B?''"i; kit'7' ^?T: THE ELEMENTS DIFFERENTIAL CALCULUS. W. S. B. WOOLHOUSE, F.R.A.S., F.S.S, &c. Partisan : JOHN WEALE, 59, HIGH HOLBORN, M.UCCC.LII. ©A , , , HtJG^ES, PRINTER, : llN&*S»9«Aa) COURT, GOUGH SaUARE. PREFACE. On first commencing to read the Differential Calculus, a subject which opens • wide field of analytical research, the student enters upon an entirely new tem of thought. In his previous investigations he has always been ; iistomed to consider quantities, whether known or unknown, as having ^ome fixed or determinate value; he has now to conceive the values of certain quantities to undergo continuous changes, and to operate upon these changes with new symbols and new processes, which in themselves have hut a remote analogy to ordinary Algebra. When two quantities, thus continuously variable, are connected by an analytical equation, and their values are therefore mutually dependent on each other, and they are supposed to be affected by simultaneous changes, it is evident that the increments will also be connected by some corresponding analytical relation. The primary object of the Calculus is to establish general methods of investigating the nature and properties of such relations when the changes or increments are supposed to be small. To effect this, it is first requisite to trace the successive values of the ratio subsisting between two increments, when the increments themselves are supposed to continuously decrease in magnitude, and to determine the limiting value of this ratio when they ultimately become infinitesimals. This ultimate or limiting value is, in fact, that which represents the ratio ^ when the increments are supposed absolutely to vanish, and it is completely defined and accurately determined by referring the successive values to the recognized law of continuity. The operation here described is the true foundation of the Calculus, and the condition of continuity, especially insisted upon in the present treatise, entirely removes from the limiting value that obscure and indeterminate character which otherwise forms an insuperable obstacle to a proper comprehension of the first principles. We recommend the student to make himself famihar with the methods of ♦' limiting ratios " and '' infinitesimals." The theory of Infinitesimals is literally that of the Differential Calculus, and the principal law which regulates this theory is directly inferred from the method of limiting ratios. The two methods are indeed virtually but modifications of the same idea. iv PREFACE. Thus, in comparing together the relative values of any two infinitesimals, the rejection of terms involving infinitesimals of higher orders is, in effect, precisely the same as that of proceeding to the ultimate ratio of the infinitesimal quantities by the method of limits, and such rejection may in reaUty be said to be the operation of cropping down the quantities to their ultimate or limiting relative proportions. The method of infinitesimals, sometimes called the method of elements, is therefore as correct in its reasonings and deductions, and as accurate in its results, as the method of limits, and, being less abstract in its nature, its apphcation, when properly understood, is usually attended by greater facility and clearness, especially in abstruse investigations. In preparing the present publication, we have endeavoured to do justice to each Chapter by restricting the applications to matters of general interest, which was considered to be essentially more solid and satisfactory than any attempt to give, within the prescribed limits, a meagre outline of a more extended variety of subjects. The first five Chapters comprise the entire theory of the Calculus as a pure branch of analysis, and the remaining Chapters exhibit the apphcations to the theory of maxima and minima, and the geometry of curve lines. The general theorems of Euler, Lagrange, and Laplace not being essentially required in the body of the work, though very important to be known by those who may desire to extend their course of reading, are inserted at the end of the last Chapter. The subjects contained in the several Chapters are treated according to the most elegant and approved methods of investigation, some of which are presumed to be new; numerous interesting examples, exhibiting their re- spective results, are inserted for the exercise of the student, and copious explanations are given of the precise nature of the principles involved in the various operations. It is hoped that these explanations may tend to obviate the pecuUar diflSculties so commonly experienced in the acquirement of correct notions, and, by making good the foundation, conduce to the rational and satisfactory advancement of the intelligent student in obtaining a knowledge of one of the greatest superstructures of the human intellect. Should this expectation be in any degree reahzed, we shall experience a cor- responding gratification. London^ March, 1852. CONTENTS. Chapter I. — Definitions and First Principles. PAGE (1) to (5.) General objects and definitions 1 (6.) Determination of a differential coefficient from an assumed * expansion 3 (7.) The principle of differentiation analogous to that of the measure- ment of the velocity of a body 5 (8.) Fundamental theory of Infinitesimals . 7 (9.) A tangent to a curve geometrically elucidates the signification of the differential coefficient 10 (10.) The leading principle of the Calculus is essentially based upon considerations of continuity 12 (11.) Lagrange's theory of DmiJerf jFMwc/fon« indicated 13 (12.) Objects of the Calculus of derived functions equivalent to those of the differential and integral calculus ib. (13.) Enumeration of the more essential preliminary notions of the Calculus 14 Chapter II. — Differentiation of Functions. (14) to (22.) Diflferentiation of Algebraical Functions 17 (23) to (27.) Logarithmic and Exponential Functions 27 (28.) Trigonometrical Functions 33 (29.) Results of elementary trigonometrical differentiations ... 36 (30.) Inverse Functions 37 VI CONTENTS. PAGE (31.) Compound Functions . . ^ 39 (32.) Implicit Functions 43 (33.) Functions of two variables 46 (34.) Functions of three variables 48 Chapter III. — Successive Differentiation. (35.) Functions of one variable 51 (36.) Changing of the Independent Variable 54 (37.) Compound Partial Differentiations may be taken in any order . 55 (38.) Functions of two or more variables 58 Chapter IV. — Development of Functions. (39.) First deduction of Taylor's theorem 60 (40.) Maclaurin's or Stirling's case of Taylor's theorem 62 (41.) Definition of an Intermediate Function ........ id. (42.) The average, or arithmetical mean, of m successive Functions is an Intermediate Function 63 (43), (44.) Theorems which limit the values of functions .... 64 (45.) Limitation to Taylor's theorem 66 (46.) Limitation to Stirling's theorem 71 (47.) Development of U = F (.r + A, y +• A:) ib. (48.) Development of tt = F( an infinitesmal of the nth. order, , an infinity of the wth order; Adx'^-^'^ Adx"" and, at the extreme limit, these become Adx'^+'* _ Bdx^ _ (9.) The method of determining the position of a tangent to a plane curve supplies an elegant geometrical elucidation of the signification of the differential co- efficient of a function. Let APB be a curve line ; P a point in the curve the coordinates of which are A D = j?, D P = 3/ ; Q another point in the curve the coordinates of which are A D' = ^ + A,r, D'Q = y +Ay; and suppose the curve to be deter- mined by an equation of the form y =fxj any function of x. Then from what precedes. Ay = Pa.2? + QAar2 + RA^3 ^ g^^. ^ = P + Q Ao? + R Aa?2 + &c. Ax In the diagram, Aa? = PG, Ay = GQ, and therefore A?/ — ^ = tan Z- 5 P G. Consequently tan A5PG = P + QAjp4-RAa?2 + &c (a) From this equation we infer that if A a? be taken less and less towards zero, the value of tan 5 P G will approximate to the differential coefficient (P) as its utmost limit. For the geometrical limit of the angle 5 P G, as Ax decreases, we may suppose the point Q to approach nearer and nearer to the point P, and watch the progress of the line rs which passes through them, or we may suppose the hne rs to turn FIRST PRINCIPLES. II gradually about the fixed point P, so that the intersection Q shall proceed towards P. The former of these suppositions will lead ultimately to an indeterminate result, whilst the latter will proceed at once to the extreme Hmit. Thus on the former supposition, when the point Q finally arrives at the point P, and the two points become one, it is evident that an indefinite number of lines can be drawn through them, and therefore that the position of the line rs is so far indetermi- nate. But on the other supposition, if the motion of rs be conceived to cease the instant the point Q arrives at the point P, it will then assume the position of the tangent II S^ which touches the curve at the point P ; and this is obviously the only position which can obey the law of continuity amongst the positions that precede it. If we now suppose the motion of r 5 to continue onward, it is evident that it will begin to intersect the curve on the other side of the point P, or between P and A, and that the positions will then have reference to negative values of A a?. The line rs will thus pass through a continuous series of positions as A a? gradually diminishes from positive to negative values ; and when Ao? = 0, though the position, as depending on the two points through which it has to pass, is then indeterminate, yet the position R S is the only one that can partake of the continuity existing amongst all the others, and the angle SPG is the only one that can partake of the continuity existing amongst the preceding and following values of that angle. Now, according to the equation (a), the series P + Qao? + Raj?2 + &C. strictly corresponds with the value of tan * P G for every value of A;r except zero ; and hence as the values of this series as A a: passes from positive to negative values are wholly continuous, and consequently, when A a? = 0, the first term P partakes of that continuity, it is conclusive that tanSPG = P=g (iS) 12 THE DIFFERENTIAL CALCULUS. which may be either considered as a fraction whose numerator and denominator are the differentials of the ordinates, or the differential coefficient of y considered as a function of a?. By this result it is evident that the differentials of the ordi- nates X, y may be relatively conceived as represented by two small coordinate lines Ym, mp terminating in the tangent at a contiguous pointy. . (10.) After what has now been explained the student will not fail to observe that the leading principle of the Calculus arises out of the following considerations : When a fraction, which in a particular case takes the inde- terminate form -, expresses the value of a quantity which we have reason to know from the nature of the subject does not become discontinuous in that case, or generally when such a fraction enters in any equation, the other terms of which are not discontinuous, the fraction is, under such circumstances, necessarily limited to continuous values, and consequently, when the numerator and denominator vanish, it must take the particular limiting value assigned by the law of continuity. It is on the ground of continuity alone that the mathematical accuracy and logical rigour of the principles and applications of the Calculus may be considered to rest. The fundamental principle of our operations, according to the theory of limits, consists in this, that if the increment of a function be divided by the corresponding increment of the independent variable, then as the increments are taken less and less towards zero, so will the quotient approximate in value to the differential co- efficient as its utmost limit. Thus the differential coefficient is that particular value of the vanishing fraction which con- forms to the law of continuity amongst the other values : and since this is the identical value of the fraction, which always enters as the subject of investigation, the truth of the principle on which the Calculus is applied, in the case of limits, may be regarded in the strictest sense, and at the same time rendered clear and satisfactory to the understanding. FIRST PRINCIPLES. 13 (11.) There is yet another mode of laying down the first prin- ciples of the Calculus, which, at the onset, has the advantage of obviating all considerations of infinitesimals and limiting ratios, so as to bring the subject within the scope of ordinary Algebra. This method, commonly called " the method of derived func- tions," is presented by Lagrange in his * Theorie des Fonctions Analytiques,' and the investigations, which in their nature are purely algebraical, are at the same time elegant, systematic and logical. In substance this method is equivalent to the following : Let h denote a small accession to the value of a variable quantity x which thereby becomes of the value x -{- h-, and suppose the binomial function f (x + A), when developed according to the powers of A, to be as in equation (1), viz. : f{x + h) ^fx -h PA -f Q/i2 + R/i^ -f &c. in which P, Q, R, &c., as before, denote new functions of x whose forms depend wholly upon that oifx. Then the coefficient P, which is identical with the differen- tial coefficient, Lagrange defines to be the first derived func- tion ; he designates it by fx^ and observes that it is quite independent of the value of A. By treating the derived function fx in the same manner, that is, by expanding f\x + K) and again taking the coefficient of A, a second derived function, designated by/"jr, is obtained; and this process is further supposed to be successively repeated to third, fourth, &c. derived functions. (12.) These definitions being premised, the more immediate objects of the calculus of derived functions are : 1 . The form of any function fx being given, to determine the forms of the derived functions, and to effect generally the form of the development of the binomial function f{x-\- h), with other problems relating to the expansion of functions. 