AN ELEMENTARY TREATISE ON THE DIFFERENTIAL CALCULUS, IN WHICH THE METHOD OF LIMITS EXCLUSIVELY MADE USE OF. IS BY THE REV. M. O'BRIEN, M.A., LATE FELLOW0 OF CAIUS COLLEGE, CAMBRIDGE, CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS; PUBLISHED BY J. & J. J. DEIGHTON, CAMBRIDGE; AND JOHN W. PARKER, LONDON. M.DCCC.XLII. PREFACE. THE method of Limits is generally allowed to be the best and most natural basis upon which to found the principles of the Differential Calculus; in the following pages this method is exclusively adopted, no use whatever being made of series in the demonstration of fundamental propositions. The following is an outline of the work, which is by no means offered to the reader as a complete treatise on the subject, but merely as an exposition of its more prominent and useful principles. In Chap. I, certain terms, afterwards to be used, are defined and explained. In Chap. II the nature of a Limiting Value is fully set forth, and the important distinction (which ought never to be overlooked) between an actual and a limiting value is pointed out and illustrated by examples. Chap. III contains a set of Lemmas, which are necessary in order to render the use made of limiting values in the Differential Calculus perfectly legitimate; and here I have endeavoured to confine myself to what seems really essential. In Chap. IV certain important limiting values are obtained. Chap. V contains the Rules for Differentiation, in the demonstration of a iv PREFACE. which Lagrange's functional notation is employed, as being the simplest to begin with. In Chap. VI the Diferential notation of Leibnitz is explained, d/ is dx defined as the quote of the differentials dy and dx, which however are not supposed to be infinitesimals, but simply two arbitrary quantities in a certain ratio. In the case of partial differential coefficients, some du modification of the common differential notation - du d, is clearly necessary: I have employed the suffix notation d.u, dyu, as being frequently employed, though not exactly in this manner. I should have much preferred the notation deu, dyu to denote partial differend,u du tials, and d- - to denote partial diferential codx dy efficients. Chap. VII relates to successive differentiation, and the change of the independant variable. Chap. VIII contains certain very important Lemmas upon which the use and application of the Differential Calculus in a great measure depends. Chap. IX contains the theory of Series, based upon one of the preceding Lemmas, without assuming that f(x + h) can be developed in the form A + Bh + C + &c...: and here I have endeavoured to shew what the real nature of a series is, and to prove rigorously the prin PREFACE. v ciple of Indeterminate Coefficients. Chapters X and XI relate to Vanishing Fractions, and Maxima and Minima, and contain some useful simplifications of the common methods. The very insufficient and troublesome criterion usually employed in distinguishing the maxima and minima of functions of two variables is not introduced. Chap. XII relates to Tangents, Normals, &c., the Curvature of Curves, and the properties of the Evolute; and here tht arrangement usually adopted is somewhat departed from, and what seems a more natural course pursued, in order to avoid certain difficulties, which I have observed very often impede the student on his first reading of the subject. In Chap. XIII the useful Polar formulce and the differentials of Areas, Volumes, &c., are deduced. Chap. XIV relates to Asymptotes. Chap. XV contains a very simple method of tracing curves. Chap. XVI relates to singular points. Chap. XVII contains the general Theory of Contacts and Ultimate Intersections: no use is made of series in explaining the different orders of contact. The remaining chapters are occupied with Elimination by differentiation, Lagrange's Theorem, the properties of the Cycloid, &c., &c. The Appendix contains Examples worked out. It was my intention to have added a few more chapters, and among the rest, one on the origin and progress of the Differential Calculus, and another on vi PREFACE. the Infinitesimal method; but from various circumstances I found it impossible to send the work to the press at the time originally promised to my bookseller, without omitting these concluding chapters. I mention this to account for the absence of allusions to the History of the Differential Calculus, which were all reserved for the final chapter, and the small number of Examples in the Appendix. Professor Peacock's excellent collection of Examples, which have been of such service to the Mathematical Student, is now out of print; but Mr Gregory's work lately published will supply its place, which contains, not only a great number of well-selected and valuable examples, but also many important explanations and theorems not to be met with in any elementary treatise. In a subject of so much importance as the present, the student ought not to confine his attention to one book or system: for a very valuable treatise bn this subject he is referred to that published by the Society for the Diffusion of Useful Knowledge. In the general plan of this work, and in several particulars, I have deviated from some of the methods often made use of, partly in attempting to put the subject in a simpler and clearer point of view, and partly in avoiding certain steps of reasoning which appear to be defective. One of these is the fallacy of establishing premises on a certain implied condition, and drawing a conclusion from them by a direct violation PREFACE. Vii of that condition. An example of this is to be found in a proof often given of the principle of indeterminate coefficients, in which the factor x is divided out of the equation Bx + Cxf + Da + &c.... =0, which of course tacitly assumes the condition that x is not zero; in this manner is obtained the equation B+Cx+Dx+ &c. o.. =0; and then by putting x =0, contrary to the implied condition, the conclusion B=0 is arrived at. Another example of this kind of reasoning is given in the first note, page 6. The assumption, that f(x + h) can be expanded in a series of the form A + Bha -- Chp &c.... seems to me to be a serious defect in the common method of establishing Taylor's Series, and thereupon the principles of the Differential Calculus. This assumption is usually justified by arguing, that if we find definite values for A, B, C, &c. it shews that the assumption is correct. Now this argument may be stated thus: " If the assumption that f(x + h) = A + Bha + Ch+ +&c. be true, then A., B, C, &c., must have definite values. But we can in general obtain definite values for A, B, C, &c.... (e.g. by the method of indeterminate coefficients.) Therefore the assumption is true." This is clearly a falacious argument, for to warrant the conclusion the first premise should have been this: .*. Vill PREFACE. "If the assumption be not true, definite values cannot be obtained for A, B, C, &c." These defective steps of reasoning and others which might be mentioned, are objectionable, not because they lead to erroneous conclusions, but because they ought not to be found in a subject like the present, in which every thing should be comformable to the strictest rules of logical deduction. M. Cauchy has done much towards the improvement and perfection of the Differential Calculus, and his writings on this, like those on the more abstruse branches of mathematics, are most valuable. In one or two places the methods I have employed in the following pages are apparently similar to those of M. Cauchy, but in reality they are essentially different: so far as I am aware I am indebted to him only for article 48. CAMBRIDGE, October, 1842. ERRATA. The reader is requested to make the following correction which is of some importance. 3rd line from foot of page 75 instead of (2) read (0). CONTENTS. CHAPTER I. PAfE PRELIMINARY Remarks. Variables and Constants. Functions, Continuous and Discontinuous. Illusory Functions. Examples...... CHAPTER II. The distinction between an Actual and a Limiting Value explained. A Tangent defined. The nature of the Differential Calculus sta te d.................................................................................... CHAPTER III. Certain Lemmas respecting Limiting Values................................. 13 CHAPTER IV. Certain Limiting Values obtained which we shall require to know hereafter................................................................... 22 CHAPTER V. Rules for finding the Limiting Value of f()f(................ 29 CHAPTER VI. The D ifferential N otation........................................................... 46 CHAPTER VII. Successive Derivatives or Differentials. Change of the Independant Variable............................................7 CHAPTER VIII. Certain Lemmas upon which the application of the Differential Calculus in many cases depends...,.................................. 69 X CONTENTS. CHAPTER IX. FAOG Application of the Differential Calculus to the Developement of Functions; with some preliminary remarks respecting series... 77 CHAPTER X. Determination of the Limiting Values of Vanishing Fractions........... 99 CHAPTER XI. Determination of the Maxima and Minima values of Functions........ 112 CHAPTER XII. Tangents and Normals to Curves. The Curvature of Curves. The E vo lu te............................................................................. 124 CHAPTER XIII. Polar Formule, Differentials of Areas, Surfaces, and Volumes......... 133 CHAPTER XIV. Asym ptotes................................................................................ 142 CHAPTER XV. On the method of tracing the general form of a Curve from its Equation.............................................................................. 149 CHAPTER XVI. Singular Points of Curves................................................. 158 CHAPTER XVII. The general theory of Contact. Intersection of consecutive Curves... 170 CHAPTER XVIII. Lagrange's Theorem. Elimination of Constants and Functions by Differentiation. Expansion by means of this Elimination......... 180 CHAPTER XIX. M iscellaneous T heorem s............................................................... 189 A PPEN DIX................................. 196 THE DIFFERENTIAL CALCULUS. CHAPTER I. PRELIMINARY REMARKS. VARIABLES AND CONSTANTS. FUNCTIONS, CONTINUOUS AND DISCONTINUOUS. ILLUSORY FUNCTIONS. EXAMPLES. 1. AT the commencement of a treatise upon any science, The nature of the Difthe reader naturally expects to find some brief statement of ferential the nature and object of that science; but in the present Calctub instance it is impossible to make such a statement, unless explained at first. the meaning of certain terms, necessary to be used, be previously explained: for the Differential Calculus, from the very first, involves notions which must be new to one who is acquainted with no more than the common elements of Algebra and Analytic Geometry; and in the following pages we must suppose the reader to have advanced only so far in the study of mathematics. We shall therefore make a few preliminary remarks, and explain certain fundamental notions, before we state what the Differential Calculus is. 2. Any quantity capable of variation is called a Vari- Variables and Con.. able. stants. Any quantity which cannot vary, or, if it can, is supposed not to vary, is called a Constant. Thus in the general equation to a right lipe, namely, y = mx + c, if we suppose the line never to change its position, 1 2 VARIABLES AND CONSTANTS. m and c are constants, and v and y are variables; but if we suppose that the line may change its position, m and c are variables as well as x and y, for they are then capable of variation. It is usual to take the first letters of the alphabet a, b, c...&c. to denote constants, and the last letters... u, v, w, xr, y, z to denote variables; but in many cases this distinction cannot be conveniently maintained, inasmuch as quantities which in one case we consider variable must often, in another, be considered constant, and vice versa. Variables are often called, Arbitrary quantities, Indeterminate quantities, Unassigned quantities. Continuous 3. A variable to which we may give in succession any continuous number of diferent values, which differ from each other as Variableslittle as we please, is called a Continuous Variable; otherwise it is said to be Discontinuous. Thus if x be taken to represent the length of any line which may be of any magnitude we please, then x is a continuous variable: but if v be taken to represent any integer, then it is a discontinuous variable; for in the former case we may give to x in succession any number of different values which differ from each other as little as we please; whereas in the latter case, though we may give i any number of different values, we cannot make them differ from each other as little as we please. The distinction between a continuous and a discontinuous variable may be briefly expressed by saying, that one admits of gradual change, but the other does not. A Function 4. The result obtained by performing a certain operadefined. The Func- tion or set of operations upon a variable x, is said to be tioal No- a certain function of x. tation explained. The words "a certain function of" are, for the sake of brevity, denoted by the letter f written before I. Thus the equation y =f (a) simply means, that y is "a certain function of" ax, i. e. that y is the result of performing a certain operation or set of operations upon x. The letter f, which we shall call the Functional Letter, is, as it were, the represen FUNCTIONS. 3 tative of the particular operation or set of operations performed upon x. Thus if f(x) = xn, f represents the operation of taking a quantity to the nth power; if f(x) = log sin, f represents the operations of taking the logarithm of the sine of a quantity; if BPQ (fig. 1) be a given curve, AM( x) MP (= y) the co-ordinates of any point P, and if we take f(x) to represent y, then f represents the operations of measuring AM equal to x along the line AX, erecting MP perpendicular to AX, producing it to meet the curve at P, and so finding the length MP or y. 5. From the definition here given of a function we may A function see that f (x) is not necessarily a quantity which changes when iSe Ots x changes; for a set of operations performed upon x may a quantity sometimes lead to a result which is the same for all values changes when the of iZ. This will appear from the following example. variable changes. Let APB (fig. 2) be a semicircle whose radius is a, C its center; take any arc AP = I, with P as center, and some given line c as radius, describe a circle cutting AB at the point Q: then we may say that AQ is a function of x and denote it by f(x); f will therefore represent the operations of measuring x along AP, describing a circle with radius c and center P, so finding Q, and therefore finally AQ. Now in general AQ or f(x) will be different for different values of x; but if we take c = a, then the circle described with P as center and c (= a) as radius will always cut AB at the center C, and therefore we shall have AQ = AC or f(x) = a, whatever be the value of x. Hence f (x) in this case does not vary when x varies. Thus it appears, according to our definition of a function, that f (x) is not necessarily a quantity which changes when x changes; and this remark is important, as will appear hereafter. 6. When we have occasion to consider several different functions at the same time, we employ different functional letters, in order to distinguish between them. The letters commonly used, in addition to f, are F, c, Al, X, and sometimes these letters with dashes, thus f', F, p(', 'j, X' or f" F", &c...&c. Thus we might put f (I) to represent x", q (~) 1-2 *4 - FUNCTIONS to represent log sin a, \f' (x) to represent y in the curve, fig. 1, &c.... &c. Functions 7. The result of performing a certain operation or set of several variables, of operations upon several variables x, y, z, &c. is, in like as well as of one. manner, said to be a certain function of x, y, z, &c., and is represented similarly by the notation f(x, y, a,...). Thus X2+ y'2 4 z, x log (y + z), are the results of performing certain operations upon x, y, z, and are accordingly said to be certain functions of x, y, z, and we denote them by functional letters written before x, y, z, thus, viz. xi y+ y f 2 (x, y, z) S log (y + z) = i ((, y, z.) Geometri- 8. We may evidently draw the curve BPQ (fig. 1) in cal representation'of such a manner that y shall be any function we please of xi; and a function, thus by means of a curve we may denote any function, and as it were represent it to the eye, which is often a very good method of illustrating general theorems respecting functions. Functions 9. In a function of several variables f(x, y, z...) it may of dependantand of happen that the variables x, y, z...are connected with each andepend- other in some manner, so that we cannot change one without ables. at the same time changing the others. Or it may happen that x, y, z...are not at all connected with each other, so that we may assign to each of them any value we please independently of the rest. In the former casef (x, y, z...) is said to be a function of several mutually dependant variables, and in the latter case f (, y, z) is said to be a function of several independant variables. Functions 10. A quantity y is said to be an explicit function of explicit and implicit. another, a, when we can state the precise operations by which y may be deduced from x; if not, y is said to be an implicit function of x. Thus if we are given the equation y5 - 34y + X= 0, we know that there must be a certain set of operations by which y may be deduced from x, but what these operations are we cannot precisely state; in such a case y is called an implicit function of x. CONTINUOUS AND DISCONTINUOUS. g 11. The functions which commonly occur in mathematical The functions which investigations are of such a nature, that they always suffer a commonly occur in gradual and not a sudden change, so to speak, when themathemavariable is gradually altered in value. Functions of this kind tics eous are called Continuous Functions. By saying that a function f(x) always receives a gradual change when x is gradually varied, we mean this; that if x be changed into a', and therefore f (v) into f ('), then f(') -f( () may be made as small as we please by taking x' - x' small enough, whatever be the value of x. That this is true for all ordinary functions, such as Ix, ax, log., sin x, &c.... &c. and all ordinary combinations of these functions, such as (ax + log sin x)", ax sin x, &c...&c. does not require to be proved, for it is quite evident. All ordinary functions therefore are continuous functions. 12. Functions which sometimes suffer a sudden change Discontiwhen the variable is gradually altered in value, are called nuous functions. Discontinuous Functions. Thus, if we draw a broken curve as in (fig. 3) the ordinate will be a discontinuous function of the abscissa: for it is evident that the ordinate will suffer a sudden change at the points P, Q, R, S, supposing the abscissa to be gradually varied. 13. We shall never have occasion to consider discontinuous functions in the following pages, and therefore we shall always suppose that the functions we make use of are continuous. Hence, if f(x) be any furction of x we make use of, and Assumption which if x be changed into x', and therefore f(x) into f(i'); wewe shall shall always assume that f(m) f(x) may be made as small kalas as we please by taking x' - small enough whatever be the respecting value of x. 14. A function f (x) is said to become Illusory when the An illusory function operations represented by f cease to give any definite result; defined. which may happen in certain cases, as we shall shew. X *' vx denotes the difference between x' and x, subtracting the greater of these quantities from the lesser; x' r x is therefore the absolute difference between x' and x without regard to sign. 6 IJLUSORY FUNCTIONS. Ex. L. 2 - X A fraction Thus, if f(m) = 2-, the operations represented by f which as- - sumes the o cease to give any definite result when x = 1; for then f(x) form orm - - 1-I 0 assumes the form, or-, which is not a definite result, 1 —1 0' as is shewn in the note*; f(X) therefore becomes illusory when -= 1. Ex.2. 15. Again, iff (x) = (1 + x)-, the operations represented (l+-x) by f cease to give a definite result when x = 0; for then when x=0. = i (1 + x); assumes the form 1i, which is not a definite result, as is shewn in the notet; f(x) therefore becomes illusory, in this case, when x = o. Ex. 3.fo 16. Again, if ABC (fig. 4) be an ellipse marked with the Case of two i intersecting usual letters, PG the line bisecting the angle SPH, and 0 0 0 5does not *That 0-is not a definite quantity appears thus. ~- according to the strict represent definition of a quotient, is that quantity which multiplied by 0 gives 0: now any any definite 0 quantity. quantity whatever multiplied by 0 gives 0; therefore 0 is any quantity whatever; i. e. it is not a definite quantity. It may be said, however, that though 0, considered absolutely, is not a definite quantity, nevertheless — 2 becomes 2 when = 1; for x_ =,+ and _ X2 - 1 2 x" - 1 - x ~X~ 1 ' x + 1 -, 2 when = 1, and therefore _ = when = 1. To this we may answer, that 2 _ is proved to be equal to + i by dividing its numerator and denominator by x-1; but we may not perform this division when x- 1 = 0, since there is no rule of Algebra which enables us to divide the numerator and denominator of a fraction by zero without altering its value; hence the equation 2 X 2 = holds only on the express condition that x does not = 1; and therefore we may not draw any conclusion from this equation which requires us to suppose X2 __ that x actually = 1: consequently, we cannot assert that 2- 1 - when = 1 be cause x2_=-+ 1 1 1 16 not a t That 1~ is not a definite quantity may be shewn thus. 1~ is that quantity definite which taken to the power 0 becomes 1; now any quantity whatever taken to the quantity. I power 0 becomes I; therefore 1~ is not a definite quantity. ILLUSORY FUNCTIONS. 7 CM (= x) the abscissa of P; then if we assume f(iv) = CG, lines becoming cothe operations represented by f cease to give a definite result incident. when,v = a. For f represents the operations of drawing MP perpendicular to CM, so determining the point P, and the lines SP, HP, and then drawing PG to bisect 4 SPH, and so finding G, and therefore CG. Now when x = a, P coincides with A, and therefore the line PG with CA, and therefore PG cannot be said to intersect AC in one point more than another; therefore CG is not a definite quantity, and therefore the operations denoted by f cease to give a definite result when x = a. Hence f (x) becomes illusory when x = a. 17. Again, if P (fig. 5) be a point on a curve PQ, RPQ Ex. 4. Case of a a right line drawn through P and any other point Q on the right line curve, meeting the axis of x (AX) in the point R: then if we joining of assume arc PQ = s, and L PRX = P (s), the operations repre- curve,when these points sented by 0 cease to give a definite result when s = 0. For are made to P represents the operations of measuring PQ = s, drawing coincide. RPQ through P and Q, and so finding the angle PRX: now when s = 0, Q coincides with P, and any line whatever drawn through P passes also through Q; therefore the line QPR does not occupy a definite position, nor is PRX a definite angle, when Q coincides with P; and therefore the operations represented by f cease to give a definite result when s = o. Hence + (s) becomes illusory when s= 0. 18. From these examples it is evident that a function may become illusory when x receives a particular value, the operations represented by the functional letter ceasing to give any definite result. In each of these examples we may easily see that if x Functions become iidiffer ever so little from that value which makes f (,) illusory, bsory only the operations represented by f do lead to a definite result. for isolated values of It is therefore only for isolated values of x that f (v) becomes the variillusory: and this will be found to be true in all cases where able functions become illusory. CHAPTER II. THE DISTINCTION BETWEEN AN ACTUAL AND A LIMITING VALUE EXPLAINED. A TANGENT DEFINED. THE NATURE OF THE DIFFERENTIAL CALCULUS STATED. The dis- 19. THE Examples brought forward in the preceding tinction chapter, for the purpose of shewing that a function may actual become illusory when the variable receives a particular value, value and a limiting lead us to make a very important though simple distinction, namely, the distinction between an Actual Value and a Limiting Value of a function. An Actual Value of a function f(x) is the result obtained by giving x some particular value, and performing upon it the operations represented by f. A Limiting Value of a function f(x) is that quantity from which we may make f (x) differ as little as we please, by making x approach nearer and nearer in magnitude to some particular value without actually becoming equal to it. That the 20. An actual and a limiting value thus defined seem limiting value does at first sight to be the same thing; and indeed, as long as the noteadefiteto operations represented by f lead to a definite result, they are quantity identical, as we shall prove presently; it is only when the when the actual function becomes illusory that the distinction between them is shewn byo real, and it consists in this, that the actual value ceases to be examples. a definite quantity, whereas the limiting value does not. This we shall shew in the case of the examples just alluded to. Ex. 1. 21. Let us consider the first example, namely A fraction which assumes the ) f ( --, (see 14). form i We have seen that the operations here denoted by f give no ACTUAL AND LIMITING VALUES. 9 definite result: when x =, o,, in other words, that the actual value ceases to be a definite quantity when x = i. But not so the limiting value; for we may divide the numerator and denominator of f(xi) by x - 1, and so arrive at the equation f(x) = -+, except when v actually = 1; now by making x approach nearer and nearer in magnitude to 1, without actually becoming equal to it, we may evidently make differ as little as we please from, and therefore the so + 1 same may be said of f(x), since it ceases to be equivalent to only when x actually =1: 2 is therefore a quantity iv+2 from which we may make f(x) differ as little as we please, by making x approach nearer and nearer in magnitude to 1, without actually becoming equal to it. And it is easy to see that there is no other quantity but i from which we may make, and therefore f(v), differ +' a t herfor f, d iffer as little as we please by making w approach nearer and nearer to 1. Hence it appears that —, and no other quantity but I, is the limiting value of f(x) when v approaches 1. We see therefore in this case that when the actual value ceases to be a definite quantity the limiting value does not. 22. We are not yet sufficiently advanced to shew that the EX. 2. same is true in the case of the second example (15); so we swointershall pass on to the third example (16). lines which become coWe have seen that the operations here represented by fincident. cease to give a definite result when X = a, i. e. the actual value of f(x) ceases to be a definite quantity when x = a. But not so the limiting value; for since PG bisects z HPS we have HG HP ae +CG a + ex SG = P' or (by Conic Sections); S nSPd. ae C G a - ex and.'. CG or f(x)= e'x, except, x actually =a. * This proof evidently fails when x=a, for then P coincides with A and the triangle HPS ceases to exist. 10 ACTUAL AND LIMITING VALUES. Now e2w may be made to differ from e2a as little as we please by making x approach nearer and nearer in magnitude to a; therefore, as in the first example, e2a, and no other quantity but e2a, is the limiting value of f(x) when x approaches a. Hence in this case when the actual value ceases to be a definite quantity the limiting value does not. AEx. 3 23. Lastly, let us consider the fourth example (see 17). A line cutting a curve in two We have seen that the operations here represented by 0 points which be- cease to give a definite result when s = 0; i. e. the actual value ncme co- of 0p(s) ceases to be a definite quantity when s= o. But not incident. so the limiting value; for let AM (= x) MP (= y) be the coordinates of the point P (fig. 6), AN(= x') NQ( y') the co-ordinates of the point Q, draw OP parallel to MN, and let y =f () be the equation to the curve: then we evidently have OQ Y'-y f(y ) -f(a) tan PRX= _ = - = OP x -x x - and this equation is true, no matter how near x' may be to x, provided x' be not actually equal to i, for then the triangle f(') -f(v) OPQ ceases to exist, and tan PRX and () - ) cease to W0 - have definite values. Now for simplicity let us suppose the curve to be a parabola having the axis of y for its axis, and its equation being IV2 accordingly y = -; then f (') - f () 1 'V - co x' + t x - - 4m oo - x 4m, except when x' =. Hence it is evident that, except when v' actually = -, we have 5 (s) = z PRX = tan-' s m o 4m Now the second member of this equation may be made to WHAT A TANGENT TO A CURVE IS. 11 differ from tan' -- as little as we please by making x' 2m approach x, or, what is the same thing, by making s approach zero; hence (as in the first example) tan-1- and no other quantity, is the limiting value of f(s) when s approaches zero. In a similar manner we might shew, in the case of other curves, that f(s) has a definite limiting value when s approaches zero. It appears therefore in this case, that when the actual value ceases to be a definite quantity the limiting value does not. 24. This last example leads us to the best and most What a accurate conception of what a tangent is. For draw the agurve is. line SPT making the angle tan- - with the axis of x; then 2m by what has been proved z PTX is the limiting value of z PRX when Q approaches P; i.e. z PRX may be made to differ from z PTX as little as we please by making Q approach P without actually coming up to it; or what is the same thing, z QPS may be made as small as we please by making Q approach P without actually coming up to it. Now this being the case, it is natural to say that the line SPT just touches the curve at the point P, or that it is the tangent to the curve at P. Hence we define a tangent in the following manner. 25. If SPT be that line to which the secant RPQ may Definition be made to approach nearer and nearer so as to make with it of a tangent. an angle as small as we please, by making Q approach P without actually coming up to it; then SPT is said to be the line touching the curve at the point P, or the tangent at P. Or to speak more briefly; If SPT be the limiting position of the secant RPQ when Q approaches P, it is said to be the tangent at P*. * This definition of a tangent seems to me to be the only accurate one that can be given, so as to apply to all cases of contact, such as contact at a point of con~ 12 WHAT THE DIFFERENTIAL CALCULUS IS. What thte 26. The example we have just been considering, will Differential Calculusis enable us now to state generally the nature and object of briefly stated. the Differential Calculus. We have seen, that to determine the position of the tangent at any point of a curve, as above defined, we have only to find the limiting value of f () f(V) when ' approaches x; it appears therefore that we may arrive at a general method of drawing tangents to curves by means of this limiting value, if we can find it in all cases. But this is a very small part of the use which may be made of this important limiting value: for there are very few branches of exact science which are not largely indebted to its assistance for the progress they have made. Indeed, without it, some of the most interesting applications of mathematics to the explanation of natural phenomena could never have been effected. Now the Differential Calculus is that branch of mathematics whose object is, in the first place, to determine a set of rules whereby the limiting value of - ( x, when I' X -.X approaches x may be found with facility in all cases; and, in the second place, to explain some of the principal uses which may be made of this limiting value in pure mathematics. The origin of the name Differential Calculus we shall presently explain. trary flexure, contact at a cusp. It certainly is not correct to define the tangent STP to be the position which the line QPR assumes when Q coincides with P, for we have seen that the line QPR has no definite position when Q coincides with P. CHAPTER III. CERTAIN LEMMAS RESPECTING LIMITING VALUES. 27. IN the preceding Chapter it was our object to shew that there is a real distinction between an actual and a limiting value in certain cases, and to state briefly the nature of the Differential Calculus. We now proceed to prove certain Lemmas, and to obtain certain limiting values which we shall find useful hereafter. But we must make a few remarks previously. 28. - When a quantity may be continually diminished, so The phrase "diminish as to become less than any specified quantity, however small, ad libi without becoming actually zero, we shall say that it may beum iedx " diminished ad libitum." We shall find this phrase convenient and perhaps less likely to be misunderstood than the words " diminish indefinitely," which are generally used in the same sense. 29. Employing this phrase, we may state the assumption made in (13) as follows: viz. By continually diminishing x' - x, when it has once Assump-d become suficiently small, we may diminish f (x') f (x) ad in Art. 13, stated libitum; f(x) being any function, and x any value of the somewhat variable. And more generally by continually diminishing differently. x - x, y' y, z' - z... when they have once become sufficiently small, we may diminish f(x'y'z'...) - f(xyz...) ad libitum; f (x y z...) denoting any function of several variables, and x, y, z... any values of these variables. Of course we here suppose that f(x) {or f(x, y, z...) if there be more than one variable} is not illusory. 14 DEFINITION OF A LIMITING VALUE. 30. And the definition of a limiting value in (19) may be stated thus: Definition If by continually diminishing x - a, when it has once of a limiting valuestated become sufficiently small, we may diminish f(x)- A ad Sdmewhat libitum; then A is said to be the limiting value of f(x), when x approaches a. It is clear that the assumption and definition thus stated are equivalent to what they were before, only that they are more exactly expressed and better adapted to the use we shall have to make of them hereafter. f(I) nA 3 31. It is important to notice, that in this definition of a does not in general di- limiting value we do not assert that f(x) - A must diminish minish with xa until when x ~ a is diminished, for all values of - a, but only when xcom has ss a has once become sufficiently small. There are many become less than a cer- cases in which f(x) - A increases as x - a diminishes, and containvaluetinues to increase until x - a has become less than a certain value, after which it continually diminishes with x - a. To avoid circumlocution, instead of saying, " By diminishing x - a, when once it has become sufficiently small, we diminish f(x) A ad libitum," we shall simply say, " By sufficiently diminishing w - a we diminish f(x) A ad libitumbn." x may be a 32. It is also important to observe, that it is not essential discontinuous vari- to our conception of a limiting value, as above defined, that x able, pro- should be a continuous variable (see 3); all that is necessary vided it be possible to iS this, that it be possible to diminish x a ad libitum, that is, diminish x-a ad to make it less than any specified quantity however small, libitum. itum. without actually making it zero. Thus if we suppose av = a + -, where n is always an integer, and therefore v not a continuous variable, we may conceive the existence of a limiting value of f(x) when x approaches a (which it does when n approaches infinity), just as well as if x varied continuously: for by increasing n, though it be an integer, we may diminish - or v - a, ad libitum;,n LEMMAS RESPECTING LIMITING VALUES. 15 which is all that is supposed necessary so far as v - a is concerned in our definition of a limiting value. But if we suppose x = a + n, where n is always an integer, then it is impossible to conceive the existence of a limiting value of f(x) when x approaches a, since we cannot make x ~ a or n less than unity unless we make it actually zero. We now proceed, as we stated, to prove certain Lemmas, which will enable us to reason more exactly hereafter, and will now serve to illustrate the nature of limiting values. 33. The limiting value of f (x) when x approaches a is Lemma I. the same whatever sort of variable x be, provided of course x ~ a may be diminished ad libitum. Let v and z be two different variables such that both v- a and z ~ a may be diminished ad libitum, and let A be the limiting value of f(x) when x approaches a. Then by sufficiently diminishing x ~ a and z - a, and therefore - v, we diminish ad libitum, f (x) A by the definition in (30), f(<v) ~ f(z) by the assumption in (29), and therefore f(z) -A. Hence by sufficiently diminishing z - a we diminish f(z) A ad libitum, and therefore, by the definition (30), A is the limiting value of f(w) when z approaches a, as well as that of f(c) when v approaches a: from which the truth of the Lemma is manifest. Thus if M be a continuous variable, and z a discontinuous Examples. 1 variable in the form a + -, the limiting value of f () when n iv approaches a, is also that of f(z) when z approaches a. Again, the limiting value of f(x) when x approaches a is just the same thing as that of f(e") or of f(tan v) when e" or tan x respectively approaches a. 34. If f(x), when expressed in terms of another variable Lemma II. z, becomes 0 (z), and if x = a when z = b; then the limiting value of f (x) when x approaches a, and the limiting value of (P(z) when z approaches b, are the same thing. For let A be the former limiting value; then, since i = a LEMMAS RESPECTING LIMITING VALUES. when z = b, it is evident that by sufficiently diminishing z - b we may diminish x - a ad libitum, and therefore f (t) - A, by definition (30), and therefore (z) - A since 0 (z) = f(x); therefore A is the limiting value of (p(z) when z approaches b, by def. (30). Q.E.D. Examples. Thus if in the function f(v) we put x = a +, and therefore f (v) =-f a + = (z) suppose; then x = a when z = co, and therefore the limiting value of f(x) when x approaches a is the same thing as that of P(z) or f (a +-) when z approaches co. e' — 1 Again, if f() = and we put e6= z, and therefore - = log z, and therefore f(x) log (z) suppose; then x evidently = 0 when z = 1, and therefore the limiting value E - 1 of f(x) or when x approaches 0, and that of (p(z) or Z —I - when z approaches 1, are the same thing. log xV We shall find this method of transforming expressions very useful in finding their limiting values. LemmaIII. 35. Every Actual value is also a Limiting value. We have defined f(a) to be the result of performing the operations represented by f upon a; therefore, by the definition of an actual value in (19), f(a) represents the actual of f(x) when x = a. But, from the assumption stated in (29), and the definition of a limit in (30), it is evident that f(a) is the limiting value of f(x) when x approaches a. Hence the truth of the lemma is evident. Of course we here suppose that f(a) is not illusory. LemmaIV. 36. If A be the Limiting value of f(x) when x ap LEMMAS RESPECTING LIMITING VALUES. 17 proaches a, f (x) has the same sign as A for all values of x taken sufficiently near a. Since f(x) - A may be diminished ad libitum by taking x near enough to a, it is clear that f(x) may be made greater or less than zero, according as A is greater or less than zero; or in other words, f(x) may be made to have the same sign as A by taking x near enough to a: and when x is made to approach still nearer to a, since we so diminish f(a) - A still more, at least for all values of x near enough to a (see 31), f (v) must continue to have the same sign as A. Hence f(x) has the same sign as A for all values of x taken near enough to a. Q.E.D. 37. If f (x) be any function of x which becomes illusory Lemma V. when x = a certain value a, and if for each value of x (a of course excepted) f (x) has only one value; then f (x) cannot have more than one limiting value when x approaches a. If possible let two different quantities A and B be both limiting values off(v) when x approaches a; then we may make f (x) differ from both A and B as little as we please, at the same time, by sufficiently diminishing - a; therefore x may be so taken, that A and B shall differ from the same quantity f (x) since f () has only one value}, and therefore from each other, as little as we please; which is absurd if A and B be two different quantities; therefore A must be equal to B. Hence there cannot be more than one limiting value. Q. E.D. 38. Hence if we can prove that A is a limiting value of f(I) when x approaches a, we are sure that no other quantity but A is a limiting value, and therefore that A is the limiting value. Hence it appears that a limiting value is not a mere Alimiting approximation, but a perfectly definite quantity; for if it value isnt a mere apwere a mere approximation, then, when we had found a limit- proximate ing value A, any quantity differing very little from A would q be just as much a limiting value as A; contrary to what has been just proved. 2 18 LEMMAS RESPECTING LIMITING VALUES. We have supposed that the function f(x) has only one value for each value of Ix; if however it has more than one, n values suppose, it is evident that there will be n different limiting values, and no more, when x approaches a. Thus if f( )=a+b e f _ ) which has two values for each value of x, then there are two limiting values when v approaches 1; viz. a + b and a - b. LemmaVI. 39. If f(x) and ( (x) be two functions, one of which, f (x), becomes illusory when x = a certain value a, and the other, p (x), does not; and if we can shew that f (x) = ( (x) for all values of x, a of course excepted; then p (a) is the limiting value of f(x) when x approaches a. For by (29), we diminish (p (Z) - (a) ad libitum by sufficiently diminishing x - a; but in so doing we never suppose x to become actually equal to a, and we are therefore sure that f(v) = p (Ix); therefore, we diminish f(x) - 4 (a) ad libitum, by sufficiently diminishing x - a; i.e. (p (a) is the limiting value of f(x) when x approaches a. Q.E.D. Cor. 1. If ( (a) be illusory as well as f(a), and if we know A to be the limiting value of i (x) when.r approaches a; then we may shew, in exactly the same way, that A is also the limiting value of f(x) when x approaches a. Cor. 2. If, instead of being able to shew that f(v) = (p (x) for all values of x except a, we can prove that f(x) - (x) is diminished ad libitum by sufficiently diminishing x - a; then the same conclusions evidently follow; that is to say; the limiting value of f(v) when x approaches a is (p (a), or A if tp (a) be illusory, Lemma 40. If f (x) be a function which becomes illusory when x = a, and if we can prove that f (x) lies between another function (p (x) and a constant A for all values of x taken sufficiently near a; and moreover, that A is the limiting * When we say that f(x) lies between p(S(a) and A, we mean that f(z) is not greater than one of these quantities and not less than the other. LEMMAS RESPECTING LIMITING VALUES. 19 value of ( (x) when x approaches a; then A is also the limiting value of f(x) when x approaches a. For since A is the limiting value of b (v) when x approaches a, ( () - A is diminished ad libitum when s - a is sufficiently diminished; therefore, a fortiori, since f(x) lies between cp (v) and A, f(x)-A is also diminished ad libitum at the same time; therefore A is the limiting value of f(x) when x approaches a. Q.E.D. 41. If U, V, W... be any functions of x, and f(U, V, Lemma W...) any function of these functions, and if A, B, C... be the limiting values of U, V, W... respectively when x approaches a certain value a; then f(A, B, C...) is the limiting value of f(U, V, Vr...) when x approaches a. For by sufficiently diminishing - a we diminish U A, V - B, W- C... ad libitum, and therefore, by the assumption in (29), we diminish f(U, V, W...) - f (A, B, C...) ad libitum; therefore f(A, B, C...) is the limiting value of f(U, V, W...) when i approaches a. Thus the limiting value, when x approaches a, Examples of U - V is A B, that of UV is AB, U A that of U is -, y I' that of (U2 + V2) sin W is (A2 + B') sin C. In the proof of this Lemma, since we only speak of the limiting values of W, V, W... we make no supposition as to whether U, V, W... become illusory or not when? = a; so that the Lemma is equally true whether they do or whether they do not. If any of these quantities, U suppose, does not become illusory when x = a, then A is of course its actual value. This Lemma evidently fails when the substitution of one or more of the quantities A, B, C... for U, V, W... respectively makes f illusory. 2-2 20 LEMMAS RESPECTING LIMITING VALUES. Example 42. The following example will shew the use of Lemma of the use of Lemma VIII. combined with some of the preceding Lemmas. VIII.combined with 2 22 2 some of the 2iz - 2-+2 preceding Let U= and V =, Lemmas. -2, an -- 6 and suppose we wish to find the limiting value of U2^ + UV + loV2. U2 + UV+ 0 V sin-{((U 7) V}, loV- U when x approaches 1. x —2 Then, dividing out x - 1, U = except when x = 1, o + I i0-2 and = - 2 when = 1. Therefore by Lemma VI. - 2 ci + 2 is the limiting value of U when x approaches 1. z" - 2z Again putting 2 = Z, V becomes - 6, and x = 1 when z = 2; hence by Lemma II. the limiting value of V z. - 2z when x approaches 1, is the same thing as that of z - 2 + % - 6' when z approaches 2. Now except when z = 2, z2 - 2 z 2 — - which = 1when z=2; z2+z-6 z+ 3 therefore by Lemma VI. 1 is the limiting value of V when x approaches 1. Hence by Lemma VIII. the limiting value of U2 + UV + U 7) VV 10V- U when v approaches 1 is 5 25k I0.5vr 4- +25 sin-' (-2+7) = 2 I10 +2 5 43. In proving the above Lemmas we have supposed, that functions become illusory only for isolated values of the variable; or in other words, that, if f(a) be illusory, f(v) LEMMAS RESPECTING LIMITING VALUES. 21 is not illusory for all the values of x extremely near a, but only for the single value a. This assumption is evidently true for all ordinary functions. We have not assumed the existence of a limiting value when the function becomes illusory in any of these Lemmas, inasmuch as they are worded in this manner; "If A be the limiting value then such and such things follow": nor shall we have any occasion to make this assumption; for in all cases where we have to consider the limiting values of functions when they become illusory, the existence of such values will be proved and not assumed. We may just observe however, that the existence of limiting values of functions when they become illusory is a necessary consequence of their becoming illusory only for isolated values of the variable, and of the assumption in (29), as it is not difficult to see. CHAPTER IV. CERTAIN LIMITING VALUES DETERMINED WHICH WE SHALL REQUIRE TO KNOW HEREAFTER. LemmaIX. 44. IF s be the length of any arc of a curve, and c S the length of its chord; the limiting value of - when s approaches zero is unity. Let PSQ be the arc (fig. 7), PCQ the chord, draw PR the tangent at P, and QR perpendicular to PR, and let z RPQ = w: then, as long as P does not actually coincide with Q, we may evidently assume that PSQ lies between PCQ and PR + QR (supposing of course that Q is taken near enough to P, so that the curve shall always bend towards the same side between P and Q); i.e. we may assume that s lies between c and c cos w+ c sin w; and therefore that - lies between I and cos w + sin w. c Now, by the definition of a tangent in (25), 0 is the limiting value of w, and therefore 1 that of cos + sin w, when s approaches 0: therefore by Lemma VII, 1 is the limiting value of - when s approaches 0. q. E. D. c Lemma Xo 45. If 0 be an angle measured by the subtending are 0 0 of a circle, the limiting value of i —, and that of -- when 0 approahsin tan i0 when 0 approaches rero, are each unify. CERTAIN LIMITING VALUES DETERMINED. 23 We may evidently assume, for all values of 0 less than 7r', that 2 0 lies between sin 0 and tan 0, 0 1 and therefore that lies between 1 and sin 0 - cos 0 Now = 1 when 0 = 0; hence by Lemma VII. 1 is cos 0 the limiting value of s- when 0 approaches 0. Q. E. D. sin 0 46. We might have deduced this result from the previous Lemma, by putting s = 20, and c (which now becomes the trigonometrical chord of s) = 2 sin 0, and therefore s 6 c sin 0 Xn -- 1 47. To find the limiting value of when x LemmaXI. x —1 approaches unity, n being any number positive or negative integral or fractional. It is clear that, whether n be positive or negative fractional or integral, it may always be expressed in the form —, where p, q, and r are positive integers. Hence we may put p-q -n - 1 I xr - 1 ZP-q — 1 a —_ = — if we put x = _ x-1: _ 1 X 1 (zP- 1) - (zI - Hence, dividing (zP - 1) (q - 1) and (r - 1) by - 1, observing that p, q, r are positive integers, we have 9'_ -1 1 (1 + z + z......p terms) + (1 + +z.....q terms) xc 1 zq 1 + o + z2...... terms except of course when z = 1, 24 CERTAIN LIMITING VALUES DETERMINED. Now when x = 1, z = 1, and therefore the actual value of the second member of this equation is evidently -- or n: n -_ 1 hence, by Lemma VI, n is the limiting value of W- 1 when x approaches 1. Lemma 48. To shew that there exists a limiting value of x1. n r 1 + - when x approaches infinity, n being always an integer. By the Binomial Theorem for positive integers we have I1\/ n 1 n.n -1 1 1+- =- _ 1 + --- + 2 —..... to (n+ 1) terms n) 1 In 1.2 X f1 \ / \/ 2\ -l 1+ 1(-1 -l) 1 (1 — (1 — 1 ) ( 2 ) (1) where r2, F3,...... and in general Fr, denote (1.2), (1.2.3),...... (1.2.3...r), respectively. Now in the series (1) all the terms are evidently positive, and when n increases each term increases, and the number of terms also increases. Moreover, this series is term by term less than the series 1 1 1 1+ 1+ - + -...to n + terms, 2 22 23 which is < + or 3. 1 —1 Hence 1 +, which = 2 when n = 1, continually inl-z~ 1 1z' A geometric series + z+z2......top terms = - Now if 1 be positive and < 1, 1- is a positive quantity, and therefore the series = --- - a positive quantity; therefore it is < 1 -, and this is true no matter how large p may be. Hence, if z be a positive quantity less than unity, 1 + z+ z2 + Z3...... continued to any number of terms is < 1-. F7-7 CERTAIN LIMITING VALUES DETERMINED. 25 creases as n increases, but never comes up to 3, no matter how large n becomes. Therefore it is clear that there must be some number between 2 and 3 to which we may make (1 + - *approach ad libitum by making n approach o, which number is the limiting value whose existence we wish to prove. 49. To determine numerically the limiting value of Lemia (1 + n)n when n approaches co. The (r + i)th term of the series (1) in the last article is r 1( -n)( -n)(1.n) ( ) 1 r ) — i Fr... n n n which is clearly <-. r Hence the (r + 1)th and following terms of the series (1) form a positive quantity less than 1 1 1 rr (r +l) +r(r+2) 1 ( 1 which is <- 1 +- +-. ) rr r r12 r which (by note, 48) is or, o(r-_i)r-i 1 — which is < - r(1. - 1 ) Hence 1 + ) = 1 + 1 - 1 -4+ ( - )( 1 --...... to r terms 13 \ n] \ nf + a positive quantity less than ---- r(r -) * By saying that one quantity u approaches another v ad libitum, we- simply mean that u, v diminishes ad libitum. 26 CERTAIN LIMITING VALUES DETERMINED. If therefore we suppose r = 12, and consider only the first seven decimal places, it is clear from the table in the note*, that 1 +- = + 1 + - 1- fn r2 n/ +A- 2-)- 1 —...... (to 12 terms; FI \ nf n) and this is true, no matter how large n may be. Now by making n approach o, the second term of this equation may, term by term, be made to approach ad lihitum to the series 1 1 1+ 1 + -+..... to 12 terms; F2 F3 which to 7 decimal places = 2.7182818 (by table, *). 1=1.00000000...... 50000000...... 2-.16666666...... P3-.04166666...... r4 =.00833333...... 1 0 0 1 3 This table is easily formed by 888.6-8 dividing 1.00000000 by 2, the re1 suit by 3, the result so obtained =.00019 8 4 1...... by 4, and so on. -8 =.00002480...... 1 I-=.00000275...... ~L.00000027...... PlO - F10 - 00000002...... Sum to seven decimal places =.7182818. CERTAIN LIMITING VALUES DETERMINED. 27 Hence if we consider only the first seven decimal places, 2.7182818 is the limiting value of 1 +-), when n approaches o. 50. In the same manner we might obtain this limiting Base of Hyperbolic value to any greater number of decimal places. It is, how- Logever, an incommensurable number like 7r, and therefore, no arith'ns; matter how far we go, we shall never be able to obtain it exactly in numbers. It is usually denoted by the letter e, and it is taken as the base of a system of logarithms commonly called Hyperbolic. For a certain reason, which we shall presently explain, logarithms calculated to base e are the most convenient to use in analytical calculations. We shall therefore always suppose that e is the base of whatever logarithms we have occasion to employ unless the contrary be specified, 51. Hence e is the limiting value of (1 + x)x when Cor. 1. x approaches zero, x being any continuous variable. For this limiting value by Lemma I. is the same thing as the limiting value of (1 + z)z when z approaches 0, where =- and n is always an integer; and this latter limiting n value by Lemma II. is the same thing as that of (1 + — when n approaches co, which = e. Q. E. D. 1 * 52. Hence, logae or g is the limiting value of Cor. 2, loga lo0g (1 + x) a ( + — x) when x approaches zero. x logr ( ) + o) o For ga (- =log,(l + )V, and by what has just been proved, and by Lemma VIII. the limiting value of this latter quantity when, approaches zero is logae. * If log, e = c, then a = e, and,'. log =nl, or C-. I = Tog~~~~ aof 28 CERTAIN LIMITING VALUES DETERMINED. Cor. 3. 53. Hence also, o or log a is the limiting value of logae ax - 1 when x approaches zero. x For, put a - 1 = z, and.~. x=loga (1 +- ); a - 1, 1 then - = 0 log (1 + ) loga (1 + z) Now x = o when z = 0; hence by Lemma II, the limiting ax - 1 value of when x approaches 0, is the same thing as that IV of when z approaches 0 which, by what has been log (1 + z) just proved, = or log a. log~e CHAPTER V..f (X') -f(x) RULES FOR FINDING THE LIMITING VALUE OF X - X 54. WE now enter upon the Differential Calculus, properly so called, which, as we have stated, is that branch of mathematics whose object is, in the first place, to determine a f(afj -f(x) set of Rules whereby the limiting value of f () f ) when x' approaches x may be found in all cases with facility; and in the second place, to shew and explain some of the principal uses which may be made of this limiting value in pure mathematics. We proceed therefore, in the first place, to determine a set of Rules whereby this limiting value may be found in all cases with facility. f(x') -f(x) Notation 55. We shall represent the limiting value of ' by which X - X vwe represent the when x' approaches x by the notation f'(x); i. e. by simply limiting value of dashing the functional letter. f(lu)f( _/~~7~1> / f~) / \ f 2 3C -fa Thus if f () = - f ( - f (V) l = 'V,V Examples. - X - -X except when v' actually = x; now the actual value of v' + x when,v' = x, is 2v; hence, by Lemma VI., the limiting value '2 -- 2 of -- when v' approaches x is 2x; i.e. X0 - * U if f (G) = G2, then f '-() -= 2o. Again if f (x) = ax + bx3, v v,v -- a +1.(v _. - 30 RULES FOR FINDING f' (x). hence, as in the previous case, we find that if f(x) = ax + bv3, then f'() = a + 3blk. And in general we may very easily shew in the same manner that if f (v) = aO + ale + ax2.....a,", then f'(x) = a, + 2ax + 3a,2x... + nax,-'l. Thus when f(x) is a rational and integral function of x, f' (x) may be derived from it by multiplying each power of x by its index, and lowering that index by unity. Origin of 56. This notation f' is due to Lagrange: the function the notation f'(x), and f'( ) was called by him the Derived Function or Derivative eth-enave of f(x), because it may be derived from f(v) by the peculiar Derivative or Derived process just stated, at least when f(x) is a rational and inFunction. I tegral function of x. The following are the Rules whereby we may find this derived function or derivative in all cases with facility. Rule 1. 57. If f (x) = a constant c*, f' (x) = 0. f(x) =c. f(X') - f(x C-C For then fv) -f - = 0, except x'= x;... lin. val. of f () f () when x' app. 'v is zero, by Lem.VI., X0 - X i.e. f'(r) = 0. Col. If f () = c + (p (x), f'(x) = ' (v). For then f(x') -f( () - (x') - (x) I -IW Z - I and.-. f'(v) = (p'(x). by Lem. VI. Cor. 1. A constant therefore added to a function does not appear in the derived function. " We have seen in (5) that a function of A' may be a constant quantity. RULES FOR FINDINGl f (). 31 58. If f(x) = c/(x), 1i denoting any function; then Rule II. f'(x) c '(x). =(cx(). For then f() - f().= c ( v) - ) (,), the limiting value of the second member of this equation when V' approaches x is cp'(<v): hence, by Lem. VI. Cor. 1, f' () = c~' (j'(). Rule ITT. 59. If f (x) = (x) _ + ((x) = X (x) 4 &c., (p, + X, &c., =(x) denoting any functions; then f'(x) = p'(x) -,/, (x) = X' (x) 6 &c. or then - f () (') - (V) (x) - () For then ' -, t ( iv SO - iv = x ( ')- x (~) =... &c. The limiting value of the second member of this equation when ' approaches;v is /' (iv) a +' (m) = x (mv) -1... &c., by Lemma VIII.; hence, f' (a) = /' (v) - +,' (v) 4- X' (iv) =... &c. 60. If f(x) =q(x).(x)x, then Rule IV. f' (x) = P' (x). (x) + 0(x).,'(x). f(^) -(x (,) For then f(v') f () MI - tt (i)') (v') - p (x) + (0') + 0 (,v) q ()') - %V (m), (x),v1 - 0. t -:(*v') -.p)(i) X+(iv) - 4,(v); now the limiting value of the second member of this equation when x' approaches w is p'(x) \,(iv) + E(v)\ x'( ), by Lemma VIII.; hence, f' () = p' (i), (m) + (p (,)) +' (,v). 32. RULES FOR FINDING f' (X). fo. (x) f6) 95(x) Cor.1. 61. If,(x) then f (X) 0s (X) f'(x)-)-9()(x) For then f(x) + Qx) = 5(x(); and, as in the rule just proved, we have f' (X) 4' (0) + f W) 4" (x) = 9'(u); whence f'(a,) = ( f x4" 4' (X) )' Qe) N'" (x). - ()) 4" (X) putting for f(x) its value. Rule V. 62. If f(x) =xX, then f'(x) = nx': '; whether n be f integral or fractional, positive or negative. f (,V) (~n>~" m For then f(x' f x" X - X (V X- L nn- _ Z if we put x'= vz. z-1l Now X' x when z = I, therefore by Lemma II. the limiting value of the second member of this equation when z approaches 1 is the same thing as that of the first member when x' approaches x; but by Lemma XI. n is the limiting z'n - i value of 1 when z approaches 1. Hence f'(x) = nmv". Examples 63. By means of these five Rules we may find the deriof these vative of any rational function of x. For example: five rules. Let f (X) = aoe+ aix + a2IV......a.,I then by Rule III. f'(x) is found by taking the derivative of each term separately; also by Rule I. the derivative of a, is 0, and by Rule 1I. the derivative of any other term a C"' is aMl x the. derivative of x"', or aM mrn lg by Rule V. RULES FOR FINDING f (X). 33 Hence f'(x) = a, + 2-am + S3at m...... naa"!. <2 -a2 da (?) Again, let f(x) = + = ( suppose; then 0'(x)) = 2X, +'(x) = 2x, and therefore by Cor. Rule IV. 2a (?2 + a2) - (X2 -a2) 2a 4a2' (J ( + a')2 (2 + a")2 Again, let f (x) = A x + Bx-3, then f'(x) =.Ax - 3Bx-4 by Rules II. III. V. Again, let f(x) = (x + a-(2) (X + X-~) = (p( X). + () suppose; then (p'(x)=2 2 - - 2 '-3, 1() = - a -; hence by Rule IV. f' (0) = 2 ( - -3) (a^ + a?-) + (' + 4- -2) (?- - )-) (For more examples see Appendix A.) 1 1 Rule VI. 64. If f(x) = ogx, then f'(x) - f(x)=loga.x log a x For then f (') - f (x) logax' - log<a 1i log,(z)- loga. = _ - g if we put! =-= 1 loga z Now when 1, heefoe by Lemma I, and Now x'= when z = 1, therefore, by Lemma II., and 1 I Lemma XIII. Cor. 2, -—. is the limiting value of loga, x f(I') -f(a) 1 f(')-f) when x' approaches x, i.e. f'(x) lo a* 3I - f' log a x 0 O 34! RULES FOR FINDING f'Qv). Cor. ~~~~~~~~~~~~~~~~~~~1 Cogn 'if a = e, then log a I=, which gives f'(v) -. f(xt)=logx. 2 by e iso It appears therefore that we get a very simple expression usually for the derivative of the logarithm of iv when e is taken as taken as the base. It is for this reason that this base is chosen in base of h ae logarithms, all analytical investigations, except when numerical calculations come in, in which case 10 is a much more convenient base. fule VII. 65. If f(x)= a, then f'(x) = log a. a f(x) = a" For then f (W).-f(v) ax' - ax tIV - CIV ( - X az 1 =ax. if we put iv'-iv=r. Now iv' =x when z 1, therefore, by Lemma II., and by Lemma XIII. Cor. 3, ax log a limiting value of Iv'f v when x' approaches i. Hence f (v) = log a. ai. f (X) C-. If f(iV) = e5, then f'(iv) = co, since log,e = 1. Rule VIII. 66. If f(x)=sinx, then f'(x)=cos x; and if f(x)=cosx, frxCOsinX then f' (x) = - sin x. For if fi(x) = sin iv f (') -f(i7) sin v - sin Iv iv-iv.Iv-ivt 2 cos sill 2 2 v -x)in - cos (i v~ v) if we put Now the limiting value of the second member of this equation when z approaches 0 is cos v, by Lemma VIII. and RULES FOR FINDING f' (X). 35 Lemma X.; therefore, since a' = when z = 0, we have by Lemma II. f'(x) = cos x. And if f(v) = cos x we find similarly! -)- Ot c ' -! 2 sin sin --- f (') - f () 2 2 0a - 0 00 - - - sin (Z + z) sin w; and therefore f'(x) =- sin x. 1 COS IX Cor. Hence if f() = cosec = s f'() = - in by f/(x)-se fc (? sin," ( 2 or cosec x. Rule IV. Cor. 1, putting ((x) = 1 and +(F) = sin x. sin x And if f(v) =sece= -, f'( = - similarly. 67. If f(x)= tanx, then f'(x)=.2 and f f(x)=cotx, Rule IX. ~67, If~x=ax t e f2Xf(x)=tanx or cotz. then f'(x) = - For if f(x) = tan x, f(x') -f (x) tan x' - tan z X X,_ 1 sin (' - x) cos a cos a -X -a 1 sin,, _= s i.-___ ---. -- if we put X' xo z.i Cos (x + v ) COs X z Hence, as in the former case, we have f'(x)= c-. COS X And if f(x) = cot x, f(V') - f( ) 1 sin (x' - x) - ( sin, sin X' -X and therefore f/'() = - sin2 x 36 RULES FOR FINDING f/ (). Cor. This Rule may easily be deduced from the preceding, This rule deducible as follows: from the preceding. sin x precedig. If f() tan =s then by Rule IV. Cor. and by Rule VIII, cos x cosv - sin x (- cos ) 1 COS2 2G COS2.2 Cos X and if f(t) =cot =.s, sin O' - sin x sin - cos x cos 1 then f'(?)= s(x = sin-.v sin i sin~ x (Rule X. 68. If f(x) = sin-x, then f'(x) --, and if or cos-1'. X.1 1 f(x) = cos- x, then f'(x) = - For iff( x)= sin-'x, and therefore x = sinf (x), we have f(i') -f(X) f(X') -f(X) iX' - X sinf (') - sinf (x) f ((V') - f (I) f(') + f(X) f( ') -f(X) 2 cos sin 2 2 1. cos {f(v) + z} sinz' f(') -f(Xv) if we put= 2. 2 Now x' = when z= o; hence, by Lemmas II. and X. we have I 1 f' o (X() =,, since x = sinf(x). And similarly, we may shew that iff (x) = cos-1 x, f'(X)= - 1 - 2 V/1 m, RULES FOR FINDIG f' (X). 37 69. If f(x) tan-' x, then f'(x) = 2 and if Rule XI. + x2 f(x)=tan-'x or cot-1x. 1 f(x) = cot-', then f'(x) = - 1 + x2 For if f () = tan-1 x, f(3') -.f() f(X) -f(?) x - x tan f(') - tan f(x) f(a') -f(v) = cosf (') cosf (re). sin {f) - f() ( = cos If(V) + -} cosf(?). ---, if we put f (x') - f (x) = x. Hence, as before, f' (() = cos2f(x) = + since tanf() = a. 1 }-,n And similarly, we may shew that if f() = cot-' (a), f'(J ) = 1 + 2x We have proved Rules VII, X, XI, directly, but they may be deduced very simply from Rules VI, VIII, IX, by means of the following Rule, as we shall shew. 70. If we are given x = 0 (y), in which case y will be Rule XII. If x=p c(y) 1 and thence some function of x, f (x) suppose; then f'(x)= =. hi to 9(yf) findff x). t For then f (x) f( = y- y_ -_ p(y) - p (y) +P (Y) - 0p (Y) y -y Now, by Lemma VIII, the limiting value of the second member of this equation when y' approaches y is, —, and 0p (y) c' = x when y'= y; hence, by Lemma II, f' = (y)' 388 RULES FOR FINDING f '(x). Rules VII. 71 This Rule is of great use in finding the derivatives duced from of inverse functions, and by means of it may deduce Rules VII, vlI. Ix. X, XI, from Rules VI, VIII, IX, respectively, as follows: by means of Rule XII. If x = logay = +((y) suppose, then y = a" =f(x) suppose; and therefore, since f'(x)= (p- and p'(y) = oga by Rule VI, (y) log a y we have f'(x) = log a. y = log a a"; which is Rule VII. If a = sin y = ( (y) suppose, then y = sin-' a =f(x) suppose; and therefore, since 1 1 f () = - = -c by Rule VIII,, ( Coy we have f'(a) = 2, since cosy V/1 _- 2; which is 2/1-,2 Rule X. If x = tan y = p (y) suppose, then y = tan-' = f(x) suppose; and we have therefore f'($) =, = Cos2y, by Rule IX. which = 1 a, since fl( (Y) 1 +.i tany = x; which is Rule XI. Rule XII. 72. We may also by this Rule obtain the following thlsedtof derivatives which it is sometimes useful to remember. sec-' x vers-'1 If x = sec y = / (y), then y = sec-' x =f(x), and we have therefore f '(c) == _ = osY by Cor. Rule VIII.= 1 qv(a? ) sin y -1 1 since cos y = Hence, if f(x) = sec- x, f' (x) = x -/x - 1 Again, if a = vers y = (y), then y =vers -' = (), and we have therefore RULES FOR FINDING f/ ( ). 39 f'(V) = - -, by Rule VIII, since versy= -cosy (Y) sin y 1, since cos y = 1 -. Hence, if f(x) = vers-' x, f' (x) = - ~2x - x2 73. If we are given two relations between x and y by RuleXIIIo the intervention of a third variable v in the form =If y=() and thence y = 0(v), and v= (x), ind(to aind f'(x)) in which case y will be some function of x, f(x) suppose; then f'(x) = q' (v) +/(x). For then we have f(v')-f(x) p(v') - Q(v) 4(') - f(,) X -v v - v i -I Now, since v' = v when x' = x, the limiting value of (v') - p (v) when x' approaches x, is the same (by Lem. II.) v - v as that of q(v) - (v) when v' approaches v, which = 0'(v); V -v hence, by Lem. VIII, the limiting value of the second member of this equation, when x' approaches x, is 0'(v) / '(x); and therefore f' () = ' (v) r' (x). 74. In the same manner we may shew, that if y = ((v), Cor. v = (u), u% = X(), and therefore y some function of I' beryofu f(x) suppose; then f'(x) = P' (v) x' (u) X'(x). And a simi- variables intervening lar result if any number of variables intervene between between i~~~yand~~~~~~~~~. ~~~y and x. y and x. For as before we have f(x') -f () _ (')- - (v) (u') - (x') - (V). X- - V U- u X -,v and.. f' (x) = i' (v) ' (u). X' (x). 