 THE DIFFERENTIAL CALCULUS. coefficient of the second term of the expansion of {y + ky is py''~\ whether p be positive or negative, whole or fractional. As, however, we propose to demonstrate the binomial theorem by means of the differential calculus, we have thought it necessary to establish the fundamental principles of differentiation, independently of this theo- rem. (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 result? by their respective signs. For let the expression be M = ato + fcy + c2 + &c. in which w, y, c, are functions of x. Then, changing x into x -{■ h and developing, to becomes w + Ah + B^^ + &c. y y + A'h+ B'h' + &c. s 2 + A"h + B"h' + &c. .-. « M + (aA + 6A' + cA" + &c.) /i + &c. .*. du = aAdx + bA'dx -\- cA"dx + &c. But Adx = dWf A'dx = dy, A"dx = dz, &c. therefore du = adw + bdy + cdz + &c. that is, the differential of the sum of any number of functions is equal to the sum of their respective differentials. (10.) We shall now apply the foregoing general rules to some examples. EXAMPLES. 1. Let it be required to differentiate the function y = 8x* — 3aP— 5x. By the rule for powers (8) the differential of 3x* is 8 X 4x^dx, and the differential of — SxP is — 3 X Sx^dx ; also the differential of — 5x is — 5dx ; hence (9), dy = 32x'dx — 9ordx — 5dx, ... $ = 32ar-'— 9x2 — 5. dx THE DIFFERENTIAL CALCULUS. 13 2. Let t/ = (a:"' + a) {3x^ + b). By the rule for differentiating a product (6), we have dy = {aP + a)d (3r' + 5) + (St" + &) d (r* + a), and (8), d {Sr' + 6) = 6xdx, d [aP -{■ a) = Sr'dx, .-. dy — {3p + a) exdx + (Sx^ + 6) SsPdx, ax 3. Let 1/ = (ax + a;^)^. The differential of the root ox + ar of this power, is orfx + 2xdxt therefore, dy = 2 {ax + x^) {a -{- 2x) dx, .'.■£ =2 {ax + x") {a-\- 2x). 4. het y = y/a-^bx~. The differential of the root or function under the radical, is 2bxdx ; hence 1 f)x % = 1 (« + bx')-^ 2bxdx = — dx, >/a+ bxP dy bx ' ' dx ^a-^-bif 5. Let y = {a + baf")". The differential of the root or function within the parenthesis, is mbaf^^dx; hence dt/ = n (o + 6x")"-' m6a?'"~' dx, ... -1 = imn (a + 6*'")""' **""'• ax x" 6. Let j( = (a + x3)2 The differential of the numerator of this fraction is 2xdx, and the differential of a + x' is 3x^dx, therefore the differential of the de- nominator is 2 (a + x^) 3x^dx ; hence (7), , _ {a+ apy 2xdx — 6x^ (a + x^)dx _ 2ax~4x^ ^ (a + x^y ~ (a + aPy '"' dy _ 2x{a — 2aP) ''' dx (a + apy ' 14 THE DIFFERENTIAL CALCULUS. 7. Let 7/= \a+ ,/(6 + ^j^ The differential of the root a + \/(6 + ^) is ^ (6 + ^) ~* d -5-, and d-^ = 7- cb ; hence ST or or ^ >/b+^ 8. Let y=Var' + Va+x'. The differential of ic^ + \/a + a^ is 2xdx + (a + ar')-2 ardar, djf X X ... ^ = = + " 9. Lety = — . Va^ + x^ — X Multiplying numerator and denominator by -v/a^ + ^ + x, the expression becomes "^ a a .-. % = d ^ + ^^ — ^ ^^+^ d VaP + a^. ar^ dy _ 2a: , Va^ + ar^ I •'•die ~ ^ a^ ^2 ^ a=» + ar' 2a: a" + 2a^ = TT + a^" Va" + x" 10. y = a?—.x'.:-^ = — 2x. 11. u = 4ar' — 2ar' + 7x + 3 .-. -j^ = 12ar' — 4a:+ 7. ax ^ THE DIFFERENTIAL CALCULUS. 15 12. y={a+ hx) x" .'. J = 46^* + Sax^. 13. y = {a -^ bx -\- ex" + &c.)'".-. ^ = w (a + 6x + car* + &c.)'^' {b-\-cx-\- &c.) 14. y = (a + 6r^)^.-.^ = ^y^+6^. I*; = ,. 4yar dy _ 6(1— ar') !»• 2^ a -f- 3 ^ ^ • • ^^ (3 + a^)2 v/x , , , c dy b , c 17. y = (ox' + 6)' + is/cf—e (x—h) .: ^ = 6111? v/a2 18. y = dy _ a:+ y/l—r" ' ' dx Vl—x'{l-\-2xy/l-r') The functions in these examples are all algebraic, we shall now consider Transcendental Functions. (11.) Transcendental functions are those in which the variable enters in the form of an exponent, a logarithm, a sine, &c. Thus, a', a log. X, sin. x, &c. are transcendental functions : the first is an exponential function, the second a logarithmic function, and the third a circular function. To find the Differential of a Logarithm.* (12.) Let it be required to differentiate log. x. Put a for the base of the system of logarithms used, and let M = 1 1 a_l_i(a_l)2 + i(a_l)3_&c.' then log. (1 + n) = M (n — i n= + i »^ — &c.) or, putting - for », X * Note (A'). t Algebra, Chap. vii. p. 219., or vol. i. p. 155, Lacroix's large work on the Dif- ferential Calculus. Ed. 16 THB DIFFERENTIAL CALCULUS. . log.(.+ fe)-log.. ^^l_ fe j^_ This is the general expression for the ratio of the increment of the function to that of the variable. Hence, taking the limit of this ratio, we have d log. X _ M ^^^ dx X ' ' ' If the logarithms employed be hyperbolic M = 1, and then d\og.x _ 1 dx X ' ' If they are not hyperbolic, write Log. instead of log. for distinction sake, then, since by putting a for 1 + » in the series for log. (1 + n), we have log. a = a- 1 -1 (a- If + 1 {a-lY — &c.=~ it follows, from the expression (1), that d Log. X _ 1 dx log. a . X Unless the contrary is expressed, the differential is always taken ac- cording 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 logarithmic function is equal to the differential of the function di- vided by the function itself. (13.) To differentiate an exponential function. 1. Let y = (f then log. y = x log. a .•. d log. y = dxlog. a, that is, — = dx log a .•. dy =" y log. a.dx = log. a . a' dx. Hence, to differentiate an exponential^ we mu^t multiply together the hyp. log. of the base, the exponential itself, and the differential of the variable exponent. EXAMPLES. 1. Let y = X {a:'' + xr) V d' — x^ .-. log. y = log. x + log. (o' + x^) + i log. (a^ -- r-), THE DIFFERENTIAL CALCULUS. 17 dy _ dx 2xdx xdx _ a^ + aV — 4x* , ' ' y X a^ + x^ c? — X' x{o? -\r x^) {a? — x") therefore, substituting for ij its value, we have, dy _ «" + «'-<^ — 4a;^ dx y/aj^ — x^ \/ a + a? + \/« — x 2. y = log. -== ;-==.• Multiplying numerator and denominator by the denominator, the expression becomes 2x y = log. == = log. X — log. (a — \/ a^ — x^) 2a — 2 \/ a^ — x^ dy _ 1 X _ a y/d^ — x^ — a^ ^^ ^ a VaF — or — a^ + ^^ x\/a? — x^\a — V d^ — x^\ — a X \f a^ — a^ \/ x^ -\- 2ax 3. y = r7=F====^ ••• log. y =i log. {x" + 2ax) — i ^xr -\- x- — X log. {3p-\-a^ — x)^ dy X + a , 3x^ -{- 2x — 1 , .♦.-^ — — ; dx — „ ^ ., ■ — dx = y or + 2ax 3 {x + x" — x) (l-^Sa) x'~{a + 2) x — a 3x{x'+ x—l) {x 4- 2fi.) ' . dy _\{1 — 3a) x^ —{a + 2) x — a\ V x ^^ Sx'^ (x' + a; — 1) 3 {x + 2a)"3' dy 4. J/ = x""^" * .'. log. y = m ^ — 1 log. X .'.~ = m ->/ — 1 dy y dx X . . -z- =m~ V — 1 — m \/ i.x"^-i-i. dx X From /-', where c is the base of the hyper- bolic system, .-. dy = c'>/~ v/ — 1 dx — e-> ^~^ v/ — 1 dar, 7. y = log. (log. x).* Put 2 for log. a?.-, y = log. «.•. dy = — but dz = d log. X ==—.'. dy = — r-^ — ••• t^ = — ; • X "^ X log. ar ax X log. x o „ . dy mn (log. x")*^* 9. y = log. J (a + X)' (a + x)- (a" + x)-"S ••• ^ = J^ j^ m' m" a -\- X a -f- X ,„ , Va + \/x dy y/a 10. y = log. .*. — ^ = . ^ Va — Vx dx (a — x) Vx 11. t/ = c^... ^ = 6** . x* (1 + log. x). 12. y = (log.)''x.-.^ = dx X log. X (log.)" X . . . (log.)""' X. ,- , v/ 1 + x^ dy 1 13. log. y = ■ .'. -i = -. ^ ^ X dx ar^ 14. y = a*', 2 being a function of X, .*. ~ = log.olog.&.a*'6^-v- * This means tho logarithm of the logarithm of x, but the notation we shall hereafter adopt will be (log.) 'a;, and which we shall extend to circular functions ; thus, instead of sin. (sin. x), we shall write (sin.)^a;, the square of the sine being written without the parenthesis, thus, sin. %. We may call such expressions a« (log.)" X, (sin.)" X, &.C the nth log. of i, the nth sine of j:, &c ■ THE DIFFEaENTIAL CALCULUS. 19 15. y = a*'""*" .-. ^ = log. o . log. b . a*'''+' . h'' +'(2^+1) dx ly dx du log. a . a ^°^^ ' 16. y = a '"s* . J — & , n dy e <'°«> ' 17. y = e ''"^-^ * .'. — = . ' ^ ' ' dx log. a; (log.)^a^ .... (log.)""' x 18. y = x'' .'.^= af'.af\l-\- log. x (1 + log. x)|. (14.) To differentiate circular functions. Let x represent the versed sine of an arc of a circle whose radius is r, then r — x will represent the cosine of the same arc, and, by trigonometry, tan. _ r sin r — x' In this expression, x is the independent variable, and as this dimin- ishes, the arc itself diminishes, both vanishing simultaneously, and the tan. . r ultimate ratio of — r- is - = 1 ; that is, the sine and tangent of an arc sm. r approximate to each other as the arc diminishes, and at length become equal. As the arc is between the sine and tangent when these be- come equal, the arc, also, must become equal to each ; therefore, we may conclude, that the ultimate ratios are as follows : tan. _ arc _ arc _ arc _ sin. _ tan. _ ^ sin. ' sin. ' tan. ' chord ' chord ' chord 1. Let it now be required to find the differential of sin. x. Chang- ing X into a; + fc, we have {Gregorxfs Trig. p. 48) sin. (x + ft.) = sin. a; + 2 sin. \ h cos. {x -\- \ A), sin. ix -\- h) — sin. x sin. \h , ■ , , v ••• — ^ — r — - = -it ■="" '^ + i *" sin. ^h , d sin. x whenx= 0, ■-- = 1, .•• ; = cos. x.-.dsm. x = cos.xdx. 1 Ai dx 2. To differentiate cos. x. d cos. X = d sin. (i * — x) = — cos. (| * — x) dx = — sin. x dx. * The differentiation of circular functions may be obtained independently of these results. See ths note (A) at the end of the volume. 20 THB DIFFEKENTIAL CALCULUS. Co7'. As d COS. = — d ver. sin. .•. d ver. sin. x = sin. xdx. 3. To differentiate tan. x. sin. .r COS. r (' : »n. '■■ — sin. x d cos. a; d tan. a? = d COS. a; COS. 'X that is cos. ^x -\- sin. -X , 1 , „ , d tan. X = :; dx = — dx = sec. ^x dx. COS. "X cos. X 4. To differentiate cot. x. dcot. X = dtan. (| * — a;) = — sec. ^(i ii' — x) da; = — cosec ^xdx. 5. To differentiate sec. x. , 7 1 sin. x , , a sec. x = d 5- = — dx = tan. x sec. x dx. cos. ■'a; COS. ^x 6. To differentiate cosec. x. - 1 COS. x , . , a cosec. x = d — = : — — dx = — cot. x cosec. x dx. sm. X sin. X Tliese six forma the student should endeavour to preserve in his me- mory. EXAMPLES. 1. y =^ sin. ^a: .*. dy = 2 sin. x d sin. x = 2 sin. x cos. a; dx =3 sin. 2a; dx, .'. ■:ir = sin. 2x. ax 2. t/ = sin. "x.'. dy = nsin. ""' xdsin. x = nsin. ""^xcos. xdx, • ' -J- = n sm. X cos. x. dx 3. 1/ = COS. mx .'. dy = — sin. mxdmx = — m sin. mxdx, dy .•.-r = — m sin. mx. dx ' 4. M = 2/ tan. of, y being a function of x, .'. du = tan. x" dy + y d tan. x", now d tan. x" — sec. "x" dx" = nx"~' sec. ^x" dx, dw diy .*. -7- = tan. x" T~ + wx"~^ sec.^ x". ax ax •' 5. « = cot. x" .*. du '= — cosec.^ x^ dx*'. Put z = x* .*. log. 2 = 1/ log. X, THE DIFFERENTIAL CALCULUS. 21 dz dx ,,77/ ^-"^ .: — = y — + log. xdij .•. as = dx^ = {]} — + 'Og- ^ dy)x^, Z X ^ du „ ,1/ , , dii. and — = — cosec.2 a;^ (^ + log. x -/) o^. rfx X dx' 6. y = xe "' * .'. dy = e "'■ "^ da; + xe "'• "^ ci cos. x = e '''"■ * (1 — X sin. x) dx, .'. -~ = e '='"" ' (1 — sin. a;), aa? J (a; e "^ ') , , , 7. y = log. (x e-^ ') .•.dy= ^^,,3.. > and d (xe- -) = gcoB. r ^j — ^ gjj^^ a:)(Za:, dy _ I — X sin. x ' ' dx X dy 8. y = COS. X + sin. x V — 1, .*. '7Z,— — sin.a;+cos.x\/ — 1 dy 9. w = cos. X -\- COS. 2x+ COS. 3x + &c. .•. -j^ = — (sin. x dx + 2 sin. 2a; + 3 sin. 3a; + &c.) 10. y = xe '^"^ ' .-.-p = \1 -\- X sec.^ xl e '="'• ". sin. ""a; dy . sin."""' a;, , ,sin.'"+' a; , 11. w = —-.'.-f- = m\ ~-—l + nl -rr-i. cos. a; dx cos. ' a;' 'cos."'*'' a; ' du lV , , dy^ 12- u = sec. 3fi .'.-r- = tan. 3^ sec. x^ a;^J-+log.a; —-J. rfa; 'a; ° dx^ (15.) In the preceding trigonometrical expressions, the arc is con- sidered as the independent variable, and the lines sine, cosine, &c. as functions of it ; we shall now consider the inverse functions as they are called, that is, those in which the arc is considered as a func- tion of the sine, the cosine, &c. A particular notation has been pro- posed for inverse functions : thus, if j/ = Fx be the direct function, then X = r~^ y is the inverse function, that is, if we represent the function that yisofxhyy = Fa-, the function that x is of y will be denoted by x = F~^ y. By thus representing these inverse functions, we may return immediately to the direct functions, considering, for the moment, F~' in the light of a negative power of F, or an equivalent to -^ ; for then x = F~' y immediately leads toy = Fx.* Thus, if * See note (B'). 22 THE DIFFERENTIAI. CALCULUS. X = log.z' y '"' y = log. X, the inverse function log."' y meaning the number whose log. is y. In like manner, y = sin.~' x means that y is the arc whose sine is x ; that is, returning to the direct function, sin. y = X. 1. To differentiate y = sin.""' x. Here the direct function is sin. y = x .♦. d sin. y = dx, that is, J J dy I 1 1 cos. t/rfw = dx .: -r- = = — — = — . dx cos.y VI — sin. =4/ ^ 1 — x" 2. To differentiate y = cos.~' x. cos.y = X .•. sin. ydy = dx .•. Ji =. — dx sm. y y/l—cos.^y 3. To differentiate y ==■ versin."' x. versin. y = x .'. sin. ydy = dx, " dx sin.y y/2x — x' 4. To differentiate y = tan."' x. tan. y =■ X .•. sec.^ Ww = dx .*. -^^ = r— = — ; -. ^ ^ ^ dx sec^y 1 + x' 5. To differentiate y = cot."' x. )t. y = a:.'. — cosec'^ydy = dx.*. 6. To differentiate y = sec."' x. , 1 , dy 1 —1 cot. V = x.-. — cosec.ydy = dx.'.-r- — s" = ; i* ^ ^ ^ £f.i! cosec. ""y 1 + x" sec. y = X, .'. tan. y sec. yc?i/ 1 = dx...^ = 1 dx tan. t/ sec. y X -v/x* — 1 7. To differentiate ?/ = cosec."' x. cosec. t/ = X .*. — cotan. 1/sec. i/(ij/ = dx .'.-j- — dx cot. J/ sec. y \. y =■ sin. X -v/x^ — 1 EXAMPLES. % _ 1 dmx VI _ ,^3;c3 * dx VI — mV THE DIFFERENTIAL CALCULUS. 23 fit I fi siTi — V^ 2. y - X sin.~' sp .'. -~ = sin. ' ar^ + x ,- and ax ax d sin. ' a^ dx dx VI— X* dy . , o , 2x2 3. y = COS."' X \/ 1 — x". Put X Vl — x^ = z — dz .'. dy = — — VI — 2^ X^ hutdz-{Vl—x' — —==)dx,iind Vl — z^ = Vl — x^ + x' V 1 — ar' ^ dy _ — 1 + 2x^ * * dx ^(1 _a;a_|-a;*) (1 —^' 4. y = tan.~' -. ^ 2 rfy _ 1 . dy _ 8 i^x ^ a^ ' ' dx 4 + x^ * 6. J/ = cot.-'(a + mx)=' .:dy = — ^_^^^_^^^^y rf (a + wx)^. J / I va n / t \ J ^y 2m (a + mx) d (a + mxY = 2 (a + wx) mdx .-. -r- = — ~ — ; 7. dx 14-(a + nix)* 6. y = sec."' — .: dy= — d — , and ^ X" ^ la x" <.^(^)" - 1 - a am dx x" x'^' * * dx ^^ _, VI + x^ , VI + r' . 7. »/ = cosec. ' . •. dy = — d — VI + x' 1 + r" 1 24 THE DIFFERENTIAL CALCULUS, — d , but a X ar X 1 ar v/ 1 + or dx, dy _ 1 ' ' dx 1 + x^' 8. y = (sin.-^ xY .'.-i^ = 2 sin.-' a? — r,. , ^ */ 1 9. y = cos.~ • ~ Vl + r' dx 1 + X- , _ . _i ,' • — X dy \ 1. y = tan. v/— -— -'' -r = — ^ ,T- 1 + x dx 2n/1 — x^ n.V = (cot.-'.)^-.| = -^oot.-.. 12. J/ = sec.-'x".'. — dx X\/ 3?" — 1 ,_ . ^ d\\ 13. « = cosec. mxr .', -t- =■ dx xVm^x^ — 1 • 1 dii ^ "^ dx \/2 — e^ (16.) In the preceding expressions the radius of the arc is always represented by unity, but, as the differentials are frequently required to radius r, we shall terminate this chapter with the several formulas in (15) accommodated to this radius. We must observe, that as y y and X are homogeneous in each of those forms, - is always a num- dv ber, so that this ratio in the limit, that is -7^, is a number. Hence, r dx must be introduced as a multiplier so as to render the numerator and denominator of each expression of the same dimensions. The for- mulas, therefore, become rdx d sin.~' X = ■ „ =• THE DIFFERENTIAL CALCULUS. 25 rdx d cos.~* X Vi'^ — ar' rdx d versin.~' x = =• V 2rx — x^ r'dx r^dx d tan.~^ X = d cot.~* X = T^ + a^ r'dx d see."' s x\/ x^ — r^ r^dx d cosec."' X = — xy/ar^ — r^ On successive Differentiation. (17.) Since the differential coefficient derived from any function of a variable may* contain that variable, this coefficient itself may be differentiated, and we thus derive a second differential coefficient. In like manner, by differentiating this second coefficient, if the variable still enters it, we obtain a third differeiitial coefficient, and in this way we may continue the successive differentiation till we arrive at a co- efficient without the variable, when the process must terminate. Thus, taking the function y = ax'*, we have, for the first differ- dy ential coefficient, -p = 4ax', as this coefficient contains x, we have, by differentiating it, the second differential coefficient = 12ax^ ; continuing the process, we have 24ax for the third differential coeffi- cient, and 24a for the fourth, which being constant its differential coefficient is 0. If we were to express these several coefficients agreeably to the notation hitherto adopted, they would be first diff. coef. -~ = 4cw^ dx d^ dx second diff. coeC — -^ = 12ax^, * It must contain the variable, unless in the single case of its being constant. Ed. 4 26 THE DIPFEllENTIAL CALCULUS. d d^ dx dx third diff. coef. -i = 24 ax, &c. But this mode of expressing the successive coefficients is obviously very inconvenient, and they are accordingly written in the following naore commodious manner : first difi*. coef. second diff. coef. third diff. coef. nth diff. coef. dx dx'' dx"* d"y in which notation it is to be observed, that d^, d^, &c. are not powers but symbols, standing in place of the words second differential, third differential, &c. The expressgions dx'^, daP, &c. are on the contrary powers, not, however, of x, but of rfx : to distinguish the differential of a power from the power of differential, a dct is placed in the former case between d and the power. (18.) The following are a few illustrations of the process of suc- cessive differentiation : I. y = if". dx d3» dPy ~d^ d'y = mx*-', = OT (m — 1) x'^S ^ = m [m — 1) (tn — 2) x~~', = m (m — 1) (wi — 2) (m — 3) x**-% rfx« &c. &c. 2. tt = yz, both y and z being functions of x, THE DIFFERENTIAL CALCULUS. 27 du dx dr-u 1^ d\ dz dy dx dr-z dx" d^z = y dx" d'^z dx" ' dx^ &c. 3. y = log. X dy dx = 1 x d^y dx" = 1 d?y dx" = 2 4. y = (f. dy dx = e' d-y = e' dx dydz dx" dydz ~d^ dyd^z zi- y dx" + 3 dx" dr" dzdPy ~d^ &c. ,dzdy_ dx^ + dx" &c. d'^y _ 2-3 dx"^ X* dhj _ 2 . 3 • 4 doc" XT' dHj _ _ 2 • 3 • 4 5^ ^ dx" d'y - ^ dx' &c. If instead of c the base were a, the several coefficients would be log. a ' a", log.^a • a% log.^a . a% log.^a • a', &c. It appears, therefore, that exponental functions possess this property, d"y VIZ. that -r^ -^ y ia always constant. 5. y = sin. x. '^y _ dx di? cos. X = — sm. X dx' dx* cos. X sm. X &c. We need not multiply examples here, as the process of successive differentiation will be very frequently employed in the next two chapters. 28 THE DIFFERENTIAL CALCULUS. CBAPTSIS IIX. ON MACLAURIN'S THEOREM. (19.) Ify represent a function of x, which it is possible to develop in a series of positive ascending powers of that variable, then will that development be where the brackets are intended to intimate that the functions which they enclose are to be taken in that particular state, arising from taking x = 0.* For, since by hypothesis y = A + Bx-{- Car'+ Da;^ + Ex* + &c...(l> ... J^= B + 2Cx + 3Dx2+ 4-EaP+Sic. ax ■j^ = 2C + 2 • ZBx + 3 . 4Er» + &c. erar 3D +2.3. 4Ex + &c. ^ _ dsP ~ &c. &c. Let, now, a: = 0, then dy rg-]=2.3D...D=-i-rg-] Ldx^-" 2-3 LrfariJ &c. &c. * This plan of enclosing the differential coefficient in brackets we shall usually adopt, when wc wish to express not the general state of this function, but that state which arises from the variable taking a particular value. What that value is will generally be made known by the nature of the inquiry. THE DIFFERENTIAL CALCULUS. 29 Hence, by substitution, equation (1) becomes » = W + [|]^+i[^F+^3[g-]-'+&c (2), which is Maclaurin's theorem for the development of a function, according to the ascending powers of the variable. We shall apply it to some examples. EXAMPLES. (20.) 1. Let it be required to develop (a + x)", the exponent n being any uumber whatever, either positive or negative, whole or fractional, rational or irrational. Put y = {a + x)" . . therefore . [j/] = a" ... A = n(a + x)"-' r-^] = na'^' dx ^ dx ^=.n{n-\){a+xr-' . . . [^-] = n(n- !)«-' g- = n(n-l)(n-2)(a + x)- [g.] = n(n— 1) {n — 2)dr^ &c. &c. Substituting these values for the coefficients in the foregoing theo- rem, there results (a + xY = a" + »a"-' x + — ^ a x^ + — ^^ — '- 2i 2 ' 3 a"-3 x^" + &c. and thus the truth of the Binomial Theorem is established in its utmost generahty. 2. To develop log. {a + x). Put y = log. (o -|- x), therefore [t/] = log. a d^i/ _ 1 • ^dar" ^ ^ rf'y _ 2.3 • trf^] "^ &c. dx ^y _ 1 ^y _ (a + xf 2 (a + xr 2-3 dx' &c. (a + X)* 30 THE DIFFERENTIAL CALCULUS. ... lo.. ia + .) = log. a+^_^+^_^+&c. dv If 7/ = log. X were proposed, then, since [y], [^r;]* &c. are infinite, we infer, for reasons similar to those assigned at art. 4, that the de- velopment in the proposed form is impossible. 3. To develop sin. x. y = sin. X .... [?/] = i = COS. X ....[-^1=1 dx dx dhi . dii^ ^ ^ = -co..x . . . .[^]=-l ^ = sin.. . . . . [^J=0 dx* Hx* -■ dhi ^dhj &c. &c. . •. sin. z = X -\ &c. 1-2-31-2.3-4-5 4. To develop cos. x. y = cos. X .... [?/] = 1 %=_.„.. . . . .[^] = o ^=— • • •[^]=-' &c. &c. x^ x^ • •. cos. X = 1 ^ + — - — : &c. 1-2 1-2-3-4 6. To develop a'. y = a'' therefore [y] = 1 ^ = *-- • • •[f] = - * A is put here for the hyperboHc logarithm of the base a, that is, for the ex- pression (a — 1) — i (a — 1)2 + J (« — 1)=' — ^- THE DIFFERENTIAL CALCULUS. 31 ^-Aa .... L^^J — -Aa . . . . Lrf^J A &c. &c. , . , AV , AV , ^ .•.a^ = l+A. + ^^ + P^3 + &c. which is the Exponential Theorem. Since A = log. a, we may give to the development the form a' = 1 + X log. a + -{x log. af + r-^ (x log. a)=^ + &c. For a: = 1, we have the following expression for any number, a, in terms of its Napierian logarithm : 1 1 a= 1 + log. a + 2^°S-^" + 2 — 3^^^'^"' "^ ^^' changing a into the Napierian base, e, we have a^ aP e- = i+a;+— + ^— ^ + &c. which, when x = 1, gives, for the base e, the value c = 1 + 1 + 2 + 2T3 + ^*^- (21.) From the development of e'^ may be immediately derived several very curious and useful analytical formulas, and we shall avail ourselves of this opportunity to present the principal ones to the no- tice of the student. If, in the development of e', we put zV — 1 for x, we shall have _ z^ zW — I z" «'^-' = 1 + ^^-1 -r^-1 . 2. 3 +1.2. 3-4 + ^^- and, changing the sign of the radical, z" z" V — I « i z y/ 1^ 1 • 2^1 • 2 • 3^1-2-3-4 *^^* If these expressions be first added and then subtracted, there will re- sult the following remarkable developments, viz. 32 THE DlFFliHENTIAL CALCULUS. W-i 4. gW-i ^ = 1 — 1 ^ + ^i — 7. ^— T — &C. 2 l-2'l-2-3-4 + , ^ o . — £ — &C. 2^'^[ l-2-3'l-2-3-4-5 Now it has been seen (examples 4 and 3) that these two series are also the respective developments of cos. z and sin. z ; hence, putting X instead of «, we may conclude that gS-V — 1 g-»V — 1 sin. X = := .... (1) 2^/ — 1 cos. X = z .... (2) where the sine and cosine of a real arc are expressed by imaginary exponentials. These expressions were first deduced by Euler, and are consider- ed by Lagrange as among the finest analytical discoveries of the age. (Calcul des Fonctions, page 114.) (22.) If for the real arc x we substitute the imaginary arc x\/ — 1 we shall have e~^ — e* sin. (a:v/-l) = ^-^==....