2. The form of a derived function being given, to find that of the original or primitive function, &c., &c. The problems comprised in the first of these are equivalent to those of the Differential Calculus ; and those of the second, which refer to the inverse operations of the Calculus, are in 14 THE DIFFERENTIAL CALCULUS. effect the same as the inverse processes of integrating differen- tials and differential equations in the Integral Calculus. And these abstract analytical problems, which embodj the essential principles of the Calculus as an instrument of investigation, are thus established without introducing any ideas relating to infinitely small quantities or limiting ratios, all considerations of small quantities being in fact deferred to their legitimate and inevitable occurrence when we come to the actual applications of the Calculus to the various geometrical and physical subjects which arise in the different branches of mathematical science. We have here given a brief exposition of the fundamental principles according to different methods of treatment, because a knowledge of each of these will be necessary to enable the student eventually to acquire a thorough command of the powerful resources of the Calculus. After a little experience he will not fail to discover that the collective reasonings em- ployed in these methods are substantially alike, and that they in reality constitute the same grand unique system of deduction, only exhibited under different points of view or modified for the purpose of more immediate adaptation to particular objects of investigation. (13.) Before entering upon the manual operations of the Calculus or discussing the practical methods of differentiating functions, we shall here concisely repeat those preliminary ideas respecting the operation of differentiation, which should in the first place be distinctly impressed upon the mind : If, when the variable quantity x increases by an increment A 07, a function u or fx increases by A i« or A (fx) ; then the *•' differential coefficient" of the function is determined by ascertaining the ultimate ratio of the increments, or the limiting ,. 1 jy j,\, r L' increment function ^u contmuous value oi the traction -. . , ^ = — or mcrement variable h,x A(fx) --^ — when the increments are supposed to vanish, and this (^u d i I X) differential coefficient is symbolized by — - or -~^ — , and sometimes more briefly by w' ox fx. FIRST PRINCIPLES. 15 If we further suppose the expansion of the binomial function f(x + Ax), according to the ascending powers of Aj*, to be /(x -f AJ-) =/x + Fax + QAx^ + &c.; then the coefficient P of A a:, exhibited by the second term, will also be the differential coefficient of the function /(x) ; that is, du ^^ d{fx) ^ ^ dx dx In these relations du and dx may be regarded as simulta- neous infinitesimal increments of i« and x ; but this idea is not always necessary, because — may be either considered as a fraction determining the ultimate ratio of two infinitesimals or as an abstract symbolical representation of the coefficient P, according to the nature of the investigation. The following examples, in which the differentials are deter- mined from first principles, will practically explain their operation. Example 1. — Let w = j?^ . then, as the equation is general for all values of x, when x becomes a? + A j? it will give {u + Aw) = (a* + Axy = ar2 + 2xAx-\- Ax^, From this take away the first value u = x^, and we get Au Au= 2xAx + Ax-' ,\ — =2a7-hAJ7. Ax This last equation is accurately true for all values of Ax, however small, and the value of 2 a? + Ao: on the right- hand side, vdll evidently change continuously as we suppose A a? to continuously diminish and ultimately to vanish. Hence making A x = and taking the Hmiting value of the fraction — , denoted by -7-, we obtain Ax dx du ^ - _ , -— =.2x or du = 2 X dx, dx which is the differential of the proposed function ?« = x^. 16 THE DIFFERENTIAL CALCULUS. Example 2. — Let w = a?^ + Sa'^x ; then, when x hecomes u + Au= (x + Ax)^ + 3«2 (^ 4- Ax) = ^2 + 3fl2^ + 3(a?^ + a^)Aa: + SxAx^ + Aa?^. Reject u =0?^ + 3«2.r, and Aw = 3 (0^2 + «2) A^ + 3 d? Aa?2 + Ax'^ .-.— = 3(^2 ^^2) ^ 3^^^ + Ax^. Ax Hence, as before, making A^ = and taking the limit, we get ^ = 3 (.r2 + a2) or du = 3 (x^ + a^) dx. Example 3. — Let u = — r > then w + Aw = — , ^ — , and — X — Ax a'^+bx + bAx a^ + bx (a^ + b^) Ax Aw = — b — X — Ax b — X {b — x) (b—- X — Ax) Au __ a^ -\-b^ " * Ax ~~' (b — x) {b — X — Ax)' Therefore, at the limit, du a^- + b^ ^ «2 + ^2 T- = 77 r^ or du = 77 dx, dx (b — x)^ (b — xy The process of finding the differential coefficient or the differential of any proposed function is called *' differentiation," and we proceed in the following Chapters to estabhsh the principal rules by which we are guided for the purpose of facilitating the actual performance of this operation on the different forms and varieties of functions. DIFFERENTIATION OF FUNCTIONS. 17 CHAPTER II. DIFFERENTIATION OF FUNCTIONS. I. Algebraical Functions, (14.) A constant quantity connected with a function by the sign of addition or subtraction will disappear after diiferen- tiation. Let M = P + c, P denoting any function of a variable x. When X becomes x -\- ^x, suppose P and u to respectively become P -\- aP, w + Aw ; then w + AM = (P + aP) + c. From this subtract w = P + c and there remains the in- crement Aw = aP. Therefore -— = — — and hence :t- = -7 Ax A J? ax ax or du = dV, in which result the constant quantity c does not appear. (15.) A constant quantity connected with a function as a multiplier or divisor will remain as a multiplier or divisor after differentiation. Let t< = c P, P as before denoting any function of a variable x; then when u, P take the new values u + Au, P + aP, we have u + Au = c (? + aP). From this subtract u = cVy and we get Aw = caP . Aw _ aP Ax Ax du dV Hence 'l~= ^ ~T~ or du = c c/P. c- ., , .. P - , du \ dV ^ dV bimilarly, if w = -, we nnd — - = - . -— or du = c dx c dx c 18 THE DIFFERENTIAL CALCULUS. (16.) The differential of a function consisting of two or more terms, connected by the signs of addition or subtraction, is found by differentiating each term separately and collecting the results with their proper signs. Let tt = P + Q + R + &c., where P, Q, R, &c. are func- tions of X ; then when x takes the value x -{- ^Xy the function %i will become w + Aw = (P + aP) + (Q -f aQ) ± (R + aR) + &c. From this subtracting the former value w=PHhQHhR± &c., we get AM = aP ± aQ ± aR + &c. Aw aP aQ aR . „ -•. — = — \^—^^ — ^ + &c, Lx l^x — Ax —• Ax — du dV dQ dU Hence -j- = —. 1 — -, h -3 h &c. ax ax — ax ~ ax ~- or du=i dV ±d(i± dV.± &c. (17.) The differential coefficient of any constant power of the independent variable x is found by multiplying by the exponent and diminishing the exponent by unity. Let u^= x^ ; then when x takes the value x + Ax, w -f- Aw = {x -{• AxY, .', Aw = (a? + AxY — x"^. To find the value of Au in powers of Ax it will be necessary to expand this binomial ; but the second term of this expansion will suffice for our present object, and this may be readily found by means of induction, independently of the binomial theorem. First, suppose the exponent w to be a positive integer. By multiplying successively by x -\- Ax, disregarding the terms which involve the second and higher powers of Ax, and in- dicating those terms by + &c., we obtain {x + Ax) z=z X + A<2? {x + Axy =a?2 + 2^Aa?4- &c. DIFFERENTIATION OF FUNCTIONS. 19 (x + A.r)3 =x^ -i-Sx^ AX + &c. (x 4- Ax) "^ ^x"^ i- 4x^ Ax + &c. &c. &c. And generally, (x + Ax)^ =z x*^ + nx^~^ Ax + &c. The value of Am is therefore of the form Au^nx"^"^ Ax 4- QiAx^ + Raj?3 + &c. where Q, R, &c. denote certain functions of ar and n. Hence — z^nx^'-^+QAx-h UAx^ + &c. ; Ax and this equation is true for all values of A x. By proceeding continuously to Aa: = and taking the limiting value of the fraction, it ultimately gives du -—-=: nx"^~ ^ or du = nx^~^ dx. ax The same reasoning and the same result also obtain when x instead of being considered the independent variable is sup- posed to represent any function of another variable. Secondly, suppose the exponent to be a negative integer, or M = ^- ^^ then m = — , m + Aw = — , ^ .^^ and x^ {x -f Axy^ 1 1 {x -^ Ax)"^ — x'^ ^u = =. (x -\- Axy x'' x"" (x + Ax)"" nx'^-^Ax^- QAx- +RAx^ 4- &c. x"" (x + Ax)"^ A?/ _ nx^"^ + QAx + 'RAx^ -\- 8ic. ' ' Ax" x"^ {x + Ax)"^ By proceeding as before to the limiting value, this gives du nx^-^ . , - , -— = = —nx"^"^ or du -= ^ nx ^^"^ dx, dx x"'^ Thirdly, suppose the exponent to be fractional, or m = 20 THE DIFFERENTIAL CALCULUS. X n ; then u^ = x'^ and nu^-^ du = mx"^~^ dx , m , au_mx^-^_ mx^-^ ___ m w~ . m If the fractional exponent he negative, or u — x ^ ; then w^ = 0? ~ *" and nu^- ^ du = — mx-^-^ dx, which in the same du m " n~^ > waygives^^=--^ The rule is therefore true for all powers, whether the expo- nent he positive or negative, integral or fractional. (18.) The differential of any constant power of a function is found by multiplying by the exponent, diminishing the exponent by unity, and finally multiplying by the differential of the function. Let u = 'P^,'P being a function of x ; then proceeding as in article (17), only substituting P in place of a?, we obtain du -— =7?P«-i sinddu = nV''-^ dF. dP As in the former case, this rule is also true for all powers, whether the exponent be positive or negative, integral or frac- tional. Cor. Hence also — = nF^~^ -r- dx dx and c?M = « P '^"^ -r- dx. dx (19.) The differential of a function consisting of two variable factors is found by multiplying each factor by the differential of the other, and adding together the two products. Let w = P Q, the factors P and Q being functions of x. When X becomes x -{■ Ax the corresponding values of u, P, Q will bew-f-Ai«, P + aP, Q + aQ respectively, and then w + Aw = (P + aP)(Q + aQ)=PQ + QaP + (P + aP)aQ DIFFERENTIATION OF FUNCTIONS. 21 .-. A« = Q AP-f (P + aP)aQ Aa? Aj- ^ ^ i^x Ilencc, making the increments vanish and taking the limit- ing values, we get ^=Q^+P^orrf« = QrfP + PdQ. CLX (iX CLX (20.) The differential of a function consisting of any number of variable factors is found by adding together the products formed by multiplying the differential of each of the factors by all the others. Let w = P Q R, a function consisting of three variable factors P, Q, R. By considering the function u to consist of two factors PQandR, we have by (19) du = R«f(PQ) +PQc?R = R(Q(^P4-Pc/Q) + PQc?R = QRrfP + RPd^Q + PQrfR. Similarly if « = P Q R S, the product of four factors, we obtain c?t/ = SJ(PQR) + PQRc^S = S(QRc^P + RP^Q + PQc^R) +PQR c + 2 c a) + (1 2 a + 6 5 + 4 c) j: + 1 8 jr2. ^,^ . a -f- -^^ 9. Differentiate u = • a — X By (21), {a — x) X dx — (a + x) X — d* 24 THE DIFFERENTIAL CALCULUS. _^ (a — 0^) dx + (a + a:) dx __ 2 a dec {a — xy^ (a — x)'^ du ^ 2 a dx "" (a — x) 2* 10. Differentiate w = V ^^-±^, or w = OM:^ . « -— 0? (a — ^)4 Here du = (a ^ x)i X ^ (a -{- x) ^ ^ dx — (a -\- x)^ X — ^ (^ •" ^) - * ^-^r (« -- x) du __ (a '— x)i (a -\- x) '~ ^ -{- (a + x)i (a — x) " i ' dx 2 (a — x) (a — x) -i- (a -\- x) a ^ 2(a — x) {a — x)^ (a + x)^ "~~ (a — x)l{a-j- x)i a (a — x) V a^ — x^ Otherwise, by squaring, we have u^ = and, by the (I —— X last example, 2udu = ^ ; {a — x)'^ du dx u (a — x)^ (« — .r)^ V a + X (a — x) ^ a^ — x^ 1 1 . Differentiate z«='v^«^ — ^2. Write u = {a^- a?3)i and by (17), (18), du = ^ (a^ — x^-)-h X — 2xdx = 7^^ ~o ' 12. Differentiate u = ^ a^ + 2bx + x^. Here u= (a^ + 2b x + x^)^ ; .-. du = i(a^ + 2bx + x^)-^ X (26e?^ + 2.r(f^) (b + .r) £?.r ~" V a^ -{- 2bx ■}- x^* DIFFERENTIATION OF FUNCTIONS. 25 3. 13. Difterentiate u = — ^^- — r, —=: ,v — ^. By (18) and (21) ^3x3 («2-f J:2)ij,^^__(^2^^2)T X 3^2^X du= ~, — ' ' dx "" x^ 3a2 x^ ,2 ^ .2 Otherwise, writing the function in the form u = («2 ^ x'^y x-^, we obtain by (19) du = x-^ X 3.rrfa: (a2 + ^3)2 + (^2 ^ ^2)T x —3^-4^ = Sdx (a2 + ^2)i {a:- 2 _ ^ - 4 (^2 4. ^2)} 3a2 = - 3a2^-4tf^ (a2 + ^2)2 _ >/a2 + j,2, X* 14. Differentiate w = _ (a2_^2)¥ X dx-x X -(a^--x^)- ^xdx UU — o ' o 0'* — a?-* _ (a^ — 0?^) c?j7 + j?^da? _ fl^^/vP (a2 _ ^2)T («2 __ ^2)1 Tx./Y» . 'vfl + «2^ — "^ a — X lo. Diiferentiate u = .. "v a -\- X + V a — X Differential of the numerator = i(a + x)-Ux-i-i{a-x)-idx "^ a + X -h "^ a — X , r — ■ CUT. 2 >/ a2 _ ^2 B 26 THE DIFFERENTIAL CALCULUS. Differential of the denominator — i {ci + ^) ~ ^ da: — ^ (a — oc)-i dx = . ^ ^ ax, 2Va'--x^ Therefore by (21) we have (A/fi5 + a?+Va — ^)3-f (V a + X — '"/ a — xY , du = -——-=: - — dx 2 (V a + X + V « — ^)2 V a^^^2 adx a (a — V a^ — x^) — dx. (a + V a^ — x^) "^ a^ — x^ x^ V a^ — a?2 16. Differentiate u = ■ JL Writing (a^ — x^)^ for \/ «^ — x^, we similarly have, bv (21), du =: (a^-x^)^x(-Sa"-x-4:X^)dx-(Sa^-4a^£c^-x'^)x -(a^-.r^)-^^ d^^ (a2-.^2)t («2_^2)f dii 17. If w = (« — ^) (6 + .2?) ; then — = a — h. 1 3 3 ^, c?w /.r + 3V 18. Ifz^=- + -^4--^; then - = -( — ~) . X x^ x'^ dx \ x"" / ,- , du a^ + 3bx -{- x^ 19. liu=z(a^- + bx + x'-)^yx;then^- ^^ du 3x^ 20. Ifz. = (2 + ^2)Vi_^3. then^=~ ^^^— -, 2.r2-«2 ^ ^^ 30^4 21. If w = ^ Va2 + ^2. thenT-= 4 / . o ' DIFFERENTIATION OF FUNCTIONS. 27 22. If„ = (^l+^;thenjf=_i^^,-r:r^. x^ ax x"^ ■2:]. If ;/ = (3 .1- - 2 «2) (a2 ^ ^2)1 . du then — =:15^'Wfl2^a:3, dx (22.) Expressions under the form of square roots are of ' cry frequent occurrence in analytical investigations, and their lifferentiation, according to art. (18), using |- for the exponent, suggests the following simple and expeditious rule : The differential of the square root of a function is fomid by aking half the differential of the function and dividing the same by the square root of the function. This useful rule may be practically applied by the student to Nos. 11, 12, 14, 16, 20, 21, of the preceding examples, and it will enable him at once to put down the final result in all ordinary cases of this kind. II. Logarithmic and Exponential Functions, (23.) The logarithmic function u = logo? depends upon the exponential relation a^ = « ^os ^ = j7. Thus if a ^^s -^^ = j?, and a log y = y^ ^e have, by multiplication, a log -^ + log 2^ ■=^ xy\ but a log ^^y) = xy, .-. log jc + log y = log {xy), which is the fundamental property of logarithms. The constant quantity a is indeterminate and may have any proposed value. It is called the base of the logarithmic system belonging to it, and, since a^ = «, it is evidently the number whose logarithm in the same system is equal to unity. Since a? = a", we have j? + Aa: = a'* + '^", and therefore Aj _ a" + -^« — G« _ ^^ a^"-l Am Am * Am In taking the limits of this equation we observe that the 28 THE DIFFERENTIAL CALCULUS. , ^ Aw 1 , , limiting continuous value of the fraction , which in Am £^x common with -— takes the form t: when Aw =;= 0, must he a Am ' function of a and independent of Am. Denoting this function by X fl, we have (1$ \ X « = Hmiting value of when ^ = dx -— = a^\a = x\a, du Again, the equation a? = «^' gives ^^ = a^^, B denoting any value whatever. Therefore x^ — \ _a'^^ — \ _ a^^ — \ This equation is necessarily true for all values of 6, By proceeding to the limit ^ = 0, m ^ = 0, the continuous values, from what precedes, obviously give X,r = MXa; , \x .*, u = log X = - — X a The value of the function X x may readily be obtained in a • X. ..• •^'-^ • .1, ^ {1 +(cr-l)}^-l series by puttmg — - — m the form -^: 1 J. Thus, by expanding according to the binomial theorem and putting ^ = in the final result, we obtain X ^ = (^ - 1) - i (^ - 1)2 + i (.r - 1)3 - i (^ - 1)4 + &c., so that the last expression for log x may be written _ {x-^\)-\{x-\f + \{x~\f-i{x-\Y^kc. ^^§^ ~ (« ~ 1) -i (a - 1)2 + I (a~ l)3^x (a- 1)4 + &c/ These equations apply generally to a system of logarithms having any value a for the base. According to Briggs's system, on which the logarithmic tables in common use have been calculated, the base a = 10, which greatly facilitates the DIFFERENTIATION OF FUNCTIONS. 29 use of the tables in arithmetical calculations which involv(» decimal numbers. (24.) If the value of a be so assigned that Xa = 1> we shall have logo: = \t, and log a = Xa = 1. This value of a will simplify the analytical relations and give the Napierian system of logarithms, of which the value of a so determined is the base. Hence it follows that the function we have indicated by X characterizes the Napierian logarithm. To determine tlie particular value of a which will fulfil the proposed condition X a = 1, instead of using the series for X a take the initial form of this function, and we have flO _ 1 hmit of — - — = 1, when ^ = ; 1 .