40 RULES FOR FINDING ' (I). Examples 75. This Rule is of great use in finding the derivatives of the use ofthisule. of complex functions, as the following examples will shew. Let f((x) - b(c + x), put c + x = v = v (,r), and.*. \/'(() = 1; then f'(x<) = /'(v) = 0' (c + x). Hence the derivative of p(c + t) is ('(c + t). Thus the derivatives of (c + xt)", log (c + ), ec+x, sin (e + x), &c,, are respectively, n (c + i)"' ---, ec+" cos (c + ir), &c., by Rules V, VI, VII, VIII. Let f (x) = (c x), put c v = (x), and.-. '(W) = c, then f'(x) = p'(v) c = civ(c). Hence the derivative of p(cex) is c'l(cx). Thus the derivatives of eCx' tan' -, sin-'-, are respectively C C C C 1 ce' & 2 i+,, 1 - by Rules VII, X, XI. Let f (v) = / (an), put iv = v = (i), and.-.,'(() = uX-i' (Rule V.), then f'(iv) = Q' (v) n" -' = n / (i") n - ' Hence the derivative of 0 (x"), is n ( v'(xn) "-'. Let f(ix) = ee' put v = e= (wc), and,*. f(x)= ev = ((v); then f'(v) = '(v) +'(x) = eV"e (Rule VII.) = ee'. ee = e +e Let f(x) = e+ee, put v = x+ e" = (Xv), and.'. f(c.) = eV= (v); then f'(v) (v) = +'( v) '( eV (1 + e') (Rules VII, III.) ee (l + er). RULES FOR FINDING f' (X). 41 Let f (x) = log (x), put v = (x), and. f(x) = logv = (v); then f'(x) = p'(v) ' () = '() = (x) v x(x) (For more Examples see Appendix B.) 76. The derivative of a complicated product may often Rule XIV. The derivabe conveniently found by taking its logarithm first. tive of a complicaLet f () a = { () } I( { () }t {x rd) *ouct.....~13 fY-') LXV-'1 found by logathen logf () m= n log ( (x) + n log f (xv) + p log X ( r)... rithims. hence, taking the derivative by Rules XIII, III, and II,.f' () m~' __) q(X) X' (a) )= + n + p f (") b (X) c(x) X () which gives f'(x) very readily, as the following example will shew. Let f ()) = e' V/1+-; Example. 1 - C% then logf( x) + I log +)-logg (1 - );.f'(IV) 1 1 2 - 1C f. () )=1+ l 6 + - 1 -- 1 - -x sin (. v/1 - v2 Let f(x)= - 2-; then logf (x) = log sin x + - log (1 - &r) - 2 log x; f' (?) - x 2 2 - ~2...: cot = cot x - --; f(X) 1 -Vv2 X2 i(1 — 2) sin /1 - x2 2-2 ) ~', / (Vt?) - ' =2 ----, - cot Xv - - 42 RULES FOR FINDING f' (X). Let f(x) = X x; then logf(t) = xlogex, and therefore we have.f' () 1 - = log o +. -; f () v'.. f' () = a (logx + 1). ft 77. * To these Rules we must add two more, which will be found of great use in many cases, and important hereafter, when we come to consider more than one independant variable. But we must first explain a certain notation which we shall make use of. Partial de- If f (vy) be any function of two variables x and y, we rivatives; what. The shall assume f'(wy) to denote the limiting value of notation q,'(ay) and Y) when v' approaches v. It is evident that 0P'(xy). Q' - iv this limiting value is the derived function of f(xy), taken on the supposition that y is constant and x alone variable. It is called the Partial Derivative off(xy) with respect to x. In the same manner we shall take f'(Xy) to denote the limiting value of f(y') -,f(,) when y' approaches y; y -y i. e. the Partial Derivative of f (xy) with respect to y, or the derivative taken on the supposition that y alone is variable. Examples. Thus if f (Xy) = 'v2 + iy + y2, then f'(xy) = 2x + y, and f'(.:y) = y + 2i. Again, if f(xy) = x sin y + yet, f' (y) = sin y + y e, and f' (x) = x cos y + e". Thus by dashing the functional letter we denote a derivative of a function of two variables, and by a dot under one of the variables we signify that that variable alone is supposed to vary. This nota- In using this notation we do not necessarily assume tion does not neces- that y is independant of x; for whether y be independsarily require that * The Articles marked thus tt - may be omitted till they are referred to, at least on a first perusal. RULES FOR FINDING f' (X). 43 ant of d? or not, we may always put v' in place of jfshould be f (X'y) -f~xy) independwe please, and so find f(x'y), and thence ant of each iv - iv other. and then, by the previous Rules, we may determine the limiting value of this quantity when x' approaches x. And the same may be said of f'(xy). The following is the Rule for which it is necessary to make use of this notation or something equivalent to it. t t 78. If we are given a relation between y and Rule XV. two other variables, u and v which are functions of x; i. e. if Sr (UV), u= U f(X), v = X(x), in which case y will be some function of x, f (x) suppose; then f' (x) = /i' (uv) Wfr' (x) + /i (U v) x' (x). f(') -f () _/ (u' V) - /(UV') q5 (uV') - qp (u v) For +, x Ml X X x - x and 1PKu'v')-5(u'v') _ ~(U'V) -(UV'). 'kk (Wv) - xJ (iv X X UUX - xU ~ (UV) - (Uv) - (UV') - - (Uv) x (I') X (x) iv-iv Llv -V X -( Now when x' approaches x, the limiting values of J(v ') (,~v) ((UV')- 4 (Uv) x _ _ x (V) and are respectively x'(w), (P' (u v), and X'(x). Also since v' becomes v when Iv' iv, the limiting value of u-u * If we suppose for a moment that v' is not x (x') but some constant, then p'(uQv') is the limiting value of when x' approaches x: there~ ~~~~p~~~~ (~u whe )' ppoacesavthre fore q)U'v' P(UV ') may be diminished ad libitumn by sufficiently diminishing x' x; and this is evidently true whatever be the value of V; therefore it must be true if we suppose v' to vary in any manner while we diminish -x; it is therefore true if v'=X(x'), and therefore by Lemma VI. Cor. 2, '(Vuv), which is the limiting value of pV'(? v'), is also that of ~(u'')-, (uv') supposing v' = x (x'). 44 RULES FOR FINDING f (X). (h(u'v) -(UV) is evidently the same thing as that of (u') - u ) which is 0' (uv). Hence we have f' (x) = )' (u ) ' (x) + ' (u ) X (V) Examples. Let q/ (fuv) = U, then q'(.uv) = vu"'- and op'(uv) =log u uv; f-. f' () = v -L1 U /I (X) + log u " X' (V). Suppose that u = xv and v = x, and. '.,'() = 1 X'(t)=1; then f' (s) = x + logx x = vx (1 + log x). Again, let c (uv) = u2 + v2 - uv; then b'Y(uv) = 2u - v c'(uV) =2v - u;.'. f' () = (2u - v) #'(i) + (2v - u) X (,). Cor. If y = (uvw) w being another function of x, i (X), suppose then by putting f(x') -f() in the form {c(Zu'V'w') - ( (uv'w')} + { (cuv'w') - 5 (uvw')} + [/(uvw') - (uvw)}. We may shew exactly as before, that f () = -' ( uvw) ' (a) + p'(u) W) X'() + cP'(uv w) [ (W). This Rule we may evidently extend to the case where y is a function of any number of functions of v. Rule XVI. tt79. If an equation be given between x and y, which of course makes y a function of x, we may find the derivative of y by means of Rule XV. For, let the given equation be + (yx) = o; in virtue of this equation y = some function of x, + (x) suppose; and 0 (yv), by substituting for y this value, becomes also a function of x, f (M) suppose: then, by Rule XV, putting y for u, and v for v, and therefore X'(v) =, we have f ()) = ' (x) +' (X) + p' (y.). RULES FOR FINDING f' (x). 45 Now, f (a) = 0, and therefore by Rule I, f' (x) = 0; therefore we have O> (.:) ' (,m) +,' (y.) = o; ( ) ) ' (y) or,, f () = ) Let the given equation be Example. 2 + y2 - axy + b =0, here, 'Q(y) = 2y - ax, 0'(y ) = 2x - ay; *'. if + (v) =y, (2y - ax) f'(x) + 2x - ay =0;, - ay.or, (' (-i) = - t y- ax CHAPTER VI. THE DIFFERENTIAL NOTATION. 80. WE have hitherto adopted Lagrange's notation, f(X) -f( ) f'(x), to represent the limiting value of,f() when x' approaches x; this notation is often very convenient, and is perhaps more easily understood than any other, at least when the student enters upon this subject for the first time; and this is the reason why we have used it in obtaining the preceding Rules. But there is a far more elegant and powerful notation, due to Leibnitz, called the differential notation, which we now proceed to explain; not however for the purpose of abandoning the former notation, for we shall often make use of it, as it is preferable in certain cases to any other. Theratio 81 In the differential notation instead of representing of the sym bols d-f (x) and dX the limiting ratio of f(- when.' approaches,, by a taken to O- - b epresent single quantity, such as f' (), we represent it by the ratio of two arbitrary quantities denoted by the symbols df(x) and db, which, for a reason we shall explain, are called the differentials of f () and v; the letter d being simply an abbreviation of the words "differential of." Definition We define df(x) and dX as follows: of df(x) and dx. df(x) df(x) and dx are two quantities whose ratio d dx is equal to the limiting value of (x- () when x' apX -x proaches x. Or more simply; df (x) and d x are two quantities such that df(x) = f(x). such that =d xW dx THE DIFFERENTIAL NOTATION. 47 In this definition, all that we say of df(x) and dx is One of the -. ' ". -, - /, quantities this, "that they are in a certain ratio;" therefore they are af(xt),te thiserefore they are df(x), dq arbitrary quantities, and we may give to either of them 'bitstey. any value we please, constant or variable, provided we df (x) give to the other that value which makes f( -f'(4). d x 82. It may be asked respecting this notation, is there Objection to this not some degree of vagueness in representing a definite quan- notation answered. tity f' (i) by the ratio of two arbitrary quantities? answered To this we may answer, that we often do the very same thing in common Algebra without any vagueness; we often m say let- represent such and such a quantity, instead of let m represent such a quantity. Thus, suppose it required to divide a quantity a into two parts in a given ratio: we might proceed thus. Let m and n be any two quantities which are in the given ratio, and let x and y be the two parts, X, qm then +y = a, and -=-; y n /m \ na ma - 1 y = a, or, y; and.. = \n 1m + n m + n Or we might proceed thus, let m be the given ratio, then iv a na - =, (m + 1)y= a; and.'. y= - -. y m+ m + It is evident that there is not the least degree of vagueness here in representing the given ratio by -, instead of simply by m *, * The chief practical advantages we gain by representing f' () by the ratio Advantages df(x) of the d seem to me to be these. We may often suppose df(x) or dx to have some notation df(x) value which will simplify our expressions (as will appear in changing the inde- dx pendant variable hereafter.) The application of Rule XIII. to complex functions is very much facilitated by using the notation dr(. We may often advantageously preserve the symmetry of our expressions by using this notation. It is almost 48 THE DIFFERENTIAL NOTATION. Origin of 83. Again it may be asked, what is the reason of using the ndf() the letter d before f(x) and x, to represent the terms of the tion ~dx 'ratio by which we denote /f( )? This question may be and of the term differ- answered by the following very brief account of the origin of ential. this notation and of the term differential. f (M') -f(x), and x'-x are the differences between corresponding values of f(x) and x, and they are often represented by a S prefixed to f (G) and x, in this manner, viz. Sf(x), S3; ~ being simply an abbreviation of the words "the difference between two values of." Now, Sf(x) and Sx become zero when v'= ix, and then their ratio X ceases to be a definite quantity; but so long as ' is not actually equal to x the ratio is a definite quantity. We may therefore conceive Sf(x) and Bx as small as we please though not actually zero, without rendering our conception of the ratio at all vague or difficult: indeed it is just as easy to conceive that a definite ratio subsists between-, /f(x) and 3x when they are in a state of extreme smallness as when they are of ordinary magnitude for our idea of a ratio is quite independent of the actual magnitude of the quantities composing it. THe differences Sf(x) and 4x when in a state of extreme smallness were called diferentials by Leibnitz (i.e. minute differences), and the symbols df(x) and dx were made use of by him to represent them; d, like S, being simply an abbreviation of the words "differential of." In all calculations into which these differentials entered he supposed them to be what are called infinitesimals, i.e. quantities so small, that they may without error be neglected compared with ordinary quantities, and on this supimpossible to represent what are called total differentials without this notation. It is peculiarly adapted to the case of definite and multiple integrals in the Integral Calculus. And it is a very expressive notation, which makes it peculiarly convenient in mixed mathematics; e.g. in the case of the principle of virtual velocities applied to an example. THE DIFFERENTIAL NOTATION. 49 position he investigated rules for finding df(x) in all cases, and made various interesting applications of these rules. In this manner it was that the differential calculus came into existence. 84. In order to explain more distinctly in what this Exampleof. the method method of Liebnitz consists, we shall apply it to the example of Leibnitz. given in (23)o According to Leibnitz, we may conceive the triangle PQO (fig. 6) to become so extremely small, that the line PQ shall coincide in direction with the tangent at P; in which case the sides of the triangle OPQ will become infinitesimal quantities; then OQ or f(x') -f(x) will be represented by df(x) and PO by dx, and we shall have tan QPO or tan PTX = QO (). PO' d Now let f(.) 4m W2 - (f + c:) (+ '- s ) then f(') -()f () or df(x) = - 2 = ) (- ) 4m 4m (2xt + d x) dx 4 m putting + + dr for Rm' Now, according to the supposition that differentials are to be neglected compared with ordinary quantities, we must consider Gx + dx to be the same thing as 2w, and therefore we have df(xt) =-dx, and -. tan PTX = ~2v ^m 2m which is the same result we obtained before. 85. Whether this is a strictly logical way of proceed- Where this ing may be fairly questioned, though the conclusion arrived method is at is true: for it is easy to see that the correctness of the Reason why the result arises from the compensation of two errors, namely, result is the erroneous supposition that the line PQ is coincident with correct. the tangent at P, which it never can be so long as the 4 50 THE DIFFERENTIAL NOTATION. triangle POQ has any existence, and the erroneous supposition that x +d dx is equal to 2x. We shall endeavour however to shew hereafter, by means of the principles already established, that this method of using differentials must always lead to correct results, certain precautions being taken; and this is important to shew, since this method is extensively used, and indeed must be used in many cases to avoid complexity. At present we shall say no more upon this subject, except just to remark, that df(v) this is the manner in which the notation d( has come do f(d) _ f(X\ to represent the limiting value of f( - ) when v' apX — X proaches x: for it is easy to see, by the example just given, that dfv(, simplified by neglecting the differentials df(v) and dx compared with ordinary quantities, will always be the same thing as this limiting value. We adopt 86. It is important to observe here, that we adopt the notation f( and only the notation f(x) and the term Differential. We do the term d v differential, not define df(x) and dx to be what Sf(x) and Sxv become but not the notion that when in a state of extreme smallness; all that we say is d f (x) and dx are in- this, that df(x) and dx are two quantities, be they small finitesimals. or large, whose ratio is equal to fJ (x), i.e. the limiting value of f() or f () - ) when (' approaches x. BO XI - O Geometri- 87. In the example just considered, if we produce PO cal representation of to any point U and draw US perpendicular to PU to meet the quanti- SPT in S; then ties df (x) and dRx. = tan PTS = lim. val. of7 f() when I' app. x, PU i -iv = f (f); hence SU and PU are the differentials of f(x) and x, since they are two quantities whose ratio =f'(x); we have therefore SU - df(x), PU = dw. RULES FOR DIFFERENTIATION. 88. Since f(x) and f'(x) represent the same thing, How f'() d 37 came to be called the we have differential coefficient. df (x) -f (x), and.. df (x) = f (x) d, dx f' (x) is therefore the coefficient by which we must multiply the differential of x to obtain the differential of f(x); on this account f'(x) has acquired the name of " the differential coefficient," and it is in this way that d (x) which =f'(x), has come by the same name. The obtaining of the relation between dy and dx is called Differentiation. 89. We may express the Rules Chap. V. in the differ- The rules obtained in ential notation by simply putting last chapter stated in the differential df(x) d9((x) d+((x) notation. dx' dx ' d for f'(x), (p'(x), 4'(x), &c. respectively, and multiplying up d. For the sake of neatness we shall put y, u, v, for f(x), /(x(), +(x), respectively: and therefore dy, du, dv, &c.... for f'(x) dx, p'(xG)dx, qI(x)dx.e.&c. In this manner we have Rule I. If y = constant, then dy = 0. Rule II. If y = cu, dy = cdu, Rule III. If y =u+v+w..., dy=du+dv+dw... Rule IV. If y = uv, dy = vdu + udv. u vdu - udv If y -, dy= O V2 Rule V. If y = x", dy = nx'1 dx. 4-2 RULES FOR DIFFERENTIATION. Rule VI. If y = logx, dy= d X Rule VII1. If y = ax, dy =log a.a~vdx. Rule VIII. If y = sinx, If y =cos X, dy = cos dx. dy = - sin xdx. Rule IX. If y = tan x, If y = cot X, RIule X. If y = sin' a If y =cos -'a,Rule XI. If y = tanQ, if y = cot-' C dx dy = ~ x dr dy =d dx d, dy =2 dx 4 -~/ -m r, dy=1 + 'F2 Rule XII. 90. Rule XII. shews that if we obtain dx in terms of dy from the equation x = j(y), the value of so obtained is the dx proper value; i. e. the same value that would be obtained if we found y in terms of x directly, and then differentiated. For by differentiating the equation x q 5(y) as it stands, we find dy 1 dx = qY(y)dy, and dx dx 5'(y) Now if f(x) be the value of y found directly in terms of x, we dy have i =f' (x); which is the same as the former value, since X2L RULES FOR DIFFERENTIATION. 53 f'(x) =,'() by Rule XII. Hence, whichever way we prof (y) ceed, the value of obtained is the same. do 91. Rule XIII. shews that if we find dy in terms of dv RuleXIII. from the equation y= 0((v), and dv in terms of dx from the equation v = +(x), and so dy in terms of dx; the value of thus found is the proper value; i. e. the same value that dx would be obtained if we found y in terms of x directly by eliminating v, and then differentiated. For by differentiating the equations y = (p(v), v = f (x), we obtain dy = p'(v)dv, dv = '(x)dx, and.'. dy- = () () (). Now if f(,) be the value of y found directly in terms of a, we have - =f'(x), which is the same as the former value, since dx f'(x) = '(v) +'(x) by Rule XIII. Hence, whichever way we dy proceed, the value of - obtained is the same. dx 92. If we use the differential notation, this Rule does RulesXII not appear to want any proof; for it is stated in that no- and XIII seem to tation, as follows. If y be a function of v, and v a func-wantno proof in the ddy dy dv.... differential tion of x, then - - -. Now, since dv divides out in notation. dx dv dx Why the second member, this appears to be self-evident. But we less they dy ought to be must remember that the - in the first member is supposed to proved. dov be derived from y expressed directly in terms of x; whereas the dy d- obtained from the second member, by dividing out dv, is dox not derived from y in this manner; and though it is easy to see that the result is the same in both cases, yet it ought to be formally proved for the sake of exactness. The same may be said of the preceding Rule also. 54 DIFFERENTIAL OF A FUNCTION For examples of the use of the differential notation, and of the Rules for differentiation in general, see Appendix (C). Rule XV. tt 93. To express Rule XV. in the differential notation, we shall assume duy to denote the differential coefficient of y on the supposition that v is constant and u alone variable, and dry to denote the differential coefficient of y on the supposition that u is constant and v alone variable; then we have du dv /'(yv) =duy,) = dvu)y, df ()=) = a % () -=d dy and f'(V) = 9dy and therefore div dy du dv dx duy dV X or dy = duy du + dydv. Hence Rule XV. may be stated thus. If y be a function of two other functions of x u and v, then dy = dyy du + d,y dv, duydu and dvydv may be called the partial differentials of Differential y with respect to u and v respectively. Hence the differential tion equal of y is equal to the sum of its partial differentials with retlie sum of its partial spect to u and v respectively. differentials. More generally, if y be a function of any number of functions of x, viz. u, v, w..., we have by Cor. to Rule XV. dy = duy du + dy dv + dwydw...... The man- tt 94. Suppose that y is a function of u, v, w,... where ner in which we U, V, W,... are variables to which we may assign any values arriveatthe we lease independently of each other; then we may suppose conception py of the dif- U, v, w,... to be any arbitrary functions of a new variable v; ferential of i a function for by altering the nature of these functions we may make OF SEVERAL INDEPENDANT VARIABLES. 55 u, v, w,... vary in aly manner we please when x varies, and of several ' 1.1 ~ * p..p,.- P ~~~~independso receive any arbitrary values: in fact a function of u, v, w, ant varia-... where u, v, w,... are any arbitrary functions of ", is just bles. the same thing as a function of u, v, w,... where u, v, w,... are any arbitrary quantities. The advantage of looking upon u, v, w,... as arbitrary functions of x, instead of mere arbitrary quantities, is this, that we thereby arrive very simply at the conception of the differential of a function of several independant variables: for, if y be a function of the independant variables u, v, w,..., we may, if we choose, consider u, v, w,... to be arbitrary functions of a new variable x, and then by the result of the previous article we have dy = duy du + dy dv + dyy dw 4-...... Thus if y be a function of several independant variables, the differential of y, regarded in this manner, is the sum of the partial differentials of y with respect to each of the variables. What the quantities du, dv, dw... are depends upon what functions we suppose u, v, w... to be of x, and what value we give to dv; if we suppose u = p(x), v = +(x), w = X( U),d, dudvdw, then du = q (x) dvx, dv = +'(x) dx, dw = x(x) dx...; and supposed hence since p, +, X may be any functions whatever, and dx tary conany quantity, it is clear that du, dv, dw... may have any stants. values whatever we please to give them. We shall in general suppose that u = Ax, v= Bx, w = Cx... where A, B, C.. are any arbitrary constants, and that dx is constant; and then du, dv, dw... will be arbitrary constants. We shall therefore assume that if y be a function of Definition of the difseveral independant variables u, v, w... then the differential ferential of of y, which we shall denote by dy, is equal to the expres- a function sion independant vandy. du + d, y. dv + dwy. dw... ables. where du, dv, dw, &c. are in general any arbitrary constants; but may also receive any variable values we please to give them. This assumption is really our definition of a differential of several independant variables; which definition the pecu 56 DIFFERENTIAL OF A FUNCTION &C. liar form assumed by the differential of a function of several dependant variables suggests to us in the manner we have explained. A total dif- dy here is often called the Total Diffeerential of y, in conferential; what. tradistinction to the partial differentials djy du, d~y du, &c. Examples. Let y =. U3 + V3 + w3 - 3uvw, then d,,y = 3 (U2 - vw), dvy = 3 (V2 - uw) d1vy = 3 (w2 uV); and.:. dy =3(u2-Vw)du + (Vu2 _ w) dv + (W2-uv)dw}. Let y = UVW, then dy= uv'vl vwdu+ulogu(vdw+wdv)l. Rule XVI. tt 95. Rule XVI. is thus expressed in the differential notation. If U be a function of x and y9 and we put U= 0, then d, U. dx + dy U. dy = 0, dyl from which equation we immediately get dxx Example. Let the given equation be U = sin x sin y - xy = 0, here d,, U = (cos v sin y - y), dy U = (sin v cos y -)v; (cos a sin y - y) d x + (sin v cos y - v) dy = 0, an d d y cos sin y - y d a sinxcosy - X CHAPTER VII. SUCCESSIVE DERIVATIVES OR DIFFERENTrIALS. CHANGE OF THE INDEPENDANT VARIABLE. 96. IN the same manner that we obtain the derivative f'(x) of any function f(x), so we may obtain the derivative of f'(x), and again the derivative of that derivative, and so on. Thus if f (a) = x', f'(xv) = nv -', and the derivative of this is n (n - 1) xn"-2; again the derivative of n (n - 1) x' — is n (n -1) (n — 2)x~, and so we may go on. The derivative of f '(,v) is called the Second Derivative of The nll, de). rivative or f (x), and is denoted by f 2 (x); the, derivative of f2' (x) Is differential called the Thtird Derivative of f (x), and is denoted by f 3 (x) coeffiient; 'what. We and so on; and in general f " (x) denotes the represent it deny,. of [the deny,. of {the denyv.... (to n derivs.)}j of f (), fn(x) is also called the nth differential coefficient of f(x). 97. If we put f (v) y, and therefore f'(xv) d y, then p (x) isre dx presented d ~ ~ ~ ~ ~ d df '(,v)- ~~~~~~~_dx ~the differf P. (T,) = ~_. ential notadx dx ~~~~~~~~~tion; on what hypothesis. Now we have seen in (80) that we may give to one of the quantities dy or dx any value we please, variable or constant; let us suppose dx a constant, then by Rule II,, d (dy -) ddy, and we l1vetherefore f 2 =V ddy2 dix} dx d x Againf (x df 2(x) ~d (xE) dddy Again '( ) - dx dx dx'v as before, and so on, and in general 58, SUCCESSIVE DIFFERENTIATION. ddd... (to n d's)y f X)= dxn To avoid the repetition of d's we represent d repeated n times by d", and thus we have f " (X) it d Xy For examples of finding successive differential coefficients, see Appendix (D). Expression 98. We have supposed dx constant for the sake of simfor f- (x) when d x plicity; if we do not make this supposition, then we have is not supposed con- dI stant. ____ ( dy d dd dx] df'(x) ________.. &c. J X)= - f _ _____=__ d dxxd dx which are the expressions for the successive derivatives of y considered as a function of v, whatever be the values of dx and dy. If we perform the differentiations, and put d' for dd, di for ddd, &c., &c., we have {dy\ dydX - dy d'x d'y dv - dy d 2X dx) dx',amd...f 2(& d and.f~(x) = df?) (d y dx - dy d'x) d' - (d'yd - dyd'x)3d 'dix (d y dx - dy d v) dx - 3 (d'y dx - dy d ') d'x d x5 and so we may go on. The expressions however become extremely complicated. In these expressions, if we suppose dv constant, we have d2X = 0, d'v = 0, &c.... and therefore d'y d3y f 2 ( =dx2 f3 () l "' INDEPENDANT VARIABLE. 59 which are of course the same expressions as in the preceding article. If we suppose dy constant, we have d'y = 0, d3y = o, and therefore f2 - ) dyd v ) dy Idd3' dx - 3 (d'x)2} dX3 d5.. and so on. 99. That variable whose differential is supposed to be Independconstant is called the independant variable; thus when we put antlehat. d2y f2"() = d, iv is the independant variable; when we put dy d'V f2(^)= - dy, y is the independant variable; and when we put f(x) = d2ydv- dy d' neither x nor y is the indXv dependant variable. 100. It is important to remember that d represents What dy dtV represents f"(x), only on the supposition that x is the independant deoenhat variable, and therefore, though we may always put d"y = we consider to be the f"(xi) dxv when v is supposed to be the independant variable, independant variwe may not do so if this be not supposed. d"y, therefore, able. does not always represent the same quantity; what it represents depends upon what quantity is considered the independant variable. Thus suppose y = "2, then dy = 2xvdx, and d2y= 2dvi2, if Example. dox be constant. But suppose that cV = z2, and;. y= 4; then dy = 43dz, d"y = 12Z2dz2, if dz be constant; now since v = z2, dx = 2zdz, and.'. 12 dz'd = 3drv2, and.'. d2y = 3di2. Hence, when x is supposed the independant variable, dy = 2 d, and when z or ix is supposed the independant variable, d2y=3dxV2: from which it is clear that d2y does not mean the same thing when x is the independant variable, and when iv is. 101. It is often necessary to change the independant variable in an expression containing differentials; this we may 60 CHANGE OF THE INDEPENDANT VARIABLE. Change of easily do by remembering always, 1st, that the independant pendant variable is that variable whose differential is constant; 2ndly, variable; dy d'y W3y how effect- 2 d3y ed. dc t d x ' d~ &c. represent the successive derivatives dxso du 2 d OB of y considered as a function of v, on the express condition that dx is a constant; and 3rdly, that these derivatives, whatever be the independant variable, are represented by dy d (d dy \d-x, dx ' dx dx - w e...&c &c.... respectively. dxt do ldx Hence, if we have any expression involving dy d2y d3y dx d'-v d3 '.."' where these quantities are supposed to represent the successive derivatives of y considered as a function of x, on which supposition x must necessarily be the independant variable: then since d d \(dx d x - dx do dxc represent the same derivatives, we may put these instead of dy d~y d~y dy d 2y d y3.. &c. respectively; dx' d x" d x;'" and then we may suppose any quantity we please to be the independant variable, since the substituted expressions represent the successive derivatives of y, whatever be the independant variable. If the expression involve dx, d dy, dy, d3 &c. but not in the precise forms dy d2y d'y dx' dx2' dx' &c CHANGE OF THE INDEPENDANT VARIABLE. 61 then we substitute in this manner d2y dx ) dy d =.d I d y did div which we put for d'2y d (dyI 3; dy d d J; dX3i dd Y..d.. which put for d3y, and so on. In this manner we may change the independant variable in any expression. The following examples will make this clear. 102. In the expression Example. d2y 2 dy y di e21 - 2 de 1 - x2 where x is the independant variable, to make 0 the independant variable, cos 0 being equal to w. dy d2y Making the necessary substitutions for -Y d- we have die die' ^(dyd d d-x x dy y 24 — - - _-. + dix 1 - i2 di 1 - <2 In this expression we are at liberty to assume any quantity we please to be the independant variable; let us suppose therefore that 0 is, and then we have dx = -sin Od, dy dy deZ dO sin ' dy d 2y sin 0 - dy cos OdO \\dic d= i d0 sin20 since dO is now constant. 62 CHANGE OF THE INDEPENDANT VARIABLE. d dy~ H e dX} _ d2y sinO0 - dy cos OdO Hence dx dO sin2& X dy cosO0 dy -x' dx sin'O0 d sin ' Y 1 -X 2 sin'2O Hence u = __ 1 de0 sin '2 in which expression 0 is supposed to be the independant variable. Another 103. Given the expression Example. (d2y - d2y d'y dy2 dy 2 dy U in which x is the independant variable; to find what it becomes when y is the independant variable. d~t'2 dC dyy ddy I dx2 J also put dd d~ dw dx and... dy = dxd dx and then we have u= s- d - -1 ~ ~ ~ ~ d dyv dy kd d xl J d x I dy- d d Y2 dv Wy" dx In this expression we are at liberty to assume any quantity we please to be the independant variable, let us therefore suppose that y is, and then we have SUCCESSIVE DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES. 63 d dy\ _ dy d"c d dyv dy d2A, since dy is now constant; d dy dc~~v d" x 3 d', v d"x d i dyv~ dx4 d~~v d X 3 dw x Hence u= (3d"x + ) d2 - + - 3 divdy dxdy dv dy' dX 2dy4 dV d cv dy I eta? d3 dxdy divdy' li \dy2 dyv in which expression y is the independant variable. (For other examples see Appendix E). The following articles have reference to the successive differentiation of functions of several variables. tt 104. Suppose that y is some function of u and v. successive u and v beino functions of v; then by Rule XV. difrentiaf(utv), uV dy = d,,y du + d'y dv; being functions of X. therefore differentiating again d2y = d (d, ydu) + d (dv ydv). Now d (d y du) = d (dy). du + du y. d2 U {d,, (d, y). du + d, (d, y). dv du + du ydlu., by Rule XV. hence assuming du2y to denote ddy, we have d (day du) = dly. duc2 + dd,, y. dudv + dyd2 u... (1); and similarly we have d(d~y dv) = d~y. dv2 + d,,dy. dudv + dyd~v... (2); and adding (1) and (2), we obtain an expression for d2y. This expression becomes somewhat simplified by means of the following article. 64 ORDER OF PARTIAL DIFFERENTIATION INDIFFERENT. Proposition- tf 105. To shew that ddy = d,d1y. d.,dy d d. y. Let y = ~ (Uv); then, by the definition of a partial derivative, if we put IJ(uV) (UV) I__ _ _ _ _ _ _ _ _ -. d y J (u'v), 'k(U'.v) may be diminished ad libitum by sufficiently diminishing u' - u, without altering v; and this is true for all + (U V'>- (ZU b) values of v: hence we may evidently diminish V -V ad libitum, by sufficiently diminishing u' - u without altering VI or v; and this is true for all values of v' and v (except of course v' actually = v). Hence, since we may make (UZ' V') - — ' (UV)) f,(u'')- xi (uv)differ as little as we please from dxf(u'v) V - v by taking v' - v small enough, it clearly must be possible to diminish d, +k (u'v) ad libitunm, by sufficiently diminishing ' - u; but d ) dv -(Iuv) d,P(u V)dd d' +F (U' v) d d d, Y; u - U hence, by Lemma VI, Cor. 2, dd,,y must be the limiting d, 0 (u'v) -lu) d, (P(u v) value of d~cp (u'v) - d~P (uv) when u,' approaches u; but, U- u by the definition of a partial derivative, this limiting value is represented by d,, jdv $(uv), or dd,,y. Hence we have dv y = dd d,, y. It appears therefore that whenever we have ddv written before any expression, we may write d~du instead of it, and vice versa: i. e. the order in which we perform partial differentiations is indifferent. Cor. 106. Differentiating the result d,,dvy = dvd, y with red4 d,". y ="d in spect to u, we have d'd,y = dudd,cdy = dd.y, putting d,d, for d,,c4,; and differentiating this result, d'd,y d ddy = d3dy similarly, and so on, and in general d"ddy = ddmity. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES. 65 It appears therefore that whenever we have d'd, written before any expression, we may write d~d. instead of it, and vice Versa. Differentiating the result dd,'y = d'd y with respect to v, we have d&,dly = dvdmd y = d~d'y, putting d'd, instead of dvd' and again dduy = d~d'd'y = dl'd'y similarly, and so on, and in general vd'dy = d"d y. It appears therefore that whenever we have dd,,d written before any expression, we may write d d,' instead of it. (Another proof of this result will be given when we come to speak of series in Chapter IX.) This result shews that when we successively differentiate an expression with respect to different variables, it is no matter in what order these differentiations are performed. tt 107. We shall now return to (104). Successive differentiad2y is found by adding (1) and (2) hence, from what tion of has been just proved, we have d2q 4 du.=_ + 2dd,-y.dudv + dvy.dv2 + dy. du + d~y. d~v, If we differentiate this result again in a similar-manner, we obtain d'y - 4. du + dvdy. du dv + s d~dvy. dudv + d4. dv', 3 dcy. dud2u + 3 d,,d,y. (d2udv + dud2v) + 3 dvy. dvd2v + duy.d'u + dy. dv'; and so we ` may- find d'y, &c.; but the results become extremely complicated, and we shall not put them down. Let y U W2 + V2 + UV, Example. then dyy=2u+v, d2y = 2, dy = 2v+u, d~y=2, 66 6SCCESSIVE DIFFERENTIATION OF FUNCTIONS dd,y = 1; and therefore we have d`2y = 2du2+ 2dudv-t 2dV2 + (Qu + v) d 2U + (2v + u) d 2V. Again d3y=O, d y=O, d d2y=o, d dvy= O; and therefore we have d3y = 3 (2du d'u + d'u dv + dv d'u + 2dv d'v) + (2u + v) d'U + (2+ u) d'v, and so we may find d'y, &c. The method here given of finding the successive differentials of a function of several variables is useful only for general purposes. In particular cases we may always find the successive differentials more readily by the common rules of differentiation. (See Appendix F where examples of this are given). Successive j t 108. If U be a function of x and y, and we put on ofan U = 0, then we have by Rule XVI. equation between y and x. d U. dw + dy U. dy o, and differentiating as in (107) we have, supposing dx constant, d2U. dX'2 + 2d,dy U. dx dy+ d1 U. dy2 +d U.d y= 0, dy. dly dlyy and so we may go on, and by this means find dy d- ' d'y &c. from the equation U= 0. Ex. Let X'-3caxy + y3 = U= 0, then d'U=3x-,3ay, d'U=6c, d U=3y'-3ax, d U= 6y9 dvdyU 3 a: we have therefore (X2 _ ay) dxv+ (y2 - ax) dy= 0...............v.. (1) 2xd 2-adxdy + 2ydyx+(y-a) dy = 0...(2), dy x'-ay from (1) we get dx Y2 axa OF SEVERAL VARtIABLES. and theni from (2) 2 X+a 2 (ay (M ay), d2y y-aie2a + 2Y +( x 2= dy2 daX x) y which Igives,d; and so we may find y (See Appendi x F.) tt 109. If y be a function of u and v, u and v being Successive differentiaindependant variables, then supposing du and dv constant (see tion of a 94), we have, as i (107) ucin two independant dy = d,,y du + dvy. dv, variables~ d2y = d2y-. dU2 +2d~d~y -du dv + d,2y. dV2, d3 y =d4'y. du' + 3d~dvy. du2 dv + 3 d~d2Y. du dv2 + d'y. dv3, We perceive that the coefficients of the terms in the second members of these equations are respectively those in the expansions of 1 + z, (I + _2), (I + Z)3 Le, A,~ B, C... be the coefficients in the expansio of (1 + z~n, and let us assume that d~y~~y~Un+Ad' d y.dull. dv+Bd~2d~y. d un -dva + then dn+ y-=d,(dny) -du +dv (dny).dv -dn+1y. du"+' + Add~y. du"'dv + Bd~'diy dl dv Now 1, A+ i, B +A, &c. we know to he the coefficients in (1 + z)n+' expanded: hence it appears that if the law we have assumed be true for n it is true also for n + 1; but we know it to be true for 1, 2, 3;- therefore it is true in general. We have therefore nn dny= d Y. du" + - d~'ld,,y.du,,' dv n (n -1) d-2d2'y. du2dV2 + d 68 SYMBOLICAL EXPRESSION OF d'y. Symbolical tt-110. This result may be briefly expressed thus, expression for dny. dny = (du. du + d,. dv)".y. By which formula, we do not mean that d5. du and dv. dv are actually quantities whose sum is raised to the nth power and multiplied into y; but simply this, that if the symbols di. du and d,. dv be connected together by the sign +, and raised to the nth power in the same manner as if they were ordinary quantities, and if y be written after each term of the result; then the expression so obtained is the proper expression for dny. Expression t 111. We may prove in the same manner that if y for day,. whenyis a be a function of several independant variables u, v, w...&c., function of then more than pe ndant dy = (d,. du + d. d d, + dw. dw...)".y variables. CHAPTER VIII. CERTAIN LEMMAS UPON WHICH THE APPLICATION OF THE DIFFERENTIAL CALCULUS IN MANY CASES DEPENDS. 112. HAVING now concluded what may be considered the first part of the differential calculus, namely that in which we determine a set of rules whereby differentiation may be performed in all cases with facility; we proceed to the second part, in which we shall shew some of the principal uses which may be made of the differential calculus in pure mathematics. But we must previously prove the following lemmas upon which the application of the differential calculus in a great measure depends. 113. If f(x) changes its sign when x passes* through Lemma the value a, then f(a) must be 0 or o XVII. For f (a) is the limiting value of f (x) when x approaches a (Lemma III), and therefore by Lemma IV f(x) has the same sign as f(a) for all values of x taken sufficiently near a; hence, if f(a) be positive, f(x) is positive for all values of x taken sufficiently near a, and therefore cannot change its sign when x passes through the value a; and the same is true if f(a) be negative. If therefore f(x) changes its sign when x passes through the value a, f(a) can neither be positive nor negative; i.e. it must be o or -. E. D. 114. It is important to observe, that what we havef(x) does proved here is, not that f () must change its sign when f(a) not necesis 0 or oo, but that if it does change its sign, f(a) must change its sign when be 0 or co. f(a) is0o 00. a By saying that, "f(x) changes its sign when x passes through the value a," we mean that when we give x any value a little greater than a, f(x) has a different sign to what it has when we give x any value a little less than a. .70 70 ~LEMMAS UPON WHICH THE APPLIGATION Thus, if f (v) = (iv - a)3, f (v) changes, its sign wvhene iv passes through. the value a, and here f (a) = 0. If f (X) =(xv- a)3'' (x changes its sg whn xv passes through a, and here f (a) = co. If f (i) = (v -- a)', f (a)= 0, but f (i) does not change its sign; and if f (v) =(- ) f (a) = co, but f (i) does not change its sign. renma 1 15. If we suppose x to increase continually., f (x) is XV1IL increasing as long as f' (x) continues positive, and is diminishing as long as f' (x) continues negative. For, by Lemnma IV f x' -f i has the same sign as iv -iv, ff'(v), for all values of iv' taken sufficiently near iv: therefore if f'(iv) he positive, f (iv) -f (iv) has the same sign as iv' - i, and therefore if iv' be > iv, f (iv') is > f (i);i e. f (i) increases when iv increases. And similarly, if f' (v) be negative, f (iv) - f (i) has the opposite sign to that of IV, -Xiv and therefore f(iv') is < f (i) if iv be > xv; i. e. f (v) diminishes when iv increase s. Hence the truth of the lemma is manifest. f (x) mea- 116. Since f (iv)-X v is very nearly the same thing sures the i'-i rate of variation of as f (iv) for all values of iv' taken sufficiently near iv, it i's f{(xp es evident that f'(iv) is very nearly the ratio of any small change with x. in f (i) to the corresponding change in iv; and therefore the greater f' (i) is, the- greater will be the rate, so to speak, at which f (i) varies as compared with iv. Hence f' (i), by its sign shews, whether -f (iv) is increasing or diminishing, and by its magnitude the rate at which that increase or diminution takes place. If we suppose iv diminish continually, it is evident that J'(iv) diminishes or increases according as f'((v) is positive or -negative. OF THE DIFFERENTIAL CALCULUS DEPENDS. 71 117. If f (x) and d (x) be any two functions of x, then Lemma f' (x) f ( x')- f ( I ' (x) is in general the limiting value of (x') -(x),p (x) < (x( - p Wx when x' approaches x. f (V') _ f (I:) f (If) -f () f - x For () ) -? q, (' ) - ~, (~) q, (.') - i, (by) a? -- and the limiting value of the second member, by Lemma VIII, is evidently q-r) in general. Q. E. D..f' (a) We say "in general," because ', may become illusory for some particular value of?, and then of course this Lemma fails. In such a case the following Lemma will take its place. f (X) Lemma 118. If a be any value of x uhich makes ',(xx) f' (x) when (x) illusory, then the limiting value of ' ) when x apf (x) proaches a is also the limiting value of () - when 5 (x) - p (a) x approaches a. f'() By the last proposition ) is in general the limiting ' (a) ve f(a') - f(a) value f(a') - f( ) when a?' approaches x; therefore fX(a')-f(a?) (a?) f (a') - f (~ ) f' (a) may be diminished ad libitum by sufficiently diminishing tZ'- x; and this is true for all values of x which do not f'(a) make ) illusory; therefore it is true if we put xt= a and sufficiently diminish x'- a; i.e. we may diminish 72 7LEMMAS UPON WHICH THE APPLICATION f(a -f () f' (x) P (a) - p (x) q (I) ad libitumn, by sufficiently diminishing - a; hence by f' (i) Lemma VI, Cor. 2, the limiting value of, is also the f f(a) f (x f f(0 _ limiting value of f(a) f() or f() f(a) when x p (a) - (is) p (v) - (a) approaches a. Q. E. D, Cor. 1. 119. If a be a value of x which makes the quantities fl (a), f (a).. f -l(a), l1 (a), 02(a)...fn'-l(a), each zero, fP (a f"(a) and if () be not illusory; then is the limiting f (x) - f (a)) value of f (x) - () when x approaches a. (x) -0(a)' For, by the Lemma, when x approaches a, f (x) -.f (a)_ f' (~) lim. val. of f () -lim. val of 0<?) - c(P(a) ^(') or of f' () -f' ( %P' (v) - ' (a) since f' (a) and )' (a) are each zero, f2 (0) -lim. val. of by the Lemma, " (c) fo (o ). f (a) or of,P2 (i) - ' (a)' since f2 (a) and 02 (a) are each zero; and so on, till we come to f n- (v) -.f",- (a) lim. val. of (a) f - ( 1) -(P)(a) which =~ (), by Lemma XIXo ^ (a) OF THIE DIFFERENTIAL CALCULUS DEPENDS. '73 _____CC f (x) - f(a) Hence f (a) is the limiting value of lp" (a) cP (x) P (a) when x approaches a. Q. ri. D. 120. If a be a value of x which makes f'.(a), f2 (a) Cor.2. f" (a) f n' (a) each zero, then is the limiting value of Fr f (x) f (a) (x) - fa) when x approaches a. (X - a)" This is easily proved by putting op (x) (x - a)" in the last Cor., and therefore c/ (x) - (/ (a) = (x - a)", /5' (a) = 0, (P'(a)= 0... 0"-'(a) = 0, /5n"(a) = Fn; which evidently gives fC"(aC) f( ) -f(a) us F for the limiting value of fx)f( when x rn2 (IV - a) approaches a. Q. D. Conversely: If a finite quantity A be the limiting value Cor. 3. f (X) f (a) of f~x) - f~a) when x approaches a, then must f (a)= 0, Of (x-a)" fP(a)TO f (a)-=...f"-' (a)=0, and f"(a)= Fn.Ai, For if we have f'(a) = 0, f2(a) = 0...f''(a) = 0, but fr"(a) not 0, r being,, any integer less than n; then the limiting value of (x) - f(a) w (a),vla -f hen x approaches a will be (x~ a) r rr and therefore that of fwi -hf(a) - am)" whica)h f f)f(a) will be I.f'( (Lemma VIII.); which is infinite, con0 Fr trary to hypothesis. Hence the truth of the Cor. is evident. 121. If f' (a), f2(a)... f"n' (a), be each zero, and Cor. 4. f" (a) not zero, then, for. all values of x taken su~ftciently near a, f (x) f (a) has the same sign as f" (a) (x - a) 74 LEMMAS UPON WHICH THE APPLICATION For, by Lemma IV, ( — f(a) has the same sign as (00-i a)fn (a) its limit for all values of x sufficiently near a, and therefore f(x) -f(a) has the same sign as f (a) (x - a). Q. E. D. Lemma 122. If f(x) be any Jfnction of x we may assume in general that x - a f2 (x- a)2 (x - a)3 f(x) =f(a) + f'(a) x f(a) + f3(a). +{f(a) + Q} (X-a)n Fn where Q is some function of x which may be diminished ad libitum by sufficiently diminishing x- a. For assume F(v) to represent the quantity I i(-a ) (.f-a)2 (a? -a)') f(x)-{f(a)+f'(a) -a +f (fa)...(a) }.. (0 then, by differentiating this expression successively and putting x = a, it is easy to see that F(a) = o, F'a) o, F2 (a) = O.... F(a) = o. Hence, by Lemma XX, Cor. 2, 0 must be the limiting value of F(x) when x approaches a, and therefore ( — ) (aX - a)" (X - a) may be diminished ad libitum by sufficiently diminishing — a a. If therefore we put F(v) ().( - a)" (( -a ) = or F() n (,v - a)' rn r n and substitute for F(x) its value (1) we have f(x) -{f(a) +f (a) +f(a) - 2 ri F2 (V - a), _I (iV - a) f'(a)~~ n Q rn OF THE DIFFERENTIAL CALCULUS DEPENDS. w-a I(X - a) or f(x) = f(a) +f'(a) +-f2) () *-a il r2 + an (a)+ Q} (~ - ay) where Q, since it = rn -(- ) is some function of x which (x - a)' may be diminished ad libitum by sufficiently diminishing x a. Q. E. D. This reasoning fails when any of the quantities f(a), f'(a)... f'(a) are infinite; for then we cannot assert that all the quantities F(a), F'(a), F" (a)... F (a) are zero; which is essential to the proof. tt 123. If M be the least value of f"(x) - f"(a), and emma N the greatest, for all values of x between a and another value b; then Q lies between M and N, for all values of x between a and b. Let C be any constant, and let us write down the ex(x - )a' pression F()-) - C ( a) and its successive differential En coefficients as follows: F(e) - C r...... (n), (v - a)" F'(x) =C C(n),,. (n - i), (V - a)" -2 nFI(x) - C r(n, (n - 2)5 F(n - 2) F- () - C ( - a)............ (1), F () - c..................... (0). Now F"() = f "(V) - f (a) evidently; hence putting C=M, which is the least value of j"' (x) - f"(a) and there 76 LEMMAS UPON WHICH, &C. fore of Fn(,) for all values of x between a and b, it is clear that if we suppose x to increase from a to b, (supposing b greater than a,) the expression (0) is always positive, and therefore the expression (1) is always increasing by Lemma XVIII, but (1) is zero when =a, therefore (1) is always positive; and therefore in the same way we may shew that (2) is always positive; and so on, and finally that (n) is F always positive, and therefore that Fn ) ---- or Q, is always greater than C, i.e. JM. In exactly the same way if we put C = N, we may shew that the expressions (0), (1), (2)... (n) are all negative while x increases from a to b, and that therefore Q is always less than N. If b be less than a we may shew in exactly the same way that when x decreases from a to b the expressions (0), (1), (2)..., and finally (n) are all negative if C = M, and positive when C = N; and therefore that Q always lies between M and N. Hence it appears that Q lies between M and N for all values of x between a and b. Q. E. D. Lemma 124. When a function f (x) becomes infinite for a particular value of x,, all its differential coefficients must also become infinite. For ( — fmay be made to differ as little as we please from f'(a) by making x approach a; hence if f(a) and therefore f(x) f() be infinite, and at the same time f'(a) finite, we may make infinity differ as little as we please from a finite quantity; which is absurd. Hence f'(a) =co and therefore by similar reasoning f2(a) = o: and f3(a) = oo and so on. This reasoning does not hold if a = co. CHAPTER IX. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE DEVELOPEMENT OF FUNCTIONS; WITH SOME PRELIMINARY REMARKS RESPECTING SERIES. WE now proceed to shew some of the principal uses which may be made of the differential calculus in pure mathematics, and in the first place its application to the developement of functions in series, 125. It will be necessary to make a few preliminary Prelimremarks -on the nature of an ordinary series of the form mark respecting o, + A- + Am2... +A +- &&c....; series. and to settle precisely what we mean by the equation f(x) = oA + A1x + A2,2...+ An,"+ &c....(1) which expresses the developement of a function f(x) in a series of powers of x. 126. In the first place we may remark that the sign= always =here, as well as elsewhere, always signifies actual equality; denot for mathematicians always consider themselves at liberty, when equality. they have two expressions of any kind connected by the sign =, to use them indifferently for each other, and to substitute one for the other in any- calculation; which certainly they have no right to do, if the sign = does not always signify actual equality. We shall therefore always use = as the sign of actual equality. 127. Secondly, by the second member of the equation (1) We must not in gene we do not in general mean a series of terms infinite in number, sal suppos, but a series of terms carried on according to a certain law a series to consist of to any number we please, with a term at the end of a dif- an infinite number of 78 NATURE OF A SERIES. terms, but ferent kind from the rest, which is commonly called a reof a finite number mainder: the former terms are represented by with a remainder. A, +- Ax + A2,... + Anns, n being any integer, and the remainder is represented by the " + &c.," which is written after the term Anx". The remainder we shall often denote by writing + R instead of " + &c." Thus when we say that 1 = + t + X2...+ Vn +-t &c.......(2) we do not in general mean to assert that 1 + x + x2... to 1 an infinite number of terms - for if so, suppose = 2 1 —,O' and then we have 1 + 2 + + +16,.. to an infinite number of terms =- 1 which is manifestly absurd. But we simply mean to assert that 1 ----- = 1 X4< +.......... (3) 1 - So where n is a number as large as we please, and R a certain quantity which must be always added to the series to make the second member of the equation equal to the first. In fact the " + &c." in equation (2), and the + R in equation (3) mean the same thing.,n+1 In the present instance the value of R is -, for it is easy to see that 1,'1n+1 1 +, +..x+ 1 -a? 1 -- and thus when x = 2 we have - 1 = 1 + 2 + 8 16... 2n - 2S1 in which equation there is no such absurdity as there is when the number of terms is considered infinite. We may Under cer 128. Thirdly, if it should so happen that R is a quantity NATURE OF A SERIES. 79 which may be diminished ad libitum by sufficiently increasing tain circumstances n, then we may assert that assert that aseries consists of an f(x) =, A + A2V + Aa... infinite number of to an infinite number of terms: or perhaps more correctly,terms. that f (x) is the limiting value of Ao + Ax1 + A_2'2....4Aw when n approaches infinity. For f (x) - (Ao + AA + Ax2...A,,x), since it = R, may be diminished ad libitum by sufficiently increasing n; and therefore, by Lemma VI, Cor. 2, f (x) is the limiting value of A, + Ac + A,22...Ax when n approaches o. When we assert therefore that f(a) = Ao + Ajx + A,'...ad infinitum, we simply mean that f(x) is the limiting value of Ao + Ax +V AX2 +... n when n approaches co. 129. When R may be diminished ad libitum by suf- Definition of a conficiently increasing n, the series verging and of a diverg-,-t A: Al^a + m2M...A.nn + X.R ~ ingseries. is said to be a converging series: otherwise it is called a diverging series. Hence we may assert that f () = A, + Axe + A2_,2... ad infinitum, when the series is converging, but not otherwise. 130. In the fourth place we may remark, that we may A function may be dealways assume m be developed in an infinite f(O) = A, + AlO + A,2... Aa +n + R, numbero different where A,, Al, A,... A are any constants whatever. For we series may always give R such a value as will make the second member of this equation to coincide with the first, no matter what values we assign to Ao, A,, A,,.A5. 80 80 ~~~PERFECT AND IMPERFECT SERIES. Thus wemay assume = ~ a? + P2 + 5x13 - 7aV4. +1,.for by giving R the value - the second member y~~~~~~~1a of this equation becomes identical with the first. We may therefore expand' any function f (v) in an infinite number of different ways in a series of the form Distinction 131. And here, in the fifth place, we may make an between a perfect and Important distinction_; namely, the distinction between what ani e-we ma call a 'perfect and anipefc develoemen fect deve- ay a I~~JU lopeent, The series A,, + Al ~ A, I.. A5,v + R we shall call a perfect developenment of f(v?), when R is of such a nature, that zero is the limiting value of when xapproaches zero; and if this limiting value, be not zero we shall call the series an imperfect developement. That we are justified in making this distinction, and regarding it as a very important distinction, is evident from the following theorem. The pi- 132. The principle of indeterminate coefficients holds ciple of indeterrninatefifl a perfect developement~, but not for an imperfect.,coefficients holds for a By the principle of ind eterminate coefficients we mean this: perfect but Y not for an' that if we have a series imperfect develope- -4,+-,v+-,x 4. R ment.a which is proved to be -zero for all values of iv~ then must A =.o,, A o,.420...*-I ~O'. Let A, + A41a + A4a?..Aa + R be a perfetsre, which i's pro ved -to be zero for all values of te;,.then must R A0 +A1a?~A+ a4K.. A4 a?5v n fl PERFECT AND IMPERFECT SERIES. 81 for all values of., except of course the value zero: but the series being perfect, the limiting value of -, when (v approaches 0 is zero; hence, by Lemma VI, Cor. 1, zero must be the limiting value of A0 + Ax + 4A,X2... AuV when x approaches 0: now if Ao be not zero, Ao + JA... AxnA AO 0 + A.. A becomes or o, ogn 0 when x =0, and its limiting value cannot be zero when M approaches zero; therefore A, = 0, and we have Ao + AI... Ax" A, + A,2x... Anx"-l tn X -1 1 except of course when x actually = 0. Hence zero must be the limiting value of this latter quantity when x approaches o, which as before cannot be true unless A = 0: and so we may go on and prove that A, = 0, A4 0o...; and lastly, that A,, + A~... Ani X n the limiting value of which cannot be zero unless An = 0. Hence the principle of indeterminate coefficients holds for a perfect series. But the principle of indeterminate coefficients does not hold for an imperfect developement. For by what we have just proved, it is evident that if A0, A,, A,... A, be each zero, then the limiting value of R when x approaches zero must be zero, and therefore the series must be perfect. If therefore the series be not perfect, A0, A,... An cannot be each zero, i. e. the principle of in. determinate coefficients does not hold. 6 82 TAYLOR S SERIES. Hence I think the importance of distinguishing series into perfect and imperfect is manifest. A function 133. Lastly, we may remark, that if f(x) be a function which has only one which has only one value for each value of x, it cannot be value for each value developed in more than one perfect series. of x can be developed For if possible let the two different perfect series feetin only peries. feseries. + A + A,2...A4An +, and Bo + B1x +B2x2...B,,iv + S, be both equal to f(x), then sincef(m) has only one value for each value of x, these series must be equal to each other; and therefore we have A- Bo + (Al- B,) + (A - B,) x2..+ (A- B,) + i~ - S = o. Now when x approaches o, the limiting values of R and of S R-S - are zero, therefore so also is that of: hence, by what has been proved in the preceding Article, we must have Ao = Bo, A1 = B, A2 = B2... A, = B,,, and therefore R = S; hence the two series are identical; and therefore f(x) cannot be developed in two different perfect series. Having made these preliminary remarks, we now proceed to shew the manner in which the Differential Calculus may be applied to the developement of functions in perfect series. Proposi- 134. To develope f (a + h) in a perfect series of powers tion. To de- of h. velope f(a+ih) in a perfect In Lemma XXI. put a + h for x, and we have series of powersofh. h h. hn f(a + ) =f (a) f'(a) +f2(a) -...(a) + Qn (1), here Q - is the remainder; call it R, then n = -; now Fn h'1 f n o zero is the limiting value of Q when x approaches a, i. e. when h approaches zero, (see Lemma XXI.), and therefore the same is true of -; hence (1) is a perfect developement. Since hn TAYLOIR'S SERIES. 8 83 Lemma XXI. fails when any of the quantities f (a),. f '(a), f '(a),.f (a) become infinite, this developemient fails under the same circumstances. This developement is known by the name of Taylor's Taylor's 1) ~~~~and MaeSeries; when a = 0 it is called Maclaurin's Series, in which1auns case it becomes series. it J~~~~~2.f () = (0)+f I(E2 Enf2 En We may write the developement, (1) thus: f (a +h) =f (a)+f(a)-~ f 2(a) +. +fn(a)- + &c. remembering what "1+ &c." means. The following are very important examples of this. de- Examples. velopement.:135. Let f (v) = x'1; and therefore Deve'lope(lnt)of then putting 1 for a in the developement (1), we find (I+ht + -nh + h a, 1 + E~~n which is the binomial theorem for any value of the index,. Let f (x) =logv, and... f Qv =x F(n - 1) ( );then- Developeputting I for a in the developement (1), we find letogf1+) 1 h2 h3 hn Let f (X) O X, and f~ f" Gv) -=knox, where k =log c, thenr Developenient of putting a = 0 in the developeme'nt (1), we find kh k'h2 knhn Ei E2 En &C 84! ERROR COMMITTED IN NEGLECTING Develope- Lt In 7\/ ment of L= sin x. and.-. f' () = sin + 2 then sin x, cosx, tan-'X. putting a 0 in the developement (1), we find h h3 h' ~~ sin h h h3 15 + (_)n h(2n+1) f -1 f -3 + 5 r on +f 1) Similarly we find C A 2 144 h2 + cos h = I + ~1 () + & c. 72 14 (2n) h 43 h5 h2n+1 tan' h — +-... + &C. 1 3 5 F(2n + 1) For more examples of the use of this developement, see Appendix G. hn Proposi- tt136. If we neglect the remainder Q in the develtion. Fn Estimation of the error opement (I) last article, the error we commit in so doing lies committed when the h" h w remainder between M and N -, where M and N are the greatest in Taylor's Fn En series is and least values of ff n(x) - f (a) for all values of x between neglected. aada+b a and a + h. This follows immediately from Lemma XXII, if we put b = a + h: for it is there proved that Q lies between M and N for all values of x between a and b (i. e. a + h), and therefore for the particular value a + h; consequently the remainder h" h" Q n in the developement (1) must lie between M u and 4" E~n ~Examples. 1iX. Let f (x) log x, a=1 h n 1 = 5; then 10 = 17(n - 1); f5(.-f5(1)=f4. (fL... I): th" (gt) l f tis the greatest and least values of this for all values of x between THE REMAINDER IN TAYLOR S SERIES. 85 1 and 1 + - are evidently 0 and r4 { - 1). Hence 10 (1+ 1 5 ' the error lies between 0 and - - and we have 5 (1ll 105/ therefore / 1 1 1 1 1 1 1 1 1 1 1 log I +- _ + _ -_ =10 10 2 10" 3 - 4 104 5 105 ~1 1 \ + an error between o and ( - 5 1 15 105' Since =.000006... this error lies between 0 and.000004. -. 0 -which has no significant digit in the first 6 5 decimal places. Ex.2. Let f(x)=a, a=l, h =-, n=5; then f (n) = l ( - 1) ( ) (L - n + 1) v-a; 1.3.5 the greatest and least values of this for all values of x between 1 and 1 + - are 0 and 10 1.3.57 i o 7 25 + -~ Hence the error lies between 0 and 1.3.5.7 f 1 y 1 25r5 IF 5 Io9 T o7 ( 1 1 — 7 i 1 /i ll' or - - -.. 156 105 115 10 86 CONVERGENCY OF SERIES. I 101 11 which is evidently, in numerical value, less than 1.000004... or - or.0000002... Hence we have 20 2/P $ ---- I 1 1 5 -- 1 7 1r5r 7 10 2 10 2'i0p 2410327 1042105 + an error between 0 and -.0000002... Ex. 3. Let f(x) ex, a = 0, h =i, n = 10; then f" (x) = do;.~ f II () - f (o) ~ i the greatest and least values of this for all values of x between e- 1 0 and 1 are e - 1 and 0. Hence the error lies between F (10) and 0, or between and 0 since e is <3; therefore Fr(i0) 1 1 1 1 e= 1 +-+$ ---FI F2 F3 F (l) 2 + an error between 0 and 2 which has no significant r7 (lo) digit in the first 6 decimal places. Condition ++137. Let u,, denote the numerical value of of conver. 1 gency. () - a)3 f (0 Fn for any value of x between a and a ~ h; then if the limiting value of u,, when n approaches co is zero, the developement (1) is converging, and we may assert that 4:.h2 f(a + h) =f(a) +fl(a) -j +fS(a) + ) a f:+ f3(aC) -rad inzfinitumn. 1 D CONVERGENCY OF SERIES. 87 For then it is clear that ({ - a)r' f n (V) -_ f n(a)} (x ) h" h" h1' and therefore —, N-, and therefore Q - may be Fn Tn I'n diminished ad libitum by sufficiently increasing n: therefore by (130) the developement is converging, and we may assert that h h2 f(a + h) f (a) + f (a) - + f (a) h3 + f3 (a).... ad infinitum. j- 138. If, for all values of n greater than a certain value A simpler condition of Un +l converr, never exceeds a certain ratio a which is less than gencyr r~ gency. Un unity, the developement is converging. U ZC /Ur+3 For then we have +l not > a +2 not > a not Ur Ur+ i Ur+ 2 > a.... --- not > a; and therefore, multiplying these in/n —1 equalities, we have - not > a-'', or u, not > an-"u.. Now Ur since a is less than unity, the limiting value of a"-r when n approaches co is zero, and therefore, zero is also the limiting value of an-',., and therefore of tu,: consequently the developement is converging. 139. By means of Lemma XXI. we arrive at the con- Taylor's clusion, that there necessarily exists a perfect developement of rived atr f(a + h) in the form somewhat differently. A0 + Ah + A2 h... + Ahn + R, provided none of the quantities f(a), f(a), f2(a)... f(a) be infinite: we shall now prove the same thing somewhat differently. 88 TAYLOR'S SERIES. By (131) we may assume that f(x) = Ao + A (x - a) + A, (x - a)2... + A ( - a)n + R....(), A0, A...An being any arbitrary constants, and R that quantity, whatever it be, which must be added to the second member of the equation to make it equal to the first. If we can so determine AO, Al,,A...A, that the limiting value of (a? - a)" when x approaches a shall be zero, then the developement (1) will be a perfect developement. Now if this limiting value = 0, it follows from Lemma XX. Cor. 3, that dR d2R d3R d 7R dox ' d32 dx3 must be each zero when x = a. Hence, differentiating (1) put in the form f(x) - Jo- A, (- a) - A,) (a - a).... - A (-a) = R n times successively, and then putting x = a, we obtain the following equations: f(a) - Ao = f'(a)- rl. A = o f2(a)- rF2. A = 0.........), fn(a)- Fn.A = o which equations give us the values of A,, A,, A... A,, necessary to make (1) a perfect developement. And conversely, if we give the values (2) to Ao, A,, A2.. Ab, we shall evidently have d dR dR dnR R = o, d 0' = ~ -'d =, when x= a; r drf<2 d x" and therefore the limiting value of (- --. when x approaches (X - a)" a, will (by Lemma XX. Cor. 2.) be zero; i.e. the developement will be perfect. FAILURE OF TAYLOR S SERIES. 89 Hence it appears (putting x - a = h) that f(a) +f'(a) +f2(a)......" f (a) + R is a perfect developement, and the only perfect developement, of f(a + h). We must except however the case where any of the quantities f(a)(a), '(a), f2 (a)...fn(a) are infinite, for then we cannot be sure that the equations (2) are satisfied by giving A,, A,, A... &c. the above values; for instance, if f3 (a) = -, we cannot assert that f3(a) - F73 3 = 0, if we give A, the value -, for then the quantity f3(a)-F3. A, assumes 1 1 the form - —, which we cannot assert is zero. 0 0 140. If f (t), and all its differential coefficients below The failure a certain one, the pth suppose, be finite, but the pth (and all of Taylor's series. above it by Lemma XXIII.) infinite; then the developement (1) holds as a perfect developement for all values of n less than p, but for all other values of n it fails. 141. When a failure of this kind takes place, it is A differengenerally possible to obtain a perfect developement containing tiaelobeff fractional powers of h after the pth term. coming infinite when x= a, indix - a cates the For let F(x) =f(x) -f(a) -f'(a) appearance r1 offractional powers in (v -a)2 (w a\'P-\ the devel-f I(a)(..a"p - (), see (123) opement r2 F(p-1) then it is clear that F (a) = o, '(a) Fo, r(a))= o.... - (a) = 0, but FP(a) =fP(a) = co; since we suppose that f(a), f'(a), f2(a)....f-(a) are finite quantities, but fP(a), fP+'(a).. infinite. 90 FAILURE OF TAYLOR'S SERIES. Hence, by Lemma XX. Cor. 2, the limiting value of F(-v) is zero, and that of F(x) is infinite, when x (,,- a)P~- (o - a)P approaches a. Now if any function b(x) = 0 when x = a, we may generally find some power of x - a, (. - a)m suppose, such that the limiting value of ('- - when a approaches a, (x O - a)* is neither zero nor infinity. Let us then suppose (x - a)m to be such a power of x -a, that the limiting value of F(x) F(x) when x approaches a is some constant C, which is (X -a)m neither zero nor infinity: then, since F (,) F (x). ( ), ( - a)P- (v - a)m a F (v) _F (v) and aL^^ (. -P, (x - a)P ( - a)" ( F ) En(d ) it is clear that the limiting values of and (v - a)P- (v - a)1 are C.o0n-p+ and C.O'-P. If m be less than p - the former of these is infinite, which we know not to be the case; and if m be greater than p the latter will be zero, which we know not to be the case: hence m lies between p - 1 and p, and is therefore a fraction. F (x) If therefore we assume ) = C + Q, Q will be some (W - a)" quantity which becomes zero when x =0, and we have, putting for F (a) its value (v - a) (x - a) f() = f (a) + f'(a) - - ( + f (a)... fP-l(a)( P (C -+ )+ (c + Q ) ( "a) which is a perfect developement containing a fractional power C (x - a)'", m being greater than p - 1, and less than p. FAILURE OF TAYLOR S SERIES. 