(3) c-^ + e^ COS. {x V — 1) = ^ .... (4) Sin. Also, since — '- = tan., it follows, from (1) and (2), that COS. C^s/ — ^ g— jV — 1 g^i^^Ti J y/ — 1 tan. X = — = = = =r * gTV-i -f e-^^/-' e^-^-^ + 1 By multiplying equation (1) by ± V — 1, and adding the result to (2), we have COS. X ± sin. X V — 1 = c±'^-' .... (6;) or if we change x into mx, cos. mx ± sin. mx \/ — 1 = c±"«^'-' .... (6), but e±""^~' is c^''-^'"' raised to the mth power. Hence this singu- lar property, viz. * Multiplying the numerator and denominator of the second member of the equation by e*^—'. Ed. THE DIFFERENTIAL CALCULUS. 89' (cos. X ± sin. X ■«/ — 1)"' = COS. mx ± sin. mx V — 1 .... (7), which was discovered by de Moivre, and is hence called De JVLoivre'a formula. If the first side of this equation be developed by the binomial theo- rem, it becomes , m(m — 1) , „ „ cos, '"x ± m cos. "*-* xp -\ ^^ cos. ""-^ xp^ it &c. p being put for the imaginary v/ — 1 sin. .r. Now in any equation, the imaginaries on one side are equal to those on the other, {Algebra) ; hence, expunging from this expression all the imaginaries, that is, all the terms containing the odd powers of p, we have, in virtue of (7), m (in — 1) cos. mx = COS. "x — r COS. "'"^ X sin.^ar + m{m — 1) (m — 2) {in — 3 n — o — A cos. "^^ X sin. *x — &c. 2 • 3 • 4 In like manner, equating sin. mx y/ — 1 with the imaginary part of the above development, and then dividing by ■«/ — 1 , we have m{m — l)(m — 2) sm. mx=m cos.*"" ' a: sin. x — oT^ cos.""" ^xsm.^x-\- &c. From these two, series the sine and cosine of a multiple arc may be de- termined from the sine and cosine of the arc itself (2ft) If in the formula (2) we represent e''^~^ by y, then c~'^~' = - ; therefore, y 1 2 cos. a: = V + ~ a y or, if in the same formula mx be put for x, we have 1 2 COS. mx = V" H — — 3 ym and from these two equations we deduce the following, viz. y^ — 2y COS. a: + 1 = .... (1) «/** — 2r/'" COS. mx + 1 = . . . . (2). Since these equations exist simultaneously, the latter must have two of its roots or values of y equal to the two roofs of the former, 34 THE DIFFERENTIAL CALCULUS. and must, therefore, be divisible by it ; or, putting 6 for mx, we have y^n _ 2y^ioS. d + 1 = . . . . (3), divisible by (4). ys — 2y cos. — + 1 But cos. 6 = COS. {& + Snir), n being any whole number, and if = 180° ; hence, making successively » = 0, = 1, = 2, &c, to n = m — 1, we have, since the first equation continues to be divisible by the second in these cases, y2m — 2?/'" cos.^ + 1 = (j/^ — 2j/ cos. + 1) 6 + 2* X(y' — 2J/C0S.— ^^+ I) 6 + 4* X {y'—2ticos.—^^+ 1) ^ + 6* X (tf — 2ycos. \r 1) &c. to m factors. The truth of this equation is obvious, for, while the substitution of 6 + 2*111' for 6 causes no alteration in the expression (3), the same substitution in (4) gives to that expression a new value, for every va- lue of n, from 7» = to n = m — 1, for the arcs — , &c. are m m all different. As, therefore, the expression (3) is divisible by (4) under all these m cheuiges of value, it is plain that these are its in quadratic factors. In this way may any trinomial of the form i/^ — 2ky'" + 1 be de- composed into its quadratic factors, provided k does not exceed unity, for then k may always be replaced by the cosine of an arc. (24.) The geometrical interpretation of the foregoing equation, presents a curious property of the circle, first discovered by De ■B 'B Moivre. To exhibit this property, let P be any point either within or without the circle whose centre is 0, and let the circumference be divided into any number of equal parts, commencing at any point A, Join the points of division, A, B, C, &c. to P, then, since in the fore- THE DIFFERENTIAL CALCULUS. 35 going analytical expression the radius OA is ex- pressed by unity, we shall have, by introducing the radius itself so as to render the terms homo- geneous, the following geometrical values of the above factors, where it is to be observed that Z POA = — and OP = y, m yim _ 2ym cos. 6 -f 1 = OF^ — 20?"" X OA" COS. m ( AOP) + AO^" t/« — 2y COS. f- 1 = 0P2 — 20P X OA cos. AOP + AO^ = PA»* 111 y^ — 2y cos. ?-±-!l -|- i = 0P« — 20P X OA cos. BOP -|- B0« = PB« m y« — 2« cos. ^-iil + 1 = 0P« — 20P X OA cos. COP + C0» = PC« in &c. &c &c Hence, OP*"* — SOP"* X 0A"» cos. m (AOP) -\- OA^-" = PA« X PB« X PC* X &.c. and this is Demoivre's property of the circle. (25.) If AOP = 0, that is, if P be upon the radius through one of the points of division A, then cos. m (AOP) = 1. Hence, OP** — 20P'" X OA"" + OA'"* = PA^ X PB" x PC^ x &c. consequently, extracting the square root of each member, OP"* ^ OA'" = PA x PB X PC x &c. If the arcs AB, BC, &c. be bisected by A', B', &c. the circumfer- ence will be divided into 2m equal parts, and, by the equation just deduced 0P=^ ^ OA^™ = PA X PA' X PB X PB' x &c. that is 0?=^"* ^ 0A=^ = (OP"* ^ OA-") PA' X PB' X &c. therefore, Qpan OA'^ __L — __- = OP"* + OA"* = PA' X PB' X PC X &c. and these are Coles's properties of the circle. (26.) If now we return to the expression (6), and suppose x = — , it becomes ♦ Gregory's Trigonometry, p. 54, or LacroLx'a Trigonometry. » V) 36 THE DIFFERENTIAL CALCULUS. 7 _1 =ca>/-i...log, n/ — 1 = ~^-^\, and From the second of these equations we get It = 2 1ogV — 1 _ log.(v/ — 1)^ ^ log.— 1 ^ _ ^^TT ^/ — l V — 1 V — 1 log. — 1. From the third , , , — * 1 flr^ I Mr' J 38 THE DIFFERENTIAL CALCULUS. dx" (1 + a:^)^ (1 + x^Y (1 + x")' ' ' Mx* ^ &c. &c. .'. y = tan. y — i tan. ^y + | tan. ^y — \ tan. 'y + &c. If 1/ = 45°, then tan. y — 1 ; .-. arc 45° = 1 — i + 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 for- mula, {Gregory's Trig., page 46,) , , , tan. a -f tan. 6* tan. (a + b)= =• 1 — tan. a tan. b for, when a + 6 = 45°, tan. (a -f- 6) = 1 ; therefore, tan. a + tan. 6 = 1 — tan. a tan. 6. If either tan. a or tan. 6 were given, the other would be determinable from this equation. Thus, if we suppose, ^ *u 1 j_ * I. -i ^^^' ^ . I « — 1 tan. a = -, then - + tan. 6=1 , .♦. tan. 6 = . n n n » + 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, 6, in terms of their tangents, may be the most convergent. This value appears to be n = 2, or n = 3 ; therefore, taking w = 2, we have tan. a = 1, tan. 6 = 1. Hence, substituting in the general development a for y and \ for tan. y, and then again 6 for y and \ for tan. 6, the sum of the resulting series will express the length of the arc a + 6 = 45°, that is arc. 45° = ^-3-^5 + 5^ " 7"^ + &C- + 1 L_+_J ^+&c 2 3 . 3^ ^ 5 . 3* 7.3^^ (28.) Another form of development, still more convergent than thisi has been obtained by M. Berirand from the formula 2 tan. a tan. 2a = — 1 — tan. "a * Lacroix Trigonometry. THE DIFFERENTIAL CALCULUS. 89 For put tan. a = \, then tan. 2a = yS_, therefore 2a Z 45°, because tan. 45° = 1 : from this value of 2a we deduce 2 tan. 2a 120 tan. 4a = — = 1— tan.='2a 119 .-. 4a 7 45°. Let now 4a = A, 45° =B, A — B = b= excess of 4a above 45°, then we have 45° = A — 6. But . UN ^ , tan. A — tan. B 1 tan. (A — B) = tan. 6 = — ; =r = — - ^ ^ 1 + tan. A tan. B 239 Consequently, if in the general development we replace yhy a and tan. y by i, and then multiply by 4, we shall have the length of the arc 4a, and, since this arc exceeds 45° by the arc 6, if we subtract the development of this latter, which is given by substituting ^i^ for tan. y, the remainder will be the true development of 45°. Thus 45° =4(i — + —- —+ kc.) ^2 3 • 6=* 5 • 5^ 7 • 5^ ^ ^ ^(— 1 1 1 &c.) ^239 3 • 239=* ^ 6 • 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 semi- circle, the following value, viz. If = 3- 141592653589793. If we take but three terms of the first and only one of the second, we shall have -r = 3 • 1416, the approximation usually employed in practice. (29.) The following examples are subjoined for the exercise of the student : 8. To develop y = sin.~^ x. , sin.^ y , 3^ sin.' y , 3^ • 5^ sin.' y y = sm. y-{-~—^- + ^ ^ ., , ^ + ^ 1-2-3 1-2-3-4.5 1-2-3-4-5-6-7 9. To develop y = cos. ~' x. .. — i cos.^ y 3^ cos.^ y , „ y = i -r — COS. « ^ 2 u &c. " ^ 1-2-3 l-2-3-4-5^ 10. To develop y = cot. x by the method of indeterminate coef- ficients, as in example 6. 40 cot. X = THE DIFFERENTIAL CALCULUS. 1 X x^ It" X 3 3^-5 3^ • 5 • 7 11. To develop j/ = (a + 6t + c^r^ -f &c.)"' {a-\-hx ■\- ex" + &c.)" = n(n— 1) (?i — 2) &C. , , , n (n — ] ) , „ -\- na"-^ c x^-\- an-3 1,3 2-3 4-n (n— 1) "-2 6c -|- no"— 1 erf a;'-J-&c. This is the multinomial theorem of De Moivre. It is given in a very convenient practical form in my Treatise on Algebra. OHAFTEH IT. ON TAYLOR'S THEOREM, AND ON THE DIFFER- ENTIATION AND DEVELOPMENT OF IMPLICIT FUNCTIONS. (30.) In the second chapter we established the form of the gene- ral development of the function F (;r + h)' We here propose to investigate Taylor's theorem, which is an expression exhibiting the actual development of the same function. The following lemma, must, however, be premised, viz. that if in any function of p + q one of the quantities p, q, is variable, and the other constant, we may de- termine the several differential coefficients, without inquiring which is the constant and which the variable, for these coefficients will be the same, whichever be variable. This principle is almost axiomatic. For as the function contains but one variable we may put p + g = x or F (p + q) = Fx, and whichever of the parts p, q, takes the in- crement h, the result ¥ (^x -\- h) is necessarily the same ; hence the development of this function is the same on either hypothesis, and therefore the second term of that development, and hence also the differential coefficient. The first differential coefficient being the same, the succeeding must be the same ; therefore generally d"F (p + q) rf-F ip + q) dp" whatever be the value of n. dq" THE DIFFERENTIAL CALCULUS. 41 Let now y = Fx, and Y = F (x + h), and assume, agreeably to art. (4) Y = y + A/i + B/t^ + C/i^ + &c. A, B, C, &c. being unknown functions of x, which it is now required to determine. Suppose, first, h to be variable and x constant, then, differentiating on that supposition, we have ^Y ^ = A + 2B/i + ZCh^ + &c. an Suppose, secondly, that x is variable and h constant, then the dif- ferential coefficient is dY dy , dA , ,dB .,dC ,^, ^ dz dx dx dx dx But by the lemma these two differential coefficients are identical, hence equating the coefficients of the like powers of A. there results that is . _ dy _ dA_ „ _ / x^ — '2ax) ... i!^! = ' 6. LetM = (cos. x) •'"■' cdu^ , V , , , sin. 'x •*» i-T-i = (cos. x) "^ • (cos. X log. COS. X ;. Implicit Functions. (39.) Hitherto we have considered 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 x, and function y, is implied in an equation between the two., and which may be generally expressed by « = F (x, y) = 0. The deductions in article (36) will enable us very readily to find the coefficient -p 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 sub- stitute y for q, we find .du. _^du du dy *dx' dx dy ' dx But here « = F (x, y) = 0, therefore \—\ = 0, for u' — u being always 0, — - — is always ; hence, du du dy dx dy dx from which equation the differential coefficient is immediately deter- minable : it is dy du . du dx dx ' dy'' 7 60 THE DIFFEREJITIAIi CALCtLUS, hence, having transposed the terms all to one side of the equation, we mxist differentiate the expression as if y were a constant, and then divide the restdting coefficient, taken with a contrary sign, by that derived from the same expression, on the supposition that x is a con~ s'anl. EXAMPLES. 1 . Let u = y^ — 2mxy -{- x'' — a = 0. da du „ dy my — x — 2my — 2x,— = 2y — 2mx • -^ — ^ dx dy dx y — mx 2. Let M = a:' + daxy + 7/^ = 0. dx dy dx ax-\- y^ If the second differential coefficient be required, we have dy dy ^,y ^ {ax + f) {2x + « ^) + {x' + ay) (a + 2y -£) dx^ {ax + t/^)2 or substituting for -~ its value just found _ 2aT/* + 6aaPy^ + 2'x'^y — 2a^xy ~{^TW _ 2xy {y^ + 3axy -\- jP) — 2a^xy _____ , that is, since aP + Zaxy -\- tf = 0, d^y __ 2a'^ xy dx^ {ax + y^Y 3. Let my^ — xy = m to develop y, according to the ascending powers of X, du du ^ „ dy y ;- = 7/, 1- = Zinf — X .'. -^ = - — ^ dx dy dx Zmy^ — x therefore, calling the successive differential coefficients p, q, r, &c. y — Smx^p — xp ^ {3my^^ xf ' THE DIFFERENTIAL CALCULUS. 5l _ 3nM/ a' ; 2°. a = a' ; 3°. a < a'. In the first case by dividing the two terms of the fraction by h'^', and then supposing h = 0, there results Fa ^ j^=v = '- In the second case the result of the same process is Fa _ A jr~ A'- In the third case, by dividing the two terms of the fraction by /t«» and then supposing h = 0, the result is Fa A >r = ^ = "- It appears from these results that the development of the numera- tor and denominator need not be carried beyond the first term, or that involvuig the lowest exponent of h,* and according as the exponent in the numerator is greater than, equal to, or less than that in the de- nominator, will the true value of the fraction be 0, finite, or infinite. We have, therefore, the following rule : Substitute a + /i for x, in the proposed fraction. Find the term containing the lowest exponent of ^, in the development of the nu- merator, and that containing the lowest exponent of h in the develop- ment 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 in- finite. But if these exponents are equal, divide the coefllicient of the term in the numerator by the coefiicient 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. (a:3 _ 3«.r -f 2a2)^ 9. Required the value of , — when x = a. * The first term wliich actually appears in the development is of course meant here. Those which may vanish in consequence of the coefficient vanishing not being considered. 8 08 THE DIFFERENTIAL CALCUIiUS, Substituting a + h for x, we have F (o 4- h) _ hi {h — ay* _ (— ah)^ + &c. / (a + h) ^^ ^3^3 _^ 3^^ ^ j^^^l (3a'hy + &c. Since the exponent of h in the numerator exceeds that in the de- nominator, we have Fa /« 10. Required the value of ~ — — — when x Vx'—a^ = a (see p. 53.) Substituting a + ^ for x. F{a + h) • (a-^h)^ — a^ -{-h^ h^ 4- &c. /(« + /») ~ h^ (2a + h)^ Fa 1 (2a/i)^ + &c. 11. Required the value of—- — , -^ when x = a. ^ (1 4- X — ay — 1 Substituting a + h for t. F (a 4- ^) h^ {2a + h)'^' -\- h h -{- Sac. f{a-\-h) (1 + A)^— 1 3/1 +&C. Fa _ ••> - ^• to i> • ^*K 1 .a{4a'+4x^)^ — ax — a' , 12. Required the value of when r (2o=» + 2x^)7 _ a — X = a. Substituting a + ii for x. F{a+h) a (8a' + l^a^'h + 12a/i2 + Ah'y — 2a' ~ afe /(o + /») (40== + 40/1 + 2^=")^ — 2a — /i ♦ To develop this according to the ascending powers of h we must write it thus: ( — a -{• A)* and apply the binomial theorem when we have the seriea ( _ fl)l 4- * a—i h 4- &c. THE DIFFERENTIAL CALCULUS. 59 which, by actiKilly extracting the roots indicated, a (2a 4- A + 2a + &c. — 2a- -/,,) 2a + h-{- h? 4a + &c. — 2a — h a(-+&c.) Fa _ 1 T 4^ *9a This example is perhaps more easily performed by diflerentiation, according to the first rule : thus F'x __ g (4a^ + 4^') "^ 4^" — a F'a ~fx ~ {2d' + 2x') -^ 2x — T • ''fa = T'x _ — a (4a^ + Ax") " ^ 32 j' + a (4a^ + 4x"')~3 Sx J"' _ (20" + 2r') "^ 4ar» + 2 (20=* + 2x^)~^ F"a _ 2 _ •■• 7^ - 1 - 2a. a (43.) Having thus seen how to determine the value of any fraction of which the numerator and denominator become each for particu- lar values of the variable, we readily perceive how the value may be found when particular substitutions make the numerator and denomi- Fa a, aator each infinite. For if —^ = ^ then obviously 1 Fa _ >^ _ ^ Fa So that if we find, by the preceding methods, the value of this last fraction, the value of the proposed fraction will be also obtained. The following example will illustrate this. 1 X tan. (- * . -) ^2 a' 1 3. Required the value of , _, .^ ^rrr when .r = a. <»<> THE mPFEUENTlAL CALCUHTS. In this case the. fraction takes the form ^ , therefore, CO 1 fi^ _ a (.r= — a") x~- F'x _ 2a^ .r" 1 : Fa: cot.<^^.l) ^'^ _ cosec^^ * • ^) ^ Fa _ ^ _ 4a /'a -rr ~ * ~ 2^ (44.) By the same principles we may also find the true value of a product consisting of two factors, which for a particular value of the variable becomes the one and the other cc. For if Fa = and /a = CO , then, „ .. Fa Fa X fa = = - -^ 1 7^ We shall give an example of this. 14. Required the value of the product (1 — x) tan. (| -jex) when a: = 1, In this case the first factor becomes and the second co . - .-. F'x X f'x = 1 1 cot. (^ * 0.-) - 2 / , \* Jx tan. (i ifx) ^2 ' ^ .-. F'a X fa=-^ = -. (45.) And finally, by the same principles, the true value of the dif- ference of two functions may be ascertained in the case where the substitution of a particular value for the variable causes each of them to become infinite. For if Fa — ao and /a = co , then 1 1 'fa~Fa_0 Fa -fa 1 Fa X fa The following example belongs to this case : THE mPFERENTIAI. CAIXTJIiUS. 61 15. Required the true valr.e of the diflference x tan. x — \'k sec. X, when x = 90^. 1 1 1 1 fx Yx ^ ir sec. x x tan. x 1 Fx X fx X tan. x X i cr sec. x by substituting for sec. x, and then dividing numerator and denominator by - ^ nc X tan. X T^, /., -T cos. X + sin. X .-. F X — fx — : .'. Fa — /a — — 1. '' — sm. X -" It should be remarked that in this, as well indeed as in the prece- ding cases, the transformation requisite to reduce the expression to the form f in many instances at once presents itself to the mind, when of course it will be necessary to recur to the preceding formu- las. The example just given is one of these instances, for since sin. X , 1 , , tan. X = , and sec. x = , the proposed expression at cos. X cos. X once reduces to X sin. X — 1 * COS. X which ia the required form. (46.) We shall terminate this chapter with a few miscellaneous examples for the exercise of the student. x" 1 16. Required the value of , when x = 1. X — 1 ^ns. n. fij^ -4— CLC^ — ^— ^OiCor 17. Required the value of 7—5 —. ; — r-^, when x = c. bxr — 2bcx + or Ans. r- 3r ^^— CLOl ^~" (VX "i" Cb 18. Required the value of , when x = a. or — a Ans. 0. * This is obtained by multiplying the last fraction above and below by a; tan. , . . sin. X . , 1 - X X Jt sec. X, then writing for tan. r, and for sec. x, Ed. 62 THE DIFFERENTIAL CALCULUS. 19. Required the value of : ^ -, when a — {ax^) * 16a X — a. Ans. — — < 9 1 X ~j~ loff. X 20. Required the value of — , when x = \. 1 —{2x — xY^ Ana, — 1. x' X 2 1 . Required the value of -— ; , when x = 1 . 1 — X -v log. X Ana. — 2. _ . , , , -tan. T — sin. T , 22. Required the value of ; , when x = 0. sin. 3r Ans.{. [ X = \ Ana. f when X = a. (x — a)t Ana. 2a^. X 1 26. Required the value of -; , when x = 1. ^ X — 1 log. X Ana. |. ax — XT , 26. Required the value °f ^4 _ ga^x + 2ax- - x^ ' ^^"" ' = "' Ana. CO . 1 X 27. Required the value of ^ , when x = 1. ^ log. X log. X Ana. — 1. 28. Required the value of — — — . tan. — , when x = a. 4 Am. . X IO£f« X — ix ~~" 1 J 23. Required the value of — -^ — -r-r ^» when x = 1. (x — 1) log. X 24. Required the value of ^ '—, when x — a „ a ( 1 — x) , , 29, Required the value of — V-r — ^^ ^'^®" x = 1. ^ cot. I /i^ (B + Ch^~^ 4- D//"^ +&C.) Putting S for the sum of the series within the parentheses, it is B-a obvious that h may be taken so small that S/i may be less than any proposed quantity A, and that therefore if h' be such a value we must have Ah'"- > S/i'^ B — a . which establishes the proposition. As S/i 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 inquire by what means we may determine those values of x which render any proposed function Fa; a maximum or a minimum. In order to do this, let x be changed into x ±: h, then by Taylor's theorem „ , , ^ ^ dy , , drri h^ (Py h^ F(,±.) = F.±^A+^— ±^^-^3 + (te* 1 • 2 • 3 • 4 * Now if 0? = a render the proposed function a maximum, then there exists for h some finite value h', such that for all the intermediate values between this and we have Fa > F (« ± /t), and, consequently, But if this value render the function a minimum, then, for all the in- termediate values of /i between h = h' and h = 0, we have Fa < F {a ± k) and, consequently. THE DIFFERENTIAL CALCULUS. 65 It has, however, been proved above, that a value may be given to h 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 k between this small value and 0, so that, for each of these values of ft, 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 + [-^] ft and— [^] ft, unless [-^] = ; we con- clude, therefore, that those values of x only can render the function a maximum or minimum which fulfil the condition ax expunging, therefore, the first term from each of the series, (1), (2), we have, in the case of a maximum, ilie condition AiJL. ± [%-!^ + &c zo . . . (3).* and in the case of a minimum, • r^J!L. + [^]-^^ + &c. 7 . . . (4). Now the former of these conditions cannot exist for any of the values of ft between ft = ft' and ft = 0, by virtue of the foregoing principle, unless [ j3-] 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 dti condition ;p = 0, those among them that also satisfy the condition — ^ Z belong to maximum values of the function, while those ful- dxr dj^y filling the condition -j^ y belong to mmimum values of the func- tion. It is possible, however, that some of the values derived from the equation -t" ~ ^ "^^.y, when substituted for x in -j-^, cause this * See Note (C). 9 66 THE DIFFERENTIAL CALCULUS. coefficient to vanish, in which case the conditions (1), (2), become and which are both impossible unless [ j-^] = 0, for reasons similar to those assigned above, and, unless, also [-7-^] / in the case of a maximum, and f^-^l 7 w the case of a minimum ; that is, on the ax* supposition that this coefficient does not vanish from the series (5) and (6). If, however, this coefficient does vanish, then, for reasons similar to those assigned in the preceding cases, the following coeffi- *^'6nt j-j must also vanish, and the condition of maximum will then d'y d?y ^^ L T~r] Z 0» and the condition of minimum [jtt] 7 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 = Fx, we must proceed as follows : dy Determine the real roots of the equation -p = 0, and substitute them one by one in the following coefficients -7^, -A^, &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 dy of the equation -1- = cause the first of the following coefficients, which does not vanish, to become infinite, we cannot apply to such roots the foregoing tests for distinguishing the maxima from the THE DIFFERENTIAL CALCULUS. 