-. a = hmit of (1 + S)^y when ^ = 0. By expanding according to the binomial theorem, we find e' ^iG-0G-2)g3^g,^ 2.3 - 1 + 1 + -^ + 2:3 + &c. Now, when B passes from a small positive to a small negative value, the value of every term of this series will evidently vary continuously, and when ^ = it gives the limiting value of 1 (1 + 6Y = 1 + 1 + I + ^ + 2^ 4- &c. = 2-7182818, &c. This arithmetical value, which forms the base of the Napierian logarithms, is usually denoted by the letter e, and sometimes by the Greek letter 6, and these symbols always represent this 30 THE DIFFERENTIAL CALCULUS. arithmetical value whenever they appear as roots of exponential functions. The Napierian system, from its greater algebraical simplicity and convenience, is also that which is generally employed in analytical investigations and formulae ; and therefore whenever a logarithmic expression occurs, the Napierian logarithm should always be understood unless the contrary is distinctly stated. We have thus, according to this system, the following rela- tions : x^ — I loff X = limit of -, when ^ = 0. 1 e = limit of (1 + ey = 2-7182818, &c. When u = log a?, the expression for — (art. 23) also du becomes -— = Wi eivins; du ^ — • ; but we shall otherwise du o o ^ determine this diiferentiation in the next article. (25.) Diiferentiation of w = log x. When 0} becomes x + Ao?, u becomes u + Aw, and we have w + Aw = log (a? + Ax) ; Aw = log (cP 4- A^) — log .r = log = log I 1 H I and, putting — = ^, we find In proceeding to the limit Aw = 0, A~ + 77 + "TT "^ ^^-^ which gives = PQR,&c.(-^ + -^ + - + &c.) which is equivalent to the formula of art. (20). (26.) Differentiation of w = a^. When X becomes a: + Ao? we have u -\- Aw = a ^ + -^-^ ; Am g-^ + a^ _ a^ a^"^ — \ ^X AcT AJ7 But (art. 24) the limiting value of the vanishing fraction , which IS of the form , is lo"; a ; therefore ^x e "" du . , , -— = log a . rt'*", or du = log a .a'^ dx. Thus the differential of an exponential quantity having an invariable root is found by multiplying together the logarithm of the root, the exponential itself, and the differential of its exponent. Hence, when a = e, or m = e', we have, since log e = 1, du -P = e^, or du = e' dx ; dx 32 THE DIFFERENTIAL CALCULUS. that is, the differential of an exponential quantity having for its root the Napierian base e is found by multiplying it by the differential of the exponent. (27.) Differentiation of w = P - 1+0? dx (I H- xy 3. If w = logo: log -^o:'; thcn -^ = Hog a? -f - ) log " * t. , I , du 1 4. It u = tan-^^ + - ; then -r-= — .-,/, , — ot- a: dx X" {\ -\- X'^) '^ du \ -\- X tan - ^ x 5. If M = tan~*a? vl + x'" '•> then -r = .. ^ — du X sin ~ ^ X G. 1{ u = X — \/l — .r^ sin -^07; then d^ ^y\-x- 7. Ifw=(2cr~ — \)sin-'^x -\- X ^yi — X'^ ; then -- = 4 .37 sin " 1 jr. ax 1^ u = x^ + (sin-^jc)- — 2 sin~^j? . .r \/l — o?-^; 1 du _ 4 x^ sm~^x djo ^\ — x'^ V. Compound Functions, (31.) If in a function u =^fx the variable x is replaced by another function (^o:, the expression u=f((l)x), which then becomes a function of a function, is called a compound function oi X. Let i/ = (l)x, so that u=:fi/, and let An, Ax, Aij denote 40 THE DIFFERENTIAL CALCULUS. corresponding increments of u, x, y\ then, as the equation Am Aw Ay Aj? Ay Aj7 essentially represents an identity, and is therefore true for all values of the increments, however small, it must evidently he true when we proceed to the limit or suppose the increments to vanish and take the continuous values of the respective fractions. Hence du dii dy du dy . dx dy dx dy dx du dy ... where -j"' ~r are the differential coefficients of the functions dy dx u =^fy, and y = (j)x. That is, according to the usual notation of derived functions, or du =zf'((j)x) 'x.dx. Similarly, if y = ^ .r, z =^y\ry, u =/^, so that the function u is of the more complicated form u =:f{yjr((l)x)}y or the function of a function of a function, it may he shown that du du dz dy , du d^ dy , __ -:= . -^^ or c?M = —.— •-/. «^ ; dx dz dy dx dz dy dx and these, according to the notation of derived functions, would be written J =/r . f y . (^'^ =/ (>/.y) ^\^x) cj^'x or du —f {-^{i>x)^^r\^^) ^xAx. In the same way the formulae may be extended to any number of superposed functions, and it is obvious that they all depend upon the following simple principle \ DIFFERENTIATION OF FUNCTIONS. 41 The differential of a function of any variable quantity what- ever is equal to the differential coeflicient of the function, with respect to that variable quantity, multiplied by the differential of the variable quantity. Thus if, as before, u=/{\Ia ((t)x)}yhy successively apply- ing this principle, we have du=f'{yl.(ct,a^)} Xd{yl.(cpx)} =f'{^(^^)} XV^'(0x) Xd(ct>x) The following examples will practically show the mode of proceeding here indicated : 1 . Differentiate u = log sin a;. By (25) and (29) we have ds'ma: dxcosx du = — ; = : = dx cot X. sm X sm X 2. Differentiate u = log;^ ^ + X (b + x) dx — (a + x) dx By (21), d i^^ ^j—^ _ (a — b) dx ^ ~~ {b + x)'^ ' Therefore by (25) we have \b + xj b-\- (a — b) dx b -\- X __ (a — b) dx {b + x)- a-\- X (a-\- x){b + x) Otherwise, since u = log (« + x) — log (b + x), we have by (25), , __ dx dx __ (a — b) dx ""fl-fj? b -i- X ~~ (a-\-x)(b -\- x) 3. Differentiate u = e"^" •*' sec x. 42 THE DIFFERENTIAL CALCULUS. Here du = secxd (e^^^^) + e^^'^^dseca:, by (19), = seca^e^^'^^dsinw + e^^^^dsecj:, by (26), = sec a: e^^^^ . dx cos .r + e^'^^^dxtonx sec ^, by (29), r= esino^ (1 + tan x sec a?) c?a7. 4. Differentiate ^« = log ( a^ a^ + a;*^ + .z-). By (22), d(VW+^ + x) = - .l^l^^ + dx ( V a-^ -\- x^ -]- x) dx Therefore by (25) we have diV^^TV^ + a:) dx du r- Va^ + x'^ + x Va-^ + x^ 5. Differentiate u = log tan e-^. Here du = d (log tan e-^) = 'J^^^^, by (25), = i(fZ!)!!^, ,y (29), =-^i^£rii±^!B!£::!),by(26), tan e""^' = — dx e-^ (tan e--*' + cot e-^). du 6. If w = 0?^ e^ino? . then - = x^-'^ (m + xcoBx) e^^^^\ dx 7. If w = 2 log sin X + cosec^ a? ; then — = — 2 cot^j?. dx du e'^^^ " ^ 8. Km = e^^^ ^*^ ; then — = dx Vl—x^ 9. If» = logfi+ iV then^=-^^ ^\x X"}' dx x{x + 1) DII 1 I HFN TT\TION OF FUN^TTOV<5. 43 10. If u = (.1- - a') log ^— ^ + 2a£; then -7- = 2 a: loe: . dx ° a — X 11. ii // = lu- u 4- 2 J" H- 2 VT + 07 -f a:-) ; c/w 1 then dx V[ + X -^x'^ 12. If u = tan-i-^ ; then — = - I — x^ dx \ -\- x'^ X fj^y^ 2 13. If ?/ = sm-^ . ; then — =-- , . ,, Ti» ,6 4-«eosa? , c?w ^/a^ — h^ 14. It w = cos "^ 7 ; then — = a -\- b cos cT ' dx a •\- h cos a: 15. If M = sm ^^ (3 J? — 4 0?^) ; then -;- = / , ^ ' c?^ V 1 — jp- , « -r/> X -1 1 ^W 1 + cT + J?" ^ -1 16. If M=r a?etan -r. then-— =— ^ — ^^""^ **'• ftj? 1 -h J?- 17. If ?/ = tan ~ ^ sin ~ ^ J7 ; then — = dx {1 + (sin-i^)2}^/i_^3 VI. Implicit Functions. (32.) The functions hitherto considered are supposed to be explicitly expressed in terms of the variable quantity involved, and upon which its value is made to depend. But a function u may have its value depending upon that of the variable x, though not expressed in any definite form, algebraical or other- wise, and perhaps not capable of being so expressed in finite terms. In fact, the relation which connects together the cor- responding values of u and x may be presented in the form of an equation, f(ii,x) = 0, / characterizing any function what- 44 THE DIFFERENTIAL CALCULUS. ever of m and x. The function u is in such cases called an implicit function of the variable quantity x. If the eq[uation f (uy x) = could be solved for u in finite terms involving x, the function u would then be exhibited as an explicit function of X ; but, as before observed, this may or may not be possible. A little consideration, however, will show that the differential of u with respect to a^ may be more directly obtained by taking the differential of the proposed equation in its original form. When X becomes x + Ax, u becomes u + Aw, and as the equation/ (w, j?) = must be true for all coexistent values of u and X, we have f{u-\- Aw, x + A,r) = 0, and f{u -f Aw, X + Ax) — / (w, x) = 0, or A/(w, 0?) = ; . A/(w,a^) ^^^ ' ' Ax This relation vdll be accurately true for all values oi Ax, and at the limit A.r = it gives •^ V^^ = 0, or df(u, x) = 0, which is the differential of the proposed functional equation, observing that u and x vary simultaneously, u being a function of X. This differential equation will be of the form "Pdu -f Q c?j7 = 0, and it will therefore give the value of the limiting ratio ~, or of the differential coefficient of u with respect to x, the same being expressed in terms of u and x: Example 1 . — Differentiate the function u when u^-2u '/«M^ + .r2 = 0. By differentiating the equation, we have . 2 ux dx 2udu — 2 ^ a^ + x^du — ~ . _ + 2 x dx =^ 0, ^ a^ -\- x'^ or2(w— \/a^ -\- x^)du— — , Lxdx = Q', Va^ + x^ du _ X dx a/«2 ■\- x'^ DIFFERENTIATION OF FUNCTIONS. 45 In this example the equation w- — 2m VoM-^^ + j?^ = q involves w in a quadratic, and may therefore be algebraically solved for u, giving u = "^a^ -{- x'^ A: a, which is the explicit form of the function u, and its differentiation will also lead to the result we have just obtained. Example 2. — Differentiate u when w' — 3 wj?^ -h 2 a?^ = 0. The differential of the equation gives Zu'^ du — Sa:^ du — Quxdx + Q x'^ dx := 0, or 3 (w- — x'^) du — 6 (ux — j?") rfx = ; du 6 (mj: — • a:-) 2 a: Example 3. — Differentiate m when x sin w — u sin j? = 1. By differentiating the equation, we have dx sin M + J7 du cos u — dusmx — udx cos ^ = 0, or {x cos u — sin a?) du — (u cos J7 — sinu) dx = ; , du ^u cos J? — sin w dx'^ X cos w — sin a: 4. If w'"^ — 3 a M j: -f a?3 = ; then — = ^ . dx u^ — a X - Tr » • r 1 A ^1, ^^ sin M — M cos J7 5. If w sin a: — J? sin M + 1 = ; then — = — . dx sm J? -^ X cos u 6. If x3 + w3__2a V^^-M- = 0; then ^ __ ^ ^ ~ ^-2:2- 1^2 dx u a + \^ x^ — u^ du 7. If w'* log M — a J? = ; then — - (/a? ^^-^(l + wlogw)* 8. If ^e" - w -h 1 = ; then ^ ^ _l! dx 2 — 9. IfM.r- (a + w)a/6"--m- = 0; then — = . r^ — ^. dx X ab" -\' u^ 46 THE DIFFERENTIAL CALCULUS. then — = — , dx Va'^ — u^ VII. Functions of Two or more Variables. (33.) het u =f(x, y) denote a function of two variables X and y. If instead of x and y varying simultaneously, x be supposed to vary alone without any change in the value of y, then )/ will be treated as the symbol of a constant quantity, and u being then considered as a function of x only, its differential or difPerential coefficient will be determined by the foregoing methods for functions of one variable. The value so deter- mined, however, as it is made to depend upon a change in the value of X without any supposed change in the value of y, will be only partial, and will not refer to a consideration of the total change of w. In order to distinguish this, the differential coefficient is usually placed within a parenthesis ; thus ( ^r ) denotes the ^partial differential coefficient, and \-i-)dx the partial differential of u with respect to x, that is, supposing x alone to change. Similarly, if y alone be supposed to vary and X to be invariable, \-r) will denote the partial differen- tial coefficient, and ( T~ ) % the partial differential of u with respect to y. These partial differentiations, as before observed, may be effected by the preceding methods for functions of a single variable ; first regarding w as a function of only one variable x, and again as a function of only one variable y. The supposition of .r or y varying separately, so as to partially differentiate the function u, is here to be received as DIFFERENTIATION OF FUNCTIONS. 4? a mere conventional hypothesis assumed for the purpose of more distinctly defining certain abstract analytical operations, to be applied hereafter. Returning now to the proposed function u z^f{x,y), when nd y respectively become x -f Aa:, y + Ay, it becomes M -h Am -=f(jo + AcT, y + Ay) ; that Am ^=-fix -f A (S> (S> - -• The same with respect to y are (?) m- (?) -• -■ the brackets indicating, as in art. (33), that the derived func- tions are only partial. But we may differentiate, in succession, sometimes with respect to one variable and sometimes another, in which cases the notation usually adopted is as follows : _ J — ) is indicated by ( ) dx \dy J \dx dy/ d d /du\ . . J. .11 / d^u \ ( — I is mdicated by ( — - — ), dx dx \dy/ xdx" dy) &c. &c. where the numerator shows how many differentiations have been taken, and the denominator shows the variables employed in the reverse order of the operations. We proceed to show that the resulting values of these successive partial derived functions are independent of the order in which the variables are supposed to change. The operation of differentiating a function ^ {x) is defined by the relation d^i^x) d . _ (l)(x + dx) — (j) (x) dx dx dx By applying this to the function w =/(^,?/), first with respect to x and then with respect to y, we have SUCCESSIVE DIFFERENTIATION. (9= _ /(j -f dx, y) —f{x, y) dx /du\ _ /(x,y + dy) —f(x,y) \dy) dy aud by again applying the same principle to these functions, we get d /du\ _ dy \dx) ~ f{i-\-dx,y -hrfy) -f(^>y + dy)- ■/(^ • + dx,y)+f{x .y) d /du\ dx \Ty) "~ dxdy f{x -\-dXyy ^dy) -f(x + dx,y)- •/•(^ ,y + dy)+f(x, y) dxdy Hence, as these expressions are alike, , we have that is, d rdu\ _ d /a dy \dx/ dx \a !)■ /dhi\ _ (dhi\ \dy dx/ \dx dy) This property is true when w is a function of any number of Variables, because when x and y alone vary, the other variables only enter in the same manner as constants, and as regards the operations performed, u may therefore be considered as a function of only two variables. Hence it follows that in calculating partial differential coefficients we may always interchange at pleasure the order in which the several dif- ferentiations are performed, without altering the results. Thus when u=f{x, y), we have also \dy dxy ■" \dx^dy)' \dy^ dx) "" V^ dyO ' and generally, when w is a function of two variables, c5 5'8 THE DIFFERENTIAL CALCULUS. / d^+'^u \__/ d'-^'u \ \dy^l^r) ~ Kdx^^^J d_ / d^'+'u \ _ / d^^'+^u \ d / d^'+'u \ __ / d^'+'+^u \ dx \dx'' dy') ~~ \dx'-+^dy')' dy \dx^' ) ^ \dx'' dy'^0' Example 1 . Let ic =^ x sin y -{- y sin x; then /dii\ . _ /du\ I — j = sin y -T y cos x, I — j = a? cos y -{- smx ; / d^u\ / d^'-u \ which two results are identical. Ex. 2. Let w = 2 ^"y^ _j_ ^Ay . then /__^__\ _ /__^i_\ _ / ^^ \ — 12 (^3 4- 2) \dy dx dx) \dx dy dx) \dx dx dy) (38.) The general property estahlished in the last article will assist us in the successive differentiation of a function of two or more variables. Let u^=f{x,y), a function of two variables ; then, art. (33), its first complete differential is in proceeding to the next differentiation it must be observed that the coefficients ( — h f — j are generally to be considered as functions of both variables, and to separately admit of being differentiated in the same manner as the original function u, by adding together the partial differentials. Thus we have /du\ _ jrf /du\ J j_ d /du\ \dx/ dx\dx) dy\dxj -\d^)''-^\d^y)^'^ /rfttX ___ d /du\ . , £ /^\ 7 \cly J ~ dx \dy) dy \dy) SUCCESSIVE DIFFERENTIATION. o!) = i^)''' + (jp)'^y- Again, if we adopt the principle of general differentiation, and suppose dx and di/ to be variable, we shall have, art. (19), ^{(e)-}='-<£)+(k)'" The sum of the left-hand members of these is the differen- tial of the value of c?w, and is therefore equal to d-u. Hence, adding together these two equations and substituting the preceding vahies of c?