91 Thus it appears that when any differential coefficient fP(a) becomes infinite, it indicates the appearance of a fractional power in the developement off(x) in powers of x - a. 142. We may generally apply Taylor's series, in the How the following manner, to determine the developement off(x), when deelopeto fractional powers appear in it in consequence of some of its be obtained when any differential coefficients becoming infinite when x = a. of the differential coefficients Put x = a + z', and f(x) will become a function of z, beomenin(p(z) suppose: then, if possible, so determine r that neither finite. (p(z) nor any of its differential coefficients shall become infinite when z = o, and this being the case we shall have, by Taylor's Theorem, (Z) = gp(o) ' + 02(o) 1... n () r + R, n being as large as we please. Now here put for z its value ( - a)y, and for (P(z) its value f(?), and we have (x - a); I = +! (0) (a a) ( a) + p'(O) 172 p5(o ) + R, which is the developement required containing fractional powers. There are also other ways of substituting for i, so as to obtain the developement of f (x) by Taylor's series. Ex. Let f(i) = sin {I + 1 + (a - 1)}; Example. here it is easy to see that f2(o) = co; therefore we shall not be able to develope f(v) in this case in integral powers of - 1. To obtain the developement by means of Taylor's Theorem, put = 1 + 2, and then f(v) = sin (2 + z' + 03) = (j), .92 FAILURE OF TAYLOR S SERIES. and it is easy to see that neither d (z) nor any of its differential coefficients become infinite when = 0: therefore we have f~m) = (0) + ) + q2(0) F2 +)3(o) + &c. ri rz 2 By actual differentiation, and putting z = 0, we find tP (0) = sin 2, q'(0) = 0, 02 (0) = 2cos 2, 0'(0) = Gcos 2...&C. and therefore, sin IM + 1 + (x-1)1} =sin2+cos2.(x-1)+cos2.(w-1)1+&c. Same ex- This method however is not generally the simplest in ample done more practice; other substitutions for iv often bring out the result simply. more readily. Thus assume iF -1I+ ("I - = Z and then fa(x) = sin (2 + z) = (z), suppose; and it is evident that neither cP (z) nor any of its differential coefficients become infinite when F = 1; i. e. when z =0; we have therefore x x2~~~~ P (z.) = P(O)$ ~)(0) l + c'(o(0)+ + &C. Now since ~(z) = sin (2 + 4 we have 4P(O)=sin2, qp(O)=cos2, c2(O)=-sin2, (0) =-cos 2... &c.; and therefore, restoring for z its value, we have f(iF) = sin2 + cos2. {(I - I) -1)+ (i. -.. if we arrange this in powers of (x - 1), expanding each term by the binomial theorem, we find the developement required. ff(e) being 143. These methods clearly suppose that f(a) is not infinite indicates the infinite. DEVELOPEMENT OF FUNCTIONS GIVEN BY EQUATIONS. 93 If f(a) = co, then if we assume appearance of ne-((S _~~~ a)S~~f( = +~ ~gative (, - a)sf() = '(p (), powers in the series. and so determine s that p((a) shall not be infinite, we may develope (p() as above in the form 1 2, (a) = A, + Al (a - a) + A (i - a)r...&c. and therefore, restoring f(f), we have f(I) = -A (I - a)-' + A1 (I - a)S+... &c. Hence, f(x) becoming infinite when = a, indicates the appearance of negative powers in the series. (For examples, see Appendix H.) ttJ 144. Suppose we have a relation between x and y, Developement of a viz. f (xy) =0, to developer y in a perfect series of powers function of x. given by an equation. To do this all that is necessary in general is, to write down to the equation f (y) and its successive derivatives, and then dy d2y to put -O = 0, and so find the values of y, d- ' &c. when doo doX2 V = 0, and thus obtain the perfect developement of y in powers of x by Taylor's Theorem. An example will best explain this method. Let the relation between x and y be Example. y3 -xy + 2 = 1. (1); then differentiating successively, we have (3y2 - a) p-y + 2 = o... (2), (3y2 ) q + 6yp2 p = 0... (3), when p, q, r... are, for brevity, put for -y dy d3y dx' d)+' d)' (3y2 _- ) r + (18yp - 3) q + 6p3 = o... (4) &c. &c. 94 DEVELOPEMENT OF FUNCTIONS GIVEN BY Hence putting x = 0, we find from (1) y3= 1;.. y= 1, (at least 1 is one of the 3 values of y);.from (2) Sp +1 =; - -; 6 2 4.*. from (3) 3q + -+-=;.'. q=9 3 9 4 2 38.. from (4) 3r +9.-+-=;.. - - = 9 9 27 &c. &c. Hence by Taylor's series, we have 1 x 4,2 38 x3 y =1 - --.. &c. 3F1 9 2 27 r3 low we tt- 145. It often happens however that we thus get inceedwhen finite values for some of the quantities p, q, r, &c., and of Teriefilss course this method fails; we must then proceed as follows. Let a be a value (or one of the values) of y when x = 0, then substitute a + uxa for y in the given equation f(y) = 0, and, if possible, give m such a value, that the limiting value of u when x approaches o shall not be 0 or co, but some quantity, b suppose. Then assume u = b + va", substitute this value of u in the equation and so determine n, if possible, that the limiting value of v when x approaches zero shall not be o or co, but some quantity, c suppose. Then assume v = c + wxP; it is clear that in this manner we obtain a perfect developement for y, viz., y = a + bam + cxrm+ + R, putting R = wx"+P, which we may carry as far as we please. Several dif- It generally happens that we obtain more than one value erent seriesf y when = 0, and more than one value of m, or of n, EQUATIONS WHEN TAYLOR S SERIES FAILS. 95 or of r, &c., which satisfies the above conditions, in which generally obtained in case we find more than one perfect developement for y. this manner. An example will be necessary to make the nature of this Example. process of developement clear. Let y3 - 3axy + X3 = o, then y =0 when x = 0; assume.'. y = uX, and.'. substituting u3xIm - 3Cauv+l + x3- 0... (1). Now here the powers of x cannot be all different, since the coefficients of them are, either constant quantities not equal to zero, or quantities which we suppose not to approach zero as their limiting value when x approaches zero: for it is clear that if all the powers of x be different, and if the equation be divided by the lowest power, and x then put equal to zero, the coefficient of the lowest power must necessarily become zero, contrary to hypothesis. Since then the powers of x cannot be all different some two of them must be the same; either the first and second, or the first and third, or the second and third. (1) Suppose that the first and third are the same; then 3m =m + 1, and -. m; = then (1) becomes, dividing out xa, u3 - 3au + a positive power of x = 0, which gives u=0 or V/3a when =0o: we must of course reject u=0, since we want only those values of u which are neither o nor o. (2) Suppose that 3m 3, and.'. m = 1, then we have u + 1 + a negative power of x = 0, which.'. gives u = co when x= o; this value of m must therefore be rejected. 96 DEVELOPEMENT OF FUNCTIONS GIVEN BY (8) Suppose that m + 1 = 3, and.'. m= 2; then we have - 3au + 1 + a positive power of x = o, which gives u = - when x = 0. 3 ca Hence it appears that there are two values of m, namely 2 and 2, which answer our purpose, and that when x approaches 0, u in the former case approaches /3a or -v/3a as its limiting value, and in the latter case -. 3a Hence, proceeding only as far as the first terms, we have the perfect developements y = 4-/3aa +x R 82 and y = - + R'. 3a To obtain the second terms of these developements we must assume m =-, or 2, and 1 ue = /3t a + vs', or u - + van respectively, 3a and substitute in the equation (1) and proceed as before, and so we may go on to any number of terms. It is often useful to obtain the first terms of the developements as in this Example, but we seldom have occasion to go any farther. The It is clear from this Example, that to determine the first method stated in terms of the developements, we have only to put y = a + u1'n general. (a being the value of y when o = 0) in the given equation, then suppose the powers of s to be the same, two by two. If, when we suppose two powers the same, any of the remaining powers is lower than them, then there is no value of u such as we want corresponding to these two powers, (as in the case where we assumed 3m = 3 in the Example). If, however, none of the remaining powers is lower than the two supposed to be the same; then dividing out the lowest power of x from the equation, and putting x = 0, we obtain a value or values DEVELOPEMENT OF FUNCTIONS OF TWO VARIABLES. 97 of u, (rejecting of course any value which = zero); and thus we obtain the developements required. (For examples and certain simplifications see Appendix I.) tt 146. If f(xy) be any function of x and y, and if we Developeinent of a put x = a + h, y = b + k, to develope f(xy) in powers of h functionof ~~~~~~~~~~~and 1~~~~k<~~. -two variand k. ables f(a+h,b+k) Putting a + h for x, we have, by Taylor's theorem, in powers of h and k. f(xy) = f(a + h, y) =f (ay) + daf(a y) h + f (a y) + &c.... (), but by the same theorem, if we put b + k for y, we have f(ay) = (a, + k) =f(ab) + dbf(ab) k kn + dbf (a b).. &c... (2). Substituting this value of f(ay) in each term of (1), we obtain the developement required. h" Now the general term of (1) is daf(ay), and by (2) the general term of this is d{d i f i h' hTm k'2 da b" f (a b) ' or dma df(ab).rhence, giving m and n their several values, we obtain f (a +h, b k) =f (ab) + df (a b) + dbf (a b) FI F1 h2 hk k2 + d2f (a b) - + ddf(a b) d + 2f(ab) - P2 a Vbf(a i 17 + 2 h3 + d3f(a b)....... - - & c............ &c............. &c.................. 7 98 DEVELOPEMENT OF FUNCTIONS OF TWO VARIABLES. Symbolical If we use a symbolical form similar to that in 110, form of the develope- this expansion may be expressed in the following manner, ment and of Taylor's {observing that r - is the coefficient of h'k"l-' in Series. rr r(n - r) (h + k)n expanded, viz: f( A){ _(dah_+dk)l (d(h+dbl) &'}f( b = edahf dbkf (ab). Taylor's series expressed similarly may be thus written f(a + h)= efh.a, f ~(a) being supposed to be the same as f(a). proofofthe tt 147. In the expansion of f(a + h, b + k) above obresult that hm kn damdy tained, the coefficient of is d}'ddf(ab). Now if we = dndc"y. FmFnn had arrived at this developement by first putting b + k for y and expanding, and then a+h for x and expanding again, it is easy to see that the coefficient of m-n would have rmFn been dbdaf(ab), hence we must have d] d f(ab) = (d' d f(ab), a result which we obtained before in (106). CHAPTER X. DETERMINATION OF THE LIMITING VALUES OF VANISHING FRACTIONS. 148. TirE Differential Calculus may often be employed Vanishing with great advantage to determine the limiting values of vanishing fractions; i.e. functions which assume the illusory 0 form - when the variable receives some particular value. 0 If f(a) =0 and p (a) =0, to determine the limiting value To deter>~f/\~~~~~ (X)s~~~~ ~mine the f(x) limiting of ) when x approaches a. value of a pa(x) vanishing fraction. Since f (a) and p (a) are zero, f) = () f() q5 (x) -) (j) - (p (a) <jff(<a?) f f (a) hence by Lemma XIX. the limiting value of ( ) is f((a) P(e) ^(a) or more generally, by Lemma XX. the limiting value of -f(I) when x approaches a is also the limiting value of f ( IV) _) f (X); if it should so happen that f,(a) is not an illusory ^ (x) r (p(a) quantity, then it is the limiting value we seek: but if f'(a)= 0 and 0'(a) = 0, then by the same Lemma, Cor. 1, f2(a) f(v) the limiting value of f ) is that of f(-, and therefore 0 2x) (;-) f'5(a), if not illusory, is the value we seek: if however f2(a) and 2S(a) be both zero, then we must try the third differential coefficients, and so on till we come to two differential coefficients which do not both vanish (nor of course become both infinite) when x = a; and thus we shall obtain the limiting value we seek. 100 VANISHING FRACTIONS. Case in If we come to two differential coefficients which both which this method become infinite when,v = a then there is no use in going fails. farther, since all the succeeding differential coefficients will be infinite also by Lemma XXIII. Howweare 149. In such a case we must proceed as follows. to proceed in such a Put a - h for x and expand f(a + h) in a perfect series f(a)+XAha+R' and Sp(a+h) in a perfect series (a)+BhP+R; then since f(a) and p (a) are each zero, we have R f(a + h) A ha + R A h = A= ha. -h (a + h) Bh- + R' "-'r B+ - Hence, since the series are perfect and therefore the R' limiting values of and - when h approaches 0 each zero, the limiting value of f(a + ) when h approaches zero is e (a + h) the same as that of ha-. -. Now if a>/3 this = 0; if B a</3 it = co; and if a =3 it =-. Thus the required limiting value is found. This latter method is sometimes preferable in cases where the former does not fail. f(o) W - a 1 Examples Let of vanishing (x) X - ax2 + a'v -a3 fractions. 0 which assumes the form - when x =a; o here f'(v) = n'11'- = nan-a when x = a, '(x) = x2 - 2a + a2 = a2 when = a; hence the limiting value of bf() when x approaches a is 'p (LV) f'(a) o - r - a-3 d) (a) 2 VANISHING FRACTIONS.:101 f(x) tan - sin wx a o 0 Example 2. Let =, which assumes the form - )p (V.) M, 0 when x = 0, f' (x) sec2 - cos x 0 -= -() =- when = 0 (p (x') 3 x2 1 1 - COSa3? cos: ~ 3 x2 150. Here we must make a remark of some importance. Important simplification we may f' ( ) often make' Suppose that we obtain ( in the form UV, and that in the proq (V,) cess. we know A to be the limiting or actual value of U corresponding to the value of x which makes the fraction vanish; then, by Lemma VIII., the limiting value we require is A x limiting value of V. Hence if in the process of obtaining f' (a) the limiting value of a vanishing fraction, we find that, (' () has any factor whose limiting or actual value we know, we may always substitute immediately for that factor its value whatever it be, and so simplify the operation. Thus in the present example the factor = 1 when COS X = 0, we may therefore substitute 1 for it, and then we have 1 _ COS3 V only to find the limiting value of - cos; i.e. of d (l - cos3 cos Cos2 o sin x or d (3 x") 2 and here again putting 1 instead of the factor cos2,v we have sin x d sin x only to find the limiting value of; i.e. of 2ox d2x COS X or, which is -. Hence the limiting value required is. 2' 102 VANISHING FRACTIONS. f(X) a- (an - n n)n Example 3. Let n, which assumes the form /. (a?) a)n 0 - when x 0, 0 f' (v) _ n-I ir 1 (WI....fl) an-' (n I-~n the limiting value of which when x=0 is evidently f 7rXa L Example 4. et) tan, which assumes the illusory p(a?) 2 form O.oo when a? 1. This is the same thing as the form o f (a). 0 as appears if we put in the form (1 - a) sin W CIV cos - 2 Now the limiting value of this is the same as that of (since sin = 1 when a = 1); -1 2 i. e. of which - sin - 2 2 The illu- 151. A function sometimes assumes the form co - co, sory form co-co is 0 the same as which is the same form as -: for let be a 0 0 f (x~) 0 (1':) the formg. function which assumes the form co - co when a= a, in consequence of f(a) and p (a) being each zero: then 1 1) _(m- f(I(V)-f(IV) VANISHING FRACTIONS. 103 0 which assumes the form - when x = o; thus the illusory 0 o forms - and o - co are identical. 0 Hence when we wish to find the limiting value of a Howtofind the limiting function which assumes the form co - co, we must reduce it value of a o function to a fraction, so that it shall assume the form -, and then when it 0 becomes proceed as above. 2 1 -a _ -- - becomes co - co when x = I; to find its Example5. IVv"s - -l 4 - 1 limiting value we must put it in the form - I, which assumes the form -, and whose limiting value therefore is 0' - -1 that of -, or -1. 2 S f(X) IV - a+ /~ - 2+ 2~ a - 9 a 152. Let ( = a + -, which assumes Example6. 1) ($V) 98/ *-a 2 Case of failure of o common the form - when x a. method. 0 Here f(a) and p'(a) are both infinite, we must therefore f (I) h -+ \/2 al put x = a + h, and then f( becomes, which (it) A/h ((2 a-+ h) h2 +/2a s= +,which = 1 when h = o, hence the required limitV/2+a - ing value is 1. f (0) sin3 /M -1 + (a - )2 Example 7. Let ) A similar <q (v) (I- 1) v/<a - 1case. vhich assumes the form - when. = 1; here f2 (1) and 25 (1) 104 ~~~~~dy.104 METHOD OF FINDING - FROM dx are both infinite; we must therefore put x = 1 + h, and then f ( OP ('becomes sin3 -Vh (h + 2) + M2 h (h + 2))h2 Now sinz'=z-f &c... Z5 sin3 zx'3 - &C.., putting z'= [3h(h + 2)~ r~ 2ij1 + &c.... we find f(x) h + ~R a perfect developement, and ( ) =2 ha + R' ditto, and.*. the required limiting value is that of which = V2. 2 h (See Appendix J.) When we ~~~~~~~~dy When we tt 153. It often happens, when we are finding d from arefinding dx dx from an an equation between y and x, as in 145, that the result comes equation it o sometimes out in the form -, which of course leaves us in ignorance assumes the o form0 as to what the true value of is. Thus suppose that we dx dy wish to obtain the value of when X =0 from the equation, dx6 X4 +( 3a2I-V2- 4a C2XY _ a B2 y2........ (1). Differentiating, we have 2X 2 2C-,.y_(rar 2oy)p 0...... (2) ( dydx 4x'~ 6ax- 4a y -(4a2 + 2a~yp()( d dx! Now putting x = 0, and.. y = 0 in virtue of (1), we have 0 o-o.p=o, orp=-, 0 thus when x = 0 we cannot find p in this manner. AN EQUATION WHICH BECOMES ILLUSORY. 105 But differentiating the equation (2) we have In such a case we d2y must differ6xv~ + 3a- 2 ap - (2a2+ a2p) p- (2 a'x 4. a2y) q = 0 (q = entiate drv' again in order to find and putting =0, and.-. y 0, here we find dy d 3a - 2a2 - (2a'~ a2p) p =0, or,? L p - 3 = 0; p=-2 \V4 + 3, and thus by differentiating twiCe we obtain p, and we find that it has two values - 2 T "/_7 and -2- 7. As another example, suppose that we wish to obtain p Another when v = 0 from the equation example. X4 + a y3 - 2 av y' - 0cax2y = 0.........(1), differentiating we find (43v- 2ay-6axy) + (3ay2 - 4ay - 3a2) po. (2), here put x 0, and.-. y = 0 in virtue of (I), and we find 0 0 But differentiating again we have - y -4(2y + 3r) p +~2 (3y- 2 x)p%- (3y2- 4(xy - 3x1) q = 0, 0 here put x =o0, and.-. y 0, and we find again p = L. 0 Differentiating again, therefore, we have 0 = - 18p - 12p9-pl 6p3 terms multiplied by Lv or y, here putting X = 0 and y = 0, we find P3 -2p2 - 3p = 0, which gives p = 0 or p = 1 aV/1+ 3 = 3 or - 1. And thus by differentiating three times we obtain p, and we find that it has three values 0, 3 and -1i. 106 METHOD OF FINDING - FROM dx Thus it appears that where the first differentiation of dy an equation fails to give us -d for a particular value of x, in consequence of all the terms becoming zero, we must differentiate successively until we come to an equation, all the terms of which do not vanish when we give x that particular value. These tt54.. If we perform these differentiations on the supdifferentiations may position that p is a constant, we shall arrive at a correct be per- result: for it is easy to see from the above examples that the suppo- all the terms obtained by differentiating p once or oftener sition that p is con- vanish when we give x the particular value for which we stant. wish to find p, and therefore do not affect our final equation, which gives us the value of p. Hence we may always differentiate on the supposition that p is constant, and this will somewhat simplify the process. A some- tt155. The values of p are sometimes found by the what dfferent method method of finding the limiting value of a vanishing fraction of findnge given in (148); thus in the first example we have for vanish-+ ing frac- 2 c3+ oa x - 2 a2y 00 tions. P 2 = when = 2as 4 'o a y O0 therefore by (148), 6V2 - 3a2 - a2cp 3-2p 6-' a - =a= - when = 0; a2+ a"p 2 -2- p.'. 2p +p2= $ - 2p, p + 4p -3 = 0, which is the result we arrived at before. Another tJ- 156. The following is an example where p assumes example. 0 the form - for other values of v and y besides zero. 0 To find p when v = a from the following equation, viz. ay2_ 2 a2y- V3 + 3aX2 = 0..... (I). Differentiating, we have (2ay - 2a) p - 6i2 + 6ax = 0. AN EQUATION WHICH BECOMES ILLUSORY. 107 When x = a, and therefore y = a, in virtue of (1), this 0 equation gives p =- therefore, differentiating again, considering p constant, we have 2ap2 - 12a. + 6a 0; which, when x = a, gives p2 - =;.. p= /3. tt 157. There are two objections to this method of find- Objections ing the values of p. 1st. It is generally very complicated and method of findingp by troublesome when we have to go beyond a second differen- successive tiation. 2nd. We have no right to assume that the terms diferentiation. d2y d'y containing r-e'... vanish; for although the coefficients cofntaese dx'ncg d,2y d'y of these terms vanish, yet, since d ' dY... may and often do at the same time become infinite, we cannot tell but that 0 these terms may assume the form 0. co or -; and therefore 0 we cannot assume them to be zero. For instance, in the first example, (153), in the result of the second differentiation we have the term (2a2x + a2y) q, which we assume to be zero because 2 a + a2y = o when = 0; but we have no right to do this, since we. cannot tell whether q is infinite or not. tt 158. The following method will be found free from A method f free from this objection, and very simple in practice, especially when it these obis ourmp jections is our object to find p for the values x = 0 and y =. explained. (1) Suppose that we wish to find p for the value x = 0 from the equation A4x + By + Cx4 + Dxy + Ey2 + Fax... &c. o..... (1). Since y = 0 when x = 0, it is evident that the value of p we seek is the limiting value of Y when x approaches x zero (see 148): if therefore we put - = u and find the limiting value of u, u, suppose, when tx approaches zero, u0 is the quantity we wish to determine. 108 METHOD OF FINDING - FROM dx Now put y = ux in (1), divide out x, and we find A+ Bu +.x (C + Du E + E ) (FT....)...... (), here let x = 0, and then u must become u0, if therefore u0 be a finite quantity we have (3).... A + Buo = 0, and.. u, B; which gives u,. If it should happen, however, that both A and B are zero, 0 the value we have here obtained for u0 assumes the form -, and therefore we cannot thus arrive at the real value of u0. But then the equation (2), dividing out x, becomes C+ Du + E + (F + &c....)... =0; here put x = 0, and therefore u = uo, and we find (supposing u0 not infinite), C D + EuDo+ E = 0; and.~. u0 - = D -; o = E /4 E? 4 E thus it appears that - has two values in this case, when x =0. dx How the In this process we suppose u, to be a finite quantity; values of o therefore, if u, admits of infinite values, this method does are to be not give them, and we are left in ignorance as to whether there are such values or not*. But we may easily determine this in the following manner: Put rx= Y instead of y = ux, and the equation (1) becomes (supposing A and B zero) 1 1 1 C - +D-+E+y(F...&c.), &c.=O; u2 U Ul' ' * In the common method of finding d'/ the infinite values are sometimes overlooked. AN EQUATION WHICH BECOMES ILLUSORY. 109 and then putting y = 0, we have 1 1 C + D- + E =0. U0 U0 Hence if E is not zero - cannot be zero; and therefore u0 1 u0 cannot be infinite: but if E = o, then one value of- is U0 zero, and therefore one value of u, infinite: if D also = o, both values of- are zero, and therefore both values of u0 infinite. U0 If C, D, and E be all zero, then in the equation (2), dividing out x2, and putting x = o, we have F + Guo + Huo2 + Iu03 = 0.. which equation gives us the finite values of u0. To obtain the infinite values we have, as before, the equation F- + G -+ H —+I=o. U0 Uo U0 It appears from these equations that there are in this case dy 1 3 values of when x = 0. If I= 0 one of the values ofdx U is zero, and therefore one of the values of u0 is infinite: if I = o, H = 0, two of the values of - are zero, and therefore U0 two of u0 are infinite: if G = 0 also, the three values of U0 are infinite: if I o and F = o, a value of uo is 0, and a value of - is o, and either of the equations becomes G + Huo = 0, o GC which gives u0 = -; hence, in this case, the three values G of U are 0, o, and - H' If F, G, H, I, are all zero, we must divide out x3 in equation (2) and proceed as before; and thus we may in all cases obtain an equation for determining all the values of u0. 110 METHOD OF FINDING - FROM dtRle fr If in general we divide out xa and the resulting equathe infinite tion for determining uo is only of the n - mtth order, then values of lo 1 immediate- m of the values of uo are infinite, as is easily seen. By rely. membering this we may detect the infinite values of uo immediately. Example 1. jt 159. Let the given equation be (see 153) x4+ 3a2x2- 4a2dxy - a2y2= 0, here putting y = ux, dividing out x2 and putting x - o, we have 3 a - 4 a2u0 - a2u0o =, or u0o2 - 4u0 - 3 0, dy which gives us the same values of - that we obtained before. dx Example 2. Let the given equation be (see 153) xC4+ ay - 2a.xy2- 3a2y =, here putting y=Luxe, dividing out x3 and putting x = o, we find au'3 - 2 a U02- 3au0 = 0, or u0o3 - 2U02~ -U0= 0 which gives us the same values we obtained before of d dx Example3. Let the given equation be y5+ ax x - b2Y y2 = o, in this case we find, putting y = ux and dividing out x3, U0Z = 0, hence, and therefore the three values of u0 are 0, 0, and co. How to tt160. We have hitherto supposed that the values of proceed ' 0 when the dy values of x and y which make - assume the form - are x = o and x and y dx 0 whichd y = 0: but let us now suppose them to be x = a and y = b, make dx AN EQUATION WHICH BECOMES ILLUSORY. III1 dy assume the then putting i = a + i, and y= b ~ y, in the given equation div, forma dyl not zero. will be the same thing as d, and = 0Y, = 0 will be the dx dy, assume the form 0; w values of iv, and y, which make, assume the form -; we dxv 0 may therefore proceed as before. If the given equation be U = 0, and if we denote by A, B, C, D, E... &c. the values of d, U, dyU, dx2U, 2dxdyU, dy U, &c. when iv is put = a and y = b; then, by 146, the result of substituting a + xi for iv and b + y, for b will be Axiv, By,~ C iv2 Divyl + Ey,2 2 &C.... 0; if therefore A and B be not both zero, we have A + Buo = o, and if A and B be both zero we have, C+ Duo+~Euo2 =0, and so on............ and we may find Ue therefore just as before. Let the given equation be (see 156) Example. U = ay' - 2 a'y - 2X3 + 38aiv = 0, dy from which we wish to find - when i = a, and.. y = b, div then d. U =-6X2 6aio = 0when x = a, dU= 2ay - 2a' = 0 when y = a, d-1 2U= -_12X + 6a 6 a when x =a, drVdyU = o, dy2 U = 2a. Hence A = o, B- o. C=- 6a, D = o, E = 2a, and therefore we have 3 + ue2= 0, and.-. ue = 4/3 as before. CHAPTER XI. DETERMINATION OF THE MAXIMA AND MINIMA VALUES OF FUNCTIONS. Maxima 161. THE Differential Calculus may be applied with and Minima values great success to determine the maxima and minima values defined. of a function, i.e. those particular values which are either greater or less than any of the neighbouring values. If f(ix) increases when tv approaches a certain value a (supposing iv to increase continually), and diminishes when i, passes the value a, then f(a) must be greater than any value of f(x) which is either a little less or a little greater than a, and is therefore called a maximum value of f(v). And if f(v) diminishes when tv approaches a and increases when v passes a, f(a) must be less than any value of f (v) which is either a little less or a little greater than a, and is therefore called a minimum value of f(i). How these 162. Now by Lemma XVIII. f(tv) is increasing or values of f(x) may diminishing according as f'(x) is positive or negative: therebeefound by fore if f(a) be a maximum, f'(') must be positive for all means of f (x). values of s a little less than a, and negative for all values of v a little greater than a; i.e. f'(v) must change its sign from + to - when.v passes through the value a: and, conversely, if f'(v) changes its sign from + to - when ov passes through the value a, f(a) is a maximulm. In like manner if f(a) be a minimum, f'(v) must change its sign from - to + when tO passes through the value a; and conversely iff'(v) does so change its sign, f(a) is a minimum. Hence by Lemma XVII if f(a) be a maximum or minimum, f'(a) must be zero or infinity: but of course, sincef'(a) may be zero or infinity without f (v) changing its sign, it does MAXIMA AND) M INIMA. 113 no llow that f (a) must be a maximum or minimum whenever f'(a) is zero or infinity. 163. Hence to determine the maxima and minima of Rile for determinfix), we must determine what values of x make f'(x) zero or ing maxima infinity, and try whether f'(x) changes its sign when x passes m ia. through each of these values; those values which give a change from + to - make f(x) a maximum; those which give a change from - to + make f(x) a minimum; and those which do not give a change must be rejected. 164. In applying this method to any example, we may Factors of suppress any factor of f'(x) which we are sure is always f-(Spay positive, or may introduce any such factor; since we are not pressed ". - *' ~under concerned with the actual magnitude of f'(x) but only its certain circumsign. This will often considerably simplify our operations, stanes as will appear. If therefore we find f'(x) in the form 0 (x). \f (x), and if p (x) be always a positive quantity, then we may put f'(x) =, (x) simply. If P/ (x) be always a negative quantity, then f'(x) will hlave the same sign as -, (x), and therefore we may put f'(x) -- (). 165. If it be our object simply to examine whether f'(x) changes its sign, and how, when x passes through a certain value a, then if <p (a) be neither zero nor infinity, we may suppose f'(x) = + (x) if ( (a) be positive, and f'() = -fi(x) if p (a) be negative. For it is clear that if p (a) be neither zero nor infinity, then for all values of x near a 0((x) has the same sign as 0 (a), and therefore 0 (). (x) or f'(x) the same sign as p (a) (v), i.e. f'(x) has the same sign as + 4, (x) or - xP (x) according as the sign of 5 (a) is + or - for all values of x near a, and therefore in examining whether f'(x) changes its sign, and how, when x passes through a, we may put f'(x) = + \ (x) or - \ (x) according as ) (a) is positive or negative. 8 114 MAXIMA AND MINIMA. We may therefore suppress all factors of f'(v) which do not become 0 or co when w = a, retaining only the signs they have when x = a. This is a very important simplification. (qu) is a 166. If (p (u) always increases when u increases, it is a maxlmun or mini- maximum or minimum whenever u, is a maximum or minimum mum when-) () ever ui is, d~ 0 (~u) d7u if v (u) respectively; for )= ' (j) -: and since ( (u) always always d increases increases when u increases, Op' (u) is always positive, and when u increases: if d) (u) du not the re- therefore has always the same sign as -, and.'. verse is the dx d ' case. (u) and u become maxima or minima at the same time. If ( (u) diminishes when x increases the reverse is the case. We often find this consideration of use in practice, inasmuch as it may in many cases be much easier to find the maxima and minima of 0 (u) than of u. We may 167. If f'(a) = 0, then by Lemma XX, Cor. 4, f '(a) determine whether has in general the same sign as f (a) (v - a) for all values of af(x) hw taken sufficiently near a: therefore if f(a) be positive f(x) changes its changes its sign from - to + when x passes through the value sign when x passes a, which indicates a maximum; and if f2(a) be negative the through a, by the con- change is fiom + to -, which indicates a maximum. If howsideration ever f2(a) = 0, let f"(a) be the first differential coefficient second or of f(x), which does not vanish when = a; then f'(x), (by higher differential Lemma XX, Cor. 4), has the same sign as f"(a) (m - a)"-' coefficient. for all values of v sufficiently near a. If, therefore, n be odd, f'(x) does not change its sign when x passes through the value n; but if n be even it does, and the change is from - to + or from + to - according as ff (a) is positive or negative. Of course we here suppose that none of the differential coefficients are infinite. Simple inspection is These considerations will enable us to determine whether often the best way f '(x) changes its sign when x passes through the value a, and and some- if so, whether the change is from - to + or from + to - only way of But this is often more easily seen by simple inspection; and making out whether indeed when f'(a) or any of the higher differential coefficients f' (x) ~ changes its are Infinite, which often occurs, simple inspection is the only sign and method we can resort to. how. MAXIMA AND MINIMA. 115 By suppressing factors of f'(x) which do not change their sign (in the manner shewn to be allowable in 164, 165,) before we differentiate f'(x), we may often find the sign of f2(a), or more properly, of that quantity which answers the same purpose as f2(a), with great facility, and avoid the necessity of being obliged to proceed to higher differential coefficients. 168. If y = b when x = a, and we can obtain a perfect When we obtain y-b developement in the form in the form R l A (x-a)+ R, we y-b = A (x - ar' + R = A ( - a) I + +R.we may immediately see whether R x = a makes then by taking v near enough to a, we may make p,, ya maxi (X - a)n mum or as small as we please, and then y - b will have the same sign minimum. as A (-a - a)"^. Therefore if m be an odd number or a fraction with an odd numerator and an odd denominator and A positive, y is <b when x is < a, and > b when x is > a, and vice versa if A be negative: in this case therefore b is not a maximum nor minimum value of y. But if m be an even number or a fraction with an even numerator and odd denominator and A positive, then y is > b for all values of x near a, and therefore b is a minimum value of y: and if A be negative then y is less than b for all values of x near a, and therefore b is a maximum value of y. If m be a fraction in its lowest terms with an even denominator, then y is impossible for all values of x less than a {or for all values of x greater than a if y - b= A (a - )"+ R}, in which case we cannot call b a maximum or minimum value of y. 169. The following examples will shew the advantages of the method of finding maxima and minima here recommended. Let f () =. (V - a), Example 1. then f'(tv) = I ( So - a)2 (8 - 5a) = 8x - 5a, suppressing the factor r' (a - a)' which is always positive see (164). 8-2 116 MAXIMA AND MINIMA. Now 8iv - 5a =0 when iv = -, and changes its sign 5a from - to + when v passes through the value - hence X = 5 a gives a minimum value of f(iv), vii. (i)' ( 5a - (a 8 8 8i/ Or thus f'Qev)= 2v - 5sa;.f'~ (iv) =2;.~. f2 (+) = positive, 88 which indicates a minimum by (167). Example 2. Let f(IV) (i- - a)'' iv2 then f'(v)= - (- 2iv-;3a) (iv- a)F — (2,v + 3a), suppressing the factor - (V - a)"' 3a SNow - (2iv + 3a) is zero when v=, and changes its sign from + to - when x passes through that value: hence x = - gives a maximum value of f(v). 2 Or thus, f'(v) = - (2ix + 3a) f ). = negative, which indicates a maximum. sin miv: Example 3. Let f(s) =, m being an integer, then ) = 2~ sin x =mcos mx sin x - sin mx cosi () sin 2 X m cos mv sin x - sin mx cos x...() MAXIMA AND MINIMA. 117 suppressing the factor si2 sin x = cos m x cos x (m tan x - tan mv). Now it is clear from geometrical considerations that there is some value of mx between 0 and -, which makes tan m x = m tan x: let this value be ma; then f' (a) = 0. Also differentiating (1), f2 (x) = - (m2 - 1) sin mx sin x, which is negative when x = a, since m2 is > 1, and ma and therefore a between 0 and -. Hence the root of the 2 equation tan m x - m tan x = 0 which multiplied by m lies T sin me between o and - makes - a maximum. 2 sin x t f(, = /sin n m 2 Example 4. Let f(x)= -(sin, sin m j then f' (x) = 2 i — (m cos mx sin x - sin mx cos x) = sin mx sin M (m cos mx sin x - sin mx cos av), sin4 07 multiplying by the factor -. Now x a (see last example) makes f'(() = 0; also sin mx sin x is positive when x = a, and may therefore be suppressed so far as this value of x is concerned, therefore f2(x) has the same value as before and is therefore negative when = a; which indicates a maximum. Again, v = - makes sin mx = o, and the product of the other factors of f' () negative; therefore by (165) so far as this value of x is concerned we may put f'(x) = - sin mx; 118 MAXIMA AND MINIMA.. f(-) = - m cos mv = positive when x = -, m 7r hence x=- gives a minimum. Example 5. Let f(V) = (I - a), then f'(x) = (. - a)-' multiplying by the factor - (iv -a) which is always positive; (..fx(~) 1, hence f'(a) = 0, and f2(a) = positive; and therefore i = a gives a minimum. (For more Examples see Appendix K.) Maxima tt 170. Let f(xy) be a function of, two independant offunctionsvariables x and y, and let f(xvly) be a maximum value of of several variables. f(a y), determined on the supposition that y is constant and x alone variable, x1 of course being some function of y; and again let f(ab) be a maximum value of f(xly) determined on the supposition that y is variable, a being the value of Xi when y becomes b: then f(ab) is a maximum value of f(xiy) when x and y are both supposed to vary in any manner. For supposing w to have any value near x,o f(ily) is >f (y); and supposing y to have any value near b, and therefore xt some value near a, f(ab) is >f(xy), and therefore a fortiori f(ab) is >f(xy). Hence if xy have any values near a and b, f(ab) is >f(xy), and therefore f(ab) is a maximum value of f((my), supposing x and y both to vary in any manner. Partial and We may call f(ab) a total maximum in contradistinction total maxima and to f(x1 y), which is only a partial maximum determined on minima. the supposition that only x varies. In the same manner if f(vy) be a partial minimum value of f(xy), y being considered constant, and f(ab) a OF FUNCTIONS OF TWO VARIABLES. 119 minimum value of f(x, y), y being considered variable; then f(ab) is a total minimum value of f(vy). Hence a maximum value of a partial maximum is a total maximum, and a minimum of a partial minimum is a total minimum. Conversely, if f(ab) be a total maximum, it must be the maximum of a partial maximum; for f(ab) must be >f(xy) for all values of x and y near a and b; therefore f(ab) must be >f(xly), y being supposed to have any value near b, and therefore x some value near a; therefore f(ab) is a maximum value of f(vry), i.e. it is a maximum of a partial maximum. And similarly, if f(ab) be a total minimum, it must be a minimum of a partial minimum. Hence by finding the maxima of the partial maxima of f(xy), and the minima of the partial minima, we find all the total maxima and minima values of f (y). And thus, by the methods already given of finding maxima and minima of functions of one variable, we may obtain the total maxima and minima of functions of two independant variables. Let f(my) = X + y4 - 4axy, Example. then to find the partial maxima and minima of f(xy) on the supposition that x is variable, we have df (xy) = 4x3 - 4 ay2, which = 0 when x = (ay2)~, and evidently changes its sign from - to + when v passes through this value; which indicates a minimum. Therefore substituting x = (ay2)3 in f(xy) we obtain the partial minimum y4 _ a y4 = F3 (y) suppose. Then to find the minimum of this, we have ' (y) = 4 y] (y - 2 a), which = o when y = 0 or 2,a: when y passes through the 120 MAXIMA AND MINIMA former value,f'(y) changes its sign from + to -, and throlgh the latter from - to +: the latter therefore makes +' (y) a minimum. Hence f(vy) has a total minimum value obtained by putting v = (ay2)~ and y = a; i.e. v = 2Qa and y 2a. (For more Examples see Appendix K.) Theequa- 171. Since f(xay) is a maximum or minimum on the tions d,f(xy) supposition that y is constant, it is clear that d f(z'y) must d=0 or become 0 or c when V1 is put for Ci, and this being true =0 or co, whatever value is assigned to y, it must be true when y= b give us thosevalues and therefore x = a. Hence if f(ab) be a total maximum or of x and y whichmay minimum, we must have make f(y) a df(xy) = 0 or o... (1) total maximum or minimum. when a and b are put for v and y. And in exactly the same way we may shew, by treating y as we have done v, and x as we have done y, that we must have dyf(xy)= 0 or o... (2) when a and b are put for i and y. a and b are therefore values of,v and y got from the equations (1) and (2) taken together; and these equations therefore give us all the values of v and y which may make f(ivy) a total maximum or minimum. The method of distinguishing maxima and minima of functions of two variables given by Lagrange is often troublesome; and since it does not include the cases where any of the partial differential coefficients of f(vy) become infinite, or where those of the second order vanish, it must be considered as very incomplete. How to find tt 172. When y and x are connected by an equation, maximum and mini- and we wish to find what values of Cv make y a maximum or mumvalues minimum, we have only to differentiate the equation to find of y, when OF IMPLICIT FUNCTIONS. 121 an equation y and proceed as before. The following example will ex- beween d x x and y. plain the process. Given y' - 4a2'xy + x4 =.., (1) to find what values of,x make y a maximum or minimum. We have, differentiating dx d y x3 - a y and. - = dx a2x - y3 3 y = - makes this zero, (at least if x be not zero, for dy o then dx = o a case which we shall consider presently); X3 now if y= we have in virtue of (1) a?' X12 X4 a- - 4a2 - + r4 = o or x8 - 3a8 ==, dividing out x4; a a or x = 3$a, and.. y = 3Wa; dy these values put in the denominator of d make it negative; therefore by (165) we may put dy dy- ay - x3; dx dy 2= ady -a — 3x dov dx negative, when we put x= 38a and.-. 0; dx.'. = 38 a makes y a maximum. Next, as to the value x=0 (and.-. y=o) we find, putting dy y =u as in 158 and dividing out V2, that u0= o, and..- =0 dcz 122 MAXIMA AND MINIMA when x = 0, therefore v = o may give a maximum or minimum of y. To determine whether it does, put y = uvm as in 145, and we have U4 4m- 4 Aa2u +1w + 1V- = 0; 4m = m+ 1 gives m =, and 4mor + 1 <4;.'. we may assume m = ~, and this gives U4 - 4a2l = o when w = o; and.u. u = (4a'2), we therefore have y = (4a'2)xA+ R. Again, 4m = 4 makes 4m or 4 > mr + 1, and must therefore be rejected. And again, m + 1 = 4 makes m= 3, and.'. m +1 or 4 < 4m; which gives us u = when = 0, and 4 a' 4as X3 ~-. Y + R. Now by (168), both these expressions for y shew that y = 0 is neither a maximum nor a minimum value of y. It appears from what has been just explained, that when we put y = Uxm for y- b = u (x - a)" if m = a and y= b be the values we are concerned with }, then we may immediately reject any value of m which is not an even number, or a fraction with an even numerator and odd denominator: bearing this in mind we may very readily obtain the maxima or minima values of y. y3 dy Lastly, m= Y makes - =co, and in virtue of (1) we 1 3 have y = 38a, I= 38ar Now assume v = 3sa + z, and y = 3a +uzm, and (1) becomes (38a + uzm)4 - 4a2 (38a + utz"') (38a + z) + (3a + z)' =0, OF IMPLICIT FUNCTIONS. 123. which, bearing in mind the remark made in Appendix I respecting the rejection of certain terms, may be written 2 1 6. 3Sa2U2z2m - 4a2uzm+l + 8. 38a- = 0. Here 2m = m + 1 gives m = 1: 2m = I gives m = -: and + 1 = 1 gives m = 0. All these are to be rejected, since they are not even numbers or fractions with even numerators and odd denominators. Hence it appears, that x = S8a makes y a maximum, and neither x = 0 nor x = 38a make y a maximum or minimum. CHAPTER XII. TANGENTS AND NORMALS TO CURVES. THE CURVATURE OF CURVES. THE EVOLUTE. THE Differential Calculus is of great use in various parts of analytical geometry. We have already seen in (26) that it puts us in possession of a general method of drawing tangents to curves. We now proceed to shew that it admits of, not only this, but many other important applications in analytical geometry. posi- 173. If a right line SPQ (fig. 6) be drawn passing To find the through any two points of a curve, and if SPT be its limiting equation to the tangent position when Q approaches P, i. e. if the angle made by PT f a urent and PQ may be diminished ad libitum by sufficiently diminishing the arc QP; then PT is said to be a tangent to the curve at the point P. Let x y be the co-ordinates of P, 'y' these of Q, Z PTX= f, then tan + is the limiting value y -- of tan PRX when Q approaches P, i.e. of, Y when x' approaches I, which, by definition of a differential coefficient, dy is -; hence do tan dy tan ke = -. Hence if x, y, be the co-ordinates of any point on the tangent PT, since it is a line passing through the point P and making an angle tan-1 dy with the axis of x, we have for its dw equation dy y,- y = - ('/ - _). TANGENTS, NORMALS, &C. 125 174. The normal at the point P is the line PG (fig. 8) Coeq which passes through P at right angles to the tangent PT: its tion to the Normal. equation is therefore dx Y' - Y - (( - ). 175. If we take s to represent the arc BP (fig. 6), s' the Proposiarc BQ, and c the cord PQ; then ds2=dx2 + dy2, s being the arc S-S - C of the ---- s- *. ---.curve. OG - IV C OG - t ) C -S (1 + ( ) since c2= (' -,)2 + (y - y):. 8t — 8? Now when x' approaches x the limiting value of - is 00 - x ds s' -s - y - that of is 1 by Lemma IX, and that of - dx c I - I dy is dy, hence by Lemma VIII, we have ds dy 2 dx V1 +, or ds2= d'+ dy2 - X - 5' - S Cor. 176. We have cos PRX - = -, and x C - s C ==ds cos/f, when x' approaches x the limiting value of PRX is f, that of ds sin. x' - d s - s is- and that of is 1; therefore s —8 ds c dx cos = d- or d=x ds cos,j. y ds /'-Y_ y'-Y s'-s Similarly sin PRX= - = Y. C — s C dy.'. sin = ds;.d. dy=dssinl/. ds ' 126 TANGENTS, NORMALS, &C. These for- 177. If it be allowable to argue according to the memulm may be very thod of Leibnitz, spoken of in (84), we may suppose Q aimply ob- taken so near P that the arc QP may be regarded as a right the method line coinciding with the tangent at P, and PO, OQ, PQ taken respectively equal to dx, dy, ds; these differentials being regarded as infinitely small: then we have immediately dx=ds cos +, dy = dssin; dy and therefore dx2 + dy2 = ds2, and = tan I. dot Which formulae are all correct results in consequence of a compensation of errors such as that shewn to exist in (85). Though this method of arriving at these formulae is not sufficiently exact for the purposes of elementary demonstration, it is useful in enabling us to remember and make out these formrule. Subtan- 178. The line MT (fig. 8) is called the subtangent, gent, subnormal and MG the subnormal, PG the normal. normal. Hence yd xt subtangent = ycot\ - dy-' dy subnormal = ytan = yd dx yds normal = ysec / = d(For examples of the use of these formulae see Appendix L.) Proposi- 179. Since =tan 1-, we have tion. dx To determine d y whether the a 1 d concavity dx d d of a curve 1d + is turned d )' upwards or FLEXURE OF CURVES. 127 1 downNow -- 2 is essentially a positive quantity; wards, to jdy \ the right or 1 + I to the left. day hence d* has always the same sign as d - dx Now if dk be positive, xI increases when x increases, and dx therefore the concavity of the curve must be turned upwards, as in (fig. 9), but if dx be negative, + diminishes when w dx increases, and therefore the concavity of the curve is turned downwards, as in (fig. 10). Hence the concavity of the curve d2 y is turned upwards or downwards according as d- is positive or negative. And in the same way it may be shewn that the concavity of the curve is turned to the right or to the left according as de de is positive or negative. 180. Hence if at any point -- changes its sign, the Proposidx" tion. To deterconcavity of the curve will change its direction, if the change mine pointe of sign is from + to - the concavity which was before turned of contrary ' -n i. i,, - flexure. upwards will begin to be turned downwards, as at the point P (fig. 11); and if the change be from - to + the reverse will be the case, as at the point P (fig. 12). A point where the concavity of the curve thus changes its direction is called a point of contrary flexure. dsy Hence, by Lemma XVII, Y must be 0 or co at a point of contrary flexure; and therefore, to determine where such points are on the curve, we have only to find what values of d2y 0x make -d 0 or co, and try whether x, in passing through 128 INDEX OF CURVATURE. d~u each of these values makes d2 change its sign or not; those dX2 for which there is a change of sign give points of contrary flexure, and those for which there is not are to be rejected. It is evident that + is a maximum or minimum at a point of contrary flexure. (For Examples, see Appendix M.) roposi- 181. measures the rate at which the tangent changes tion. To deter- dp minethe its position as we go along the axis of x, and of course its curvature magnitude depends in part upon what position the axis of x at any point d of a curve. measures the rate at f de uofe occupies with reference to the curve. dmeasures the rate at curvature. which the tangent changes its position as we go along the curve, and its magnitude does not at all depend upon the position of the co-ordinate axes, but solely upon the nature of the curve: the greater therefore the curvature at any point, d+ d, the greater will be and vice vers. d may therefore be ds ds called the Index of curvature at any proposed point, since it indicates the rate at which the tangent changes its position as we go along the curve; i. e. the degree of curvature of the curve at each point. Proposition 182. If PS, QS, fig. 13, be two normals to a curve at respecting the quang the points P and Q, and if p be the limiting value of PS tity P. when Q approaches P, and c the chord joining PQ; then p is the limiting value of when Q approaches P. c c sit S For S sin S S PS sil S sin PQS S Now by sufficiently diminishing PQ we may evidently make L PQS differ from - ad libitum; therefore when Q approaches P, 1 is the limiting value of sin PQS; 1 is also the limiting INDEX OF CURVATURE. 129 sin S value of by Lemma X, and we suppose p to be the limiting value of PS. Hence the limiting value of - when S Q approaches P is p. Q.E. D. 183. Let + and \' be the angles which the tangents at Cor. Index of P and Q make with the axis of, and s, s' the lengths of the curvature arcs BP, BQ; then since /' - f evidently = S, we have is equal to vidently~I ' we 1b f'-/-+ ZS c P 8s -s c s- 8 Now when Q approaches P, the limiting value of Yis d S 1 s sS is d~s that of- is -by what has just been proved, and that of --- is 1 by Lemma IX: hence, we have s - s. ds p i.e. the index of curvature =-. P p is called the radius of curvature for reasons we shall hereafter explain. 184. We may find p in terms of x and y as follows: Proposi'To find the dy value of p we have t = tan-; see (173). at any point d CT of a given ~curve. (dy) d t).. d = _ _ d dy 1+d a2 (dy\ = ds d since d 2+ dy2= ds2 (175) Hence, since - = -, we have p ds 9 130 THE EVOLUTE. 1 d'2 dy _ dd2y - d2dy ( p ds' \d ds3 If we suppose x the independant variable, this becomes 1 dx2 d2y dxd2y p ds3 dx ds 3 *...... (2). From the expressions (1) or (2) we may find - when we p know the curve. Since -- = -, it is clear from (179) that p is positive or ds? negative according as the concavity of the curve at the point P is turned upward or downward. (For examples, see Appendix N.) Proposi- 185. Let P' (fig. 14) represent the limiting position of tion. To deter- the point S in fig. 13, when Q approaches P; then PP' is the mine the co-ordi- quantity we have denoted by p; let AL (= a), LP' (= 3), be nates of the the co-ordinates of P', PK parallel to LM. Then, since PP' extremity of p, t evidently makes an angle - - with the line PK, we have a = x -p sin (1) 3 = y + p cos j. which equations determine the position of the point P' corresponding to the point xy of the curve. Proposi- 186. It is evident that the point P' changes its position tion. To find the and traces out some curve when the point P is made to move cedour along the curve BPQ; we may find the equation to the curve by P' thus traced out as follows: when P longthe Since p, +, and y are all certain functions of x which we cBUVp may find when the curve PQ is given, it is evident that we may eliminate x, y, +, p, from the equations (1), and so find a relation between a and 3 which is independent of x, and therefore holds for all positions of the point P. This relation is therefore the equation to the curve traced out by P'. This curve has some curious properties, as we proceed to shew. THE EVOLUTE. 3 131 187. Differentiating the equations (1) we find Remarkable properties of da = dx - -p cos +Idxk dp sin 'Jthis curve. dj3= dy -p sin xfd4, + dpo cos 4P.. d s dxNow p Cos xkd+P = - T- d'q1 (183, 1-76), dx, and p sin xfrdxk dy. similarly; hence dac = - dp sin4, d/3 = dp cos~r from which equations we find dcc da + d/32 d p2.......(3). 188. Let B'P'Q' (fig. 15) be the curve traced out by First Pro. then the equation ~~~~~~~~~~~~perty. P'; henthe quaionto the tangent to this curve at the The normal point P' is PP'touches the curve d/f3 traced out Yi - - x aby P' at dac the point or yi -/3=-cot (.QI1-a) by () Now this is evidently the equation to the line PP' since that line passes through the point P", and makes an angle 4, + - with the axis of xv. Hence the normal PP' touches the curve traced out by P" at the point P' as we have represented in the figure. 189. Again, if a- denote the length of the arc B'P' of Second Propry the curve B'P'Q' measured from some fixed point B', we have Thoe curvtedo-=V'da 2~+dI32 = 4dp by(3): tae u by ~~~~~~~by the endl of a string since p evidently decreases wheni o- increases in consequence unwound off the of the way we have measured a-, we must take the lower curve,of these signs., and therefore we have B'P'Q'o da T+ dp = 0; 132 THE EVOLUTE. and.'. d (B'P' + P'P)= o, i.e. the length B'P' + P'P = some constant, C suppose. Now let one end of a string whose length is C be fastened at B', and the other end to the point of a pencil; and the string being full stretched let it be wound on the curve B'P' till it touches it at P', and therefore coincides with P'P in direction: then since the length of the string is C, which =B'P'+ P'P, the pencil point will be just at P; and this is true whatever point of the curve P' may be. Hence if we unwind the string off the curve B'P', the pencil point will trace out the curve BPQ. n1'P' is It is on this account that the curve B'P'Q' is called on this account the evolute of the curve BPQ. called the evolute of BPQ. (For examples see Appendix 0.) .CHAPTER XIII. POLAR FORMULJE, DIFFERENTIALS oF, AREAS, SURFACES, AND VOLUMES. WFE now proceed to investigate certain formulae analogous to those of the last chapter, supposing the curve referred to polar co-ordinates.:190. Let APQ (fig. 16) be any curve referred to polar Proposico-ordinates; -SP = r, PSA= 0; SQ = r', QSA= 0' chord To findi PQ =V, arc AP=8, arc AP'= s' then.9'-8 S -S C Now C2 -r Q + r -2rr' cos (0' 0) -(r' r)2 + 4rr' sin 2 z, putting for brevity Z and co ' =12 sin2 V. s- Ir - rs ( sin x) Hence + _ ~iii rr' 0' 0 C /0- 0I Let 0'. approach 0, and therefore z, zero; then. the limiting value Of is dST. that of is 1, that of sn is also 1, and, that of r' is r:therefore we have ds ' + 2 o s r.2d dO \dO +/ 2o. 1345 POLAR FORMULX. This result 191. This result we may also prove as follows. deduced from the By (175) ds V/dx2 + dy2', rectangular formula. but ==rcosO, y = r sin 0; dv = dr cosO. - r sin 0 d9, dy= drsin90 + r cos 0 d8; da + dy' di2 + r2 dO'; and.-. ds = Vdr2 r2dO2. Proposi. 192. Draw SY perpendicular to the line QPT which tion. To find p passes through the points P and Q, and draw PO perpenthepepn dicular on dicular to SQ, then by similar triangles the tangent from S. OP, rsin (0' - 0) SY= SQ. PQ c sin (0'-O0) O'-0 s'-s -Pr. ' 0' - 0 ls Now let 9' approach 0; then the limiting position of the line QPT will be the tangent at P; and therefore if p be the perpendicular from S upon that tangent, the limiting value of SY will be p. Hence, by Lemmas IX and X, we have 2 2 p r ds Cor. 193. This result we may also prove as follows. This result deduced The equation to the tangent is from rect.. h angular formule. dy Y, - y ('Vj m), d x and the perpendicular p upon this line from the origin is dy dx ~~~xdy -ydx dd, I OI~ dI —dor,since ds= Vdx' + di'; (d ~ds POLAR FORMULAX. hence, since y = r sin 6, a r cos 6; and therefore dy = dr sin 6- + r cos 6 dO, dx = dr cos 6 - r cos 6 d6, we have xdy - ydx = r2 dO, i'2 dO and therefore p = ds OP r sin (6' - 6) Proposi194. Sin OQP =- = tion. C C To find the sine and cosine of the r sin (6 -) 6' 6 ' angle un____________ ______ der the tan6S - s c gent and radius vector, viz. 4. Now let 6' approach' 6, then if / be the angle which the tangent at P makes with the radius vector SP, the limiting value of OQP is q. Hence we have rd6 sinc= ds de 2-de~ Hence cos ~= 1/ - (rd6)2 ds2 by (191). r dO Hence tan = dr 195. Since p r sin /i, we have from this Cor. p hence r'de found, p d as before. 136 POLAR: FORMULE Proposi- 19. tnOP=OP 'rsin (O' -O) tion.19. tnOP=~= To findOQ '-o(O. ) tan (p inde- - pendantly or 2Z0 of the pre- Here put =,and cos (0' -0) =1-2sin~z ceding2 articles. then tan OQP- = sIn(0 -1.- r -i- 2,r sin2,r sin (0' - 0) 0' - 0 r r sin~ + r sin Z Now let 0' approach 0, and therefore z zero; then the sin (0'- 0) r'r sinz limiting values of ~,~,~,,and sin ',are dr respectively 1, 1, and zero; hence we have r rd0 tan =dr dr dO Cor. 197. The line drawn from S perpendicular to the radius To find thevetro e a polar s vetrt eet the tangent is called the polar subtangent; it tangent. evidently is equal to r.2 dO r tan, or d These for. 198. If it be allowable to argue according to the method provdlb of Leibnitz, spo~ken of in- (84), we mnay, as in (177), suppose Leibnitz' method QP =ds, QO dr, OP =rd0, zOQP~q and hence we get 2 2 ~r dO rd0 ds=V2 2dO0~dr2, sin = tan - r'dO t dO p= r sin,~= polar s'ubtangent = r tan di' POLAR FORMULAE. 137 This method of arriving at these formulse will enable us to make out any of them very readily, in case we forget them. (For Examples of the use of these formulae see Appendix P.) 199. If + be the angle made by PT with we have + = 0 +,, and..-.d = d0 + dp. SA, then Proposition. To find p in terms of r and p. Now p = r sin r, and... dp = drsin(P + rcos(pd d 'rdO rdr = dr -- + d d ds ds rdr rdr =-T (do + d) ds r dr. d r _rd, since (183); P ds p rdr hence p =- dp 200. We have as in (199) rdO = p + 0= tan-1 + 0, dr) hence differentiating, supposing dO constant, we have rd2r d, = d 0 d t dr'2 +.1'k r Proposition. To find p in terms ofr and 0. 2dr2' - d2p + r2dO = -- d --- — -- r, since ds' = dr2 + r2 d0 ds2 - 188 POLAR FORMULIE. hence - d s =d d+11 dO (2dr' - rd2r + r'd8') ds' or dO (dr' + dS' - rd~r) Cor. 201. This result may also be arrived at thus. The same result de. ds3 from We have by (186) =dydx d'd formuhe. Now dy = drsin 8 + rcoA 18, d2y = d r2sin8 + 2 drcos8d8 - rsin8d 8, dx = drcos 8 - r sin 8 d 8. Hence d'ydx = (drd'r - Srdrd 82) sin 8cos 8 + 2 dr 2cos28. d8 0- r (d'r - rd 80) sin28. do. Now d'xdy is evidently got from this by putting ~ - 8 2 instead of 8; hence we evidently, have, putting sin 8 for cos 8, cos 8 for sin 8, - d8 for d8, and subtracting d'ydx - d'xdy = dO (2 dr2 - rd2r + r'd8'), ds' and.~. p~~~~~d.3 and'.. P- d (2dr' - rd'r + r 2d8) Proposi. 2021 If p increases when r increases, it is easy to see that Tfind the concavity of the curve is turned towards the pole, and if whether the p diminishes when r increases, it is turned from the pole. concavity ofthecurve Hence, by Lemma XVIII., the concavity of the curve is is turned to or from the dp pole. turned towards or from the pole according as T is positive or negative. (For Examples see Appendix P.) The use of the following formulae respecting the differentials of areas, surfaces, and volumes cannot be shrewn without the assistance of the Integral Calculus. DIFFERENTIALS OF AREAS AND VOLUMES. The Integral Calculus is the inverse of the Differential Calculus, its object being to find the quantity from which a given differential is derived. When the student becomes acquainted with it, he will perceive the importance of these formulae. 203. Let BPQ (fig. 17) be any curve, AM= x, MP= y, Proposition. AN = x', NQ = y', the co-ordinates of any two points P and Q To find the of it. Let A denote the area BPM and A' the area BQN; diffenretial draw PO and QR parallel to MN to meet NQ and MP in O and R. Then area VPQN lies between MRQN and MPON, i. e. A'- A lies between y' (x'-,v) and y (x' - i); 4A'- A * - lies between y' and y. 'C - O Now y'= y when x'= x, hence by Lemma VII. y is A'- A the limiting value of, when x' approaches x, i.e. dA d-= y, dA - ydx. 204. Let V and V' be the volumes generated by the revo- Proposition. lution of the areas BPM and BQN (fig. 17) round AN, then To find the differential vol. gen. by MPQN lies between vol. gen. by MPON and ofaolidof revolution. vol. gen. by MRQN, i.e. V'- V lies between Try2 (x?'- ) and Ty'2 (('- ); V'- V '., lies between wry2 and ry'2; dV.'. as before d-= r y2, d V = 7ry2dx. 205. Let S and S' denote the surfaces generated by Proposi. tion. the revolution of the arcs BP and BQ about AN (fig. 17); To findthe let s denote the former arc and s' the latter; produce POdifferential 140 DIFFERENTIALS OF AREAS of a surface to U and QR to T, so that PT and QU shall each be equal of revolu- tion. to the arc PQ. Then we may evidently assume that surf. gen. by PQ lies between surf. gen. by PT and surf. gen. by QU, i.e. S'- S lies between 2Qry (s'- s) and rry' (s'- s), S'-IS. lies between 2 ry and 2ry'. dS Hence as before = 27ry, or dS=2 ry /dZ2+ dy2. s S Pionposi- 206. Let APQ (fig. 18) be a curve referred to polar tion. To find the co-ordinates, SP= r, SQ = r PSA = 0, QSA= 0': let A differential of a polar and A' be the areas ASP and ASQ; describe the circular area. ea arcs PO and QR round'S as center. Then area SPQ lies between area SPO and area SQR, It (of '- ) r.' (o '- ) i. e. - A lies between ----- and 2(; 2 2 A'- 4 2 r2.'.,- 0 lies between - and-. 2 2 dA r2 Hence as before =. dO 2 Thesame 207. This result may also be obtained as follows: draw result de- PM perpendicular to SM, then SM = x and PM = y will be duced from the rect- the rectangular co-ordinates of P, let B be the area APM, angular then by 203 formula. dB= ydx, but B = area SPM - area SPA xy _ A; xdy + ydx dA = ydx; dA dy - ydx '~, -d ~c2 AND VOLUMES. 141 Hence as in (193), r2 dO dA == 208. The result is of importance. 2dA=xdy -ydx. 2dA = xdy - yd,,v CHAPTER XIV. ASYMPTOTES. An Asymp- 209. IT often happens, in the case of curves which have tote, whatinfinite branches, that when the point of contact is moved off to an infinite distance from the origin, the tangent remains at a finite distance, or, to speak more accurately, approaches a certain limiting position which is at a finite distance from the origin. In such a case the tangent or rather its limiting position is called an asymptote. Proposi- 210. The equation to an asymptote, when one exists, tion. To deter- may be thus determined. mine the equan he equation to th t the tangent, which may be written in an asymptote of a the form curve. dy dy Y = d %o + Y - d x, y d1 dy y _du put - =, and therefore d - - it' x dx x" dx and it evidently becomes (y dud\ 2du /y,\= + t v -) x- d put moreover 1 i, 2.jdz o = -, and.'. d = and the equation reduces to the following form [/ du\ du yi= u-z- I Now if the curve be of such a nature that when xv approaches co, i.e. when z approaches zero, the limiting values ASYMPTOTES. 143 du of u and - are finite quantities A and B; then the limiting dT form which the equation to the tangent assumes is y, = Am, + B, which therefore is the equation to the asymptote. 211. It is very easy to find A and B in all cases from Howto find 1 A and B. the equation to the curve, by putting x = -, and y =-in that equation, and so finding a relation between u and z: then A being the limiting value of u when z approaches zero may be found by putting z = 0, and B, which is the limiting value du of - when z approaches zero, may be found by differentiating the equation and putting z = o. Let x3 + 3axy - y3= 0 be the equation to the curve; then Example. putting -~ y = - and multiplying by z3 we find 1 + SaUz - 3 = 0 and differentiating Sau + (3aw - S2t) = 0; du in these equations put z=0, and.. u=A, =B, and we find dx 1 - A3, SaA - AB = 0;.A. A=1, B= X, oe and the equation to the asymptote is therefore y = m + a. 212. If A and B turn out to be impossible or infinite, There may then there is no asymptote corresponding to an infinite value of be ante asymptote the abscissa: but there may be one corresponding to a finite corresponding to a value of x, for the point of contact may go off to an infinite finite value distance from the origin for a finite value of x, y of course must beh becoming infinite. Such an asymptote cannot be found by the foundw. otherwise. 144 ASYMPTOTES. method just explained, since in that method we suppose x infinite; but it is easy to see that whenever the ordinate becomes infinite for a finite value of Ov, it becomes an asymptote, and therefore if a be a finite value of x which makes y infinite, the line whose equation is x = a is an asymptote. Therefore all we have to do in a case of this kind is to put y =- in the equation to the curve, and then make z = o; and if this gives us a finite value of x, a suppose, x = a is the equation to the asymptote. Example. Let the equation to the curve be y2 - -ay3 =, U 1 putting y -, X = - and multiplying by z3 we find u2 - 1 - 2au2Z = 0, du and differentiating (2u - 4auz) -- -2au 2 = o, du in these equations putting z=o, and.. u =A, - =B, we find A2- 1 = 0, B - A= 0,... A=4 1, B=L =a, hence there are two asymptotes corresponding to = co whose equations are y = + a, and y= -.* - a. Moreover, putting y = - in the original equation and multiplying by z2, we find i - tv3z2 a = 0, and therefore when z = o? =2a, which is a finite value; hence there is another asympfote whose equation is v == 2a. ASYMPTOTES. 14;5 When therefore our object is to find all the asymptotes of a curve, we must not forget to try whether there is one corresponding to a finite value of X. 213. We may consider asymptotes somewhat differently, Asymptotes considered and perhaps more simply, in the following manner. somewhat differently. By putting - for z, and - for y in the equation to the Z %- du curve, we find a relation between u and a; and if u and dz have finite values A and B when z = 0, we may by. Taylor's Theorem assume that u = A + Bz + R; where R is some quantity, such that zero is the limiting value of - (i. e. of Rxg) when - approaches zero, i. e. when x approaches infinity. Therefore, restoring x and y, we may assume that y = A + B + Rw. Now let BP, fig. 19, represent the curve, AM= x, MP y: also let B'P' be the line whose equation is y'= A x + B, where y' =. MP', then PP' =y - y' = R; therefore, since zero is the limiting value of Rx when x approaches co, PP' may be diminished ad libitum by sufficiently increasing m. Therefore the line B'P' continually approaches the curve BP as we go off to an infinite distance from the origin, but never actually meets it, though we may make PP' as small as we please. A line thus circumstanced is called an Asymptote, and we may find its equation just as before, since A and B are the same quantities as before*. 