67 minima, because the true development of the function for any such value of a: 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 /», which ought to enter this is odd or even. In a case of this kind, therefore, we must find, by actual involu- tion, extraction, &c. the true term that ought to supply the place of that rendered infinite in Taylor's series for x = a. If this term take an odd power of A, or, rather, if its sign change with the sign of A, then X = a does not render the function either a maximum or a mini- mum ; but if the sign does not change with that of /», then the value of a: renders the function a maximum or a minimum, according 6is the sign of this term is negative or positive. To illustrate this case, suppose the function were y = b -{- {x — a)* ..-^-3(x-«)3 dhj _ 10 _ I dy Now the equation T" = gives x = a, so that if any value of x could render the proposed function a maximum or a minimum, this most likely would be it. By substituting this value of x in ■j-j the result is infinite, and we cannot infer the state of the function from this coefficient ; therefore, substituting a ± /i for x in the pro- posed, we have F (a it ;i) = 6 ± A* and, as h^ obviously changes its sign when h does, we conclude that the function proposed admits of neither a maximum nor a minimum value. Again, let y — h -{■ {x — ay dy _ 4 I 68 THB DIFFERENTIAL CALCULUS. _, . dy , , . , <^y ^ The equation -^ = gives x = a, a. value which causes -r-j- to become infinite ; therefore, substituting a ± /i for a: in the proposed, we have F (o ± /i) = 6 = /i3 . i . . . and, as the sign of /i=* is positive whatever be the sign of A,, we con- clude 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 deyelopment is possible, as far at least as the first power of /(, but we cannot say that among those values of x, which would render the coefficient of this first power iiifinite, there may not be some which cause the function to fulfil the conditions of maxima or minima ; therefore, before we can conclude dy in any case that the values of x, deduced from the condition j^ — 0, comprise among them all those which can render the function a maximum or minimum, we must examine those values of ^arising from dy the condition -y- = co by substituting each of these ± hfor x in the proposed equation, and observing which of the results agree with the conditions of maxima and minima in (47). (52.) If the function that y is of x be implicitly given, that is, if u — ¥ {x,y) = 0; then, by (39), we have, for the differential coefficient, dy du . du dx dx dy ' ' ' ^ ^* dy du and therefore, when -7- = 0, we must have j~ =^ 5 hence, the values corresponding to maxima and minima, are determinable from the two equations* dy . du * Other values may be implied in the condition - = co , which leads to — = 0, but to ascertain wliich of these are applicable would require us to solve the equation for y. THE DIFFERENTIAL CALCULUS. 69 ^ = o\ ' ' • (2)- dx * Having found from these values of a: that may render y a maximum or a minimum,* as also the corresponding values oft/ itself, we must dry substitute them for x and y in j-j, when those values of y will be maxima that render this coefficient negative, and Ihose will be mini- ma that render it positive. But those values that cause it to vanish, belong neither to maxima nor to minima, unless the same values , d'y Qause also -7^- to vanish, and so on. dx-^ The second differential coefficient may be readily derived from (1), for, putting for brevity we have ^ 4- ^ ^^ — M (— 4- ^ ^^ d'y ^ dx dy ' dx dx dy ' dx M which, because =^ = 0, becomes for the particular values of x re- sulting from this condition, d'y _ d'u ^ du '-rf^-l"-L^J • Lrf^J • • • ^^■>' d^y By differentiating the above expression for -7-j we shall find and so on. (63.) Before we proceed to apply the foregoing theory to exam- ples, we shall state a few particulars that may, in many instances, be serviceable in abridging the process of finduag maxima and minima. * Gamier, at p. 271 of his CalcuL Differential, says, that, by means of the equations (2) " on obtient les valeurs de x et 1/ par lesquelles F (x, y) devient ou peut devenir maximum ou minimum ;" but this is evidently a mistake, since, by hypothesia, F (x, y) is always = 0. 70 THE DIFFERENTIAL CALCULUS. 1. if the proposed function appears with a constant factor, such factor may be omitted. Thus, calling the function Ay, the first dif- ferential coeflicientwill be A -^, and A -^ = leads to -^=0,also ax ax dx — — = leads to ^ = 0, so that A may be expunged from the dx dx function. 2. Whatever value of x renders a function a maximum or mini- mum, 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 ; more- over, if the rational function be = 0, when a maximum, the corres- ponding value of the variable will be inadmissible in any even root, because the contiguous values of the function must be negative. 3. The value x = cb can never belong to a maximum or minimum, inasmuch as it does not admit of both a preceding and succeeding value. EXAMPLES. (54.) 1. To determine for what values ofar the function y ■= a^ ■{■ ly'x — c^ x" becomes a maximum or minimum, dx dx" From the second equation it appears that, whatever be the values of dy X, given by the condition -^ = 0, they must all belong to maxima. From 6^ — 2c^x = 0, we get x = — -j ; hence when X — —-r- .•. r/ = a* + -— r, a maximum. dy The equation -r- = cc would give, in the present case, z = oo, a value which is inadmissible (53). tion THE DIPPBRENTIAJL CALCULUS. 71 2. To determine the maxima and minima values of the func- y = 3oV — b'x + c* putting ax ax' ga^x' — b' = .'.X = ± — 3a Substituting each of these values in -^ we infer from the results that when X = — . . . . y = c , a mm. 3a ^ 9a b'' , , 26" X = . . • . y = c^ + , a max. 3a ^ 9o 3. To determine the maxima and minima values of the function y = A/2ax. Omitting the radical du u = 2ax .'. -T- = 2a, ax as this can never become or co , we infer that the function has no maximum or minimum value. 4. To determine the maximum and minimum values of the function y = \f ^(^3? — 2ax^. Omitting the radical and the constant factor 2a (63), M = 2ar' — a^, ..._=4ax-3x»,^=4a-6a:, 4a .*. X (4a — 3a;) = .*. x = 0, or x = -^. Substituting each of these values in -j-r-, the results are 4a and ax* — 4a ; hence when X = . . . y = 0, a minimum. 73 THE DIFFERENTIAL CALCULUS. _ 4a _ 8 ^ - y • • 2/ -3 If, instead of the preceding, the example had been „ • • J/ — o «^ maximum. y = \/2aar* — 4aV, we should have had du (Pu = 6x — 4a. 4a -T~ = ^3^ — 4ax, -j-^ = 6x — 4a. X (3x — 4a) = 0, .'. X = 0, or X the same values as before ; but the first corresponds here to a maxi- mum, since it makes - — negative ; this value, therefore, must, by (53), be rejected. If, indeed, we substitute ± ^ for x, in the pro- posed function, it becomes y = V— 4a^h' =F 2a/i^ \yhere 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 =z a-\- \/d^ — 2a-x + ax^. If t( is a maximum or minimum, y -r- a will be so ; therefore, trans- posing the a, cuid omitting the radical (53), u = a^ — 2a^x + ax^ -J- = — 2a'' + 2ax, r-— = 2a, ax air .: — 2a^ + 2ax = .*. x = a, .*. when X = a . . . y = a, a minimum. 6. To determine the maxima and minima values of the function ' (a - x)^ In solving this example we shall employ a principle that is often found useful, when the proposed function is a fraction with a denominator more complex than the numerator. Instead of the function itself we shall take its reciprocal, which will give us a more simple form, and it is plain that the maxima and minima values of the reciprocal of a THE DIFFERENTIAL CALCULUS. 78 Function correspond respectively to the minima and maxima of the function itself. Omitting, then, the constant a^, and, taking the re- ciprocal, we have a^ — 2ax + x" a? ^ , u = = 2a + X X X ' ' dx x^ ' dr^ x^ a" , d-iL . 2 .-. — -V + 1 = .-. X = ± a .-. [ j— ] = ± -, ar aJT a hence x = a makes m a minimum, and z = — a makes it a maxi- mum, therefore when X = a . . . y — co , a maximum, x = — a . . ' y =^ — i^j^ minimum. 7. To determine the maxima and minima values of the function V =b + 1/ {x — ay. Omitting b and the radical u = (x — ay ,,_=6(x-ar,— ==4.5(x-a/ .-. 5 (x — o)" = .-. X = a .-. f-j-^] = 0. As this coefficient vanishes, we must proceed to the following, which however all contain x — a, and therefore vanish, till we come to -r-r = 2 • 3 • 4 • 5 ; as therefore the first coefficient which does dxr not vanish is of an odd order, the function does not admit of a maxi- mum or a minimum. 8 To determine the maxima and minima values of the function dy d?y 1 ■£ = X' (1 + log. ^). 5;^ = ^ I- ^- (» + log. ^Yl 10 74 THE DIFFEUENTIAL CALCULUS. The factor x' can never become 0, therefore (1 + log. x) =^ .-. log. X = — 1. 1 .*. a: = c~^ = - e . r^] = (!)■. . 1 .'. when X = — , x* = ( — ) , a minimum. 9. To determine the maxima and minima values of y in the function u = aP — 3axy -\- y^ = du - = 3ar'-3ai/.-. (52) x' — 3axi/ + t/^* = 0. r" Sr" — Say = o^ ''' V =^ ~^ •'• ^ ~ 2«'ar' = .-. X = orx = a 3/ 2 ••. (52) (Py ^cPu ^ du X* 20,' ^ = - or a 3/2 — 1 .•. when X = .... 1/ = 0, a minimum. X = 0^2 .... 1/ = a ^4, a maximum. 10. To divide a given number a, into two parts, such that the product of the mth power of the one and the nth power of the other shall be the greatest possible. Let X be one part, then a — x is the other, and t/ = x" (a — x)" = maximum, .'. -p = mx"~' (a — x)" — nx" (a — x)"-' = x^' {a — x)"-' \ma — {m -\- n) x\ = 0, .'. X = 0, or o — X = 0, or THE DIFFERENTIAIi CALCULUS. 75 ma — {r.i + n) a; = 0, which give ma X = 0,x = a,x = ; . m + n The first and second of these values are inadmissible, because the number is not divided when x = or when x = a. Substituting the third value in — i- = x""' (o — x)**"* \{ma — {m -]- n) xy — m{a — xY — nx-| oar we have ^^^ ^ - W"-^ [« - ^T-' ^» [« - ^T + ^^\ which is negative because each factor is positive, hence the two re- quired parts are ma , na , . , , and — -j- — bemg to each other as m to n. m + » 7n -\- n Cor. If TO = 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 n M = arP (a — x) = A maximum, ' du , , ^ = xP-' \pa—(p -jr \)cr\ ^ 0, pa .-. X = — or pa — (p + 1) X = .-. X = — ij — . P + 1 This last value substituted in _ = r^-^ ^pa —{p+i)xl—{p-\-l) X'-' causes the first term to vanish ; the result is therefore negative, so pa ma , ^ • , ^ that X = ~ = ; — corresponds to a maximum value of «, p + I m + n and therefore (53) to a maximum value of m" — x" {a — x)". Another easy mode of solution is had by using logarithms, for it is 70 THE mrFERBMTlAL CALCULUS. obvious that sinco the logarithm of any number increases with the number, when this number is the greatest possible, its logarithm will be so also. .'. in log. jj + n log. (a — x) ~ max. du Ml n da X a — X \ > j tn + « as before. The expression for the second differential coefficient is — (tn + n) showing that the foregoing value of x renders the logarithmic ex- pression a maximum. 1 1 . To divide a number o, into so many equal parts, that their continued product may be the greatest possible. It is obvious from the corollary to the last example, that the parts must be equal, for the product of any two unequal parts of a number, is less than that of equal parts. Let X be the number of factors,, then,^ 0- = o, maximum, ... log. 0- = = a: log. {-) = a maximum,. ••• log.-_l = a X = log. -'1 = e.-. a X —-* e hence the proposed number must be divided by the number e = 2-718281828. 12. To determine those conjugate diameters of an ellipse which include the greatest angle. Call the principal semi-diameters of the ellipse a, b, the sought semi-conjugates x, x' and the sine of the angle they include y. Then {Anal. Geom.) * There is obviously no necessity to recur to the second differential coefficient to ascertain whether this value render the function a maximum or a minimum, since it js plain that there is no minimum unless each of the parts may bo 0. THB DIFFERENTIAL CALCULUS. 77 T^ + «" = a^ + 6= .-.x = ^^a? ■{■ y — ir ah xxy =^ ao .'. y =^ — — '' ^ xx' , ab . •. t/ = : = max. ^ xV a' + b^—sr' Omitting the constant 06, inverting the function (ex. 6.) and squar- ing, we have u = oV + b^ar^ — x* = max. du ... — = 2o='a; + 2b^x — 4x' = 0, ax fjfi 4- A3 3 1 '2 2 2 The first of these values is inadmissible, from the second we find that hence the conjugates are equal. For the second differential coefii- cient we have -r^ = 2a=^ + 2b- — Ux" This being negative, shows that x = V corresponds to a maximum value of m, or to a minimum value of j/, so that tho conju- gates here determined, include an angle whose sine is the least pos- sible; 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 & into two parts, such that the product of the nth power of the sine of one part of the >«th power of the sine of the other part may be the greatest possible. Let X be one part, then & — a; is the other, and sin. "a; . sin.*" {& — x) = maximum, . •• » log. sin. X + m log. sin. (^ — x) = maximum, n cos. X in cos. {& — x) sin. X sin. (d — x) 78 THK DIFFERENTIAL CALCL'LUS. .'. ntan. {6 — x) — m Ian. x, .'. n : HI : : tan. x : tan. {6 — x), .'. n -}" ni : " — "* : : tan. x + tan. {6 — x) : tan. x — tan. (6 — x), : : sin. d : sin. {'2x — 6),* .-. sin. (2x — 6) — ■ sin. 6, n + m which determines x. 14. Given the hypothenuse of a right-angled triangle to deter- mine the other sides, when the surface is the greatest possible. Call the hypothenuse a, and one of the sides x, then the other will be Va' — ^ and the area of the triangle will be ^ %/ a^ — r* = maximum. .•. u = a^x* — X* = maximum. t^" « o . 1 ^ r. « .•,-r- = 2a-x — 4:X = .'. a: = or X = — r-. ax V2 Substituting the second value in =z 2aF— 12x2 the result being negative, shows that the above value of x corresponds a to a maximum. Therefore the required sides are each —r~. V 2 1 5. To determine the maxima and minima values of the function 1/ = x^ — ISx^ + 96x — 20. when X = 4t . . . . y = 356 a maximum. X — 8... .7/ = 128 a minimum. 16. To determine a number x, such that the ath root may be the greatest possible. Ans. x — R— 2-71828 .... 17. What fraction is that which exceeds its nith power by the greatest possible number ? m— 1 2 Ans. \f — . m * Dr. Giegory'i Trigonometry, p. 47, Equation (S). THB DIFFERENTIAL CALCULUS. 19 18. Given the equation y^ — 2mxy + ar^ = a^, to determine the maxima and minima values of y. When X = ma y a a maximum, v/ 1— TO- VI— ni"' X —- — ma y — a , a minimum. V 1 _ ,rt3 •' ^ i — m^ 19. Given the position of a point between the sides of a given angle to draw through it a line so that the triangle formed may be the least possible. Ans. The line must be bisected by the point. 20. The equation of a certain curve is a^j = ax^ — x^ required its greatest and least ordinate s. When x = |a . . . . J/ = maximum, at = . . . . t/ = minimum. 21. To divide a given angle d less than 90° into two parts, x and t — X, such that tan." x . tan."* (^ — x) maybe the greatest possible. n — m tan. (2x — &) = i tan. 6. 22. To determine the greatest parabola that can be formed by cutting a given right cone.* SCHOLIUM. (65.) It will be proper, before terminating the present chapter, to apprize the student that in the application of the theory of maxima and minima to geometrical inquiries, care must be taken that we do not adopt results inconsistent with the geometrical restrictions of the problem. We know, indeed, from the first principles of Analytical Geometry, that when the geometrical conditions of a problem are translated into an algebraical formula, that formula is not necessarily restricted to those conditions, but, in addition to all the possible solu- tions of the problem, may also furnish others that belong merely to the analytical expression, and have no geometrical signification.! If, * It will be shown hereafter that a parabola is equal to J of a rectangle of the same base and altitude, \ See the Analytical Geometry. 80 THE DIFFERBNTIAL CALCULUS. therefore, among these latter solutions there be any belonging to maxima or minima, they are inadmissible in the application of this theory to Geometry. The following example is given by Simpson, at art 47 of his Treatise on Fluxions, to illustrate this. -S^F From the extremity C of the minor axis of an €lhpse ,y |\ to draw the longest line to the curve. Suppose F to be jj the point to which the line must be drawn, and call the ' abscissa CE,t then the geometrical restrictions of this variable are such that its values must always lie between the limits x = and x = 26, a and b denoting the semi- axes. By the equation of the corve. EF' = f = ^i2bx-:c') a" ... CF^ — w — ^ + IT (2^^ — xF) = maximum. du „ , , "^ d^ , d'—b^ and since -r-^ = 2 (1 — y^) it follows that the foregoing expres- sion for x renders u a maximum for all values of 6 less than a, and a minimum for all values of b greater than a. Hence if the relation n-h between a and b be such that — r^ may exceed 26, the analytical expression for CF will admit of a maximum value, although such value, not coming within the geometrical restrictions of the problem, a^b is inadmissible. If the relation between a and 6 be such that — j-„ a-' — 6^ = 26, that is, if a^ = 26^ the solution will be valid, and in the ellipse whose axes are thus related CD will be the longest line that can be dravm from C, agreeably to the analytical determination, and the solution will always be vahd if the axes of the ellipse arc related so a^b that p; is not greater than 26, which leads to the condition 26^^ a" — 6" not greater than a^. THE DIFFERENTIAL CALCULUS. 81 CHAPTER VII. ON THE DIFFERENTIATION AND DEVELOPMENT OF FUNCTIONS OF TWO INDEPENDENT VARIABLES. Differentiation of functions oftivo independent variables. (56.) Let z = F {x, y) be a function of two independent varia- bles ; then since in consequence of this independence, however either be supposed to vary, the other will remain unchanged : the function ought to furninsh two differential coefficients ; the one arising from ascribing a variation to x and the other from ascribing a variation to y, y entering the first coefficient as if it were a constant, and x enter- ing the second as if it were a constant. The differential coeffi- dz cient arising from the variation of a: is expressed thus, — ; and that arising from the the variation of y thus, — ; and these are called the partial differential coefficients, being analogous to those bearing the same name considered in chapter IV. We have seen, in functions of a single variable, that if that variable take an increment, and the function be developed, what we have called the differential coefficient will be the coefficient of the first power of the increment in that de- velopment ; so here, as will be shortly shown, the partial differential coefficients are no other than the coefficients of the first power of the increments in the development of the function from which they dz are derived. As to the partial differentials they are obviously — dx and -J- dy and hence we call -r- dx -^ -r- dii the total differential ay dx dy "^ of the function, that is, dz = -r- dx + -r- dy, dx dy and we immediately see that this form becomes the same as that 11 82 THE DIFFERENTIAL CALCULUS. given in chapter IV. for the differential of F {x, q) as soon as we suppose 7/ to be a function of x, for we then have dz . dz dz dy ^ux dx dy dx^ as indeed we ought. In a similar manner, if the function consist of a greater number of independent variables as « = F (x, y, z, &c.) we should necessarily have as many independent differentials, of which the aggregate would be the total differential of the function, that is du =^ -r- dx + -r- dy -\- ^- dz -\r &c, dx dy dz Hence, whether the variables are dependent or independent, we infer, generally, that The total differential of any function is the sum of the several partial differentials arising from differentiating the function relatively to each variable in succession, as if all the others were constants. We shall add but few examples in functions of independent varia- bles, seeing that the process is exactly the same as for functions of dependent variables. d {x -^r ]}) = dx -^r dy d . xy ^ ydx + xdy 1 X _ ydx — xdy y ¥ ay as^dy — ayxdx Vxr' + y'' (^ + 2/')* , . , X ydx — xdy d tan.~' - = y f -V 3? d y _ yd^ — ^y^dy — xdy 3j/2— X {^Mf — xf d. a' b^ c' = a'b^ c''{dx log. a ■\- dy log. h-{-dz log. c) d log. tan. - = ^^-^ — ^^y = 2 {ydx — xdy) y o ■ X X „ . 2x y sm. - cos. - w^ sm. — y y '^ y dy' = f log. ydx + I/*-' xdy. (57.) If the function that x, 2/ is of 2 is given implicitly, that is by the equation then but (39), THE DIFFERENTIAL CALCULUS. 88 « = F {x, y, z) = 0, ■du^^ „ , .du. ^dx^ du du dx dz dx du du dy dz ' ■? = « dy , Au , du dz^ , , ,du , du dz _ ^dx dz dx' dy dz dy' -^ Thus : let Ax" + Bi/=' + Cc^ _ i = o, .-. du = {Ax + Cz^) dx + (By + Cz^) dy = 0. (58.) If « = Fz, z being a function of x and y, the two differen- tial coefficients are (33) du _ du dz du _ du dz dx dz ' dx* dy dz ' dy and the total differential is, therefore, * The brackets arc employed here for the same purpose as at (37), viz. to im- ply the total differential coefficient derived/ram u, considered as a function of a single variable. This form it will be necessary to adopt whenever m contains, besides x, other variables that are functions of x, provided we wish to express the total coeffi- cient with respect to a:. No ambiguity can arise from our calling these same coef- ficients partial in one sense, and total in another. They are partial coefficients in relation to the whole variation of i«, but they are total coefficients as far as that variable is concerned whose differential forms the denominator; and it may be re- marked here, once for all, that when we enclose a differential coefficient in brack- ets, we mean the tot(d differential coefficient to be understood, arising from consi- dering the function, whose differential is the numerator, as simply a function of th« variables whose diflerentials form the denominator. 84 THE DIFFERENTIAL CALCULDS. - du dz - , du dz dU = -r- . -r- dX -{- -r- . -f- AV. dz dx dz drj Now it is worthy of notice, that the ratio of the tieo partial differ- ential coefficients is independent of F, so that this may be any func- tion whatever. Thus du , du du dz _ du dz dz . dz dx ' dy dz' dx ' dz' dy dx ' dy which is an important property, since it enables us to eliminate any arbitrary function F of a determinate function y'(,r, ij) 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 as- certain whether an expression containing two variables is a function of any proposed combination of those variables. For, calling this combination z and the function «, 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 dz _ dx ' dy dy' dx exists. For instance, suppose we wished to know whether m = a?