( — ), c?( — J, we obtain The process of differentiation may be successively carried on to higher orders in precisely the same manner, so as to deter- mine general expressions for d'''u, d'^u, &c. ; but as the fi^rmulse for the higher orders become rather cumbrous and are seldom required, it will not be necessary to give any of them here. If the variables x and y are independent of each other, and their values admit of being connected by a relation of the lorm y = a a? + jS, so that we may consider both of them to increase by constant increments ; then dx and dy=zadx may be both supposed to be invariable. On this hypothesis, d'^x = (), &c. and c?-y = 0, &c. and the expressions become u=f{xyy)y &c. &c. &c. 60 THE DIFFERENTIAL CALCULUS. Here the numerical coefficients will be found to observe the same law as those of the binomial theorem ; and the nth differential may be put down as follows : d-u = (p^\ dx^ + n (-J!^\da:-'di/ \dx'y Xdx^'-^dy) ^ The successive differentiations of a function of any number of variables may be determined in the same way as the pre- ceding. Let uz=f{x, y, z) be a function of three independent variables, and suppose y = a a? + ft 2* = a' ^ + /3', so that x, y and z may severally increase by constant increments ; then we find u-=f{x,y,z), ''"=(5)''' + (|)''''+(s)'''' &c. &c. &c. CHAPTER IV. EXPANSION OF FUNCTIONS. I. Functions of One Variable. (39.) Let u =f{x) denote a function oi x, and, h denoting a finite quantity, let the binomial function f(x-\-h) when expanded in terms involving the integral powers of h be supposed to be f(x + h) =zf(x) + PA -f Q^2 + R7,3 ^ g^c.. EXPANSION OF FUNCTIONS. 61 ill which P, Q, R, &c. are new functions of x to be determined from fix). It has been shown, art. (6), that the coefficient P of the second term of this development is the differential coefficient of the function /(^r), and is therefore to be obtained at once by differentiation. The other coefficients Q, R, &c. may be similarly determined by means of successive differen- tiation. Thus, by differentiating successively the above form of expansion, we get the following equations : /(^-f A) = P-f 2Q^ + 3R^2^_&c. f{x^h)= 1.2 Q +2.3R/i+&c. f"{x + h)= 1.2.3 R + &c. &c. &c. As these must be true for all values of h, by supposing the coefficients P, Q, R, &c. to be finite in value, and making ^ = 0, we obtain, f{x) = P, f{x) = 1 .2 Q, /"(^) = 1.2.3 R, &c. &c. ; Hence the expansion of/(x -f h) is, fix + h) =f(x) +f{x) \ +f"{x) ^ + f"(x) ~ + &c. = w H dx du h d'^u h- d^u h^ 1 '^ dx'^' 1.2 "^ dx^ ' 1.2.3 "^^''•' which is Taylor's theorem, and is one of considerable import- ance. In deducing it we have in the first place assumed without proof that the function is capable of being developed in the proposed form. The mere fact of obtaining an intelligible result will, however, be sufficient to establish the truth of this supposition. We have also necessarily assumed that all the coefficients 62 THE DIFFERENTIAL CALCULUS. P, Q, U, &c. should be finite, as the reasoning evidently ceases to be conclusive when any of these coefficients become infinite in value. When one of these coefficients becomes infinite in value, we shall find that all the coefficients which succeed it will also be infinite in value. Whenever this happens, which can only be in particular cases and for particular values of x, Taylor's theorem is commonly said to fail; but it may in such cases be more properly said to be inapplicable, in conse- quence of the impossibility of exhibiting the complete expan- sion of the given function in the required form for that par- ticular value of X. We shall hereafter give a more satisfactory investigation of the development in a modified form, so as to obviate any want of generality or of logical accuracy that would otherwise be experienced in the many important appli- cations of this celebrated theorem (40.) By making ^ = 0, Taylor's theorem becomes f(h) =/(0) +/'(0) - + /'(O) ^ + /'"(O) ^ + &c. Or, substituting x for h, fix) =/(0) +/(0) ? + /'(O) :^2 + /'"(O) 1^ + ^^-^ which is generally known as '' Maclaurin's theorem,'^ and is useful for the expansion of functions in powers of the variable. Professor De Morgan has observed, that Maclaurin was anticipated in the use of this theorem, and it has in consequence been latterly called '' Stirling's theorem ; " but of this it may be remarked, that it is an obvious and very easily deduced particular case of Taylor's theorem, of still earlier date ; being, in fact, merely the development of f{x) considered as a binomial function /(O -f- x), II. Theorems which Limit the Values of Functions, (41.) Let/(.r'), f{x + h) be two values of a function which varies continuously between x and x -\- h\ then if aiiy value of EXPANSION OF FUNCTIONS. 63 X between x and x + hhe substituted in the proposed function, the result will be an intermediate function. For example, the functions f{x -[:\h),f{x^ i^),/(j? + — ^ ^ ) are all intermediate functions with respect to f{x) and/(j? + h) ; but it does not necessarily follow that their values are arithmeti- cally intermediate between f{x) and f{x + h) unless the function between these limits either continually increases or continually decreases. If, however, x be supposed to vary continuously and to take every possible value from xio x -\-hy and V, V denote respectively the greatest and least values of the function between those limits, then the value of every intermediate function will obviously be comprised between V and V, (42.) When a variable x takes m progressive values o^j, x^^ x^ Xyn, let the corresponding values of a function u =f(x) be denoted by u^,u^,u^ m^^; then if the function be continuous in value from u^ to Um we shall have where 6 is some arithmetical value between zero and unity, so that the value of 6m is between 1 and W2, and ?/^;„ is a function oi X intermediate with respect to u^ and M;„. Let V, V denote the greatest and least values of the function u when x is supposed to pass continuously through every value from x^ to x,ni so that u-^y il^,u^ u^ are severally comprised between them, that is, less than V and greater than V ; also let the sum of these m functions be denoted by m (u), then V -f V + V &c. to m terms = mV (1) u^ -{- u^ + u^ +Um = m(u) (2) V -f v -f t? &c. to 7/1 terms = mv (3). On inspecting these we observe that the terms of (2) are severally less than the corresponding terms of ( 1 ) and greater than the corresponding terms of (3), and therefore the total 64 THE DIFFERENTIAL CALCULUS. ' value of (2) is less than that of (1) and greater than that of (3). That is, the value of (u) is comprised between V and V, and is therefore a value of the function between these values. Hence, as V and v are each intermediate with respect to Wi and t«^, (u) must necessarily be the value of an inter- mediate function with respect to u^ and Um, and may therefore be represented by u em, B expressing a numerical value between zero and unity. It will be observed that the basis of this proof is the evident proposition that when, with respect to certain functional limits, a value is arithmetically intermediate it must also be functionally intermediate, provided that the function is con- tinuous between the stated limits. (43.) Let fix) be a function of x, continuous and finite from to Xy and which vanishes when j? = ; then will / W = ^f\Bx), where 6 is some arithmetical value between zero and unity. Suppose X to be divided into a number (m) of parts, each equal to dxy so that mdx-= x, the number m being indefinitely great and dx indefinitely small. Then, according to the first principle of differentiation, /(O + dx) -/(O) ^ .,^Q dx f{dx + dx) -f(dx) =zf(dx) dx f{2dx + dx)^f(2dx) ^f'^^dx) fjSdx + dx) --/(Sdx __ -, .^ ^ V dx &c. &c. f{mdx)-f{(m-])dx} dx =-^ {("^ - ^^ ^^^* Hence, observing that m dx = x, the sum of these equations. EXPANSION OF FUNCTIONS. 65 according to (42), gives or, since /(O) = 0, /(^O = m dxf'{6x) = xf{ex). Cor, If a function /(jt) be continuous in value from to ^r, and also vanishes at each of these limits, so that /(O) = 0, /(x) = ; then, by the preceding theorem, ^f\ex)=f{x)=0; ,'. f{dx) = 0. That is, if/(x) vanishes at both of the values and x, the derived function or differential coefficient /'{x) will vanish at 6x, some value between and x. (44.) If/ (A) a function of A together vrith its first w derived functions be finite and continuous from to A ; and if more- over the function and the first w — 1 of these derived functions severally vanish when A = ; then where 6 is some positive arithmetical value less than unity. Let h be supposed to be constant and x variable, and assume F(x)=zh''f(x)-x''f(h), Then, since 'F{x) vanishes when x = and x = hyit follows from the corollary to (43), that the derived function F(x) = h^'f^x) -nx»-^f{h) will vanish when x=:6^h=.h^, where h^ is some value between and h. But since, by hypothesis, /'(O) = 0, this derived function Y{x) also vanishes when j? = 0. Hence again, as the function Y{x) vanishes when x ^=Q and a: = A ^ , it follows from the same corollary, that its derived function F"(^) = h^'fix) - ;i (« - 1) x«-2/(/i) 66 THE DIFFERENTIAL CALCULUS. j! will vanish when cc = hr^, some value between and h^. But ^ since, by hypothesis, f"(0) = 0, this function F"{x) also vanishes when cc = 0, Hence, as before, will vanish when a^ = h^, some value between and h^. By pursuing this process we shall evidently find that vanishes when jt = ^„, some value between and kn-i- That is, substituting for x this last value, A"/^")(A„)-1.2.3....»/(A) = 0; where hn is some value between and h, which may therefore be designated by 6h, 6 being an arithmetical value bet\^'een zero and unity. Hence we have which is a further extension of the theorem of art. (43). Since h^h^ >^2 ^^s ^^-i ^^n it follows that as the order n advances, the value of hny or of ^„, diminishes* III. Limitations to Taylor's Theorem. (45.) Let R(^) be a function of h which represents the sum of all the terms after the first in the expansion of the binomial function /(a? + h) ; that is, let fix + h) =/(x) + R (h), and suppose h alone to be variable ; then the values of R(/0 and its differential coefficient or derived function R'(/i) will be R(A) =f(x + h) -fix) W{h) =f{x + A). Therefore as the value of R(/i) vanishes when A = 0, if the EXPANSION OF FUNCTIONS. 6? function f{x) be continuous and finite from x io x -\- h, we have by the theorem of art. (43), or the more general theorem of art. (44), R(^) = k R'{6h) = h/'(x -h Oh), the vahie of R'(^^) being expressed by substituting 6h for h in the value of R'(A) ; .-. f(x + h) =f(x) + hf'(x + 6h) (1), wliich is the development made complete in two terms. Let now R(/i) be a function of h which represents the sum of all the terms after the two first in the development of the binomial function /(j? -f K) ; that is, as suggested by equation (1), let f{x + h) => 4- hf\x) + R(A), and, as before, suppose h alone to be variable ; then the values of R(^) and its derived functions will be R(y^) =/(^ + A)-/(^)-V'W R'(A) =f\x + h) -f\x) R"(/0=/"(^ + ^). Therefore as the values of R(^), R'(A) both vanish when ^ =1 0, \if{x),f(x) be continuous and finite from j? to j? -f A, we have by the theorem of art. (44) RW = ^ R"(^/0 = :^/'V + ^A) ; .-. f{x + h) =f{^) + hf'ix) + ^f{x + eh) (2), which is the development when made complete in three terms. Again, let R(/i) represent the sum of all the terms succeed- ing the three first in the development of/ (a? -f h) ; that is, as suggested by equation (2), let fix + h) =f{x) 4- hf\x) -f ^/%^) + ^W ; then the values of R(^) and its derived functions will be 68 THE DIFFERENTIAL CALCULUS. R(h) =/(^ + h) -f(.v) - hf'ix) - ^/"W R'(^) ^f{x + h) -f\x) - hf\x) R"(/i) =f\x + h) -f"{x) Hence, as the values of R(7i), E/'(A), R/"(A) severally vanish when A = 0, iif{x), f'(x), f"{x) be continuous and finite in value from .r to a? + ^, we have by the same theorem, art. (44), ^^^^ = 1:2:3 ^'"^ ^^^ " 1X3^"^^' -^ ^^^ '' r. fix + h) =f(x) + hf(x) + ~^-^f"(x) which is the development completed in four terms. In like manner, so long as the functions are continuous and finite in value, may the binomial function f(x + h) be com- pletely exhibited in any number of terms. Thus, let 'R(h) be a function of h which expresses the exact residue of the development after the Jirst n terms, so that fix + h) =f{x) + \ f\x) + ~ f"{x) + —rxx) Then the values of Pt(A) and its derived functions will be EXPANSION OF FUNCTIONS. 69 lV{h) =/[(x + h) -f\x) - j/"(a') - ^/'"W hn-2 1.2.3.... n-2^ ^^^ R"(/0 =f'\x + h) -/" W - J/'"(x) - r:2:3T::T^=3/"'-^w &c. &c. &c. R(«-i)(A) =:/('»-i)(a: + /i) -/(«-i)(.r) R(n)(^) =/(«)(^ + h). Therefore, when h vanishes, R(0) = 0, R'(0) = 0, R"(0) =0, R('»-^)(0) = ; and hence if /(x), f'{x), f"{x) f^''^\x) are severally continuous and finite in value from a? to a? + hy the function R(A) fulfils the conditions of the theorem of art. (44), which gives The development in Taylor's series, when made complete in n -f 1 terms, is therefore fix + h) =/W + \ fix) + ^/"(^) + ~f"i^) where 6 is some positive numerical quantity, the value of which is undetermined further than that it is contained between the limits of zero and unity. We are hereby enabled to affix corresponding hmits to the completion of Taylor's series after any number of terms ; but it must be remembered, art. (41), that the value of /^"^j: + BK), though functionally inter- mediate, is not necessarily contained arithmetically between f^^\x) and/('»^(a: + A). Let V and v denote the greatest and 70 THE DIFFERENTIAL CALCULUS. least values oi f^^\x) whicji occur from x to x H- 7i, then we conclude that, by stopping at the Jiih term, the final correction, to make the value of the development exact, will always be comprised between y~2 ^ ^^^ f2 n ^' This formula is Lagrange's limitation to Taylor's theorem, and it should be remembered that the conditions on which it depends are, that the n -\- I functions f{x), f(x)y f'(x), f"{x) f^^K^) inust be severally continuous and finite in value between the limits x and x -\- h. It is not aifected by any of the subsequent functions f^^'^^\x), f^'^'^^\x), &c. becoming discontinuous or infinite, and it is true when stopped at any number of terms, provided only that the functions are so far continuous and finite. Thus we may have &c. &c. &c. which equations admit of being made exact by values oi 6^, ^2> ^35 ^^'j ^^ch less than unity, so that x + 6h is in every case comprised between the limits x and x -\- h. By equating each of these values off(x + h) with the next, we deduce the following relations, /'(^ + e,h) =f(x) + lf"{x + e,h), &c. &c. &c. /(--^) (x + Bn-x h) =f(n-i) (^) +^/(.) (a; 4- 6nh) ; and from these we infer that, when h is small. EXPANSION OF FUNCTIONS. 