214. We may determine the asymptotes of a polar curve Proposias follows: tion. * See note, page 154. 10 146 ASYMPTOTES TO POLAR CURVES. To deter- Let the equation to the curve be put in the form mine the asymptotes to a polar 1 curve by' r =........... 1 means of J (O) what has been proved. then, since t =r cos 0, y = r sin 0, we have, putting - for x, and - for y, Z. ( c.....0 (1) u- = tan 0.., (2); rcos0 cos " du d tan 0 1 and.'. d-~ =3/and d f(d tan0 f' (0) cos 0 + f(O) sin0 ( Now let a be a value of 0 which makes r infinite, and therefore f(0) (which =) =0; then 0 = a makes z = 0 by (l)*; and therefore if A and B be du the values of u and d when = o, we have by (2) and (3), d z A = tan a, B = f'(a) cos a and therefore the rectangular equation to the asymptote required is 1 y = tan a. o + 777-, — tan a +f, (a) cos a or x = — _ - if a=-; see note. f'O 0 * Except when a = 2, in which case z assumes the form ~ when 0= a, and its limiting value when 0 approaches a is -f();.. - makes r and thereioe f (1) fore y infinite; therefore by 212 x =- is the equation to the asymptote. W(i) ASYMPTOTES TO POLAR CURVES. 147 ~215. We may also obtain this result as follows. Sanmeresult obtained inLet a be the angle which the asymptote makes with the dependantprime radius; then it is evident that 0 = a ought to makely. r infinite, and therefore f(a) must be zero. Let the rectangular equation to the asymptote be y = tan a ( + c)............ (2), which, putting r' cos 0 r' sin 0 for x and y, becomes c sin a sin (0 - a)' r f e na ene ci. hence - = c sin a. r sin (0- a) ( Now the limiting value of - when 0 approaches a ought to be unity, from the nature of an asymptote; but by the usual method of vanishing fractions this limiting value is c sin af'(a). Hence we have c sin af'(a) = i; and.-. c sina =f (a) The equation to the asymptote is therefore {substituting for c in (2)}, 1 y = x tan a + cos af (a) We have here obtained the rectangular equation to the asymptote, because it is easier to make use of it than the polar. a Let r -= be the equation to the curve; here Example l. 0 0 1 1 f(0) =; and.-. a = 0; also f'(0)- -;.. f'(a); a a a therefore the equation to the asymptote is y = a, which represents a line parallel to the axis of x at a distance a above it. See fig. 20. a (e' - 1) Let r = (a hyperbola referred to focus), Example 2. 1 - e cos 0 148 ASYMPTOTIC CIRCLE. I -- e cos 0 here f()= - a (e2 - i) and.'. cosa =-, sin a /1 -- = --- e e2 e tan a = ve - 1. e sin 8 1 Also f'(O)= (; - f(a)= ~ a -l) ' av /- 1 hence the equation to the asymptote is y = A / /e_- 1. hx. ae %6/e - 1, b or y= A - (x + ae), a, which shews that there are two asymptotes making angles b b tan1 -, and -tan' - with the axis of x, and meeting it at a a a distance ae behind the origin. Asymptotic 216. It sometimes happens that r assumes a finite value, circle, what. c suppose, when we put 0 = co in the polar equation to a curve: it is easy to see that in such a case if we describe a circle round the pole with radius c, we may by continually increasing 0 make the curve approach as near as we please to this circle without ever actually meeting it. Such a circle is called an asymptotic circle. Example. Let r; then 0 = co makes r = a; therefore by r + 0 continually increasing 0 we may diminish r~ a ad libitum, and therefore make the curve approach as near to the circle as we please without ever actually meeting it, since it requires an infinite number of revolutions of r to make r = a. The circle in this case is an exterior asymptote, since r is evidently always less than a, (see fig. 21). aO If - - be the equation, r is always greater than a (at least when 0 is taken large enough), and therefore ~the circle is an exterior asymptote. CHAPTER XV. ON THE METHOD OF TRACING THE GENERAL FORM OF A CURVE FROM ITS EQUATIONS. IT is often necessary to make out and trace the general form of a curve from its equation, without actually calculating its exact dimensions, or ascertaining the precise positions of its remarkable points: we now proceed to state how this may be done in most cases. 217. If we can find y in terms of a, it is easy in general How to to trace the general form of the curve by determining the trache n curve when values of v which make y=0 or co, the corresponding values wecanCfind y min terms dy of x. of d-, and the signs which y has between its zero and indx finite values. We may do this in the following manner, viz. In one column write in order the values of ix which make y = 0 or co, and in addition to these, the value xv =o: opposite to these values, in another column, write down the corresponding values of y, and between each two put the sign which y has between them (y will always have the same sign between each two of these values, since it can only change its sign in passing through 0 or co): and in a third d 7y column put the corresponding values of Y. By means of div such a table it will be easy, in most cases, to trace the general form of the curve, as the following example will shew. 218. To trace the general form of the curve whose Example. equation is y --; a, - a, here y is zero when v = 0 or - a, and infinite when v = a: hence the values of,' to be written down are -a, 0, a, C; 150 TRACING OF CURVESo the corresponding values of y are 0, o, co, co: and by differentiating, or rather by the method given in the note * we shall find that the corresponding values of -y are -1 d x 0, co, cot; also when x is <- a the sign of y is +, when How to find * To find the value of the differential coefficient of f(x) when x= a, supposing f'(a) very a to be a value of x which makes f( )=0 or cs, we have only to divide f(x) by readily when f(a) -a and then put = a: for iff(a)=0, the limiting value of f() when x ap=0 or cc. x -— a proaches a isf'(a) by LemmaXIX.; and iff(a)=cc, f*-a becomes cc when x=a, and thus gives the proper value of f'(a), which by Lemma XXIII. we know to be co. In either case therefore, if we divide f(x) by x- a and put x = a, the result will be the proper value of f'(a). Thus let f(x) = - - a; then if we divide by x + a and put x =-a we obtain immediately f'(-a) =-. If we divide by x and put = 0, we findf' (0)= 0. If we divide by x-a and put x= a, we find f'(a) = c. To find y t The values of y and -y when ==co may be easily found thus.,dy d and dwhen Suppose y to be in the form of a fraction -, put u in the form xm (A + R) and = c. v in the form x' (B + R'), where R and R' are quantities which vanish when x =co; then -y='-1^ B and therefore, if m =n, y = when = co:; if m be < n, y = when = co; and if m be >n, y = cc when x= cc. It is always quite easy to put an algebraical function of x in the form x"' (A + R) by examining which term contains the highest power of x, and making that power appear as a factor of the whole. Thus a2x1 + a? /x2 -a2 + a+a2 X a2 (a + a\ a + a) =2(2a + R). dy To find -y when x=cc; it is easy to see that if m-n be >1, supposing A+R dy dy A y='xl-tB+ R then = co when x = c; if m-n= 1, then dy = when B +R' dv dx- B = co; and if m-n be <1, then = 0 when x= o. So then we may very readily dy find y at the same time that we find y corresponding to the value x =c. dx X2 +.a For example, if y - we may immediately put it in the form a x —a TfACING OF CURVES. 151 is >-a when x is as follows. and <o it is -, when x is >0 and <a it is -, >a and <co it is +. Hence the table will be -a 0 a y 4 -0 0 CO CO dy dx 0 co.Co Y 7jI_ a +R' (a ) x --— 8 dy.. y co, and x=c when x=co. dx For a second example, let y = x --; IVy - a/ then y = ( -a = x R I at 1+A..y = co, and -= 1, when x co; a2 VX2 a2 For a third example, let y = b) (- ); (00 - b) (ar - c) ' xa2 I a2 X2. 1 a2+Rt then y= (6 + R'; I. O and when R.'. y = 0, and -y = O, when x = s. ddv 152 1TRACING OF CURVES. From this table it is immediately evident that fig. 22 represents the curve. For if we take AX, AY as axes, AB = a, AC = - a; then the table shews that at the point C the curve passes from the positive to the negative side of the axis of ax at an angle tan- (- -), as is represented in the figure; that it touches the axis of x at A, but does not cross it; that it goes off to an infinite distance when m = AB; and when x becomes greater than AB it appears on the positive side of the axis of v, evidently in the form FPG, for y is infinite when i = AB, becomes and continues positive when xR is greater than AB, and becomes infinite again when *i is infinite; y therefore must decrease as x gets greater than AB as far as a certain point after which it must increase again in order that it may go off to infinity when x becomes infinite without becoming negative; from which it is evident that the curve must be in the form FPG. Since y and dy - are both infinite when x = co, they must be very large dx when x is either a very large positive or a very large negative quantity, and therefore the curve must run off to infinity on the positive and negative side of the axis of y in the manner represented in the figure. Represen- 219. In the plates marked M the Student will find some tation by figures of of the different cases which may occur in tracing a curve differenth represented by figures, to which he will find it useful to refer. cases which may occur. Under each figure is set down the corresponding line in the table from which the figure is deduced. By means of this plate the use of the table will appear evident. To deter- 220. Thus we are able to trace the general form of the form ofthe curve; but this method does not always shew us the points cureselyre of contrary flexure of the curve; to determine which we must we must resort to the method given in (180). Thus in the present examine whether it case, differentiating y twice, we get has any points of d2y (i - a) + 2 a contrary - 2 - flexure. dx' ( -a)3 the only real values of ix which make this =0 or o are iz = a, = a - 2Aa or -a (o, -): so that there can be only two TRACING OF CURVES. 153 points of contrary flexure; which indeed the figure shews, for there must evidently be a change of flexure somewhere between C and A, since the concavity of the curve is turned upward at C and downward at A; also there is evidently a change of flexure at B. There is no change of flexure elsewhere, and therefore the figure represents the precise form of the curve. 221. In order to determine still more minutely the form We may find the of the curve, we may find the points where the ordinate is maximum a maximum or a minimum; to do this in the present case and minimum ordiwe have nates, in dy m3 - a - _a termine the - ~ a - form of the d3 a (,aO - a)2 curve more minutely. and therefore, when y= 0, we have dx - a -- a2C2 = 0, which gives,: = 0, and x = - / + a2, or (1 /5); 2 4 2 which three values correspond to the points A, Q, and P. 222. When the curve has asymptotes, it is often useful Asymptotes are useful in to find them; for they enable us to see more clearly the nature givingusan of the infinite branches of the curve; for example, take the idea o the manner in equation which the branches (aM' + a) of a curve y= - (? go off to (I - a)' infinity. Forming a table as in the former case, we find xample dy tPI Y -a 0 0 + a To:~X t $c 154 TRACING OF POLAR CURVES. Since* y --- = - (1 +- &c.) = x + 4a + R, aso where R becomes 0 when x = co, the curve has an asymptote whose equation is y a= + 4a. Hence, take AC (fig. 23) = a, AB = a, AN= 4a, and draw the line DMNG, making L450 with AX, and the line BF perpendicular to AX. Then DMNG is-an asymptote to the curve; the curve comes from infinity below the axis of a?, touches it at C, but does not cross it, crosses it at A at an angle 45~, goes off to infinity as it approaches the line BF, appears on the positive side of AX when it passes BF, and then turns off towards the line DMNG as its asymptote. (For more Examples, see Appendix S.) How to 223. We may trace the general form of a curve referred curve from to polar co-ordinates in a similar manner, by putting down the equation values of 0 which make r zero or infinity; but as there are often no such values, or only very few of them, a table of those values is not in general sufficient to indicate the form of the curve. We may in most cases perceive the form of the curve by considering whether r increases or diminishes as it turns round; and this we may see, either by simple inspection, dr or by finding - and examining whether its sign is + or-, which will tell us whether r increases or diminishes with 0. There is no use however in putting down the values of dr dy as we did those of d in the case of rectangular curves: d0 do but it will sometimes be advisable to put down the signs which 4 It very often happens, as in the present example, that we can immediately put y in the form A + B + R, where R becomes 0 when x = co; in such a case we have no need to resort to the general method of finding asymptotes given in the preceding Chapter. TRACING OF POLAR CURVES. 155 dr do has, in order to see whether r is increasing or diminishing dr~ with 0. When r is negative, -d =+, indicates a diminution in the absolute numerical value of r, and vice versd. It is sometimes necessary, in order to make out properly the nature of the curve, to put down a few of the values of r 'ir $7r Wr $W corresponding to such values of 0, as 2 7r r. - -,... &c. When 0 = a makes r = 0, it is clear that there is a tangent at the origin making an angle a with the prime radius. When 0 = a makes r - oo, we must find the corresponding asymptote. It is usual to neglect the negative values of r in tracing polar curves, but there is no reason whatever to justify this omission, and we shall therefore always consider these values to be, as they really are, just as important as the positive values. 224. In measuring positive and negative values along r, Positive it will save mistakes, to suppose an arrow-head fixed upon the tive values radius vector to revolve with it as 0 increases; then this arrow- ofr, how measured. head, supposing it to point in the positive direction when 0=0, will always continue to point in the positive direction, whatever be the value of 0. Thus, in fig. 24, let 0 = 7 and then from S to P is the positive direction, and from S to P' the negative: in fig. 25, let 0 = - +4, and then from S to P is the positive direction: in fig. 26, let 0 = +,and then from S to P is the positive direction: and in fig. 27, let 0 = - and then 4 from S to P is the positive direction. Thus we see that the positive direction with reference to a revolving line is not a fixed direction in space, but depends on -1.56. TRACING OF POLAR CURVES. the angle at which the radius vector is inclined to its original position. The principle on which what we have just stated depends belongs to a different part of elementary mathematics, and therefore we assume it here without explanation. Represent- Bearing these considerations in mind, it will in general be figures of easy to trace the form of a curve from its polar equation, as different cases. will appear by the following examples. The Student will find in the plates marked M a representation by figures of some of the different cases which may occur, and the corresponding line of the table under each: to which he will find it useful to refer. Example 1. 225. Let r = a be the equation to the curve. Here then is no use in forming a table; we see immediately that as 0 increases, r continually increases; when 0=0, r =0; and when 0 is negative, r is negative. Hence the curve passes through the pole and touches the prime radius when 0 = 0, and then continually recedes from the pole as r turns round. When 0 = a negative angle, ASP suppose (fig. 28), then r is negative, and therefore must be measured in the direction SP'. Hence the curve is evidently of the form represented in the figure, the dotted part corresponding to the negative values of 0 being similar to that corresponding the positive. 02_ -] Example2. 226. Let r = a Here d = a (1 + - which is essentially positive: hence r always increases with 0. When 0 = 0, r = co, and by the method explained in the last chapter y = a r is the equation to the corresponding asymptote. Forming a table, we have 'TRACING OF POLAR CURVES. 157 d r dO 0 r 7r 7W 7r 0 - — a + + 2 2 0 co...... asymp. y=-a71 -+ 3SW 5w7 - -~a 0 2 6 + + Hence the curve has the form represented in fig. 29; the dotted part corresponding to the negative values of 0, -and 37rra 5 rr a SB = a-725, SC SD =3 2' 6 A reference to the plates M will help the Student to make out the forms of curves from tables of the corresponding values of the co-ordinates. The curves whose equations are y2 (x' - 4 a?) = X2 (x2 -a2)......(42), y2 (ai - 4a') = (.v - a2)........ (43), r = aeo sin 0.................... (44), r = aeo sin 20.....(.... 5)...,(0500 r - a ee sinl 2 r =a sin 30........... (47), r = a sin-. 2.0 r.(49), S are respectively represented by figures 42, 43, 44, &c... &c. CHAPTER XVI. SINGULAR POINTS OF CURVES. IN the preceding Chapter we shewed how the general form of a curve may be traced: we now proceed to shew how its remarkable points may be detected and examined. Proposi- 227. To investigate the nature of a curve in the imtion. mediate vicinity of any proposed point of it. Let the equation to the curve be Y =f (); let the abscissa of the proposed point be - = a, and let x, be any value of x a little less than a, and vx2 a little greater. T'hen we shall consider several different cases which may occur. Case l. 228. We shall in the first place suppose that f(x,) and f(x,) have one real value each, and only one, and this being assumed: (1) Let f(a) = co, f(xi) = positive, f(x2) = positive, then the ordinate belonging to the abscissa a is an asymptote, and the curve lies above the axis of x on each side of it: therefore fig. 50 represents the curve, where AM is the abscissa a, and MP(= co) the corresponding ordinate. If f(V,) = negative, and f(xv,) negative, fig. 51 represents the curve. If f(xl) = negative and f(x,) = positive, then the curve lies below the axis of z on the left side of MP, and above it on the right, and therefore fig. 52 represents it. SINGULAR POINTS OF CURVES. 159 If f(x,) = positive, and f/(2) = negative, fig. 53 represents the curve. (2) Let f(a) be a finite quantity; take MP =f(a), fig. 54, and draw the line O'PO parallel to AM; then f() -f(a) represents the distance of any point on the curve above O'PO. Hence If f' (a) = o, f(ix) -f(a) positive, f(x,)-f(a)= positive, the tangent at P coincides with MP, but the curve lies above OPO on each side of MP; therefore fig. 54 represents its form. If f'(a) = co, f(xi) -f (a) = negative, and f () - f (a) =negative; then fig. 55 represents the curve. If f'(a) = co, f(,) - f(a) = negative, and f(V)2) -f(a) = positive; then the curve touches MP at P, lies below it on the left side of MP, and above it on the right; therefore fig. 56 represents it. If f'(a) = co, f(xi) -f(a) = positive, and f(x,) -f(a) = negative then fig. 57 represents the curve. If, however, f'(a) be a finite quantity, draw the line T'PT (fig. 58) making an angle tan-'f'(a) with the axis of o; then this line is the tangent to the curve at P, and, its equation being y' -f(a) =f'(a) (xv - a), y - y' or f(x) -f (a) - f'(a) (v - a) is the distance of any point of the curve above this line. Hence, if for brevity, we put S (x) =f () -f(a) -f (a) (,v - a), it is evident that: If qp (i) = positive, and (p ()) = positive, the curve lies above the tangent T'PT on both sides of MP, and is therefore represented by fig. 59. If ) (x,) = negative, and p (x,) = positive, the curve lies below T'PT on the left side of MP, and above it on the right, and is therefore represented by fig. 60. 160 SINGULAR POINTS OF CURVES. If q5 (x,) = positive, and ( (xv) = negative, fig. 61 represents the curve. These are all the cases that can occur when y has one real value, and only one, for each value of x. Case 2. 229. Let us now in the second place suppose that f(xl) has no real value, and f(x,) two real values, and only two; and this being assumed: (1) Let f(a) = o, and both the values off(x2) positive: then MP is an asymptote to two branches of the curve, both on the right side of MP; therefore fig. 62 represents the curve. If both values off(%2) be negative, fig. 63 represents the curve. If one value be positive, and the other negative, fig. 64. (2) Letf(a) be a finite quantity, and f'(a) = o, then: If both values of f(x2) -f(a) be positive, two branches of the curve touch MP at P, but do not go below O'PO, nor appear on the left side of MP: therefore fig. 65 represents the curve. If both values be negative, fig. 66. If one value be positive and the other negative, fig. 67. (3) Let f (a) be a finite quantity, and f'(a) be so also, then: If both values of (x2) be positive, two branches of the curve touch the line TPT at P, but do not go below it, nor appear on the left side of MP: therefore fig. 68 represents the curve. If both values be negative, fig. 68, his. If one value be positive, and the other negative, fig. 69. If we suppose that f/(r) has no real value, and f(x,) two real values and only two, then it is evident the curve in each case will be exactly similar in form to what it is when.f(x,) SINGULAR POINTS OF CURVES. 161. has no real value, and f(x2) two real values; it will be merely reversed in position: thus, instead of fig. 68, we shall have fig. 70; and similarly the other figures. These are all the cases that can occur when f(x) has two real values, and only two, for some values of xa and no real values for others. 230. It sometimes happens that f (m) and f (,) are both Conjugate impossible, no matter how near xz and x2 may be taken to a, points. and yet f(a) a real quantity; in such a case the point P is a point belonging to the curve, since its co-ordinates satisfy the equation to the curve, and we define the curve to be the assemblage of all the points whose co-ordinates satisfy that equation: but no points in the immediate vicinity of P belong to the curve; P therefore is an isolated point of the curve, completely detached from the other points. Such a point is usually called a conjugate point. 231. If f(,v) and f/(x) have each several values while Multiple f(a) has only one value, then several branches of the curvepoints. must meet at the point P; in such a case P is called a mnul tiple point. We shall not extend this enumeration of cases any farther, as it will be easy, after what has been explained, to make out the form of the curve in any case that may present itself. 232. The following are examples of the cases that we Examples. have just discussed. a3 (I) y=a+ (-a ( - a) Here f (a) = co, f () = positive, f(x) = positive; therefore fig. 50 represents the curve. x - 2a (2) =Y = - 2 Here f(a) = co, f(vx) = negative, f(te) = negative; (fig. 51). 11 162 SINGULAR POINTS OF CURVES. (3) y=~ IV - a Here f(a) 7, f(xj) = negative, f e,)= positive; (flg. 5) _(4) y = a + aA (xv - a)1. Here f(a) = a, f'(a) = co f (x1) -;f(a) positive; f(x,) -f(a) = positive; (fig. 54). (5) y= a + a (x - a)A; (fig. 55). (6) (x - a)' a Here f(a) = a, f'(a)= 1, / (,)= positive, / (x,) positive; (fig. 59). (x -- a)3 (7) y=x+ -a)' (fig. 60).. (8) {y(x_- a)-a'}'=a (x- a)3. a2 Solving this, we have y = - Wax - ai. tX - a Here therefore f(a) =, f(x,) is impossible, and f(x,) has two real values both positive*; (fig. 62). (9) (y - a)' (ax - a')= a. a' Solving this, y = a A= - Here f (a) = co, f(x,) is impossible, and f (x,) has two real values, one positive and the other negative; (fig. 64). (10) (y' - a)'= ax -aa Solving this y = a V/ax - a, " For when x'nearly =a the first term is very great compared with the second,. and therefore y has the same sign as the first term, SINGULAR POINTS OF CURVES. 163 andd a d 2 x/ax - a2 Here f(a) = a, f'(a) =o, f(,VI) -f(a) is impossible, and f(x,) -f(a) has two real values, one positive and the other negative; (fig. 67). (n)) a (y _ X)2 (X - a)3. (x - a)# Solving this Y a1=6 and dy 1a 2r - Here f(a) = a, f'(a) = 1, / (x,) is impossible, and (P (V2) has two real values, one positive and the other negative; (fig. 69). (12) y = x n a' (x - a) a (x - a)'. Here f(a) = a, f'(a) = co, f(xv,) is impossible; f(x) - f(a) = (x - a)A I c ai - (x - a)') Now (x, - a)i is very small, and. 6. a1 a' (is - a) is positive whether we take the + or the -. Hence f@2v) -f(a) has two real values both positive; (fig. 65). ( )2 XL!-C) (18) y=x+ a 6 a a aa Here f(a) = a, f'(a) = 1, f (xj) is impossible; P (i) - (x - a I)2{Qv (X - a)} a2 sveysalad. (in2 -a)i Now (in - a)' is very small, and =6 a1 is positive whether we take ithe + or the -; hence 0 (in) has two real values both positive; (fig. 68). (14) y= a 6 (x - a) Vx'in-2a. Here f(a) = a, and f (,v,) f nv) are both impossible, since v/ - 2a is impossible for all values of i less than 2a; 1 1-. SINGULAR POINTS OF CURVES. hence the point whose co-ordinates are x = a, y= a is a conjugate point of this curve. (15) (a _( - a) - 0 -,) (' a )a2 Solving this y a A (- a ) - ( -a) a Here f(x) has two different real values for every value of x except x= a; hence the curve has two branches which meet at the point whose co-ordinates are x= a, y = a. Since f'(a)= - 1 the tangents of the two branches at the point P make angles 450 and - 450 with the axis of x; the branches therefore cross each other at right angles at the point P; the concavities of both turned downwards since f2 (a) is negative; (fig. 71). (16) y = + (a? - a) sin-. Here an infinite number a of branches cross at the point (x = a, y = a). Cusps. 233. The curve at the point P in (fig. 54, 65) is called a cusp from its pointed form: in (fig. 69) it is called a cusp of the first kind, and in (fig. 68) a cusp of the second kind. Singular Points of contrary flexure, cusps, conjugate points, mulpoints. tiple points, &c. are called singular points. How the 234. The existence of a point of contrary Jlezure is existence of each of d2y these sin. indicated by - changing its sign. (See 180). gularpoints d x is indicated. cated 235. The existence of a cusp is indicated by the curve not crossing a right line which does not coincide with the tangent. If - be a finite quantity at a certain point, and if dm the curve does not cross the ordinate at that point; or, if dy= c at a certain point, and the curve does not cross dx the line drawn through that point parallel to the axis of x; then there is a cusp at that point. SINGULAR POINTS OF CURVES. 1-65 236. To determine whether the cusp is of the first or How to de termine second kind we must find qp(x) (see 228), and arrange it in wether ascending powers of x - a; suppose that the result of the point is a cusp this is of the first kind, or p (x) =A ( - a)" + B (x- a)" + &c. not (x - a)" {A + B ( - a)n"-" + &c.} Now when x - a is very small, A + B ( - a)-m + &c. will have the same sign as A, therefore P (2x) will have the same sign as A (x, - )m. Hence if m be a fraction with an even denominator (of course reduced to its lowest terms) A (x2 - a)n will have two real values with opposite signs, and therefore the two values of 0p (,2) have opposite signs, and therefore the cusp is of the first kind. But if m be an integer or a fraction with an odd denominator A (r -a)"' has only one real value, and therefore both values of 0((x2) have the same sign as A, and consequently the cusp is of the second kind represented by fig. 68, or 68 bis, according as A is positive or negative. Of course we here suppose that J(x2) has two and only two real values, and $ (x1) no real value. If it be q((xi) which has the two real values, all that we have just said is equally true; only the cusp in each case will be reversed in position. We here suppose also that f'(a) is not infinite. If it Cusps, be, the cusp will be of the first kind, if f(rv,)-f(a) and f(ea)=o, f(2s) - f(a) have each one real value of the same sign; and how disa cusp of the second kind, if f(x2)- f(a) has two real values, and f(x) -f (a) no real value, or the reverse. 237. If the quantities f(a), f'(a), f 2(a), &c. have each Iff(a), one real and finite value; then when x is taken sufficiently (a'), &C near to a, we have.....have each one real finite -x a- 2 (x - a)2 value, the f(A ) =f(a) +f (a) +f2 ( -- +... ad infnitum, point {x=a,: 1 +I =f (a)} cannot be which gives one real value for each value of x. asingular -16-6 166 ~~SINGULAR POINTS OF CURVES.,point of any! Under such circumstances we have therefore the case where kind except one of Con-f (a) and f'(a) are both finite, and f (x,) and f (x,) have each trary flexure. one real value: the curve therefore must assume a form similar to one of those in fig. 72 in the immediate vicinity of the point P: consequently P may or may not be a point of contrary fiexure, but it cannot be a cusp, or a multiple point, or a conjugate point. Hence if f(a), f'(a), f2 (a,) c~c. have each one real and finite value, the point, whose co-ordinates are a and f(a) must be either art ordinary point or a point of contrary fleceure: it cannot be a singular point of any kind eaxcept -one Of contraryfJlecevure. At all sin- 238. Suppose that the equation to the curve is an gular points except one algebraical equation between ve and y cleared of radicals of of contrary the for flexure two r conditions hold, when A +Ba +Cy +Dce2l +Exy +Fy + G X3... +Pyp=0, the equation to the curve is which for brevity we shall represent by algebraical and cleared U O.() of radicals. then to find the successive differential coefficients of y we have the following equations got by differentiating (1), viz. dy U -y+ d~U= o.(2),...... dx d2y dy d y 2 dU.:y+ d 2U., YI d2ydU.-+CV d'U0.() d+U. 3+d T. dx &e....(4), dx' I &c.....&c.... From these equations we may find the values of dy dx'3... &C. corresponding to any values a and b of xe and y which satisfy ~(1):~ for by substituting a, and b for xe and Y SINGULAR POINTS OF CURVES. 167 in (2), (3), (4), &c. we find the corresponding value of d d 2d from (2), and then that of from (3), and then that of 73~~~~~~~d y dY from (4), and so on. dx" Now it is evident from the form of the equations (1), (2), (3), (4), &c. that if dyU is not zero, we thus obtain a single dy d2y d3y finite value for each of the quantities -, dY' d~...for dx dxa da) then we evidently find dy a quantity which cannot be infinite* dx dyU dJy a quantity which cannot be infinite dx2 dyU and so on. Hence it appears by the preceding Article that if dyU be not zero, the point (a b) must be either an ordinary point or a point of contrary flexure. And the same may be proved in exactly the same manner of dU. Hence at all singular points except a point of contrary flexure we must have dxU = o, dyU = 0, U = o being an algebraical equation cleared of radicals. 239. If therefore the equation to a curve can be reduced Hence we may find all to the form of a rational and integral algebraical equation thepointsof U= 0, and if we determine those points whose co-ordinates a curve which may satisfy this equation, and moreover the equations dU = o, be singular. dyU = 0; then no other points of the curve but these can be singular points of any kind, except points of contrary flexure. To determine whether such points are really singular points, and the class to which they belong to, we must examine them separately; and first, we must determine how many * The numerators of these fractions cannot be infinite, because U is a rational and integral function of x and y. 168 SINGULAR POINTS OF CURVES. branches of the curve meet at each of these points, which we may do as follows. Tondeter- 240. Let x( =a and y =b be the co-ordinates of any mine how many one of these points; then since the substitution of a and b branches of acurve for x and y makes pass through a dU =o and dyU=o, which dU=O, dy dyU=0. we must proceed to determine - as in (153, &c.), and we shall dex dy arrive at an equation for determining - of a higher order dx than the first. If this equation has n real roots, then n branches of the curve and no more meet at the point (a b). If any of these roots be equal, then the corresponding branches have the same tangent at the point (ab). If when ( is a little greater than a, y has n real values, and when x is a little less than a only n- 2m real values, (for an odd number of real roots cannot disappear) then 2m of the branches do not appear on the left side of the ordinate b, and each two of these branches must therefore either touch the ordinate b as in fig. 67, which will be the case with those branches for which - becomes co when x =a; or else they dx must meet the point (a b) in the form of a cusp as in (fig. 68) or (fig. 69), which will be the case with those branches for which dy does not become infinite when x = a. dx If the 2m real values of y are wanting when x is a little greater than a, the same is true, only the cusps, &c. are reversed in position. dy av root If the equation for determining d3 have no real root (a b) is a conjugate point. Themethod 241. We may often very readily find the nature of a of ex.pansion in curve near a proposed point (ab) by expanding y- b in powers oft4enell of -a, by the method given in (145): each different expansion SINGULAR POINTS OF CURVES. -169 will indicate a different branch of the curve, and will shew able us to make out how it lies near the proposed point. the nature of a curve We may also by the same method make out the nature neara pro*1' J~posedpoint. of the infinite branches of a curve, by putting - for r, and expanding y in powers of z. And thus we may often make out of a curve when we cannot solve its amples see Appendix V.) very readily the form equation. (For Ex CHAPTER XVII. THE GENERAL THEORY OF CONTACT. INTERSECTION OF CONSECUTIVE CURVES. Different 242. LET PQ, PQ', PQ", (fig. 30) be three curves having contact. a common point P, the co-ordinates of which are AM = a, MP - b; and let the respective equations to these three curves be NQ =y =f (), NQ'=y'= ](), ( NQ"=y"= x()); th QQ () - f (x) then QQ' q(- ) (I) ) Which, since the three curves meet at P, and therefore f(a) = (a) = ((a), assumes the form - when x = a. 0 Hence, by (148), when x approaches a, we have in general, QQ",+(a) -f'(a) li. val. of QQ ('(a)-f'(a) Now suppose that (P'(a) =f'(a), while +'(a) does not f'(a); then this limiting value is infinite; therefore when the point N approaches M, QQ" gets continually larger in comparison with QQ', and may be made as many times larger than it as we please by sufficiently diminishing MN; and therefore the curve PQ' must lie infinitely closer to PQ in the immediate vicinity of the point P than the curve PQ" does. Hence it appears that if, for the same abscissa, two curves have the same value, not only of y, but also of -,, they not ORDERS OF CONTACT. 171 only meet, but also lie infinitely closer to each other in the immediate vicinity of the point of occurse than they would do dy if they had not the same value of d at that point. But suppose that +'(a) -f(a), and ('(a) -f'(a) are both zero, then by (148) we have in general, when v approaches a, QQ +2(a) -f (a) lim. val. of QQ = (a) -pf2(a). QQ' q)(a) -f (a) Suppose here that +2(a) =f2(a) while \2(a) does not =f2(a), then this limiting value is infinite. We may therefore conclude just as before, that if for the same abscissa two curves have the same values, not only of y and d-, but also d2 of then they lie infinitely closer to each other in the immediate vicinity of the point of occurse than they would d2y do if they had not the same value of d xY And in general, if q: (a) -f (a), '(a) -f'(a).-..n"-l(a) _ fn-1 (), (a) - f (a), 4(a) - f (a) '.... n"'(a) -f'-'((a^), be each zero, we have, when x approaches a, QQ,,' (a) _ f n (a) lim. val. of QQ () QQI ' V (a) -f (a) Suppose here that +8(a) f"(a) while "n(a) does not = f(a); then this limiting value is infinite. We may therefore conclude that if for the same abscissa two curves have the dy d"y dxy same valtzes of y d —, d... d —9 then they lie infinitely closer to each other in the immediate vicinity of the point of occztrse than they would do if they had not the same value of dny d x" 172 ORDERS OF CONTACT. Orders of Hence it is that when two curves meet, and at the point contact depend upon dy the differ- of occurse have the same value of they are said to have ential co- dx d 2 efficients., the ar e id th efficientscontact of the first order, if moreover the same value - dxZ contact of the second order, and so on; and in general, if dy dry dny they have the same values of, d.. -, they are said dx9 dxo2 dxr1 to have contact of the nth order. onant of 243. If p (a) -f (a), '(a) - f'(a) *.. (a) -f (a) be orderisac- each zero, and f"n+(a) -f"+l(a) not zero, then by (121), companied \ with inter- p (x) -f(x-) has the same sign as section, whereas \n+1 cwheteof as X + (a) - f (a) (x - a)n+l contact of an odd order is not. for all values of x sufficiently near a: now if the contact be of an even order n is even, and n + 1 is odd, and therefore ~?(Ze) -f(x) changes its sign when x passes through the value a; therefore, at one side of the point P, (p(x) is greater than f(x), and at the other side less; i. e. the curves cross each other at P, as in fig. 31. But if the contact be of an odd order, n is odd, and n + 1 even, and therefore QQ' does not change its sign when x passes through the value a; i. e. the curves meet each other without crossing at the point P, as in fig. 32. Hence contact of an even order is accompanied with intersection, but contact of an odd order is not. What has 244. It is evident that all we have just said is equally been said is true for true whether the co-ordinates be rectangular or oblique. oblique and Moreover it is easy to see that if we refer to curves to any polar coordinates new axes of co-ordinates, the degree of contact is not altered. also. The order It is clear also that what we have said applies equally to of contact is not dr d2r dnr changedby polar co-ordinates; i.e. if -, have the same a change of dO d dO co-ordinates. values in both curves at the point of occurse, the contact will be of the nth order: for then it is easy to see, putting dy d'y dn' = r cos, y = r sin, that will also have the same values in both curvesd the same values in both curves. ORDERS OF CONTACT. 173 245. To make a line whose equation is Example 1 A right line. y = Ax B......... (1), have contact of the first order with a given curve whose equation is y =f(x), at a given point whose abscissa is a. y must = yi when = a; therefore we have f(a) = + B...... (2). dy dy, must = when x = a; therefore we have d-v dv-d f(a)...... (3). (1), (2) and (3) give y -f (a) f' (a) (a - a), which is the equation required. It coincides with the common equation to the tangent, see (173), and hence the ordinary tangent has contact of the first order with the curve. If we wish to make the line (1) have contact of the second order with the curve, we must put y = Y when V= a: d x2 dax2 but this gives f2(a) = o, and this equation of course cannot hold except at particular points of the curve. Hence a right line cannot have contact of an order higher than the first with a curve except at particular points of it, determined by the d2y d y equation d = 0 (or in some cases -- = co, see note ). 246. Suppose that the curve is referred to polar co-Example, ordinates, and that its equation is r =f(0). The equation to A right a right line being y = tan a (x + a), putting x = r, cos 0, ferred to polar coy r sin 0, becomes ordinates a sin a sin (0 -a) d'2y d2x * When d 2-, =d C (y being the independant variable) will be a in certain cases, and then there will be contact of the second order, 174 ORDERS OF CONTACT. then1 sin (0 - a) dr, cos (0 - a) then - = - r, a sin a r, dO a sin a drv and. - - rl cot (0 - a). dO Now suppose that the line has contact of the first order with the curve at the point whose polar abscissa is 0 =, dr, dr then when 0 = 3 we must have r, =r and d - d a sin a ~ f(/3) = sin (3 )' and f'(/3) = -f() cot (3 - a). From the second of these equations we may determine a, and then from the first a, and thus the polar equation to a line touching a curve given by a polar equation at any proposed point may be found. Since /3 may have any value we have in general f'(0) = - f(0) cot (0-a), ta — f(O) rd0 or tan (a - 0) = (0) - dr which coincides with the result obtained in (196). Since a - 0 is evidently the angle under the tangent and radius vector, i.e. (P. The order 247. The order of contact which we may make two of contact whichwe curves have with each other depends in general upon the may make number of constants we have at our disposal; if there be two curves have with n + 1 constants disposable we may so determine them that the contact shall be of the nth order; for then we may, by giving them proper values, satisfy the n + 1 equations dy_, dy d2y, d2y dny dy y d y d dX - dv' dx - dx " d which equations must be fulfilled to produce contact of the nth order. Thus in the most general equation to the right line, y - A x + B, there are two disposable constants A and B, and CIRCLE. OF CURVATURE, l75 therefore we may in general make a right line have contact of the first order, but no higher, with a given curve at a given point of it. In the most general equation to a circle, viz. ( - A)2 + (y - B)2 R2 there are three disposable constants A, B and R, and therefore we may in general make a circle have contact of the second order, but no higher, with a given curve at a given point. 248. To determine A, B, R, so that the circle Example 3o A circle (,- A)2 + (y1- 3)2 R} tuch a proposed curve with shall have contact of the second order with the curve contact of 2nd order. {y = f(,) at the point whose abscissa is a. We have (X - A)2 + (y1 - B). = R..2 (1), - A + (y, - B) d (2) )-~, + ^ =o ^(2), I + (d ] + (y- -B) d2y-= o e.. (s). Now since the circle touches the curve with contact of the second order at the point whose abscissa is a, if for brevity we put f (a) = b, f' (a) = p, f2 (a) = q must have dy, d2y, y, =, -,p,. -q when = a, and therefore putting x = a in the equations (1), (2), (3), they become ( - A4)2 + (b B)= R.^ (l), a A + (b -B)p =. (2), +p2 + (b - B)q=o... (3). (3) gives h - B 9-_ q 176 ULTIMATE INTERSECTIONS. then (2) gives a - A = +p P and then (1) gives R = (1 + p), q which equations give us B, A, and R, and thus the circle is completely determined. Circle of This circle is commonly called the osculating circle, or curvature Its radius the circle of curvature. Since a may belong to any point is theindex whatever of the curve, we may put x instead of a, and. of curva-., ture p. See dy d2 y (181.) y', d ' and d2 instead of b, p, q respectively, and then we have +1 +f { ( (dxt _: ds3 d'y dx dy' dx' hence it appears that R = p (see 184), and hence the point P' (185) is the center of the osculating circle. It follows therefore that the limiting position of the intersection of two consecutive normals to a curve when they approach each other is the center of the osculating circle. We may also arrive at this result by means of the following article. Ofthe lti- 249. Let the equations to two curves be mate intersection of curves. f (xy a)= 0...... (1), f (ya')= 0.....(2), (2) differing from (1) only in having a' in place of a, a and a' being two constants to which we may assign any values we please. Then if we determine x and y from these two equations taken together, the resulting values will be the co-ordinates of a point of intersection of the two curves. Now this point, which we shall call P, must be some definite point as long as a' is different from a, but if a' becomes ULTIMATE INTERSECTIONS. m177 equal to a, it ceases to be a definite point, since then the two curves are identical. x and y therefore become illusory when a'= a, but of course they have some limiting value when a' approaches a as in the example (22). We may determine the limiting values of x and y as follows. By Lemma XXI. the equation (2) may be put in the form f(vya) + {f '(wya) + Q} (a' - a) = 0, (see 77) where Q is a quantity which may be diminished ad libitum by sufficiently diminishing a' a. In virtue of (1), and dividing out a' - a, this equation becomes f' (ya) + Q 0. (3). Of course we necessarily suppose in this process that a' does not actually = a. From (1) and (3) we may determine iv and y, which will of course be different for different values of a'. Let a' approach a; then, in virtue of (1) and (3), the limiting values of f(xya) and of f'(ivya) + Q must be zero; if therefore x, and yi be the limiting values of x and y, since that of Q is zero, we have by Lemma VIII. f(vYyEa) = 0, and f'(ivya) = 0, and from these equations we may determine x, and y,. Since v, and y, are the limiting values of x and y when a' approaches a, it is clear that we may diminish the distance between the points (my) and (1, y) ad libitum by sufficiently diminishing a - a. The point xiy, is therefore called the ultimate intersection of two consecutive positions of the curve (1), and its co-ordinates are the values of x and y which are obtained from the equation f (vya) = 0, and its partial derivative with respect to a, viz..f'(y) = 0; a is called the variable parameter of the curve. 12 178 ULTIMATE INTERSECTIONS. Example 1. 250. To find the ultimate intersection of two consecutive positions of a line which always includes equal areas between it and the co-ordinate axes. x y Let - + 1 be the equation to the line, and 2c the constant area; then af = c, and the equation to the line becomes 07 ay -+ - - 1 =0......(1) a C a being the variable parameter. The partial derivative of this with respect to a is w y +- = 0......(2) a2 c from these equations we easily get Ov X a - + — 1 = 0, or <v= -; a a 2 and then y=-. 2a The value x = - shews that the line is bisected by the 2 point of ultimate intersection. We may verify the correctness of this result thus. Let a' be another value of a, then the equation to the corresponding line is iv ay - + - 1 =0......(3), a c (3) a'- (1) a gives ~ (.'2 - a2) _ -a or y = -—; and therefore the limiting value of y when a' approaches a a is -, and therefore by (1) that of x is - 2a c2 ULTIMATE INTERSECTIONS. 179 251. Hence we may find the ultimate intersection of Example2 two consecutive normals as follows. The equation to the normal to a curve y =(x) at the point whose abscissa is a is f'(a) y -f(a ) + v - a = 0............ (1), taking its partial derivative with respect to a we have f'(a)yf() \y - { f'(a)}- 1 = 0.....(2), and the values of x and y got from these equations are the co-ordinates of the point required. These values are Y f()+ {f( a)} I + 1. f2(a) =, _ f' (. {W- f (a) 2 + Comparing these values with those of A and B in (249) we see that the point of ultimate intersection of two consecutive normals is the center of the circle of curvature. 252. We may find the locus of all the points of ultimate How to find the intersection of the set of curves represented by the equation locus of the points of f (xy a) = 0, ultimate intersection by eliminating a between this equation and its partial deri- crves~f vative f'(xy&) o. Let us take the first example, last article, in which Example. f(vya) = - + -y 1 0, f' (xya) - - +- = 0 a C a c Eliminating a, we find a = 2x, and.~. xy=c, which gives a rectangular hyperbola referred to its asymptotes. 12-2 CHAPTER XVIII. LAGRANGE'1 S TIIEOREM. ELIMINATION OF CONSTANTS AND FUNCTIONS BY DIFFERENTIATION. EXPANSION BY MEANS, OF THIS ELIMINATION. THE following theorem, due to Lagrange, is often found useful in expanding functions which are given by equations. Lagrange's 253. Suppose that u, y, z, x are variables connected Theorem. by the equations U = f (y). (1),+y z~x 9(y). (2),. to expand u in a series of powers of X. To do this, we shall find d' u as follows: y is a function of the independant variables z and x, and we have, by differentiating (2) with respect to z and x, (P (y) d'y1(y) + x/ '(y) dy; and.~. d,y + "V (P(y).-y + x P'(y) dz a h d. d,, + "V q5'(y)... hence, dy = P(y) d,y...... (4); and.d. d~u =f"(y) d.y =f'(y) cp(y) dy. (5). Now if (y) denote any function of y, dx {'J'(y) d.y} = 4/(y) d~y d.y + xF,(y) d.d.y = dz {(y) dxy} dz{xf(y) (y) d-y1, by (4). Hence supposing that (i) =f'() 0 (y), and differentiating (5), we have LAGRANGE S THEOREM. 181 d/2u = d [f'(y) { (y) 2dzy], and again d'u = dd.[.............. = dW [f'(y) p(y)3dzy], similarly, and so on; and finally, d u = dn-1[f'(y) {qS(y) }dy]. Now when x = 0, = y, and by (3) dcy = 1; therefore, when x = 0, we have dzun = dz-I[f' () {(f) "] Hence, by Taylor's series, if for brevity we put f'(z)= V, (z) = Z, we have since u =f(z) when = 0}, u =f(z) + VZ. - + d,(VZ2). + dZ(VZ3). + dz-'(VZn)- + &c.... which is the developement required. The following example will shew the use of it. Let u = eY, y = x + wey. Example. Here f(y) = 5Y;,..f'(z) or V=, p(y) = eV;.. y (z) or Z ez;.. d;n- (VZ") = dz-1 l (n+'1)}, = (n + 1)"- I (n+ ), which = 20e2z, eZ, 42e4z, 535Z... when n = 1, 2, 3, 4,... &c. respectively; hence x] w3 4 u = ez ez + 4e2e4z + 53e5Z. + &. When we are given an equation between and y Eim 254. When we are given an equation between a and y Eliminacontaining any constants a, a, a3... a, suppose, we may tiants fbyeliminate these constants by differentiation, as follows. Let differentia. the given equation be tion. U=o......... ), 182 ELIMINATION OF CONSTANTS and let it be differentiated n times, and so give the following equations; viz. dU=.........(2) d2 = 0......... (3), dnU = o......... (n + ); then from these n + 1 equations we may in general eliminate the n quantities ac, ac, c... a., and so obtain an equation dy d2y d y between B r, y -, -... - not containing any of these dTx dx dan constants. Example. Let the given equation be y - a#x + ab = O........ (1), differentiating twice, we have p - 2Cax + ab = 0......... (2), P = dyY q 2 = O...d......... (3); d,2. hence, 2a = q, and therefore, by (2), ab = qxt -p; and therefore, by (1), - + qx Px = 0, 2 or y + — - O 0; which is an equation between x, y, p, and q, not constants a and b. A number 255. It is evident that there be n + m of different equations a2... a, a,,... a,,n in the equation U=o, we may be formed in any n of them we please between the equations this way. U = 09 du = 0 containing the constants a, may eliminate BY MEANS OF DIFFERENTIATION. 183 d2u = 0, dnu = 0. m + n.m -{*n - 1... m + 1 Now there are m + (= N* suppose) 1............... m different combinations of the m + n constants a, a2... &c.; therefore there are N different ways in which we may form an dy d"y equation containing a, y,...- and m of the constants; and consequently we may in general obtain N different equations involving mV y d and m of the constants from tin in dxn the given equation. In the example just given, if we eliminate a between (1) Example. and (2), we find a = P from (2); 2 - b and therefore, by (I), poa (r - b) Y- 2-b 0.....* (5). Again, if we eliminate b, we obtain y -pv + a2 = 0......... (6), and thus we deduce from (1) two different equations (5) and (6), each involving 'x, y, d-' and one of the constants. 256. We may eliminate the constants a1, a, a3... an Thiselimifrom u = 0 somewhat differently, as follows, nation m be effected somewhat Solve the equation u = 0 for al, and let the result be differently. u, = a1, then, differentiating, we find du- = 0, in which a, does not appear. Solve this equation for a,, and let the result be u,= a2, and then differentiating, we find dut 0, in which neither a1 nor a, appear; 184 ELIMINATION OF FUNCTIONS and proceeding in this manner we may eliminate al, a,, a3. successively. Examtiple. Thus in the example just considered, if we solve for a we find Y,' - b a therefore differentiating we have p (&v - bxt) -y (2w - b) = 0, (('-< bv)2 or (IV - b) p - y (2w - b) =0, which equation does not contain a. Again, solving for b, we get -X2_ - -2y - = b; wp - Y and therefore differentiating we find (X2q - 2y) (xp - y) - (*2p - 2xy) x q = 0, or xqy - 2y (p - y) 0, or t2y = 2 (xp - y) = o, which is the same equation as before. This method is generally longer-than the former. Elimina- 257. In the same way that we eliminate constants from tions of functions equations between two variables I and y, we may eliminate from equa- functions from equations between three variables x, y, n. For let u = 0... (1) be an equation between I, y, and z, containing the n functions Ib (v1) 02(V2), (PA3()... (q(vn), v1 v2... v being any functions of x y z. Then in virtue of (1) we may consider z as a function of the two independant variables x and y, and we have by successively differentiating with respect to I and y the following set of equations: u = 0, BY MEANS OF DIFFERENTIATION. 185 dxu = O, dyU = O, d,2U = O, ddyu = o, du = 0,............................. &c.... d"u= 0 d, u =., dym-d yu = 0, (m + 1) m in all 1 +2+3... + m, or -(+ equations. Now in these equations we evidently have involved the functions 1 (VI), yP2 (V2)),jj e n() )'I (V 1), X 2 (v2),............ - n (vI) 'lm (V1), ppm (V,),............ 0 (V), in all m n different functions. (m + 1)m Hence if (m 1)m be > zmn, or = mn + 1 at least, we 2 shall be able to eliminate all these functions from the equations, and so form an equation between v y ', dxz, dyz, dz,... &c.... drmz, d~-ldyz... dy"nz, containing no trace of the functional letters (PI, b, 3... *,, If n = I the condition (+) > mn becomes 2 m+ 1 - - > 1, and.. m>. If n = 2 it becomes mn + 1 -— >, and.' m>3 and so on. Hence to eliminate one function it will be necessary in general to go beyond the first order of differentiation; to eliminate two functions we must go beyond the third order; and so on. Often however it happens that, in consequence of the peculiar form of the equations, the elimination may be effected without proceeding so far; as the following examples will shew. 1.86 DEVELOPEMENT OF COS Vb 0 Example. Let the given equation be u = p ( + y) +,y y 0... (I), then du = p(p (? + y) + p'( (x + y) + y= o... (2), dyu = q0 (x + Y) + z ' (x + y) + x = o... (3), where p = dz, q = dyz, (2) and (3) give (p - q) P (a + y) + y - - o, which by (1) give - y z y =y 0. - q Thus we obtain from (1), by differentiation as far as the first order, an equation which contains no trace of p. Example2. Let the given equation be (0 ( + Z) + ' (y + ') = 0... (1), then (1 + p) ('(x + ) + p '(y+)... (2), q (' + ) + (1 + q) q+ (y + Z) = 0... (3), (2) and (3) give immediately 1 + _ p - or I +p + q = 0, q +p from which q and qJ are eliminated. Develope- 258. This sort of elimination is often useful in exment may be effected panding functions, as the following example will shew. by elimi. nating functions. To develope cos (m cos-~ x) in powers of v, assume Example (OS y = cos (m cosl ')... (1), d y m then. - sin (m cos-' xv)... (2); N//1 i V IN POWERS OF 0OS 0. 187 m. 1 _ — m sin (m cos'1 v) = 0; dxm.. differentiating again, VI — d2y O dy m2 v/1 _ --- t2 + cos (m cos-' x) = 0; fdo, - V v d 2/1 - m or multiplying by V/1 - x, and putting y for cos (m cos-1.), d y d2 Y dyy ( -7a - -7 - ~...=. dcx2 dd d I? Having thus eliminated the functions cos and cos-' by differentiation, we shall be easily able to develope y in the powers of x by assuming y = Ao + Ax + A,2 '.. Ajc" + A,+1n+ + + BA+2"-2.-... &c. Substituting this for y in (3) we find for the coefficient of 0n (n + 2) (n + 1),n+2 - n (n - 1) A, - nA, + m2A,,, which being put equal to zero gives n2 - 2 (A n + l) (n + 2) from which we evidently get - =2,(- m) (2 m - m) 2 r4 ~ 12 - m2 (12 _ m) (32 m2) A3 = 1 A --- A... [3 F5 To determine A0 and A1, put x = 0, and.'. y = Ao and y = A,, in (1) and (2), and we find dxv l 2r4 +- 1 A0 = cos (m cos-') = cosm r 2r + 1 A, = m sin m, 188 DEVELOPEMENT OF COS M 0 IN POWE RS OF COS 0. Hence if for brevity we assume 2 2" W~ w4-92 2 (M2 _ 2) M2- 42) F2 f4 F6 V=m _X _ M (MIn _ 12) X+m (m2 - i2) (in2 - 92)X &C. 173 F5 we have 2r+ 1 2 r+ 1 Cos(m cos-'iv) = U cos mw~ +V~sin M7r which is the general expansion of the cosine of a multiple arc in powers of the cosine of the arc. CHAPTER XIX. MISCELLANEOUS THEOREMS. 259. IF y = uv, u and v being two functions of x, Theorem of Leibnitz. then If y== UV to find dny- du. +n dn1U v+n (n-I)dn2u.d'v..+ u d"v. d (U v). 1 F For we have dy =du.v + u.- d~v, d2y d2u -v + 2du. dv + ud'v, d3Y = d'u. v + 3d 2U dv + Sdu. d2V + ud V, Let us assume that d'y = d~v. u + AAd1-1v. du + B,,dn-2V. dlu...... then we easily find by differentiating, which shews that if the coefficients of d~ly be the same as those of (I +,_)f expanded, the same will be true when n + 1 is put for n; but we have sbewn that this law holds when n = 1 or 2 or 3-; therefore it is true in general, and we have d"y = d~uv+ ~u. v + (n - 2 ~U. d2V...+udv 1 P2 190 th II DIFFERENTIAL COEFF~ICIENT This is called Leibnitz' Theorem, and is often of use in finding successive differential coefficients. nth differ- 20 oW ential co- 260 Tofind the nth diJerentiat coeficient of f (x) if efacienrt of f (x) = q (a + b x + cx2). cp (a-tbx + cx').. We have f v + ~h) = (A+ Bh + ch) (...(), if A = a + bx.+ C, X2 B =b~+ 2cw. Now the general term of / (A + Bh + ch'2) expanded in powers of Bh + Ch2 is (Bh + ch"r Fr and the general term of this expanded in powers of chM is Bil - SCS (Pr(a) s.F(r -s) +s In this term let us suppose that r + s = n, and it becomes B' r - a 7- r orr(A) Bh-e-r h-= U,.h", suppose. r/r()1 (n - r) 17 (2 r - n)A'=Uhspoe Now s is evidently not > r, and not <0, i. e. n - r is not n > r and not < 0, i. e. r is not < - and not > n. Hence the sum of the terms formed by giving r all possible values is (Un + Un - + U,... -.2 U.) h' if n be even; 2 and (U,, + U, -1+ U,,... -2 U,) h if n be odd; and hence it follows that the coefficient of h" in the second member of (1) expanded in powers of h is U, + U,,-, + Ufl,2... &C. o or - 2 '2 OF THE FUNCTION P (a + bx + c a). 191 Now the coefficient of h' in the first member of (1) ex1 panded is, by Taylor's Theorem, rf o(v); hence we have fn() = rn (u, + U,,i + U,_... U,or) 2 2 where (b + 2eC)2r"nCn- - U, = q (a + bx + c)2) ( )-n F(n -r) r (2r -n)' which gives us the differential coefficient required. 261 Let 0(a + bm + c2) =,/1 + + (1 + x + 2) (, + Example of this method. then q) (a + bx + cx2) 1. 3 5... (2-) 1 I. - -; ( ) - (- i),'- (i + a, + )~ -(; 1 3 r -.3.5.. 2r 1,=(l~'( F(n- ir(2r —n)' nU=135... (2n - ) ()n(1 +I) r n U-l = 12-2 1 2 (1+ + 2) + -nl+ (1 + 2X)n-2, n.(n -l)(n -2)(n -3) 1.3.5...(2n - 5) rn U., = '-. (- 1)n-3 (1+X + x +,)a-"l+2 ( + 2a)"4. ee, if f or brevitw1 + ( -+,e Hence, if for brevity we put =, we have (1 +e 2ha)v 192 2THE CYCLOID. 1.8.5...(2 n- I) (1~r,2w)?? f"(ce)=2? (- 1)'.( +c f ~~2n + X +,-n") 2 n- I 2n (n -!2) (n - 3) 9n (2n - 2) 2 I I --- — ---— z + Z + &C. j 1 2n-1 1. 2 (2 n - 1) (2 n - 3) n n-i as far as Z2 or Z 2 Example 2. Let (P (a + bx + Cx') = log (a + b.- + X2) then ePVr (a + bcv+ cr') = (- l)'' (r - 1) (a + bv + _e2)-r; r~~u r 1) + X2 3 m)-r 2r-~2 )1-n U1. = (a + bc (b + 2e) r~ f() n (- r). F(29- n) (+ceK 2. 2n-4 The Cycloid. Generation If a circle NPS (fig. 73) roll on a straight line AX, a of the cycloid. point P in the circumference of it will trace out a curve of the form APBX, which is called a Cycloid. It is a curve of some importance in Mechanics. Equation We may find its equation as follows: to cycloid, Let A be the point on the line AX with which P originally coincided, take A! ( = ce) MP ( = y) to be the co-ordinates of P; draw NC from the center C to the point of contact N of the circle, join CP and draw PO parallel to MY, let a denote the radius CP, and 0 the angle NCP; then, since in the rolling of the circle every point of the arc PN has coincided with every point of the line AN, it is evident that AN = arc PN = aG; therefore we have e = AN - MN = aO - a sin 01 y=CN - CO = a -a cos f. (i) y = avers 0; and 0. = vers a XI SCELLA,NEROT S TIE OREMS. 193 also, sin 0 = - a - y.. i =vers- = s /2ay - y 2........ (*), a which is the equation to the cycloid, The point B at which 0 = wr, which is called the vertex Equation of the cycloid, is often taken for the origin, the line BY' tavered parallel to AX for the axis of y, and the line BX' perpendicular to AX for the axis of x. We may easily find the equation to the cycloid referred'to these axes by putting in the equations (1) y'+ AX' or y'+ 7ra for x, and BX'-a/ or 2a- x' for y, which gives us y + 7ra = a (0 - sin 0), 2 a - v' = a ( - cos 0); which, if we put 7r + 8' in the place of 0, become y'= a (0' + sin 0) ' (l (1- cos ') and therefore as before (suppressing the dashes), y = a vers -1- + 2ax - x2......... (4), a which is the equation to the cycloid referred to the vertex. To determine the tangent and normal at P. Tangent or normal to a cycloid. From the equations (1) we find d = a (1 - cos 0) dO, dy = a sin Od; dx a (1 - cos 0) dy a sin 0 PM - = tan PNX.; MN 1.3 194 MISCELLANEOUS THEOREMS. therefore the equation to the normal at P is Y1 - y = tan PNX' (cx - x), which is the equation to the chord PN. Hence the chord PN is the normal at P, and therefore the chord PS, which is perpendicular to PN, is the tangent at P. Radius of To determine the index of curvature at P. curvatureo It follows from what has been just proved, that if xf be the angle the tangent at P makes with the axis of x, = - - PSC = — 2 2 2 ds 2ds P= d' d Now ds2 = d V2 + dy2 a2 {(1 - cos 0)2 +sin0}0 dO2 = a2 (1 - cos 0) d0 = 2ayd02; '. p= Say, which gives p. Since PN2 =SN.NO = 2ay, this gives p=2 PN; therefore, if we produce PN till NT = PN, T is the center of curvature corresponding to P. Evolute. Let U (= /), UT(= a) be the co-ordinates of T, then MN= NU = a sin 0, and UT -PM= a(1-cosO);.'. =a (0 + sin 0), a =a (1 - cos 0), MISCELLANEOUS THEOREMS. 195 and.-. 1=avers-' - +V2aa - a 2 a Comparing this with equation (4) we see that the locus of T, i.e. the evolute is a cycloid A TVXW, as represented in the figure, being equal in magnitude to the original cycloid, but having its position altered, its vertex being at A, and its cusp being at V. From the equation (1) we have Lengrhe of a cycloid. ds Vdxv2 + dy= V2a2(I - cos0) dO'.0 = 2a sin - dO 4dacos -) 2 dS+ 4a cos - = 0 and.'. s + 4a cos - must equal some constant, C suppose. Now, assuming s to represent the arc AP it is evident that when 0 = o, s = 0, therefore we have 0 + 4a = C,.~. C=a; 4a(1 - cos -) which gives the length of the arc AP in terms of 0. 0 When P is at B, 0= =w, and O. cos - = 0 hence the arc BP= 2 2 chord PS, length of the arc AB is 4a. Arc BP = arc AB - arc AP = 4a - 4a I - cos - 0 =4a cos -, 2 196 MISCELLANEOUS THEOREMS. Now chord PS = 2 a cos PSN = 2 a cos -; 2.'. arc BP = 2 chord PS. as'=8a'. If we denote arc BP by s', then since chord PS = /SN. OS = V2a', we have S'2 8ax', which is the relation between the length of the arc and the abscissa measured from the vertex. APPENDIX. (A.) (1) f(IV) =2- J -,( (I) f(Z)=^^=^ + l _ ( suppose, then q'(x) = 6x, if'() = 2,v; = 6fa (a' + I) - 2a (S2 - 5) (2 + l)2 16x (a2 + 1)2 + - ( + a ) (2) f(x)=,2 + 1 ( IV + l' + (&2 + -+ )2~ + I m — 1 (3) f(s) = -. + - + -2 - 1 s + 1 - 4w 2 4- 8 + 3 + + f5' () ($2 _- 1)2 (OG + 1)2 _ )2. '3. (4) (- =f + -— 2 --- + + +. (5) f () (x + ) () ~ suppose, x~ —3 $.va-3 then f' (x) - ( - )2 - (P'(V) = 2m (_a + 2) + (a?2 - ) = S32 + 4a - 1; (...f (s~+4~'-1)(,-s)- (2_)(x+2) 2)3-7 -2_12x+5 (.?0_ 3)2> (^-3) 198 rAPPENDIX. (6) f() = (2 -( 3) (6)~ ( ((a - 1) (V - 4) (7) f(x) =^8-^ x2+ -1 a2 w7 + 1i' f'() = 3s +,,-5 + _ _g (8) (.- 0 a, n 2nal-1 a? (ta + a) 2 (B.) (1) f(= )= = (), Z =+ e=, () then f'(ai) =- '(z) 4'(Q) = (2) f(a?) = tan z = 0 (Z), z = m tan-ix = _, (a), cos Z 1 + 2a' (3) f() ()log = (z), z = sin x + tan a= V(v), then f'()= '(z) '()= 1 ( sr+CO cos_ 2x (4) f() - cos -1'= (z), z = e sinv (), then f'(x) = /.'(z) xf'( x)= 1 X( i) then f//oG) '(Z) ^(t) g = *6X(sin + cos,). (C.) (1) Y = / --, — (1 _,a)-, dy = d(7. (I -,,v)- + d ( 21 - 2)-, APPENDIX. 1t99 - d(1 - 2)- = - (1 -l2~)- )d = (1- 2)- d x; dy _ ( ) - '2 - + (1 - 2)\- = 1 dx -V (1 -v X2){' Or thus: log y = log x - I log (1 - 2); dy _ dv_ - 2 Id 1 dv y x 1 -,2 v - v2i dy y 1 1 d dx 1 v -7 (1 - X2)' (2) y = log (cv sin c + cos2 c) d (x sin x + cos2 x) dy — -sid sinx + cos2 X dc. sin cx -+ x cos xcdcv + 2-cos x d cos cx v sin mV + cos2 x dy sin w(1 - 2 cos x) +cx cos v dc x sin x + cos2 X (3) y = (tan c)ec; 1... log y log tan x, cos ca dy d cos 1 d tan x = -- log tan c + ---- y cos cos tan v dy /a sin x 1 =.y{ 2 -2- log tan c +sin os2 c dx cos og sin x cos x (4) y = log = log x - log (a2 + c2); dx 2cvxdx dy a2 + 7v;' dy a2 dm c' (a2 + cv2) 200: A PPENDIX. ex sin.x- (5) y= tan-1 ( —i —) +1 + e< COS <? then log tan y = x + log sin m - log (1 + e' cos x); dtany dy cos xdx c (cos a - sin w) dx dr + tan y cos y sin y sin 1 + ex cos x dy (sin x + cos x + e"x. -- =cosy siny. s -' dx v(1 + ecos x) sin tan y ez sin x Now sin y = - - /1N + tan2y /1 + 2 e cos X + e" 1 (1 + e' cos ) COS y - V/l + tan2y \/1 + 2e cos x + e"; dy ea (sin +- cos + c") d' 1 + 2e cos X + Or thus: dr e sin x dy - where = 1 + + c 1 + os? and therefore, d (e sin ). (1 + e" cos M) -ex sin x d(d cos x) (1 + eC cos x)2 e (sin X+ cos ) ( +e cosx) - 'sinx e (cos t-sin ) d; (1 + e cos x)2 dy 1 '" (sin O + cos x + e') d' 1d + (z (1 +e" cos V)2 6x (sin i + cos x,+ e') 1 +2ewcosx+ cs+ (X -Vi - X)\ (6) y = log, xc + VI - = log (V - v 1 _ t2) - log (<,v + -(va); APPENDIX, 201'. dx+ 0/1 - w..dy -= - _ X —v1 d xdx dx -- / 1 -,V2; X+ \/1 —; dy 1 " d - - jx + / - Z x- v/ I- _2 ( — //1 - - (P - \/ 1 - x2 x + \/1 - x,11 VI _ (2 (2xI i) (7) y = r n 1 - Vs,?n 1 - I... Sm then y = and 1-y y - y2 = xm;. dy- 2ydy 2 = mx-ldx, dy xm-"1 dx 1 - 2y 1 - xm I - -1 --- I- - ---- *aM (8) y = log (x + sec x + sin-1 x), I + d sec x +d sin-' dy =,v + sec x + sin-l B (9) 1 y = tan' 1 -.', log tan y = log x -- log (l -- x2), dy d dx dx sin y cos.y X -- - sin y cos y/ t 1 - x^ 202 APPENDIX. (10) y-=og l+sin 1 - sin 0 = 2 log (1 + sin 0) - 2 log (1- sin 0), - cos 0d0,cos OdO dy = + 1 c + sin 1 - sin O' dy cos 2 1 de 2 cos2 0 cos 0 (11) y log tan (- d - + - dy 1 di cos wx 4 4 2 /2_ 2 + b cos Ilog cos ny = log (b + a cos x)-log (a + b cos x) {n = a/-bl; d cos ny o usin nydy - -asin xdx+ bsinaxdx cos ny cos ny b + a cos x a + b cos x.. -^=-cot ny sin andv n b + a cos - asi + bcos \4 2/ \4 \22 dyy sin d(b + a cos (a + b cos b + a acos (12) Y a2 - b2 a + b cos.v log cos ny =log (b + a cos x) - log (a + b cos x) n a=c- 2 d cos ny u sinnydy a sin xdx b sin mxd or + cos ny cosny b - acos a+bcosx dy 1 a b dv. -- n=- cot ny sin ^ -x -- acosv a + b cos, — = cot ny sin x (b 4 a coS x)(a + b cosx) b +a cos x Now since cos ny = a + b cos IV APPENDIX. b + a cos x 203.'