* -f- 2x^f + ?/* is a function o^ z ^= oc^ ■{• y^. Here ^« . ■, I . » ^^'* . o I . -^ o,z ^ dz - = 4.^ 4- 4:ry^- = ^x^y + Ay^,- = 2y,-= 2x; du dz du dz ,,■,,. n\ ^ , ^ n , . t^ ^ dx dy dy dx ^ j ^ j v :; n j consequently, since the proposed condition exists, we infer that u is a function of x. We shall now proceed to apply Taylor's theorem to functions of two independent variables. Development of Functions of two Independent Variables. (59.) In the function z = F (x, y) suppose x takes the increment h, the function will become F (a; + h, y), y remaining unchanged, since it is independent of x, then, by Taylor's theorem, dz d-z h^ d^z h^ &c. . . . (1). THE DIFFERENTIAL CALCULUS. 85 But if y also take an increment k, then z will become dij dif 1*2 dif 1 • 2 • d dz so that in the expression (1) we must for -j- substitute dz d^z d?z dz djj If F _LS. ^' , o dx "^ c/x ^'+ rfo; •1-2'^ dx • 1.2-3 + ^^' dz d^z d^z d^z "^ ' dy "* dy' Jf_ _J_f_ ^-^ d^ dFz cPz d^z ' dy ' dw^ k^ ' dy^ P 1 ± h J ±— 4- i_ L fop dx" ^ dar" '^^ dr* •1-2^ dx" •l-2-3^*^*^' and so on. Before, however, we actually make these substitutions, we shall, for abridgment, write dz drz d'z ^ ' dji _^z__ ^' df di^z dydx dx ' dy^ dx dx S 7 ^yidxP d'z for •' dxP this last expression implying that after having determined the qth differential coefficient of the function z relatively to the variable i/, the j9th differential coefficient of this is taken relatively to the other variable x. Hence, the result of the proposed substitutions in (1) will be . ¥{x-\-h,y^h) = 86 THE DIFFERENTIAL CALCULUS. ax dz dy k + d'^z h^ dx^ * 1 • 2 dydx dh ¥ dtf '1-2 c/y dx^ d'z d^f dx i.^z df 1 •2-3 hh~ 1 • 2 ¥h 1 -2 1-2-3 + &C. The general term of the development being di/* d.i'' ' (1 • 2 . . . 5)(1 -2 . . .pY If in the proposed function z = 'F {x, y) we had supposed y to vary first, then, instead of (1), whe should have had ^■2 . d'z P + &c. . . . (2). But, if a: take the increment h, z will become dz , , d'z h^ z + — h •+■ . dx ^ d3^ 1-2 + d'z df' l-2'3 + &C. dx' 1 • 2 • 3 and, therefore, we must substitute, agreeably to the foregoing nota- tion for dz dy for dz . d^z , dy dxdy di'z dh + d'z da^ dy * 1 • 2 dx'^ dy 1 • 2 • 3 + &c. df dPz _fz__ ^^ d'z df dx dy^ dx^ df h^ d'z K' 1 • 2 dx^dy'' ' 1 • 2 • 3 + &C. for df d'z d^z , , d^z h + Iv" + d'z df ' dx dy"^ '" ' dir dy' * 1 • 2 dx' df ' 1 • 2 • 3 and so on ; so that the development would be F {x -\- h, y -\- k) = + &c. THE DIFFERENTIAL CALCULUS. 87 ,dz , dx dz dx -t- d"-z Iv" dor 1 . 2 d'z dxdij d'z hk df 1 . 2 + d'z h^ dx' ' d'z 1 •2-3 ]rk dx^ dy ' d'z dx dif ' d'z 1 • 2 1 • 2 df 1 • 2 -3 + &C. hP . h'i the general term being d'-^z ~dxP df ' {1 -2. . .p) (1 -2 .. . q)' As this development must be identical with that exhibited above, we have, by equating the like powers of h and A;, d'z _ d^z d^z _ dPz dijdx dxdij dy dx^ dx^ dy and generally d^-^z _ dP-^z dif dxP ~ dxP dy'' ' we conclude, therefore, thatif we first determine the gth differential coefficient relatively to the variable y, and then the pih. differential coefficient of this relatively to the variable x, the final result will be the same as if we first determine the jpth differential coefiicient rela- tively to Xy and then the qth differential coefficient of this relatively to y ; so that the result is the same in whichever order the differen-. tiations are performed. (60.) We see from the foregoing development, that the partial differential coefficients of the first order are the coefficients of h and k, the first power of the increments, so that the term containing these first powers is in this respect analogous to that containing the first power of the increment in the development of functions of a single variable, and, by a very slight transformation, it will be seen that the same analogy extends throughout a 1 the terms of the two develop- ments. For the development just given may be put under the form Fix + h,y + k) = z +(^^ + 1*) ^dx 88 THE DIFFERENTIAL CALCULUS. + _J_ f^ 7,3 . 2 '^'^ hk 4- ^^ k") , 1 ,d^2 ,„ , „ d^z ,,, , „ rf^2 , ,, , d^z J.. 2-3 VjH du^ dij dx dy^ dx' ' + &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 : dz — -^dx-\-~dy. . . . (1), dx dij the coefficients y-, -^, being functions of x and y, we have dz _ d'z d'z dx dx^ dxdy dz _ d^z d'^z dy dydx dif and, consequently, In like manner, these coefficients being functions of a: and y, we have d'z _ d'z d'z dx^ dx' dj^ dy d'z d'z d'z dxdy dx^ dy dx dy^ , d'z d'z , d'z ■ d ' —r^r = , ., , + — r-, — dy dy^ dy dx dif so that '^"y- ■ ■ '■ ■ (^'' and so on ; the numeral coefficients agreeing with those in the cor- responding powers of the expanded binomial. (61.) Having now applied Taylor's theorem to functions of two THE DIFFERENTIAL CALCULUS. 80 Variables, we may equally extend Maclaurin's Theorem. For, if in the foregoing development, we suppose x and y each = 0, the de- velopment will become that of the function F {h, k) according to the powers of h and k ; or, substituting x and y for the symbols h and k, since these are indeterminate, we have The principles by which we have thus extended the theorems of Taylor and Maclaurin are sufficient to enable us to extend these theorems still further, even to the development of functions of any number of variables whatever, but this is unnecessary. It maybe remarked, however, that if we wish to develop a function of several variables according to the powers of one of them, it may be done independently of any thing taught in this chapter ; for, if all the varia- bles but this one were constants, the development would agree with that already established for functions of a single variable, and, as these constants may take any value whatever, they may obviously be replaced by so many independent variables. We shall give one instance of this extension of Maclaurin's theorem to a function of two independent variables, choosing a form of extensive application and of which the development is known by the name of Lagrange's Theorem. (62.) The function which we here propose to develop according to the power of x, is M = Fz, in which z = y -\- xfz, z being ob^ously a function of the independent variables x and y. We shall first develop z = y -{- xfz according to the powers of x : this development is by Maclaurin's theorem dz d'z x^ , d^z T x^ ' = W + ts^ - + [5?] —2 + fe-] FFTa + «"=• and if we denote according.to the notation of Lagrange the successive differential coefficients of/z, relatively to x hyf'z,f"z,J"'Zf &c. we shall have 12 90 THE DIFTERENTIAL CALCULUS, dz &c. &c. Consequently, when a; = 0, W =y ^'d^^~^-d^J'J df ^da^^-^^^jy^ ^ ^ • ~di, d^ ^ &c. &c. Hence n J u -yjj ^-y ^y 1 . 2 ^ dy' 1 • 2 • 3 + &C (1). Now, instead of this development, we should obviously have obtained that of Fz = F (]/ + xfz), if in place of ^ and its differen- tial coefficients we had employed Fz and its differential coefficients. We should then have had u = F {y + xfz) . . therefore . . [ m ] = Fy du _du dz du _ dFy dx~ds'dx *■ dx^~ dy ^y d'u_ d^u dz du dPz r^i —^Fl-f f \a 4. d?~'d^^dx^ "^ dz' dx" '-(/a^-' dy" ^^^ "^ dFy dFy d.{fyf ^ ^•'dj^fy^' dy ' dy dy &c. &c. HeQC« THE DIFFERENTIAL CALCULUS. 91" Tz = F (y + xfz) = Fw + — -^ fy •- + ; . + and this is Lagrange's Theorem.* From this remarkable expression, which includes that marked (1), other forms may be readily deduced as particular cases. Of these the two following are the most important. Put a? = 1, then the formula (1) becomes z = y-\-fz = y-{-fy H hLLL. . L — \dAL . 4- H i J a JiJ-r ^^ 1 . 2 ^ dy'' 1 • 2 • 3 &c (3), and the formula (2), d.Fy "*• dy ^^y^ 1 „ = F(y+/.)=F, + -^//,+ ^_._ + ^.^iJyf , la-^ — • TT^ + &^ (4). (63.) We shall terminate the present section with one or tAvo ex- amples of the application of these formulas, referring the student for more ample details on this subject to Lagrange's Resohilion des Equations J^Tumeriques, note xi. ; and Jephson's Fluxional Calculus, vol. i. EXAMFLES. 1. Given ^ — 52 + r = 0, to develop z according to the pow- ers of r. Since here2 = --] ,wehavey = -, fz = - 2^ .: fy =^ - (-Y q q -^ q'J q J^ q^ q> * For another and very complete demonstration of this theorem see note (Bl at the end. M THE DIFFERENTIAL CALCULUS. .difyy_ 1 d.t/_ 6 3 dFjfyY _ 1 ^Y _ 9 /^l/' _ di/ 9^ ' dy q^^ ' df q^ ' dy^ f ' dy Hence, by the formula (3), we have, by putting for y its value - = !:4.i rl-i-—!— '" I- ^ • ^ li + fe q q^ 1 • 2^' 1 • 2 • 9" 2. Given the radius vector of an ellipse, viz. {Anal. Geom.) 1 — e^ r = a . 1 + e COS. u to develop r", according to the powers of cos. w. Since r = o (1 — c-) — e cos. w . r, we have, by putting y for 0(1 — e^) and ar for — e cos. cj, Fr = F (t/ + x/r) = F (y + X • r) = (y + X • r)". Hence, by the formula (2) dy' X dy ^ x^ '" = J/"+%- 2'- 1 + dy -1^ + dy ^ x' = y- + ny . I + n (n + 1) y" . ^p-_ + n{n-{- l)(n + 2)t/".^-^+ &c. = a- ( 1 -' e^)" ( 1 - "^^^^^ + ^ ^^ "^ ^^ C COS." W — n(n+l)(n + 2) , , fTY-l '^ <^os- <^ + &c. 8. It is required to revert the series a + I3z + ys? ■{■ dz' + &,c. = 0, THE DIFFERENTIAL CALCULUS. 93 that is, to express the valufe of z in terms of the coefficients. Here 2 = — 1--|- (7 + ^^ + &c.) = 2/ +/^ therefore, by the formula (3), ^ d.^,(r + 5,/ + &c.)^ ^ d^ "I, (r + ^2/ + &c.)^ J 1^ (7 + % + &c.) (5 + 2sy + &c.) -^ 6 |j (7 + 5y + &c.)=' + &c. + &C. where t/ = ^ . consequently 4. Given 1 — z + az = to develop log. z, according to the powers of a. log. 2 = a + 1 o^ + 1 a^ + i a* + &c.* * This we know from other principles ; for, since the proposed expression re- dncestoz = — — - .•. log. z = — log. (1 — a) and this, in the hyperbolic system, is equal to the above series. (See the Essay on Logarithms, p. 3.) 94 THE DIFFERENTIAL CALCULUS. CHAPTER VIII. ON THE MAXIMA AND MINIMA VALUES OF FUNC- TIONS OF TWO VARIABLES, AND ON CHANGING THE INDEPENDENT VARIABLE. (64.) It remains to complete the theory of maxima and minima by applying the principles established in Chapter VI. to functions of two independent variables. The same character belongs to a maximum, or minimum function of two variables that belongs to a maximum or minimum function of one variable, that is, the maximum value exceeds the contiguous va- lues of the function, and the minimum value falls short of them. Hence, if 2 = F {x,y) be any function of two variables, wliich becomes a maximum for cer- tain particular values of them, then h and Jc being finite increments, however small the condition is that, between such finite values and 0, we must always have Flx,9j-]>F[x± h,y± kl and, consequently, (60), (±£*±?/)+*(^*'±^^"+^*=)+^-<''-* If, therefore, of the small values which we suppose h and k to take, h be the smallest, a part of k maybe 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 '- dx dx ' dar^ dx dy dy'^ This condition being similar to (1) art. (49), we infer, by the same reasoning, that dx dy * This is the manner in M^hich analysts have agreed to express an isolated ne- gative quantity ; which must necessarily have resulted from the subtraction of a greater from a less quantity. It is not, however, to be inferred that a negative quantity is less than zero, as the above expression indicates, as such supposition would be manifestly absurd, Ed. THE DIFFERENTIAL CALCULUS. 95 which cannot be for both the signs ± unless *=0,^ = 0....(l). ax: ay By continuing to imitate the reasoning in (49), we find that these same conditions must exist for all the values of the variables that ren- der the function a minimum. ' , Hence (49), we have, in the case of a nuiuHatnn, the condition ^ ^da^ dx dy dif and in the case of a minimum, d-z d-z , d^z -,,,„, o SO that, supposing these first terms do not vanish for the values of j: and y given by (1), the condition of maximum is and the condition of minimum, d-z d'z d'z '-dx^ dy dx dy^ In either case, therefore, the expression within the brackets must have the same sign independently of the sign of the middle term. To determine upon what other condition this depends, let us represent the expression by A ± 2B + CorA(l ± 2^ + ^). B^ B^ Adding — — — = to the quantity within the parenthesis, its JO. A. form is B , . C B2 A((i± _)=+___). Now this expression will always have the same sign as A provided C B^ C has, and that "x > Ti' *^^* ^^' ^^ 7 ^^ °^ AC — B^ 7 0, be- cause then the factor of A will be necessarily positive. Hence, be- side (1), the condition that a maximum or a minimum may exist is 96 THE DIFFERENTIAL CALCULUS. and we are to distinguish the maximum frcm the minimum by ascer-- taining whether the proposed values of x and y render ^ZOor/O, or, which amounts to the same, whether ^ZOor/O. d?z "3 _ 12a — 2 [p'"^} ~ L p" J 5"^ ... 3 [p"2] = 12a.-. [//'] = ± 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 f = {x-aY(x-^b) to determine the values of -z-, when x = a, ax dii. 2. Given (^y __ xf — {x — ay {x — b)=0 dy iPi/ to determine the values of -j- and -irr, when x = a. _dy d^ii We here terminate the First Section, having fully considered the various particulars relating to the diiferentiation of functions in gene- ral. THE DnFTERENTIAL CALCULUS. 113 SECTION II. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE THEORY OF PLANE CURVES. CBAFTSn Z> 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')i {^"i !/")> i^^ ^"y plane curve, is {Anal. Gcom.) V' — '/" y—y' = '^'^],"{^ — ^')^ y' — J/", being the increment of the ordinate or proposed function corresponding to x — x" the increment of the abscissa or independ- ent variable. The limit of the ratio of these increments, by the prin- ciples of the calculus, is -y-, ; that is to say, such is the representation of the ratio when x — x" = 0, and, consequently, y' — y" = 0. But when this is the case the secant becomes a tangent. Hence the equation of the tangent, through any point (x', y') of a plane curve, is dy' y—y''='db'i''--^') • • • • (!)• dii' It appears, therefore, that the differential coefficient -i-, for any proposed point in the curve has for its value the trigonometrical tan- gent 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.) By means of the general equation (1) we can always readily de- termine the equation of the tangent to any proposed plane curve when the equation of the curve is given, nothing more being required than to determine from that equation the differential coefficient. 15 114 THE DIFFERENTIAL CAtCTTLUS. dy' for the ellipse. We here have to determine -— from the equation Suppose, for example, it were required to find the particular form 3re have to determine - d and which is dy' _ BV dx' ~ Ay therefore the equation of the tangent is y—y =— Ay ^''■~"''^* {x, y) being any point m the curve, and (x, y) any point in the tan- gent. 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" + Bx'tj' + Ca;'2 + Dy' + Ex' + F = 0,. * we have 2Ay'^ + Bar' -^ + By' + 2Ca;' + D -^ + E = 0^ dy' _ 2Ca;' + By' + E •*' Ix' 2Ay' + Ba;' -\-~I) so that the general equation is 2Cx' + By' + E y-y =-2Ay' + Bx'+^^^-^^- (78.) As the normal is always perpendicular to the tangent, its general equation must be, from (1), !/-2/'=-^C^-^') t.-..(2). ♦ The general equation of lines of the second order in its most commodiou* form is t/'J =: mx' + nx'\ from which, by differentiation, we have dy' m-{-2nx' 'd^'~ 2y' and the equation of the tangent to a line of the second order is thereforo , m + 2nx' dx' ^ dy'^ ' Ed. THE DIFFERENTIAL CALCULUS. 116 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 subtan- gent T^ {Anal. Geom.), .'. T = — ^ .... (3). dx' If, instead of the equation of the tangent, we put y = in that of the normal, then the resulting expression for x — x' ^vill be the value of the intercept between the normal and the ordinate y\ that is, it will belong to the subnormal N,, .•.N,=,'|....(4). As to the length of the tangent T, since T = -s/i/^ -f T/, w« have, in virtue of (3), dx'^ Also, since the length of the normal N is N = V y''' + N'^ w© have, by (4), N = !/' V (1 + ;g[) . . . . (6). The foregoing expressions 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 compli- cated 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 {x', y'), we may, by putting successively x = and t/ = 0, in the equa- tion (1), determine the two points in which the required tangent ought dy' to cut the axes of the coordinates and then draw it through them. If-—-. or 116 THE DIFFERENTIAli CALCULUS. is at the proposed point, the tangent will be parallel to the axes of x, because, as remarked above, -r-, is the value of the trigonometrical ax tangent of the inclination of the linear tangent to the axes of x, and for this reason also the tangent will be parallel to the axes 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 .^ ^y' _ P ' ' 'dx' 2y' ' -^ 2y' p Hence, having drawn from P, the perpendicular ordinate PM, if wo set 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 determuie the subtangent and subnormal at a given point (x'j y') in the parabola of the nth order, represented by the equation y = aaf dy' ax .-. T, = t/' -r nax"-^ = -, N, = y' nax'"*-^ = na'aP"-^ or —-, 3. To determine the subtangent at a given point in the loga- rithmic curve. The equation of this curve related to rectangular coordinates is y = a% which shows that if the abscissas x be taken in arithmetical progres- sion, the corresponding ordinates y will be in geometrical progres- sion, so that the ordinates of this curve will represent the numbers, the logarithms of which are represented by the corresponding ab- THE DIFFERENTIAL CALCULUS. 117 scissas, a being the assumed base of the system. Hence, calling the modulus of this base m, we have, by differentiating (13), dy _ 1 drx~my' .-. T, = y - y Hence the remarkable property that the subtangent is constantly equal to the modulus of the system, lohose base is a. 4. To determine the subtangent at a given point in the curve whose equation is a^ — 3axy + y^ = 0. Here A=3^_3.,-3„| + 34^ = 0. •** dx' .-. T, = ay — X- y aa/* y^ — ax y ay — X * 6. To draw a tangent to a rectangular hyperbola between the asymptotes, its equation being xy = a. T, = x'. 6. To determine the subtangent at a given point in a curve whose equation is y"' = aif, which, because it includes the common parabola, is said to represent parabolas of all orders. T^ = ^x'. n 7, To determine the subtangent at a given point in a curve whose equation is x" y" = a, which, because it includes the com- mon hyperbola, is said to belong to hyperbolas of all orders. T^ = ^x'. m (80.) If the proposed curve be related to polar coordinates, then the expressions in last article must be changed into functions of these. If the curve AP be related to polar coor- dinates FP = r, PFX = CO, then if PR be a tangent at any point, P and PN the nor- mal, and if RFN be perpendicular to the ra- dius vector FP, the part FR will be the polar tubtangentt and the part FN the polar sub- 118 THE DIFFERENTIAL CALCULUS. normal. When the pole F and the point P are given, it is obvious that the determination of the subtangent FR will enable us to draw the tangent PR. The formulas for transforming the equation of a curve from rec- tangular to polar coordinates, having the same origin, are {Anal: Geom.) X = r cos u,y—'r sin. u, x^ -\- y'^ = t'^, and the resulting equation of the curve will have the form r = Fw, or F (r, u) = 0, in which we shall consider w as the independent variable. Now RF = PF • tan. Z P = ** tan. Z P, but by trigonometry, tan. cj — tan. a tan. Z P 1 + tan. w tan. a that is, since dy . w tan. a = -y- and tan. u = - ax X tan. Z P = " f X ax therefore r times this expression is the value of the polar subtangent. dy But the differential coefficient -p, which implies that x is the princi- pal variable, ought to become, when the variable is changed to Ur (66), dx dy dv dv dx -n du dui ■r- = rr- -^ -T- -'• tan. z P = — r 7- dx d(ji aw dx . du X _i_ „ — — du *' dox Also, from the above formulas of transformation, dx dr . dy dr . = — — COS. u — r sm. w, -^ — = -7— sm. w + r cos. w du du du du • *. H —, — = ** ~; — sin. w cos. u — r^ sin.^ u •^ acj du ^y ^^ ■ L ^ a X -r— = 5* —, — sm. w COS. w + r* cos." w au du ■THE DIFFERENTIAL CALCULUS. 119 dx dr „ o • X = r -7— COS. w — r* sm. w cos. u dcjj du dy dr . . , „ . « — ^ = r -5 — sin.^ '^ + IT sin. w cos. u whence dx dy „ dx . dy dr x-f- = —ir',x-r-+ y-r~— ^ -T- dcj aw aw aw aw consequently, tan. / P = -^ .-. RF = -7- = subtangent. dr dr dw dw Also FN = = r^ -T- = = subnormal. FR dr dw dw (81.) We shall apply these formulas to spirals, a class of curves always best represented by polar equations. 8. To determine the subtangent at any point in the Logarithmic Spiral, its equation being r = = a dr w w — log. a a = dw m J . a a = -^— I /^ ^ m I / .'. IT — -7 — = mr = i dw 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 equal to 45°, because tan. ^ P — 1. Since, by the equation of this curve, log. r = w log. a, it follows that, if a denote the base of any system, the various values of the angle or circular arc w will denote the logarithms of the numbers represented by the corresponding values of r. Hence, the propriety of the name logarithmic spiral. In this curve tan. / P = r — — — = a — = m ; dw m hence the curve cuts all its radii vectores under the same angle. 120 THE DIFFERENTIAL CALCULUS. 9. To determine the subtangent at any point in the Spiral of Archimedes, its equation being r = au dr . dr „ „ • •• -r— = a .'.T^ -7- —r- = aijf = rw = T, aw aw so that FR is equal to the length of the circular arc to radius r, com- prehended between FR, FA ; when, therefore, u = ^ir^ the subtan- gent equals the length of the whole circumference. The spiral of Archimedes belongs to the class of spirals represented by the general equation r = aw". When n = — 1, we have rw = a, and the spiral represented is called the hyperbolic spiral, on account of the analogy between this equation and xy = a. It is also calle^ the reciprocal spiral. 