71 ^1 = "2» ^2 — 3» ^3 = 4> • • • • ^« = ;^l' and they will seldom in any case differ much from these values. (46.) By making j- = in the formula (w), Taylor's theorem with limits becomes /(/,) =/(0) + \f{0) + ^^/"(O) + 7^3/"'(0) or, substituting x for hy /(x) =/(o) + -/'(o) + -/'(o) + Y:^^rm and this equation, which is necessarily exact for some value of B less than unity, is the corresponding limitation of the theorem of Maclaurin or Stirling. The conditions essential to this theorem are, that the functions /(.r), f'{x), f"{x) y('»)(a:) should be continuous and finite in value from to a?. This theorem may also be put under the form W,= M0 -f x/dM\ aT- /d^u\ x^ /d^u\ l\dx)o \.2\dx'-)o "^ 1.2.3 V'TVo x"" /d''u\ \.2,,.rAdx^)ex IV. Functions of Tico or more Variables. (47.) Let M = F {xyy) be a function of two variables, and let it be required to expand Y{x ■{• hy y -\- k) in powers of h and k. Take k =z ah and put U = Y(x-hh,y ■^k)=iF{x-\'hyy-h ah). rhen, by supposing k alone to vary, U may be considered as a function of one variable h, and expanded in powers of h by Stirling's theorem, art. (46). When h becomes h + dh, the function U becomes F (x + h -{- dhy y -\- ah + a dh), and this 72 THE DIFFERENTIAL CALCULUS. form is identically the same as if we had supposed x to become X + dh and y to become y + a dh. Therefore, substituting dh for dx and a dh for dy^ in the formula -=0-^(f)* we find the differential of U = F(^ + ^, y + «A), with respect to A, to be ■•■^^=(S)-(f) <■'■ As this value of — - must be a function of a? + A, y -f a/i, it may evidently be again differentiated by applying to it the same formula (1). Thus d^d\] _ / d d\j\ / d_ d\]\ dh Idi "" V^ Ih) ^ \dy 'dh ) ' that is, operating on the preceding value of — as indicated on the right hand of this equation. In the same way, treating this as another function of a? -j- A, ?/ -\- ahy and again employing the formula (1), the process may be carried to any order of differentiation ; and we shall obtain generally dh (n). EXPANSION OF FUNCTIONS. 73 in which the numerical coefficients are those of the expansion of (1 -f xy. Now, by StirUng's theorem with Hmits, art. (46), we have A'» 1.2.. in which expansion the function U and its differential co- efficients are the values when A = 0, excepting the last, in which h takes the value 6h, But when A = 0, functions of jc -^ h, y -{- ah become corresponding functions of x, y, and U, and its differential coefficients with respect to x and y become the same as if the function u had been employed ; also when h becomes 6h, functions oi x -{■ h, y -\- ah become corresponding functions oi x •\- Oh^ y -\- 6 ah. Hence substi- tuting the values according to the preceding expressions (1), (2), {n)y and observing these transformations, we have for U the following development : « + i{(S)-(i)} -S{(S) -(a) -'($)} "*■ 1.2....W \\dx^') "^ ^''\dx''-^dy) , n(n-\) , / d^u \ "^ 2 ""Xdx^-^dy^) /d''u\ 1 the value of the term exhibited in the last three lines being taken when x and y become x ■{- Oh, y -\- Oak, where ^ < 1. D 74 THE DIFFERENTIAL CALCULUS. By substituting k in place of its value ah, the formula becomes U = F (^ + y^, y + ^) = 1.2 «\ Xdlx^'J \dx^-^dy) . ^iji^l) / d-u \ + 2 ^ \dx-Hyy +^""(3-^)1^+^^ (48.) In the formula just determined make ^ = 0, 2/ = 0, and afterwards change h into a: and A; into y ; then « = F(^,3^) = .o + ^(g)^ + y(|)^ + 1.2.... w 1 '^'^ V5^7 "*" '^ ^'^ ^ V5i^^=^/ -'•(?-:)}« where we have to make x, y each = in the several functions, except in the term which occupies the last two lines, where they are to be replaced by 6x, Oy, 6 being < 1. Note. — It may here be remarked with respect to expansions EXPANSION OF FUNCTIONS. / ,) generally, that if the wth or limiting term decreases without limit as n increases without limit, the development may be then continued without introducing any limiting term. (49.) If in Taylor's theorem we make h = dx, it becomes , X ./ X df{x) dH(x) d^f(x) , fix + dx) =f(x) +-Zi^ + -^ 4- ^J^ 4- &c.; that is, if w =/(j?), du d'^u d^u U =f{x + dx)=u^-^—^ ^-^ + &c. This formula represents in a simple form the most general theory of expansion, and may be extended to the expansion of a function of any number of variables, under the following general enunciation : * Let u ^f{x, y, s, &c.) be a function of any number of variables, and let hx, by, bz, kc. denote arbitrary increments of the respective variables. Suppose the function U =f{x ■\-bx,y-\- by, z + bz, &c.) to be partly expanded, and denote by bu the terms which involve the first order of the increments bx, by, bz, &c. Then x -\- bx, y -\- by, z -\- bz, &:c. being substituted for X, y, z, &c. in the value of bu and the result again partly expanded, denote by 5-w the terms which involve the second order of the increments. And again, the same substitutions being made in b'^u, and the result expanded, denote by b^u the terms which involve the third order of the increments, &c., &c. Then will and the values of bu, b'^u, b^u, &c. may be determined by successively diiferentiating the function u-=f{x, y, z, &c.) on * This theorem was first announced by the author in the Appendix to the * Gentleman's Diary ' for the year 1835. 76 THE DIFFERENTIAL CALCULUS. the supposition that dx, di/, dz, &c. do not change, only writing 5^, Si/, bz, &c. in place of dx, dy, dz, &c. ; also the series may be stopped at pleasure by substituting x + 6bx, y + Bby, z + 6dz, &c. for x, y, z, &c. in the last term, 6 being < 1 . By making x, y, z, &c. severally = 0, and writing x, y, z, &c. in place of bx, by, bz, &c., the result will be the expansion of the function u ■=. f{xy y, ^, &;c.) in powers of the variables. The preceding developments may all be deduced from this general theorem. Exam^^lea, 1. Expand/(a: + A) = (a? -f A)** by Taylor's theorem. Since /(.r) = x'^^ we have by successive differentiation f\x) = n x''-\ f"{x) =n(n — l) x""-^, f"(x) = w (» - 1) (w - 2) a?»-3, &c. Hence, by the theorem, art. (39), (x + hy ;^ ;c^ + 7 x^-^h + ^^^1^^ x^-^ h^ i 1.2 . w(w — l)(w— 2) + -^ j-2 3 -^""-^h^ + &c., which is the formula of the binomial theorem. 2. Expand log {x + h). Here /(a?) = log x, and by differentiation f{x) = x-\ f"{x) = - \,x-\ f"'{x) = 1.2.a:-3, &c. Therefore, by the theorem, h h^ h^ A^+h)=logix + h) = \os(x) + -__ + A._&c. which is divergent and inapplicable when ,r < A. If we employ the theorem with the limitations, art. (45), we shall obtain log (x 4- h) = log (x) + JZfsl EXPANSION OF FUNCTIONS. 77 which expressions will be strictly accurate with values of 6 between the Hmits of zero and unity. Let a? = I, then By the first of these expressions it follows that the value of log(l 4- h) is comprised between - and ; and by the second the same value is comprised between the narrower limits A — TT and k 2 2(l-hA)2 3. Expand the function w = sin a? in powers of x by Maclaurin's theorem. By differentiation, du d^u . d^u -- = cos X, -riy = — sm x, -r-r, = — cos Xy dx dx^ dx'^ = sin X, -Yi = cos Xy &c. d^u . d^'u --—. = sm X, -r-r dx"^ ' dx^ which, when .r = 0, respectively become 1,0, — 1, 0, 1, &c. Therefore by the theorem, art. (40), x'^ x^ „ Or, by the theorem with limitations, art. (46), sin j: = J? cos 6x = x ^ •— sin d^x ; where ^^ < ^ < 1, and which may be similarly expressed in any required number of terms. 4. Expand u = cos Xy in powers of x, _ du . d^u Here -r- = — sm x, -—^ = — cos Xy dx dx^ d^u . rf-^M -T-^ = sm Xy — -J = cos J*, &c., dx"^ dx^ 7> ■T>* qnO 78 THE DIFFERENTIAL CALCULUS. which, when x = 0, become 0, — 1, 0, 1, &c. ; ... cosa: = l---fj;^;3-^~&c. Or, with the hmitations, cos 07 = 1 — a:sin6x = 1 ■— ■— cos S^x = &c. 5. Expand w = e* = log~^a? in powers of x, du d u By art. (26) we have -— = e', -—t, = e*, &c., which, when •^ dx dx" ^ = 0, severally become equal to unity. .-. ^^ = l + Y^ Also, with the limitations, e* = 1 + ^ e^^ = 1 + 7 + —^ e^/* = &c. 6. Let u=z xy 2, and expand U = (^ + S.r) (2^ + hj) (z + 52r) by the general theorem of art. (49). By operating upon u=^ xy z with the symbol S in a manner analogous to successive differentiation, and supposing hx, by, ^z to be invariable, we have u = xy z bu = y z8x -{■ zxby + xydz b^u = 2 X dy bz -j- 2 y dz dx -{- 2 z dx Sy d^u =. Gdxbydz, which substituted in the formula ^ 1 ^ 1.2 ^ 1.2.3 • we obtain {x + Bx) (y -{■ dy)(z -\-dz)=xy z -\- (yzdx + zxdy -\- xy dz) -j-(xdydz + ydzdx + z dx d7j) + dx by bz, which may be verified by multiplication. EXPANSION OF FUNCTIONS. 79 (oO.) In the series for e', example 5, replace a? by a? V— 1 ; then X- p4 = *-t:2 + l2X4-^^- + (^-1^3 + ^'=-)^^'' that is, examples 3 and 4, ^« vIT __ ^Qg^ _|_ ;^_ J gjj^^ . . . (1). In -this equation replace xhj — x, and we have also e-'^^ = cos 07 — V -- 1 sin a: (2) ; rV_i I -_^\/_i C03JC = sm J? = 2 (3), 2V-I which are Euler's formulae. Again, replacing a: by mo? in (1) and (2), g±m*vZl _ cos mo? + V — 1 sin mo*. Hence, as ^±^^^^-1 = (e±*^- i)^, we have cos mx + V — 1 sin mo? = (coso? + V — 1 sino?)*" (4), which is De Moivre's formula and is true for all integral values of m. When expanded by the binomial theorem, by equating separately the real and the unreal portions, we may obtain from it the trigonometrical values of cos wo: and sin/^/.i in powers of cos x» sin x. In (4) replace 0: by 0: + 2 ttt, r denoting any integral number; then (cos 0? Hr V — 1 sin x)^ = cos (mx + 2 rmir) Hh V — 1- sin {mx + 2 rmii) .... (5), 80 THE DIFFERENTIAL CALCULUS. which is the complete form of equation (4) and is now true for all values of m^ whether integral, fractional, real or unreal; and hoth sides will now always contain the same number of identical values.* From the preceding values of cos a?, sin x^ equations (3), it is evident that all the trigonometrical functions of x may be expressed in algebraical functions of the exponentials e*^-^ aDde-'^^"^ , CHAPTER V. INDETERMINATE FORMS. (51.) When a function for a particular value of the variable assumes any one of the forms ^, ^, X 00, 00 - 00 ; 00, ooO or 1±», 00 the function, absolutely considered under this singular form, becomes then essentially indeterminate and admits of having any value whatever assigned to it. But if the proposed function represent a quantity which varies continuously so that the function up to the particular value of the variable is subject to a condition of continuity, its value will evidently be determinable in a manner analogous to that by which we obtained the differential coefficient of a function in art. (6). I. Functions in the Form of Fractions, fix) . . (52.) Let u = —T—^ be a function of x which becomes - ^ ^ Y (x) when .r = «. It is evident that this will arise from the in- corporation of certain vanishing factors in both numerator and * An investigation of the general theory of exponential and imaginary quantities arising out of this last equation is given by the author in the Appendix to the ' Gentleman's Diary' for 1837. INDETERMINATE FORMS. 81 denominator. Suppose the resolution of these factors to give Y{x) "" (^-a)'*Q' where P and Q are of finite value when x = a. Then by division we should have and when j: = a, this would obviously give for the required value, P if iw > w ; — if m = w, or CO if m < w. The ehmination of the vanishing factors will in most cases be facilitated by substituting a + ^ for x, so that x -— a-= h. The form of w will then be a function of h which becomes - when A = 0. By expanding, if necessary, the numerator and denominator of this function in ascending powers of A, and dividing by the power of A which is common to them both, and afterwards making A = 0, the result will be the required continuous value of the proposed vanishing fraction when J = a. (53.) The continuous value of the vanishing fraction may be otherwise determined by ascertaining in a different manner an expression of its value in a continuous form for values of x contiguous to J? = a. Thus when x takes the value a -f A, we have by Taylor's theorem, art. (45), observing that /(a) = 0, F(fl) = 0, f{a + A) /(a) -h \ f\a -h Qh) f {a + Sli) F(a-fA) F(a)-f-yF'(a + ^A) Y\a-^6h) This equation is necessarily strictly true when A is of any value, however small, positive or negative, and if /'(«)> D 5 82 THE DIFFERENTIAL CALCULUS. F'(«) do not both vanish or become infinite, the fraction on the right hand will be continuous in form when h vanishes ; therefore, making ^ = 0, we obtain, for the continuous value, r(«) Y\a) ^ ^* But if /'(a), F'(a) both vanish, by extending Taylor's series to another term, we shall have /(g + h) _ f{a) + \f'{a) + f^ f"{a + 6h) F (a + ^) F(«) + ^ F(«) + ^ F'(a + 6h) _ fja + eh) F"(a + 6h)' Hence, if /"(a), F"(a) do not both vanish or become infinite, we obtain, by making A = 0, r(a) F"(a) ^ ^' By proceeding in this way, we similarly find that if the numerator and denominator with their first n — l differential coefficients, Viz,f{x),f{x),f\x) /(^-i) (^), andF(j:), F'(a?), F"(a?) p^^-^) {x) severally vanish when.r = a, and the ^th differential coefficients/^'^) (a?), F^**) {x) do not both vanish or become infinite, then the continuous value of the fraction will be F(a) F(^)(a) ^ ^' (54.) Suppose the numerator and denominator of the func- that it becomes of the form — . Then by expressing the function by the reciprocals, thus. fix) tion '—-{ to be both of them infinite in value when ^ = a, so F(«) . 1 INDETERMINATE FORMS. 83 it will become of the form - . Therefore by equation ( I ) we get, by differentiating the numerator and denominator, - J>L /(a) _ {£WJ_'^ f/(a)-l^F'(a) F(a)~_ /'(g) IF («)//'(«)' { /(«) }' which gives /(a)^/'(«) F(a) F'(a) ' This being the same as the equation ( 1 ) before obtained, we conclude that the mode of operating in this case is identical with that already indicated when the function is of the form-. Thus, if after n— 1 differentiations the fractions ^tt- > f^y f^y .^S;Sg severally become of the form — or -, and if \^, s , [ does not become of either of those forms ; then, according to equation (n), f{a) _ r-\a) F(«) F ('»)(«)* (55.) We have therefore the following rule for determining the continuous value of a fraction which for a particular value of the variable becomes of the form- or — : — Divide the dif- 00 ferential coefficient of the numerator by the differential coeffi- cient of the denominator for a new fraction, in which substitute the given value of the variable. Should this latter fraction 00 still assume the form - or — , the same process may be sue- 84 THE DIFFERENTIAL CALCULUS. cessively repeated until one or both of the numerator and denominator ceases to vanish or become infinite in value. Example 1 . — ^When a? = 0, find the continuous value of 1 — cos 07 __ sin 2 ,r Here /a? = 1 — cos j?, F(.r) = sin^^r ; and by differentiation, f'{x) sin ,r 1 F '{x) 2 sin X cos x 2 coso? which, when a? = 0, gives \ for the required value. Example 2. — When ir = 0, required the value of , . ^ ^ logsm2x — 00 Since /( J?) = log sin a?, Y{x) = log sin 2xy we have _, co^x _, . ^ 2 cos 2x fx = -: — , F(^) = . ^ ; sm ^ sm 2 J? . y'(^) cos^ sin2.r F'(.r) 2 cos 2 .r * sin a? * When J? = 0, the first factor of this expression is determi- cos J7 , , ,, T p sin 2a? ^.