. cotny = b + a cos cx n sin w /(a + b cos x)2 - (b + a cos?)2 dy 1 dx a + bcos (D.) '(1) y VE dn y = e (1) y e, =C~.d = one?. d n' (2) y = ", d n =( (m-) ( +-...)(-n+)m~n. (3) y= log d x,-1 ddv" (4) y = sin (x + a), (5) y = tan x, dn y _.(9 Z w + dx" 2 d2y d =X 2 sin x cos3,% d3y 2 6sin2 dx3 cos2 + cos4 4 6 Cos'2a COS4 &c. &c. Hence assume that in general d2n-ly A, Bn C,. _. -+-+.*..+ dwaen-~ cos 2v cos4 Cosa x and we find d2"y 2 A4 sin x 4 B, sinx 6 C, sin x dx2n - cos3x cos5x cos7x 204 APPENDIX. d2n+y 2A,4 4 B, 6C, d + cOs'? os 4 a co + Cos6 *a 2. 3An sin2 x. 5 B sin2 6. 7 C, sin2x + 4 + - + 8 COS at COS6 a COS8 I - 2,4 2. 3A, - 42 B, 4. 5Bn - 62C, + + 8 COS2a COS4 X COS8 Hence we have A,+1 = - 22A., B,+1 2.3A, - 42B, C,+1 = 4. 5 B - 62 C... &c. &c. From these equations we may find by successive substitution the quantities A,, B,, &c. The general expression for dY is very complicated in d aG this case. (6) y= E;,. v/_( 4. 1)c, dxy daa d2Y = (a + n) exe (7) y = 2" ~, = (C, + mrn -1) 6T da? d2= t 2m atit + m. (m - i). a-?2} 6X d X2 ' APPENDIX. 205 By Leibnitz' Theorem, see Chap. XIX., we obtain immediately, putting u = e, v = m d"y n + n n (n - 1) = {m + - -mm-X + m (m - ). x-"..}. (8) y = exsinx. We find similarly by Leibnitz' Theorem, dny nn n (n -1) d =Y" e { sin x + - cos? - sin N ---F cos x +.... [ 1n.(n 1) (n - 1).(nl(n - s) (sin (n n(n - l) (n - 2); i Now 1 V/- 1 - /2 (cos r sin r /.. ((1 + /- )"+ - 2(cos - = sin 4 4 dny n n IT n mr\ '.. d /-22 e" sin cos -- + cos x sin -- dx" 4 4 n = 2J2esin (1+- Y) Or thus more generally and simply. Let y E= 6 sin (x + a), 206 APPENDIX. then dY -_ e {sin (a + a) + m cos (,C + a)} dma -— 3 sin (0 + a +;), cos p if we put n = tan/3; ~. in a similar manner we find, d'y d 3 = 2 sin (t + a + 2/); dX' cos /3 d y e6m and.'. in general d - sin (s + a + n3); d cx cosO 4 (9) y = tan-1 x, then = cosy, dx d2 y dy d = - 2 sin y cos =y sin 2 ycos y y+ 2y, d y + c siy + i. dy == -2cosy + - )siny + sin 2 ca- cosy cosy =2 cos (3y + r) cos2y Now if u = A cos (ny + 3) cos"y, du A5 \.,n dy n = {cos (ny + 3)s si+n (ny + s3) cos y cos"-'y d n A cos (n +l)y + + )}cos"+ly; from which it is clear, that if we assume d " )ony+( Cv xG" APPENDIX.20 207 the same law will hold when n + 1 is put for n, but we have found this law to be true for 2 and, teefrits true in general; and it gives us the nth differential coefficient of tan-',v. (E.) d s3 ()Having given u = ~ d'y dx where dx is constant, and ds'2 = dx2 + dy2; to find what u becomes if s be made the independant variable. We have d'y =dxd L \dxl d'y dv d'y dv - d yd'x.. 1) Now since ds' = dxe + dy2, and ds is constant, we have 0 = dxd'x + dyd'y... () (1)2 + (2)2 gives, observing that dX2 + dy' =ds2, (d'y dx-) 2 =_ (d'y)' + (d 2X)'. d S3 Hence u =Vdy'+(') in which expression a is the independant variable. ds3 (2) Given u = - yx where dv is consta nt, and x'= a cosn t, y =b- sin nt tto make t the independant variable, dx = - na sin ntdt, dy 2i cos ntdt, ds2 = n2 (a' 2sin' nit + b'cos' nt) d t2, dy __b dx cot nt, 208 APPENDIX. drdy = dd ( ady i b n dt dxd'~Y = dav2d n nasinntdt dlw) a sin' nt = n3abdt3; (a2 sin2 nt + b2 cos2 nt) ab (F.) (1) y = e tan u + log (u + v), d2y = eVtan u+ --- + + -- (u + V —) [os-2 +u I u + v) 2 2 ev sin u 1 + -3- - 7 -- 2} dU2* Cos3 U (u + v) J (2) y - + v +; put u+v =, then y = u + v;,-. dy - ( log u + v logv) d + wz-'ldu+ + xzvd-' dv, and.; since dz = du + dv, dy = {u+ log + v+" log v + (u + v) u+-} d + {u+v log + vu+vlogv + (u + v)vu+v-l dv. (3) ysin x+a siny- y =o, dy ycosx + sin y - y + (sin + cos y - ) - = dx -ysin+ 2(cos +cosy -1) d ms-. (dY) d2y + (sin x + x cosy - ) y = 0, &c.... &c., whence dy dY, &c. may be determined. dm dw, APPENDIX. 209 (G.) (1) Let y = sin-,; dy 1 I.34 then d- + - + 2- 4 doo V/!, ^2 2. 4 ---- ' + &c.; '2 4.4.. 2n dl+' y 1... 2n - 1 d —. = -------. rFn + powers of r d a'+1 2.4... 2n 1.3...2n- 1 =- In, when, = 0. 2.4... 2n Hence the general term of the expansion of y in powers of,v is 1. 3... 2n - l 2'L 1 2.4... 2 n + 1 and.. since sin-l,== mnrr when O= 0, we have a _v' 1, 1. 3aI, sin-a == nr + + -... &c. 1 o3 2.4 5 (2) We may often obtain developements of this kind more simply in the following manner. Assume y = A + SAx: + A~'v2... &c., dy and A.' + 2....I + 3A3V < Comparing this with the above value of we find dr immediately 1.3 A, = 1, A2 = o, 3= o ~,, = o, -... &C. Also.A = the value of y when vG = o, which is mnr hence y = m7 + - + - + &c.... as before. 1 1 14 210 APPENDIX. (s) jy = sin cv.........(........ ). Assume y = A, + A, 'v + Av.............(2), then from (1) we have - sin = -4, - A4v - AJ2M......(3), and from (2) we have d2y 1 ~2 A, -t- 2.3 A~lx: t- 3.4 A~cV2......(4). d A2+23-3V+3;A Now AO=y when cv = 0;..O 0; and A, dy os when v = o;.. dcv and therefore, comparing (3) and (4), we have 1 A3 1 Ad =0, A3= 4, A 4=o, A 5 4 ]-... X X" X' sinc - + &c. ri 13 Es (4) if we differentiate the result just obtained, we find X2 X4 coscv= l - -+-. &c. 172 174 (5) If tan y = a tan x, to expand y in powers of cv. (6) If sin y = a sin (y + cv), to expand y in powers of cv. (LI.) (1) To expand cot v in powers of v. cot (v = c= when cv = 0; therefore there are negative powers in the developement, and Taylor's series fails. We may proceed as follows: APPENDIX. 211 2 4 - +. - cos - 12 + r4" cot, - sin v IV X3 sin ~ cS tV3 1 - -.... &c, ~ 1 P2 assume this to be equal to -(A + Bx + CX..); then we have (G + Bv + Ct2...)(l 1 -~..)= - F73 Fo2 "' or A B + (C - x,... = l-, 1 1 o. A =., B==o, C= — =- ~' —,.'. cot =- (1 -,-.. &c.). (2) Let y = (e' - dy I Here d-= e' (e' - 1)- when x = 0 therefore fractional powers occur in the developement, and Taylor's series fails. We may proceed as follows: a; a - = +-r-.. y= ( + V.. =b (1 +th&... &c ), by the binomial theorem. 14-2 212 APPENDIX. (3) y 1+ log (e~ i\uv), HZ-ere -=, and y + +x+-... &c d cv'C e The following rem-ark is of some importance, as it will often enable us to simplify the process explained in (145). Suppose that we wish to expand y in powers of x, having given a relation between y and v; then if y = a when x = 0, we assume y = a + ux"', as is explained in (145). Now suppose that the result of this substitution comes out in the formi UjVZl' + U2, - ~,,,V + 0, and suppose that we can see by insfpection that one of the indices r, must be greater than another ir, whatever value m may have, (m of course is always positive); then we may immediately ileject the term U2,'2., since it cannot be one of the terms containing the lowest power of v., and therefore cannot affect the result, as is evident from the example in (145). (1) Let y' - v2y' + acivy - adi`0; here y=0 when v=0; assume... y = uiv!; then u4 aim 2 ~2 aux1 — &+2 Now here 9m + 2 m List be > m1- 4- 2 whatever m, be, and m + 2 must be > 2; hence we may immediately reject the terms u' 2m+2 and attx7+; and.,. we have simply,LlW., -a2,Vs = 0, which gives 4m = 2, and.. m =- and..u=a 2/a, and:. y=6 A/a.x ~ R, APPENDIX. 213 Of course we must not reject these terms. if we proceed to find the second and higher terms of the expansion. (2) 1 (2y2+ axy' -a2'V3O=; u4 4m~ 2 U2nfl+2 +2 u'2,ni+ I - a2 ';3 or since Qm + 2 is >2 m + 1 u4x -au2 2mn+1 a2'3 Suppose that in is > 1, then 4m I's > 2m + 1, and 2m + 1 is > 3, which cannot be; therefore m is < 1;I' is >'2 m+ I; a2xv3 is to be rejected, and.'. we have 4m 2rn+ 1, and.'.- m= —,and.. Va (J.) ()Let y =xlog x; to find the limiting value of y when x approaches zero. x 0 We have y=- - when wv=0; log x by Lemma XX., y and d d log ie have the same limiting value when x approaches 0. dx Now -V (log V)dlo y and - have the same limiting value when x approaches 0; 214 APPENDIX. therefore, multiplying each of these quantities by '-, x and Y - y have the same limiting value when Ad approaches zero, and therefore 0 is the limiting value of y. Here we assume, that if any functions U and V have the same limiting value when x approaches a, the same is true of WU and WV, W being any other function; which is evidently true, since if A be the limiting value of U and V, and B that of W, then by Lemma VIII, BA and BA are the limiting values of WU and WV. (2) Let y = ve't'; to find the limiting value of y when x approaches o. e- 0 y = = when x = c 1 de -,~. y and -- have the same limiting value when x apd - proaches co. de-,~ y Now = ivx e - 1 e-x d - x.'. y and e-' have the same limiting value when x approaches co therefore the limiting value of y is 0. (3) Let y = (sin x)tanx; to find the limiting value of y when x approaches 0. logy = tan x log sin = -- zx log z, COS a? if we put sin x = z; hence, since z approaches zero when x does, and since the limiting value of cos is 1, and that of zlogz zero, the limiting value of logy is zero, and therefore that of y is unity. APPENDIX. 215 (K.) (1) To inscribe the greatest rectangle in a semi-circle. Let CM= x, MP= y (fig. 33), then 2wy is the area of the inscribed rectangle; hence we have d(2xy)= 0; and.-. d (2y2) = o; or d I2 (a2 -2) 0;.2. 2a2 - 4?3 = 0, a a which is satisfied by x- and x = 0. x = evidently gives a maximum. (2) To inscribe the greatest semi-ellipse in an isosceles triangle. (fig 34.) Let CM=, MP = y, then if CE =h, CF= k, we have a2 b2 h -- k = -; y a2 b2.*. h2+,..- ~ (1). ''h" + I...... Also 7rab is to be a maximum; and.. d(ab) =0;.. bda + adb; acda bdb and by (1), - + =; a b bbk' a i' which, along with (1), gives a and b. (3) To find the greatest triangle that can be inscribed in a given circle. ABC (fig. 35) is a maximum, supposing AB invariable, if AC = BC, as may be easily shewn. Therefore any two 216. APPENDIX. sides of the maximum triangle must be equal; therefore it must be an equilateral triangle. (4) To find the greatest quadrilateral figure that can be circumscribed about a circle. Wre may prove similarly that it must be a square. (5) To find the greatest triangle which has a given perimeter. (6) To find the greatest polygon that has a given perimeter. Suppose all the sides but three invariable. (L.) (1) The normal to any curve is in general the greatest or least line which can be drawn to a given point to it. Let vry be any point on the curve, x'y' any other point, and R the distance between them; then R 2= (t-,a')2: ( -y)2, RdR = (, -,;v') d,, + (y - y') dy, but dR = 0 if R be a maximum or minimum; and therefore xV - + (y' - y) = 0, which shews that x'y' is a point of the normal at xy. (2) To draw a tangent to a given curve, cutting off the greatest or least area from the space included between the co-ordinate axes, The equation to the tangent being (y' - y) d, - (x' -,v) d y = 0 the portions it cuts off the axes are ydx- - vdy a xdy - ydxm d -a and a =dwt dy APPENI)1X. 217 an a/3 ((y d - xd y)2 and - or - 2 2dxdy is to be a maximum. Therefore, considering dx as constant, we have (ydx.- xdy )2Y or y =0,, dy - 2 (ydo - zdy) ovdfydy - (yfdsc - xdy)2d~y or _ 0, dy ydx - xdy which gives 2x - = -a; a 2 this equation shews that the portion of the tangent intercepted between the co-ordinate axes is bisected at the point of contact when it cuts off a maximum or minimum area. (3) In the ellipse, if we assume x = a cos t, then we have y = b sin t, in virtue of the equation y2 w_ b-2 = 1 a2 Hence we may always suppose in the ellipse, that x = a cos t, and y = b sin t. If we describe a circle on the major-axis as diameter, and if we produce the ordinate MP to meet this circle in Q, then it is evident that z QCA4 = t, for CM x cos QCA = - = cos t, MQ a This transformation is often useful. Differentiating, we have dx = - asin tdt, dy = b cos t dt; 21.8 APPENDIX. ds = Va' sin 2t+ b' cos't. dt, b tan 4' = - - cot t, a cos xe A/1 +- cot2 t Equation to tangent is x cos t y sin t a b (4) To find -the locus of the intersection of the tangent and a perpendicular upon it from the focus of an ellipse. The equation to the tangent is b b y - -cot t. x + a sin t The equation to the perpendicular upon this from the focus is y = tan t. (v - acu), which equations may be put in the form ay sin t + bx cos t ab, by cos t - av sin t - a e sin t; therefore, squaring and adding these equations, (IV + y') (a' sin2 t + b' cos2 t) a2 (b2 + a'e2 sin' t) - a (b' cos' t + a" sin' t). Since a2e'- a - b..'- ~y'=a'; hence the locus required is the circle described on the major axis as diameter. APPENDIX. 219 (5) To find what relation must hold between a and /3 in the equation 1 + = 1... (I). a,3 So that it shall be the equation to a tangent to the ellipse 2 2 2 + 2 = 1... (2). a2 b2 It is easy to shew that the equation to the tangent to (2) is X1'V YMY - b2 a-+ = Comparing this with (1) we have a.- b.V y x a y b a a b - a2 b2,'. by (2) + = which is the relation required. (6) To find the curve in which the subnormal is a constant c. dy We have subnormal= Y = c; 2.. yydy - 2cdcv = 0;.'. d (y2 - 2cx) = 0. Now in general if du = o, u must be a constant; y-2 -- 2 c. + some constant, which is the equation to a parabola. (7) To find the curve in which the subtangent is twice the abscissa. 220 220 ~~~~~APPENDIX. We have yd - 2x; dy dx dy IV y d (log x.- 2 log y) = 0; logvx - 2? log, y =constant == log C suppose; =C, which is the equation to a parabola. (8) In the ellipse, if n be the normial and p the perpendicular from the center on the tangent, jjn = b2. Fr y dx - x dy ds d ~ ~ ~ d dy b2C2 d y b2xV = y~ ~,since = — y + dx a2y b'. (m~.) (1) Let the eqUatio n to the curve he a')y = ( ax3 + a2av2 dx2 the sign of fy is + when x is < a, -when x is > a and < 2a, and + when x is > 2a. APPENDIX. 221 Hence the concavity of the curve is turned upwards when i is <a, downwards when iv is between a and 2a, and upwards when iv is > 2a. And there are two points of contrary flexure, one when iv = a, and another when v = 2a. (2) Let the equation to the curve be then d' y 2a'2(v + 2a) C.d X2 (X -4a) ' Here there is a change of flexure when iv passes through the value - 2a, but none when iv passes through the value a. (N.) (1) To find p in the ellipse. We have v = a cost, y =sinD, {page2l7 (2),( d s" = divd'y - dyd dV (2 2i"t - 2 COS2 t)~ (a2 sin2 I + b2 o1) a b sin2t + abcos2t 1a2 2 av2 (2) If n be the normal, ds n = ydiv ndiv a dsr d = — ndt; Y b 1 a3 n3 ab h3" h" 22'2 APPENDIX. The negative sign indicates that the concavity of the curve is turned downwards, in which case ~ is negative. ds (s) To find p in the parabola. Let us use the equation x2 Y=, 4m then dy = dx; 2m dx2 and d2y 2m (dx being constant); 2m (1) To (1) To -. We have Hence in is3 y - = 2m 1+-I. vdd2y my (0.) find the evolute of the ellipse. a = x - p sin \, /3= y + p cos x, ds'?dy or a = x - dd_ dxd y - dydx ' ds d =+dxdy- ddyd x' the ellipse, a a= a cos t - (a' sin2 t + b2 cos2 t) b cos t (21..) = - (a2 - a sin2 t - b ) cos) o t a a2 - 6b2 == ---- cos, t. a, APPENDIX. 223 Similarly /3 - sin3t b Hence, since sin't + cos2t=1 we have a b when a, = a -b'and b1= ba2 which is the equation to the evolute of an ellipse. It is represented by UVWY, (fig. 36). (2) To find the evolute to the parabola. We have = 4m' and supposing dv constant, ds'dy a d2 ydx' also ds' =2 ( + dv dy - div d'y= -; 2m 2 m 3~~~~~~~ /- m _ (4m 'a4m V 4 2 a3 4 224 APPENDIX. i 1 or /3 = 2rn + 7 a 31, which is the equation to the evolute of a parabola. It is represented by UWV (fig. 37). (P.) (1) Let the equation to the curve be r = ael then dr= 1ma e"' dO, rdO tan =- = - m. dri Hence in this curve, which is called the logarithmic spiral, the tangent always makes the same angle with the radius vector. ds2 = 2 d02 + dr2 = a2e2 m (1 + m2) dO2;.'. ds - a / 1- + m. emOd = o, /' + mu d (. ' -- -- e' o) = 0; %//1 + 2.. - a n --- —- = some constant, C suppose: m if s = 0 when 0 = 0, then.\/1 + m.- t - a C;.. =, ----- ( - a?). D/, (2) The polar equation to an ellipse may be put in the form cos2 0 sin2 1 + -...................(1); a' b',r APPENDIX. 225 (I 11' 1 dr 1 ( cos 0 sin 0 = - cot..(2) Now (1) gives us 1 ( 2 — )Cos" - 2 —, 2.2 a 2 f(21-) sin 201= - 1 \a2 77b2 therefore squaring each side of (2) and substituting these values, we have (;;)-/1 ' I - 1 _ coto/, (r2 a) (b2 r2) a2 b2 or r4 (a2 +b2) rsi _ 0, sin ~2 1 which equation gives us r, the length of the diameter which makes an angle 5 with the tangent at its extremity. Hence, since conjugate diameters are parallel to tangents at their extremities, the two values of r got from this equation are the lengths (a'b' suppose) of the two conjugate diameters which make an angle /p with each other. We have therefore a's b = a2 + b2 a2 bI a-'2 b'2_=a --- or ab' sin = ab, sin2 p two well known properties of the ellipse. (3) In the spiral r = aeme r2dO r2 d ds -= 1 +m. rdO' -*. p - =_.-.* (1); dr.. dp = 1-; 15 226 APPENDIX. ' dp If 0 be the extremity of p, and SO = ri, then r2 p2 + r- 2rp cos SPO = p2 + r2 - 2pp = s 2,;2. i. 91: g'r. Also if p, be the perpendicular from S on PO, we have 2 2 2 m ri A/1 + m2 ~P. P; = %/1 eq-m Now r, and p, are evidently the radius vector of the evolute and perpendicular upon its tangent; hence, comparing this relation between p, and rI with (1), it follows that the evolute to a logarithmic spiral is a similar spiral. In general we may find the equation between p, and ri by eliminating p, p, and r, between the equations P = f(r') rdr r = dp f (-fr) r2 = 2+ p2 2pp p i2 + p2 = 2. p =f(r) being the equation to the given curve between p and r. In this manner we may obtain the equation to the evolute of a spiral. .227 APPENDIX. (S.-) (1) Let the given equation be 21 (x -a)' (x+ 2b). ay = v +b I ay = =L(V~- a) x+ 2b; then we have the following table. (? - 2 b - b 0 a y 0 I mposs. co 4: 0 4:= 0 dy dxv co co h \,/ a_+ 2 b co by fig. 38., AD = 2b. 'Hence the curve is represented AB =a, AC =b, (2) Let the given equation be a (v- a)' See fig. 39., AB= a, AC a. 228 APPENDIX. (U.) (1) To determine the position and nature of the multiple points of the curve, U = y4 + 4 - 2a2y2-_ 2a2x + a4 = 0... (), d U = 4 -3 - 4a2Xw = o. @@4@@@(2), dyU= 4y3 - 4a2y = O which equations are satisfied by any two of the values o = or - a, and y = or: a. Now if. = o, y = - a in virtue of (1), and if m = L-a, y= o or aV/2.a. Hence the only values of x and y which satisfy (1) and (2) are = 0- 0 -o;O () - )x =) - a) (~) (j), (7),,, )' y= aa y = -0 a y = y= Now d2 U = 12<2 - 4a2, ddyd U = o, dy2 U 12y2 Hence, using the notation i A = o, B = o, C = - 4a2 if x - 0, D = 0, E = - 4a2 if y = 0, Hence for the values (a) - 4 a2. in (161), we have or 8a2 if x = - -a, or 8a2 if x = - a. and (/), we have dy 1 -4aa2 + 8a2u2 =, and. UQ or d == dv '2/' APPENDIX. 229 and for the values (y) and (S), we have dy 8a2 - 4a2u = 0, and.*. u, or d" = /2. dx Hence if we take AB = AE = AC = AD = a (fig. 40), there is a double point at B, at C, at D, and at E, as is represented in the figure. This is an example which may be very easily solved by the method in (156). For in this case P =_. -;?3 - a2y 3 X2 - a3x22 - a2 ' p (3y2_ a2)p.*. if =, and y = a, p2, and if - =: a, and y o, 2 = 2. In general the method given in (159) ought to be used, dy only when we wish to find dy for the values x = o, y = o, da2 or when more than two differentiations are necessary to find dy dy in which cases it is simpler than the common method. (2) Let U = ay - bX2y + x4 = 0... (1), then dU = - 2bxy + 43 =... (2), dyU Says - b,2 = o........ (3). (2) gives x=o or y b ' if x = 0 y= 0 in virtue of (3) and these values satisfy (1), 2 ) becomes if -= V (3) becomes 230 APPENDIX. 12a all- bv2 = 0, b3 b which gives 2 - and.-. y 12a 6 a these values do not satisfy (1), and are therefore to be rejected. We have therefore to consider only the values x = o, y = 0. Hence putting y = uxm in (i), we have au" - bau + v = o, and.-. au3- bu =o; dy -0 or _ dx ~~~a which indicates a triple point at the origin. (3) To examine the nature of the curve y5 + axv4 - b2ixy2 = 0 in the immediate vicinity of the origin. Assume Y = uxm', then ux 5m + a4 - b2 u2 2?n+l = 0. Suppose 5m = 4;.,. m =, and.~. 2m + 1<4; which wvill not answer. Suppose 5m = 2m 1; I.. m=.4, and.'. 4>2m + I; which will answer, and gives U5 -b2 U2 =0, and.~. u = M. Suppose 4 = 2m 1;.. m=.4, and *~. 5m>4; which will answer, and gives a - b'2u2t= 0, and.?. b Hence we have y b vixa + R, APPENDIX. 231 and y = a _ iA + R' b The former of these represents the portion PAP' of the curve (fig. 41), and the latter the portion QAQ'. We rejected the value 5m = 4, because it would give u = co when x?= 0; this value belongs to the infinite branches of the curve; for putting 5m = 4, and.. m - and assuming v7=-, we have 1 1 (u5+ a) - b2U' 13- o, or u5 a- b which gives the limiting value of u a2 when z approaches 0;.. y O ~ t -7 f 14 — a~xb + R where R o, when x =co; V'y = - as very nearly for large values of x; dy 0 0 when x = cc, as is represented in the figure. dcc Plazte -F 23 2; 29. -P 1 26' lf.5 p2+ I*? 27 32..: 3f1 '' 3 ~3> / y. _... ~..... 34. 33. C X: C m- R 37. 3. vi?( y y N A "I c. 3zs., O'Brien's CCLcut'us. Ptate j. T 01Briien's CcZcu1ts..P /%JF.:2: 61. 62 (0. 0? a f T y r x V. A x Y / 70. -P~. 72' 72. x st x: s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ M etcalfe & P alm e r, Camr b:r'id ge. OBriens Cacuzdus. PlZote 7-. x y t 0 -0 +~ 0 \ +c O -m, a + a 0 cc 4 - + 0 0 -t 0 o co - a o o _.1 i: aC 0 00 _ ios.s -- a 0 o -l 0 C a + 7. 0 CO +F cc cc -+ _. 03 oo ' imposs: a 10 0 + a 0 0 inposs:. I., Ct 1/ / I 0 imwross: cl co 4t o 0 Itmtoos5: bi-mtposs c,0 cc.4- + cc 0 CO tmposs: Ct c O C 4+ Metcalfe & Palmer, Cambridge. PLaL e X. S-r - - dr -+ 0 0 00 00t T s B S.B=a-, 1-s -AL _ 2 0 o oo asymptotei _2 + y=a 2 -r o + It 0 o oo asyn-o+te o Y-__ 0..+ r~- + 2 + 4-. 2 \ 0 vnpooss 0 0 0 oo o o I\ b + Bo 0 A DIGESTED SERIES OF EXAMPLES IN THE APPLICATIONS OF THE PRINCIPLES OF THE DIFFERENTIAL CALCULUS. DESIGNED FOR THE USE OF STUDENTS 3a tfte (tmbersitp. BY JOHN HIND, M.A. F.C.P.S. F.R.A.S. LATE FELLOW AND TUTOR OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE. CAMBRIDGE: PRINTED BY J. SMITH, PRINTER TO THE UNIVERSITY; AND SOLD BY DEIGHTONS, STEVENSON, NEWBY, HALL, JOHNSON AND HATT, CAMBRIDGE; TALBOYS, OXFORD; WHITTAKER & CO.; AND FELLOWES, LONDON. M.;DCCC.XXXII, ADVERTISEMENT. TiE Collection of Theorems and Problems, submitted to the notice of the public in the following pages, is intended for Practice in the Applications of the "Principles of the Differential Calculus" as laid down in the Author's Work upon that subject; and they have been distributed into Chapters agreeably to the plan there pursued, with occasional references to the particular Articles upon which their solutions depend. Although many original examples are introduced, it may be necessary here to observe that by far the greater portion of them have been selected from the Works of different Authors who have written upon the subject, and from Papers proposed at the public and private examinations in the University. The order observed in the arrangement of each Chapter is generally the same as that of the Articles in the corresponding Chapter of the Work referred to, each individual question comprised in every Chapter being supposed capable of solution independently of those which succeed it: and to the whole has been prefixed a Summary of the principal Articles contained in that Work, for the purpose of enabling the Student to recur with greater facility to the Exposition of any Principle that may be required. CAMBRIDGE, Feb. 20, 1832. CONTENTS. INTRODUCTORY CHAPTER. PAGE THE Method of Limits.................................... 1 DIFFERENTIAL CALCULUS. CHAP. I, Differentiation from first Principles............................ 5 CHAP. II. Differentiation of Algebraical Functions of one principal Variable 6 CHAP. III. Differentiation of Exponential and Logarithmical Functions of one principal Variable........................................ 11 CHAP. IV. Differentiation of Trigonometrical and Geometrical Functions of one principal Variable........................... 17 CHAP. V. Successive Differentiations, Elimination of constant &c. Quantities, and Applications of Maclaurin's and Taylor's Theorems... 22 CHAP. VI. Indeterminate or Vanishing Fractions................... 37 CHAP. VII. Maxima and Minima of Functions of one independent Variable... 42 Vi C ONTE E N T S. PAGE CHAP. VIII. Tangents, &c. to Curves referred to rectangular Co-ordinates...... 49 CHAP. IX. Tangents, &c. to Curves referred to polar Co-ordinates.......... 71 CHAP. X. Singular Points of Curves referred to rectangular and polar Co-ordinates............................................. 86 CHAP. XI. Tracing or Describing of Curves referred to rectangular and polar Co-ordinates........................................ 104 CHAP. XII. Differentiation, &c. of Functions of two or more independent V ariables............................................... 115 CHAP. XIII. Maxima and Minima of Functions of two or more independent Variables................................... 125 CHAP. XIV. Tangents, &c. to Curve Surfaces and Curves of Double Curvature 130 CHAP. XV. Miscellaneous Theorems and Problems......................... 146 A SUMMARY Of the zmost important Articles contained in the Principles of the Differential Calculus. INTRODUCTORY CHAPTER. METHOD OF LIMITS. ARTICLE PAGE 1. DEFINITION and Examples of Limits.... 1 2. To prove that the limits of the ratios subsisting between the sine and tangent of a circular arc, and the arc itself, are ratios of equality. 3 of equality........................... 3 4. To find the Circumference of a Circle.................... 4 7. To find the Area of a Circle............................... 5 11. To find the Surface of a right Cylinder.................... 6 13. To find the Content of a right Cylinder.......... 7 15. To find the Surface of a right Cone. 8 20. To find the Content of a right Cone........................ 10 23.,To find the Surface of a Sphere............................ 12 27. To find the Content of a Sphere........................... 13 DIFFERENTIAL CALCULUS. CHAP. I. 1. Definitions and Preliminary Observations................... 14 CHAP. II. Differentiation of Algebraical Functions of one principal Variable. 20. If u and v be two functions of the same independent variable 'x, such that for every value that can be assigned to it, u =v, du. dv then will d — d-, and d v................. 25 dx dxr' Viii SUMMARY. ARTICLE PAG 23. If we have it = p - q + r - &c., wherein p, q, r, &c. are all functions of the same independent variable x, then will du dP dq dr dx dx d-Tx dx and du=dp-dq+dr-&c................ 26 24. If we have it pq, p and q being functions of x, then will du dp dq W- + p$P, and du = qdp + pdq........ 27 27. If u =V, where p and q are both functions of x, then will q du 1 dp dq qdp -dpdq dx dx dx q2.. 29 31. If u = ptm where p is any function whatever of x, then will du 1dp) dir nMp M-1 d- -, and du = mpm-ldp, whether m be positive or negative, integral or fractional. 30 CHAP. IIJ. Differentiation of Exponential and Logarithmical Function8 of one principal Variable. 34. If it = aP, where p is any function whatever of x, then will du dp =:0 log a aP and d-l = log a aPdp, dx dx where the logarithm of a is taken in the system whose base is 2.71828 &c. represented by e...........................- 38 9. If u =t log p, where p is any function of x, then will du _ dp_ d - M.f -,and du=M dx pdx p where M is the modulus of the system of logarithms used.. 40 42. If it = p9, where p and q are both functions of the same independent variable x, then will du = dq dp an p Sb d+ qldp 41 dl plog p W- + qpg- jx-, and du p opdq+ p4 44. If ut = log of log p, which is usually written nt = log2p, then will dplogpdx anddu= 1dp...42 dx - plogp dx' 10. P..... i'x -SUMMARY. CHAP. IV. DiJfferentiation of Trigonometrical and Geometrical Functions of one principal Variable. ARTUCLE PAGE 47. If u sinp, where p may be any function of x whatever, then will co)du dp then will =cosp j-, and du=COS p dp. 406 56. If s be the Arc of a Curve referred to the rectangular coordinates x and, then will ~\,'1+ (dy\,2 and ds =d + dy2. 53 ax V \dXJ 57. If S be the Area of a Curve referred to the rectangular co-ordinates x and y, then will dS d- = y, and dS = ydx. 58. If Z be the Surface of the Solid generated by the revolution of a curve whose co-ordinates are x and y about the axis of x, then will dl/ dy ds -dx \r + d~x dx anddS=2rmydX 1+()2=27rnYds......... 54 V/ \dxj 59. If V be the Volume or Content of the Solid generated by the revolution of a curve about the axis of x, then will d V =?r2, and d V = y2dx................... 55 61. If s be the Arc of a Curve referred to the polar co-ordinates O and r, then will ds /2, /dr\2 and ds = dO 4/r+(56 dO dO '~' ev'dO 62. If S be the Area of a Curve referred to the polar co-ordinates 0 and r, then will dS 1=r2, anddS=1r dO................. 57 b x SUMMARY. CHAP. V. Successive Differentiations, Elimination of constant, (cc. Quantities, and Maclaurin's and Taylor's Theorems. ARTICLE PAGE 65. General Explanation, Notation and Successive Dlfferentiations............................................. 59 67. Derivatives of various Orders, and Elimination of constant, exponential and transcendental functions................ 64 69. Maclaurin's Theorem, and its Applications......... 68 74. Taylor's Theorem, and its Applications.................. 83 83. Given the differential coefficients belonging to the equation u =f(x), to find the differential coefficients belonging to the corresponding equation x = ().............. 100 84. If u be considered a function of y at the same time that y is considered a function of x, it is required to find the differential coefficients of u considered a function of x........ 102 87. Method of Fluxions................................... 107 91. Method of Indivisibles................................ 114 92. Method of Infinitesimals............................. 116 95. Method of Derived Functions........................... 120 98. Residual Analysis.................................... 128 CHAP. VI. Indeterminate or Vanishing Fractions. 100. General Explanation of the subject..................... 130 103. To find the true values of functions which for particular values of the principal variable assume the form...... 133 CHAP. VII. Maxima and Minima of Functions of one independent Variable. 109. General Explanation of terms, &c...................... 147 110. To determine when a function of one independent variable is a maximum or a minimum, and to investigate a criterion for distinguishing the one from the other................ 150 SUMMARY. xi ARTICLE PAGIK du 112. To determine whether all the roots of the equation d- = 0, necessarily render the function u either a maximum or a minimum........................................ 157 CHAP. VIII. Tangents, 8yc. to Curves referred to rectangular Co-ordinates. 126. General Introduction to the subject............... 180 128. To find the angles which a right line, cutting a curve in two points, makes with the co-ordinate axes.............. 182 129. To find the angles which a right line, touching a curve at any point, makes with the co-ordinate axes.............. 183 133. To find the angles in which a curve intersects the co-ordinate axes................................ 187 134. To find the rectilineal angle in which two given curves intersect each other................................... 189 135. To determine the points in which the tangent to a curve intersects the co-ordinate axes.................... 191 140. To find the equation to the tangent at any point of a proposed curve........................................ 200 143. To find the equation to the tangent of a proposed curve, which shall make given angles with the axes.......... 202 144. To find the equation to the tangent of a proposed curve, which shall pass through a given point.................. 203 145. To find the points in which the normal to a proposed curve intersects the co-ordinate axes........................... 204 148. To find the equation to the normal at any point of a proposed curve......................................... 208 151. To find whether a proposed curve admits of a rectilineal asymptote, and to determine its position........... 210 152. To find the equation to the rectilineal asymptote of a curve 215 155. To find the position of a curve at any point with respect to its rectilineal tangent.................. 219 159. Definitions, &c. of the Orders of Contact................. 224 162. To determine the nature and conditions of the contact subsisting between a straight line and any proposed curve.... 226 xii SUMMARY. ARTICLE PAGI 163. To investigate the nature and circumstances of the contact subsisting between a circle and a given curve............. 227 165. To find the nature and circumstances of the contact subsisting between a proposed curve and a parabolic curve of any order......2.................................... 229 167. To ascertain generally the nature and circumstances of the contact subsisting between any proposed curve, and a curve of the second order............................ 231 170. If the contact between two curves be of an odd order, they have only contact at the point of concourse, but if it be of an even order, they have both contact and intersection.......................................... 233 176. To find expressions for determining the radius of curvature of any proposed curve, and also the chords of curvature parallel to the co-ordinate axes........................ 236 184. To find the equation to the curve which is the locus of the centres of the osculating circles at the different points of a curve............................. 2 185. The radius of the osculating circle at any point of a curve is a tangent at the corresponding point of its evolute..... 255 186. The radius of the osculating circle of a curve varies by the same differences as the corresponding arc of its evolute.. 257 CHAP. IX. Tangents, 8-c. to Curves referred to polar Co-ordinates. 190. General Introduction to the subject..................... 262 191. To find the angle which a straight line, cutting a spiral in two points, makes with the radius vector................. 263 192. To find the angle in which a straight line, touching a spiral, cuts the radius vector................................. 264 196. To find the magnitude of the subtangent, and to draw a tangent to a spiral.......................... 267 199. To find the magnitude of the subnormal, and to draw a normal to a spiral................................ 269 200. To find whether a spiral admits of a rectilineal asymptote, and to determine its position..................... 270 SUMMARY. Xiii ARTICLE PAGE 201. To determine the position of a spiral with respect to its pole or radius vector............................... 272 202. Definitions, &c. of the Orders of Contact................ 274 203. To find the conditions necessary that a circle may have contact of the second order with a proposed spiral..... 276 205. To find expressions for determining the magnitude and position of the circle of curvature to any point of a spiral, and also the chord passing through the pole.............. 277 206. To find the equation to the evolute of a polar curve..... 278 CHAP. X. Singular Points of Curves referred to rectangular and polar Co-ordinates. 214. Points corresponding to evanescent ordinates............. 286 215. Points corresponding to infinite ordinates......... 288 216. Points corresponding to equal ordinates................. 290 217. Points corresponding to maximum and minimum ordinates 292 218. Points of Inflexion or Contrary Flexure............. 295 226. Isolated or Conjugate Points.......................... 308 229. Points of maximum and minimum curvature............. 312 231. Points in which two or more branches intersect each other 316 234, To determine the multiple points of a curve formed by the intersections of its different branches..................... 320 235. Points in which two or more branches touch each other... 325 236. To determine the multiple points of a curve formed by the contacts of its different branches........................ 326 238. Points of Reflexion, commonly called Cusps............. 327 243. To determine the nature of any proposed point of a curve whose equation is given............................ 336 251. To find whether a polar curve has a point of Inflexion, and to determine its position.......................... 342 CHAP. XI. Tracing or Describing of Curves referred to rectangular and polar Co-ordinates. 252. A variety of Curves traced or described...................0. 344 Xiv SUMMARY. CHAP. XII. Differentiation, 8yc. of Functions of two or more independent Variables. ARTICLE PAGE 256. General Introduction and Notation..................... 370 259. Given u =f (x, y), to find the developement of u' =f(x+h, y y+k), h and k being any two indeterminate magnitudes........ 373 263. To find the successive differentials of a function of two variables expressed generally by u =f(x, y)............ 380 269. To find the developement of a function u' of three variables x, y, z independent of each other, corresponding to their values x + h, y + k, z + I respectively................. 385 276. Elimination of constant quantities, and of functions whose forms are indeterminate................................ 393 281. Lagrange's Theorem, and its Applications................ 399 282. Laplace's Theorem................................... 412 CHAP. XIII. Maxima and Minima of Functions of two or more independent Variables. 283. To investigate the conditions necessary that u = (f(x, y) may be a maximum or a minimum, and to find a criterion for deciding whether it is a maximum or a minimum..... 415 286. To investigate the conditions necessary that a function of three variables may be a maximum or a minimum, and to deduce a criterion for deciding which it is............... 424 CHAP. XIV. Tangents, 4c. to Curve Surfaces and Curves of Double Curvature. 290. General Introduction to the subject..................... 428 291. To find the equation to a plane, touching a given surface at any assigned point.................................. 429 294. Of all the straight lines that can be drawn from the point of contact in the tangent plane, to find that which is inclined to the plane of xy at the greatest angle............. 434 296. To find the equations to the normal of a curve surface at any proposed point.............................. 436 SUMMARtY. XV ARTICLE PAGE 301. To differentiate the volume and surface of a solid, bounded by the co-ordinate planes and a curve surface whose equation is z =f(x, y).................................. 439 303. To investigate the analytical circumstances of the contacts and osculations which may exist between two surfaces defined by given equations........................... 443 307. To determine the radius of curvature of the section of a curve surface made by a plane passing through the normal at any point..................................... 449 310. To determine the normal sections of greatest and least curvature at any proposed point of a curve surface.... 451 312. To express the radius of curvature of any normal section, in terms of those of the normal sections of greatest and least curvature..4.................................. 452 315. To express the radius of curvature of the section of a surface, made by any plane, in terms of the greatest and least radii of curvature belonging to the same point... 454 317. To find the equation of a normal plane to a curve surface, through a point whose co-ordinates are x, y, z, the section of the surface by this plane having dy=mdx..... 456 319. To determine a paraboloid which at its vertex shall osculate with a proposed surface at a given point...... 458 321. To find the directions on a curve surface in which the consecutive normals may intersect each other....... 459 323. Conical Surfaces and their equations...........461 327. To find the equation to the conical surface which shall envelope another given surface...................... 465 328. Cylindrical Surfaces and their equations.............. 470 330. To find the equation to the cylindrical surface which shall envelope a given ellipsoid....................... 473 331. Surfaces of Revolution and their equations............... 475 333. Annular Surfaces and their equations.................. 477 335. Developable Surfaces and their equation............ 480 336. Curves of Double Curvature........................... 482 337. To find the equations to the tangent at any point of a curve of double curvature.......................... 483 Xvi SUMMARY. ARTICLE PAGRK 341. To find the equation to the normal plane, or the plane which is perpendicular to the direction of a curve of double curvature at any point...................... 485 342. To find a surface of given species which shall osculate with a curve of double curvature at any point......... 486 CHAP. XV. Miscellaneous Theorems and Problems. 344. It is required to prove the following formula: m m ((m - ) - 1)( _2) &c. MM L(m - $+ - (m - & continued to m terms = 1.2.3. &c m.................. 489 345. To investigate the law of the formation of the coefficients of the terms of the Multinomial Theorem, by means of the Differential Calculus.........4................... 490 346. It is required to decompose m --- into a series of fractions having simple or quadratic denominators......... 491 347. To find the sum of n terms of the series 12 sin + 32 sin 30 + sin 5 &............ 492 348. It is required to find the form of the functionf, so that f (x + ) +f (x -- h) =f (x)f (h)........... 493 349. To find the particular value of the fraction m am (F x) n+ (F x)- +- &c. (f x)n + ( f2X) + &c. when such a value is assigned to x, as causes each of the terms to vanish...494 351. To find the shortest straight line that can be drawn through a given point, between two other straight lines at right angles to each other.......................... 495 352. Of all triangles having a given base and vertical angle, to find that whose area is the greatest...............496 353. To find the radius of a circle such that the segment of it, formed by an arc of given length, may be the greatest possible....................................... 497 SUMMARY. XVii ARTICLE PAGE 354. To find the greatest quadrilateral figure that can be formed by four given straight lines taken in a given order........ 497 355. Of all spherical triangles upon the same base and having equal perpendiculars, to find that which has the greatest vertical angle................................. 498 356. To determine the greatest curvilineal figure of a given species which can be described in, and the least which can be described about, anothei given curvilineal figure....... 499 357. To draw a tangent to a curve defined by an equation between two straight lines drawn to two given points. 502 358. To draw a tangent to a curve defined by an equation between a rectilineal ordinate and a curvilineal abscissa... 503 359. To draw a common tangent to two curves defined by given equations........................................ 504 360. To find the locus of the intersection of the tangent with the perpendicular, drawn upon it from the origin, in any curve................................ 506 363. To find the locus of the intersection of two tangents drawn to a curve after some determinate law................... 508 364. To determine the nature of a curve which shall always touch any number of straight lines drawn after a given law 510 365. To determine the nature of the curve which shall always touch any number of curves of a given species, described according to a given law............................ 513 366. To find the length of the evolute of any proposed part of a curve whose equation is given...................... 514 367. In a polar curve to find the locus of the point of intersecsection of the tangent by the perpendicular, let fall upon it from the pole.......................................... 515 371. To find the locus of the point of intersection of the polar subtangent and tangent.................................518 372. To express the cosine of the multiple of an arc in descending powers of the cosine of the are itself................. 518 373. To find the content of the greatest parallelopiped that can be inscribed in a given ellipsoid.................. 519 374. To find when the volume of the pyramid, formed by the tangent plane to a curve surface, and the three co-ordinate planes, is the least possible............................ 50 C Xviii SUMMARY. ARTICLE PAGE 375. To investigate the nature of the surface which shall envelope any number of surfaces of a given species, described after a given law......................... 521 377. To express the radius of curvature of any section of a curve surface, in terms of its partial differential coefficients 523 378. To find the radius of absolute curvature of a curve of double curvature, the arc of the curve being considered the principal variable.............................. 524 CORRECTIONS To be made in the PRINCIPLES of thse DIFFERENTIAL CALCULUS. t'age 9, line 4 from bottom, for log (4), read by (4). 12,.. 2 from bottom,for 4-rr-.AN=47rIP2, reaed2xrA N=7rAP 2. 17,... 15 from top, for Intranscendental, read Interseendental, and the same in line 6 from the top of the next page. d3U h2 dnu f2 89,... from top, foar a, read &C. l d x3.2.3 xn IL1.3 c. i read diith3_ +&. + /i WP T. 1~2.3 axnn 1. 23.3. &. n 89,, 2 from top, for ax, read e + 7~'. d2P d2P 134,... 7 from top, for a read u ( a "(12 a~~ (IdX2 aX2 195,... 10 and 13 from top, Cubical and Semi-cubical miust change places. 19.5, L.. 4 from bottom, for AT=-,,x, read A T - (m +1) x, and in the next two lines, for AW= (m + 1) y, read AW 1 y, and for LW= —mny, read L W 200,... 2 from bottom, for x = 0, read x'= 0, and for, axis of x, read axis of y. 224, l.. 1 from top, for Orders of Contact, read Contact and Osculation. 233,... 9 from bottom, for even read odd, and in the next line but one, for odd read even. 267,... 1 from top, for a9, read a2. 270,... 2 from top, for P x 7-SP cos SPG, read P x = SP cos SPG. X 1 383,... 5 from top, for 2- read y and in line 3 from bottom for 3; / X'- 2 1 "-~ -- sec, read -C sec2 409,..~ 4 from top, for 24, read 2'6. 417,.. 8 from bottom, for be- ag, read a g - be. 429,... 6 from bottom, for x z and y z, read yz and xz. 442,... 7 from bottom, for XA M l- and Z/P =@, read XAM (p and Z A P -; and in the next line but two, for r d p, read r dO, and fin' ~ sin O dO0, read g sin G d p. 447,.. 2 from top, Jbr 1 + P2 + (z'- y) R, read 1 + P 2 + (z'- y) R = 0. 489,... 10 from top, for mm-4, read Xm-4. 493,... 5 from top, for cos 50, read sin 50. 496,... 6 from top, for (a tan 0 + b2), read (a tan 0 + b)2. d it du 499,... 3 from top, for d-, read 519,... 10 from bottom, for Cube, read Parallelopiped, and the same throughout the Article. 524,... 7, from bottom, for tangent, read normal, and tie same in tie next line. EXAMPLES IN THE DIFFERENTIAL CALCULUS. INTRODUCTORY CHAPTER. 1. WHAT are the inferior and superior limits of a+-bx a2 +bmx-c an a +b -9 - - and - -? cx+d' x' —eox+f c" - e A a ad b c +b Answer. a and a; -- and - — an d a; or o accordd c f 1-e as a is greater or less than c. 2. The convex surface of a right cylinder is equal to the area of a circle whose radius is a mean proportional between the height and the diameter of the base. 3. In a right cylinder, the radius of the base: twice the height:: the base of the cylinder: its convex surface. 4. The convex surfaces of right cylinders are to one another in the compound ratios of their heights, and the diameters of their bases. 5. The contents of right cylinders are to one another in the compound ratios of their heights and the areas of their bases. 6. The convex surface of a right cone is equal to the area of a circle whose radius is a mean proportional between the length of the side and the radius of the base..7. The convex surfaces of right cones are to one another in the compound ratios of the lengths of their sides and the diameters of their bases. A 2 8. The convex surface of a right cylinder: the convex surface of a right cone of the same base and altitude:: height of the cylinder: half the side of the cone. 9. The convex surface of the frustum of a right cone is equal to a circle whose radius is a mean proportional between the side and the sum of the radii of the bases. 10. The contents of cones are to one another in the compound ratios of their heights and the areas of their bases. 11. The surface of a sphere is equal to the convex surface of its circumscribed cylinder. 12. If a sphere and its circumscribed cylinder be cut by two planes parallel to the base, the intercepted parts of the surfaces of the sphere and cylinder will be equal. 13. The whole surface of a hemisphere is equal to three times the area of its base. 14. The convex surfaces of spherical segments are as the corresponding segments of the diameters and the diameters jointly. 15. The surface and solidity of a sphere are each two thirds of those of the circumscribed cylinder: required a proof. 16. The surface of a sphere is twice as great as the convex surface of the inscribed equilateral cylinder. 17. The whole surface of the inscribed equilateral cylinder is three-fourths of the surface of the sphere. 18. The whole surface of a cylinder circumscribing a sphere is twice as great as that of the inscribed equilateral cylinder. 19. The convex surface of any segment of a sphere is to the convex surface of the inscribed right cone on the same base as the side of the cone to the radius of the base. Compare them also for the hemisphere. 3 20. If the inscribed cone be equilateral, the surface of the spherical segment is twice as great as the convex surface of the cone. 21. If a right-angled cone circumscribe a hemisphere, the surface of the hemisphere: the convex surface of the cone:: 1: /2. 22. The surface of a right-angled cone circumscribed about a hemisphere is twice as great as that of the inscribed cone, whether the convex or whole surfaces be taken. 23. The surface of a sphere: the convex surface of an equilateral inscribed cone:: 8: 3. 24. The whole surface of an equilateral cone inscribed in a sphere: the surface of the sphere:: 32: 42. 25. The surface of a sphere: the whole surface of the circumscribed equilateral cone:: 22: 32. 26. A sphere is equal to a cone whose height is the radius and base equal to four great circles of the sphere. 27. A hemisphere is twice as great as its inscribed cone. 28. A right cone, sphere and cylinder of the same base and altitude are as the numbers 1, 2, 3. 29. Sectors of spheres are to one another in the compound ratios of their radii and convex surfaces. 30. Find the content and surface of the middle zone of a sphere. 31. If the altitude of a cylinder be equal to the diameter of its base, the whole surface is equal to six times the area of the base, required a proof. 32. If the bases of a cylinder and of a cone have the same radius as a sphere, and each of their altitudes equal to the diameter of the sphere, the solidity of the cone is equal to the excess of the cylinder above the sphere. 4 33. The content of a hollow sphere is equal to the frustrum of a cone whose bases are equal to the exterior and interior surfaces of the solid, and height is equal to the difference of their radii. 34. A sphere has to its inscribed equilateral cylinder, its inscribed equilateral cone and its circumscribed equilateral cone, the ratios 4//2: 3, 32: 9 and 4: 9 respectively. 35. The right cylinder and equilateral cone circumscribing a sphere are to one another as 2: 3, whether we consider their convex surfaces, whole surfaces, contents, bases or heights. 36. It is required to divide the area of a given circle into n equal parts by means of concentric circles. 37. Divide the area of a circle into n equal parts whose perimeters shall be equal. 38. Divide a given sphere into n equal parts by means of concentric spherical surfaces. 39. The whole surface of a cone is three times as great as the area of the base; find the vertical angle. 40. Find the radius of the sphere which can be inscribed in a given cone. 41. Compare the sum of the contents of all the spheres that can be inscribed in a cone, having their centres in its axis, with that of the cone itself. 42. Divide a right cone into two parts, whose contents shall be in the ratio of m: 1, by a plane parallel to its base. Also, that their convex surfaces may be in the same ratio. - CHAP. I. MISCELLANEOUS EXAMPLES. PROVE from first Principles the truth of the following formulae, as in the examples of Article (18). 1. If u-=aX2 +bx+ c, then =2ax+b. dx a b du 2 a b 2. If zu= - + cx, then d -- - + + c. 3. If u=-ax- bx2 +c, then-= - - 2b, dx 2 2 4. If u = a2 (b + x)2 - b2 (a - )2, then dut d =2 (a+b) {ab+(a-b) x}. dx du -ax 5. If=/1 u_ 2, then - = du 6. Ifu= ax2+2b +3c, - --- e dx V/ax 2+2b+3 c ' - x du 1 7. If u=, then _ -- /dx + 2)' _____2 du x (4a3 +x3) 8. Ifu= 9 then Va - o 2 (a3 + x3)j 9. If = -- then - =e e'+ 1 d7 (ex + 1)2 du 10. If u = sinl.cos 2,. - cos x cos2 - c 2 sin x sin 2, d z~ CHAP. II. MISCELLANEOUS EXAMPLES. BY means of the rules laid down and observations made in Articles (20)-(33), it is required to prove the following formula. du 1. If u =a'-b +, then- =3axc -2bx. 2. If u = axm -ba-l + cen-2 + +c, then dx du =qmaancl - (Sm- 1) b' S-'2 +(m -2) C m-3 3. If u = (2n - 1) C2n+1 - (2n + 1) x2"-1, then du _ = (4- 1) 2n-2 (2_ 1). 4. If u = (ac2n"- bx + c)m, then,du -i= n-n (a?2n b2n + c)mb-1 (2 an _ b) cn-1. dx' du 5S. (. )T(-= (1 + 02), -= -(1 - 2 + 3 2). dxv 6. If = (x - 1)4 (v + 2)', then dx j=(7v+ 5) (c-1)3( + 2)2. 7. If u = (a + x) (b + 2 x) (c+- 3x), then du d- 182 +(l2a +6b+4c) +3 ab+2ac+bc. dcx a+ 3x du a-b 8. If = --, then - --- b + x dx (b + )2 9. If = =- +- then —=3 - - (IV dvc \YJ / 7 to. if u - am du 2 m anaM, - fO. If mu = -- am, then aM' + a d -c (xm + am)2 If U=X2-a2 then d 12 a (a2 - 2) 11. If u -- a), then- =: --- --— ) (a2+ai2)3 T dX- (aO 2)4 a+5 bx du 20 b' 12. If =, then = — (a + bi)5' dx (a + ba)6 a + 3bx2 du 12b2X3 13. If u( 23= b then -(a (a + b2)3' dcx (a + baX)4 3 1 1 5 14. If = ax2 -bx2- + c7- ex-S2, then du i 1 1 3 7 d- =- p3aai2-b^^ a? 2 +5ea2 dx -- 1 3 2 1 du 9b2 4cx 3 15. If uZ= a + bx2 + cXa3 2 31 21 d 2 {a +b2.+ c cx32 1 2. 1 1 4 dz 2 7 ( 2,2. -3*3 ) 16. If u={ 1 -x +3-, - -5 a ~ -— 6 ---' _.2 -' dx 15 ~~4, du 17. If u=-(3 -2) (1 + ).- =6x/-^ 5 dx 18. a- du a-3x 19. If u=(a2 + x)) v/a du _,(a2 — 3X) l9 Iv/a,_$ ' 20. If u= (2 a2 + 32)(a,a 2), d -= 15 r32-/a — 2 21. If U(2 ad+a2)a2+i2,_ 1 3a'+xt dc. 2 v/ /a2 2 3 xX du 4 b2x 21. If u = (2ab 4- c ) ( ~ + 2 -' a2 2.= =22. If u(bx-3a)(a+b )3, = 421'. ~^22. If d S)(a+3 (ab 5b )3 8 23. If u =(3b -2a) (a +bx), d = 1b /ax + b dx 2 - du 24. If uz=(s b 2- a) (+ b2)2, =- 35b23 (a+bw2)2. 25. If u = (8a2- 4abx + 3b2) a+2b, du = L5bX dx 2V/a+bx 26. If u=$ x(a2'+2)Va2 _ -.s2, du a4- + a-x2- 4 4 dx V/a2 x2 ^wx,du a 27. If u: b,= then -= Va- b dx (a- bx2)' 28. If u= -_, then d-u x d x x2 V/a_ x2 x2 du ($ a — 2 ) x29. If u _a, then - = (- 2) A/Va-x dx 2(a- -x) 30. If (a+bx2) du 3a(a +bx2) 30. If u =, then = -- xo d x x4 2a + bx du b2x 31. If u= - then - = /a4 +b7 dx 2 (a+ bx)2 a 2- 2 bx du a2 32. If u=, then = a — 'V/ax _ dbx 2(ax?+ bxS2)' (x + a) du x -- 2 a /-a 33. If u =, then - =x- a V/'x-a dx x-a x-a x4 - 4a2x~ —8a4 du 3S5 34. If u =, then -—. Va2 - x dx (a2 - ) 3ax + 2bx3' du 3a2 35. If u =, then - (a + bx2)2 (a + b2) 2 9 b + 2cx du 4ac-b2 36. If u=. Va+b4 +cX2 dx 2 (a+bv+c 2)2 2x2 - a2 du 3a4 37. If = V --- —/ =.a 2 ca? dv 4 2a2+ 2 x- 1r du - (a?5 + 4 —3 +41) 38. If u=-= x+i, - 2 (-x+ 3_1) + ~ d,, (X + 1)2V/4 4+ If We _ _ a2 39. If u = - G V/a + v2 a3 -- W3 di (a 2 _ x2)' d 41. If u= \/ \/a2 _ 2 ' du a2 (a2 + 4x)2) dx x 2 (a2 + X2) l~ 3 a26? (a+) (a2 2)3 du a a2 - a + X2 dxvj (a2- 2) 2a'e-Gx ~Ja2+ 2zdu 42. If = -, ' -= ^/^ _ 3( a2x (a - x) /a2+2 (a3+a 4 2/a ( +, ( 3 + 3)! / 1+ 2 d q +2-2 _-4 43. If u=m ---- du 1 _ <a2 (1- 2,)/1 -44. If u= 5 (- -1)3 +21 (-1)2+3 5x} /x-1, then d u 35 x3 2 -1 1 -- + ~V/l-x2' 45. If u=, t then t/1+2 - /1 _ 2 du dx V/1- 4 (- V/ -I 4) du u-2 ax 46. If ux- ax -b= o, then d-= — dx 2bu —. B 10 4du u (a- 9,u) 47. If uw (a - x) = a3, then d- - d- B (a- x) du X (2u+Sx) 48. If u3 - ux - 3= then = d x 32u -- 49. If (2 + 2)3- 4a2u22 =0, then du S 3u4 + (6xc2-4a2) u2 + 3x d o \3 u + 3+(6a2 -4a2) 2 + 341 * 50. If u2 + x = 2 a V' u, then du x {a- 'V Zu, dta? ^u a + A/M2 u2 51. If: = (a + u)b v2 - 2, then du u (a + u) (b2 _ u2) drv x (ab2 — u3) 52. If u =-. + x2 + I3 + a4 + &c. to n terms, then du x"Z- 1 -- =1 + + xS + X + &c. to n terms -- dx x-.1 53. If u = v/2 + + + a2 + v&. ininf.: then du 2 d? / — /4x + 4a + ' 54. If u =V 2+ 2 ax + \V2 2+2 ar + &c. innf.: due 2 (iv + a) d. - V/42 + 8aiv+ 1 CHAP. Ill. MISCELLANEOUS EXAMPLES. IT is required by means of the Articles (34)-(46), to deduce the following results. 1. If u = a, - - then d- +-) log a. aCL dx a'v du 2. If u == ( - 1) a, -ax log a - (log a - 1) a,. 3. If u = ex"', then = exrn-i (m + z). dx 4. If u=ex (x3-342 + 6 x-6), x =U 3e" do a du a2"+ 1 5. If u= — log a a' a"' —I' dx (a -1)2 x- 1 d u (2 + 1) e 6. If u= e+ (+1 + 1 d. (x + )." f u e du - ex ++1 7. If ubrz' ~ e'+1 ex. ex - 1 du e" 8. If u=, ' - -- (e.1)V,e 1I + i dx (2e-x + ) e9 If u=e v - - (= 2 -2ex 1-x' dxl (l1x)\/1l-2 12 /&2 du (X - e-1) then- 2 10. If et r then da ( -1) V 11. If Ve X 7Z-+' then d U (x4+2) e dx ( 2- x + 1) V/A4 + + 1 ' du _/7 12. If u -2ev(-3.+6- -6), thenL d 13. If + T= ve T --— ' d v + 14. If u=2-log(x+)l) -lg(x+)l ' -2+2 du X 15. If u =log(- +,) then d+ - ( 3 du i-atd' 16. If u=log(x + 5) — log( + 4), dx - ( +x4) (x-4)2 then - x 18. If u=log ( )+ log )t dut 6 dxwA-2 0G3 _ x2 +x s 19. If u-(.2 a2) log ( ). + X then dZ a + x ^^. =2logV loCdav &\,ffL cI 13 a b a + b ao (a +, 20. If u= + log the a +x b +x a-b b v th du (a - b)2 dx (a + )2 (b + ) 21. If u = 3 { (logx)3 - (log x)2 + 2 log - } then -= 3 x (log X)3. dxd du 1 22. If u= log (x + 2/2 + ), then = /a + 23. If ~ = log (1 + a + V/) — ), = /2 a + * dx4 Id1 ac y v^2-"^ —^.25. If u=log a//+,2 + x2 du 2 a2x then = --- a4x dov a4 —_ 4 + I 3-en du 2X2 26. If u =log / --- then - - xs3- 1s ' 6 dx- 1 27. If u= log ( -i V I _ <2 d u x +V 2/1 x O2 dGx (22 -l) (- x2) du 2 28. If u=log tg hen = 3x/( 3(- "-) Va 2 o 2 _a du 29. If Vu = log -a — then d Ifu^lo+a dx 2a = a - 1 Oa 2 v2t2 + a2 30, If u =log V/x2+a2-, Va2.aY + Cx' du then dx 14 + / 2/ 3-_ 1 du X2 31. If =log --, then er - V/e3-1 ( d, 3 - 13 I/X2+4x+s dzu 32. If u=log V++S d-u x + 2 dx + 6 +n 11 6 6 -x+ 6 (X + 2)2 dzu.x V33. If /log -/)3 dx (x+l)(x+2)(x+3) 34.If u=log 2+X-2V/1+x+x<2 d1 07 du 35. If u =log ----- _ I - - a 1 +2 '/T du 1 +/ -a2 36. If u=log +bx ' d+ - x/a a-bw dx a-b. x -x 37. If ==log/a+bx-v/a dn ^ V/bn dx 2' a + bxn' 38. If =log i+/J +b du / b '/ax +&2/b b dx ax ba' Vl+'\/+T-l dxl 1 9. It ~==log _- _ +_ d =a du - 39. If u =log -- _ V\aa+x-Va-xG dx x\/a2-X2 \/1+ 2+ 1-2 du 1 / 41. If u = log -- - = a l+x /1 - G2 d 1 - v 1 -7 G2' 42. If ze = log -)-2 ---- (1 —, then du 1 / 3 dxr 1+x i-x+2 15 20 f I + A/2 du 1+a2 /+. 43. If u =log 1- x 'V d a 1-x dx 1 —2 ~ -x42 44. If u = v +aa ~ + a2 log ( + v + a), then du ---- dx \ a 1 Ib (c+bc3 du 3a_ 45. If= - -u- log, _ 3 a+b 3 a \ - - da - (a t b+b T' 46. If u=/a2+T2+a og Va, + then V a2+ 2 +a du 2V/a2 + 2 dx x _ _m _ 2 2 47. If =2 log I - -, log then 1 -VA V <1-~ du 1 _ 1 _ -.- log \ 1) a 82 + a 2 a a + V/2 +,x 48. If u= -- - log -----, then x"2 du 2aC dcv cx: '/a' +- xc a 2 /'/a2-j+ a - - 49. If u=2 + log, then du - %1 + a. >2 da <tc 1 Cv -ta+ O. If t = --— / - + log, + ---, then du 1 ' d -t d 2V/1.v + x 16 O 'x.- 1 3 V/I-J - I 51. If ~ = t -- lox then V\/I +, 2 V/l+ + du 1 d, x2(I + x) 52. If = (log m- ) vaS + 22- a log v/ +' a 2 v2 +V' 2 + a du x logx. dx - \/'a2 + -2 53a. If 2i / / a u+ 2 /a^V-a x2 -\/ 2a 53. If = 2 log --- 2 V + 2\/ ax - a2 +\ //2 a V+2/ - du 1 + ~Va + 2 X/ _2, then = - dv Va - v - VI 54. If u= 2 (a2 + 2)2 log - b (a2 + b2)a +X 2a2 <2,2 b21 a2+b2+V + (/ a2 + b) (aa+~)i(aaP~b~ig + _a b2 V/;~+~~ 9 (- a2+2) — (a+) log x- - -b du _ 2 - b2_ dx = x a b2 1 du_ 55. If u= _ log(dv/l-72+V/-1), -= v — +V \/aa2 - u2- a + / 'a - u2 '56. If- = log -----, then a U du u dx - /a2 q_ 2 CHAP. IV. MISCELLANEOUS EXAMPLES. IT is required by means of what has been said in Articles (47) -(54), to establish the following results. 1. If u = sin mx, then m- cos mx. du 2. If u = - sin x cos x, - -2 sin" 2. du 4. If u = tan xG- G. then - = tan2. 5. If u = cot x - tan x, then — = e dx (sin x cos x)" du 3 sin3 x 6. If u =sec3 —3 sec x, then - =dr= cos4x d cos X 7. If = V/cosec x then -= - o _ dx 2(sin x) du 8. If u=3Sx+ 3 cot — cot3x, - =s3 cot4x. du 2 9. If = 2x sin + (2 - 2) cosx, x=2 sin x. dC 10. If U = (2 + sin2X) cos du - - - sin3X. dx C 18 dx I1. If U = (2 + s cos'X) sinox, dx = 15 sin~x cos3x. 1 - cos x du sin x (2 - cos x) 12. If ut -, then - = cos~? dx cos3 2-3cos2x du C cos3 2 - -3 cos2 v du, 3 cos3x 13. If u - - =. -.in sin x dr x sin~ x cos3 — 3 cosx -3x sinx du 2 cos4x 14. If u=- - 2 sin s - dc sin x sin x du cos3x - sin x 15. If U = - - = 1 +tan x d (cos x + sin )2 du 16. If u = ec~ sin x, = eC~s (cos2x + cos x - ) dx 17 du ec - COS x- Cos2x) 17. If u = esec cotx, = se co - wxC Cos x - COCS3a d cos cos 18. If u=emm cosm-, - me' cos' x(cos - sin x). du 19. If u-=xmnesinx _ =,m-leSinx (e + - x cos X) dx du 1 20. If u=log (xeC~x), d= sin. d x'X du 21. If = eax (a sin - cos x), = (a2 +1) e sin x, 22. If u=d e log 1 -sinx log x}. /1 + sin'ax 'du 2 sin- Ix cos x 23. If u=logV 1si ' - =-1sin2nI -- i tandx -- sin2x [ / —tan x du 24. If u — Iog, Imv-t 0 ^1_^_1 ta tan $J 19 du 25. If u =2log sin + cosec"w, - =-2cot3. dx du 26. If u = 2 log tan c + sec2, = 2 sec3 x cosec x. dcv du 2 (4 sin 2x + 1) 27. Ifu =4logtanv7 —cot2cx, - =I. 2(in2+ dx - (sin 2g)2. 2 3 cos x x 28. If u =s + 3 log tan, then Sin cosiin2 2c du 2 dx - sin3 x cos2 X 1 -Z2 d 2 29. Ifz = sin —, 1 + 2 dx 1 + x2' du 1 30. If u = cos-' (2 - 1), - = x du 1 31. If u -= sin-l - = /, q-X2 dx 1T +X 24 — d1 ud 42 32. If u= cos -- c'4 +1 d- < -- x l 2 dx 1 -\ 35. Ifu~ot', /14 du4 I / cx du 2 34. If u=tan- --, -~ =. T' d2(1 2x + c') /- du 1 35. If zu=cot-1 ' // — -. tt dx 2 -_ V/ + +2- 1 du 1 36. If ==tan-l d (-2)0 dx 2 (1 +- 2) I ddu 2 37. Ifu u=cosec-'1, 2 tVl -2 d x2 38. I f. sfil 1 2 V ab du I 38. If u s111-+b V2\/a-b a + bx2 ' dx a+ bx' 20 2 1 /b du 1 39. If u -- tan-1 - = V~h / /ab- 2' dwa /ax-bx2 1 2a-bx du 1 40. If u == os -c -: = /a bm ' dxV N/bx -a 1 2 (b2-a) du 1 41. If uZ —vers-1 - - = 20 &/ca b2 2v 1 a-b6- dz 1 42. If u = cos-1 -- - - /ab a+b4 d x (ab + ) V< b + cxm du A/4 ac - b2 43. If u ==tan-1 = 4ac - - 2 d 2(a+b+ 2) biv-2a du V/a 44. Ifusi ==slin-' - - duc 3 45. If = sin-' (3- 413) =d./ da? x du $ 46. If u = tan- ( --, - '1I-2) d/ 1 — a2 47. /1 ~ a? u 1 I - 47. If u=cos1l ' d - =1 - 2 1 —ax2 dx 1 —aw2 1 —x2 ag bgX2 du xbfg\ ag2 48. If u- = sin-l -/ v bf + bg dm (f gx'),\/a b2 /b249. If u~tan-1 C^^^^-} ^ Va2 - b b + a cosx v d a — b cos v dled x ssin~- \ 50. If U == (sin- X13l)2 d - n- l $ --- 2 du 2 s1 ^ dx 51. If ufx- s du 1 sin 1- (~ 52. If 21 == --,,- 3 sin-lX3, 2 3 21 53. If u xetanl I ++ eta x sin1 /- du sin- x 54. If uz= S +logV1 - dx = _1 ^2)i-,/V^2 *&' dx (1-)O2 55. If u = (2x-tan-'x) tan-lx -log (1 + 2), then du t2 tan-lx dx 1 +x2 1 1 + V2 +x2 1 % 1 ' 4V/2 l- V/2? ~x2 2V/2 1- duz 1 da 1 - sin ( - m + n) x sin (l + n- n) x i-r1 —n l+ r + m - d x dt 1 58. Ifu = tan'-sin1-, - a dT - x\ 2 /a' 2{l+ sin-) 59. Given the differential coefficient of vers = sinx, it is required to deduce those of sin x, cos x, tan x and sec x. 60. Given the differential coefficient of sec x — tan x sec, deduce from it the differential coefficients of sin x, cos x, &c. 61. From the differential coefficient of each of the Trigonometrical functions, it is required to deduce those of all the rest. 62. Having obtained the differential of the arc of a circle, infer those of the sine, cosine, tangent, &c. of the arc. CHAP. V. MISCELLANEOUS EXAMPLES. I. Successive Differentiations as in (65) and (66). 1. IF u=-ax3-bx2 + cx - e, then d2 u d3u -- 6a x-2 b and - =6a. dx" dWX 2. If u-m =r nm+l- (m + 1) xm + 1, then d22 dx'u d2 =m (mn- 1) {m "'- (n- 1-) mj-2. 3. If u=x4 + 2a'3 - 2a + 2a 2 a3x - a4, then d3au C d4u -- = 12 (2 + a) and = 24. d2 u 4. If zt = ex, then --- = (x + 2) e, d xd3u d4u = ( -+3) e",.(x+ )e., &c. d V3 d.Z 4 d'u 5. If =X 2 log 2, then -- =3+ 2log, d3u d4u — =2 — 2x- — 1. 2 -2, &c. dx'" d xi4 6. Ifu = 1 - 2 sin" then - = - Cos 62 d X2 dau d4u a -- =in x, - COS= u, &c. d'u 7. If u==x sinx, then = - 3 sin - x cos x 23 8. If u = ev cos x, then =- e" - sin x do2 d3u d4u = — 2e' (sin +cbs ), d4 =-4e cos x =-4, &c. dx3 dx4 d3u 9. If u = sec x, -- = sec x tan x (6 sec2 X- 1). dx3 d2u 10. If = u tan, - 2 = 2sec2 (1 +- tan ). 11. If z= e"coti, d2- (2 cot3 x - 2 cot2 x +3 cot X-2). d52G 5 dxo 12. Ifu=sin-xd -- 9(1 —x2) -7 9 + 2. 5. 9,2 (1 2) 2 +3. 5. 7(1 -- 2) 2 d4u 3. 23S 3.24 3 13. If u = tan-Ix, == - -' d-4 (1 + <22) ( +,2)4 d3u 4 14. If u = (1 + 2) tan-l x, d -( divx (1 + xv)' d2u 1 1 15. If = sec L d -i-2 a= 2 v — i1 (72) 1 d4U 24 - 240<v:2 +-120X4 16. If^u=1~ 2 di4 (1 + ~2)5 1 dn" 1.2.3.&c.n 17. If = -,; =( --- )" d2 _ u ( - 2v) 18. If uZ -2t u+a2=o, = )d2 2vux 19. If aut'-xu-a-=, d -- ( 3u dIu 20. If u = (a- )"', find the value of dx 21. If u = x'a, find the value of. diva 24 d'u 22. If u = x'v log x, find the value of dx7 d"zu 23. If u =- xr sin x, find the value of d nU" d"~u 24. If u = eX cos x, find the value of 25. If u = em, sin x, it is required to prove that d-U eMV sin (x + no)) dru emv sin (x + '"P),where d = cot.'m. For the solution of the last six examples see Ex. 6. of Article (66). II. Elimination of constant Quantities and irrational or transcendental Functions as in (67) and (68). 1. If u + ax + b=o, the two derivatives of the first du du order are u-.~ +b=o and +a=O; and that of the ddv d x d2U second order is x - 0. d X2 2. If x2 - au + a2 = b2, the two derivatives of the first du du du. order are x-a — =oandxvFu ---= VuX- x2b2 ----; and dx dx dx-v du~ d2 U, the derivative of the second order isdu! - d- = 3. If c be eliminated from the equation 2 + y2 =cX, dy then y-px=xX/1~1 2, if p dx 4. If Z1- -y2+ y 1 - x2= a., it is required to prove 1 p3 that V1 +2 + a/\ m2 V I - -Y 25 5. If /1 -2 + /1 - y2 = ( -y), then will 1 p v1 _,a2 vi *' 8y2 6. If (w-y)(- (2a+ - y)3b4, then will (a- - + 2y)p =a + 2 x -y by the elimination of b. 7. If (y - v)m-1 (y + )m+l = a2n, then will 1P(1) y ) 8. If y =ax + c /i + x2, then will (1 +,2) p = + ty. 9. If y2 = a ( + +) -2 b (+ 2 V1 +s2) y2 then will py= a+ - + x I +?2 10. If xe =- c, then will xy + y2 =pw2. 11. If c(x +y)=-ey-' then x+y+1 =p( +y-1). 12. If ( +y) log - =xe;, then will C _y pxy=y + (x + y)2e 13. If log ay + - =0 then will py + +2y =0. a x +y 14. If log- =sin-", then will pax-y= -- /a?- y. a s /. 15. If tan- y= a log then will x? b (x - ay) p=ax +y. 16. If ay2+2xy-bab2=0, then will (xy - b2) p + by - y2=, xy +ay2 — (I +ay)p-=o and (p-y)3 o. 17. If y= xea, then will pwy (1 +log. D 26 18. If - = ae2x+-(2B2+2x +i), then will p+2y-=x2y" y 4 a-+ bx a2 19. If x2y —x=a tan, then will p + y2 + 0. 