10. To determine the polar subtangent at any point in the hy- perbolic spiral. T, =a. 11. To determine the polar subtangent at any point in the spiral — L whose equation is r = aw ^ . T = 2aw^ = — . r 12. To determine the polar subtangent at any point in the para' — bolic spiral, its equation being r = aw 2. T =^ a" ' 13- To determine the polar subtangent at any point in a spiral whose equation is (,^ _ ar) w^ — 1 = 0. ^' 3 • (r' — ar)'^ Rectilinear Asymptotes. (82.) A rectilinear asymptote to a curve may be regarded as a tangent of which the point of contact is infinitely distant, so that the determination of the asymptote reduces to the determination of the TPHE DIFFERENTIA-L CALCULUS. 121 t&ngent on the hypothesis that either or both y' — 0, x' = 0, 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 we have, by making successively y = 0, x = 0, the foTlovving ex- pressions for the parts of the axes of x and y, between the tangent flnd the origin, viz. x' — ^mdy'—x'-^. . . .(1). dii' ^ dx' ^ ' W If for a; = CO both these are finite, thsy will determine two points, one on each axis, through which an asymptote passes If for a: = cd the first expression is finite and the second infinite, the first will de- termine a point on the axis of x, and the second will show that a line through this point and parallel to the axis of y is an asymptote. If, on the contrary, the second expression is finite, and the first infinite, the asymptote will pass through the point in the axis of]/, determined by the finite value, and will be parallel to the axis o(x. When, however, asymptotes parallel to the axes exist, they may generally be detected by merely inspecting the equation, as it is only requisite to ascertain for what values of a-, y becomes infinite, or for what values of?/, x becomes infinite. Thus, in the equation xi/ = a, X = 0^ renders j/ = cjo » and y = renders a; = co , therefore the two axes are asymptotes. Again, in the equation bx* cfiy^ — ''f ^ — ^-p" = 0, or y^ = — ; U' XT it is plain that x = db a renders y = oo, 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- ponding to X = 00 . If both expressions are 0, the asymptote will pass through the origin, .... diy' and its mclmation d, to the axis of x will be determined by -rr- ==* tan. 6. 16 122 THE DIFFERENTIAL CALCULUS. If for J/ = 00 one or both of the above expressions are finite, there will be an asymptote, and its position may be determined as in the foregoing cases. EXAMPLES. (83.) 1. Let the curve be the common hyperbola, of which the equation is b a dy _ bx " ^ a ^/ x' — a? hence the general expressions (82) are x^ — a? _ a" X X and 6 a? ba X 1 -J both of which are 0, when a? = oo ; hence an asymptote passes through the origin. Also dw , 6 1 -r-= ± — . Ax a ^/ a^ which becomes ± — , when x = oo , therefore, this being the tangent a of the inclination of the asymptotes to the axis of x, they are both rep- resented by the equation y = ± - X, ^ 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 if = mx + nx^, THE DIFFERENTIAL CALCULUS. 123 y 2tf _mx + 2na^ — 2f _ m + 2«a; m + 2nx mx dy m + 2nx dx ^ mx + 2niP _ 2tf — mx — 2nx' _ mx ^ ^dx ^ 2ij 2y 2s/wx + ju-^ Dividing numerator and denominator of each of these expressions by jj, they reduce to m , ♦»» and - + 2» 2 n/^ + n •* X and these, when x = oo , or indeed when y = cc , become m and 2» 2Vn Hence the curve will have asymptotes, provided n be neither 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 hyperbola is the only Une of the second order which has asymptotes. (84.) 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 w, and w may be found from the equation of the curve, so that there will thus be determined 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 w is finite, for r = oo . If, for r = oo , w is al.«o 00 , 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 a by the equation of the curve w = - = 0, when r = 00 , therefore the asymptote is perpendicular to the fixed axis at the distance a from the pole. Neither the logarithmic spiral, nor the spiral of Archimedes have an asymptote. 124 THE DIFFERENTIAL CAtCULCr.- ■4. Let the spiral whose equation is C.2 _ 1 1 _ U-' be proposed, M'hich admits of a rectilinear asymptote, because w= I renders r = co . Ths direction, therefore, of the asymptote is ascer- tained, and consequently the direction of the infinite radius vector,- sinco they must be parallel. It remains, therefore, to determine the subtangent, or distance of the asymptote from the pole ctr _ 2au)-^ _ 2r^ d(ji (1 — u~~)^ au^ » _^ rfr _ ^ 2r^ _ "'■^^ _ ** _ rp du ou' 2 2 ' because w = 1 when r = co . (85.) 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 a' though w 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, moreover, the value of r for w = co be greater than the value of v for every other value of w, the spiral will be included within its circu- lar asymptote, but, otherwise, it will be without this circle. 5. Thus in the spiral whose equation is (r* — ar) cP — 1 = or w = — •«/ »*^ — ar cj is infinite when r = a, and for all less values of r, u 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 V ar — r* then also r = a gives w = oo , but for all other real values of w, r is less than a, so that this spiral is enclosed by its asymptotic circle, the radius of which is a. 4' Co THE DIFFERENTIAL CALCULUS. 125 6. To determine the rectilinear asymptote to the logarithmic curve. The axis of a?. 7. To determine the equation of the asymptote to the curve whose equation is The equation isy =^ x -\- \ a. 8. To determine the rectilinear asymptote to the spiral whose equation is , I r = ow - . The fixed axis is the asymptote. 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 asymp- tote. (86.) 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, i/ ; let also BD be a tan- gent at B, and BC any chord, then if BE, ED are parallel to the rectangular axes, BC will be the in- crement of the arc s corresponding to BE = h, the increment of the abscissa x. Now, putting tan. DBE = a, we have and ED = /ia .-. BD = n/ /i2 + k'a? BD + DC _ ^ h^(^\ + oC') -f /la— CE BC x/ A^ + CE'' CE This ratio continually approaches to t— or to unity as h diminish- es and this it actually becomes when h = 0. Consequently, since the arc BC is always, when of any definite length, longer than the chord 126 THE DIFPERENTIAL CALCULUS. BC and shorter than BD + BC,* it follows that when h = that the ratio of the arc to either of these must be unity ; therefore . , ,. . arc BC arc BC chord BC in the hmit -r— 3-^7; = 1 •*• — r — -^ , = l^ chord BC h h but chord BC n/ /i2 + CE^^ n/j CE^ h h h' ' and CE is the increment of the ordinate y corresponding to the incre- ment h of the* abscissa x ; hence, when h = 0, the ratio becomes ^ ^ ^1 + -^ = 1 dx ' daP ds V df dx dx^ If any other independent variable be taken instead of x, then, denoting the several differential coefficients relatively to this new variable by ((ic), (dt/), {ds) we have (66) At the point where -^ = 0,— = 1, or (ds) = {dx). CHAPTER II. ON OSCULATION, AND THE RADIUS OF CURVATURE OF PLANE CURVES. (87.) Let 2/=/p. Y = Fx, be the equations of two plane curves, in the former of which we shall suppose the constants a, 6, c, &c. to be known, and therefore the curve itself to be determinate ; while in the latter we shall consider the constants A, B, C, &c. to be unknown, or arbitrary, and there- * See Young's Elements of Plane Geometry. THE DIFFERENTIAL CALCULUS. 1'27 fore the species only of the curve given. The constants which enter into the equation of a curve, are usually called the parameters. If, now, X take the increment h, and the corresponding ordinates y% Y' be developed, we shall have, by Taylor's theorem, Now, the parameters which enter (2) being arbitrary, they maybe determined so as to fulfil as many of the conditions dy_dY d^y _dn ^^-^'d^-^'d^-d^'^^ ^^^' 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, 6, 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 a;. Other corresponding terms of the two series may, indeed, be rendered also identical, but this can take place only accidentally, not necessarily. Hence, whatever par- ticular value we now give to x, the resulting values of the corres- ponding coefiicients 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 particuleir 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 coeffi- cients become infinite, (1) and (2) will fail to represent the true de- velopments of the ordinates y', Y' at the proposed points. Neverthe- less, 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 development, 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, h may be taken as small as we please ; for if »t — 1 terms in each are identical, we may represent the difference of the two developments by 128 THE DIFFERENTIAL CALCULUS. A„ A'^ + S — {k'X' + S') (4), where S, S' represent the sums of the remaining terms in each series after the nth. Hence, ^ being the highest power of A. 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 A„ h to become greater than all the other terms in (4), and consequently, for this small value, A„ h"- — a; }/ 7 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 developments approach nearer to identity, for all values of h be- tween some certain finite value h' and as the number of identical leading terms become greater. When the first of the conditions (3) exist, the curves have a com- mon 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 para-^ meters are all determined agreeably to the conditions (3). No other curve of the same species can, from what is proved above, approach so nearly to coincidence with the proposed, in the immediate vicinity of the point of contact, 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. (88.) It appears, from what has now been said, that there may be different orders of contact at any proposed point. The two first 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 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 osculating curve will have the highest order of contact^ THE DIFFERENTIAL CALCULUS. 129 at any proposed point, in a given curve. If the curve, given in species, has n parameters, the highest order of contact will be the » — 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 predicted of any proposed point, although we see it is possible for such points to exist. (89.) At those points in the proposed curve, for which Taylor's development 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 con- tact 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'„) A"" + S — S' (5,) A„ A'„ being here the n — 1th differential coefficients. If these are odd, the contact is of an even order, also a being odd, h"" will have contrary signs for h = -\- h' and h = — ft', and therefore, since for these small values of ft, the sign of the whole expression (5) is the same as that of the first term, the differences of the ordinates corresponding to a? + ft, and to X — ft, will be the one positive and the other negative, so that the two curves must necessarily 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, a: + ft, will, for a small value of ft, have the same sign, whether ft be positive or negative ; so that, in this case, the curves do not cross each other at the point of contact. (90.) The student must not fail to bear in remembrance, that the proposition just established, comprehends only those points of the proposed 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 suppositicm 130 THE DIFFERENTIAL CALCULUS. that the true development, within the limits, proceeds according to the ascending integral and positive powers of h, that the foregoing conclusions respecting the signs of the difference (5) can be fairly drawn. (See note C.) (91.) From the principles of osculation now established, it is evi- dent 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 coincide 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 proposed 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 simpUcity 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 ra- dius the radius of cimature, and, from what has been said above, it follows that the determination of the curvature at any point in a pro- posed curve, reduces itself to the determination of this radius : to this, therefore, we shall now proceed. Radius of Curvature. FROBLEM I. (92.) To determine the radius of curvature at any proposed point of a given curve. The general equation of a circle being (^x~uY+{y—l3y = r\ it becomes determined as soon as we fix the values^f the parameters a, jS, r, and these may be determined, so as to fiilfil any three inde- pendent conditions, but not more. In the case before us, the condi- tions to be fulfilled are those of (3) art. (87), that is to say, putting THE DIFFERENTIAL CALCULUS. 131 p', p", &c. for the successive differential coefficients derived from Y = Fx, the equation of the given curve, the conditions to be fulfilled are y ^Ux ^'dx" ^' in order that the resulting values of a, ^, r, may belong to the equa- tion of the osculating circle. Now dy _ X — a dy- _ 1 (^ — o-Y r^ , di f^ij^'t? ^^^ W—^f" iy — ^r hence the three equations for determining a, (3 and r, are (X — a)+l>'(2/ — /3)=0..,.(2), From the second equation {x-o,r = p'^y-^r. Substituting this in the first, ip' + 1) (y ~ I^T = r"' Adding this last to the third, there results y-^ = -'-^' ? which, substituted in (2), gives X — a — p" ■ Consequently, a = x . p" + 1 y+ p" da^' r : P"- These equations completely determine the osculating circle, when- ever the co-ordinates x, y of the proposed point £ire given. Should this point be such as to render p = 0, then the expression, for the radius of curvature at that point, becomes 182 THK DIFPERENTIAL CALCULTTS. p" w dor' But when p' = 0, the tangent at the proposed point must be pa- rallel to the axis of x (78), or, which is the same thing, the axis of y must coincide with the normal ; hence, under this arrangement of the axes, x = at the proposed point, and therefore Should p" = at the proposed point, r will be infinite, whether jj' = 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 "y^ will lie on contrary sides of it, as in the annexed diagram (89), that is, supposing p'" is neither nor CO . Ifp'" = 0, and the next following differential coefficient nei- ther nor C30 , the contact will be of an order which is unaccompanied by intersection. 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 inquired into, when we come to speak of the singular points of curves. (93.) By referring to equation (2) above, which has place even when the contact is but of the first order, we learn that the centre (a, |8) 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 cur- vature, to a few particular cases. EXAMPLES. (94.) 1. To determine the radius of curvature, at any point in a parabola. THE DIFFERENTIAL CAIiCULUg. 133^ Differentiating the equation of the curve, y^ = 4mx, we have, 2yp = 4m .•. p = — 2yp"-{-2p- = 0.'.p" = ^ = -^ = r — (See Anal. Geom.) 4nr As the expression for the normal dimini^es with x, the vertex is the point of greatest curvature, r being there equal to 2m, or to half the parameter. 2. To determine the radius of curvature at any point in an el- lipse. By differentiating the equation ay + 6V = oFl^, we have a)fp' + b^x = .'. p' = ahjp" + ay2_j. 5a = o.-.p „ _ 6' + ay a^y ay ''•*" p" ay * 6^ a'b* " ^^' From this expression, others occasionally useful may be readily derived. Thus, since {Anal. Geom.) the square of the normal, N, is b* — - ar" + «^, therefore. « a*N^ = 6V + aV .-. r = ^ = -^ N^ . . . (2). Again, since {Anal. Geom.), aN = 66' .-. r = ^ . . . . (3). 134 THE DIFFERENTIAL CALCULUS. At the vertex r = — = semiparameter {Anal. Geom.) 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 *• as a function of X, the angle inclu- ded between the normal and the transverse axis. For this purpose we have since ar^ = o^ (1 _ |!.) and f = TS^ sin.=^ a^ b^ ' .•.NM1-(1— ^)sin.^|=-^ but {Anal. Geom.) .■.N=-^ I a (1 — e^ sin.2X)2 = ^ N3 ^ ^1^ a (1 — e^) a (1 — e^ sin. ^X) ^ ( 1 — e^ sin. ^X) ^ (95.) Since, in the ellipse, the principal transverse is the longest diameter, and its conjugate the shortest {Anal. Geom.), it follows from (3), that the curvature - is greatest at the vertex of the trans- verse, and least at the vertex of the conjugate axis. At the former fe2 a^ point r = -T , and at the latter r = -j-. The present is a very important problem, being intimately con- nected with inquiries 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, I, for these lengths may, without error, be considered to coincide with the osculating circles through their middle points ; and since THE DIFFERB9rTIA.L CALCULUS. 135 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, but g (1 - e') _ «(l-e^) R = ^^ '. — - and r = -^ (1 — e2sin.2L)2 (1 — e==sin.2/)2 therefore, since r»R = Mr, we have m M (1— e^sin.^L)^ (1— e^sin.^/)^ or m\(l — — y sin. &))''+ (y COS. w — a) (p„ COS. w — 2/?,sin.ft» — ycos.w) + {p, sin. 0) + y COS. &))2-j- (y sin. to — 0) (p„ sin. a + 2/), cos. lo — y sin. to) = ... (3) If from the two latter equations we determine the values of y sin. w — (3 and y cos. w — a, and substitute them in (1), we shall obtain the following expression for r in functions of y and its differential coefficients, viz. _Jf + p^ .... (4) ■f + ^f'-m, But we shall arrive at this expression more readily by first deducing from the equations «• y = Y sin. u, x = y cos. w the differential coefficients —^ = y COS. u -{-p, sin. w = (dy) -T— = — y sin. w + p^ cos. w = (da?) -T-^ = — y sin. w + 2p^ cos. w + p^^ sin, w = (cFi/) cPx -5-5- = — y cos. w — 2p^ sin. u + p^^ cos. w = (d^a?) and then substituting them in the general formula (1). Since (80) the expression for the normal PN is N - y^ + p^K we may put the above expression for r under the form _ W ''""^M^P? — y/?, ■ * ■ ■ ^^^' 5. To determine the radius of curvature at any point in the loga- rithmic spiral dry a y du m m dPy _ y 'd^~~^ ~ ^"' 140 THE DIFPBRKNTIAL CALCULUS. Hence (y^ _1_ „ 2^2 1 I 1 I I y v/ 1 -I- (art. 80) = y cosec P. tan.- 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. w ± 1 3 (5 ± 4 cos. u)- .*. r = • 9 ± 6 COS. cj CHAPTER III. ON INVOLUTES, EVOLUTES, AND CONSECUTIVE CURVES. (100.) If osculating circles be applied to cwr?/ point in a curve, the locus of their centres is called the evoliUe 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. 131, containing the variable coordinates a, /3 of the centre. As these three equations must exist simultaneously for every point of contact [x, y), the two quantities x, xj may be eliminated, and therefore, a resulting equation obtained containing only a and j8, which equation therefore will express the general relation between a and jS for every point {x, y) ; in other words, it will represent the locus of the centres of the osculating circles. Or, representing the equation of the proposed curve by t^ = Fx, we shall have to eliminate x and y from the equations (p. 131) THE DIFFERENTIAL CALCULUS. 141 y = F^» when the resulting equation in a, /3 will be that of the evolute. EXAMPLES. (101.) To determine the evolute of the common parabola f = 4mx.'.p' = — .'.p" = — — .'. 1 -{- p^ - y^ + ^"^' = 1 + ^,^ = _i * * '^ y^ x^ p'' 2m . y^ , « , « a — 2m .*. a = ar + ■—- + 2m = 3a; + 2m .-. x = ^ = y 2m 7/=* _—f_ — 2:r^ ^ _ wijS^ ^ Am^~y y^ T~ ''' ^ ~ ~^ ... -f 111,2 .•./3^ 27 (a — 2m)^ 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 be- comes The locus of which is called the semicubical parabola. It passes through the origin because ^ = when a = ; 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- sists of two branches symmetrically situated with respect to the axis of a? or of a, and lies entirely to the right of the origin, for every posi- tive value of a gives two equal and opposite values of (3, and for negative values of a, (3 is impossible. It is easy to see, there- fore, that the form of the curve is that represented in the margin. 2. To determine the evolute of the ellipse. By example 2, page 133, we have 142 THE DIFFERENTIAL CALCULUS. b'x „ b* P — 2-' P = —3 Now, since, by the equation of the curve, .'. a'f + b'x' = 6" {a' — c'x') or = a" {¥ + c^)' c^ being put for a^ — 6^. Hence, by substitution, Substituting these values in the equation of the involute, we have c c a^b^ or, finally, dividing all the terms by — , we obtain for the evolute the A' C3 equation If a = 0, then (3 = ± -j-, so that the curve meets 6 the axis of y in two points, c, rf, equidistant from the origin 0. If /3 = 0, then a = ± — , so that a it also meets the axis of a; in two points, 6, a, equi- distant from 0. If a is numerically greater than — theordinatesbe- come imaginary, and if ^ is numerically greater than -r-the abscissa becomes imaginary ; therefore the curve is limited by the four points a, 6, c, d, and touches the axes at those points. It consists, there- fore, of four breinches symmetrically situated as in the figure. THE DIFFERENTIAL CALCULUS. 143 3. To detemiine the evolute of the rectangular hyperbola, its equation between the asymptotes being ocy = a^. The equation of the evolute is 2 2. 2 flS (a + /3)3_(a_/3)^ =-. 43 THEOREM. (102.) Normals to the curve are tangents to the evolute. Let the equations of the curve and of its evolute be y = Fx and (3 = fa, then differentiating the equation (2) p. 131, considering a, ^ as va- riables as well as x, y, we have ,_| + p..(,_« + ,-=_p.f = 0, but (130) Hence, by substitution, ^+p'jf = ax ax dB da d^ 1 — y, .'. -f- — ^ or -J- = - = ' ^ fequa.2,p. 131). dx, dx da p a — x -^ ' dB Now -T— expresses the trigonometrical tangent of the angle be- tween the axis of x and a linear tangent through any point (a, j8) of the evolute, and ;- expresses the trigonometrical tangent of the angle between the axis of x and a normal at any point {x, y) of the involute ; but this normal necessarily passes through a point (a, (3) of the evolute, and, therefore, in consequence of the above equality, it must coincide with thq tangent at that point. THEOREM. (103.) The difference of any two radii of curvature is equal to the arc of the evolute comprehended between them. Differentiating the equation THE DIFFERENTIAL CALCULUS. on the hjrpothesis that a is the independent variable, we have but by last article y-f3 = ia:-a)^ and _(._„K^ +„=.*...(.). Dividing (2) by the square root of (1) we have that is (86) da? da ds dr ■ — =-j- .'. — s =r ± a constant, da, da. for otherwise there could not be — = -r-. da da Hence if r, v 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' 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 direc- tion 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 (102) 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. THE DIFPERKNTIAL CALCULUS. 