„ ate and is ;:; -^ = i ; but the other factor — : still 2 cos 2x sm* maintains the indeterminate form - , and its numerator and denominator must therefore be again differentiated, giving 2 cos 2i27 = 2. The value of the proposed expression is COSO? r r r therefore ^ x 2 = 1 . Example 3. — When a? = ao , determine the continuous value g* 00 of -^ = — , the exponent m being a finite integer. f(x) e^ 00 Here we have i^rri = — :7 = — > when j? = oo , F(J7) 0?^ oo' f'{x) e' 00 ^ •1,, ^ = r = — , when ^ = X , F'(a?) m^'^-i cx)' &c. &c. &c. INDETERMINATE FORMS. 85 i. \-l = , = GO , when j? = co . Y^'^^x) 1.2.3 m The sought value is therefore infinite. . «ri , - 1 — J?*" m 4. When x = 1, then :; = ,t- = — gj? e<* 5. When a? = a, then = - = e'*. J7— a 6. When j? = 0, then = - = log 7. ^•r — e~ 7. When j? = 0, then- 8. When j? = 0, then sin J7 J7 — sin cT __0 _ "" " = 2. _ ""0" 1 2.3 „^ ^ , tan J? — sin a:- 9. When a: = 0, then : = - = 3. X — %\VlX jpj: X 10. When j? = 1, then -—^ = - = - 2. 1 -h log a: — 0? , , -nri /^ .1 lo?; cot J7 00 11. Whend?= 0, then -^ = =-1 12. When a? =0, then log X — 00 cos aJ7 — COS /3j7 __ ___ a^ — |9^ cos aj^ — cos ^wC a^ — 6^ II. Functions in the Form of Products, (56.) Again, \iY{x)f{x) be a function of j7 which, when j: = a, becomes X x , it may be diiferently expressed, as follows : Since, when j; = a, F(x) = 0, /(x) = x , the former of these will assume the form -^ and the latter will assume 86 THE DIFFERENTIAL CALCULUS. the form — , and either of them may be evaluated by art. (55). Also, if F(a?)— /(a?) be a function of w which, when x= a, becomes of the form oo — oo , it may be expressed thus : _1 1__ FW/W which, when a? = «, will now become -, and may therefore be I evaluated as before. " Example 1 . —When x =z -, required the value of /'l>_^'\tan^ = X 00. In this example we have / 1 j tan 07 = __ ?^ cot 07 77 . . When 0? = — , this expression assumes the form - , and its value is hence found to be 2^ __2 TT 77 2 cot 07 — COSeC'^07 77 X 1 Example 2. — When o? = 1, find the value of , , log 07 log X = 00 — 00 . _ X 1 07—1 Here , , = -. , log 07 log 07 log 07 which, when o? =1, takes the form -, and its value is there- fore found to be ^- 1 _1 _ ^_i log 07 1 INDETERMINATE FORMS. 87 3. When.r = 1, then 1 = 00 —00 = ^. 4. When j? = oo , then e-'log x = x cao = 0. 5. When j: = 0, then xlo^a: = x — oo = 0. X 1 6. When x = \, then 7—1 log J? 00 — 00 = ■ 7. When J? = 0, then 1 8. When J7 = 0, then — x" ortanj? 1 — =00 -oo = .i. = X —00 = ■ III. Functions in the Form of Exponentials, {o7.) The general exponential function u = ¥(x)-^^^ may for a particular value of x become one or other of the forms QO, 00 «, 1±», 0±«, 00 ±». Only the first three of these are indeterminate in their character : the other two are determinate, and their values are evidently 0±"= / ^ 00^ \qo Since u = Y{xy^^\ we have logw=/(a?)logF(a?) = logFW 1 Therefore, referring to this expression for log w, 00 when u is of the form log u is of the form Hence the value of log u may be determined by art. (55), and thence the corresponding value of w. 88 THE DIFFERENTIAL CALCULUS. Example 1. — "When a? =. 0, find the value oi x' = 0^. Here u = .r*, and logw = x\o^x = 2§Ji, When a? = 0, this expression for log u takes the form , 00 and hence, by differentiation, its value is found to be 1 loff X X l0gW=-^=: —-=—^7 = 0; .-. W=l. 2. When ^ = 0, then a?«i°'^= 0^ = 1. 3. When ^ = 0, then (cot^)«i°'^= ooO = 1. 1 4. When a? = oo , then j^iog^-^ = oo^ = e. 5. When a? = 0, then (1 + mj?)-^ = 1 «= = e'". _L 1 6. Whena?= 1, then,r^-'*'= I"= ~* IV. Exceptions to Taylor's Theorem* (58.) In art. (39) allusion has been made to the existence of certain functions, to the development of which Taylor's theorem ceases to be applicable for particular values of the variable, in consequence of the differential coefficients or derived functions becoming infinite in value. Let ^^t{x)hQ a function of a?, and suppose a given finite value o to be a root of either of the equations ^(^) = «' W) = '-' then it may be shown that ^ (x) will be of the form ylr(x) = (x-arct>ix) (1), the function /(a? + h), and if /(x) be a minimum we must hsLYe/(x)' ^/vhenf(x -f- h)—f(x)< continues to be positive, neither J (^ changes its sign. But, art. (45), f(x + h)-^f(x) = hf(x + eh). If the first derived function/' ( when — =f(x) passes from < . , a mmimum j dx 1 -- to -f- . In the case f{x) — 0, by extending Taylor's series to another term, we have fix + K)-f{x) = ^f"(x + eh). Here again, if /"(a?) be supposed not to vanish, the value of A may be taken so small that f"(x + 6h) shall not change sign when the sign of h is- changed. As h^ is necessarily positive the value of /{x -f h) —f{x) will have the same fixed alge- braic sign as f'{x -f 6h) orf"{x); and therefore the function will be a maximum a minimum )when^=/'Wis l^^g-tive, J dx"^ I positive. Again, suppose that a value of x which mokes f(x) = also causes several of the subsequent derived functions /"(jt), /"'ix)y &c. to vanish, and let/'*)(cr) be the first that does not vanish. Then, art. (45), f(x + h)-^f(x) = ^-^^/(n)(^ + ^A). Asf("^\x) does not vanish, it is evident, as before, that a value may be assigned to h so small that/('»)(j: + dh) shall not change its sign when that of k changes. The effect upon the sign of A** will however depend upon whether the number n be odd or even. Thus we find, If n be an odd number, f(x) is neither a maximum nor a minimum, unless y*^"^a: passes through - . 100 THE DIFFERENTIAL CALCULUS. If n be an even number, /(x) is a ( ">a^i'""'° I if ^" =/(n)., is / negative, -^ ^ '' I minimum J ^^» -^ \ positive. (65.) The nature of the preceding relations, which constitute the theory of maxima and minima of functions of one variable, may perhaps be made more familiar by the following simple considerations : As the derived function — =zf{x) represents the limiting dx ratio of the increment of the function to that of the variable, and as a decrement is indicated by a negative increment, let the variable x be supposed to increase continuously ; then the value of the function /(a?) will increase vf\iQuf{x) is positive and decrease when/'(a7) is negative. But iif{x) increases up to a certain value of j? and afterwards decreases, it will oddently pass through a maximum value, and if it decreases and afterwards increases, it will pass through a minimum value. The function will therefore pass through a maximum or a minimum value whenever the value of the first derived function — =.f{x) passes from -f to — or from dx — to + respectively. After determining the values of j- which make/'(j?)= and - — = 0, this last simple criterion, which is that first ob- tained in art. (64), will generally be sufficient to distinguish the maxima and minima values, if any exist ; and then it will be unnecessary to proceed to any derived functions beyond The process is also sometimes facilitated when the function admits of being reduced or simplified by first multiplying or dividing it by some constant, raising it to some power, taking the logarithm, or performing some other operation according to the particular form of the function under consideration, the only restriction being that this preparation of the function MAXIMA AND MINIMA. 101 should not disturb the general relations as to corresponding maxima and minima. (66.) The different cases specified in art. (64) may also be characterized geoihetrically by making the variable a: the abscissa, and the function /(r) the ordinate of a curve line, of which the equation is y =f(x). Fig. 1. 1. If for a value of x which makes f(^)=zO, the value oif'{x) is negative, or if the first of the successive derived functions that does not vanish be of an even order and its value negative, the " d -^ corresponding value of the functional ordinate will be a maxi- mum as represented in fig. 1 . 2. If for a value of .r which makes /'(a?) = 0, the value of f'{x) is positive, or if the first of the Fig. 2. successive derived functions that does not vanish be of an even order and its value positive, the corresponding value of the functional ordinate will be a minimum ... ^ D j: as represented m fig. 2. 3. If for a value of x which makes/'(a:)= 0, also/"(j:) = 0, and the value of/"'(x) is positive, or if the first of the successive derived functions that does not Fig. 3. vanish be of an odd order and its value positive, or if the first of the derived functions that does not vanish be of an even order and its value passes through — from — Qo to -f 00, the corresponding value of the functional ordinate will be neither a maximum nor a minimum, and will y be of the kind represented in fig. 3. 4. If for a value of x which makes /'(a:) = 0, B\so/"ix)= 0, and the value off"(x) is negative, or if the first of the successive derived functions that does not vanish be of an odd order and its value negative, or if Fig. 4. ^ ^ lOL' THE DIFFERE-NTIAL CALCULUS. the first of the derived functions that does not vanish be of an even order and its value passes through - from -f oo to — go , the corresponding value of the functional ordinate will be neither a maximum nor a minimum, and will be of the kind represented in fig. 4. 5 . If for a value of x which makes — — = 0, the value of f{x), as X increases, passes from + oo to — 00 , or if for a value of x the first of the successive derived functions f{oc)y f\x), &c. that does not vanish is of an odd order and its value passes from + go to — oo , the corresponding value of the functional ordi- nate will be a maximum as represented in fig. 5 or fig. 1 . 1 Fig. 5. A 6. If for a value of x which makes /'(*) =•- 0, the value of f(x), as X increases, passes from — oo to -h 00 , or if for a value of x the first of the , derived functions /'(.r),/" (a?), &c. that does not vanish is of an odd order and its value passes from — oo to + oo , the correspond- ing value of the functional ordinate will be a minimum as represented in fig. 6 or fig. 2. Fig. 6. Example 1. — Divide a number a into two parts, such that their product shall be the greatest possible. Let X be one of the parts, and a ^ x the other ; then f{x) = X (a—x) = ax — x^is required to be made a maximum ; .*. /(x) = a — 2x put = 0, gives x = \a. When x is less than \a the value off'(x) is +> and when x exceeds ^a the value off'{x) is — ; hence, when x passes through its value, f'(x) passes through + —, which indicates that the value of the function first increases and then decreases, and therefore passes through a maximum, the number being then equally divided. MAXIMA AND MINIMA. 103 Example 2,— U u =f(x) = 2x^-'9ax"--\- \2 a'-x -- 4 a'^ ; then ^=f(x) = 6x'—\Sax-\'l2a^- = 6{x-a)(x-2a)=0 gives r = a and x = 2a. When x passes through the first of these values, f'{x) passes through + — , which indicates a maximum, and when x passes through the second value, f^(x) passes through — -f, which indicates a minimum. Therefore, when x = a, f(x) = a^ a maximum, and when X = 2a,/{x) = a minimum. Ex, 3.— If « = i + (x — a) 3 ; du i- then r =/'W = i (<^""^)' = gives a? = a, and as x passes through this value, /'(a:) passes through — +, which indi- cates a minimum of the kind represented in fig. 2. Ex. 4.— If M = b + (^ — a)^; then—- = f\x) =z\{x—aY =0 gives x = a, Ks x passes through this value, f{x) passes through + + and does not change sign. The value of the function therefore first increases, then just ceases to increase, and again increases. It is hence neither a maximum nor a minimum, but of the character shown in fig. 3. Ex.b.—liu = b -f(a?-a)^; then — = f{x) = \ (x — a)"^, which = ao when j? = a, and ax as X passes through this value, /'(jf) passes through — oo -f , which indicates a minimum of the kind represented in fig. 6. Ex. 6. — Required the height (x) at which a light should be placed above a table so that a small portion of the surface of the table at a given horizontal distance (a) shall receive the greatest illumination from it. If (j) denote the angle under which the rays of Hght meet the given surface, the degree of illumination will vary as the sine of this angle directly and the square of the distance (r) inversely. 104 THE DIFFERENTIAL CALCULUS. Butr^ = a^ + x^Sindsmd>=z-=z _^ ; .-. ?__ 3 must be a maximum ; or, taking the logarithm, the value of logo*— flog (a^-\-a;^) must be a maximum. Denoting this last function by u, we have which = 0, when ar = a Vi» and as ^ passes through + 0—, the value of the function is then a maximum as required. 7. If w = -o^^-TT * ^^^^ when j: = a, « = | a maximum, and when j?= — a, w=~|^a minimum. 8. Of all rectangles of a given area, a square exhibits the least perimeter. 9. If w = 0?^ — 3 ax^ -\- 4a^ ; then .r = gives u = 4a^ a maximum, and x = 2a gives m = a minimum. 10. If M = —^ ; then when x = e, u=z- a, maximum. X e JL ^ i. 1 1 . If tt = x^"^; then x — e"^ makes u = e^^ a maximum. 12. If w = (a + ^)(6 + ^)' then X = \/ab makes u = / / , /^xg a maximum. 13. If w = cos^o^sin x; then cos^o? = f, sin^a* =^ give « = i tt; V 3 a maximum and a minimum. II. Functions of Two Variables. (67.) Let u z=:f{xy y) be a function of two variables x andy. "When the value of w is a maximum we must have f{xy y) MAXIMA AND MINIMA. 105 >/(*^ 4- ^» y + ^*)j ^^^ when it is a minimum we must have JX^i y) Kf{x -\- h^y -\- k), and in either case this relation must remain unchanged whatever may be the algebraic signs of A and X: = a A. Therefore, for all combinations of values and algebraic signs that can be given to the small quantities h and k •= ahy if for brevity we put f{x -^h,y + a h) -fix, y) = Sw, the value of the function u will be a maximum 1 f continues to be negative, a minimum > when bu < continues to be positive, neither J [ changes its sign. But, art. (47), we have '"=H(£)-(S)};: Bh eah. When the value of this expression continues to be of the same algebraic sign, the value of the factor contained between the brackets, which corresponds to x -{- 6h, y -\- 6 ah, must change sign with h, and this change of sign must occur when A = 0, or when x + 6h, y -\- adh become x, y. Therefore, as the value of a is arbitrary, we must then have 0=»' (£)=«• dyj unless one or both of these partial differential coefficients should pass through the value - with corresponding algebraic signs. These two equations or conditions will determine the particular values of the variables. To ascertain further regarding the algebraic sign of the value of 6m when | — j = and \ r) = 0, let the expansion of f{x -j- hf y + a A), art. (47), be extended to another term; then, as the term of the first order in h now vanishes, we obtain £ 5 106 THE DIFFERENTIAL CALCULUS. If the second differential coefficients do not severally vanish and their relative magnitudes be such that the value of shall not vanish but continue of the same sign for all values of a, it is evident that k may be taken so small that the value of bu will always have a corresponding sign, which will not change wdth that of h. For brevity let this expression be denoted by (A)-f-2(c)a4-(B)a2; then when a = its value will be A, and, when the arbitrary quantity a, which is unrestricted in value, is made indefinitely great, its algebraic sign will be determined by that of B. The differential coefficients represented by A and B must therefore have like signs, and for all other values of a the expression must retain the same sign. By putting the expression under the equivalent form, ■{(-r)'^'-^"-} it becomes evident that it will necessarily have the same sign with the coefficient A when the value of A B—c^ is positive, or AB > c2; that is. This is Lagrange's Condition of maxima and minima, and when it is satisfied the value of the function u will be a maximum f -p/ \\ (df^vX . J negative, a minimum \ ^ ^ "~ \J^^) \ positive. If (A) and (B) or | — ^ j and ( y-^ ) l^ave different signs, or if Lagrange's Condition be otherwise unsatisfied, the function is neither a maximum nor a minimum. Also if the values of X and y which make (~j = 0, /— j — O should happen to MAXIMA AND MINIMA. 