20. If y =etan " then will p (1 +2)= ( + ( + 2) e2x+ eTM +1 21. If y= e2x -- then will p= 1 — y2. 22. If y =a sin (x -b), then q+ y =o, if q= d2. dw2 23. If y = aev + be-", then will q - y o. 24. If 2y = ae + be-'" +c, then will q =/ab + p2. 25. If y =log sin (ax + b), then will q +p2 + a2 = 0. 26. If aey = X2- b2, then will p - q =p2x. 27. If y2+-ax +b=2cx log x, then qy +p2x=c. max 28. If y= mx log, — then mqx=3 = (y -p x) b (1 - ax) 29. If y = log (ev + e -), then will 2 x =log ( y and p2 + q-. 30. If y = esee ", then will y = (1 - x2) q — px. 31. If y=- e cos, then will q -2 p + 2y = o. 32. If y = ev sec x, then will (p + y) qy =p3 + y3. 1 p 33. If -, then will q 2+p=4y3. X V2/2 (y 4 a 4)5 34. If (I -x) y = ax +bx+c, then will da3y 3(15 -I2 y4. q= o, if y a= -. - 35. If y=e (a+bxw+cx2), then r- 3q +3p -y=Oo 27 36. If y=ae + be- + c sin (x +e), then will s - y =o, d4y if s - -. 37. If y = ew (a + bx) + c sin (x + e), then will 38. a,, /3 7' 39. nate c. -s- 2r +2q —2p+y=o. If y = ae"" + be~ + ce' l + &c. to n terms, where &c. are the n roots. of 1, then will d - = o. If +(-c)(y+ c, it is required to elimi If yx + (1 - c) (y + x) =- c, it is required to elimi 40. Eliminate a and b from. the equation ax2 + by2 = ax + by. 41. Eliminate sin w and sin 2 from the equation y = a sin x + b sin 2.. C 42. From y=aek+ bel - - cos nx, it is required to qmn eliminate the terms involving a and b, when k and I are the roots of the equation z2 + mz + n2 = o. 43. Eliminate the constant quantities and exponential function from the equation y = e' (a + bw + cx2 + &c. + IZ"'1). III. Developement of Functions, Sc. as in Articles (69)-(86). 1. If = (a — X)4, then u= a4 -4ax+ 3 6a2'' -4a + x4. 2. If u= (a2+-2)5, then will u = al'T+ -8 5asx2 l10a6x4 + 100a4xO 6 + 5a2XS8 + lo,1 3. If u= 2/1 +, then will 1 1 <'2 3 X3 3.5 x4 u=l + -x- --- + &c. 2 2" 1.2 2 1.2.3 24 1.2.3.4 4. If u = a2 (1 - 2), then will a2 62X 6 a2 X 6. 1 1 a2 6. 1 1. 16ax U-aI?+, - + ---- + ------ + ------ +.&C. 5 5.10 5.10.15 5.10.15. -0 28 5. If u =(a + x) -, then will 1 4x o102 20x 35 x4 - = + -'+ -- &c. a4 a a6 a' a8 6. If u = (a + x)~, then will u=-a multiplied into 1 c 1.2 x2 1.2.5 x3 1.2.5.8 t4 +23 --- + - C. 3a 3"2 lo2a2 33" 1.2.3 a - 3" 1.2.3.44a J axc a2X 2 a3 3 7.f it = e-a, then u =1 - + -- --- + &c. 71 12- 1.2.3 8 If u = log (a + bx), then will b x b22 b3 x3 b4 x4 u = log a + — - & alo~ a1 2 a 3 - 4 9. If log -ta b then will \a-boGJ (b b3 3 b' x b5 7 t7 u=2 + 5, + &c. \a 43 3 a 5 c 7 / 10. If u = sin (a + bx), then will 2 3 usina+bc 1 1.2 2.3 1 102 1.2.3 11. If u -cos (ax -b), then will cos b -a sin b - - cos b --- -a3 sin b - - &c. 1 1.2 1.2.3 12. If u==sin- -, then will: — - -+ + &c. a a 1.2 a x 2 X3 24 5 13. If u =tan, then u= - + - + - + &c: 1 1.2.3 1.2.3.4.5 14. If =tan- -, then will a 00: IV<?3 C5 G7 u5. -- --- + -- - &c. l.a 3. a3 5. a3 7. a1 15. If u=esi, then u + - &C — 1 1.2 2.4 29 g 2 16. If u=cosmx, then u=1- -- +(3rm-2) m — -&c 1 2 1.2.3.4 17. If Z sec, then will x2 5 x4 61 X6 u --- + + --- — + &c. 1.2 1.2.3.4 1.2.3.4.5.6 18. If u- log sec x, then will x2 24 6X6 u = -— +- - + &c..2 1.2.3.4 1.2.3.4.5.6 ew2 ex4 19. If u = eos, then will u = e --- - - &c. 1.2 1.2.3 2 X 4,X3 20. If zu e sec, then u =- -t x+ -- + + &c. 1.2 1.2.3 21. If u=asi " then will 2 3 u= + log a + (log a)2 - +{ (log a)3 + loga} 1 + (log a +4 (log a)2} +&c. 22. Expand to six terms by means of Maclaurin's 1 -x J Theorem, 23. Develope tan (a + bw) in a series of the form A + Bx + CX2 + &c. by Maclaurin's Theorem. 24. Expand log cot (450~- - ) in a series ascending by powers of w. 25. Expand etan to four terms by means of Maclaurin's Theorem. 26. Develope eC~o(S+-) by Maclaurin's Theorem up to the term involving x4. 27. If sin y = m sin x, express y in a series of the form A < + Bx2+ CX3+&c. 28. If tan y=m tan x, expand y in a series of the form 4 +B + C" (-+ &c. 30 29. By Maclaurin's Theorem expand sin (a+b4+cx2+&c.) into a series of the form A + B + 2 + &c. 30. Having given 1 + x sin y=tan y, find y in terms of x by Maclaurin's Theorem. 31. If x = m tan ( - nx), prove by Maclaurin's Theorem Z. m sin 2z that when - is very small, x -- very nearly. z 2 mn + cos2 32. Expand ea+b+x +&c. by Maclaurin's Theorem. 33. Prove the following formula: G,aG3 2 5 XI ' cot x.= - - - -2 - - &c. 3 3. 5 32. 5.7 32. 5. 7 1 a? 14x a 34. Shew that cosec x = - 4 -- --- + &c. x 1.2.3 1.2.3.4.5.6 35. By Maclaurin's Theorem find the developement of in descending powers of x. a3- 1 36. Apply Maclaurin's Theorem to express the arc of a circle in terms of the secant. O^ QG3,GX4 37. If my -xy=m, then y=1 + --- + -&c 3m 3 4m 3'm4 38. If my3 X3y- x3 = 0, then will m4 3m7 12m10 y= - - - &c. 39. If y3 - 2 y2 + x2y _ a3 = 0, then will a3 2a6 7 a9 Y -+ 5 — +- +&c. cy" XX X0X 40. If y_-3y+ _x=o, then y == -+ + a &c., 3 34 36 also, y = V/3/s- - W- &c. 6 81 1 2 4 and y = - - c' - -- &c. - l 1.2.$ e 3 31i 41. If y3 + a2y-2 a3 + axy- x3 = then will x x2 a 2 a2 y=a- - + -— &c. and y=x+ + - +-&c. 22 26a Y 9x 42. If y3 -ay + ay - 2X3 =, then will a a2 a3 __ _ 2 y=x I -- + -+ - - + &c., y-=a+ - + -- + &c. 3 3x 81 *2 2 8a <3 4 5 X6 and y = --- &c. and a a5 43. It is required to prove that 1 1 1 1 1 log ( ) = - (=) + 3 ( &C. 44. Find the developement of u- = e6e to five terms. 45. Given the differential coefficients of it is 1 + 9 required to deduce the differential coefficients of log 1 + x 46. From the differential coefficients of cos x, it is required to derive those of log cos x. 47. Given the differential coefficients of log x, to deduce those of log of log x, or of log2 X. 48. If u =f (), it is required to prove that xdu x 2du (aI d3 f(2t) =f (X) + d + +.2.3 +&c. vv2 dvl.^d 1.2.3 d^ xdu X2d2u Xa3d3 f (0) =f (X) - - + -2- - + &c. Oda t 1.2 dx" 1.2.3 dxa 49. On the same hypothesis will there result f =fsG) + du x+ d2u X" f) (~) dt6 r d+ Jr&C.: Z) ) d x 2 P 1.2.22 /a?\ du u d2u X2 f^ =f (.)d X'2 da 2 1 2.22 50. On the same supposition willf (mat) =f (v) xdu ixd2u 3d3u + (m,-1)-_ + (m — 1)2 - + (m_-1)3 -3 d +&c. f(?+ )-]Tf ')+ fd r1 + d1. 2 (1 + x)2 d3u x3 d 3 1. 2. 3 (1 ++ ) 1 (+27 ) dl 1+ -+ 1.2da? (+) 1.2.3d3 il-\-)+ 51. If u =x, then will 1.2 (xo + nx)n or (O + n m X1?? = Sxn + mx- nx + -v — M-2n2xS + (m - 1) (m 2) m-3?3+&C. 1.2.3 52. If = log, then will log 2 = log x+1 -- + - &c.: and log mn = log + (n1 - ) ( -1)2 + - (m- 1)3 - &c. 53. If u = sin x, then will iV2 sin x x3 cos x sin 2 x sin = - xx cos x- -- &c. 1.2 1.2.3 and sin nam (- 1)'2. (m 1)3a3 sin x + (m — 1) x cos x --- -- sin x -- cos -- &c 1.2 1.2.3 54. It is required to prove that (em -r=e) m 1 ~+ - (e + 2m +. ) - 1.2.33 e em. 55. If u =f (x), it is required to shew that (.) f( ) df ( + 2h) 2h df (x+3h) 32^'h &c dx 1.2 dxv 1.2.3 33 56. It is required to prove by Taylor's Theorem, that h h2 tan (x + h) = tan x + sec2xR - + 2 sin x sec3x - + &c. 1 1.2 57. Shew by the same Theorem, that 1 h x h2 sin-' (x + h) = sin- x (- 2- -+ -- sin'(+/,)=sin'+ (1-2 1 (1 (-,2) 1.2 2x2 + 1 h3 3.x (22 + 3) h4 + + - - 2.3. + &c. (1-)1 3 1 2.34 58. By the same means it is required to prove that 1 h 2x h2 tan-' ( + h) = tan- x + 2 1 +xI (1 qx)~1.2 2 (32-1) /3 2.3. 4x (2- 1) I4. _ &cd (1 + 2)3 1.2.3 (1 + 2)4 1.2.3.4 dnu 59. Given u = (a - bx)n, to find the value of"d-. dxu 60. Given u = (I- -)", to find the value of ddlu 61. Given u = (2e + v2)m% to find the value of d v, dn~ u u 62. Given u = (1 + -2) + 3"2), to find the value off x. d oo 63. Given u=(1,b )', to find the value of -- 1 dn~u 64. Given u - (a2s q- 2)m, to find the value of ~~1 d~~du 65. Given u = - m, to find the value of 65. Given u --- to find the value of - ( d dw t 66. Given u. =, to find the value of (1 -- 2,- + s3x2) d+,n E I d"u 67. Given 1,to find the value of 5 (a+bx +cx')' dx' For the solution of the last nine Examples, the reader; is referred to Ex. (2) of Article (82). 68. If (y' ax) - ay + x' = where x is the indedx d x pendent variable, then will (x- ay) - -ax + y = 0, where y dy is the independent variable. dy /dy\3 d'y 69. If - - I- I=X -Y where x is the independent dX -dx d I dx'2 variable, then will x - + d 1, where y is the indedxy2 d y pendent variable. 70. If Y, +2(xd =Y, where x is the independent X2 d'x /x\' d d2X dX x 3 dxt variable, then will y + y - = 2, where y is the indy' Ydy we i dependent variable. d'2y dy tday\2 71. If x +x - =0, where x is the ind x dx \dxj d'x fdx\' dx dependent variable, then will x — + -) = x where ay \y dy' y is the independent variable. d2y X dy 2 I 72. If (x +a) a + x 0, where x is the dx' b \dx dx independent variable, then will d'x (dx)2 x (dx) (,,v + a) -d-j + -I = ~0, dy dy b \dy where y is the independent variable. 73. If (dy' ~ dx9 + a dx d y = 0, whe-re x is the independent variable, then will (d y2 + dx')- adyd'x =0, where y is the independent variable, 35 d'x i/dX\' dX 74. Iff + - =0, where y is the inded y2 x d~ y/ d d 2y d y d \"=, 2e pendent variable, then will x - (d 0, where x dX2 dx \dx/ is the independent variable, 75. If x be the independent variable in d3y d'y dy d'- 3- 3 - - = 0, dX3 d XP dx find the equation when y is the independent variable. d 3y d2_ 2dy 76. Transform al - bxd + cx —x'y=0, where dX' dx' dx x is the principal variable, into a formula where y shall be the principal variable. 77. If x be the principal variable in d'y d'y d2y dy - 2- - 2 3+ 2 -- 2 - y = 0, dX' dx' dx' dv find the corresponding equation when y is the independent variable. The last ten Examples are dependent upon the principles explained in Article (83). 78. If x V - ay = 0, where x is the independent variable, dx dy then will T - ay = 0, where z = log x is the independent variable. 79. If 0 be the independent variable in d2y dy -- +2yo +secC20=0, d 02 d thnwil d'y dy then Will (1 +x2) + 2 (gX+ y)- + 1= 0, wherex tanO0 dvx dv is the principal variable. 36 d2y x dy y 80. Change - + - =0, where x is the de2 1-x dx 1 - v2 independent variable, so that 0=cos-lx shall be the independent variable. d4y 4 d'y 12 d2y 24 dy 24 81. If d - + --- — + y = 0, where dxv4 z d x3 xZ~ dx x dx i4 x is the independent variable, find the equation wherein z=logi is the independent variable. d2 y dy 82. Reduce the equation a + b - + cy- 0, where d ci dx_ dx is considered constant, into an equation wherein V/dx2 + dy2 is constant. 83. If a v/(d V)2 -+ (d y) = ds', where ds = V/dv + dy' is invariable, it is required to find the corresponding formulae when x and y are respectively the principal variables. d2x d3x 84. If ydx be invariable, and for —, 39 &c., there d '' di"v be substituted - p2_-qy be substituted - - y, y -, &c., a homogeneous function of an order above the second, wherein neither x nor y is thS independent variable, will become a function of y and its differential coefficients relatively to x: required a proof. 85. Apply the formula d d dy to find the diffIrendx dy dx cos x ( cos x) tial coefficients of u = (ax) csx and a(Cs) 86. From the same formula, if y = a (2 cos 9 - cos 20) and x = a (2 sin 0 - sin 20), deduce the values of dy and y. 87. If =f(td and ), it is required to express 87. If v =f (t) and y= (t, it is required to express dy d2y *' d2, &c. in terms of t. dTX d"' The last ten Examples are illustrative of the principles explained in Articles (84) and (85). CHAP. VI. MISCELLANEOUS EXAMPLES. 1. IF Zu=. ---...; when x = 1, 'tu3. X3- 22 + 2 — 1 3- 19 + 30 7 2. If u-= - x+; when xc=2, u=-7, and xa-~ 2-2 9x + 18 5 4 when x= 3, u =. 3 1 + x — x 2 -- x3 3. If u= - + hen t 1+2 2 + 2 -2- +x 2 z3 wh + =1. x —5 x2 + 8x - 6 4. If uz= 3; when xv= 3, u =5.. -I2= -6 + x 1 2 6.-6m2,51 1 2 5. If --- — 5; when - x= + _n, e= —, m4- 4x3 + 8x2- 16x + 16 6. If u= --- when x=2, u='8. x- 6xa + - 3lSx- 12x t — 4a x ~3x3 —732-7 ~27x -18 7. If u=- -x- when x-, u=10, *4 3X3 - 7 x22 + 27X -18 1 and when x=-3, u = -. 10 5 a 4- a a -4 '- a4x - a5 8. If u= +a <I + 2ax3 + 2a2x3+. 2 a3 x + a4' when x= -a, u= — 2a. a (T +- c)-2 acx / 9. f'l=,?- when t = c,? t b (,2 + cO) - 2 bcx b 38 52 1 1 10. If u==; when x1, =. - -. 2-1 X-1 2 a - v/an - x 1 11. If u=; when =0, u - 0n n an-1 i^a T 4-x- 2a 12. If u= '~/a+3 — X/-/-a 13. If u= - ---- ---; when x=a, u=- Va+2x-x /3a e e2x + 1 - e" - 13. If u= 2; when x = O, =-1 e2' — 1 x e2x + e - 2 e2+2 e 1 14. If u=; when x=o, u=-. (e- 01)3 6 an_ arn 15. If u = -; when x = a, u =a". log (n)- log (0 x 1 16. If u= -; when a=l, u=-. x- 1 log x e21 — 1 17. If u=e - ( ---; whenx=0, zu==2. e" log (1 + x) If e' - log (1 + c) 18. If u=; when =o, = 0 1. x2 alogv -- 19. If u= -; when x=, u=loga-1. log tx 20. If u=(1 -x) log(1. —x); when = l, u=0. cos x - cos 2x 21. If u=; when x=o, u=-. COS 7 - COS 3 22. If u2= (sin X)2 + Sin X-i wlen x = 300, u 3' 2(sin X)2 - sin x -+ 1 22. If ==; when x=o30~,= - $. 2 (sin )'" - 3 sin 3a + 1 23 If u tan x + sec x- 1 2a3. If n=; when x=c u =. tan 3a - sec x -+- 1 39 sin Ix+u (sin ) sin2 + 24. If uz= ( -- - -; when lv=O, u=4. 1 - cos x tan x- sin x 25. If u= -; when =0, u=-' (sin X)3 1 +cos 27rx 7T2 26. If u= 2e; when x=, =-. Ife = e2-~x e -_ esinx 27. If u= -; when = 0, u= 1. x - sin x.fel - e-,)2 28. If u= -; when x=O, = 2. 1 1 X COS < 29. If - - -; when a=0, uz= sin 2X 3a 1 1 l 30. If u --; when =0, u=-. 2 X2 2x tan x 6 X2 - a2 '7r X 4 31. If = -- tan -; when =a, u= - - x2 2 a Ir Wxa 1 2 32. If = sec -log -; when x =1, -—. 2 Bx 7r tan (a+ s.) - tan (a - x) when x;33. If ~== -; when m =0, tan~1 (a + x) -tan- (a -x) = (a2 + 1) sec2 a. 34. If u= sec-) vers 27rV; when x=1, u=8. -l 35. Ifu -m tan - sinx; whenx=-, u=-1. 2 '2 cos'- (1 -- x) 36. If u=-; when x=,0 u=1. v/2X — x2 40 37. If u = -.- -r; when =1, u = -1 2X2 (1- ) 2 tan r 38. If _b_ log tan x 38. If u=log ta --- — -; when x = o, u =1. log tan 2 x a 39. If u=2 tan —; when x= oo, u=a. a4 sin ax - b sin bx a - b 40. If u = -;when x = o, u= - C' sin cx - e' sin ex c -e 41. If= -; when x-o, u-=12. log (l+ l) x 42. If u= -- ~; whenx=o, u=1.,v2 (x - l) 6 (aT - a-) -- m- a 43. If u=;- when x- - a, u -. (I a-2-a)- 2 1 - x 2+ -log x 44. If u; when- v=l, u= +=1. 4 If a-2w4-aa2 316 45. If u= ----; whenx=a, u=-. a - / 2 a<2 -9 aw 2 46. Ifu= ---- -; whenm= a, u=. 48. If u= a-vx-aloga+alogx a8. Ifx/ u=2 -; when x=a, =-1, 49. If u/ (v -c)V 'v-V — x.. +. wh^c 41 50. If u =JZ a. /a2 + ax+ - / + 2 a -t u = -; when x= 0, /a + - - \/a — x a /4 a 34 + 4x -. ax- a2 51. If u -; when = a, = V'2a +2x~-a-x x/2 a2 + 2G ax -- a xt-a+ ~/2a-2=a2; 52. If u=; when =a - u=l.,/8 2_ a2 a - V- _ a2 - ax - X2 + Ga 53. If u= --- when c=a -- / -a2:,/3 a3 -2 x4-ao /ax4 54. If u= -; when x=a, a -/ax2 81 u= - a, by the methods of Articles (103) and (104). 56l. / + a - \/a a- + Va2-a 55. It = I —; when x= AX -a2+a3 - aZ+7a 56. If u =....; 2 a. t=-0. = a, Va2x6 + 15a45<?3-V/15a4x?6 + 17ax: 160 when = a, u — =, — by both the methods. 693 a 57. In y4- a2y2 + 2a22- x4 = 0; when = 0 and y 0, the values of Y are + \/2. 58. In (y 2+ a)2=2 (a2+2 ax -x2); when x=0 and y=0, dy the values of _,2)r; when and o the values ofn - are + - and o o, values of 2a r e+ a d. CHAP. VII. MISCELLANEOUS EXAMPLES. 1. IF w=a2 -2,ax+b2; when x =a, u will be a minimum,. whose value is b' - a. 2. If u-X2+ ax +b'; when x= '-a, u=Vb.-+a' will be a minimum. 3. If9x-X2 ~24x + 16; when x=29,~ u=36, a maximum; when x = 4, u =32, a minimum. 4. If U=x'-_18X2+ 96x-20; when x=4, u=140, amaximum; when x = 8, u = 108, a minimum. 5. Ifu /- 28ax+ 8ax-9q6a'x~.+48a4; when x=a, u=iia4, a minimum; when x=2a, U=i6a4, a maximum; and when x =4a, u =-1i6a a mninimum. 6. If u =12x'- 45&4 +40x3; when x=,u=7, a maximum; when x = 2, u =-16, a minimum; -and when x = 0, u = 0 is neither a maximum nor a minimum. 7. i-f u=&6-6ax'+9ax'x+a6; -when x=o,u=a6, a minimum; when x = 2 a, u = 17 a6,- a Maximum; and when x=3 a, u =a, a minimum. 8. If u=10x6-12x'+1.5x4-2O'o0+2o; when x=0, u =20, which is neither a maximum nor a minimum,; and when x =1, u =13, a minimum. 9. If u= a-2b~ A-x when 6 + n /m a2-b' nb~ + (ma'-b2) (in- n2) m, m in-n' u =Mi a maximum, or a minimum, according as m is less or greater 43 than n', provided m- n2 and ma' - b' have the same sign; and when b n ma2 -2 nb- (m a -b2) (m- n) 0 = - - - - -- mm m -n m a minimum or a maximum on the same hypothesis: also, when m-n2 and ma - b2 have different signs, the values of x and u become impossible, and consequently by (115) there can be neither a maximum nor a minimum. a a If m=2, n=1 and b=o; when =-, = -- -a 3a a minimum; and when x = -_/, - = a maximum. 5/V,/ 2 x 10. If u = 24x'A - 135x4 + 2803 - 270x2 + 120x + 180; when x-=, 2u=1993, a maximum; when x=2, u7=188, a minimum: and when x =1, t=199, is neither. 11. If u = (x - a)"; when x = a, u =, which is a minimum if n be even; but neither a maximum nor a minimum if n be odd, as appears from (112) and (113). 12. If u=x2 (a-.)3; when x=0, u=0, a minimum; when x = a, u =, which is neither a maximum nor a mini2a 108a5 mum; and when x =, u =, a maximum. 5 3125 13. If u = (x - 1)4 (x + 2)3; when x= -2, = 0, which 5 is neither a max inm nor a minimum; when = -, 112"."7 i24. 9'3 u = a maxinmum; and when = 1, u = 0, a minimum. 7 14. If u=xm (a- x)"; when x=0, u=0, which is a minimum when m is even, but neither a maximum nor a minimum when m is odd; when x = a, u = 0, which is a minimum when n is even, but neither a maximum nor a. Cb qa m?z am + n ma mmnfar minimum when n is odd; when x= —, = ---m + n (m + n)+)m which is a maximum: as appears from (114). 44 a + a + ITx" 15. If u -----; when x=o 0 = ua, a minimum; a+x and when x= a, u = - 3 a, a maximum. 16. If u= ( +); when =-2+/7= 1+7 =2.220 O1 +1^2)2 1 16 17-7 ~/ a maximum; when x 7= -- /7, u = = —.0950, 16 a minimum, and when. = - 1, tu = o, is neither. (x + 2Q3 +7 17. If u= ( + 2); when x= - 5, u - —, a maxi(X + 3)2 4 mum; when x=-S, u= 3 o, a maximum; and when x=- 2, = o, which is neither a maximum nor a minimum. ( -1)" n+m 18. If u=; when x-= —, and ma and n (x +I)fl' n-imn are both positive whole numbers of which n is the greater, t n mn -m n-r u =- )-,7 a maximum; but if n be less than m, m \ 2 =u = - -) which will be a maximum or a minimum, nn n-m_ according as m - n is an odd or an even number. If m =3, and = 2; when x= -5, u= -13-1, a maximum. -2 x-,v + 1 \ 19. If u= +-; when =0, u= -- l a maximum; X +x —1 and when x-c =, 2u 5, a minimum: for these see (123). 2x -- 3x 7+ 2' a minimum; and when x= =-2/, u - 12 /2- -179 a maximum: as may be proved by (123). 2 1. I f u = ( x ' ~ 1 ) 21. I f u= -;2; when x=, u =2, a maximum; and when x = - 1, = - 2, a minimum: see Article (123). 45 22. If u =, —; when = — 2 = + a maximum and a minimum; and when = + (1 p ), u = ~ 2, a maximum and a minimum: see Article (123). (a + x) (bc), - 23. If _(+ )) (b +) (+) (b+x) (a-()- (b- )-; (a- )) (b - ) - a + V-b 2 when x = /ab, u = - ~a minmum; \V^/ ' a - Vbinu \/a-/b\ when x== /a- b u - (-<'7 t, a maximum. a IV " a a 24. If = -; when x=-, u==e, a maximum. \x)~ e 25. If u=V/a +-~+ ^- + -); when x = /ab, u = 2-la (a +- b) a minimum; and when x = - x/ab, u =2 a/ (a + b), a maximum. 26. If u=; when x e, u =e, a minimum. log x \ 2 a maximum; and when x=sin -l - -, u- a minimum., -r ~-a 28. If u = cos " s (a - ); when x = m + 1 M_, (7+ m4a\ u = cos -- (7r - a) cos' +, a maximum. m+1 /nf 1 b3 29. If u = a secx + b cosec; when sin =- i- -2 2 V a3 -+ 53 2 2 2 u F +(a3 + b3)b a minimum and a maximum. 46 30. If u = sin x vers; when x= 3w, u = v, a maximum; and when x = o, u = 0, which is neither a maximum nor a minimum. 31. If u = sin x cos2 x; when x = 900, ' = 0, a minimum; and when x=cos- /3T, 9u=2V 3, a maximum. 32. If u =cos xsin (a - x); when x=a + 45~, u = -2 ( -sin a), a minimum; and when x = a - 45~0, u-= -(l sin a), a maximum. a+ 33. If u = e' cos (x - a); when x = a +, u= e V/2 57~ a+V2 e25 W a 34. If u=;when -=a —, u=V /2ea+T, a e x 7r r- n5X sin (x-a) 4 minimum; and when x = a+ - u e 4 T a maximum. 4 35. If u2-x2u+ X —x= 0; when x -— l, u=1,' a maximum; and when x= -1, u =, which is neither a maximum nor a minimum. a 36. If 3 - a2rx + x= 0; when x= - - =- a V/3 3a -a 2~ a minimum; when x = u, - = a, a maximum. V/ 3 32 37. If u2 + 2X2u + 4x -- 3 0; when x== —, u= 2, a maximum; and when =1, u= - 1, which is neither a maximum nor a minimum. 38. If 4 -4xum+x4+2=+; when x=l, u=1, which cannot be considered either a maximum or a minimum, because the values both immediately preceding and following it are impossible; also when x= — 1, u = -1, to which the same observation applies. 47 39. If u y/a +x-2x; when xw= oc, u= 0,:which, according to the criterion, is neither a maximum nor a minitmum; but since it cannot become negative, u =, may be considered a minimum. 40. If u"q+mx + a2 b + q-nx2 = o; when x 2nb + m\/n2 + (2n- 4n) zna when- -x = n (m2-4n) - mb - 2 /nb' + (m - 4n) na u ' --— n ---, a maximtum; m" -- n 2nb-m/nb 2 + (m2 - 4n) na2 and when x = 2 - n (m - 4n) -mb + 2V/nb2 + (m - 4n) na2 = ---, a minimum. m - 4n b 1 If m = o, = —, and u = + -- Vnb — 4n'2a2, which 2n - 2n are respectively a maximum and a minimum. If n = o, = oo, and u = —, which is neither a maximum m nor a minimum. m a + b2 m a2" _b2 If m= 4n, x= - mb, and u=,b which mn b -m b is a minimum. 5 41. If uu = 2x — x2 (1 - x)2; when x = 1, u = 1, a maxi9 2869 mum: when x= -= u= --, a minimum of the first kind; 25 3125 and when x =1, u = 1, which is also a maximum of the second kind. 42. If =cos x +cos2x+cos 3x; when =O, u=3, a maximum; when x cos -1 - - u= 27 -\ 6 27 a maximum. 48 43. If u4-4a2uz+x4;o; when x=a/3, u=a/i27 -a maximum; and when x = - a/3, - a x/27, a minimum. a2 a - a u = + /a2 - ab, which are both minima. _ /l + ~ +2 a\/ + /1 + -2a Vt 45. If = --; when IV1 + + 2 a\/ - V/1 +t - a\/,tV' 1 + /-a2 a x -= 1u --— 1, which is a minimum. a 46. If a = +1a-; when=- log (a —)og a log a 1+ log {(a - 1) log a} ilo =a= cga, a minimum or a maximum aclog a cording as a is greater or less than 1. 47. If u = a (cos )2 +b cos (a —) }2; when a a - tan a) = -1 tan tan at 2 ^ \a+b I f ^a i lfa b 2V a a -b u= a {cos [- I tan1 ( b tan a } +b cos[ + Ltan1 b tan a a maximum. L2 " \a + b 48. If t = (tan x)1 {tan (a- x) }'; when a a.- ) \ x = - f - tan( -- tana, u= tan - + tan -— tana)} x { [2 2 \nm+n J {a [ ~ an (r a)n, aam a, m aximum t - an — tan-1 --- tan a a maximumi \ i.22 \m-}n J CHAP. VIII. MISCELLANEOUS EXAMPLES. IT is required by means of Articles (126)-(189) to effect the solutions of the following Theorems and Problems. 1. If x and y be the co-ordinates of a parabola, whose latus rectum is a, corresponding to the arc s, then will ds N/a + 427 1 2+y - I Ja+4x - ixa2+4y2; d-~ =2 + -4y2 ds /a+4x 1 / — and - == = a2 + 4y2 dy a a 2. In a common cycloid, the radius of whose generating circle is a, it is required to prove that ds /2a d2a dax X dy 2a-x 3. If C2y =x-, be the equation to a curve; then will the secant, passing through two points whose abscissa are a and b, intersect the axis of x in an angle whose trigonometrical tangent a2 + ab + b2 is -: required a proof. c2 4. Find the angles in which a secant drawn through two given points of the curve whose equation is y =ax2 - bx +fc, intersects the co-ordinate axes. 5. Determine the angles in which a secant to any curve of the second order intersects the co-ordinate axes. 6. Find the angle contained between the two straight lines drawn from the vertex of an ellipse, to the extremities of the latus rectum and axis minor. 7. Find the angle which a straight line, equally inclined to each of the co-ordinate axes, drawn through the focus of an ellipse or hyperbola makes with the distance from the centre. G 50 dy 8. If p=; then may the inclinations of the rectilineal dx tangent to the axes of co-ordinates be determined from the p - 1 equations X= sin - P -= os- //1 pc2 lCO + / 'p2 9. What are the inclinations to the axes, of the rectilineal tangents at the extremities of the latus rectum of each of the conic sections referred to rectangular co-ordinates? 10. At what angle does the rectilineal tangent of the curve whose equation is ay =-a2x + x, meet the axis of x, at a point where x a? 11. Find the angles which a tangent to the cycloid makes with the axes; and prove it to be parallel to the corresponding chord of the generating circle. 12. Determine the inclinations to the axes, of a tangent to the trochoid whose equation is y= ma vers-' - + -/2a -, a (m +-1) a at a point whose abscissa is 13. If any ordinate of the curve whose equation is y' —3axy- =- 0 be drawn; prove that the sum of the trigonometrical tangents of the angles which the lines touching the curve at the points of intersection make with the axis of x =0. 14. Find the angles at which the cissoid of Diodes is inclined to the axes, where it intersects the generating circle. 15. Determine the points in a circle at which it is inclined to the axis of x at the angles 30~ and 60~; and find the distance between them. 16. Required the inclination to the axis of x, of the curve whose equation is log (a 4 2- y2) = + log y, a a aa at a point where y= -7 V/2 51 17. In the curve whose equation is al3y= (b3-a-3), it is required to find the points at which the rectilineal tangents make with the axis of x, the angles of 30%, 45~, 60' and 75~. 18. If -+ -=1, be the equation to a curve; it is ay bc required to draw a tangent to it, and to find the points at which it is parallel to the axes. 19. In the curve whose equation is ay= VG/'a - Ga, find the points at which the tangents are equally inclined to both the axes. 20. Shew that the rectilineal tangent to the curve whose equation is (cX -2) y = ( - 1) (x- 3), can never be equally inclined to both the axes. 21. Prove that the curve whose equation is-(a2+c2) y=b2x, intersects the axis of x at the origin of co-ordinates in an b7 angle whose trigonometrical tangent is - 22. Required the angles in which the curve whose equation is y2 + ay = ax - x, intersects the co-ordinate axes. 23. Find all the angles of intersection with the axes, of the curve whose equation is y = a sin x + b sin 2 cx 24. Find all the angles in which the curve whose equ/ation is (x y2)2 =2a2 (x2 -y2)) cuts the co-ordinate axes. 25. Prove that the curve whose equation is 4by=4abx - (a2 - 1) x2, cuts the axis of x in two angles supplemental to each other. 26. The equation to a curve being (a-'x)2 (x2+y2) = -ay2; it is required to find the angles in which it intersects the axes of x and y. 27. Two given rectangular hyperbolas have a common vertex and their axes perpendicular to each other: find the angles in which they intersect each other. 52 28. A tangent is drawn to a given ellipse which cuts the circle described on its axis major at an angle of 450: determine the co-ordinates of the points in which it intersects the circle, and thence shew that no such line can be drawn if the eccentricity be less than 29. The centre of an ellipse coincides with the vertex of a common parabola, and the axis major of the ellipse is perpendicular to the axis of the parabola; find the eccentricity of the ellipse when it cuts the parabola at right angles. 30. A parabola and hyperbola have the same vertex and principal axis; it is required to draw a tangent to the former which shall cut the latter in a given angle. 31. The vertex of a parabola is A and the co-ordinates of any point in it are AM, MP: in MP a point Q is taken always equidistant from A and P; it is required to find the angles in which the locus of Q cuts the axis and the arcs of the parabola. 32. Find the angles in which two concentric ellipses having their axes in the same directions intersect each other; and explain the result when the ellipses become similar. 33. The focus of a given parabola is situated in the centre of a given circle; find their angles of intersection and the length of the portion of the corresponding tangent of the parabola included within the circle. 34. Find the same when any given point in the axis of the parabola coincides with any given point in the diameter of the circle, their principal axes being in the same directions. 35. Prove that the rectilineal tangent to the cissoid of Diodes cuts off from the axes of x and y, portions represented ax a x \2 by- and -a respectively. 3 a —x 2a- x 36. In the curve whose equation is ym_= axt- + m, the portions of the co-ordinate axes cut off by the rectilineal tanax ay ent are ax and — ay: required a proof. (m-1 a- 1a +-mx m (a - x) 53 37. Find the distances from the origin at which the rectilineal tangent of the curve defined by the equation ay"+"= bx (a + x)', intersects the co-ordinate axes. 38. Determine the points in which the rectilineal tangent to the curve defined by the equation x'y2- (a2- x2) (b -x)2, intersects the axes. 39. In the logarithmic curve whose equation is y-=a", the distances from the origin at which the rectilineal tangent 1 - x log a meets the axes of x and y are and ( - log a) y, log a and the subtangents are and x y log a: required a proof. log a 40. Find the points of intersection and the angles of inclination, of the rectilineal tangent to the co-ordinate axes, when x2 + y2 = 2/2- y is the equation to the curve. 41. Determine where a rectilineal tangent to the quadratrix of Dinostratus defined by the equation y = (1 - x) tan I r x, intersects the co-ordinate axes. 42. Find the subtangents in the quadratrix of Tschirnhausen whose equation is y = sin -7rx. 43. Determine where the rectilineal tangent to the cycloidal curve defined by the equations x= a (1 -cos 0) and y = a (mO + n sin 0), intersects the co-ordinate axes. 44. Find expressions for the subtangents of the cardioide defined by the equations x =_ a (2 cos 0 - cos 2 0) and y = a (2 sin 0 - sin 2 0). 45. If the equation of a curve be xy (x + y) = a3; it is required to shew that the rectilineal tangent meets the axes at distances from the origin represented by 3a3 3a3 -^- - and 2wY + y2 2xy + x2 46. If the focus of a parabola be the origin of co-ordinates, it is required to prove that the perpendicular from it upon the tangent = 2 /a2 (x"a + y2). 54 47. Find the values of the perpendiculars from the vertices of an ellipse and hyperbola upon the tangent at any point; and determine what they become for a circle and equilateral hyperbola. 48. Required the same from the centres and foci generally, and for the above-mentioned particular cases. 49. Find the parts of the tangent of an ellipse intercepted between the point of contact and the co-ordinate axes through the centre: and thence prove that their rectangle is invariable when the ellipse becomes a circle. 50. Determine the same in an hyperbola generally; and deduce a similar conclusion when the hyperbola becomes equilateral. 51. If a diameter and a tangent at the extremity of the latus rectum of a parabola be considered the co-ordinate axes; it is required to find the points in which the tangent to any other point meets them, and also the perpendicular upon it from the origin. 52. In an ellipse referred to a pair of conjugate diameters, one of which bisects the angle between the semi-axes, it is required to find the subtangents and the lengths of the corresponding tangents. 53. Determine the angles in which the rectilineal tangent to the cissoid of Dioces, at points whose subtangent is equal to 2a, intersects the co-ordinate axes. 54. In the trochoid whose equation is y=ma vers — + 2/2ax —G, a find the subtangents when the tangent is equally inclined to the co-ordinate axes; and determine for what values of m this is possible. 55. The equations to the tangents of an ellipse and hyperbola whose centres are the origins, are a2yy' - b2x'= + a'2b2; and the portions which they cut off from the axes of x and y a2 b2 are and - respectively: required, proof. Y2 55. 56. Find the equations to the tangents of an ellipse and hyperbola in which the vertices are the origins of co-ordinates; and determine the portions they cut off from the co-ordinate axes. 57. The tangent to the cissoid of Diodes is defined by the equation (2a —x)iy'== (3ax' —xx'- ax) x: required a proof. 58. Prove that the equation to the tangent of the conchoid of Nicomedes is x2 V/ b2 x3 /b2_ x y' + (abb2 + x3) '- -ab - b2 X2 + a = 0; and find what this becomes when x = a = b. 59. The witch is defined by the equation xy=a/ a/a-; it is required to prove that the equation to its tangent will be 2 x/ax- xYy' + a (ax' + 2 %2 - 3 ax) =0, which cuts the co-ordinate axes at distances from the origin < a (3 a - 2v) expressed by - (3a- 2x) and a( - a 2,/ax — x 60. Find the equation to the tangent at any point of a cycloid: prove that it is parallel to the corresponding chord of the generating circle, and determine what portion of it is intercepted between the co-ordinate axes. 61. Find the equation of the tangent to the companion of the cycloid defined by the equations x = a vers 0 and y = aO. 62}. Required the equation of the tangent of the curve defined by y= (V/a -- /x)); and by means of it prove that the sum of the portions of the co-ordinate axes cut off by the tangent is of an invariable magnitude. 63. Find the equation to the tangent of the curve defined 2 2 2 by ax3 +y3 = a3; and from it shew that the length of the tangent intercepted between the co-ordinate axes is invariable. 64. Determine the equation to the tangent of the curve df+nby 'l-a~2~ —ly2 _//a"- y" 2 d-i by + \/Ga -- y - a + defined by --- -- log; and thence prove a y 56 that the part of it between the point of contact and the axis of x is of invariable length. y2 - a2 a a 65. If the equation of a curve be = --- + -log-; 4a ~2 y find the equation of its tangent, and thence prove that the difference between the tangent and subtangent belonging to the axis of I is invariable. y 66. Of the curve defined by ea =x- 2 a2, find the equation to the tangent; and thence prove that the sum of the tangent and subtangent varies as the rectangle of the corresponding co-ordinates. 67. Find the equation of the tangent to the curve defined by (m, —)m-lym = mnam-l'x: and thence shew that if AD be drawn from the origin perpendicular to the axis of x and meet the tangent PT in D, ADm = a"-'AT. 68. Required the equation to the tangent of the curve whose equation is v V/2 (IV2 + y2) = a (x2 - y2); and find where it intersects the axes of co-ordinates. 69. In the curve whose equation is (1 + log x) y = x, find the equation to the tangent; and thence prove that at any point, the subtangent: the distance of the intersection of the tangent with the axis of x from the origin:: the abscissa: the ordinate. 70. From a point in a curve whose co-ordinates are x and y, a straight line is drawn making with the curve an angle whose tangent is a; prove that the equation to the line will be a+p, ( dy y' — Y - ('V - x), where p. 1 - ap d 71. If straight lines be drawn perpendicular to the major axis of an ellipse at its extremities; find the points in which a tangent at any point of the curve meets them, and prove that the rectangle of the parts cut off is invariable. 72. In the curve defined by log v2 + y2 = a tan-1 Y find the equation to the tangent; and prove that the'angle in which it intersects the line drawn from the origin to the point of contact is invariable. 73. Draw a tangent to the curve whose equation is 2amym l= y2m - a2m; and prove that the distance of the point of contact, from a given point in the axis of x, varies as the segment of the said axis between that point and the tangent. 74. If any tangent be drawn to an ellipse, and four perpendiculars be drawn to the axis major, from the centre of the ellipse, the two extremities of the axis major and the point of contact, these four perpendiculars are proportionals. 75. Find the general equation to the tangent at any point of a curve when the co-ordinates are oblique; and apply it to an ellipse where the co-ordinate axes are parallel to any pair of conjugate diameters. 76. If a and b be the semi-axes of an ellipse or hyperbola; the product of the tangents of the angles at which any system of conjugate diameters are inclined to the axis major is b2 b2 -a2 or -: required a proof. 77. If a system of conjugate diameters of a conic section be produced to meet a tangent whose position is given; the rectangle of the parts of the tangent intercepted between those diameters and the point of contact, is independent of their position: required a proof. 78. The equation to a straight line drawn from the point a, j3 of a curve whose equation is y =f(x), perpendicular dx ie to its tangent is y'-,f3=- (ad-a): required a proof. dy 79. In the ellipse and hyperbola referred to the centres and principal axes; prove that the equations to the perpendiculars upon the tangents from the points a, /3 of the curves are (' - a) aly + (y' — ) b2 = 0. 80. The point of intersection of the rectilineal tangent with the perpendicular upon it from a point a, 3 of any curve H 58 2x +a —p(y-P) may be found from the equation x'= - 2 1 q-p" required a proof. 81. If a perpendicular upon the tangent from the focus of an ellipse be drawn; prove that the distance of the point of their intersection from the centre of the curve is invariable. 82. Given the length of the perpendicular from the centre of an ellipse upon its tangent; to find the point of contact. 83. In an hyperbola, given the length of the perpendicular from either focus upon the tangent; to determine the point of contact. 84. In the lemniscata of Bernoulli whose equation is (x2 + y2)2 =2a2 (X2 _ y2), find the equation to the rectilineal tangent which shall make equal angles with the co-ordinate axes. 85. Required the equation to the tangent making equal angles with the axes of a curve whose equation is x3 + y3 = a; and find where it meets them. 86. In the curve defined by x2y2 = (a2 - _2) (x - b)2, it is required to find the equations to the tangents inclined to the axis of x at angles whose trigonometrical tangents are a2 + -- 1: and to determine the points of contact. 87. Required the equations to the tangents inclined at angles of 30~ to the axes of the ellipse defined by the equation y2 — 2y + 3X2 - 2y - 4x + 5 = 0; and find their points of intersection with the axes. 88. Tangents to an ellipse, from a given point a, /3 without it, may be drawn by means of the equations a2y2 + - ^2" = a2b and a2cy + b2ax = 2b; required a proof. 89. It is required to prove that the equation to the straight line joining the points of contact as determined by the expressions in the last example is a203y -t b2aXGa'b2 59 90. The equations for determining the tangents to an hyperbola, drawn from a given point a, t3 without it, are a^y2 2- 2bX= ' and a2ty a bC a a2b2; and that of the straight line joining the points of contact is a23 y- bax = - a2 b: required a proof. 91. From a given point without a circle, two equal straight lines may be drawn to touch it: prove this, and find their inclination to each other. 92. If from any point in the directrix of a parabola, two tangents be drawn to the curve, they shall be at right angles to each other: required a proof. 93. From a given point in the axis of the curve whose 2 2 2 equation is (a)3 + (by)3= (a2 - b2)3, it is required to draw a tangent to it; and to find the limitations under which this is possible. 94. From a point whose co-ordinates are a and 2a, it is required to draw a tangent to the curve whose equation is 27ay2=4 (x-2 a)3; and to determine its length. 95. Find the angle included between two tangents to an ellipse drawn from a given point without it; and determine the distance from its centre at which this is a right angle. 96. Find the portions of the axes, through the vertex of an ellipse or hyperbola, cut off by the normal at any point of the curve. 97. Find the points where the normal to a cycloid meets the co-ordinate axes, and its lengths intercepted between them and the point of contact. 98. Determine the portions of the axes, through the centre of an ellipse or hyperbola, cut off by the normal at any point; and prove that the rectangle of the parts of the normal intercepted by them is invariable. 99. Find the equation to the normal of the curve defined by x" +y"= a (x - y); and determine the position of a fixed point through which it always passes. 60 100. Required the equation to the normal of the conic section defined by y2 = mX + n a2; and thence deduce the values of the subnormals on the axes of x and y for the parabola, ellipse and hyperbola. 101. Find the equation to the normal of the curve IV2 y2 defined by - + - =1; and determine the distances from ay bx the origin at which it intersects the co-ordinate axes. 102. Determine the equation to the normal of the curve defined by 2y = a (ea-+ e -a); and the portions of it intercepted between the curve and the axes of co-ordinates. 103. Prove that the equation to the normal of an hyperbola between the asymptotes is abxy, - x3x- a2b2 + x4 = 0; and thence construct it. 104. Find the equation to the normal of the cissoid of Dioces; and thence determine where the normal at the point of its intersection with the generating circle meets the axes. 105. Required the equation to the normal of the curve defined by y2 = ax - + x which shall make equal angles with the axes of co-ordinates. 106. Determine the equation of the normal of a conic section which shall pass through a given point a, 3; and find the points of intersection with the axes. 107. The equation to the straight line drawn from the point a, /3 of a curve perpendicular to the normal is Y,-A= d (dx - ) required a proof. 108. The intersection of the normal with the perpendicular upon it from any point a, f3 of the curve may be p2a+ x-p (3 —y) determined from the equation,= 1+ required a proof. 109. If a straight line be drawn from the origin of the co-ordinates to any point of a curve, another straight line 61 equally inclined to the normal will cut off from the axis of x a portion represented by 2 (X + ) (p -Y) required a J 2px —(1 - p2) y proof. 110. Determine the limits of the length of the normal in each of the conic sections, by means of the general equation. 111. Express the co-ordinates of a point in any curve in terms of the angle which the normal at that point makes with the axis of x. 112. If 0 be the angle made by the normal with the major axis of an ellipse; then will the co-ordinates from the centre be expressed by a2 cos 0 b2 sin 0 and /a2 cos20 + b2 sin 0 va2 cos20 + b2 sin20 required a proof. 113. On the same supposition, it is required to prove that the subtangent and subnormal are represented by b sin 0 tan 0 b" cos -y, * -- and, V/a2 cos2 0 + b2 sin2 0 V/a2 cos2 0 + b sin2 0 114. The same notation being retained, the tangent and normal will be expressed by b2 tan b2 and,/a2 cos 0 + b sin2 0 /a2 cos2 0 + b2 sin2 0 respectively: required a proof. 115. If the equation of a curve be a2y= b2 (2aax + 2i), and the notation of Article (151) be retained; then will A W= a and AZ= b: required a proof. 116. If y3 a2 -- 3 be the equation of a curve; then will AW= -- a and AZ=-a: required a proof. 117. If (x - 1) y = ( + 1) x be the equation of a curve; then will A W= 2 and AZ ==2: required a proof. 118. If (v - a) y2 = (i + a) x2 be the equation of a curve; then will AW=a and AZ= + a: required a proof. 62 119. If ym - axm-l_ -xm = 0 be the equation of a curve; a a then will AW=- and AZ -: required a proof. m m 120. In the curve whose equation is aym + f= bxm (a + x)" it is required to prove that n n AW= -- a and AZ= (a+n-lb)T^+n. m+-n m+ n 121. If the equation to a curve be x3 + y3 = a3; then will A W= 0 and AZ =0, and the asymptote passes through the origin, making 135~ with the axis of x: required a proof. 122. If (a - x) y2 = (a + x) x2 be the equation of a curve; then will AW:= -a and AZ= + co, and the asymptote is parallel to the axis of y: required a proof. 123. If (a2 - x2I") y2 = (a2 + V) be the equation of a curve; then will AW= +a and AZ= + co, so that the asymptotes are parallel to the axis of y: required a proof. 124. If the equation to a curve be (a+c+x) y = b (c+x); then will AW= co and AZ=b when x is infinite, and the asymptote is parallel to the axis of xB: also, A W= a+ c and AZ =-co when y is infinite, and the asymptote is parallel to the axis of y: required a proof. 125. If y2= a be the equation of a curve; then iv-a will AW= co and AZ= + 1, or there are two asymptotes parallel to the axis of x and equi-distant from it: again, A W= a and AZ = o, or there is another asymptote parallel to the axis of y: required a proof. 126. Apply the principles of Article (151) to determine the position of the rectilineal asymptote of the curve whose equation is x5 + a3xy - y 0. 127. Find the equation to the rectilineal asymptote of the hyperbola defined by the equation = hyperbola defined by the equation - - = -1. b' a' 63 128. If the equation of a curve be xy=aV/a —x2; then will y' = - co or x' = 0, be the equation to its rectilineal asymptote: required a proof. 129. The equation to the rectilineal asymptote of the curve defined by ( - 2) y = ( - 1) (x - 3), is y' - x' + 2= 0 required a proof. 130. If the equation of a curve be (x + 1) y = ( - 1) x; then will the equation to its rectilineal asymptote be y' - x' + 3=0: required a proof. 131. If the equation of a curve be 2 - y — a2 = 0; it is required to prove that the equations of its rectilineal asymptotes are y' + x' = 0. 132. The equation of a curve being y - a2 + x3 =- o; it is required to shew that the equation to its rectilineal asymptote is 3 (x' + y') - a= 0. 133. The equation of a curve is ay3 +x3y —ax30; then the equation to its rectilineal asymptote is y'+ a=0: required a proof. 134. If the equation of a curve be (x2- 1) y = (2 +- 1) x; it is required to shew that the equations to its rectilineal asymptotes are a' = + 1 and y' = x'. 135. If the equation of a curve be y3-2Xy2+ x2y-a3-o; then will those of the rectilineal asymptotes be y' - a =o and y'=0o: required a proof. 136. The equation of a curve is xy2-y=ax3+b x2+cx+e; then the equations to its rectilineal asymptotes are y =x /a ' - + y = -x /a, —, and a'==0: 2V/a 2\/a required a proof. 137. Find the equations to the rectilineal asymptotes of the curves defined by the following equations; (1) 4y2(a- _ 2)=(2a2 _2)2: (2) X2y2-a2 + (a-b)2: (3) (2a a?- X2)( a2 + y2) = 22: (4) (x +1)2y= (a-1): (5) (a2 +2) y2 (ax — a2)2: (6) y-_ 4 —a2?y=0. 138. The hyperbolic asymptote of the curve whose equaa2 tiondis x4 - y4 + a2y = 0, is defined by y' = x' +: required a proof. 139. The hyperbolic asymptotes of the curve whose equation is y3-2Xay2'+xy-a'=o, are defined by the equations y-xs - \/a / O: required a proof. 140. In the curve whose equation is x y2 _ a2 y = 3 +- 2 a + x + b, the hyperbolic asymptotes are defined by the equations 1 1 y = '+1+ — and y'=- - - -: 2 x 2x' required a proof. 141. In the curve whose equation is y3- Samy + = 0, it is required to prove that the rectilineal and hyperbolic asymptotes will be defined by the equations x'+ y'+ a =o and x'y' + a'2 + ax' + a2 0, respectively. 142. The curve whose equation is y2 - 2ay + 3x2- 2y - 42y? + 5 =0, is concave or convex towards the axis of x, according as the greater or less value of y is considered: required a proof. 143. In the curve whose equation is (x - 1) y = (x + 1) x, it is required to prove that the branches have their concavities upwards or downwards according as the abscissa is greater or less than 1. 144. The curve whose tangent is of an invariable magnitude is always convex towards both the co-ordinate axes: required a proof. 145. The branches of the curve whose equation is (a + w) y= a2 + ax + x2, are convex towards both the coordinate axes: required a proof. 65 146. Prove that the curve defined by the equation (y - )3a =- a - 2 as + a,2, is always concave towards the axis of x; and find how it is situated with respect to the axis of y. 147. Required the direction of the curvature of the curve whose equation is (a2 - v2) y = a3. 148. Find whether the curve whose equation is (y - b) =- (x - a)2, is convex or concave towards the co-ordinate axes. 149. Find the direction of the curvature of each of the curves defined by the following equations: (1) a2y =b2( (-2a): (2) ( -) )y2 =(a+ ) 2: (3) (+ a)y2=b2 (w+ 2a): (4) y3+e=3axy: (5) y4+2 a (y2)3 (6) (2+ y2)3 P-4a22y: (7) XyS = ab2 —b (2a + b) X + (a 2b)x2- -3: (8) y4 + 2a (o - sa) y2 _ 2aa+t3 + a2r2 = O: (9) x4- ax2y - 1a2y2- aY3 0. 150. Determine whether the curve whose equation is (a- )2 ('2 - y2) = a2y2, is convex or concave towards the axes, at the points where the values of x are a and 3a. 151. If yoccxm in one curve and yocxa in another, m being greater than n; it is required to prove that the former curve always falls between the latter and its rectilineal tangent. 152. Determine the conical parabola which shall have contact of the second order with an ellipse at a given point, and have its axis parallel to the axis major of the ellipse. 153. Define the circle of curvature of the curve whose equation is 2y = ex + e-'; and prove that its radius is equal to the normal. 154. Determine the position and magnitude of the circle )f curvature, when the equation of the curve is x2- ax = ay _y2^ I 66 155. If x', y' be the co-ordinates of the circle of curvature at a point x, y of any curve: it is required to prove that the equation to the circle is (( - q +(l +P2) 2 + (y -) q -(1 +p2)}2(l p2)3 156. Required the equation of the circle of curvature of the curvedefined by the equation - = cos (-) a a 157. Determine the equation of the circle of curvature of the curve defined by x = sec 2y; and thence find its magnitude and the position of its centre. 158. Find the equation to the circle of curvature at any point of the tractrix. 1dy / s2 159. If ym- =V/a - ym be the differential equation of a curve, it is required to shew that the radius of curvature at any point is m times as great as the normal. 160. Determine the radius and chords of curvature of the curve, whose ordinate is equal to the circular arc of which its abscissa is the versed sine. 161. Required the radius and chords of curvature parallel to the axes, of the curve traced out by taking a portion of the radius of a circular arc drawn through its extremity, always equal to the sine of the arc. 162. In any curve, find an expression for the distance of the centre of curvature from any given point a, b; and also from the vertex. 163. Determine expressions for the portions of the axes of any curve included within the circle of curvature; and apply them to the case of the common parabola. 164. Apply the expression for the chord of curvature through a given point, to find the diameter and principal chords of curvature in each of the: conic sections. 67 165. If 0 be the angle which the normal at any point of a curve makes with the axis of x, then will the radius dy ddx of curvature at that point be expressed by d or --- d sin 0 d cos 0 required a proof. 166. Apply the expressions above written to prove that in an ellipse the radius of curvature is equal to a2b2 (a2 cos2 0 + b2 sin2 0)1 167. Determine the magnitude of the radius and the co-ordinates of the centre, of the circle of curvature in each of the curves defined by the following equations: (1) y3=a2X: (2) (1+ -2)y=x: (3) ay = (b3- 3): (4) a3 =, (v + b)2: (5) a(y-b)2 = x(< -a)2: (6) ay2= ' (x + b)2: (7) (a-?) y = (a + ) 2: (8) 2y2= 2 (a2- i2): (9) 3 + y=a3: (10) y=sinx: (ll) y=tan x. 168. Determine the radius of curvature of the curve whose equation is y3 = ax2 - x3, at the points where x =o and X _ a. 169. Find that point in a parabola at which the curvature is one fourth part of the greatest curvature. 170. If a circle of curvature to the vertex of a parabola be described, and another circle touch that and the. arcs of the parabola, and so on continually: it is required to prove that the radii of these circles are as the odd numbers 1, 3, 5, 7, &c. 171. Find the points in which the circles of curvature at the extremities of the principal axes of an ellipse intersect each other, and also the angles at which the circles are there inclined to each other. Is the intersection always possible? 68 17s. Find the evolute the he curve defined by the equation y2- a a2 =0; and determine whether it meets the involute. 173. The evolute of the curve whose equation is a2y2- b2, 2 + ab2 b= -o 2 2 2 is defined by the equation (aa)3 - (b(3)3 = (a2 + b2)3: required a proof. 174. The equation to the evolute of the catenary is a /+,\/a 2 4a2 aV/a2-4a 2 3= a log (a+V2-4-) - a 2 a, 2a if the distance of the vertex of the catenary from the origin of co-ordinates be a: required a proof. 175. The evolute of the tractrix is defined by the equationa a log (+V - a; required a proof. 176. Find the equation to the tangent of the curve 2 2 2 defined by (ax)3 + (by)3 = (a2 b2)3; and prove it to be a 'x0 y2 normal to the curve defined by + Y = 177- Required the equation to the tangent of a cycloid, and prove this line to be perpendicular to the tangent of its involute at the point of concourse. 178. Required the length of any portion of the arc of the evolute of the common parabola. 179. Determine the length of any portion of the arc of a common cycloid, the radius of whose generating circle is given. 180. In the curve whose equation is ay=_x/A 2-a,2 it is required to prove that the square of the distance of any point in the curve from the origin is equal to half the rectangle of the corresponding abscissa, and the line intercepted between the vertex and the normal. 69 181. In the curve whose equation is m (e -L- 1) x=-y9 it is required to prove that the portion of the ordinate included between the curve and a straight line through the origin making equal angles with the axes varies as the cotangent of the curve's inclination to the axis of x. 182. Describe a circle with a given centre so as to touch a given parabola; both when the point is within and without the parabola. 183. A straight line drawn through the focus of any conic section to the point in which a diameter meets the directrix will be perpendicular to a tangent at either extremity of that diameter: required a proof. 184. If from any point in a line parallel to the axis of a common parabola, two tangents be drawn to the curve; it is required to prove that the sum of the cotangents of their inclinations to the axis of x is invariable. 185. If y=mx+csin- be the equation to a curve; a it is required to shew that the sum of the tangents of the angles in which two ordinates at the distance a from each other cut the curve, is constant. 186. If with a radius equal to the line joining the extremities of the axes of an ellipse, a concentric circle be described, two tangents drawn to the ellipse from any point in the circumference of this circle will be at right angles to each other: required a proof. 187. If with the co-ordinates of any point in an elliptic quadrant as semi-axes a concentric one be constructed, this will be touched by the chord of the first: required a proof. 188. Shew that the ellipse whose equation is Y2- 4 a + 22 = 0, always cuts at right angles the parabola whose equation is y2=m (a —x), whatever be its latus rectum. 189. Prove that the part of the tangent of an hyperbola intercepted between the asymptotes is equal to the diameter to which it is parallel, and is bisected by the point of contact. 70 190. If two equal parabolas have a common axis, prove that a straight line touching the interior and bounded by the exterior will be bisected in the point of contact. 191. Two parabolas having a common axis, it is required to find a point in one of them, from which two tangents drawn to the other shall include a given angle. 192. A curve is constructed by cutting off from the ordinate of a circle a portion equal to the difference between its abscissa and ordinate: find the angles in which it intersects the diameter and the circumference of the circle. 193. If a straight line equal to the sum of the semi-axes of an ellipse have its extremities in the axes and cut the curve, and the parallelogram of which this is the diagonal be completed; it is required to prove that the line drawn from the angle of the parallelogram to the point of intersection will be a normal to the ellipse. 194. If between a rectangular hyperbola and its asymptotes any number of concentric elliptic quadrants be inscribed, the rectangle of their axes will be invariable: required a proof. 195. The ordinate MP of an ellipse whose major axis is AB is bisected in Q and AQ is joined and produced so as to meet the tangent at B in T: prove that the straight line TP will be a tangent to the curve. 196. An ellipse and hyperbola being constructed on the same axes, if from any point in one of the curves two tangents be drawn to the other, the straight line which passes through the points of contact will be a tangent to the first curve: required a proof. 197. AP is a portion of a common parabola, PT a tangent at P, PG a normal and TR a perpendicular to the axis at T: prove that if GP be produced to meet TR in R, GR will be- equal to the radius of curvature at P. CHAP. IX. MISCELLANEOUS EXAMPLES. BY means of the principles laid down in Articles (190)(212), it is required to prove and solve the following Theorems and Problems. 1. If 0 and r be the co-ordinates of the logarithmic spiral corresponding to the arc s, it is required to shew that d s/ 1 q+ (log a)= aOV/1 + (log a) =r/ + (log a)2; d- log a) dO dr log a 2. In the lemniscata of Bernoulli whose equation is r2 a cos20, it is required to prove that ds a a2 ds 1 a2 d - -; and d0 ~ /cos 20 r dr sin 20 /ai3. In the spiral of Archimedes whose equation is r = aO, it is required to prove that the tangent of the angle between the radius vector and a straight line through two points whose (2 a —/) sin (a-/3) abscissae are a, 3 is expressed by (2a- - )os (a-a), the (2 a - /3) cos (a-3) - a said radius vector belonging to the abscissa a. 4. In every curve referred to polar co-ordinates, it is required to establish the formulae rd0 dr dO sin P= -- cos P= - and p=r2. ds ds ds 5. The equation of the reciprocal spiral is r = a0-1; then, if the notation of Article (193) be retained, tan P=- - and or ar p= - -: required a proof. Iat& + r2 72 6. If the equation of the spiral be r = at"O; it is rem + 1 r Iy\ T m quired to prove that tan P= - and p = 2 - m \a 2 2 vN 2am + rm 7. If the equation of the lemniscata be r2 =a2 cos 20; then will r2 r3 tan P - a and = p - a/ 4 I a2 required a proof. 8. In the cardioide whose equation is r-=a (1 - cos0), prove that tan P= - and p - r \/2 ar- r2 2a 9. If the spiral be defined by the equation r = a sin 2 0; ~~r r2 then will tan P= —, and p=; required 2Va-r/2 V4a — 3r2 a proof. 10. The equation to a spiral being r=asinm0, it is required to prove that r 92 tan P= and p = m / a2-_ 9 - ~m2 a2 _ (m2 _ l) r2 11. In the spiral defined by the equation r2= a tan 20, it is required to prove that a2r2 a2r3 tanP= P - and p - a4+r4 n ' a8d= + 3ars+8 +r84+ 12. If the equation of a circle, of which the fixed axis is the chord of a quadrant, be r = a (cos 0- sin 0); then will r r2 tan P= - and - -; required a proof. V2 a2- r2 a '2 13. The equation of an equilateral hyperbola from the centre being r2= a sec 2 0, it is required to prove that a2 a2 tan P= -7 - and p -: also that P= _ r-20 and /r4- _ a p = a/cos 20. 73 14. The equation of a rectangular hyperbola referred to one of its asymptotes being r2 =2 a2 cosec 2 0; it is required to 2a2 2a2 prove that tan P= - and p= -: also that P= ir - 20 Vr 4_ 4 4 r and p=aV'2 sin 20. 15. The involute of a circle being defined by the equation V/r2 - a2r - sec-; it is required to prove that tan P= and p= r2 a-a2 a 16. If the equation of the helicoid parabola be (a-r)2=b20: 2 (ar-r 2) 2r2 (a -r) then will tanP= - and p= reb2 2V(a - r)2I2+ b4 quired a proof. 17. The equation of the trisectrix being r=a (2 cos 0 + 1); it is required to prove that tan P= - _ and,\I+ a22rar- 2 18. If the equation of a polar curve be r= a (sec 0-tan 0),; 2ar 2a r22 prove that tan P and p a2 +r Va/a4 + 6 d2r2 + r4 19. In the elliptic spiral defined by the equation Va 2- 20 = a cos'/ a ar it is required to prove that sin P= a and p = \lb -f r V/6 + 2Z 20. In the hyperbolic spiral defined by the equation _ a2 b2+Va2 b2 -r2 Va2P - b b log b br it is required to prove that sin = and p v K 7 r2 ci2-r2 74 21. In the spiral whose equation is = /a2 b b2 /r2+ aV/2-b2 0 = b log - -, r b br it is required to prove that sin P = - and p = r 2 Va2~+2 -a ++ 22. If a spiral be defined by /ac+b0 = a sec- a a ar then will sin P= and p = /*-: required a proof. Vr2 2 — Vr2 — b2 23. If (r - a) 02= r be the equation of a spiral, it is required to prove that ar2 ar sin P= and p= V/a2r+4 (r-a)3 v/a2r + 4 (r-a)3 24. Find the values of tan P and p in an ellipse and hyperbola referred to the centre, focus and vertex as a pole. 25. In any conic section, given the angle included between a tangent and the diameter through the point of contact, to find that point. 26. Required the same, when the angle included between the tangent and the distance from the focus is given in each of the conic sections. 27. The polar equation of a straight line being r sin (a + 0) = a sin a; it is required to prove that P = r - a - 0 and p = a sin a. 28. Determine the values of tan P and p in the spiral defined by the equation r = a log tan (45 + 0). 29. In the spiral whereof the equation is (ac+rn) p2=r9+2, it is required to prove that tan P= (-) \al 30. Find the values of tan P and p in the spiral defined by the equation r =a (1 + 2 sin 20). 75 31. Required the values of tanP and p in the curve whose polar equation is r = a sec 0 tan2 0. 32. One of the polar equations to a rectangular hyperbola being r = (cosec 0 —sec 0); it is required to find the corresponding values of tan P and p. 33. In the rectangular hyperbola as defined by the equation r cos 20= 2a cos 0, it is required to find the values of tan P and p. 34. If the notation of Article (196) be retained, it is required to prove that in the spiral of Archimedes whose equation r2 is r = a, the polar subtangent ST= - and the polar tangent a 35. In the circle referred to a point in its circumference 2 when r = a cos 0, it is required to prove that ST = - a and PT= 36. If a spiral be defined by the equation r =asec0; it is required to prove that r2, r"2a2 ST= 2 and PT=r V r /r2 r2 2 37. If r= a0m be the equation of a spiral; it is required rO r to prove that ST= - and PT= + 02. m m 38. If a spiral be defined by the equation a + V\a- T2 0 = log --- it is required to shew that ST= -/ ---ar 'and PT a T/a2 -, a2 2a 76 ar 39. If the equation p belong to a polar curve; ar / 2 - ar2 then will ST=- and PT= r Va2 - b6 _- r a2-b — r2' required a proof. 40. Find the values of the polar subtangents and tangents in all the conic sections referred to the centres, foci and vertices: in the cissoid of Diodes and in the conchoid of Nicomedes. 41. If a line be drawn through the focus of an ellipse making an angle 0 with the major axis, and tangents be drawn at the extremities of this line; it is required to prove that these tangents will include an angle p, such that (1 -e2) tan qj=2e sin0. 42. If a and b be the semiaxes of an ellipse, and 0 and 0 the angles which any two conjugate diameters make with the major axis; prove that a2 tan 0= b2 cot. 43. A line drawn parallel to the focal distance of an ellipse or hyperbola, through the centre and meeting the tangent, is equal to the axis major: required a proof. 44. Prove that the tangent to any point of a conic section makes equal angles with the focal distances. 45. In the spiral defined by the equation r = a sin 20, if a straight line be drawn through any point parallel to the polar subtangent, it is required to shew that the length of it intercepted between the rectangular axes through the pole is invariable. 46. If from any point without a conic section, two tangents be drawn to the curve and two straight lines to the foci: it is required to prove that the angles between the tangents and straight lines will be equal. 47. If a be the radius of a circle and 0 the angle at its centre, then may a tangent be drawn to it by means of the equation r' cos (0- 0') = a, if r' and 0' be its polar co-ordinates: required a proof. 48. If qp be the angle which a tangent to any point of the logarithmic spiral makes with the fixed axis; then may 77 this angle be found from the equation tan ( -0) log a =1 required a proof. 49. If the notation of Article (199) be retained, it is required to prove that in the reciprocal spiral the polar sub92 r2 r\/a2 + r normal SG= - - and the polar normal PG= a a 50. If the equation of a spiral be r=aOn, then will 22 1 m-1 / ia SG = marr n, and PG = r / 1 + Mn2 (): required a proof. 51. In the spiral defined by the equation r= +-, it is required to prove that SG -- V/a d r2 and PG= - 2a-r'. a a 52. In the spiral whose equation is r cos3O=a, it is required to prove that f 2 / 2 2 3rG\ —$/ - a'3 r,\9ra- 8a ^SG^= 3/ a /-8and PG= r/ 1 1 a3 a3 53. If a spiral be defined by the equation r=a- -b02 62 and (ar-r2)2'+b then will SG= - and PG= re2(a-r) 2 (a -r) quired a proof. 54. In the involute of the circle whose radius is a, it ar r2 is required to prove that SG=- and PG 55. If the equation of a spiral be r2 -a2 cos20; it is required to prove that SG= - and PG=-. r r 56. If a spiral be defined by the equation r= a sec 0; it is required to prove that SG = _- /r_ - a8 and PG = - a a 78 57. The equation of the parabola being r = a cosec2 -0; it is required to prove that SG= -r / and PG=r \/ a a 58. In any conic section, if normals be drawn at the extremities of a chord passing through the focus so as to intersect each other; the sum of the squares of the parts of the normals between the curve and their intersection will have a given ratio to the square of the chord: required a proof. 59. The notation of Article (200) being retained, if the equation of a spiral be r = a + b sec 0, it is required to shew that L LSX= 900 and ST= - b. 60. If the equation of a spiral be rs =a2 tan 20; it is required to prove that z LSX-=45 and ST=o. 61. If a spiral be defined by the equation r2 = a2 sec 20; it is required to prove that / LSX==450 and ST= o. 62. In a spiral defined by the equation r cos 0+a cos 20=0, it is required to prove that z LSX= 900 and ST= a. 63. If the equation of a polar curve be r cos 2 0 = 2 a cos 0; a then will Z LSX= 45~ and ST= —: required a proof. V/2 64. The equation of a polar curve being r cos 0 = a cos 20; it is required to prove that / LSX= 900 and ST= - a. 65. In the curve defined by the equation r (sin3 0 + cos3 0) = a cos2 0, it is required to prove that Z LSX= 1350 and ST= - a 66. If a curve be defined by the equation r2=a2(tan20-1); then will z LSX= 90 and ST = + a: required a proof. 67. If the equation of a curve be 2r (sin3 0 + cos3 0) = a sin 20; then will z LSX-=135~ and ST=-: required a proofe V/2 79 68. If a curve be defined by the equation r3 (cos3 0 - sin3 ) = a3; it is required to prove that Z LSX= 450 and ST= 0. 69. In the polar curve defined by the equation r2 cos2 0 (tan 0 - 1) a2 (tan 0 + 1), it is required to prove that z LSX- 90~ and ST= + a give two asymptotes: and z LSX= 45~ and ST= O, give a third. 