146 On Consecutive Lines and Curves. (104.) Every equation between two variables aiay 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, these constants are fixed and determinate. If, however, the equation con- tains an arbitrary or indeterminate constant a, then, by assuming dif- ferent values for a 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 = a', another, intersecting this in some point (x', y'), may be determined from a new condition a = a' + ^ ; and if ^ 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 {x, 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 mathe- maticians call consecutive curves, are, in reality, coincident curves, and the point which has been denominated their point of intersection may be determined as follows : (105.) Let F{x,y,x') = (1) represent any plane curve, x being a parameter, and for any inter- secting curve of the same family let x' 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, we have, by Taylor's theorem, F(x,j/,x'-i-;i)=F(:r,i/,a:')+ ^^^ /^ + 'PF{x, y, x') h? ■■ . —17^ — r:^ + ^^- but F (x, y, x) = 0, therefore 19 146 THE DIFFERENTIAL CALCULUS. F (.r, y, X' + h) __ (IF (t, y, x') drF (y, y, x') h ^^ h ~ dx' dx'^ 1-2 hence, when the curves are consecutive, that is when /t = 0, we have the following conditions, viz. F {x, y, x') = \ dF (^, y, X') ^ ^ J .... (2) dx * to determine x 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 y' = Fx\ and any point in the normal by (ar , y), we have for the equation of the normal y — y' = — -r{^—x')oT {y — y')p' + x — x' = 0. This corresponds to the first of equations (2), x' being the parameter ; hence, differentiating with respect to x of which y' is a function given by the equation of the curve, we have {y-y')p"-p"--^=o p" + 1 ••• !/ = !/' + P" P" hence (92) consecutive normals intersect at the centre of curvature. (106.) If we eliminate the variable parameter x' by means of the equations (2), the resulting equation will belong to every point of m- tersection given by every curve of the family F{x,y,x,x') =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 all the intersections, and we may show that at these same inter- sections this locus touches every individual curve in the family. The equation (1), where x' represents a function of x, i/, determined by THE DIFFERENTIAL CALCULUS. 147 the second of the conditions (2) in last article, is obviously the equa- tion of the locus of which we are speaking, and the same equation, when «' takes all possible values from to ± go, 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), x' being constant, we have , du . , du . du = -T- dx + -T- ay = 0. dx dij 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 _ but by the condition just referred to -j-^ = at the point where the curves whose equations we have just differentiated meet; hence, since at those points each of these equations give the same value for -^, it follows that they have contact of the first order; we infer, dx 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 the entire family of curves represented by equation (1), 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. (1 07. ) 1 . To determine the curve which touches an infinite series of equal circles, whose centres are all situated on the same circum- ference. Let the equation of the fixed circle be ^2 ^ y>2 ^ ^'2^ then, for the coordinates of the centre of any of the variable circles, the expressions will be x and \/r"^ — x'^. 148 THE DIFFERENTIAL CALCULUS. SO that the general representation of these circles will be (x _ x'f + (y— Vr'-' — x'y—r" == = « (1), x' being considered as an arbitrary constant. If, however, x' be con- sidered not as an arbitrary constant, but as a function of x and y, du fulfilling the condition -y-, = 0, then, by the preceding theory, (1) tix will represent the curve which touches all the circles in those points where each is intersected by its consecutive circle. Hence, differ- entiating (1) with respect to x, we have -^, = — {x — x)-\- — - — x=^ ax ^ Vr — X' ... _ X Vr'^ — x'^'+x'y = '/x ''' ""' ^ N/a^ + / This, then, is the function ofx, y, which, substituted for x, in (1), gives the equation of the locus sought. The result of this substitu- tion is x^ + y^ — 2r' Va;' + y" + r" = r", or, extracting the root of each side. >/a^ + y^ = r' zL r .-. x^ + y^ — {r ± ry, an equation representing two circles, whose radii are respectively r' + r 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 6, and, taking its sides for axes, we have, for the equation of every variable line, T/ = ax + /3 . . . . (1), and, putting successively y = and a? = 0, the resulting expressions for X and y denote the sides of the variable triangle, including the -4 1, -t^— \— A-^W" '■ THE DIFFBRKNTIAL CALGVLVS. 149 given emgle, so that these sides are and [3; hence, calling the constant surface s, we have /S^ . /S^sin.^ s = sm. e .'. a, = ; 2a 2s hence the equation (1) is the same as y = — "^-^^ x-\-^ . . . . (2), where (3 is considered as an arbitrary constant. But if for this arbi- trary constant we substitute the function of a^, arising from the condition dy Tg- = 0, then (2) will represent the locus of the intersections of each variable line, with its consecutive line, which locus touches them all. Differentiating them with regard to ^, we have {3 sin. 6 , ^ s — x+ 1 =0.-. ^= ^— , s X sm. & this substituted in (2) gives^for the equation of the sought curve or rather _ 8 hence the curve 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 diameter that the diameter makes with the tan- gent : required the hne touching them all? Am. They are touched by a point, viz. the focus, in which therefore they all meet. 150 THE DIFFERENTIAL CALCULUS. CEAFTER IV. ON THE SINGULAR POINTS OF CURVES, AND ON CURVILINEAR ASYMPTOTES. jyitiltiple Points. (108.) 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 in- quire how, by means of the equation of any curve, these points, if any, may be detected. JVfultiple points of the first species. When the curve has multiple points of the first species, we readily arrive at the means of determin- ing their position from the consideratioa that at such points there must be as many rectilinear tangents as there are touching branches, and, dii consequently, as many values for ^, the tangent of the inclination ax of any tangent through the point {x, y) to the axis of a- ; so that the equation 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 dx dy O' so that no systems of values for x and y can belong to multiple points of the fiirst species, but such as satisfy the conditions dx dy 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 ascertamed by proceeding as in (41), and those systems only will belong to multiple points of the THE DIFFERENTIAL CALCULUS. 151 first species that give multiple values to p'. Let us apply this to an example or two. EXAMPLES. (109.) 1. To determine whether the curve represented by the equation ay^ — ar'y — bsP = 0, has any intersecting branches At the points where branches intersect we must have 3^" {y + b) = 0, 'Saf — 0^ = .'. x = 0,y — X = y/ 3ab% y = — 6 ; this second system of coordinates do not satisfy the proposed equa- tion, and therefore do not mark any point in the curve ; the first sys- tem, which is admissible, shows that if there exist any multiple point it must be at the origin. Hence, to ascertain the true value of p' at this point, we have, by differentiating both numerator and denomina- tor in the expression __ 6x (y + 6) + Sx'p' _ "~ •- 6ayp' — 3ar -" ~ _ My + 6) + 12p'x -\- 3x^p" _ 66 n=,y^ L 6ayp" + 6ap'2 — 6x ^ 6a[ p'f"^^^ ^ a' therefore, as this has but one real value, the curve has no intersecting branches. 2. To determine whether the curve represented by the equation x'* + 2a2fyf — ay^ = has intersecting branches ^ = 4x{x^+ay) = 0,~ = a {2x^-3^) = 0. I 152 THE DIFFERENTIAL CALCULUS. There is but one system of values that can satisfy these three equa- tions, viz. so that if there are intersecting branches they must intersect at the origin. To determine, therefore, whether at this point p' has multiple values we have [Pl 4x {jr + ay) -, _ _ Gr" + 2ay + 2 axp' _ '- 3ayp' — 2ax / ~0 - 4a [p'] 3a[p'Y — 2a .: 3a [p'Y — 6a [pq "= ... [p'] = or [/] = ± V 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 x, and the tangents to the other two are symmetrically situated with respect to the axis of ?/, since they are inclined to the axis of X, at angles whose tangents are -\- V 2 and — y/ 2. (110.) Should the values ofp' corresponding to any values of ar 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 h, the development of the ordinate will agree with Taylor's develop- ment, as far, at least, as the second term for all values of h, between some finite value and 0, it follows that all the corresponding ordi- nates between these limits must be imaginary, so that the proposed point is isolated, having no geometrical connexion with the curve, although its coordinates satisfy the equation. Such a point is called a conjugate point. (111.) 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 THE DIFFERENTIAL CALCULUS. 153 tipplication of the differential calculus, and independently of all con- siderations about the failing cases of Taylor's theorem, except, in- deed, those connected with the theory of vanishing fractions. We shall now seek the analytical indications of JVLulti'ple Points of the Second Species. (112.) The second 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 equa- tion of the 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 ^, 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 of some finite value, would each have but one value, and, there- fore, different branches could not proceed from the point {x\ if). But we have no means of ascertaining « 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 equation of the curve be e^ y ~ {x — ay V X — 6 + c, we at once see that x = a destroys the radical in y and p', that re-ap- pears in p" ; therefore, at the point corresponding to this abscissa, these will be but one tangent, and yet two branches of the curve proceed from it on account of the double value ofp". Hence the point is a double point of the second spe- cies, the branches have contact of the first order, and, because p' = 0, the common tangent is parallel to the axis of the abscissas ; 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 x a radical is multiplied by the factor {x — a)"*. If ni = 1, the branches of the curve intersect at the point ■whose abscissa is x = a, because then p' at that point takes the same 20 154 THE DIFFERENTIAL CALCULUS. values as the radical, but if m > 1 then the branches touch, because then the radical is destroyed in p' for .t = a ; in both cases the index of the radical will denote the number of branches which meet ui the point. Such, therefore, are the geometrical significations of the cases discussed in (75) and (76). Ctisps, or Points of Regression. (113.) A cusp or 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 of the common tangent, is called a cusp of the first kind, and that in the second figure, where the branches are both on one side, a cusp of the second kind. (114.) 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 poini, then the ordinates in the vicinity, corresponding both to x + h and to a; — h, would be both possible or impossible at the same time. We are not, however, to infer that when the adjacent ordinates are real on the one side of any point, and on the other side imaginary, that a cusp necessarily exists at that point, for it is plain that the same analytical indications are furnished 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 parti- cular points of curves to be able to distinguish the point which limits the curve in the direction of the axes from cusps. (115.) Now at the limits, the tangents to the curve are parallel to the axes, the limits are therefore determined by the equations -^ = ao ax and ^ — ^1 ^^^ *h®y fulfil, moreover, the following additional con- THE DIFFERENTIAL CALCULUS. 155 ditions, viz. 1°, the ordinate or abscissa, whichever it ^—^ may be, that is parallel to the tangent, immediately be- yond the limit, must be imaginary ; but if it be ascertained 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 i^//' this, viz. that immediately ivithin 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 discovered by seeking among the values which satisfy the equations -j^ = and -^^ =00 , those which (XJ[/ (XX do not fulfil both the foregoing conditions. Let us illustrate this by examples. EXAMPLES. (116.) 1. To determine whether the curve whose equation is {y — bf={x-aY has a cusp at the point where the tangent is parallel to the axis of?/. By differentiating dy _ 2 X — a 'dx~2,' {y —b y this becomes infinite for y = b, therefore the point to be examined is (a, 6). In order to this, substitute a ±: h for x, in the proposed equa- tion, and we have, for the contiguous ordinates, y = b± h^, which is not imaginary either for + ^ or — A ; 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 156 THE DIPFERENTIAL CALCULtlS. tj — a = {x — b)^-\-{x— b)* has a cusp at the point where the tangent is parallel to the axis of if. Here the coefficient -~ becomes infinite for x = b, therefore the ax point to be examined is (6, a). Substituting 6 + ft for x, we have y = a + h^ -\- h*. 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 Hmit is fulfilled. To the right of this or- dinate the two values of y, corresponding to a value of ft ever so small, are both greater than y = a, so that the condition 2° is not fulfilled, the point (6, 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-xy = {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 (6, a + 6). Substituting b -\- h for a:, 2/ = (a + 6) + ft* + ft, ^ negative values of ft render this imaginary, therefore the *\^ condition 1° is .fulfilled; positive values give two values 3, for y, and as ft may be taken so small that ft* may exceed 3 ft, and since, moreover, the two values of ft* are the one positive and the other negative, it follows that the real ordinate contiguous to the point has one value greater, and 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, therefore, lies to the right of the ordinate, through x = b. (117.) 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 form y'^'^h^ Aft"+Bft^ + &c. ax THE DIFFERENTIAL CALCULUS. 157 and the corresponding ordinate of the tangent will be hence, subtracting this from the former, we have A = A/i" + /3/i^ + &c. (118.) Now in order that the pouxt (x', y') may be a cusp, this dif- ference 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 h may be so small that A^ may exceed the sum of all the following terms, h must have two opposite values ; hence, a must be a fraction with an even denominator, and, conversely, if a be a fraction with an even denominator, the point {x', y') will be a cusp of the first kuid. Hence, at such a point, -7^ is either or oo : if /3 >2, and 00 if^ < 2. (119.) In order that the cusp may be of the second kind, both values of A must have the same sign there, for A cannot admit of opposite values of the same value of h, consequently a must in this case be either a whole number, or else a fraction with an odd denomi- nator ; and conversely, if a be either a whole number, or a fraction with an odd denominator, the point (x, 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. (120.) 4. To determine whether the curve whose equation is y =^ X ± x^ has a cusp. Here y is possible for positive values of x, and imaginary for all negative values ; hence there may he a cusp at the origin. To as- certain this, put h for x, in the equation, and we have, for the con- tiguous ordmate, the value y = h ± h^. I 158 THE DIFFERENTIAL CALCULUS. /^X The coefficient of /i being 1 = -j-, we see that the tan- gent 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 first kind. 6. To determine whether the curve whose equation is 5 y — a = a? + hi? + cx^ has a cusp. Here ?/ is imaginary for all negative values of .r, therefore the point (0, a) nmy be a cusp. Substituting h for x, we have t/ = o + /i + i/t' + cli^. 1^/ As before, the tangent is incUned at 45° to the axes, and, since the exponent of the third term is a whole num- ber, and the whole expression admits of two values, in consequence of the even root \i^ , it follows that the proposed point is a cusp of the second kind. The branches are situated to the right of the axis of ?/, because h must be positive, and they are above the tangent because h]^ is positive. 6. To determine whether the curve whose equation is (2y + .T + \f= 2(1— x)** has a cusp. Here values of x greater than 1 are obviously inadmissible, and to this value of x corresponds ?/ = — 1 ; hence the point having these coordinates may be a cusp. Substituting 1 + fe for x, we have therefore the tangent to the curve at the proposed point has the tri- gonometrical tangent of its inclination to the axis of x equal to \ , and since the fraction f has an even denominator, the point is a cusp of the first kind. Because /i is negative, the branches are to the left of the ordinate to the point which \\ is below the axis of x, because this ordinate is negative. Points of Inflexion. (121.) Points of inflexion have been defined at (92), and we have THE DIFFERENTIAL CALCULUS. 159 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 nih, provided the nth be odd and becomes neither nor go . The simplest indication therefore of a point of inflexion is [-rj] — 0, and [ -fj^] neither nor co ; such indications, however, cannot be fur- nished by any point at which the tangent is parallel to the axis oft/, since in this case [y^] and all the following coefficients become infi- nite. 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 diftcrential coefficients beyond the first. The only general method of doing this is that which we have already em- ployed for the discovery of cusps, and which consists simply in ex- amining 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 characteristics common to both, and to the limiting points of curves as already hinted at in (114). But the characteristic property of a point of inflection is, that the adjacent ordinates on each side are the one greater and the other less than the ordinate at the point. This pecuharity distinguishes a point of inflexion from a limit, inasmuch as at a limit the ordinate immediately beyond is imaginary ; and it distinguishes it from a cusp of the first kind, in- asmuch 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 ex- ist, or to find what points are given by the conditions ^ = Q =Oorcc, or, which is the same thing, what points are given by the separate conditions. 160 THE DIFFERENTIAL CALCULUS. P = 0, Q = 0, we are then, by examining the course of the curve in the vicinity of each point, to determine to which of them really belongs the charac- teristic of an inflexion. Thus the means of distinguishing points of inflexion being sufli- ciently clear, we shall proceed to a few examples. EXAMPLES. 1. To determine whether the curve whose equatio n y = b + {x — ay has a point of inflexion where the tangent is parallel to the axis of a:. Here p' =zB(^x — ay, and when the tangent is parallel to the axis oi" x, p' = 0, .: x = a and 2/ = 6, at the proposed point. In the vicinity a; = a + A, ... ij = b + /i^ 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 y'^ = x^ or 5. y = x^ has an inflexion at any point. 2 1. 3 * » i^ 3*3* this becomes co for x = 0, therefore a point of inflexion may exist at the origin. Putting h for x we have y = h^, 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 ori- gin. Also the equation of the tangent being ?/ = -I x^, the ordinates corresponding to x = ± h are both less than those given by the above equa- tion ; 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 J THE DIFFERENTIAL CALCULUS. 161 y — X = (x — a)^ has a point of inflexion p' = i + f (a: — «)^p"=l•f (^ — «)"' this becomes infinite for x = a, therefore a point of inflexion nwy exist at the point (a, a). In the vicinity of this point x = a + ^i .5. .♦. 1/ = « + /i + h^t 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 ± h, one, viz. y = a -^ h, is less than that of the curve, and the other greater ; hence the curve bends, as in the figure. On Curvilinear Asymptotes. (122.) 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 dis- tance.* Hence, since the expression for the difference of the ordinates cor- responding to the same abscissa in two such curves becomes less as the abscissa becomes greater, and finally becomes 0, when the abscissa becomes co , it follows that that expression can contain none but negative powers of x, without the addition of any con- stant quantity. For, if a positive power of x entered the expres- sion for the difference, that expression would become not but oo , when X = CO ; and, if there were a quantity independent of x, the dif- ference 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 A = A'x~" + B'x"^ + C'x"y + &c (1), or when the general expression for the difference of the abscissas cor- * Spirals meet their asymptotic circles only after an infinite number of revolu- tions ; these we do not consider here, having examined them at (85). 21 162 THE DIFFERENTIAL CALCULUS. responding to the same ordinate is A = A'y~"' -\- B'y"^ + C'y~^ + &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 sup- posed to give the difference (1) there be substituted another, which would reduce that difference to B'x~l^ + C'x~'y + &c. this new curve would be an asymptote to both, and would obviously, 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 dif- ference (1) to c'x~y + &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 em expression of the form y = Aa^ + Bx* + . . . . A'x~"- + B'x~^ + &c (3). admits of an infinite number of asymptotes. Since the general expression for the ordinate of a straight line is y = 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 y = Ax + B+ A'x'"- + B'x~^ + &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=±^{jp — Ay = zp -? a: q= i ABx"' + &c. 