107 cause the second dilFerential coefficients I -t-^ ), I . , j, I y-o ) to vanish, it may be shown, as in art. (64), that a maximum or minimum value of the function will require that the first set of differential coefficients that do not vanish be of an even order. III. Functions of Three Variables, (6f^.) Let u =f{/(j? + A, y -h A-, 2 + /), and when it is a minimum /(a?, y, z) <,f{x + h, y -f /:, j + /), where the symbols A, k-= ah and l^= fih denote small changes in the values of the variables. As in the last article, the values of x, y, z which maintain either of these relations must be found amongst the systems determined by the equations (£)=->• e7)=«' ($)-"■ excepting, as before, the occurrence of infinite values. If the second differential coefficients do not vanish, h may be taken so small that the value of bu =f{x + hyy ■\- ahy z -\' /3A)— /(j7, y, z) shall have the same sign as the expression (d'^u\ /d'^u\ , /d'^u\ , ^ / d'-u\ ^ ^f d'u\^ \dxdy) and not change its sign when that of h changes. For a maximum or a minimum therefore it will be essential that the value of this expression be either always negative or always positive, whatever values be given to the arbitrary quantities a and /3, which are wholly unrestricted. To facilitate the determination of the requisite conditions amongst the coeffi- cients, let the expression be more briefly denoted by e = (A) + (B) «2-f (C) ^24. 2(a)«3 -f 2(6)/3 + 2(c)a ; 108 THE DIFFERENTIAL CALCULUS. and by putting it under the equivalent form it is obvious that it will always have the same sign with the coefficient A, provided that the value of (AB — c^) a^ + 2{Aa—bc) a/3 + (AC— 6^)/32 be always positive, and this will be thecase when AB-c^ and (AB-c2)(AC-52)-(A«-5c)2 are both positive, or AB >c2 and (AB — c^) (AC — P) > (Aa — bc)^. There are therefore two conditions of maxima and minima, viz. \dx'0\dyy^\dxdy) dx^)\dyy \dxdy) J \\dx^)\dzy \dx dz) i / /d'-u\/ d^u \ / d'-u Y d^u xy I \dx^)\dy dz) \dx dz/\dx dy J j * When both of these conditions are fulfilled, the function u will, as before, be a maximum a minimum {"w=(S)"{;s;;"r (69.) The conditions may be otherwise obtained in a symmetrical form, and the extreme value of e determined as a maximum or minimum value of a function of two variables a, 0. Thus we have /J^ = 2(Ba + «/3 + c)=0....(l) f^ = 2(C^ + aa + b) = 0..,. (2) (£)=-. (©=-■ 0-S)-- * The first of these conditions is as essential as the second, although it is commonly neglected by writers on this subject. MAXIMA AND MINIMA. 109 Hence (G7) if BC > a^ the value of f will be a maximum j .^ ^ g ^^^ (. ^^^ f negative, a mmimum J L positive ; so that if this value have the same sign as A, B, and C, all the values off will have the same sign. From equations (1) and (2) the values of a and /3 which determine this value of c are __ ab — Cc Q _ ac — Bb "'" BC-a2' ^"^ BC-«2 • For simplification, previous to the substitution of these values, multiply equation (1) by a, equation (2) by ft and add the results, and Ba^ + C^- -+- 2aa^ + 6^ 4- ca = 0. These terms being therefore omitted in the expression for e, it becomes f = A + 5/3 + ca, in which, now substituting the particular values of a, (3, we get ' BC-a^\ BC " CA "" AB "^ ABC/ ^^* When this extreme value of e is of the same sign as A, B, and C, we have therefore the symmetrical condition _a2 _ J^ ^c^ 2abc ^ ~ BC CA AB ABC > ^ • • • (4)- Also, putting cos=^ = ^. cosV=^. cosV'= |1 . . (5), the value of e becomes f = 4^(l-cos2, 0, which divided by the positive factor A^BC gives (4). Also when the values fulfil the condition (4) and any one of the three conditions AB > c^, BC > «^ AC > ^2, the other two will necessarily follow. In conclusion, it may be as well to observe that the conditions and criteria of maxima and minima here investigated, though occasionally indispensable, are not often required, as the general PROPERTIES OF PLANE CURVES. Ill circumstances are in most cases sufficiently indicated in the nature of the problem, and it is then only requisite to solve the equations ( y ) = ^^m 7" ) " ^M 7; ) = ^> ^or the determi- nation of the variables. ^- B CHAPTER VII. PROPERTIES OF PLANE CURVES. I. Quadrature and Rectification, (70.) The theory of plane curve lines forms a leading subject in Analytical Geometry of Two Dimensions, and the investi- gation of the various properties is generally found to be con- venient and symmetrical when the positions are referred to rectangular coordinate axes. In the annexed diagram let Ox, Oy represent the positive directions of the axes; then, OD = ^, DP = y being the two coordinates of the point P, the curve which is the locus of P is determined by an equation y = {x), orMy)=0. ^^ ^ ^' ' Suppose X and y to receive the increments ^x and Ay, and let the new coordinates OD' =. x -{- £^x, D'Q = y -\- Ay de- termine a second point Q, so that DD' = PG = Aa? and GQ = Ay. Then if A denote the function which expresses the value of the area contained between the ordinate, the curve, and the axis of x, the curvilinear area between the two ordinates DP, D'Q will geometrically represent the value of aA, and it is evident from the diagram that this value of aA will be comprised between the two rectangles yA*r and (y + Ay) Ax, being greater than one and less than the other ; aA . .*. — is comprised between y and y -f Ay. Hence, proceed- Aj: 112 THE DIFFERENTIAL CALCULUS. ing to the continuous values at the limit when Aj? = 0, we obtain — = y, or (/A = ydx. As this relation must correspond with the differentiation of A as a function of x, it is evident that the determination of A from it will be the inverse process to that of differentiation. This inverse process is called Integration, and is usually indicated by prefixing the symbol yj thus A ^fydx. The method of obtaining the value of this integral is the province of the Integral Calculus ; and, when taken between given limits, it will express the area contained between the corresponding ordinates. (71.) Again, let it be required to express, by means of infinitesimals, the area contained between the curve, two given ordinates yoj yrm and the axis of x. Suppose a number m — \ oi equidistant ordinates i/,, y^, 2/3 • • • • 2/m-i to be inserted between them, and let dx be the common difference of the abscisses x^, x^, x^ Xm. For brevity let {y^ yi) denote the portion of area contained between y^, y^, the axis of x and the curve, and the same for the other ordinates. Then it is evident that (y 0I/1) ^^^^ ^^ comprised between y^dx and yidx iyiVi) ,> » y* Vidx „ y^dx {y^Vz) y> yy » ^a^*^ yy y^^^ &c. &c. &c. (ym-\ym) » j> M ym-ldx „ ymdx. Hence, if 2y dx = y^dx + y^dx -f y^dx + ym-i dx, the sum of these relations proves that the total area (y^ ym) will be comprised between ^ydx and 7,ydx -f (ym — yo)^*^* PROPERTIES or PLANE CURVES. 113 If we now suppose the number m — 1 of intermediate ordinates to be increased without limit, dx and (y^* — yo) ^* will decrease without limit, and therefore "lydx will approxi- mate to the proposed curvilinear area as its utmost limit; that is, A = 2ydx. But we have seen that this curvilinear area is expressed by the integral yyc?a7. Therefore fydx = 2ydx, Hence it appears that every integral /y c?x expresses that value to which ^ydx approximates as its ultimate limit, on increasing indefinitely the number of subdivisions dx, both being estimated between the same limiting values of x. This character of an integral presents to the mind a clear view as to the result of a process of integration, and the area of a curve offers the most simple geometrical representation of the pro- cess. When dx is taken indefinitely small so as to be con- sidered as an infinitesimal, called an element of x, each of the terms ydx of 2ydx is a. similar element of the area; and we have shown that the nearer the values of these elements are taken to zero, the more accurately will they represent the relative changes of their respective primitive quantities, and the more accurately will a succession of them compose those quantities so as to form a continuous result. The idea of elements greatly facilitates our reasonings in the higher applications of the Differential and Integral Calculus, and gives to the mind the most ample scope in geometrical and physical researches, whilst a strict adherence either to the principle of derived functions or to what is usually called the theory of limits, which some authors rigidly contend for, would render many investigations exceedingly cramped, and others almost impossible. (72.) If a right line rs which passes through the two points P and Q be supposed to revolve about the point P so that the intersection Q with the curve may proceed towards P, it has 114 THE DIFFERENTIAL CALCULUS. been shown, art. (9), that when the point Q arrives at the point P or when the distance PQ becomes an infinitesimal, the corresponding continuous position of the line rs will ultimately coincide with the tangent TP which touches the curve at the point P, and that the infinitesimal line PQ becomes then an element of the arc of the curve. These considerations are equivalent to that of conceiving the tangent to be a line which passes through tw^o points of the curve that are infinitely near to each other. Let s denote the length of the arc from a given point in the curve to the point P ; then will da:, dy, and ds symbolize the relative infinitesimal values of PG, GQ, and PQ. But PQS = PG^ + GQ^ ; and 5 —f\/dx^ + dy'^ =J*dx A/ ^ + T^ * When y is known as a function of x, explicit or implicit, this expression serves to determine the length or rectification of the curve ; but the inverse operation of integration, indi- cated by/, will require the aid of the integral calculus. II. Tangent and Normal. (73.) Let 0) denote the angle PTD or the inclination of the tangent with the axis of x; then, from what precedes, we have, as before deduced in art. (9), tan o) = -p . ax If a, /3 be the coordinates of any point in the tangent PT, this gives /3 — y _. dy . a — X dx therefore the equation to the tangent is T dy ^ * A O D N The normal PN being perpendicular to the tangent, if a', /3' PROPERTIES OF PLANE CURVES. 115 be the coordinates of any of its points, its equation is hence Hence i£ p denote the perpendicular OH from the origin upon the tangent and p' = VII that upon the normal, we shall have xdy —ydx , xdx + ydy Also, if a", p" be the coordinates of any point in the line OH drawn through the origin perpendicular to the tangent, the equation to this line is dy Again, since tan o) = ^, and ds'^ = dx^ -f dy-, we have cos 0) = — , and sin oo = -^ ; ds ds PT = = tangent = 2/ sino) yds dy PN = = normal = COSo) yds DT = subtangent = --^ = ^ , tan o) dy DN = subnormal = y tan © = '^—r^ . ^ dx (74.) When the equation of the curve is of the form u = f{x, y) = 0, the differential elements dx, dy will be connected by the corresponding differential equation Therefore the elements dx, dy^ and ds will have the same 116 THE DIFFERENTIAL CALCULUS. mutual proportions as the respective quantities \dy/ \dx/ V \dxj \dy / and by replacing them by these quantities the preceding relations, and any formulae involving the ratios of the elements, will then become adapted to the case in which y is an implicit function of x. The equation to the tangent^ under this form, is thus (£)(-') + (r:)<«-^' = °- and it is therefore to be practically obtained by this simple rule : Differentiate the given equation of the curve, u =/(a:, y) = 0, and write a — ^, /3 — y in place of dx and dy. Also the equation of the normal is (^(•■-')-(S)<^-^)=»- Example, — The equation to an ellipse when referred to its centre and principal semidiameters a, 5, is —- + ^ = 1 . By differentiating, this gives -^dx + ^dy == ; dy ____b^ ds _ s/^^^^TT^ dx a^y dx a^y ds _ sja'^y^ + h^x'" dy "~ h'^x tangent = ^ ^^2 ' normal = ^—r, , h'^x d^ -To—, and subnormal = b^x a^ subtangent = — -7—- , and subnormal = -^x- PROPERTIES OF PLANE CURVES. 117 Also, the equation to the tangent is and the equation to the normal is /3' = a2-62. III. Asymptotes. (75.) Two curves or a curve and straight line are mutually asymptotic when they continually approach indefinitely nearer and nearer to each other, hut do not meet at any finite distance. By an asymptote to a curve we generally understand a straight line, such that if it and the curve be indefinitely continued they will thus continually approach each other but never meet. It may therefore be considered as a determinate tangent to the curve when the point of contact is removed to an infinite distance. The position of the tangent to the curve is geometrically determined when the intercepts OT, O^ of the coordinate axes are known. In the equation of the tangent, art. (73), make /3 = 0, and we shall find the intercept of the axis of x, between the origin and the tangent, to be* ydx xd y—ydx Also, by making a = we similarly find the corresponding intercept of the axis of y to be xdy xdy — ydx 0o = O' = y- dx dx * In the diagram, OT being in the contrary direction to Ox must be accounted a negative quantity, and equal to OD— DT. 118 THE DIFFERENTIAL CALCULUS. If, when 0? = CO or y = oo , either of these values of qq and /3() should be finite, the curve will have one or more asymptotes which will thence be determined. When a^ is infinite and ^^ finite the asymptote is parallel to the axis of x, "When qq is finite and /3q infinite the asymptote is parallel to the axis of y. When qq and /3q are both finite the asymptote passes through the two determined points T, t. When the values of a^ and 0q are both = the asymptote passes through the origin, and its direction vdll be determined by the value of — when a: = oo or y = oo , But when the values of a^ and jSq are both of them infinite y the tangent is at an infinite distance from the origin, cannot be constructed, and is not an asymptote. The asymptotic branches of the curve will, with few ex- ceptions, be analogous to one or other of the forms exhibited I in the annexed diagrams, and will only differ with respect to relative situation. These diagrams, for example, may be considered to represent the general features of the respective curves determined by the equations When the axes of coordinates or lines parallel to them are asymptotes to a curve, the circumstance will at once be indicated as follows : If, when y = 0, d? = 00 , the axis of x is an asymptote ; and PROPERTIES OF PLANE CURVES. 