0 70. The spiral being defined by the equation r cos - = a; it is required to prove that z LSX=m- and ST=ma. 71. If the equation to a curve be r sin (m+1)0=a sin mO; it is required to prove that sin2 z LSX= and ST=a m + m+ -t- 1 7r (m +1) sin m+1 72. It is required to draw asymptotes to the polar curves defined by the following equations: (1) r2 sin0=22 a: (2) b = asec-1: b (3) =a (cosec 0- sec 0): (4) r(cos0 +cot0) ==2 a (5) rsin 20 =a sin 0 sin 30: (6) r (e -1)= a (e + 1): (7) r/b2 cos20-a sin20=ab: (8) r(02-1)=a02. 73. In the reciprocal spiral whose equation is r=a0-1, the arc of the curve is always concave towards its pole: required a proof. 74. If r = a"- be the equation of the logarithmic spiral, it is required to prove that the curve is always concave towards its radius vector. 80 75. The equation of a circle being r = b cos 0 + V/a2 -b2 sin 0; it is required to find when the circumference is concave or convex towards its radius vector. 76. Find whether the spiral defined by the equation r-=aO, is convex or concave towards its pole. 77. Determine whether the curve whose polar equation is r = a (cosec 0- sec 0), is convex or concave towards its pole. 78. If the centre of the generating circle be the pole, the equation to a cycloid is r2= a2 (1 + 02+20 sin 0): find the direction of its curvature. 79. Find the direction of the curvature of the curve defined by the equation r = a (cos 0 - sin 0). 80. Determine whether the curve defined by the equation r cos 2 = a cos 0, is convex or concave towards its pole. 81. Find the direction of the curvature of the curve whose polar equation is r2 -2 a2 cosec 20. 82. Prove that the curve whose polar equation is r cos 0 =a cos 20, is always concave towards the pole. 83. Shew from the equation r cos 0 =2 a sin2 0, that the arc of the cissoid of Diodes is always concave towards its. radius vector. 84. If the notation of Article (205) be retained, prove that in the spiral of Archimedes, the radius of curvature (a2+r2) 2r (a2+r2) CP= - and the chord of curvature PV-= (a+2) 2 a + r2 2a2+85. In the reciprocal spiral whose equation is r = a -, it is required to prove that CP- r (a2 + and PV 2a (a2+r) a" a2 81 86. In the lemniscata of Bernoulli whose equation is a2 r2= a2 cos 2, it is required to prove that CP= - and 3r PV- = r 87. In the cardioide whose equation is r = a (1 + cos 0), 2 r —4 it is required to prove that CP= - -r and P= - r. 3.3 88. Determine the radius and chord of curvature of the curve whose polar equation is r2 = a2 (tan0 - 1). 89. Determine the radius and chord of curvature of the curve whose polar equation is r = a cot 0 cosec 0. 90. Find the circle of curvature of the polar curve whose equation is r2 cos40 = a2 cos 2 0, and the magnitude of the chord passing through its pole. 91. Determine the radius and chord of curvature of the curve whose polar equation is r2 (e - 1)2 = 4a2e. 92. Find the magnitude of the radius and the position of the centre, of the circle of curvature of the curve whose equation is r = a (cos 0- sin 0). 93. Determine the radius of curvature of the curve whose polar equation is r cos 2 0 = 2 a cos 0; and find where the centre is situated in terms of the co-ordinates. 94. Express the radius and chord of curvature of the curve defined by r = a sec 0 tan20, in terms of the polar co-ordinates. 95. Determine the magnitude and position of the circle of curvature of the curve whose equation is r = a (1 + 2 sin -0), at a point where 0 = 45. 96. The equation of a curve being rsi (+1) sin ( ) sin mO, it is required to find the radius and chord of curvature at a point where 0 = 90~. L 82 97. If the equation of a curve be r = a (cos 0 + 2 0 sin 0), it is required to find expressions for its radius and chord of curvature at any point. 98. Find expressions for the radius and chord of curvature of the curve defined by the equation r (1 - e sin 2 0) = a sin 0: and determine the points at which they are equal to a and 3. 99. Prove that the radius of curvature at any point of a parabola referred to the focus is equal to half the polar subnormal. 100. If the equation of a curve be /2r-a r-c a 0= -- + cos-1 a r it is required to prove that the curvature at any point varies inversely as the radius vector. 101. Apply the expressions investigated in Article (205) to determine the radii and chords of curvature of all the conic sections referred to their centres, foci and vertices. 102. If the equation of a spiral be r = a0-"; it is required to compare its radius of curvature with the normal, and its chord of curvature with the radius vector. 103. If u denote the inverse of the radius vector of a spiral, p the perpendicular upon the tangent, y the radius of curvature and 0 the spiral angle: then will 1 [zdu\2 1d2 + U2 and- =p3U3 + d2) required a proof. 104. Express the radius of curvature at any point of a parabola in terms of the angle which it makes with the fixed axis. 105. If the ordinary notation be retained, it is required to prove that -= P when the circle has contact of the yth drrr third order with the spiral. 83 106. The equation to an epicycloid being e r2 -a2 e (r2 a2) 0 = tan- tan -tan- 1 —, a e2 r a (e- r)' e2 (r. - a2) it is required to prove that p2 = e (- a, where e= a+ 2b, -ae" if a= the radius of the base and b = the radius of the generating circle. 107. It is required to find the evolutes of the epicycloid and hypocycloid. 108. Find the evolute of the spiral defined by the equation r= aC0. 109. If the equation of a spiral be r- a=aO+a sec' -, a it is required to find that of its evolute. 110. If 0oc -, it is required to prove that the polar subtangent at any point: the corresponding circular arc whose radius is r:: m 1. 111. If Oocr, the number of revolutions made by the radius vector varies as the square root of the polar subtangent: required a proof. 112. If a spiral be defined by the equation r2 cos 2 0= a2, it is required to prove that the sine of the angle at which the radius vector is inclined to the tangent oc. 113. Given the radius vector at any point of a parabola and the angle it makes with the curve, to find the position of the vertex and the magnitude of the latus rectum. 114. If - lo =log gsin (a- 0 be the equation 7f a 7r of a polar curve, it is required to shew that two tangents drawn to the curve at the extremities of a line passing through its pole will always include the same given angle. 84 115. In any spiral, if 0 be the point of intersection of perpendiculars to two consecutive radii vectores through their extremities, it is required to prove that SP2 =SO.Sy. 116. In the focal distance SP of a parabola, SQ is taken equal to the rectangular ordinate MP: it is required to draw a tangent to the locus of the point Q. 117. If Q be a point so taken in the radius vector SP of a parabola that SQ is equal to the perpendicular upon the tangent from the focus: it is required to find the values of tanP and p in the locus of Q. 118. Prove that the spiral whose equation is (Or- 20) r=2a cosO, cuts off equal arcs from all circles passing through its pole and having their centres in its axis; and that the distances from the pole at which the spiral cuts the axis are as - - 1 3 5 &c. the tangents of the angles in which it cuts it being as 1, 3, 5, &c. 119. Determine the asymptotic circle of the curve defined by the equation 0 v/ar - = 1. 120. In any polar curve, if r be considered the independent variable, it is required to prove that the radius of curvature is expressed by p + d20 dO /d\0) r- +2- r2I + - dr2 dr \drJ 121. If the arc s of the spiral be taken as the independent variable, it is required to shew that dO ds d )2 ds2 rdo ds2 85 122. If neither 0 nor r be considered the principal variable, it is required to prove that {.+ (dr\ CP- = d. /dr\2 drd20 d2r ^4-r- _ +r- - -- - r + \d0) do d02 d02 and PV= -.dd 2 dr\2 d r d20 d 2 r 2 Tl ( r — 1 d - -d --- \d0] d do0 ddO 123. It is required to apply the criterion deduced in Article (211) to ascertain whether the spiral whose equation is r (e +e-0) =2a, is concave or convex towards its pole. 124. If the tangent to a spiral make an angle ( with the rectangular ordinate, and p be the perpendicular let fall upon it from the origin: it is required to prove that ds d2p + dab =P + d2 where the upper or lower sign is to be used according as the curve is convex or concave towards its pole. CHAP. X. MISCELLANEOUS EXAMPLES. BY reference to Articles (213)-(251), the solutions of the following Theorems and Problems may be effected. 1. The equation of a curve being y- 1 = ( -2)3, it is required to prove that it cuts the axis of v when = 1 and tan X= 32: and the axis of y when y= -7 and tan Y=- 2. The curve whose equation is ay2= 2 ( -b), meets the axis of x when x = 0 and x = b, and corresponding to them tan X-= - and tan X= oo: required a proof. a 3. If a curve be defined by the equation, (a -) = y (a + y), it meets the axis of x when x = and =a, so that tan X = + 1: and it meets the axis of y when y = O and y= -a, so that tan Y= + 1: required a proof. 4. The curve whose equation is V-y = a + x, meets the axis of x when x = - a and tan X = 0: required a proof. 5. If the equation of a curve be a3y=XI4-bX3-bb2V2; then will it meet the axis of x when x = 0 and x = b (1 + 5), tan X=o and tan X=; and the axis of y when y=o and tan Y = oo: required a proof. 6. It is required to prove that the ellipse defined by the equation y2 - 2Xy + 3$2 + 2y- 4 -3 = O, meets the axis of v when = 9(2 + /13) and tan X= (5 T+ A13): also the axis of y when the values of y are 1 and -, and tan Y= and 2, 87 7. The parabola defined by the equation y2 -4xyy+4 x2 -3 +\/41 -8y+3x -2=0, meets the axis of x when = — 8 -41 + 13\/41 and tan X= -- 6-: and the axis of y when y=4+ 3\/2 64 96 + 78 \/2 and tan y= -96 + 78 -: required a proof. 41 8. Determine where the curve whose equation is y3-X3 + a3 = 0, meets the co-ordinate axes, and the angles it makes with them. 9. Find where the curve whose equation is (a-x)y2 = (a + ) X2, intersects the axes of co-ordinates, and the angles in which it cuts them. 10. The curve whose equation is ay2 (x - a) =-e, has an infinite ordinate when = a, which is an asymptote to the curve: required a proof. 11. The curve defined by the equation a2 y = a4-b4, has an infinite ordinate through the origin, which is an asymptote to it: required a proof. 12. If y (a + c + x) = b (c + x) be the equation of a curve, it is required to prove that when = - (a + c), the ordinate parallel to the axis of y is infinite, and becomes an asymptote; and when y = b, the ordinate parallel to the axis of x becomes infinite, and is then an asymptote. 13. The equation of a curve being (x +a) ys= b2 (x+2a), it is required to show that when = - a, the ordinate parallel to the axis of y is infinite; and when y= + b, the ordinates parallel to the axis of x are infinite. 14. Find the positions of the infinite ordinates in the curves whose equations are y = tan x and y = sec x. 15. The equation of an ellipse being 2y2 -4y+5a22-3S=o, it is required to prove that the two values of y are equal to one another when x =O and when v = 1: that the two values of B are equal to each other when y = 1 (2 + V/l-); and that the corresponding ordinates are tangents to the curve. 88 16. In the curve whose equation is y4 -.2axy2 + x4 = 0o it is required to prove that two values of y are equal when _x =O and x = a; and that the corresponding ordinates are tangents to the curve. 17. In the curve whose equation is y4-96a2y2 + 100a2x2 - = o, the values of y become equal when x = +6a and x = ~ 8a, and the corresponding ordinates are tangents to the curve: prove this, and shew that the values of x can never be equal to one another. 18. If y2 = a2 - 2 be the equation of a curve; then, when = 0, y= -~ a, a positive and negative maximum; and when y= o, = + a, a positive and negative maximum: required a proof. 19. If (y-a)2 - (x - b) = c2, then will the values of y be a c, a maximum and a minimum when x=b: and the values of x will be b + c, a maximum and a minimum, when y =a: required a proof. 20. If the equation to a conic section be y2-2exy +x2 -ay=0, it is required to prove that when x=0, y=O, a ae a minimum; and when x= 1-e2 Y 1 — a maximum l-e~ 1 -- e~ a a also, that when y = )' - = ), a maximum; 2(1 - e) 2 ( - e) a a and when y= x= - a minimum or ne2 (1 - e) 2 (1 +- e) gative maximum. 21. In the curve belonging to the equation (x2 +y2)y = a (x + y)2, it is required to prove that when x = 2 a, y = a, a maximum ordinate; and when x = oo 9 y = a, which is neither a maximum nor a minimum. 22. In the curve whose equation is x4'- a2 S~ a2y2-=0 a it is required to shew, that when x = -~~, y= + a, which are maxima; and when y = 0, x = a, which are also maxima. 89 23. If the equation of a curve Fe (y-a)2 (=x- b 2 it is required to shew that when x=lb, and y=a+ b2 j 3 - SV_ the upper is a maximum and the lower a minimum ordinate. 24. Determine the maximum and minimum values of theco-ordinates of the curves defined by the following equations: (1) ax —ay-X=+y (2) x4y-a ('-y'): (3) X'y2=(X- 9) (X2-2)2: (4 ay'=(x'-a')': (5) y'(a -x) =(a-2x) X': (6) a y=x(x'+y'): (7) (x'+y2) =a2 2 +2by2: (8) (y2-x2)2-ax4 +bX2: (9) a'y2=x'(2a-x): (10) a'(x'-y2)2=(X2 -i2)3: (ii) y~e-2aa)=x(x- a): (12) (2ax_-') y= (a XX2)2: (13) (2a-x) y2=X(a+b- x)2: (14) (' + y'2)2: 4a'xy: (is) (y + _)2 = 2 ax - x2: (16) x' + a3xy=y': (17) (a _V)2y2 = X2 (3 a2 ax _ x2): (18) (a 2-x)2) y = X 2a-2a' + +a2X. 25. Prove that the curve defined by the equation y =ax + bX' - has a point of contrary flexure corresponding to b ~~9 abc +2~b' X = - and y= 27c SC 27 C2 26. The curve whose equation is y=&I 4-12 X3 + 48x2- 64x, has points of infiexion corresponding to x = and y= - 16: also to x =4 and y =0: required a proof. 27. If the equation of a curve be a4y=3"-35-Vx4 + 140ax' - _ 240 a3xX, then corresponding to points of inflexion, the values of x are a, 2 a and 4 a: required a proof. 28. In the curve whose equation is y' = x3- a3, it is required to shew that when x= 0 and y = - a, also when x= a and y=0, there are points of contrary fiexure. 29. The equation of a curve being y = b +2 (w - a)3, it is required to prove that there is a point of inflexion when X= a; and to verify it by the substitution of a + h for x. M 90 30. If the equation of a curve be ay= - /2ax- 2: then corresponding to x=0 and y=o, as also to x la (3-1i) and y =-a x/6 v/ — 9, there are points of contrary flexure: required a proof. 31. In the companion of the cycloid whose equations are x = aO and y = a (1 + cos 0), it is required to shew that there is a point of inflexion when x = 7r a and y = a. 32. The quadratrix of Tschirnhausen being defined by the equation - = sin 2, it is required to prove that there a 2a' are points of contrary flexure corresponding to x= =2ma and y=-a sin mwr= O. 33. The syntractrix is defined by the equation b + V^^ -/ -y = a log + —y -/b2- y; Y and it has points of inflexion corresponding to /alVa-b+/ab-b /a-b db a a =alog //' a - b + /ab / and- b // 2aVn V/a 2a-b 2a-b required a proof. 34. Of the Witch defined by the equation xy2 + a2x = a, the points whose co-ordinates are x =- a and y=, are points of inflexion: required a proof. 35. If a curve be defined by the equation y=aec~s, then will its inflexions correspond to cos = — ( — 1 /5) and y=ae(-'+-"): required a proof. 36. If a vers y =x be the equation of a curve, its inflexions correspond to x = a and y = (2m + 1) ~: required a proof. 91 37. If the equation of a curve be y a vers 2a, then will its points of inflexion belong to x = (2m+ 1) ~ and y=a: required a proof. 38. It is required to ascertain the positions of the points of inflexion in. the curves belonging to the following equations: (1) a2y = br a: (2) a'y= 2X(a2-_2): (3) 4I a2y2+x2y2: (4) (X2+y )2=ara2+b2y2: (5) (a2+x')y, a': (6) ay2=a2(a -X): (7) (a-'v)y2=a2x: (8) =ayx=av- b)Qx- c): (9) (a-x) 2 y'=(2ax _V)2: (10) (a - X)y2-a2r(2a- x): (x1 ( - a) yS=(a - x) x (1!2) (a )yy= c~-m)2 (is) logy=aX: (14) y=x?-tcosx: (15) a2y2=,4_(b2+c2) aI2+b2c2: (16) a' y3 + b3&'= a' b3 (17) (2a - x) (XI +y2) b2,V~ (18) X4 _ d2X2 + a~y2 = 0 (19) (2 a -X) _V (X a) X_ 3 a) 39. In the curve whose equation is ay2 - X3+ ba X=0, it is required to prove that the origin is a conjugate point. 40. If the equation of a curve be (a - e) y2 = X"-6 b, prove that the origin of co-ordinates is a conjugate point. 41. If the equation of a curve be iy' = (x -9) (X - 2)2, it is required to prove the existence of a conjugate point corresponding to x =2 and y =0. 42. The equation of a curve being a (y - a)' = a' (w - a), it is required to prove that there is a conjugate point when x=0 and y=a. 43. If the equation of a curve be a (y - b) = (a- C)3, it is required to prove the existence of a conjugate point when x c and y = b. 44. In the curve whose equation is ay2 = (b + X)2 it is required to shew that there is a conjugate point when M= -b and y =o. 92 45. The curve whose equation is y = (x + 2) /x+ + 3, has a conjugate point corresponding to = - 2 and y =: required a proof. 46. In the curve defined by the equation y + a =-(a - b +?) V/-b, there is a conjugate point when = - (a - b) and y = - a: required a proof. 47. If the equation of a curve be y + a = (x-b) /-x- 2b, it is required to shew that when x =b and y= — a, there is a conjugate point. 48. It is required to prove that the curve whose equation is a2 (y - b)= (2cx - 2)(c + x)2, has a conjugate point corresponding to x = - c and y = b. 49. If the equation of a curve be y2 - 1 = ( V + 1)V/i — 1, there will be conjugate points corresponding to = - 1 and y= + 1: required a proof. 50. The curve defined by the equation (y- 3)6 = (- 2)2 (1 -_ )3, has a conjugate point when x= 1 and y = 3: required a proof. 51. The origin of the co-ordinates in the curve whose equation is (y2 — aa2)2 — =a +2 a22 -3$a2Xy, is a conjugate point: required a proof. 52. It is required to prove that the curve defined by the equation a (y2- b2) = (x2 _- c) V/C2 - 2 2, has four conjugate points corresponding to = + c and y = + b. 53. It is required to determine the positions of the conjugate points in the curves defined by the following equations: (1) ( —a')2 + (y2 b2)2 (2) a (y — a)-2 v2 (x-b)' (3) a3(y+a)2=( -2b) (i-b)4: (4) (X + a) y2 = ( - b) i: (5) a5(y-2a)2=(iv-3b) ( —b)6: (6) i (2 + y2) = 3- 2: (7) xy' + 3 = 7i2v- 15X +9: (8) X -av2y-ay3+a2y2=o. 54. In the curve whose equation is y =a2 +ax, it is required to prove that the curvature is a maximum when = -a and y=o. 93 55. In the logarithmic curve whose equation is y= a', it is required to prove that the curvature is a maximum when 1 V/2 log a 56. It is required to prove that the maximum curvature of the curve, whose equation is a3y =x(b3-)3), takes place b 3b4 when xa= -and y= —. >~4 443 a 57. The equation of a cycloid being y = a vers' -+ V 2ax- V2, a it is required to prove that the curvature is a minimum when -==o, y=0; and a maximum when x-=2a, y=7ra. Y- -y 58. In the catenary whose equation is 2 - =ea + e, it is required to shew that when = a and y=0, the radius of curvature -a a minimum. 59. Of the curve defined by the equation V/a (y b) = \I' (x - a), it is required to prove that the maximum curvature corresponds to x =0 and y =b. 60. If the equation of a curve be (a - x) y2 =(a -- ) X2, it is required to prove that the curvature is a maximum when x-= -a and y=0. 61. The equation of a curve being a2 (y - a)2 = (2 ax- X2) (i - a)~, it is required to shew that when x- =o and y = a, the curvature is a maximum: when ix=-a and y=a, the curvature is a minimum: when xv=2a and y=a, the curvature is a maximum. 62. Determine the points of maximum and minimum curvature in the curves defined by the following equations: (1) y- 2 ( 1-2): (2) 4y+ — 2 '-4=0 (3) b'y =b'(v a): (4) a( ( a(-y)= xy: 94 (5) y-x+X3=0: (6) v/y=V/'~+/V: (7) ( 2S+y)2 = a2y: (8) by2 = a-by: (9) (et-1) e = e-+ 1: (10) V/xy-a +: (11) a3 y=- x4-bx73-b2X2: (12) (a2 + 2) y= a,: (1S) (2a-x) y- =x(a+b _-)2: (14) a2V2y2 =(a2+y2)3. 63. In each of the examples just given, prove that the order of contact subsisting between the curve and its circle of curvature at the points there determined exceeds the second. 64. Prove that the curve whose equation is 4 - a2iV2 + ay2 =0, has a double point corresponding to x = and y =0: and find the directions of its arcs. 65. The equation of a curve being (a - x) y2 = x (b - w)", it is required to prove that there exists a double point corresponding to x = b and y = 0: and to find also the directions of its arcs there. 66. Prove that the curve defined by the equation (a2 v2) y2 = x, has a double point at the origin of co-ordinates: and find the directions of its branches. 67. The curve whose equation is a2(y -a)2 = (2 _a - 2) ( - a)2, has a double point corresponding to x = a and y = a: required a proof, and find the directions of the arcs which form it. 68. In the curve whose equation is (a2-x2) y2=(a?+X ) X2, there is a double point at the origin: prove this, and find. the directions of the branches. 69. A curve being defined by the equation y- - =( — b) x/ - c, it is required to prove that it has a double point when x=b and y = a: and to find the directions of its branches. 95 70. If the equation of a curve be (y 2)2 ( - l)2, it is required to prove that it has a double point corresponding to x- 1 and y = 2: and to find the directions of the arcs forming it. 71. In the curve whose equation is ay2 =x (a + j)2, it is required to shew that a double point corresponds to x= a and y=O: and to find the directions of its branches. 72. The equation of a curve being xy2 = a ( -- b)2, it is required to prove the existence of a double point corresponding to x = b and y = 0: and to find the directions of its branches. 73. The curve whose equation is y4 -a2 + ay 2 a2- x4 = 0, has a double point at the origin of co-ordinates: prove this, and find the directions of its branches. 74. If the equation of a curve be y4 - -y3 - 12xy2 + 16y2 + 48xy + 4 2- 64x=0, there will be a double point corresponding to x = 2 and y = 2: prove this, and find the directions of the branches. 75. The equation of a curve being y4+ x4- 2ay2 2b2x2 + b4= 0, it is required to shew that the curve has two double points corresponding to x = + b and y = 0: and to find the directions of the branches. 76. The curve whose equation is y4 + - a2 y2 2 a2 X2 + a4 = 0, has two double points when x =0 and y = + a: and also two double points when x = + a and y =0: required a proof, and find the directions of the arcs forming them. 77. The curve defined by the equation vy2 +t3 - (a + 2b) 2 + (2ab + b2) - ab2= 0, has a double point corresponding to w=b and y=0: prove this and find the directions of the branches. 78. The equation of a curve is X4 -2 av3 + 2 a2z y ay2 = 0; 96 it is required to shew that there are double points corresponding to x = o, y =0, and x = a/2, y = 0: and to find the directions of the arcs by which they are formed. 79. The curve whose equation is (x"- a2)2= aay2 (2y +3a) has double points corresponding to x- =a, y=0: mi= -a, y =0 and x =, y = - a: required a proof, and find the directions of the arcs at those points. 80. Prove that the curve defined by the equation ay3 - bX2y + x2 = 0, has a triple point at the origin of the co-ordinates, and determine the direction of its arcs. 81. The curve defined by the equation (y2 + a x)2 = x2 (a2 + 2 as.- x2), has a triple point at the origin of its co-ordinates: prove this, and find the directions of its arcs there. 82. If the equation of a curve be (y2 -,2)2 = a x3 -- 2 bx y2 it is required to shew that the origin is a triple point, and to find the directions in which the curve proceeds from that point. 83. Of the curve defined by the equation y4 - 2x2y2 + 4 + 2 axy2- a 5a3 = 0, the origin is a triple point: required a proof, and determine the directions of its branches there. 84. In the curve whose equation is y4+r4-4ayx3+2ayy2+2ay3 a3+8 a 2y-4a2 Xy-8y a' +2a4=0, it is required to prove the existence of a triple point when.x= -a and y =a. 85. In the curve whose equation is I4 —ay'3 +- 2cax'y + 4a3 + 3a2y2 +4a2xy' + 4a2' 2 a3y =0, it is required to prove the existence of a triple point corresponding to x - = -a and y =-a. 97 86. It is required to prove that the origin itw'the curve whose equation is (xi" + y2)a = a (X2 _ y2)2, is a quadruple point and to find the directions of the arcs meeting there. 87. -In the curve whose equation is y-a = (-b)2' x Cit is required to prove that two *branches of the curve have contact of the first order corresponding to x = b and y1= a. 88. In the curve defined by the equation Y - a=Q (- b)3 VvX - c, there will subsist a contact of the second order between two branches of the curve corresponding to = b and y = a. required a proof. 89. Determine the positions and natures of the multiple points in the curves defined by the following equations: (1) (x'+y')1=4a'xy: (2) y'(2ax-a ')=(ax-x')': (3) (a+a)2y2=x2 (3a2_2ax-x 2): (4) (I-x)y2=x (2-x)2: (5) y2(2a-x) =a (b22ax+a<): (6j Y 2 ((b-)2: (7) y3 - ax2 X3 = 0: (8) a (b+y)2=,X2 (c +): (9) (aA +yy) (2a- x)=b2a: (10) a2 ( y2 - = X2) y2: (ii) ay2 -_X3v —b2=0: (12) ay +y — bx&'=0: (13) ay2-2aby-x=~civ2-ab': (14) 2am(a2+y)=a(x2-y2): ('15) (y/2- X2)2+6ary26as-7ax-4a2 a2a?2-20a'x+8a4 —0: (16) y' + my2 - 8ay" - 4aa Xy2 -4a X22y~+ 1Qa2y2+ 16a' 2y+5 a2 2 - 12a y3 —14-'14 — 3a4"= 0. 90. Prove that the semicubical parabola whose equation is ay2 =, has a cusp of the first species at the origin, the tangent of which is the axis of x. 91. The curve defined by the equation a (y-a) = ( has a cusp of the first species corresponding to iv = a and y = a: required a proof. 92. It is required to prove that the curve whose equation is a (iv y)2 - (, + y)3, has a cusp of the first species at the origin, and to find the direction of its rectilineal tangent, N 98 93. The curve defined by the equation a ( +y- 1)= (x-y + 3), has a cusp of the first species corresponding to A v= - 1 and y= -2: prove this, and find the direction of its rectilineal tangent. 94. The equation of a curve being a ( -y + ) = ( + ( y- 5), it is required to prove that there exists a cusp of the first species when x=2 and y=3: and to find the direction of its rectilineal tangent. 95. It is required to prove that the curve whose equation is x (y - )2= (2- )3, has a cusp corresponding to x =2 and y = 1: and to find the direction of its rectilineal tangent. 96. Prove that the curve whose equation is (m2 + y2) ( - y) a (i + y)2 has a cusp of the first species at the origin of co-ordinates: and find the direction of its rectilineal tangent. 97. In the curve whose equation is a3 (2y + )2 =2 (2b -,), it is required to prove the existence of a cusp of the first species corresponding to =2 b and y= -b; and to find the direction of its rectilineal tangent. 98. If y=a cos" 1 ( ) — +/a-v' be the equation of a curve: prove the existence of a cusp of the first species corresponding to x ---0 and y =wa: and find the direction of its rectilineal tangent. 99. The equation of a curve being (y - - a)2 = (_-a)9,< it is required to prove that there is a cusp of the first species when x = a and y = 2 a: and to find the direction of' its rectilineal tangent. 9 9 100. The equation of a cur ve being. jtis, required -to shew that there is a cusp of the first species corresponding to v 1 and y =0: a nd to find its tangoent. 101. The - equation of 'a curive be~ing -'ay (x - a), it is required to prove the existence of two cusps of the first species corresponding to v a and y =_ 0: and to find the directions of their tangents. 10,2. The curve defined by the equation a2y2 2 a%,2y = 3x'~bx3, has two cus ps ''of the~_ flirst species corresponding to x0 Y 0, ad x b y= required a proof: and find. the" positions of:their tangents..103. The equation of a curve being 4 ay= (X - 2 it is required to prove the existence of two, cusps of the-flrst speces crresondig to v =+~a and y=0: and, to fi nd thedirections of their tangents. 104. Linthe curve defined by the equation, 5 'y-a=bi,2 ~"e- i)2, it. is required to -prove -the existence of a cusp of -the second 'specie s correspon'ding, to x =1 and y =,a +b: and to find the position of its' rectilineal tangent. 105. If the equation of a curve be (2a+ _ - X2 =X Ga )2 ~it 'is required to priove the existence of a cusp of the second species whrien x =I1 and y =- -I1: and to find the direction of its- taget-irt i00 10&6d A curve being defined by the equation y- 2 = (1 + x + ), it is required to prove that there is a cusp of the second species belonging to - = 0 and y = 2: and to find the directior of its tangent. 107, If a curve be defined by the equation y + + 1 = - S_2x2 there will be a cusp of the second species corresponding to =0 and y= -1: prove this, and find the direction of its tangent. 108. in the curve defned by the equation V - axay - asvy2 + ~a2y2 =0 it is required to prote the existence of a cusp of the second species at the origin of co-ordinates and to find the direction of its tangent, 109. Determine the positiotis and species of the cusps belonging to the curves defined by the following equations: (1) (y_ )2 =: (2) (y= XI2 = b) (3) (y T b 7 )3 3 (4) (y-a)=r ( —b)2: (5) (2y-+ m + 1)2 = 4 (1 - <,)5, (6) (y —m-a)"2= (-a)3 (7) (y -— a)2= (aX2 + b- )2: (8) a (y- )"= (,-1).: (9) ( )3 )210) (y-a() y=( b)2: (1i) y-a=b (s-~)+*-a) a)' (12) '3==a9-2+.3: 2 2 2 (IS) 2 (y2 + l) =2 + (4 - b ( - = (2a: 2 2 2 2- 2 (105) ( (b)y) - =(a- 2): (a6) (y+ - -)(y ) {18) + aa a (17) i2+2: y _+!- ae2 -a Sax yG - a c23 = +: (18) t'4 Y - _f x --- y 2 ta x2.yy2 - ay U 0.O 101 110. If the equation of a curve be 4+ x3y xy2 + ay 2 y + ay02 + a'2y = O, it is required to prove the existence of a point of complete Embrassement at the origin. See the Diagram of page (326). 111. In the curve defined by the equation a3y2 _ 2 ab2 y =, the origin of co-ordinates is a point both of inflexion and osculation, commonly called a point of OsculinJfexion: required a proof. 112. The curve whose equation is 4- b xa3y2, has at the origin of the co-ordinates a point of Rebroussement with a double inflexion; required a proof. 113. The curve defined by the equation cy2 = (a + b) cy c33-aba + cx33y23 has a multiple point at the origin of co-ordinates: find its nature and circumstances, 114. The curve whose equation is a2y3 -a by2 + a22y + X5 = 0, has a point of Osculinflexion at the origin: required a proof. 115. In the curve defined by the equation y5+a 4b2'xy"2 there is at the origin of co-ordinates a point of inflexion with a cusp of the first species required a proof. 116. The curve defined by the equation y4 + x4+2 bx2y=0o has at the origin of co-ordinates a triple point with a cusp of the first species: required a proofd 117. The curve whose equation is xa5 - ay4 + 2b xy + b"y2 = 0, has at the origin of co-ordinates a triple point with a cusp of the second species: required a proof, 102 118. Thfe curve ywhose equLation is X ay' + t:ba3 y ~ -~b~x2= ha's'at 'the origin' of co-ordinates a triple- point with a point of;.EmbrassemIent:~ required' a proof. 119. The-curve defined- by the equation IT a1 a -y4L a. Xy -p a-'2 has -at ilhe'origin of co-"Ordinates, a triple point withi a point of E~mbi'ssernent: required a proof. 120, The curve whose equation is a'y" - b?y ~ xV 0, has at thie origin of co-ordinate's a triple" lpoinit with a point Of Rebrottssemiert: required aproof. 1 21. In the curve whose equation is x6~2 a~x~y - by'0 there is at the origin of d'o-ordinates a triple point- with a point of Rebroussernent: required ai proof. 122. In -the, curve whose- equatio'n is it is required to find the nature of the point correspondings to x= b. 123. Find the nature ofe the point corresponding to =1and y -, in -the curve whis eution 'is 124. If a curve be defined by the1- equation y-a= e-i4+(eit is' required to' find t he nlgaure, of the point. belongingf to:, =b. 125. Find the nature of the point corresponding to X r- in the cuL'rve 'whose equation is g a '=e +ib (x+ (1)2 + C('V + (12 103 126. Required the magnitudes and positions of the greatest and least radii vectores in the curve defined by the polar a equation r = - e co 1 + e cos 0 127. Find the positions and magnitudes of the greatest and least radii vectores in the curve whose equation is r2 - a2 sin 2 0. 128. Determine the maximum or minimqnum radius vector in the curve whose polar equation is r cos 0= a (1.-cos 0). 129. Find the magnitude and position of the maximumi or minimum radius vector of the Lituus, whose equation is r"0 = C2. 130. Determine whether the spiral whose equation is r (e0 + e-) 2a, admits of a maximrnm or minizmum radius vector, and decide which it is. 131. Find the singular points of the spiral whose equation is r=a sin 20. 132. Determine the singular points of the polar curves whose equations are r'- a2 cos2 0 + b2 sin' 0. 133. Find the positions and natures of the singular points in the curve whose polar equation is r a ( + cos 0). 134. In the polar curve defined by the equation 2r (sin3 0 - cos3 0) = a sin 20, it is required to find the positions and natures of the sin-, gular points. CHAP. XI. MISCELLANEOUS EXAMPLES. 1. IT is required to trace or describe the common parabola whose equation is y2= 4ax. See the second Diagram of page (304) with the axes of x and y interchanged. 2. Trace the parabolic curve whose equation is a y3 = v4. See the second Diagram of page (304). 3. Determine the form of the curve defined by the equation ay2 = 3. See the Diagram of page (338). 4. Required the form of the curve whose equation is a2y = 3. See the Diagram of page (303). 5. Trace the curve defined by the equation y3 = a2. See the Diagram of page (303) reversed. 6. Trace the curve whose equation is (y- a)2=4a ( - a). See the second Diagram of page (304) with a new origin and the axes interchanged. 7. Find the form of the figure belonging to the equation a (y + b)3 = (x - a)4. See the Diagram just referred to. 8. The equation of a curve being a (y -b)2= ( - c)3, find its figure. See the Diagram of page (338) with a new origin and parallel axes. 9. What is the form of the curve defined by the equation a2 (y - b) = (x + a)3? See the Diagram of page (303) with a new origin and parallel axes. 10. Trace the curve whose equation is x4 - bx3 = a3y. See the Diagram of page (346) inverted. 11. Find the figure belonging to the equation 4 + a3y + b3 = 0. See the Diagram, last referred to, reversed. 12. Trace the curve defined by the equation 4y = x?3 32 - 4x + 12. See the first Diagram of page (302). 13. Find the form of the curve whose equation is y = 3 + 2# - x - 2. See the Diagram, just referred to, reversed. 14. Trace the curve defined by the equation y+-1 = — (1 + 4x —44x2). See the Diagram, last referred to, inverted. 15. Determine the figure of the curve whose equation is a2y + c2 --;.3 = 0. See the second Diagram of page (302). 16. If the equation of a curve be y3- a-y2= b", find its figure. See the Diagram, last referred to; with its axes interchanged. 17. The equation of a curve being 4 (y -1) = x2 (2 - 5), it is required to find its figure. See the first Diagram of page (304). 18. If the equation of a curve be y4-13y2= 36 ( -1), it is required to trace it. See the first Diagram of page (304) with interchanged axes. 19. The equation of a curve being 9 (y — ) = -03, it is required to describe it. See the first Diagram of page (306). 20. The equation of a curve being 4 (2y -a)= c3 (x2 -5), it is required to determine its figure. See the Diagram, last referred to, inverted. 21. If the equation of a curve be 6 (2y +1) = X2 (4 6 X2 + 11), it is required to find its figure. See the Diagram of page (307). 0 106 22. The equation of a curve being y6-9y4+26y2=24(l -X)9 it is required to trace it. See the Diagram, just referred to, inverted. 23. Trace the curve defined by the equation ix2y=a2 (vx-y). See the Diagram of page (287). 24. Describe the ellipse whose equation is y2 - 2y+ $3x2 10 + 12 =0. See the Diagram of page (291). 25. Trace the rectangular hyperbola whose equation is xy=a (x + y). See the first Diagram of page (289) with a new origin and parallel axes. 26. The equation of a curve being y2 (1 + i) = x2 (1 - x), it is required to trace it. See the second Diagram of page (289) reversed. 27. If the equation of a curve be x4- a2 + a2y2 =0, it is required to trace it. See the Diagram of page (294). 28. The equation of a curve being (y2+ X2)2= 2a2 (y2_ X2), it is required to trace it. See the Diagram, just referred to, with axes interchanged. 29. Trace the curve whose equation is (y - 2)2= x (x - 1)2. See the Diagram of page (317). 30. Trace the curve defined by the equation a(y -b)2 =(-2a) ( - a)2 See the Diagram just referred to. 31. Find the form of the curve whose equation is a3y +x (b + x)3 - 0. See the Diagram of page (348) inverted. 32. Describe the curve whose equation is a'3y = (b + x)3. See the Diagram, just referred to, reversed. 33. Trace the curve belonging to the equation a'x =y (y-b)3 See the Diagram, last referred to, with its axes interchanged. 1o7 34. If a curve be defined by the equation a3y = x2 (b + a-)2, it is required to trace it. See the Diagram of page (349) reversed. 35. Trace the curve whose equation is a3y + (a2+ b)2= 0. See the Diagram, last referred to, inverted. 36. The equation of a curve being y -3 = a3, it is required to trace it. See the Diagram of page (350) reversed. 37. If the equation of a curve be 3+y3+a3= 0, it is required to trace it. See the Diagram, just referred to, inverted. 38. The equation of a curve being (s + 2) y + (~ + 1) ( + 3) = 0, it is required to describe it. See the Diagram of page (352) reversed. 39. If the equation of a curve be (x + 2) y= ( + 1) (x+3), it is required to trace it. See the Diagram, just referred to, inverted. 40. A curve being defined by the equation (1 2) y = (1 +a 2), it is required to trace it. See the Diagram of page (353) reversed. 41. If a curve be defined by the equation y3 s3- a2x, it is required to trace it. See the Diagram of page (354) reversed. 42. Trace the curve whose equation is y- 1=(a- 1)V2/-!2. See the Diagram of page (357). 43. Trace the curve whose equation is (2 + y2)= a2xy. See the Diagram of page (294) with its axes changed through half a right angle. 44. If the equation of a curve be (1 + x) y-= (1 - ) w, it it required to trace it. See the first Diagram of page (364) reversed. 108 45. Trace the curve whose equation is (c + ) y= (-1) aSee the Diagram, just referred to, inverted. 46. The equation of a curve being y3 a3- +ax2, it is required to trace it. See the second Diagram of page (364) reversed. 47. If the equation of a curve be y3+-a3+ aa2=o, it is required to trace it. See the Diagram, just referred to, inverted. 48. A curve being defined by the equation y4 + (a2 + 2) y2 2 a2 2 it is required to trace it. See the first Diagram of page (365) with its axes interchanged. 49. Trace the curve defined by the equation 4aZxy- (X2 + y2) (v + y)2. See the Diagram, just referred to, with its axes changed through half a right angle. 50. Trace the curve whose equation is (2~ - A) y2 = _ (_m - I) (a2) See the first Diagram of page (366). 51. Trace the circle whose equation is (y- a)2 = a2 - x2 finding the position of its centre and the magnitude of its radius. 52. Describe the equilateral hyperbola as defined by the equation (y- a)2 = X2 -a2. 53. Trace the rectangular hyperbola as defined by the equation x (a - y) = y. 54. It is required to trace the circle as defined by the equation x2 + y2 = aix + by. 55. Trace the common parabola by means of the equations /y + V/ = V/a. 56. Trace the common parabola whose equation is (y- )=3y. 57, Trace the ellipse as defined by the equation y-2 2 xy + 3i,2 + 2 y - 4 3 =0o 109 58. Describe the hyperbola belonging to the equation ya - 6vy+,-2+2y-6x+ 5=0. 59. Trace the curve defined by the equation ay4 - ay4 _4+ b8 o. 60. Trace the Cissoid of Diocles as defined by the equation ( - y) (x2 + y2) =a (x + y)2. 61. Trace the curve defined by the equation a2y2 + 2y2 = a4. 62. Let it be required to describe the curves defined by the following equations: (1) y2 (a- x)= 3: (2) xzy2+ a2x = (3) ay2 = 2 (b + ): (4) a4 = ay2 + 2y2: (5) + by3=a y: (6) ay2=2 ( - b): (7).(y + )2 = (2 a- )y: (8) (a2- 2) y2 = 4: (9) v (_2+y2)=a (2_y2): (10) (2 a_-)=) (a-_)2 (1i) (a2-X2) y2=(a-_)22: (1) ( a+ )2y2=(a2+a)2) a2: (13) (a- )2 ( + y2) a2y2: (14) ay2=(2 + a2): (15)?2=y2=+(a+) (a- )2 (16) (a2- y2 )y2=(2a2-a?2)2 (17) a2y-2y =a3: (18) (X?2 —3?+2)y=x —3. (19) (ex -l) y = 2 e: (20) (x -1) = e + 1: (21) ay2 —bwy= 2(,v-b): (22) 4+2a?2y=ay3: (23) 4 + y4=2a y'2: (24) (x- a)y2=(2a - ) 2 (25) (?2+y2) y2=(ax+by)2: (26) (x+y)2 (e'+y2)=4a2a: (27) (X2 y2)2 =-a2a2 + b2y2: (28) a33+ b3 Y= ab3: (29) (? + a) y2 = 3-a3: (3o) y2 ( - a)=a - b 3: (31) a2y =4-b4: (32) ay2 (x- a)=a4: (33) aw=b(y2+ y): (34) a2y = 2 (b-a ): (35) a2fy=X (b2 2): (36) y ( _2-b2) =,2( -a) (37) (y - )4 = —a 4: - (38) (y - 1)2 (2 + 3) t2 (39) axy = x3 -3: (40) m22 = (? _ y2)2 (41) (Xa2 ) -y=b2: (42) 2^=a2 -=: 110 (43) (45) (47) (49) (51) (53) (55) (57) (59) (61) (6:3) (65) (67) (69) IVy2 =(x-9)QvX- 2)2: (46) a2 (iv2 _ y2)2 =( (2 + y2)3: (48) ay 2=x (b 2+ 2): (5o) (x 2+ yC) y=a (X +y)2: (52) _V)2 = a2 ~ (54) (x + a) y2 =b2(x + 2a): (56) =v+ 4 2a y2 — b Xv2y: (58) y =sin X: (60) y =sinwX+ 2sin 2: (62) y =Cos-,(I1- X): (64) ay=- sin2 x: (66) y=e ox:(68) 5 y = bv+ ( - a)-: (70) (y,2 - 2)3 -33 iv (a2 + iV2) y = 3 ( + 1)2 y =(iv- 1 i 2: y3 - y= b v3: ay 3+ b X3 =abxvy: (xv - a) y2=(xv+ a) 2 (a -x)y2= a-2 (Xvy+ 1)2=(2 1 i-)3 y = sin iv + cos X y =sec x-"tan x: y =tan-'(a + ): y =(log X)3: 5 y b+ ci2 +I (x - a)2:~ (72) (29a +iv)y2= (2 a -v)(a +iv 63. Investigate the' figure of the curve belonging to the. equation y4-ay- 2aiv2-.' v+ a,4= 0 and -find the posi-' tions and natures of all its singular points.' 64. Trace theI curve defined by the equation (y - a X)2 = &2(4 a2 —2): and find the position s and natures of all its singular points. 65. Determine the figure of the curve whose equation is /-2aiv+ 42 is )= and explain, the natures of its singular points. 66. 1 Trace the curve and find its singular points when the equation is (y2 - a2)2 = aiv2 (2ivx + 3a). 67. Trace the curve whose equation is a~'1y =x (x - a)', both when m~ is an even and an odd number: and find the numbers and natures of its singular points. Ill 68. Investigate the form of the curve whose equation is y = 2 sin 2 - sin x: and find after what values of x the form of the figure recurs. 69. Let it be required to trace the curve whose equation is 4 r- a x2y + by3 =o, distinguishing particularly its singular points. 70. Trace the curve belonging to the equation a3 (y - )= ( - b)4 (C-c), explaining fully the natures of its singular points. 71. If the equation of a curve be a' (y - a)2 = ( - b)' ( - c), it is required to trace it, and to distinguish fully the natures of of its singular points. 72. Trace the curve belonging to the equation a3 y2 -2 a b '2 y = - 5; and particularize the nature of it at the origin of co-ordinates. 73. Investigate the figure of the curve belonging to the equation a4 + x4 + y = 2 a' (X2 + y2): and find the natures of its multiple points. 74. Trace the curve whose equation is (y2 + x2)2 - 6acy2 = ax2 (2x - a):.and find the natures of its singular points. 75. Trace the curve whose equation is y2 = a ( - b) ( - c); and distinguish the different cases depending upon the relative values of b and c, regarded as possible and imaginary. 76. If the equation of a curve be y'=y2 (a-x) (x-b) (x-c), it is required to trace it, and distinguish the various forms of which it admits. 77. Investigate the figure of the curve called the Trident, as defined by the equation cxy = ac3+ bc2 2+ c + e: and distinguish the different cases originating from the forms of the roots of the equation a '+b2 +- C + e = O. 112 78. Investigate the form of the curve expressed generally by the equation xy2 + ay = b + c a: and distinguish particularly the cases wherein c is positive, negative and evanescent. Sir Isaac Newton in his Enumeratio Linearum tertii Ordinis, designates the curve belonging to the equation last given, by the name of the Hyperbolism of an Hyperbola, Ellipse or Parabola according as c is positive, negative or evanescent; and the reason he assigns for it is the following: Solving the equation proposed with respect to y, we have -a /a 2+4ba +4ecx2 y — Y - 2z and if the denominator of this expression were an invariable quantity, it is obvious that the equation would belong to an hyperbola, ellipse or parabola, according as c is positive, negative or evanescent: whence, if such constant quantity be replaced by the variable quantity 2x, the conic section becomes hyperbolized by having in each case an infinite branch at the origin of its co-ordinates. 79. Trace the curve defined by the polar equation r cos2 0 = a sin 0. See the second Diagram of page (304). 80. Find the figure of the polar curve defined by the equation r92 cos3 0= a" sin 0. See the Diagram of page (303). 81. Trace the rectangular hyperbola as defined by the equation r2 sin 20 =2a". See the first Diagram of page (289). 82. Investigate the figure of the curve whose equation is r" cos4 0 = d cos 20. See the Diagram of page (294). 83. Find the figure of the curve defined by the equation r=- a sec 0- b. See the Diagram of page (312). 84. Describe the figure of the curve whose equation is r = a2 sin 20. See the Diagram of (294), with the direction of the fixed axis changed through half a right angle. 85. Trace the figure of the reciprocal spiral whose equation is r-=a0-~. 113 86. Determine the figure of the logarithmic spiral whose equation is y = a. 87. Describe the lituus as defined by the equation r20=a2. 88. Trace the circle as defined by the equation r = a (cos 0- sin 0): and find the values of its maximum and minimum radius vector. 89. Describe the figure of the rectangular hyperbola as expressed by the equation r = a (cosec 0- sec 0). 90. Trace out the figure of the cissoid of Dioces by means of the equation r cos 0= 2 a sin2 0. 91. Find the form of the cardioide by means of its equation r = a (1 + cos 0). 92. Required the form of the figure called the trisectrix, whose equation is r = a (2 cos 0 + 1). 93. It is required to trace out the figure of a rectangular hyperbola by means of the equation r cos 20=2 a cos 0. 94. Required to trace out the figure belonging to the equation r sin (a + 0) = a sin a. 95. It is required to give graphical representations of the curves defined by the following equations: (1) r=aO: (2) r=asin20: (3) r2=-a2 tan 20' (4) r= a (sec - tan 0): (5) r=acos 0: (6) r=a sec 0 tan2 0: (7) r=atan0: (8) r= a sec 0 (9) r2=a2(tan2 0-1): (10) r=a(+2 sin 10): (11) r= a log tan (45 + -0) (12) r(e + e- )2 a: (13) r2(e —1)2=4a2eO: (14) r(0 2- )=a02: (15) r sin 20=a sin 0 sin 30: (16) r (cos 0 + cot 0) =2 a. 96. Trace the curve whose equation is y3 -3 axy + 3 = 0, 3ay by assuming y =, and therefore x = 32. P 114 97. By means of the same assumption, determine the form of the curve whose equation is y5 = aX4 + m5. 98. After the same manner it is required to trace the curve defined by the equation x5 + a3xy-y=5 0. 99. Apply the same principle to the describing of the curve whose equation is a + -xy2 +3 - y3= o. CHAP. XI. MISCELLANEOUS EXAMPLES. BY reference to the Principles laid down and exemplified in Articles (256) -.(282), the following Theorems and Problems may be solved. 1. Given u = x2 (a + y)3, to find the corresponding value of u' =(x + h)2 (a +y+k)3, by means of the Theorem of (259). 2. Apply the same Theorem to find the values of u' when u = 3 logy and u =?2 sin2y. 3. Also, to find the values of u' from the equations u =,I2 (log y)" and e = e" (sin y). 4. Again, to determine the values of u' if u=sinxec~v, u = log sin ey and u=e"Ysin (x +y). 5. It is required to verify the formulae established in Article (261), in the following instances: V2 CC 2_ Y2 (2) = _ ---.-: () =+=- — y: x~ +2 2 2y ( 3 ) US = 2 2 ( ) e (5) = = + -- (6) 2 _ --- (7) u= e''Y(e +eY) (8) = log (t+ Y) / ' -y (9) = ^er" (Io) u sin- \x x " so~ 116 (11) u=logx x2y (12) u=logtan - y y (13) _u=,x sin y-y sin. (14) U -log(15) = (sin )COS: (16) u=cos{cos + cosyl. Xm 6. If u=-, it is required to prove that xm-1 du =- T (mydx-n xdy). y"91 ay axydx - ax2dy 7. If = 2 --- then will du =- -y -ad y 2 Txy (ydx —xdy) 8. Itf U 2 ( then du 2 2) 4_ y ItV + (+ ) ^^^ 9. If u=log /, then du = y - mdy,\/,_ - 2 y/ 2 -V ydx yV2-dy 10. If u=tan-'-, then will duz d -= dy y 72 2+ y22 11. If u=sin-' N/ du= yd-mdy +' (x + y),/2 y( -y)' 12. If u=tan -, then will duc- yd.-dy sece f y Y2 y 13. If u=sin-', then will du= yd - y yV/y2_ 14. If u=xY, then will du-y x'-ldx +xYlog dy. 15. If u= sin +yn sin, then we shall have du = (sin y + y cos x) dx + (sin x + x cos y) dy. 16. Let it be required to find the total differentials of the following functions of two independent variables: 117 (1) tu= — 3 ay +y3: (2) u 2y + 3y3, + - y2: (3) u= + y +/2y + y2 (4) u = 2++y2+a V2 -y2: xf: x2"3 + y3 (5) c=- -: (6) z +: x+y x-Y v Y?+ x27-^ (7) ( = _ (8) y= ~ + 9 Y/-2 Y2 y -2/ + (9) u =log (X +,/x2 + y2): (10) u=log (x y-V/2y2-1) (11) u=sin ---: (12) u=tan-1 -- x+f y - y xy (13) =cotx': (14) u=sin-l Vx2+y2 (15) - y= Y - - lg (: (16) = sec- sec-1 a —y gy (17) u=\/x2+y2+a tan-' - (18) u=sin-1 (cosx —siny). Y 17. If uZ==yz, then it is required to prove that du = yzdx + xzdy + xydz. 18. If u= xy —xz+yz, it is required to shew that du = (y- z) dc + (x + z) dy- ( -y) dz. 19. If u = --, it is required to prove that du= - (yzdx+2z2dy - 2,vzdy -+ydz - 2yzdz). y1 xy ax2dy+bxzdy+byzdx-bxydz 20. If u= -- du= - ax+bz (ax + bz)2 21. If u = /x2-y2+z2, then d= — ydy zd /x -- y + z 22. If u=a( (x) + by, then du=-a' (x) dx + bdy 23. If u = x2( (y) + cx, then will du = {2xP (y) + c} dx + x2: ' (y) dy. 118 24. If u = x (x + z) + y (y + z), then will du = (2 +z) dx+ (2y+z) dy+ (x +y) dz. 25. Required the total differentials of the following functions of more than two independent variables: / y-f-% 2 < 2 (x + Z) y xyz (3) Y=: (4) -= - —: y+z — y+z (5) u = log (x + y)2 (y + z): (X2 + y2 s x22 t x yz (7) u=- 22 (8) u= t+-+ - (9) ( xy, Y ) (10) u=(y y ') 26. If u - X3 -2y + ay2 + y3, it is required to prove in accordance with Article (272) that dzu du d2u d2u d2 2 6u== Idx 2+2 d-dyy + dy2; d 2 dxy dyd 27. It is required to verify the same formula in the fol lowing instances: (1) u=x +y2:,3 _ y3 (3) a= - -: x +y_ (5) u=xa/2axy+yy: (7) u=sin N/': 1 —y in +n (9) u=log sec-' / '+y (11) e"=x-y+~/Ro-y+-y2.: (2) u=( + y) (I3- y3): x2 _ y2 (4) u 2 () (2 + y2)2 (6) u =x - y + x/x'2 + y2 3 ' (8) u _tan-. +y X + y (10) U= --- sin (x - y) (12) u -e Ytan- (x +y). 119 28. If (x + y + z) ez = c, prove that by eliminating, the constant quantity and exponential function we shall have dx +dyt+dz + (x +y +,) dzO=. 29. if z -F y t = (y + z), it is required to prove that (yz +z2) dx + zt- xz) dy +(xy -yt) dz-+ (y2 +yz) dt =. 30. If u =f (z) and z = (x, y), it is required to prove du du_ dz dz that -- -; and thence to shew that the indedx 'dy dx dy terminate function f of a determinate function z of two variables x and y may be eliminated. 31. If the indeterminate function be eliminated from the x equation z - i-f (ay - bx), it is required to prove that a dz bdz dx dy 32. If z'= then will px +qy O, where P zand q = 33. If z =aVX2~2+j (f) then pv +qy ='a 'V2 y2. 34. Ifz=xy+P(k ) x thenwillpxz+qyz=xy. 35. IfZ= ~ ( +logx), then will ap -qx =. a 36. If z=- 1ogy+44 x,, then willnpx +mqy=a37 IfZ (X + y)mp95(X2 _y2), then will py +qx m%. 2 38. Ifx S+, y), then Will qXy =pX2 +y2. 3X 120 39. If x= b - m then willp + qy + =o. 40. 'Ifz=jS -(Y ), then will 2p y +q(4y2)=0 41. If 2 _ 22 == 2 z (c4 —) 9 then cap - qx- + vy= o. 42. If a?2 + y2 + 2 = (ax + by + t), then will (y - b)p - ( - ) q=b-ay. 43. If x'-' (x +y + z)= (y) = then will y. m (X +y+Z)= -px- qy. 44. If =ea f (x - y), then will a (p + q) = 45. If Z22=x2+ (-), then will px +- qy z= 2. 46. If z = e yf(x + y), then will (p - q) (x + y) z, 47. If sin' -1 =log - (y2 + 2), then will /y2 + z2 px +qz + y= o. 48. If = tl.f (;' ), then we shall have dz dt 49. If (m - 1)4- = XI' (, -) 9 then will + t I y +t+ t ) t 50. If = ( -y)2 ( + y + t), (x _ t)2 (X + y + t) then will p (t + y) + q (t + x) + v (- + y) = o 51. If y +t= y t ( =y (x — Z y), (xy - xt)}, then will p +q ( +t) + ( - +y)= t+y. 121 52. Required to eliminate the indeterminate function 4 -x2y2) from the equation y = ) - ' Xa}2+y 2 X ' 2 + y2 y2>\X2Sy2/ m 53. Eliminate the indeterminate functionf (X2 2) from the equation z(p (y) =f x2 +y): and shew what the result becomes when P (x y) = sin xy. 54. Eliminate the indeterminate function from the equation - =f ( -- ): and find what the result becomes when -x - a Ya =/ = 3y = 0. a x f(y) 55. If z= - r- + (y), it is required to prove d2z that rxy + 2py = a, where r = -2 56. If z =i 73y2 +f (x) + (y), it is required to shew d2z that s - x2y = 0, where - = s. dox dy ay log x f (x) 57. If y= y g. + + b (y), it is required to mt+1 ym prove that sxy + mpx — ay= o. 58. If =- 4( + y +- z)2 +f(y, z) + (x, z), it is required to prove that s2 = + y + 59. If = 2y2t2 + F (, y) f (x, t) + c (y, t), it is d3z required to prove that dddt =-yt. d- dy dt 60. If =F (y +ax) + f (y+a x)) + p(y + bx), then d3z d3z d3z d3Z -(2a+b) (a2+.2ab) - a2b = 0. dx3 drXdy dxdy2 dy3 61. Let it be required to eliminate both the indeterminate functions from the equation z = xf (- +q () - Q x Q 122 62. Eliminate the indeterminate functions from the equation z = xm F () -.Y 63. From the equation z = F(y + ax)f(y-ax), it is required to eliminate the indeterminate functions F and f. 64. Eliminate the indeterminate, exponential, &c. functions from the following equations: Cy (1) -Z=^Y +f: (2) z - +f(ax -by): \x/ a (3) z x 2 +f (y + log x): (4) z=ysin-1 + (y) (5) z -=f (y)- y/ '2 (6) log z-f (y) - (7) (-y) Z=e-yf(xy-y)': (8) sin =f (z + \/ -: (9) z=fl (log +2 /y): (10) z=log-l{y- +- L (11) z=/ 2 + y2/f {tan1 - log 2-2+ y: y (12) Log z=b (y-a ) +log {f(y+-ax)+ P(y-ax)}: (13) = ( )log (x2 + y2) - - (+6xy2-6y3 tan' - 6y 9y y + ~ (2 + 3y2)f (X) + (y): (14) p+aq=f Y() ((15) ap-bq=f(ay+bx). (16) p = qf (: (17) P+ q=(p -q) f (): (18) z=V/yf (+P (xy)' (19) z=e'{(y —ax)+0 y}. 65. From the equation z = cx +f (fv + y), it is required to prove that r2 - t2 = rt - s2. 66. From the equation - y + ay =0, it is required to find the values ofy2 and y2 by Lagrange's Theorem. 123 67. From the same equation determine the values of the functions y-m, log y, sin y, el y, log (2y ) and sin (1-. 68. It is required to find the sum of the mth powers of the roots of the equation a - by + cy2 = 0. 69. From the equation 1 -y + y2 =0, find the values of the functions y", y-", ey, log y, sin y, sin-l y and tan ) 70. Given the equation xy" - y + a =0, to find the values of y2, y-2, y-, eY, log y, sin y, cos- y and tan-' 3y. 71. Find all the roots of the equation y3 - 9y +28 = o, by means of the Theorem of Lagrange. 72. Given the equation 1 - y + ay= 0, to find the values of yl, y~-", log y, siny and sin-' 2y. 73. Apply Lagrange's Theorem to deduce sin y from the equation y= 1 - x sin y. 74. Find the values of u, sin u and sin mu in terms of 0, from the equation 0 = u + e sin u. 75. Required the value of y from the equation I + x sin y = tan y, by means of Lagrange's Theorem. 76. Apply Lagrange's Theorem to find the value of z in terms of y, from the equation ay3 4 zy' — a 23 = 0. a(1 - e2) 77. From the equation r — a ---- e' it is required to 1 + e cos 0' find the values of e and log r in terms of the rest. 78. Apply Laplace's Theorem to find the value of ey from the equation y =log (z? +x sin y). 79. If u2 = (x - a)2 + (y - b)2 + (z - c)2, it is required to prove that ( ) ( ) d2 (I) + + - ---- =o., dx' dy2 dz' 124 80. If u be a homogeneous function of x y, z, &c. of rn dimensions, it is required to prove that (M- 1) du= rdp +ydq + &c. where p, q, &c. are the partial differential coefficients of the first order relatively to x, y, &c. 81. Having given nt=0-2e sin O+1 e' sin 2-0- e3 sin 30, it is required to apply Lagrange's Theorem to prove that *O = nt + (2e 1e') sin nt +e2 sin2nt e3 sin 3nt, where the powers of e higher than e" are neglected. CHAP. XIII. 113Y means of the Principles laid down and illustrated in Articles (283)-(989), it is required to solve the following Theorems and Problems.. -if U=x2y+xy 2- axy, it is required to prove that when x= a and y=4a, the value of U =O, which is a minimum. 2. If u==x'-3axy+y', then will x = a and y = a render u= - a, a minimum: also, if x=O and y=0, then u=O, which is neither a maximum nor a minimum: required a proof. 3. It is required to prove that the function u = a'bx+ a2cy-b X2y cxy, admits of neither a maximum nor a minimum. 4. In the equation u = X'y3 + x'y4 - ax'y", it is required afi to shew that u = -, is a minimum when x =I a and 4~32 - y =la. 2 ~ 5. If u a2u-a#y - 2a Xy + X44y -_2ax'y2 ~ "y2 +'~x'y3, it is required to prove that corresponding to - a and y =a the value of u is a maximum, andfind it. 6. If u = ey + Xy 3- a2X2 - a-y2, it is required to shew that neither a maximum nor a minimum can take place. 7. If u=X2y4- xy - nax2 - ay2_3 ax y + 2 a2~V2a2y, it is required to prove that corresponding to x = 2 a and y 2 the value of it is a mtaximum, and find it. 126 8. If u =2a'xy - 3ax'y - ay' + Xly + wy3, it is required to prove that when x=a and y-1 athe value of u a='. 31 4~ a minimum: when x=4a ard y -ia, the value of u = a, a maximum: when x=-I-a and y'= -'-a, the value of u =1a4, a maximum: and when x= a and y -a, the value of u - a, a minimum. 9. Given ut=x4+ y4-4axy', it is required to shew that when x=a~- and y=24a, the value of u= - a, a minimum. 10. If 0u =y2 (a - x - y)3, it is required to prove that when x= a and y a, the value of u=4 a6, a maximum. 11. If u 'y (x4 + 2 y3 - a)', it is required to prove 13a that corresponding to x= - and y the value 17 n7 of u is a maxmimum. 12. Given u=y4-S-y'+18y'-8y+x'-3x2-3x, it is required to shew that when x= 1- <2 and y =2, the value of u=3+ 4 V a maximum; and when x=1 +V'2 and y= 2+V', the value ofu = - 6 - 4 N/2, a minimum. 13. Given u = y' - 8y' + 18y'2 -8y -_x' + 3x2 + 3, it is required to shew that when x = 1 + - and y = 2, the value of u= 13~4V4\2, a maximu: and when x=1 - and y = 2 + 2, the value of u = 4 - 4 V2, a minimum. ao a3 14. If u=xy + - + -, then if x= a and y=a, it is X Y required to prove that u = 3 a 2 is a minimum. 15. If u =sin x + sin y + sin (x - y), it is required to prove that when x = 6o0 and y = 600, the value of a = -i <p3 a maximum. 16. If u = sin x sin y sin (x + y), it is required to shew that when x = 600 and y = 60', li = 3 V3, a maximum. 17. Find whether the function u =x2y 3(a- 2x -Y)" admit-s of a maxinqmu or a mrni'mum, and determine its value. 127 18. Determine whether the function zu=x3+ay2-bxy+cx, admits of a maximum or a minimum, and find its value. 19. Given u ==/(x-a)2+y2+b2 + /(x-a)2+y-/32, to find whether u admits of a maximum or a minimum, and when possible, to determine its value. 20. Determine the different maximum or minimum values of the function u = x' + y2 + b + Va2 - (x - c)2 Y, and apply the criterion to decide which they are. 21. Find what maximum or minimum values the function u = x4 + y4 - ax2y - axy2 + c2x2 + c2y2 admits of, and decide which they are. 22. In the equation u = a sin x -+b sin y - c sin (x - y), it is required to deduce the equations by means of which a maximum or minimum value of u may be ascertained. 23. If u-=xy /a'b2" —a2 - by2", it is required to find its maximum or minimum value, and to apply the criterion for deciding which it is. 24. If u = a xy2z3- -_ Xy23 - xy3z -_ xy2z4 it is required to prove that when 'x I= a y = a and a =3, the value of 7 fa\ u= -, a maximum. \77 25. Given u =ax2y3z4 —x3y z4 _ —,2y4z4-x 2y35, it is required to prove that when x=aa, y=-a and z=2a, the value of u = a maximum. 26. It is required to prove that the function u = a - bxy + cxz + y, admits of neither a maximum nor a minimum. 27. Required the values of x, y and z, which shall render xyz the function u = -xy a maximum (a + x) (x + y) (y + z) (z+ b) or a minimum. 128 28. It is required to find the maximum or minimum values of u = (-) () - subject to the condition expressed by the equation xo4 + y = c". 29. Find the maximum or minimum value of - = m cos2 x - n cos2 y, when the equation of condition is x + y = a. 30. Required the maximum and minimum values of U t sinm x cos" y, when x and y are connected by the equation x - y = a. 31. Required the maximum or minimum value of u = (mx? + n) (ny + m), subject to the equation of condition am 'bnY = c. 32. Find the maximum value of = xyz, subject to the condition expressed by the equation x - y + z = k. 33. It is required to determine the maximum and minimum values of u in the equation - + - + ( ' X yl \X1 under the condition that x, + y14 + zn = kn. 34. Determine the maximum value of u=- y2z3, when x + my2+ n z3 = a, is the equation of condition. 35. It is required to find the maximum value of u = x4 y, when x, y and z are connected by the equation of condition x2 + 2y3 + z4 = a. 36. Required the maximum or minimum value of u = axa + byP + cczY, when xm' + yt' + zY' = k is the equation of condition. 129 37. Find the maximum or minimum value of u-=x'y, subject to the condition expressed by the equation abY = - k. 38. Required the maximum or minimum value of = (x + a) (y + b) (z + c), the equation of condition being abycCZ — k. 39. Determine the maximum or minimum value of U = (a - 1) (by -1) (CZ - ), when the equation of condition is axbycz =k. 40. It is required to find the maximum or minimum value of u = (. when the equation of \ oo I \x \ %j condition is xwyzay = k. 41. What is the maximum value of the function u = oyz-y^tc, when the quantities x, y, z and t are subject to the condition expressed by the equation x + 2y + 3z + 4t = a? 42. If x + y + z = w, it is required to find the value of - = sin x sin2 y sina z, when it is a maximum or a minimum. 43. Find the maximum or minimum value of = sin x sin y + sin x sin z + sin y sin z, when the equation of condition is x + y + z = 45~. 44. Determine whether the function xm log y sin z, admits of a maximum or a minimum, when the equation of condition is x y z - - + -=1. a b c 45. In the equation u = a — (X2 y2), it is required to prove by the substitution of o + h and o + k in the places of x and y, that when x =O and y=o, the value of u =a is a maximum. R CHAP. XIV. MISCELLANEOUS EXAMPLES. BY means of the Principles laid down and exemplified in Articles (290)-(343), the solutions of the following Theorems and Problems may be obtained. 1. If the equation of a sphere be x2 + y2 + z2 = a2, then will the equation to its tangent plane be xx' - yy' +zz' =a2 required a proof. 2. Determine the inclinations of the tangent plane last found to each of the co-ordinate planes. 3. Let it be required to construct the plane touching a sphere by means of the equation above found. 4. To a given sphere it is required to draw a tangent plane which shall pass through a given point, and to shew how far the position of such plane is indeterminate. 5. Required the equation to a plane passing through two given points and touching a given sphere. 6. Draw a tangent plane to a spheroid defined by the V2 y2 2 equation + -Y + - =1: shew how it may be constructed a2 a2 c and determine its inclinations to the co-ordinate planes. 7. To the surface last mentioned it is required to draw tangent planes which shall pass through one and two given points. 8. A conical surface whose vertex is the origin of coordinates being defined by the equation = e/22+ y2,it is required to prove that its tangent plane will be defined by the equation zz' =e2 ( x' +yy'). 131 9. Construct the tangent plane thus defined, and prove that it always passes through the origin of co-ordinates. 10. It is required to prove that the tangent plane last found always cuts the axis of z at an angle whose tangent is e: find also the angles at which it meets the co-ordinate planes.,2 y2 2 11. The equation of an ellipsoid being - + - + - =1, a2 b' C2 it is required to prove that the sines of the inclinations of its tangent plane to the co-ordinate planes are 7 y % __ —_ _ _ --- and x2 2 z A2 2 y z2 X2 y2 z2 a2 a c4 ab b T + + - + 12. In the same surface, it is required to shew that the length of the perpendicular upon the tangent plane from the origin of co-ordinates is 1 a2 2 2 13. In the same surface it is required to shew that the area of the tangent plane intercepted between the co-ordinate ab 2 c2 b2 e / y2 planes is represented by --- + 4 + Xy a z4 14. In the same surface, it is required to prove that the volume of the pyramid formed by the tangent plane and 1 a2b2c2 the three co-ordinate planes is represented by 6 -6 xyz 15. The equation of the elliptic paraboloid being X2 y2 a b it is required toprove that the equation to its tangent plane is 2Z ' 2yy a b 132 16. Let it be required to construct the plane last defined: to find its inclinations to the co-ordinate planes: the perpendicular upon it from the origin: the area of the tangent plane, and the volume of the solid included between it and the coordinate planes. 17. The equation of the hyperbolic paraboloid being x2 y2 =- -, it is required to prove that the equation to its a b 2Q x' 2yy' tangent plane is - - z'=. a- b 18. Construct the plane last determined: find its inclinations to the co-ordinate planes: the perpendicular upon it from the origin: the area of the tangent plane, and the volume of the solid comprised between it and the co-ordinate planes. 19. The hyperboloid of one sheet being defined by the IV2 y2 2 equation + y2- - =1I it is required to prove that the b" - - cS Z equation to its tangent plane is - + -b c2 -=1. 20. It is required to construct the plane last defined; to find its inclinations to the co-ordinate planes: the length of the perpendicular upon it from the origin: the area of the tangent plane, and the volume included between it and the coordinate planes. 21. Of the hyperboloid of two sheets defined by x2 y2 2 aq- b-L - - ---- 1 +a2 b c it is required to prove that the equation to the tangent plane is mm' yy' zz' + V — = — 1. a2 6 2 c2 22. Construct the plane just determined: find its inclinations to the co-ordinate planes: the length of the perpendicular let fall upon it from the origin: the area of the tangent plane, and the volume of the solid included between it and the co-ordinate planes. 133 23. The cono-cuneus of Wallis being defined by the equation /a + Y =1, it is required to prove that the \/ a2 - -2 b equation to a plane touching it is b (2 (a —2 _ 22) y + bsxzz = a2bx + (a2 - 2). y a y V/'2Z2 24. In the groin whose equation is - + - = it 0a b a is required to prove that the tangent plane will be defined by x y ZZ a the equation - -- + - -. a b a a 2-_~ 2/a —z2 25. It is required to construct this plane: to find its inclinations to the co-ordinate planes: the length of the perpendicular let fall upon it from the origin: the area of the tangent plane, and the volume of the solid comprised between it and the co-ordinate planes. 26. If a surface be defined by the equation xy =a, then will the equation to its tangent plane be- + + — = 3: x y z required a proof. 