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 c^}ied parabolic curves. THE DIFFERENTIAL CALCULUS. i6d The common hyperbola we see by the above equation admits of TJ the two rectilinear asymptotes y = ± j- x, and of an infinite num- ber of hyperbolic asymptotes. As an example of this method of discovering rectilinear and cur- viUnear asymptotes, let the equation my'^ — xy^ = mx^ be proposed. The development of y in a series of descending powers ofx is (Ex. 9, p. 62,) y = — m r — &c. •^ x^ therefore the curve has one rectilinear asymptote, parallel to the axis of X, its equation being y = — wi ; the hyperbolic asymptote next to this, and which lies closer to the curve, is of the fourth order, its equation being yx"^ + mx"^ + m'' = 0. Again, let the equation of the proposed curve be b y ~ = bx-' + &c (1), also, since b^ b^ x^ -> a^ = — .-. .r = a + J- . — «-^ + &c (2). if ^ a ^ From (I) it appears that the curve has a rectilinear asymptote, coin- cident with the axis of x, its equation being ?/ = ; the hyperbola whose asymptotes coincide with the axes is also an asymptote, its equation being xy = 6. From (2) it appears that the curve has another rectilinear 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 radical, in the proposed equation, to admit of either a positive or a negative value, then there will be two rectilinear asymp- totes, parallel to the axis of y and equidistant from it, as also two hy- perbolic asymptotes, symmetrically situated between the axes. 164 THE DIFFERENTIAL CALCULUS, SECTION III. ON THE GENERAL THEORY OF CURVE SURFACES AND OF CURVES OF DOUBLE CURVATURE. CnilFTEH I. ON TANGENT AND NORMAL PLANES. PROBLEM I. (123.) To determine the equation of the tangent plane at any point on a curve surface. Let (x'j y\ z',) represent any point on a curve surface of which the equation is z = 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 ofxz, zy ; their equations are z — z' = a{x — x') \ ^ij^ } (2), y = y and z — z' = b{y — y') X = x' But since these are tangents to the plane curves, which are the sec- tions through {x', y', z\) parallel to the planes ofxz,zy, therefore (77) dz , , dz' Moreover the traces of the plane through the lines (1), (2), upon the planes of xz, zy, 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 THE DIFFERENTIAL CALCULUS. 165 same in the plane as in its traces, it follows that the equation of this plane must be z-2'=p'(x—x') + q'{y-y') .... (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 a? we have, by putting s = 0, in (3), P' tan. mc. ^. (124.) If the equation of the surface is given under the form u = F{x,y,z,)=0 . . . . (4), then the expressions for the total differential coefficients derived from tt, considered as a function, first of the single variable x, and then of the single variable y, are (57) .du^ du .du , _ f. dx dx dz ,du _ du .du , _ - ^dy^ dy dz^ from which we get the values du du , dx , _ dy du du dz dz hence, by substituting these expressions in (3), the equation for the tangent plane becomes /v <^M , , ,. du , , ,. du ^ ,^. (.-.') 5^+ (.-.■) 3J+ fa-*') ^ = . . . . (5). PROBLEM II. (125.) 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, perpendicular to the plane (3), and passing through the point of con- tact (ar', y\ z'.) ^f'' + q" -q' + 1 Vf' + 9" 1 + 1 16B THE DIFFERENTIAL CALCULUS. New 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 — y' + q'{^ — ^') = o] which together, therefore, represent the normal. (126.) If we represent by a, /3, y, the inclinations of this line to the axes of x, y, s, respectively, then {Anal. Geom.) cos. a COS. ^ = COS. y ^^,2 + g'2 4- 1 (127.) If the equation of the surface be given under the form (4), last problem, then, in these expressions for the inclinations, we must, instead of p' and q', write their values as before determined from that equation. If, for brevity, we put _ 1 the expressions for the cosines will then be du - du du cos. a = V -r-, COS. p = 1) -r-, COS. y = t) -j-. dx dy dz As every plane which contains the normal Une must be perpendicular to the tangent plane, it is obvious that there exists an infinite number of normal planes to any point of a surface. PKOBLEM III. (128.) To determine the equation of the tangent line to any point of a curve of double curvature. We have already indicated {Anal. Geom.) how this equation is to be determined : Let THE DIFFERENTIAL CALCULUS. 167 y=fx,Z = Fx .... (1) be the equations of the projections of the proposed curve, on the planes o£xy, xz, and let (»', y', z',) be the point to which the linear tangent is to be drawn, which point will be projected into {x, y') emd {x', z',) on the plane curves (1), therefore tangents through them to these plane curves will be represented by the two equations these, therefore, together represent the required tangent in space. PROBLEM IV. (129.) To determine the equation of the normal plane at any point in a curve of double curvature. The equation of any plane passing through a proposed point is {Anal. Geom.) A{x — x')-\-B{y-y') + C{z — z')=0 .... (1), and for the traces of this plane on the planes of xy, xz, we have, by putting in succession z = 0, y = 0, the equations A C z — z' = —-^{x — x)-)r-^y', but since these two traces are respectively perpendicular to those marked (2), last problem, B , C _ , hence the equation (1) becomes x — x'+p' {y _ 7/') + 5' (5 — 2') = . . . . (2), which represents the normal plane sought. 168 THE DIFFERENTIAL CALCULUS^ GBAPTER ZX. ON CYLINDRICAL SURFACES, CONICAL SURFACES, AND SURFACES OF REVOLUTION. (130.) These surfaces have been considered in the Analytical Geometry, and the general equations of the two first classes have been deduced, on the hypothesis that the directrix is always a plane curve. We shall now suppose the directrix to be any curve situated in space, and investigate the differential equations of these surfaces, as also ©f surfaces of revolution in general. Conical and Cylindrical Surfaces. PROBLEM I. To determine the equation of cylindrical surfaces in general. Let the equations of the generating straight line be "~ X = az -]- a \ ( a = X — az ,-v y = bz+ ^] •*• \^=y — bz'''' ^*^' and the equations of any curve in space considered as the directrix, F{x,y,z)=0,f{x,y,z)=0 .... (2). Now for every point in this directrix, all these equations exist simultaneously ; moreover, the constants a, 6, are fixed, since the inclination of the generating line does not vary, but the constants a, ^, are not fixed, since the position of the generating lines does vary. If, then, we eUminate x, y, z, from the above equations, there will enter, in the resulting equation, only the constants a, 6, and the indeterminates a, /3, hence, solving this equation, for /3 we shall get a result of the form (3 = (pa; consequently, if we now substitute in this the values of a and /3 given above, in terms of x, y, z, we shall have this general relation among these variables, viz. y — bz = cp (x — az) = . . . . (3), which is the equation of cylindrical surfaces in general, the function

' * ' ^ ■'* R= — =C These are called the principal radii of curvature at the proposed point, and the sections themselves the principal sections through that point. (167.) If we know the principal radii and the inclination

0, and contrary signs if r'i' — s'^ < ; this last condition, therefore, exists in the case just considered. (162.) 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 ar, 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.) r and R being the semi-parameters of the sections of the paraboloid on the planes of xz, yz. Now the equation of a normal section of this paraboloid, by a plane whose inclination to that of X2 is . . • . (A). dx" dy' h^ J 188 THE DIFPERENTIAL CALCULUS. The coordinates {x, y, z,) of the intersection of the proposed nonnals will be obtained by the combination of the four equations (1) and (3) in terms of x\ y' z', which are fixed, and of the increments k, h. But from four equations three unknowns may be always ehminated, and the result of this elimination will be an equation between the other quantities ; hence then there exists a constant relation between the increments k, h, when the normals intersect, these increments are therefore dependent; consequently the y, x, of which these are the increments, must be dependent ;* therefore when the normals are consecutive, that is, when ^ = 0, the equations (3) become dA dA dy' _ dx' dy' ' dx' i r„,v dB dB_ 4^ dxf dy' ' dx' :\- or, by substituting for A and B their values (1), 1 + P' ( P' + 9' ■^) + (=' - ») Cr' + »■ -^j = . . . . (4), from which, eliminating zf — 2, we have the following equation for • . dy' determmmg -^7 ((1 + q'^) S' -p'q't') ^ + ((1 + q") r'-il+ r)t') g- - (1 +p'=)s'+p'gV = . . . (6). This being a quadratic equation furnishes two values for — the tangent of the incUnation of the projection of the line of curvature through {x', y', 2'), on the plane of x?/ to the axis of x. Hence, there are two directions in which lines of curvature can be drawn through any proposed point, and if in (6) we substitute for p', q', &c. their general values in functions of x, y, that equation will then be the dif- ferential equation which belongs to the projections of every pair of * If this should appear doubtful to the student, its truth may be shown by re- moving the axes of a;, y, to the proposed point, in which position k, h, 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. THE DIFFERENTIAL CALCULrS. 189 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. (166.) 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 coordi- nate 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 ^ + tzzl ^_i^0 ..(7) do^ ^ s' ' dx " • • • • UJ. therefore, calling the two roots or values of -r^, tan. (p and tan. 9', we have, by the theory of equations, tan. 6 tan. d' = — 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. Moreover, the equation (7), if divided by-^^ = tan.^ 5 becomes identical to equation (6), page 183, 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 coordinate planes 2' = 0, therefore the equation (4) or (6) gives 1 tan. d r' + s tan. a s' + /' tan. & but if the plane of X2 coincide Avith a plane of principal section, it will, as we have just seen, touch the line of curvature, and then ^ = 0, so that 1 1 r t and these are precisely the expressions found at (152), for the two radii of curvature of the principal sections at the proposed point, in ♦ Since tangent f and tangent ^' are the roots of equation, (7), and — 1 is their product, recollecting that tangent X cot. = radius' = 1, whence 0' is the complement of ' The last equation is that of a plane, passing through the point (a, (pa), or centre of the sphere ; it is, moreover, perpendicular to the tan- gent to the curve (1) at this point, for the equation of this tangent is (/3' — (3) = (p'a ((pa' — (pa), and that above is t,_'^+9'^ + p"(y-^)+9"(z-7)=0 (5), mp" + nq" = . . . . (6). All these six conditions, therefore, must exist for the contact at the point {x, y, 2,) in the proposed curve to be of the second order ; and as the equations (1), (2), of the touching curve, contain six disposable constants, viz. a, f3, y, r, m, n, all these conditions may be fulfilled, but no more ; hence, the circle, determined agreeably to these con- ditions,'will touch the proposed curve more intimately than any other, that is, it will be the osculating circle. From equations (4) and (6) we get (/' p" m = , » = ~ , q'p" — p' q" q'p" — p'q" hence, equation (2) becomes x — a + ^-^ — {y — 13) — -^^ — (^z — y)= 0, IP —pq qp —pq or ^-y- ^'^"T^"^" (^-a)+^(y-/3) (7) hence, the three conditions (2), (4), (6), determine the plane of the osculating 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 — o') r ^-"^ M ^y-^ = M—^ _ {m — p')r where M is put for the expression x/ \ {np — mq'f + (w — qf + (m — p'Y\. Substituting these values in (5) we have, for the radius of the oscu- lating circle, (1 -t- p'^ + 9"=) M {n — q')p"—{m—p)q;'' Hence, putting for m and n the values already deduced, and restoring the value of M, we have THE DIFFERENTIAL CALCULUS. 221 {i+P^ + q-'r V \p"^ + g "' + {p'q — qfYY B = v4- (1 + p'^ 4- r) \f—p' {p'p" + q'q ") \ y = - + (!+?' + g'') |g" — q (p'p" + q'q ") \ p"' + q"+ ip'q'-q'p'r (202.) The expression for r may be rendered more general, by- considering the independent variable as arbitrary ; in which case we have (66), „ ^ {dry) {dx) — {(Px) (dtj) „ _ jdFz) (dx) — (fe) (dz) ^ {d^^ ' ^ {dxf • Also (198) {dxY ^^P ^ '^ ' hence, making these substitutions in the above expression, we have ((?5)» *"" V \ (dx) {dh))-(ily)(a^)\'+(dz)(a-^x)-{dx) {a^z)\'+{dy) («^2)- (rfs) (ct^y) f | (203.) If it were required to determine the circle having contact of the first order, merely with the proposed curve, only the conditions (1), (2), (3), (4), must be satisfied; the conditions (2),. (4), deter- mine the plane of this circle, that is the tangent plane, but as the condition (4) leaves one of the constants m, n, arbitrary, the tangent plane is not fixed, but may take an infinite variety of positions ; but as it must necessarily pass through the linear tangent, which is fixed, it follows that a plane through this, and revolving round it, is a tan- gent plane in every position, in one of which it touches the curve with a contact of the second order, and thus becomes the osculating plane. (204.) There is another method of determining the equation of the osculating plane, very generally employed by French authors ; they consider a curve of double curvature to have, at every point, two consecutive elements, or infinitely small contiguous arcs in the same plane, but not more, the plane of these elements being the osculating 222 THE DIFFEREJfTIAL CALCULUS. plane at the point. The process, then, is to assume the equation of a plane through the point x — x'-\-m{y—y')-^n{z — z') = 0.... (1), and to subject it to the condition of passing also through the points {x + dx', y' + dy', z' + dz% and X + Idx' + d\v, y' + 2%' + dhj , z' + 2dz' + d^z'. 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 + dx, y + 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 a?' + A x', where Ax' means the increment of x, then substituting x -\- A x' for x', the equation (1) becomes X — x ■\- m {tj — y) + n {z — z') — (Aa?' + inAy + wAs') = . . (2), which, in virtue of (1), is the same as Ax' -j- mAy' + nAz' = 0, or Ay' . Az Suppose now that these two points merge into one, that is, let Ax' = 0, then • > + "'|- + »^ = ----(«); 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' THE' DIFFERENTIAL CALCULUS. 223 hence, dividing by Ax', and supposing this third point to coincide with the former, that is, supposing Ax' = 0, we have the new condition 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. (205.) By whatever process the osculating plane is determined, the radius of the osculating circle may be easily found from consider- ations different from those at (201). 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 intersec- tion of this normal plane, with its consecutive normal plane ; hence, if this line be determined, the combination of its equation with that of the osculating plane, will give the point sought. Now (189) the line of intersection of consecutive normal planes is x — x'-^-p' {y—y')+q' {z — z') = \ p" iy — y') + q" i^ — ^') —p" — q"-i = oi therefore, the centre is to be determined by combining these equa- tions with that of the osculating plane, viz. being precisely the samfe equations as those employed before, for the same purpose. If the origin be at the point, and the tangent be the axis of ar, then x', y', z',p', q', are each ; therefore, the equations of the line of intersection are q" . 1 X = 0,y = —2-z +— , and the equation of the osculating plane p"z-q"y = 0; this, therefore, is perpendicular to the line of intersection. {AncU. Geom.) (206.) The expressions in (201) for the coordinates of the centre 224 THE DIFFERENTIAL CALCULUS. of the osculating circle will become very simple by introducing the substitutions furnished by art. (202); the results of these substitutions will be the independent variable being s. (See JVote D.) PROBLEM II. (207.) 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, corres- ponding to the equations (2) art. (199), 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 {x — ay+ {y—^y- + {z — yy = r' .... (1), and the particular sphere required will be that whose constants are determined from the following differential equations : x — a + p'{y — l3) + q'{z — y) = . . . . (2) P" {y-(3) + q" {z-y) + l-\-p"+q" = (3) p'" (!/ - /3) + q'" (^ - 7) + 3 W + "/3 + q"y = 1, p'"/3 + q"'y = 0, _ q" _ p'" *'• ^ ~ p"q"' —f'q'"'^ ~ q"p"' — q''p" ' TttE DIFFERENTIAL CALCULVll. 225 iience, by substitution in (1), r ^/ p"'a + q'"- p"q"'-9"p"' (208.) 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 developable surface which they generate, and the intersection of any characteristic with the consecutive characteristic will be a point in the edge of regression, corresponding to the given point on the pro- posed curve. Now equation (2) above being that of the normal plane, this point is determined 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 regres- sion of the developable surface generated by the intei sections 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 x, y, z, 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 a, ^, y, which will be the equations of the edge of regression. PROBLEM III. (209.) To determine the points of inflexion in a curve of double curvature. Since a curve of double curvature as its name implies has curva- ture 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 but one projection of the tan- gent crosses 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 teuigent 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 (201) which fix the osculating plane, 29 226 THE DIFFEREKTIAL CALCULUS. we must at a point of simple inflexion have the additional condition arising from differentiating (6), viz. mp'" + nq" = 0. Eliminating — from this and equation (6) vi^e have p"q"' - q"p"% which condition renders the expression for the radius of spherical 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 deter- mine this the curve must be examined in the vicinity of the point. As to points of double inflection,, it is evident from what has been said (121) with respect to plane curves that such points must fulfil the conditions p" = or CO , g" = or CO , and these render the radius r of absolute curvature infinite or 0. Evolutes of Curves of Double Curvature. (210.) In speaking of the evolutes of plane curves we observed (103,) that the evolute of any plane curve was such that if a string * The French mathematicians consider a point of simple inflexion to be that at which three consecutive elements of the curve lie in the same plane. In a recent publication from the university of Cambridge the author has attempted to deduce the above equation of condition, by viewing the point of inflexion after the manner of the French. He has however confounded the consecutive elements of a curve with what the same writers term consecutive points ; moreover, after having estab- lished the conditions necessaiy for the plane z = Aa; + Bj/ + C, passing through one point (r, j/, r,) in the curve, to pass also through two points consecutive to this, viz. the conditions ^=A + B^.|^=A+Bf?i dx ax ' ax ax where y„ z„ belong to one of tlie consecutive points, it is inferred that dx^ dx^' dx^ dx^ an inference which is quite unwarrantable, and which cannot exist unless the plane pass through/owr consecutive points instead of three. THE DIFFERENTIAL CALCULUS. 227 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 con- tinued 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. (211.) 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 this 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 cir- cumference. 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 places of any two are given. (212.) Now, as respects curves of double curvature, we have seen 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 (205) 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 (190) a developable surface. (213.) Suppose now through any point, P, of the curve a normal plane is drawn of indefinite extent, the characteristic or line of polef> 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 228 THE DIFFERENTIAL CALCULUS. 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 correspond- ing 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 normal plane being in every position tangent to the surface through- out 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 de- scribe the curve of double 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 ori- ginally drawn was quite arbitrary, the proposed curve has an infinite number of evohdes 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, (214.) Should the curve which we have all along considered of double curvature be plane, then, indeed, since the characteristics are THE DIFFERENTIAL CALCULUS. 229 ^11 parallel, and perpendicular to the plane of the cun'e, the line PQ once perpendicular will be always perpendicular to the chajacteristic, 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 per- pendicular to the original characteristic, but is inclined to it at an an- gle 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 cyUnder, 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. (215.) 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 sur- face without twisting,* it will trace one of the 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 evo- lute. If the curve be plane, 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. (216.) Having given the equations 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 sur- * This is what I understand Nonge to mean, when he says {Jlpp. de VAncd. de Gdom. p. 348,) "si I'on plie librement sur cette surface le prolongement de cette tangente." It seems not improper to call such lines placed on a developable sur- face developable lines, and those which form curves on the developed surface hoist. ed lines. Of these two species of lines all the former are evolutes, but none of the latter are. 230 THE DIFFERENTIAL CALCULUS. face, the equation of this surface must be common to them all, and we have already seen (194) how the equation of the surface is to be determined, so that it only remains to find for each evolute a particu- lar equation which distinguishes it from all the others, and determines its course on the developable surface. In order to this let us consi- der that each evolute must be such that the prolongation of its tan- gent 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 projection of the evolute ; therefore, considering the plane of xy as that of projection, we have, for the tangent at any point {x', y') in the projected evolute, and, since the same line passes through a point {x, y,) in the project- ed involute, its equation is also Y — y' = yLz:yfx — x') ,.,M. - v'—y . dx' x' — ar ' hence, combining this equation with that of the developable surface, determined agreeably to the process pointed out in article, 194, and eliminating x, y being a given function of x^ we shall have two equa- tions in x\ y', z', of which one will contain partial differential coeffi- cients of the first order, and which together will represent all the evo- lutes. 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 which the differential equation mentioned is deducible ; this primitive equation will involve an arbi- trary 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. THE DIFFERENTIAL CALCULUS. 231 CHAFTER VII. MISCELLANEOUS PROPOSITIONS. PROPOSITION I. (217.) To prove that the locus of all the linear tangents at any point of a curve surface is necessarily a plane. This property we have hitherto assumed ; it may, however, be de- monstrated as follows : Let the equation of any curve surface be « =/(^'2/) • • • • (1)' X and y being the independent variables. Through any given point on this surface let any curve be traced, then, the projection of this curve on the plane of xy will be represented by y = (px . . . . (2), which will equally represent the projecting cylinder ; hence the com- bination of the equations (1), (2), completely determines the curve, and its projection on the plane of a;^ may be found by eliminating y from these equations ; the result of this elimination will be the equa- tion z=f{x,C!)x)=-^x.... (3), therefore, since the linear tangent in space is projected into tangents to these two curves (2), (3), its equations must be — X) \ — x')) where a;', ?/', 2', are the coordinates of the proposed point on the sur- face. Now —7— is the total differential coefficient derived from the ax function s = / (a?, t/), in which y is considered as a function of x given by the equation (2), that is 233 TUB DIFFERBNTIAI. CALCULUS. d^x _ cdz^ __ ^ ^ , d(px ITx te« ~^ ' ^ dT' hence, by substitution, the equations of the tangent in space become' y—y --rfT"^''— "^^i .-.' = ip' + ,''£)ix-x')]""^'^' Now, to obtain the locus of the tangents whatever be the curve through the point {x', tj', z'), we must eliminate the function (par, on which alone the nature of the curve depends. Executing then this elimination by means of the equations (4) and there results for the required locus the equation z—z' =-.p' (^x — x') + q'(y — y'), which is that of a plane. PROPOSITION II. (218.) Given the algebraic equation of a curve surface to deter- mine whether or not the surface has a centre. That point is called the centre which bisects all the chords drawn through it, so that if the equation of the surface is satisfied for any constant values x\ y\ z', it will equally be satisfied for the same val- ues taken negatively, that is, for — x', — y', — z', provided the ori- gin of coordinates be placed at the centre, so that if no point exists for the origin of coordinates, in reference to which the equation f{x,rj,z,) = of the surface remains the same whether the signs of the variables be assumed all + or all — , then we may conclude also that no centre exists. The mode of proceeding, therefore, is to assume the indeterminates x^, J/,, 2 , for the coordinates of the unknown centre, and to transport the origin of the axes to that point by substituting in the equation of the surface x -]- x^,y + y, z + 2 , for x, y, 2. This done we may readily deduce equations of condition which will give the proper val- ues of X, j/^, 2,, 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 the coefficients' THE DIFFERENTIAL. CALCULUS. 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 equa- tion of the surface be of an odd degree, then we must equate to zero the coefficients of all the even powers and combinations of a*, ?/, 2 ; so that only odd powers and combinations may effectively enter the equation, for then whether the variables be all + or all — the function f(x, y, s,) 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 -{- x^,y+ y,? 2: + z^, for x, y, z, in the equation of the surface. For if we conceive these substitutions made in the function /(a?, »/, z), we may consider the result as arising from ^/' !//' ^/» taking the respective increments x, y, z, and we know that every such function may by Taylor's theorem be developed accord- ing to the powers and combinations of the increments, and that the several terms of the development consist each of the partial differen- tial coefficients of the preceding term, the first being/ (a:^, ?/^, zj. Hence, if the coefficients of the first powers of x, y, z, are to be re- spectively zero, then we have to equate to zero each of the partial coefficients derived from u^ = /(^,» y,» «,>) = 0, or, which is the same thing, from u=f{x^ y, z,) = the proposed equation ; if the coeffi- cients of the second powers and combinations of x, y, z, are to be ren- dered each 0, then we shall have to equate to zero each partial coef- ficient derived from again differentiating, and so on. As an illustration of this, let the general equation of surfaces of the second order Ar' + Ay -\- A"z' + 2Byz + 2B'zx + 2B"xi/ ) _ ^ _ ,, + 2Cx + 2C'y + 2C"z + E J -"-«... (1) be proposed, then the degree of the equation being even, the coeffi- cients of the odd powers of the variables in the equation arising from putting X -\r x^, y -\r y^fZ -\- z^, for x, y, z, 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 differen- tial cofficients to 0, that is 30 234 THE DIFFERENTIAL CALCULUS. ^ = Ax + B'z + B"y + C = ^ ^ = A'v + Bz + B"x + C = V . . . (2). dy ^ = M'z + B'z + B« + C" = dz ^ The values of x, y, 2, deduced from these equations are the coordi- nates x^yy^, 2 , of the centre. These values may be represented by _N' _N' _N" ^ - D ' ^' ~ D ' ^' ~ D * where D = AB'' 4- AB"" + A"B"2 — AAA" — 2BB'B", so that the surface has a centre if D is not 0, but if D = and the numerators all finite, the surface has no centre, and, lastly, if D = 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 of the Analytical Geometry. PROPOSITION III. (219.) To determine the equation of the diametral plane in a sur- face of the second order which will be conjugate to a given system of parallel chords. Let the inclinations of the chords to the axes be a, /3, y, then the equations of any one will be x = mz -]r p^y = nz-\- q . . . . (1), where cos. a cos. ^ m = , n = . cos. y cos. y For the points common to this hne and the surface we must combine this equation with equation (1) last proposition, and we shall have a result of the form Rz^ + Ss -f T = . . . . (2), which equation will furnish the two values of 2 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, THE DIFFERENTIAL CALCULUS. [ 285 2 = — ^.•. 2R2==S + . . . . (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, dtt. du dx du dy du dz* dx dz dy dz dz du , du , du = m—-\-n — + -j- = Q,.... (4 , dx dy dz where p and 5, the only quantities which vary with the chord, are eliminated ; hence, this last equation represents the locus of the mid- dle 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, m (Ax + B'z + B"y + C) + n (A'^ + Bs + B"x + C) + A"z + Br/ + Bx + C" = 0, or (Am + B' + B"n) x + (A'n + B + B"m) y + (A" + Bn-I- B'm) z + Cm -f C'n + C" = 0. PROPOSITION IV. (220.) A straight line moves so that three given points in it con- stantly rest on the same three rectangular planes ; required the sur- face 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 x,y, z, and the invaria- ble distances of this point from the three points resting on the planes of yz, xz, and xy, X, Y, Z. The coordinates of these three points will be In the plane of yz, 0, y\ z' xz, x", 0, z" xy,x'",y"'0. X y — y' _ Y Z y -y — y'" X z~z' z Y Z — z" z 236 THE DIFFERENTIAL CALCULUS. 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 . . (1). X Y Z But the part X of the moveable straight line comprised between the generating point {x, y, z,) and the point (0, r/,, z^,), resting on the plane of y, z, has for its length the expression or J2 X^ ^ X'' • • • • ^^' but from the equations (1) !/ — !/' _ y z — z' _ z_ X Y ' X Z ' . hence, by substitution, (2) becomes \. ^— -\- =1 X*" Y^ 2^ ' 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 JVf. Dupin, in his Developpements, p. 342, whence the above solution is taken. PROPOSITION V. (221.) 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 tan- gent makes with the horizon a greater angle than any other tangent to the surface drawn through the same point of the curve. Now, a» all the linear tangents through any point are in the tangent plane ta THE DIFFERENTIAL CALCULUS. 237 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 determuie the curve to which the linear tangent at every point i» always perpendicular to the horizontal trace of the tangent plane ta the surface through the same point, or, which is the same thing, the projection of the linear tangent on the plane ofxy must be perpen- dicular to the trace of the tangent plane. Now the equation of the projection of the linear tangent at any point is and, by putting z = in the equation of the tangent plane, we have for the trace in the plane of xy the equation — z'=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' ' dx " 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 primitive 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. PROPOSITION VI. (222.) The six edges of any irregular tetraedron or triangular pyramid are opposed two by two, and the nearest distance of two op- posite edges is called breadth; so that the tetraedron has three 238 THE DIFFERENTIAL CALCULUS. 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 sura of the reciprocals of the squares of the heights. Let the vertex of the tetraedron be taken for the origin of the rec- tangular coordinates, and let also one of the faces coincide with the plane of xz, then the coordinates of the three comers of the base will be 0, 0, 2', I x", 0, z", I x"\ y"\ 2'\ and the equations of the three edges terminating in the vertex will be y = X y = X _2/" s» y = "^''- 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 Arr + Bt/ + Cz = 1 I Ex + Fy + Gz = 1 I la; + Ky + L2 = 1, they must be drawn so as to fulfil the conditions (See Jlnal. Geom.) Gz =1 Ea;"'+F«/"'+G2"' = l Ex" +Gs"=0 dz =1 Ax" +C2"=1 Ax""+By"'+Cz"'=0 These conditions fix the following values for A, B, C, &c., viz Ix" +Lz"=l Ix"'+Kj/"' + L2"'=l Lz' =0 - A--4 z ' x" X z X y z X y x"'z" y z G = ir, E = _-4-,,F=4 + z x z y X y z L = 0, 1 = 4, K y z Hence, calling the breadths B, B', B", we have {Anal. Geom.) ■ = A='+B2+C2 :E3+F2+G==: _ (y'V — y"'z"Y-\-{x"'z" — x"'z — x'V")^+ {x"y"y {x"y"'z'y {y"'z"Y-\- {x'z + x"'z" — x"z"'y + {x"y"y {x'y'zf THE DIFFERENTIAL CALCULUS. 839 4-,= F+K=+L.-(!'-^'>'+(^"^-^'"^'>' B"3 {x"y"'z'y Hence ^ + ^ + ^ = I {z'—z"')x"+x"z"\^+ {y"'zy-{- {x"—x"'yz'^ / • ^^ i/ ^ ^ ^'^ Again, the expressions for the heights or perpendiculars demitted from each of the points (0,0,0); (0,0,/); (x", 0, s") ; {x"\y"', z'"), upon the plane which passes through the other three are, severally, {Anal. Geom.) (z" — z')- y'"^ + \ {z" — z) x" + (2 — z") a;"'P + {y"'x"Y .. {^Y'^Y {z"y"'Y + {x"z"' — x"'z"Y + {x"y"'Y {y"'z'Y + {x"'z'f {x'z'Y ' -L + JL + JL+ ' • • 112 ^ Wa ^ IT//2 T^ {z"—z'Yy""'+\{z"'—zf)x"+{z'—z")x"'\^+ \ 2{3^yy+{y'''zy-^{x''2'''—x'''zfy-i-{y'''''-{-x'''''-\-x'''')z'''i ■r- {x"y"'zfy . . . {2), which expression is the same as that before deduced, and thus the theorem is established by a process purely analytical. This remarka- ble property was discovered by JVT. Brianchonj and formed the sub- ject of the prize question in the Ladies' Diary for 1830 : a solution upon difierent principles may be seen in the Diary for 1831. END OF THE DIFFERENTIAL CALCULUS. NOTES Note {A), page 19. The expressions for the differentials of circular functions are all readily derivable, as in the text, from the differential expressions for the sine and cosine. We here propose to show how these latter may be obtained, independently of the considerations in art. (14). By multiplying together the expressions COS. A + sin. A %/ — 1, cos. Aj + sin. Aj V — 1, the product becomes cos- A cos. Ai — sin. A sin. Ai, = (cos. A sin. Ai + sin- A cos. Ai) V — 1- But {Lacroix's Trigonometry.^ cos. A cos A, — sin. A sin. Aj = cos. (A + A,) ) ,, . cos. A sin. A, + sin. A cos. Ai = sin. (A + Aj) | * ' * w' hence the product is cos. (A + Ai) + sin. (A + Ai) %/— 1, = cos. A' + sin. A' ■s/— 1.* Consequently, the product of this last expression, and cos. Ajj + sin. A2 -s/ — 1, is cos. (A' + A2) + sin. (A' + A2) n/'^^oT,! = cos. A" + sin. A" y/— 1, the product of this last, and COS. A3 + sin. A3 \/ — 1, is COS. A"' + sin. A'" v/— 1. * Writing A' for A + A ,. + Writing A" for A' -f As &c. Ed. 31 242 NOTES. Hence, generally, (cos. AH- sin. An/ — l)(cos.Ai+sin.AiN/— l)(cos.A2+sin.A2V--l) &c. cos.(A-|-Ai+A2+A3+&c.)sin.(A + A, + A2+A3+&c.)N/— 1. Supposing, now, A = Ai = A2 = A3 = &c. this equation becomes (cos. A + sin. A \/ — 1)" = cos. nA ± sin. nA V — 1, or, since the radical may be taken either + or — (cos. A ± sin. A V — 1)" = cos. nA ± sin. nA \/ — 1, which is the formula o^ Demoivre, n being any whole number. n Put a = — A then m (cos. a ± sin. a s/ — 1)" = cos. ma ib sin. ma \/ — 1, = COS. nA ± sin. nA V — 1 = (cos. A ± sin. A \f — 1)", therefore, extracting the mth root, n n COS. — A ± sin. — Av/ — 1 = (cos. A ± sin. A s/ — 1)"* which is the formula when the exponent is fractional. Having thus got Demoivre^s formula, we may immediately deduce from it, as in art. (22), the series cos. wA = cos. "A — — - cos."~^ A sin.^ A + &c. sin. nA=» cos."-' A sm. A ^ -^ ^cos."-'Asm.3A+ &c. Let n = i, sin. A = = A .*. nA = f = any finite quality a:, hence, by these substitutions, the foregoing series become cos.x=l-3-:^ + p-^-3-^-&c. consequently, d sin. X = (1 — - — - + :; — — &c.) dx = cos. xdar, \ t a 1.^.0.4 NOTES. 243 I had intended to have given here another method of arriving at the differentials of the sine and cosine, and to which allusion is made at page 41, but, upon close examination, I find that the process I had then in view -is liable to objection, and is therefore best omitted. Note {B),page 91. Demonstration of the Theorems of Laplace and Lagrange. Let it be required to develop the function u = Ys where z = F (t/ + xfz). By differentiating the second of these equations, first relatively to • tCj and then relatively to y, wc have | = Fto+./.)(l+./'.|). Multiplying the first by -j- and the second by ^, and subtracting, there results dz f dz _ dz _ dz but since u or "Yz depends only on z, we shall have du _ 1 dz du ^ dz dx dx^ dy dy therefore, eliminating Y'z, we get du dz du dz _ dx dy dy dx ' dz or putting for -7- its value (1), and making for abridgment /z = Z, * At page 89 we put/'z to represent the differential coefficient of/z relatively to *; here the same symbol denotes the coefficient relatively to z. 72 ^-.. '^ri"vy^/,r 244 NOTES. dz this last expression becomes divisible by -r- and reduces to dy ^_ydu Tx-^d^j (^)' so that we may always substitute for ~ the quantity Z -j-. If we differentiate the preceding equation relatively to x, we shalli obtain du dZ-r which are the di as ds NOTES. 251 same as those of -j-j, -y^, -7-^, remain unchanged. Therefore CtX (JLX 0tX this second member being substituted in the expression for cos. w leads to a result of the form r = K cos. w, K being a constant expression for all the curves on the proposed sur- face which touch MT at the point M. Put now in this expression cd =0, then r' becomes r, therefore r = K, consequently r = r cos. w, which result comprehends the theorem of JVlewsmer, since, if the curve MG' is plane, its plane will coincide with N'MT, and the angle w 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 d, la Ge- ometne, p. 268. We may take this opportunity of remarking that, in our investiga- tion of this theorem, at p. 182, it might easily have been shown, with- out referring to article 86, that -— = 1 + tan.2 6, dxr because, by the right-angled triangle, x" — x^ sec.^ d = a;^ (1 -f tan.^ H) .'. -7-:r = 1 + tan.^4 = -—-. dar dxr Note (E), page, 106. The erroneous doctrine adverted to at page 106 is laid down also by Lacroix, in his quarto treatise on the Calculus, vol. 1, p. 340, from whom, indeed, Mr. Jephson seems to have adopted it. The princi- ple 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-meme imagi- naire," It is very remarkable that analysts should have hitherto held such imperfect notions respecting the failing cases of Taylor's theo- rem. NOTES BY THE EDITOR, Note (A') page 15. As the Algebra here referred to may not be in the hands of the student, we shall find the differential coefficient of a logarithmic func- tion, by previously obtaining that of an exponential one, which is the course pursued by most writers on the calculus. Let u = a' . . . . (1), in which if a; be increased by h, we shall have u' = a^ = a" X aK Now in order to develop the last factor of this product, we suppose a = 1 + 6, in order to subject it to the influence of the Binomial Theorem, we shall then have ji /, I ina t I Li I '*' ^ — 1 LI I h h — 1 h — 2 t^={l + bf = l + hb + -. —^ .b^^--. — 2— • — 3— h' + &c. The multiplication indicated in the second number of this being executed, and the result ordered according to the powers of h, repre- senting by sh^ the sum of all the terms containing powers of h above the first, we shall then have 6^ 6^ 6^ a*= 1 + h{b — — + ___ + &c.) + s/i2 ^ O 4 Both members of which being multiplied by a^, designating the coeffi- cient of h within the parentheses by c, we shall then have a' X a'' =u' = {1 + ch + sh?) a'. The primitive function being taken from this, leaves u' — M = ca'h + a'sh?, whence U tt , , — r — = ca' -f- a'sh. NOTES. 253 which, when h = becomes ^ = 00-.... (2), where 1 2 3 4 ^ ^ It is thus perceived that the differential coefficient of an exponen- tial function, is equal to that function multiplied by a constant num- ber c, which is the above function of its base. We have from equa- tion (2), du = ca'dx, and we perceive from equation (1) that log. u = a;, whence d log. u = dx'^ eliminating dx between this and the last, we have du =■ ca'd log. M, and ,, du I du d log. u = —^ = — .—. ca" c u The differential coefficient therefore of a logarithmic function is equal to the differential of the function divided by the function itself, multipUed by the constant — , the modulus of the system whose base is a. The modulus of the Naperian or Hyperbohc system of Loga- rithms being unity, we have d . lu = — , u lu representing the Naperian Log. of «. NoTis (B') page 21. The leading part of article 15, in regard to the notation relative to inverse functions, though very plausible, is nevertheless calculated to mislead the student. For in the equation x = F~' y, expressing the function that x is of y, the direct function being y = Fa:, the symbols F and F~' should not be considered as quantities or operated upon as such, since they here stand in place of the words a function of, the forms of both functions being different. 254 NOTES. Note (C) page 65. Article 49 should have commenced with the equation y = Far* and though the succeeding articles are full and ample on the subject, it may not be amiss to present the maxima and minima characteris- tics of functions in less technical language. Remembering the note page 63, let u =fx be the proposed function to ascertain whether it admits of maxima or minima values ; and if so, by what means they and the variable on which they depend may be discovered. In the proposed function if the variable x first increase and then decrease by any quEuitity h, we shall then have u =fx .... (1), and by Taylor's Theorem, , ^, , ,, , dull , dFu h"" , dhi h^ , d'u 1.2.3. 4 + &c (2), " — ff j\ — du h d^u Ir d?u ¥ . d'^u u -J{x — li)-u — —--Jr-^Y72~d^ 17273 "^"d^ ^' + &c.' (3). 1.2.3.4 Now, in order that the function u under consideration may attain a maximum or minimum value (2) and (3), must be both less, or both greater than (1), and as h may be assumed so small that the term containing its first power may be greater than the sum of all the suc- ceeding terms, (2) will be greater than «, while (3) will be less. Since the first differential coefiicient has different signs in the two developments, the function therefore cannot attain maxima or mini- ma values, unless this coefficient becomes zero. The roots of the equation -p = 0, will give such values of x as may render the func- tion a maximum or a minimum; such values of the variable being NOTES. 255 substituted in the second differential coefficient -j-j- if these render its value any thing, we are certain the function may become a maxi- mum if that value is negative, or a minimum if it be positive ; for in the first case (2) and (3) are both less than ii, and in the second they are both greater. But if the same values of x render the second dif- ferential coefficient zero, as well as the first, we readily see that the third differential coefficient, must also become zero, in order that the function may admit of maxima or minima values : because this coef- ficient has different signs in (2) and (3), we then substitute the same values of a: in the fourth differential coefficient, which has the same sign in (2) and (3), if these render it negative we shall have a maximum value of the function, and if positive a minimum value ; but should this coefficient also vanish with the preceding ones, the next must be examined, and so on. In order therefore to determine the values of x, which render the proposed function a maximum or minimum we must find the roots of the equation — = 0, and substituted then in the succeeding differen- tial coefficients, until we find one that does not vanish ; if this be of an odd order, the roots we have employed will not render the function a maximum or minimum, but if it be of an even order, then if this coeffi- cient be negative we have a maximum value of the function, but if posi- tive a minimum value. THE END. FOURTEEN DAY USE RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. F?gC^D LP ^r-K 19 \%l MA\ REC'D LD 2:0d'S7BR nEf'P ^° APR 2 4 ' 64 -2 FM OGTSlVd^"^ 3Apr'61JR REC'D LD MAR2U 1961 l63ct'S''WT ■C ' -' LU' O C T 2 ; j6? LD 21-100m-2,'55 (B139s22)476 General Library University of California Berkeley I -v^ ■7 I