27. Construct the plane whose equation is just found: find its inclinations to the co-ordinate planes: the length of the perpendicular let fall upon it from the origin: the area of the tangent plane, and the volume of the pyramid formed by it and the co-ordinate planes. 28. Determine the equation to the plane touching the surface whose equation is (a + y2) = a3: construct it, find its area, and the volume of the pyramid formed by it with the co-ordinate planes. 29. Find the equation to the plane touching the surface whose equation is - + - + = 1: construct it and determine all its circumstances. 134 30. If A2+ By'+Cz- +D=O, be the equation to a curve surface, it is required to prove that the equation of the tangent plane is Axx' + Byy' + Czz' + D=o0: and thence to construct it, and determine all the circumstances. 31. Draw a tangent plane to the ellipsoid at the point in the plane of xz where z- — /, and find its inclination to the plane of xy. 32. Draw a plane parallel to 'a given plane, so as to 2 2 2 touch the surface whose equation - b- - = 1. a2 b2 c2 33. Draw a tangent plane to the surface defined by the equation ax — by2 + 2c = 0, and determine all its circumstances. 34. Find the equation to the plane touching the surface defined by the equation x2 +y+ a2 + =b2, and determine all its circumstances. 35. Find the equation of the tangent plane to the surface defined by the equation (az- cx)2 +(bz-cy)=2-e (az-c) (az-cx), and prove that it always passes through the same fixed point. 36. The equation of a surface being (x - a )2 + (y - b)2 = e (x - a ), it is required to find the equation to its tangent plane, and thence to prove that it is always perpendicular to some fixed plane. 37. Draw a tangent plane to the curve surface whose equation is 12y - bxzz + b (zy - cx) = e (a2 -- z), and determine all its circumstances. 38. Determine all the circumstances of the plane which touches the curve surface defined by the equation a xy + b2 z =0. 39. Draw a tangent plane to the surface whose equation is ( - 1) (y - 2) ( - 3) (( - 2) (y - 3) (z - 4), and determine all its circumstances. 135 40. Draw a tangent plane to the surface whose equation is z =f (2 +y2), and find the portions it cuts off from the co-ordinate axes. 41. Apply the equation deduced in the last example to construct the plane touching at any point the surface whose equation is (V/ + 2 — c)2 = a2 - 2. 42. The equation to a curve surface being z2_ C2 = 2 (2y) it is required to prove that the portion of the axis of z cut off by the tangent plane at any point varies inversely as z. 43. Adapt the result of the last example to the cases of the ellipsoid and hyperboloids of one and two sheets. 44. If bz- cy =f(bo- ay) be the equation of a curve surface, then will its tangent plane at every point be perpendicular to the same fixed plane: required a proof. 45. The equation of a curve surface being x - a \ o - a) it is required to draw a tangent plane to it, and to prove that it will always pass through the same fixed point. 46. If log T ) = d (), be the equation to a curve surface, it is required to prove that the plane touching it at any point cuts off from the axis of z a part propor1 tional to - 47. The square of the perpendicular let fall from the centre of an ellipsoid upon the tangent plane at any point, is equal to the sum of the squares of the projections of the three semi-axes upon the normal at the same point. 136 48. If the equation of a curve surface be n(a + y+ z s + 2) = (/), it is required to prove that the tangent plane cuts off from the axis of z, a part proportional to the distance of the point of contact from the origin. 49. Find the equations of the line of greatest inclination in the planes touching the ellipsoid, the elliptic and hyperbolic paraboloids, and the hyperboloids of one and two sheets. 50. Determine the equations of the normal to any point of a sphere, and prove that it always passes through the same fixed point: find also its length. 51. Find the equation to the normal of the spheroid \2 2 [ 2 whose equation is -) + -( + - = 1 prove that it always passes through the same fixed line, and determine its length and the angles which it makes with the co-ordinate planes. 52. Find the equations to the normal of an ellipsoid, and deduce the equations for drawing a normal to it which shall pass through a given point. 53. Determine the equations of the normals of the hyperbolic paraboloid, the hyperboloids of one and two sheets, the groin and the cono-cuneus of Wallis: and find all the circumstances of each. 54. Find the equations to the normal of the curve surface defined by the equation (x2 + y2 + 2 - a2)2 = 4b2 (y2 + z2), and determine its lengths to the co-ordinate planes, and the angles it makes with them. 55. Determine the equations of the normal of the curve surface whose equation is y=f( 2+y2), and find all its circumstances. 56. The equation of a curve surface being bz - cy =f (bv- a'y), 137 find the equations to its normal, and thence deduce all its circumstances. 57. Draw a normal to the curve surface whose equation is Z- c- b -- a== ( — ) a9 and determine all its circumstances. — a v -a58. Draw a normal to the curve surfaces included in the general equation ( - _a)2 + (y-_)2 +Z2= (-a) ( + - + b ) + and point out its peculiarities. 59. Determine the angles which the normal to the curve surface whose equation is Xdx + Ydy + Zdz= o, makes with the co-ordinate axes. 60. The equation of a curve surface being x2 + 2y2 + 3z2 = a, it is required to find the equation to its normal, to determine its length, and to prove that it always meets the plane of xy in the point of intersection of the lines drawn from any two of the angles of the triangle AMN, to bisect the opposite sides. 61. Determine the equation to the tangent plane and also to the normal of the surface defined by the equation z =a tan' (-, and point out their peculiarities. 62. Draw a tangent plane and a normal to the curve surface whose equation is a = (sin-i )+y2 \/xeq y+ y2' 63. Draw a tangent plane and a normal to the surface whose equation is (a - + b)2 + (bz + c)2 2 + y,a and determine all their circumstances. 64. Determine the tangent plane and the normal in the surface of the second order as defined by the equation ax2 + by-2 + CS2 + ax + fy - yz = o. S 138 65. Two equal paraboloids of revolution having their axes in the same straight line, it is required to prove that the tangent plane to either makes with the other a section whose projection on a plane perpendicular to the axis, is always of the same magnitude. 66. Determine the radius of curvature of the section of an ellipsoid made by a plane passing through the axis of z, and bisecting the angle between the axes of x and y. 67. In the elliptic paraboloid it is required to determine the radius of curvature of the section made by a plane passing through the axis of z, and making an angle of 300 with the axis of x. 68. In a paraboloid of revolution, it is required to find the radius of curvature of any section passing through the axis of z. 69. Determine the radius of curvature of a section of the surface whose equation is (x —a)2 + (y- b)2 cz, made by a plane passing through the axis of z: and prove that at any point the sum of the curvatures of any two sections at right angles to each other is invariable. 70. The hyperboloid of two sheets being defined by the X2 y X2 equation -b -b c- = 1, it is required to find the radii of curvature of its sections made by the planes of mv and yz respectively. 71. Find the normal sections of greatest and least curvature passing through the vertex of the elliptic paraboloid whose X2 Y2 equation is z =- + - and their radii of curvature at any a b point. 72. Determine the same in the hyperbolic paraboloid defined by the equation ab = bx2-ay2, and shew that the sections are at right angles to each other. 139 73. Required the sections of greatest and least curvature through the axis of z in the cono-cuneus of Wallis, whose equation is + = 1, and determine their radii at V '2 _ 2 b any given point. 74. What are the sections through the axis of z, of greatest and least curvature in the surface defined by the equation xyz = a3, and what are the magnitudes of their radii? 75. The equation of a surface being z (x2+ y2) =a3, it is required to express the radius of curvature of any normal section through the axis of z, in terms of those of the sections of greatest and least curvature. <a3 y3 3 76. If the equation of a surface be -3 + - + it is required to exprjss the radius of curvature of any normal section through the axis of z, in terms of those of the sections of greatest and least curvature. 77. Express the radius of curvature of the section of an ellipsoid at the extremity of the axis of z made by any plane, in terms of the greatest and least radii of curvature belonging to the same point. 78. If S, and A2 be the minimum and maximum radii of curvature at any point of an ellipsoid defined by the. 2 y2 z2 equation - + + = i, it is required to prove that they a2 b2 C2 are the roots of the equation 2z + b2+ - X2 _ Y-2 Z + abc) x hX2 where X is the length of the perpendicular let fall from the origin upon the tangent plane. 79. It is required to shew from the last equation, that the sum of the greatest and least radii of curvature varies inversely as the length of the perpendicular upon the tangent plane, so long as the distance of the point from the centre remains invariable. 140 80. In the same surface, it is required to prove that so long as the distance of the tangent plane from the centre is invariable, the rectangle of the greatest and least radii of curvature is constant. 81. Determine the points in the surface of an ellipsoid at which the curvature of every normal section is of the same invariable magnitude. 82. Find the radii of greatest and least curvature at any point of the curve surface defined by the equation xyz = a3. 83. Required the same for the elliptic paraboloid whose X2 y2 equation is z=- +. a b 84. In any curve surface, the radii of greatest and least curvature will become equal in sign and magnitude when the 1 p-' 1 -_ +q' conditions expressed by the equations = - = +r S t are fulfilled: required a proof. 85. If the radii of greatest and least curvature, having different signs, become equal in magnitude, it is required to prove that (1 + p2) t + (1 + q') r = 2pqs. 86. It is required to prove that the. surface whose equation is xS2 - y2 - ( - C)2 = 0, is at the same time a conical surface and a surface of revolution, by satisfying the general partial differential equation of each species of surface. 87. The equations of the curve of double curvature formed by the intersection of the surface of a sphere with that of the cylinder whose diameter is its radius, being y2 2=2ax- -x2 and z2=4a2-2ax, it is required to prove that the equations to the tangent line are yy + x = a ( + x') and zz'= 4a2 - a (x a'). 88. Shew how the line determined in the last example may be constructed, and determine where it intersects each of the co-ordinate planes, and also the loci of these intersections, 141 89. In the same case determine the length of the tangent intercepted between the point of contact and the plane of zy, and also its inclination to that plane. 90. The equations of the helix being - cos - and -= sin - a b a b it is required to determine the equations of its tangent line, and thence to construct it and find all its circumstances. 91. Draw a tangent line to the curve of double curvature formed by the intersection of a sphere and cylinder whose equations are 2 + + y 2 = a2 and (x- c)2 + y2 = b2, and determine all its circumstances. 92. If the axes of two cylinders whose equations are z2 + z2 = a2 and y2 + z2 = b2, intersect each other at right angles, it is required to draw a tangent to the curve formed by their surfaces, and to determine all its circumstances. 93. If the axes of two right cones whose equations are a+ =m m/y2 + and b + y = n\/ 2, intersect each other at right angles, it is required to draw a tangent to the curve of double curvature thus formed, and to determine all its circumstances. 94. The equations of a curve of double curvature being 2 + y2 -+ z2 = a nd tan-' - = m cos- -, it is required to draw O Ca a tangent line to it, and to ascertain all its circumstances. 95. Draw a tangent to the spiral of Pappus as defined by the equations z Y x - Ccos, - - = sin ( and - = cos dP, a 4 r v/a2- V 2a and determine all the circumstances attending it. 96. The equations to the screw of Archimedes being y 2 ax - 2 and z = b vers-l -, it is required to draw a tana gent line to it, and to ascertain all its circumstances. 142 97. It is required to draw a tangent line to the curve of double curvature defined by the equations yXv/as sin-1 -sn y- _ 1 a a a y= a X_ and —je a+e f f and to determine all its circumstances. 98. Draw a tangent line to the curve of double curvature formed by the intersection of a right cone with the cylinder erected on the radius of the base as a diameter, and determine all its circumstances. 99. Determine the equation of the normal plane to each of the curves of double curvature mentioned in the last twelve examples and construct it: 100. Find the equation of the osculating plane in each of the curves of double curvature just referred to, and to its intersection with the normal plane. 101. In any curve of double curvature, if a, 3, 7 be the co-ordinates of the centre of spherical curvature, it is required to prove that they may be determined from the three equations dy da- ( — y + (/3 -) d = 0,+ d,2 d2 1 (dy)2 (d) 2 ( ~-y) + ( 3 I ) d = 2 + -d z w y) +(7-^:7-1 iXj + 2IWJ d' dz f. 3^ d dy dx y d dx dC2 and the radius of the sphere from the equation 2 =(a )2 + ( - y)2 + (y - Z)2. 102. If the proposed point of the curve be the origin of co-ordinates and the tangent line be considered the axis of x, it is required to prove that a =0, d3z d3y dX3 dx d2zy d3 d d2zdy' 7 d2z d3y d2y d3z' d 2 d w2 d2 da Yx d w d x2 d x3 dx d 143 /d3y2 Id3X 2 and S-2= (d2y d'3 d2X d 3y2' dx2 dx3 dW2 dx3) 103. If the absolute centre of curvature be the point where the intersection of two consecutive normal planes meets the osculating plane, it is required to determine the coordinates of this point, and thence the absolute curvature. 104. If the arc s of the curve be considered the principal variable, and a, J, 7 the.co-ordinates of the centre of absolute curvature, it is required to shew that d2x ds2 /a-a= 2 2 /d 2 d 2 (ds2 + \ds2) + \d) d2y ds' W _+ [_ +. v (d )2 (d2y)2 (d2 \2' d2z ds2 7/d^^\2/~d2y\2 /Cry\d2 \ds2) + (da) + (d and -_ 2 (d2 ()2y2 d2+ 2 idds] + \+ ds2 + 105. If the point under consideration be regarded as the origin of co-ordinates and the tangent line as the axis of xa, it is required to prove that d2y d2z a ds 3-28 d s2 2/ + {ds) [ds2) + \\d$ 144 and 2 f= d2y 2 \ d 2 2 ds2) + ds2 106. If the equation of a curve surface be of the form / du\2 (`du\2 /du\ 2 u =f(x, y, ) = 0, and 2= (- + - + (, and VlaJ v\dyI kd v7 the notation of Article (297) be retained, it is required to I du 1 du 1 du prove that cos a =- - cos =- - and cos y=- - d x JL dy / u dz 107. Determine the equation to the curve traced out upon any of the co-ordinate planes by the intersection of the tangent line to a curve of double curvature whose equations are y2=2a — 2 and 2 = m (x y2). 108. Prove that the differential of the area of the surface traced out by the ordinate z in any curve of double curvature -= z/da2 + dy2. 109. Trace the surface whose equation is x 2 + y2 = a2, and determine the positions of the tangent planes at the points which it cuts the co-ordinate axes. 110. Find the form of the surface defined by the equation - a + (Y — + (z- ) = 1, determining the a b C natures of its principal sections. 111. If an ellipsoid be cut by a plane passing through the origin perpendicular to the plane of xy, it is required to prove that the normals to the surface drawn from every point of the intersection of the plane with the ellipsoid will cut the plane of xy in a straight line, and to find the equation to that line. 112. If the partial differential equation of a curve surface be z = a + px + qy, it is required to shew that there is some point from which all the perpendiculars, let fall upon the tangent plane at every point, are of an invariable length. 145 113. The straight line joining any two points of the surface of an ellipsoid is bisected by a plane passing through the centre of the ellipsoid and through the line of intersection of the tangent planes at those points: required a proof. 114. Draw a tangent plane and a normal to the curve surface defined by the general equation 3- ai=f (b —a; X - a f x-ay and determine all their circumstances. X2 y2 z2 115. The equation of a surface being - + - - - =, it is required to prove that the surface of the right cone defined 72 2 2 by the equation - + Y -=- will be asymptotic to it. 116. Required the asymptotic surfaces to the hyperbolic 3X2 y2 paraboloid whose equation is = - -. a b 117. It is required to determine the directions of the convexity and concavity of the curve surface defined by the equation ax2- by2+ 2cz = 0, with respect to each of the co-ordinate planes. 118. The equation to a curve surface being (a22 + by + 22 c2z2) (O2 + y2 + Z2) - a2 (b2 + c2) 2 - b2 (a2 + c2) y2 - c (a2 + b2) 2 + a2b2c2 = 0, it is required to determine all its circumstances; and to apply the results to the cases when a = b and a= b c: also when c=O and b=c=0. 119. Determine the conditions necessary to be fulfilled in order that two curves of double curvature may have with each other contact of any specified order: and apply the result to the case wherein one of them becomes a straight line. T CHAP. XV. MISCELLANEOUS EXAMPLES. BY means of the Principles laid down and exemplified in the various Articles of the Work, the solutions of the following Theorems and Problems may be effected. 1. Find the sum of the areas of all the circles that can be inscribed in an equilateral triangle touching its sides and one another, and compare it with that of the triangle. 2. If tr and r, be the radii of two spheres inscribed in a right cone, so that the greater may touch the less and also the base of the cone: it is required to prove that the 5 content of the cone will be expressed by - 3. Compare the sums of the surfaces and volumes of all the spheres which can be inscribed in a right cone with those of the cone itself. 4. Differentiate from first principles the following functions: a2 - = V ---2a, 2 u 2e — am'~, v — Cae a" + oAt ex -1' cos 2x = sin rax, u= --- and u =tan(2 x + 2). sin x 5. If the sines of two arcs be always in a given ratio, it is required to shew that their evanescent increments are in the ratio of their tangents. 6. If the ratio of the cosines of two arcs be given, what will be the relation between their evanescent increments? Similarly, of the rest of the Trigonometrical functions? 147 7. If cs 2(1- c due (/ r - d2 7- If u = cos-l --, then will = ) l+ v dix (1+< )2 required a proof. 8. If u= /_ tan- (tan /, it is required to /a6 b a d u 1 prove that= - dx a cos2 x + b sin2 9. If u = ax + b log (a cosx + b sin x), it is required to du (a2 + b2) cos X prove that -= dx a cos x + b sin x 10. If u = tan x sec x +log (tan x + sec ), it is required d u to shew that - =2 seca3. dx 11. Find the differential coefficients of log (a - vx) ~u = log (a + x) log (a - x) and ue = log (a -- ) 1 +2ivZ/1 — 2 12. Differentiate the functions u = log and A/ 1-2 =tan~- (-~7%/ ~: and find the values of the diferential coefficients when x= 0 and = 1. 13. Find the successive differential coefficients of u = ee6. 14. If u = -,% it is required to prove that d'u =- 1 - d t == X 32(1 +log ) (1 log )3}. 15. From the following equations, sin (p = m sin q', cos 9' 1 x = m cos qd' - cos p and y = m- - 7 *COS 20 COS 0 d2 X d y it is required to find the values of x + d and y + -w 5 when d4P" 0d" 148 16. If two distances u, and u, from a fixed point be such that u 2= -a2, it is required to compare their evanescent increments or decrements in any position. 17. Corresponding to the extremities of the latus rectum of a common parabola, it is required to find the ratio of the rates of increase of the abscissa and ordinate. aa-n x- n 18. If u =,a it is required to shew that when a-a -=a, the value of u=aan( -- +loga}. a J sin mO0-cos(m.90-0)nO 19. Find the value of u= si- co.- when sin mO+cos (n. 90~+m0) 0 =0, m being a whole number of the form 4t + 1 and n of the form 4v + 3. 20. If 2y4 - Sy3 + X2y2_-aSx =0 it is required to find the values of d when x = o and y= c. dxv 21. In the equation x4y4- 2 aCX2y4 + ay2- a8 =0, it is dy required to find the values of - corresponding to =0 and y= 00 22. If y = (a + bx) ex + m sin ax + n cos x, it is required d4y d3y d2y dy to prove that d -2 2 z-2- d+Y=O. dx4 dX' dx' d 1 1 1. 23. If the equation of a curve be - -+- it is y acn (n required to prove that an x A T -= AMn+, where AM =, MP=y and PT is a tangent. 24. Describe a parabola which shall touch a circle at a given point, and have its axis coincident with a given diameter of the circle. 25. If y and y' be the radii of curvature of a curve and its evolute at corresponding points, it is required to prove that 149 y' 3spq2 - r (1 +p2) = Pq 2 — (+ ), where p, q and r are the first, second 7y 2 and third differential coefficients belonging to the point in the curve. 26. If q, r, s and t be the second, third, fourth and fifth differential coefficients, it is required to prove that in all curves of the second order 9q2t -.45qrs + 40r3 =0. 27. Apply Lagrange's Theorem to find a series for v in terms of nt as far as e3 by means of the equation (1- e) tan2 =(1 +e) tan2, where t = u - e sin u. 28. If a number a be divided into such a number of parts x that the continued product of the first, the square of the second, the cube of the third, &c. shall be the greatest possible, it is required to prove that log = --. x?+2a 29. Into how many parts must a given number be divided, that their continued product may be the greatest possible? 30. Given the three prime factors of a number, required the indices when the number of its divisors is the greatest possible. 31. Given the three prime factors of a number, to find the indices when the sum of its divisors is the greatest possible. 32. If m be an even number, it is required to prove that 27r 4w7 2 - 2 cos- 2 - 2 cos - m 2 m m Adn _ + __ + &c 7 _ xm I -- 2 2 r 47qr 1 -2 cos - + x2 1 - 2X cos — + x m m to 1m terms. See Article (346). 33. Decompose -- into a series of fractions as in t t x m + the last example, both when m is odd and even. 150 - 34. If f denote any function whatever, and we have f (x + h) =f (x)f (h), it is required to prove that f (h) = eah. See Article (348). 35. Find the form of the function f so that we shall have f ( + h) —f (-h) =f () f (h). 36. Required the form of the function f such that f (x) f (x- h) = f (h) ^,-,.,,,. (x + a') (s' - a2)37. Required the value of u = -(3- + a) (x 3 a) when (3cT - a s) (x" - a3) = a, according to the principle laid down in Article (349). 38. Determine the arc of a given circle, when the rectangle of its sine and the excess of its sine above the cosine is a maximum, and verify the result. 39. In Article (351) it is required to find the straight line which drawn through P, shall cut off the segments BT and BV, so that their sum shall be the least possible. 40. Retaining the same figure, find the position of the line TV so that the area of the triangle TBV shall be the least possible. 41. Of all triangles under a given perimeter and one determinate side, prove that to be the greatest in which the two indeterminate sides are equal. 42. Of all triangles upon the same base and of a given area, it is required to prove that the isosceles triangle has the greatest vertical angle. 43. Find the greatest of all triangles having equal vertical angles and equal distances between that angle and the bisection of the opposite side. 44. Of all equiangular parallelograms inscribed in a given triangle, the greatest is that formed by the bisection of the sides: required a proof. 151 45. Given the difference of the angles at the base and the radius of the inscribed circle, to find the triangle whose perimeter is a maximum. 46. Find the area of the greatest rectangle that can be inscribed in a given triangle, so that one of its sides is parallel to the base of the triangle. 47. A circle and an ellipse having the same major axis, it is required to compare the areas of the greatest rectangles that can be inscribed in them. 48. It is required to prove that of all triangles described on the same base and having equal altitudes or vertical angles, the perimeter of the isosceles is the least. 49. Of all triangles upon the same base and of equal perimeters, it is required to find that which has the greatest area. 50. In a given line, it is required to determine the position of a point at which another given line subtends the greatest possible angle. 51. Find the greatest rectangle that can be inscribed in a given segment of a circle. 52. Draw a tangent to an ellipse so that the part of it intercepted between the axes produced shall be a minimum. 53. Find the shortest line which can be drawn touching a given ellipse and intercepted by the tangents at the extremities of its axes. 54. Prove that the perimeter of an equilateral triangle inscribed in a circle is greater than the perimeter of any other isosceles triangle inscribed in the same circle. 55. Find the greatest isosceles triangle that can be inscribed in a given circle. 56. Determine the points of a given ellipse in which the sum of the conjugate diameters is the greatest or least possible, and distinguish the maximum from the minimum. 152 57. Determine those conjugate diameters in an ellipse or hyperbola whose rectangle is the greatest possible. 58. In a given ellipse, it is required to assign the positions of those conjugate diameters which include the greatest and least angles. 59. It is required to determine the magnitude of the greatest parabola that can be made by cutting a given cone. 60. Inscribe in a portion of a parabola cut off by a double ordinate the greatest possible rectangle. 61. In a given parabola, inscribe another parabola whose area shall be the greatest possible, the vertex of the inscribed parabola being any given point in the axis of the other. 62. Of all the straight lines which can be drawn from the vertex of a given ellipse to the circumference of its circumscribed circle, find that of which the portion intercepted between the circumferences of the ellipse and circle shall be the greatest possible. 63. Two circles of given radii intersect each other, find the longest straight line which can be drawn through either point of intersection, and terminated by the circumferences. 64. In an ellipse, it is required to find the point at which the normal makes the greatest angle with the line drawn from it to the centre. 65. If M be any point in the diameter of a circle whose centre is C, PMQ a chord and CP be joined; prove that the chord PQ is a minimum and the angle CPQ a maximum, when PQ is perpendicular to the diameter. 66. Perpendiculars A a, Bb are drawn from the extremities A and B of the axis major of a given ellipse, and through a given point P in its circumference is drawn a straight line aPb intersecting them in a and b respectively: prove that the rectangle A a. Bb is a maximum when aPb is a tangent to the ellipse at P. 153 67. Two given points A and B are situated on the same side of a given straight line: find a point P in the line such that AP + BP shall be a minimum. Also, if the straight line move parallel to itself, determine the locus of the point P. 68. Given the length of a chord drawn through the vertex of a common parabola; find its latus rectum when the corresponding area APM is the greatest possible. 69. Find the point in the diameter of a given circle, through which an ordinate being drawn, the mth power of the ordinate multiplied by the nth power of the less abscissa shall be a maximum. 70. In a given ellipse it is required to find a point from which straight lines drawn to the extremities of the axes shall contain the greatest angles. 71. If P and Q be corresponding points in an ellipse and the circle described upon its major axis, it is required to prove that the angle contained between the tangents at P and Q will be a maxvimum when AM= a {l- A/ b { a + b 72. For the same point P it is required to prove that a b CM= a a-, MP=b b — c + b a b CP, MT=-b'/ -, TAP -b: a+-b a+b and if CF be drawn perpendicular to the tangent at P, CF= V/a b, TF = a and PF = a - b a maximum. 73. Prove that the area of the triangle formed by the tangent and the axes produced is a minimum corresponding to the co-ordinates a / — and b /2 /./2 2 74. Through a given point in the diameter of a given circle it is required to draw a chord which shall form with the U 154 lines joining its extremities with either extremity of the diameter, the greatest possible triangle. 75. Through a given point in the surface of a right cone it is required to pass a plane, so that the area of the elliptic section shall be the least possible, and to find the limitations of the problem. 76. Determine the greatest ellipse that can be formed by cutting a right cone, and find the limitations to which the problem is subject. 77. From the, extremity of the minor axis of an ellipse it is required to draw a line such that the part of it cut off by the periphery may be the longest possible, and to determine the limitations of the problem. 78. In a given straight line it is required to find the equation which determines a point P, such that straight lines PA and PB being drawn to the given points A and B on opposite sides of it, the sum of mxAP and n xBP shall be a minimum: and to compare the angles made by AP and BP with the proposed line. 79. Two points are taken in a wall and joined by a string which passes about a corner of it, prove that this string is the shortest possible when the parts of it make equal angles with the line which forms the corner of the wall. 80. A circle being given and a point without it, it is required to find the circle of which this point is the centre when the segment of it inscribed in the first circle is a maximum. 81. Determine the point in a curve whose equation is am-x= ym, to which a line must be drawn from the vertex making the greatest angle with the curve. 82. If the semi-axes of an ellipse be 2 " and xt, it is required to find the least area the ellipse admits of. 83. The corner of the leaf of a book being doubled back so as just to reach the other edge of the page, it is required to find where the length of the crease will be the least possible. 155 84. Inscribe the greatest trapezium having two of its sides parallel in a given parabola, and compare its area with that of the parabola. 85. Find the radius of the circle, such that the versed sine of an arc being given, the are itself may be a minimum. 86. Two points move at the same time from two given positions at given rates in given directions, find where their distance from each other will be the least possible. 87. Required the triangle of a given perimeter which has the greatest possible area. 88. Find the greatest triangle that can be inscribed in a circle of given radius. 89. The three angles of a right-angled triangle are at given distances from a fixed point, find the greatest area the triangle can admit of. 90. It is required to describe the greatest quadrilateral figure in a circle, and the least quadrilateral about it. 91. Given one of the angles and the perimeter of a plane triangle, to find the sides when the area is the greatest possible. 92. Given the three sides of a triangle, to find a point such that the continued product of the three perpendiculars let fall from it upon the sides may be a maximum. 93. Given the four sides of a trapezium taken in order, it is required to find its area when the sum of the squares of the diagonals is a maximum. 94. In a given triangle it is required to determine the position of a point from which if perpendiculars be drawn to the sides, the sum of their squares shall be a minimum. 95. To find a point within a given triangle, from which if lines be drawn to the angular points, the sum of their squares shall be the least possible. 96. Find a point within a given trapezium from which perpendiculars being let fall upon the sides, their sum shall be of a given magnitude and their continued product a maximum. 156 97. Find a point within a given quadrilateral from which lines being drawn to the angular points, the sum of their squares shall be the least possible. 98. Required the position of the point in the line joining the centres of two given spheres, from which the sum of the spherical surfaces visible shall be a maximum. 99. Required the same when the sum of the corresponding spherical segments is the greatest possible. 100. Among all the angles contained in a given parabolic segment, determine that which is the greatest. 101. Corresponding to a spherical surface of given magnitude, it is required to prove that a hemisphere is greater than any other segment of a sphere. 102. Let it be required to inscribe the greatest cylinders in a right cone, a sphere and a paraboloid: and to compare their volumes in each case. 103. Inscribe the greatest cones in a spheroid, and a paraboloid, the axes and vertices of the solids being coincident. 104. Of all cones inscribed in a given sphere, determine that which has the greatest convex surface. 105. In a given segment of a sphere, inscribe a paraboloid so that the content shall be a maximum, the vertex of the paraboloid coinciding with the centre of 'the base of the segment. 106. Find the dimensions of the greatest cylinder that can be cut out of a solid formed by the revolution of a curve about its axis, of which the equation is a'xn -yml+1 and the length of the axis = b. 107. The volume of a cylinder being given, it is required to find whether the convex and entire surfaces admit of a maximum or a minimum. 108. A cylindrical vessel of given thickness is required to be of a given capacity, find the least quantity of material whereof it can be made. .157 109. Describe the greatest parallelopiped in a given sphere, and the least parallelopiped about it; and compare their volumes and surfaces. 110. The surface of a right cone being given, find its dimensions when the volume is the greatest possible. 111. Given the volume of a right cone, required its dimensions when the whole surface is the least possible. 112. Determine the rectangular parallelopiped of a given volume, which has the least possible surface. 113. A certain quantity of material is to be formed into a spherical segment of given thickness, what will be its dimensions when the included volume is a maximum? 114. A given sphere is to be formed into a solid composed of two equal right cones on opposite sides of a common base, so that the surface is a minimum: find the dimensions of the cones and compare their surfaces with that of the sphere. 115. In a right-angled spherical triangle is given an angle to find the sides, when the difference between the hypothenuse and the side adjacent to the given angle is a maximum. 116. If two sides of a spherical triangle be given, it is required to find when the area will be the greatest possible. 117. If the hypothenuse of a right-angled spherical triangle be given, find its dimensions when the sum of the cosines of the sides is the least possible. 118. If the sum of two sides, and the included angle of a spherical triangle be given, it is required to determine when the sum of the remaining angles will be the least possible. 119. Determine that point in the arc of a quadrant from which two lines being drawn, one to the centre and the other bisecting the radius, the included angle shall be the greatest possible. 120. Of all isoperimetrical polygons having the same number of sides, the greatest is that which is equilateral required a proof. 158 121. Describe the least paraboloid about a given cylinder, when their axes are coincident in direction. 122. Find that point in the surface of a given paraboloid through which if two planes be drawn, one perpendicular to the axis and the other perpendicular to the generating parabola and parallel to the axis, the sum of the areas of the sections shall be the greatest possible. 123. On a given triangle a pyramid is to be constructed of a given volume; required its dimensions when the surface is the least possible. 124. Find that point in the surface of a spherical triangle, from which if straight lines be drawn to the angular points the pyramid thus formed shall be the greatest possible. 125. Given the greatest and least slant sides of an oblique cone to find the diameter of the base, when the volume is the greatest possible. 126. Required the area of the least isosceles triangle that can circumscribe a given circle, and compare their perimeters. 127. Find the content of the least cone that can circumscribe a given sphere, and compare their surfaces. 128. It is required to inscribe the greatest parabola in a given isosceles triangle, the axis of the parabola coinciding in direction with the line bisecting the vertical angle of the triangle. 129. Find the area of the least triangle which can circumscribe a given segment of a parabola. 130. Determine the content of the greatest paraboloid that can. be inscribed in a right cone of given dimensions. 131. Find the content of the least cone that can circumscribe a paraboloid of given dimensions. 132. Determine the dimensions of the least paraboloid that can circumscribe a given sphere. 159 133. Required the least triangle which can circumscribe any given right segment of an ellipse. 134. Determine the dimensions of the greatest spheroid and hemispheroid that can be inscribed in a given right cone. 135. Determine the least triangle that can circumscribe any segment of an ellipse. 136. Find the content of the least cone that can be circumscribed about any right segment of a spheroid. 137. Find the least triangle that can be circumscribed about an ellipse of given dimensions. 138. Required the dimensions of the least ellipse that can be circumscribed about a given trapezoid. 139. Inscribe an ellipse in a given parallelogram so that its area may be the greatest possible. 140. Inscribe the greatest ellipse in a given segment of a circle, one axis of the ellipse being parallel to the base of the segment. 141. In a given ellipse inscribe the greatest semi-ellipse, the extremity of the major axis of the inscribed figure touching the extremity of the axis minor of the given ellipse. 142. Inscribe the greatest spheroid in a given paraboloid, and compare their axes and volumes. 143. If a curve be defined by the equation amx'= by", where x and y are the distances of any point of it from two given points, it is required to draw a tangent and a normal to it. 144. If m =n = 1, and the curve become a circle, it is required to determine the position of the tangent, and to shew that the normal always passes through the same point. 145. If the arc of a cycloid be considered a curvilineal abscissa and the corresponding ordinate be always equal to it, it is required to draw a tangent to the curve, and to find where it meets the co-ordinate axes. 160 146. It is required to draw a tangent to two given circles whose centres are at a given distance from each other. 147. An ellipse and a circle given in magnitude and position have their axes in the same straight line, it is required to draw a straight line which shall touch them both. 148. A given ellipse and parabola have their principal axes in the same straight line, it is required to draw to them a common tangent. 149. If perpendiculars be drawn from any given point in the circumference of a circle upon the tangents, it is required to find the locus of their intersections. 150. It is required to find the nature of the line traced out by the intersections of the tangents to a parabola with the perpendiculars let fall upon them from the focus. 151. Determine the locus of the intersections of the perpendiculars from the focus of an ellipse with the tangents at every point of it. 152. Determine the locus of the intersection of the' tangent to an hyperbola with the perpendicular let fall upon it from the centre. 153. Find the equation to the curve traced out by the intersections of the tangents to the conic sections with the perpendiculars let fall upon them from any given point. 154. In each of the conic sections a straight line being drawn from the focus to any point in the curve, it is required to find the locus of the intersection of a perpendicular to this line through the focus with the tangent. 155. Find the locus of the point from which two tangents drawn to a circle given in magnitude and position, shall always include the same given angle. 156. Find the locus of the centre of a circle touching a given line and always passing through the same fixed point; 161 157. Determine the locus of the centre of a circle which shall touch a given straight line and a given circle: also when it touches two given circles. 158. Required the locus of the intersection of two tangents to a given parabola, when the angle included between them is invariable. 159 Required the locus of the intersection of two tangents to a parabola inclined to its axis at angles, the product of whose trigonometrical tangents is invariable. 160. Determine the locus of a point from which two tangents being drawn to a given parabola, their sum or difference shall be invariable. 161. Find the locus of a point from which two tangents being drawn to a given parabola, their lengths shall have an invariable ratio. 162. Required the locus of the point from which the rectangle of two tangents, drawn to a given parabola, shall be of invariable magnitude. 163. Determine the locus of the intersections of the tangents to an ellipse or hyperbola, which shall be inclined to the principal axes at angles, the products of whose trigonometrical tangents are invariable. 164. Required the nature of the loci, when the tangents to the ellipse or hyperbola intersect each other at right angles. 165. Find the locus of the intersections of two tangents to an ellipse or hyperbola, when the sum of the angles made by them with the principal axes, is equal to a right angle. 166. Required the loci of the intersections of the tangents to an ellipse or hyperbola, which are parallel to conjugate diameters. 167. Determine the locus of a point from which tangents drawn to a given ellipse, shall have an assigned ratio. X 162 168. Required the locus of a point, from which tangents being drawn to a given ellipse, they shall have an assigned sum or difference. 169. Required the nature of the line which touches all equal circles having their centres in the same given straight line. 170. Determine the nature of the line which touches all circles passing through a given point, and having their centres in the same given straight line. 171. Of a right-angled triangle, if the sum of the sides including the right angle be the given magnitude a, it is required to prove that the curve to which the hypothenuse is always a tangent, will be defined by the equation v/ + y = 5/a. 172. Find the nature of the curve to which the hypothenuse of a right-angled triangle is always a tangent, when the difference of the sides including the right angle is invariable. 173. Required the nature of the curve which a straight line of given length having its extremities always in two straight lines at right angles to each other, continually touches. 174. If PT be a tangent to the curve AP, and AD be drawn perpendicular to the axis of x through the origin, it is required to find the nature of the curve wherein A Toc AD1..175. Required the nature of the curve formed by the intersections of consecutive normals to all the points of a given parabola. 176. Prove that the curve which the normals to every point of a given cycloid perpetually touch is a similar and equal cycloid in any inverted position. 177. If a straight line be given in position, and lines be drawn intersecting the given line so that the perpendiculars to them from the points of intersection shall pass through a given point, find the curve to which these lines are always tangents, 163 178. In two straight lines at right angles to each other two points are given in position, and two other points at equal distances from them are taken: it is required to find the nature of the curve to which the straight line joining the latter points is always a tangent. 179. If normals and tangents to a parabola be always drawn at the extremities of any parameter, it is required to find the loci of their intersections. 180. Determine the nature of the curve which shall include all isosceles triangles of a given area, having their bases and perpendiculars coincident in directions. 181. Two straight lines ate drawn through two given points whose equations are = m and x= n, perpendicular to the axis of x, and their rectangle is of invariable magnitude: it is required to find the curve to which the line joining their extremities is always a tangent, both when they are drawn in the same and in opposite directions. 182. From two given points two straight lines are drawn parallel to each other so that the difference of their squares is of invariable magnitude, it is required to find the nature of the curve to which the straight line joining their extremities is always a tangent. 183. Required the nature of the curve analogous to that of the last example, when the sum of the squares of the lines cut off by the tangent is invariable. 184. Two points are assumed in two given lines and a straight line intersects them so that the rectangle of the distances of its intersections from the said points is invariable: required the curve to which the straight line is always a tangent. 185. Three points A, B and C being given in position, and AB, AC being drawn, a straight line TV meeting these lines is so situated that the rectangle BT x C V has to the rectangle AVx AT an invariable ratio: required the nature of the curve to which this line is always a tangent. 164 186. Required the nature of the curve which shall cut all curves of a given species, so that the tangents at the points of intersection may pass through the same given point. 187. Apply the result of the last example when the curves of given species are a series of parabolas defined by the equation y2=atx. 188. Required the curve when the curves of given species are a series of ellipses, having one of their principal axes invariable. 189. Find the nature of the curve when the curves of given species are a series of ellipses, having their principal axes in a given ratio. 190. Determine the nature of the curve when the curves of given species are cissoids of Diodes, having their axes coincident in direction. 191. Find the nature of a curve which shall cut all curves of a given species, so that the tangents at the points of intersection may be of an invariable length: and apply the result when the curves proposed are a series of parabolas. 192. Determine the nature of the curve cutting all curves of a given species, so that the tangents at the points of intersection may touch a given circle. 193. Determine the nature of the curve which shall bound a series of circles, whose respective radii are the successive ordinates of a common parabola. 194. Find the nature of the curve which shall touch all parabolas whose areas are equal, and — whike —pats —hrough~-a givexl -.pntpo-id have their extreme ordinates coincident in direction. 195. If the intersection of two straight lines, which contain a given angle, be upon a line of any kind given in position, and one of them pass through a given point, required the nature of the curve to which the other is a tangent. 165 196. Apply the result of the last example to the cases, wherein the line of the proposed kind becomes a straight line and a circle. 197. Let AEC, BFC and EF be three straight lines given in position, and from the points A, B let straight lines be drawn to any point in the straight line EF meeting AC in P and BC in Q: then it is required to find the nature of the curve to which the straight line joining P and Q is always a tangent. 198. Prove that all the straight lines defined by the equation y - cx=-a/1 + c2 by assigning different values to c, are tangents to the circle whose equation is y = /a -- 2. 199. Determine the nature of a curve when its tangent is defined by the equation (c —l)x -cy=a - b, whatever be the value of c. 200. Find the curve to which the straight line belonging to the equation y = c (x - a) + b\/l + c2 is always a tangent, whatever be the value of c. 201. Required the curve to which the straight lines whose equations are y = c (a- x) +_ / 2 + c2a are always tangents, whatever value be assigned to c. 202. If the equation to a straight line be - + -1 a /3 1 1 1. where a and /3 are subject to the equation -- =, it is required to prove that the curve formed by the intersections of all such lines will be a circle whose equation is x2 + y2 = c2. 203. If an infinite number of straight lines defined by the equation ay + 3x = a/3 be drawn, subject to the condition expressed by the equation a' + /3" = c', then will the curve whose 11 n n equation is xn + + yn+1 = C + 1, touch them all. 204. Determine the nature of the curve which shall bound all the parabolas expressed by the equation y = ax- (1 + a2) xV2, whatever value be assigned to a. 166 205. Find the nature of the line which shall touch the curve defined by the equation y2 =ax - a2, whatever be the value of a. 206. If any number of concentric ellipses have their axes coincident in direction and their sum =2 a, it is required to 2 2 2 shew that the curve whose equation is x3 + y = a3 will bound them all. 207. If the sum of the squares of the axes of any number of concentric ellipses, having their axes coincident in direction, be invariable, it is required to prove that a straight line will bound them all. 208. The equation to a series of ellipses being /W\2 fy\2 (a)+ =1( ' subject to the condition expressed by the equation a" + /n = c", it is required to find the equation to the curve which touches them all. 209. Required the nature of the curve which shall touch all the curves defined by the equation (x2 + y2)2 = a2x2 + /3y2, wherein a/3 is invariable. 210. Determine the nature of the curve which shall touch all the curves defined by the equation y =ax - (a + 4) a2, whatever be the value of a. 211. Given the focus of a parabola and a point through which it passes, to determine the locus of the vertex. 212. Given the focus of a parabola and the position of a tangent, to determine the locus of the vertex. 213. Find the locus of the foci of all parabolas having a given vertex and touching a given straight line. 214. Given one focus of an ellipse, a point through which it passes and the position of a tangent at that point, to determine the loci of its vertices. 215. The centre of an ellipse and its axis major being given, it is required to find the loci of the extremities of the 167 minor axis, when the ellipse always passes through a fixed point. 216. Given one focus of an ellipse, a point through which it passes and the magnitude of the major axis, to determine the locus of the vertex. 217. Given the centre of an ellipse, a point through which it passes, and the position of a tangent at that point, to find the locus of the focus. 218. Ellipses having a given centre and eccentricity touch a given line, it is required to find the loci of the extremities of their major axes. 219. Given the position of the centre of an hyperbola and the magnitude of its major axis, to find the locus of the focus when it always passes through a given point. 220. Given the position of the vertex of an hyperbola, a point through which it passes and the position of a tangent at that point, to determine the locus of the focus. 221. Given the vertex of an hyperbola, a point through which it passes and the magnitude of the major axis, to determine the locus of the focus. 222. Find the locus of the centre of an ellipse which touches two given straight lines at two given points. 223. If two sides of a triangle be divided into the same number of equal parts and the points of division of one side beginning from the base be joined with those of the other side beginning from the vertex, find the curve in which the intersections of every pair of consecutive lines taken in this order are situated. 224. If on two sides of an isosceles triangle equal parts be assumed, it is required to find the curve in which the points of intersection of every pair of lines joining the successive points of division are situated. 225. Determine the locus of the vertices of the osculating parabolas to a given curve; and prove that for the cubical parabola, it is a curve of the same species. 168 226. If tangents be drawn from a given point to each of a given system of curves: shew generally how to determine the curve which is the locus of all the points of contact: and apply the method when tangents are drawn from a given point to a system of concentric similar ellipses. 227. By means of the equation to the tractrix, it is required to determine the length of the catenary. 228. Determine the length of any portion of the curve 2 2 2 whose equation is (y + x)35- (y.- )3 ==a3, by means of the rectangular hyperbola referred to its asymptotes. 229. Required the length of that part of the evolute of an hyperbola which is included within the branches of the hyperbola, and determine the angle in which the two curves intersect each other. 230. From the equation between polar co-ordinates, it is required to prove that the locus of the intersections of the tangents to a parabola with perpendiculars upon them from the focus is a straight line. 231. Find the equation to the locus of the intersection of the tangent to a circle with the perpendicular let fall upon it from any point in the circumference. 232. Required the same when perpendiculars are drawn upon the tangents of a rectangular hyperbola from the centre. 233. In the involute of a circle where the given point is the centre, it is required to prove that the corresponding locus is the spiral of Archimedes. 234. It is required to prove that in each of the conic sections, the locus of the extremity of the polar subtangent from the focus is a straight line. 235. Shew that the extremity of the polar subtangent of the logarithmic spiral traces out the involute, and that of the polar subnormal the evolute, of that curve. 236. Determine the curve from every point of which perpendiculars drawn to a given parabola are equal. 169 237. If several chords be drawn through the focus of a conic section, and through the points where they intersect the curve, tangents be drawn, it is required to prove that the locus of the corresponding intersections is a straight line. 238. Determine the curve from every point of which perpendiculars drawn to a given ellipse or hyperbola are equal. 239. A straight line being drawn from the centre of an ellipse to meet the tangent and parallel to the distance from the focus, it is required to find the locus of their intersections. 240. Inscribe an ellipse in a given triangle so as to touch two sides in given points. 241. Inscribe in a triangle an ellipse of given area, so as to touch one side in its middle point. 242. Among the several parabolas which may circumscribe a given triangle, to determine that whose segment cut off by any side of the triangle, shall be the least possible. 243. In an ellipse or hyperbola, it is required to determine the locus of the extremity of the polar subtangent, the centre being assumed as the pole. 244. A right line being given in position, it is required to find the points in it, from which two tangents drawn to a given parabola, shall contain the greatest or least possible angle. 245. Find the equation of an ellipse touching the three sides of a given triangle, and shew how far the curve is indeterminate. 246. In the circumference of the circle described on the major axis of an ellipse, it is required to determine that point from which two tangents to the ellipse shall include the least possible angle. 247. Describe the least parallelogram about a given ellipse, and compare the areas of the two figures. Y 17o 248. One circle rolls upon another equal to it, it is required to determine the locus of any point of its circumference. 249. Of two ellipses having at first their vertices coincident and their axes in the same directions, one rolls upon the other, it is required to find the loci of the vertex and focus of the moveable ellipse. 250. From the vertex of a parabola a straight line is drawn inclined at an angle of 450 to the tangent at every point: find the equation to the curve which is the locus of their intersections. 251. To determine the greatest ellipse inscribable in a given triangle touching one of the sides at a given point. 252. Prove that the line, joining the points of contact of rectangular tangents to the cardioide, passes through a fixed point. 253. Determine the locus of the intersections of rectangular tangents drawn to the cardioide. 254. If a circle be inscribed in an ellipse and a common ordinate to the minor axis be drawn, it is required to prove that the locus of the intersections of the normals at the corresponding points of the ellipse and circle is a concentric circle whose radius is equal to the sum of the semiaxes. 255. If about a given point D the angle BDC revolve, cutting the straight lines AB and AC given in position in the points B and C: it is required to prove that the curve which is always touched by the straight line BC is a conic section. 256. If tangents be drawn making given angles with the axes of all conic sections having the same given foci, it is required to find the curve which is the locus of the points of contact. 257. If a and /3 be two sides of a triangle subject to the condition expressed by the equation a + a,3= c2, it is required to find the nature of the curve to which the third side 171 is always a tangent, when the angle included between a and )3 is of invariable magnitude. 258. Lines being drawn to cut the three sides of a triangle, so that the continued product of the parts estimated from the angles is invariable; it is required to find the curves to which these lines are always tangents. 259. A series of harmonic curves with the same co-ordinate axes and equal maximum ordinates, have tangents drawn to them at the points where they meet the axis: it is required to find the curve which is formed by the continual intersections of these tangents. 260. In a given parabola, it is required to find the locus of a point which shall always divide the normal into two parts having a given ratio. 261. Determine when one proposed function is a function of another involving the same principal variables. 262. Of all sections of a paraboloid of revolution made by planes passing through a tangent at its base, it is required to find that whose area is the greatest. 263. Inscribe in a portion of the elliptic or hyperbolic paraboloid cut off by a plane parallel to that of xy, the greatest parallelopiped. 264. Draw a plane through a given point equidistant from three co-ordinate planes at right angles to one another, forming with them a pyramid the area of whose three faces shall be the least possible. 265. A given square is placed in a spherical surface and four planes are drawn through the centre of the sphere and the sides of the square: determine the surface cut off. 266. If x and y be the co-ordinates to any point P in the plane curve APQ, x + h the abscissa belonging to any other point Q in the curve and AP=s; prove that the surface of the solid generated by the revolution of the arc PQ about the axis of,, is equal to d (I d2td h2 yds 2 'e + h + d t 1 2 + &c. }, where u = d 172 267. Draw a plane which shall touch two spheres given in position and magnitude, and shall pass through a given point. 268. Find the equation of the curve surface traced out by one of the angles of a square in constant contact with a spherical surface, without sliding upon the surface. 269. If three tangent planes to an ellipsoid be mutually at right angles, it is required to prove that their point of intersection will trace out a concentric spherical surface. 270. There is an indefinite number of spherical caps, having equal surfaces but different radii and a common tangent plane at their poles: it is required to determine the surface in which the circumferences of all their bases are situated. 271. What surface will be always touched by a series of planes so drawn that the rectangles of the perpendiculars let fall upon them from two fixed points shall be invariable? 272. If two concentric surfaces of the second order, have the same foci for their principal sections, it is required to prove that they will every where intersect each other at right angles. 273. If three planes mutually at right angles to each other, touch a plane curve of the second order, it is required to prove that the locus of their intersections is the surface of a concentric sphere. 274. Required the nature of the surface touching a series of planes which cut off from a given right cone, a portion towards the vertex whose volume is invariable. Similarly of a parabola of revolution. 275. Determine the nature of the surface which shall envelope all equal paraboloids of revolution passing through a given point and having their axes parallel to a given line. 276. If the equation of a surface be (;)+ d" ( y ) 173 subject to the condition expressed by the equation an Jr (3 + 3n = yf: required the equation to the surface which shall touch all surfaces possessing this property. 277. Required the nature of the surface touching all such surfaces when the equation of condition is a 3y=c 3. 278. If an oblate spheroid revolve about any one of its diameters, it is required to determine the nature of the surface which shall envelope it. 279. If all possible ellipsoids be described of which the semiaxes a,/39 y, are coincident in directions, and subject to the conditions expressed by = 7- -: it is required to deterp a c mine the surface which shall touch them all: and to find the equations of the curve in which it is touched by any one of the ellipsoids, and of the curve which is the locus of the intersections of all such curves. 280. Required the greatest possible equilateral triangle, which shall have its three sides passing through three points given in position. 281. From two points given in position, it is required to draw to a point in a given plane, two straight lines whose sum shall be the least possible. 282. Of a right cylinder there is given the length of a line drawn from the circumference of one of the ends to the centre of the other: it is required to find the dimensions, when its volume is the greatest possible. 283. Determine the position of a luminous point, in the plane passing through the centres of three given spheres, so that the sum of their illuminated surfaces may be the greatest possible. 284. Required the shortest straight line which can be drawn through a given point, and terminated by two straight lines whose equations are given. 174 285. Required the shortest distance between two straight lines whose equations are given. ~286. Find the shortest distance between the diagonal of a rectanglar parallelopiped and one of the edges which it does not meet. 287. Apply the theory of maxima and minima to determine the shortest and longest straight lines that can be drawn, from a given point, to the surface of a given sphere. 288. Perpendiculars being let fall from a given point in the axis of z on tangent planes to the surface whose equation is x2+ y2 =az, determine the locus of their extremities: and shew that, when the given point is the origin of co-ordinates, the intersections of the locus with the planes of xz and yz are cissoids of Diodes. 289. If A and B be two given points in the circumference of a given circle whose centre is C, and the tangents AS, BS intersect in S: it is required to draw a third tangent to the circle cutting the former in P and Q so that the rectangle CP x CQ may be the least possible. 290. If a logarithmic spiral roll upon a straight line, it is required to find the nature of the curve traced out by its pole. 291.' A straight line AB is drawn perpendicular to the axis of a right cylinder, and another PQ is traced upon its surface: also planes revolving about AB cut the cylinder in elliptic sections and the line PQ in points P, P', &c.: and tangents to these sections in their respective planes are drawn at P, P', &c.: it is required to ascertain whether these consecutive tangents intersect one another: and if so, to find the locus of their consecutive intersections. 292. Through the focus of a given parabola, describe any circle to cut the curve, and from any point in that part of the circumference which is exterior to the parabola draw two tangents to the curve: then it is required to prove that the 175 chord which connects the two points where the two tangents cut the circle is a third tangent to the parabola. 293. If A and B be two fixed points and P a point in any curve such that AP + BP is a minimum: it is required to shew that the straight lines AP and BP make equal angles with the normal at P. 294. If from two given points, two straight lines be drawn to a point in a given curve of any order so that their rectangle may be the greatest or least possible: it is required to prove that these lines will have the same ratio as the cosines of the angles which they make with the curve at the point of intersection. 295. If (- = secmO be the equation to a spiral, it is \a/ required to prove that the equation to the spiral which is always touched by a perpendicular through the extremity m of the radius vector of the former is (-) = sec ( + a \m+l 1 296. There are three geometric series, whose first terms are 1, 2, 3, and if the fifth, eleventh and thirteenth terms of the first, second and third series respectively be added together, the sum amounts to 101: it is required to find the three series such that the sum of the fourth, sixth and seventh terms respectively may be the greatest possible. Lately published by the same Author, THE ELEMENTS OF ALGEBRA. SECOND EDITION. Price 12s. 6d. THE ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY. SECOND EDITION. Price 10s. 6do THE PRINCIPLES OF THE DIFFERENTIAL CALCULUS. SECOND EDITION. Price 16s. All designed for the Use of Students in the University. Also, in the Press, THE THEORY OF EQUATIONS, &c. &c. THE PRINCIPLES AND PRACTICE OF ARITHMETIC, Intended for the same purpose.