119 if, when j: = 0, y = oo , the axis of y is an asymptote. Such is the case with the curve whose equation is xy ■=■ a^. If, when y = i, a? = 00 , a Hne parallel to the axis of a?, at the distance y ■=. b^ is an asymptote ; and if when x =^ a, y = 00 , a line parallel to the axis of y, at the distance a: = a, is an asymptote. Such is the case when the equation is xy — ay — hx ^=^ 0. In other cases the position of the asymptotic tangent, if any such exist, will be ascertained by determining as before the values of the intercepts a^ and jS^. (76.) The practical calculation of the values of a^, /3q and of the equation to the asymptote may be considerably facilitated by putting the expressions under the following form : © . '($} «0- /jx' ^0- .jx- ^.^ . y— 3 dy - JN ow smce — --- = — , where a, jS are the coordmates of any point whatever in the tangent, if when a? = oo , y = oo this tangent be an asymptote and pass at a finite distance from the origin, this point can be taken so that a and ^ shall be both finite, and the relation then gives - = --. Let therefore ~ X ax X = t and - = r ; then /3f^ = — - , and the equation to the tan- X ^ dv ^ dy gent when it becomes an asymptote is y •=• ^^ + ~ x = ^Q + tx. Hence the following easy rule : In the given equation of the curv e substitute a? = - and V y = -, and, after reducing the equation so obtained in t and v, determine from this equation the values of /„ and /3q = — 120 THE DIFFERENTIAL CALCULUS. when V is made to vanish ; then, if the value of ^q he finite, the equation to the required asymptote is If hy making ^ = go we ohtain a finite corresponding value of V, this will determine an asymptote parallel to the axis of y at the distance jr = — V Example 1. — Let the equation to the curve he xy^ay—bx = ; then substituting - and - for x and y, and reducing, we V V obtain dt at + b t—avt-'bv=:0, /3 = 1- Therefore, making v = 0, we get ^o = ^ ^"^ Po = ^» ^^^ the equation of the asymptote is y = b, indicating that it is parallel to the axis of x at this distance. By making t = co we get t? = -; .'. a? = a is another asymptote and is parallel to the axis of y, Example 2. — Let y^ + x^—axy = ; then substituting as before we get ^ ^ dt at ^^ + l—atv — Oy 3= -— = dv 3t'^—av Hence making v = we obtain ^q= — 1 and ^q = ~"o' o and the equation to the required asymptote is therefore a 3. The curve {x + \) y ■= {x—\) x has an asymptote de- termined by the equation y = j?— 2. 4. The curve y^ — ajr^^^s — Q j^^s an asymptote deter- mmed by y = - —• ^• 5. The curve y^— 2xy^ + x'^y = a^ has two asymptotes, PROPERTIES OF PLANE CURVES. 121 viz. the axis of x and the Ime y = x, which makes equal angles with the coordinate axes. 6. The curve xy'^ — y =^ x^ •{• 2ax'^ '\- bx -\- c has three asymptotes, viz. the axis of y and the two lines y =. x -\- a and y=i —x-a. IV. Circle of Curvature, {77') A tangent to a curve may be conceived to be a line drawn through two of its points which are indefinitely near to each other ; and these points being considered as the extremi- ties of a differential element of the curve, it is evident that the first differentials of the coordinates which appertain to the tangent will correspond with those of the curve at the point of contact. Similarly, the circle of curvature or the osculating circle may be conceived to be that circle which passes through three consecutive points of the curve which* are indefinitely near to each other, the position and magnitude of a circle being determined when three of its points are known. These three points being considered as the extremities of two successive differential elements of the curve, it is evident that both the first and second differentials of the coordinates which belong to the circle and curve must correspond at the point of contact. Let J?", y" be the coordinates of the centre of the circle, and x—x", y—y" will be the two lines drawn from it respect- ively parallel to x and y and terminating in the circumference at the point of contact ; hence, denoting its radius by /5, its equation is Now since this circle corresponds with the curve at two other points contiguous to the point of contact, we may dif- ferentiate twice and consider the first and second differentials of the ordinates x, y as agreeing with those of the curve. F 12*2 THE DIFFERENTIAL CALCULUS. Hence differentiating, observing that in proceeding to these points x", y" remain invariable, we get r/.r(a— y')-f-rfy(y-y") = 0, r-x (^-x") + rf-y (y-y") + rf.^^ = ; where ds^-^ = dx--^ dir, art. (72), 5 denoting the length of the curve. The first of these two equations rei]uires the centre of the circle to be situated in the normal, and the second com- pletes the determination of its position. Thus, from the two equations we deduce ,_ , _ —dyds^ __/.__ dxds^ "" dyd^jt-dxdY ^ ^ ^ dyd^x-^d^td^' Therefore, substituting these values in the equation p^ = {x—x")- -h (y— y")-, we find d^ ^ dyd'^x-dxdhf' Having proceeded on the principle of general differentiation in obtaining this expression for the radius of curvature, wo may hereat\er assimie an independent variable at pleasure. If we consider the axis of x to be horizontal, the value of tl c radius will be post tire when the convex side of the curve i^ presented tiptcardsy and it will be nepatire when the convex side of the curve is presented dotcntcards, (78.) The value of the radius of curvature may be otherwise determined by conceiving the centre of the circle to be the intersection of two normals dra\>-n from two points which are indefinitely near to each other. Let PR, PR be two consecutive normals meeting in R, the centre of curvature, the element PF of the curve being ds. Let also two tangents be supposed to be drawn at P and F, the former making an angle u) with the axis of x. Then, as <» is decreas- ing, the angle included by the tangents will he— dm, and this must e\*identlv be the same as that included bv the normals. or PiMiirB cCRTvs^ Wd tore thoB FE = FR = ^ FF =: da^ md PRF = - diy^= &-. we ^gre X i.jTTT^ (^%«fi»^ dEri^)^ to this^ the mii& is &f 124 THE DIFFERENTIAL CALCULUS. and, making $ the independent variable, this becomes which is a symmetrical form of expression for the radius of curvature. Example 1. — Find the radius of curvature at any point in an ellipse whose equation is -^ + ^ = 1 . Making x the independent variable, we have dx a^-y dx^ ~ a^y'^ ' Example 2. — In the cycloid, taking the vertex as the origin of coordinates, y = \/2ax — x^ + a vers~^ - ; a dy _ A /2 a — X d^y _ ^ dx ^ X dx^ x\/2ax — x^* ,', p = 2\j2a{2a — x). Example 3. — In the parabola y^ = 4m.r, Example 4. — In the rectangular hyperbola, referred to its g asymptotes, 2xy=: a^, pz= --^ r being the line drawn from /If* r a the origin to the point in the curve. Example 5. — In the conjugate hyperbolas -g — Tg = d: ^* PROPERTIES OF PLANE CURVES. 125 Example 6. —In the catenary y = -(e^-fe ^J, p = — ^. ^ ^ 1. Example 7. — In the hjpocycloid x^ + y^ = «^> J. P = — 3(aary)3. V, Evolute and Involute, (79.) If we suppose the point P to pass continuously through every point of the curve, the corresponding positions of the centre R of curvature will trace out another curve. This curve, which is the locus of the point R, is denominated the evolute of the proposed curve, and conversely the proposed curve is its involute. If the normal PR be supposed to move along with the point P, it is evident that the locus of the consecutive intersections R will be that curve to which the normal is always a tangent. This is rendered still further evident by considering it inversely : thus, by supposing a tangent to roll over a curve line, its successive indefinite inter- sections will obviously be the points of contact and therefore trace out the same curve. Hence a tangent drawn to the evolute at any point coincides with the radius of the osculating circle drawn to the point of contact. The equation of this tangent, art. (73), gives dy^x - of') - dx"(y - y") = 0. Differentiate the equation (^-/')' + (y -/)' = />'> supposing x", y", and p to vary, and we have (dx - dx") (x - y') -f (dy - dy") (y - y") =pdp; but, x", y" appertaining to the normal of the curve at the point .ry, we have by its equation dx(x-x")-\-dy(y-y") = 0, 126 THE DIFFERENTIAL CALCULUS. which rejected and the signs changed, we get dx^'ix — y') + dy'' (y — y") = ^ pdp. From this and the preceding equation to the tangent to the evolute we find dx" y-y'=-pdp—^. where ds"^ = dx"'^ + dy"^, s" being the arc of the evolute from any given point. These values of ^ — x" and y — y" being substituted in the equation p'^ = (x — x")^ + (y — y")^, we get p2=p2g or ds"^ = dp^; .-. ds^' = dp where pq is the radius of curvature corresponding to the given point from which s" is estimated. Hence the length of the arc of the evolute between any two points is equal to the difference between the radii of the corresponding osculating circles. From this elegant property it follows that the original curve may be described by the unwinding of an inextensible thread from off the evolute. Thus if the normal or radius of curvature AQ be conceived to be a thread extending round the evolute QR, it is obvious that by unwinding this thread, keeping AQ always stretched, the point A will trace out the curve AB, and the unwound portion of the thread having passed from AQ to PR, the intercepted arc QR of the evolute will be equal to PR— AQ. Considering the evolute as a primitive curve, its involute is thus described. (80.) For the determination of the equation of the evolute PROPERTIES OF PLANE CURVES. 127 to any proposed curve we have, art. {17)9 the following ex- pressions for the coordinates of the point R or of the centre of curvature, viz. „ __ dy ds^ __ I !^ dyd^x — dxd^y ds „ _ dxds^ __ dx ^ ^ -^^'~ dyd'-x-dxd^y'~^'^^Ts' or, making x the independent variable, rfyS dx d^y dx d'^y dx^ dx^ By means of these and the equation of the curve AB, if the ordinates xy and their differentials admit of being eUminated an equation will thence be found expressing the relation between x" and y", and will be that of the evolute. Let the equation of the evolute be given to find that of its involutes ; then since p = p^ -\- s" and dp = ds", the values of X — x", y — y", art. (79), give x^x"- (po + ^") g', y = 2/'^ - (po + y') gj', which being calculated in terms of ar" and y", if these variables can be eUminated, the resulting equation in x and y will be the required equation to the involutes, p^ being an arbitrary constant. Example 1 . — Determine the evolute of the Elhpse whose equation is X- y^ 1- — = 1 ^•2 ^ U1 — Taking x as the independent variable, dy h^x d'^y b^ dx a'"y dx^ a-^y^ 128 THE DIFFERENTIAL CALCULUS. -x^, and y" = X JL tion the required equation of the evolute is Example 2. — The evolute to the parabola y^ =z Amx is the semicubical parabola 27 my"^ = 4 (^ ■— 2m)^. Example 3. — The evolute to the rectangular hyperbola xy = «2 is (y/ + y/)! _ (y/ __ ^^/>)f _ (4^)1^ 2 2 Example 4. — The evolute to the hyperbola ^ __ ?^ =z= l is (ay')"^- (hy")i = (a^ + 52)*. Example 5. — The evolute to the cycloid y = \j2ax — x^ - a vers~^ - is a inverse position. -h a vers~^ - is a cycloid equal to the original one, but in an a VI. Position of Convexity. (81.) As before, let © denote the angle which the tangent to the curve at the point xy makes with the axis of x ; then, art. (73), tan 0) = -p- . ax For the purpose of conveniently expressing the relative positions, let the axis of x be considered to be horizontal, and that of y vertical, the positive direction of x being to the right hand and the positive direction of y being upwards. Then the tangent being supposed to be drawn in the positive direction with respect to the axis of x, its inclination (w) with the horizontal will be upwards 1 , dy . ( positive, , ^ . } when tan a> = -- is < downwards J ax I negative. PROPERTIES OF PLANE CURVES. 129 Now, when the curve at the point P, as in the diagram, has its convex side upwards, the angle o) thus estimated will evidently decrease as X increases; .*. will be ne- dx g alive. Also, when the convex side of the curve is downwards, the angle it is presented < / dx^ L positive J t downwards. In a similar manner the position of convexity with respect to the vertical will be determined by the algebraic sign of d tan G) i» 1 7 . J — - — , or 01 dy d tan o) ; and dy dy d-y . J positive 1 . . f to the right hand dx dx'^ \ negative J \ to the left hand. VII. Poin ts of Inflexion, (82.) When a curve is convex downwards, or in any other direction, and becomes afterwards convex in the opposite direction, it must have passed a point of contrary flexure in the vicinity of which the curve vrill resemble the middle turn of the letter S. In passing through one of these points, the second differential coefficient — ^, which determines the posi- dx^ tion of convexity upwards or downwards, must change its algebraic sign, and its value must therefore pass through Oorl. The condition for determining a point of contrary flexure or point of inflexion is therefore -4^ = or 00 . dx^ F 5 130 THE DIFFERENTIAL CALCULUS. Diagram 1. If the value of — ^ at this point pass through — + , the inflexion will be of the character represented in diagram 1 ; and if it pass through +0 — , it will be as exhibited in diagram 2. These two forms will represent all cases of inflexion if they are only placed in difl*erent positions with respect to the coordinate axes. It is also obvious that the value of the angle <», which the tangent RS makes with the axis of x, will be a minimum in diagram 1, and a maximum in diagram 2. The expression, art. (78), for determining the radius of curvature p, contains — ^ in Diagram 2. the denominator. Therefore when — ^ dx^- passes through and changes its sign, the value of the radius p will also change sign by passing through -. Hence the reason why the formula referred to expresses the value of p when the convex side of the curve is upwards, and gives to p a negative value when the convexity is downwards. Also as these radii are drawn in opposite directions, the centres of curvature being on opposite sides of fhe curve, this is in strict conformity with the usual geometrical interpretation of the symbols + and — . 1 Example. — The Witch xy =^2a{2ax — x'^y has two points 3« 2 — of inflexion determined hjx= Tr^ V^^r^a V3. z o (83.) Note. — When the equation to the curve is given in the implicit form u =-f{x, y) = the values of the difl*erential coefficients. dy d^y , —V, of y with respect to x, used in the dx dx^ preceding formulae, arts. {7^) to (82), will require some PROPERTIES OF PLANE CURVES. 131 preliminary calculation. The consideration required for this may be obviated by expressing the formulae in terms of the partial differential coefficients of the function u=f(x,y). To effect this, the successive differentiation of the equation u = 0, art. (38), making x the independent variable and d-x = 0, gives (dhi\ J^\dy (d^\ df (du\ d^_ \dx^J ^ \dx dy) dx ^ \dr,y dx^ "^ \dy) dx^ ' which are the relations connecting the values of -J- and -_ V ax dx" with those of the partial differential coefficients of u. Hence we obtain (du\ ^ — V-r/ dx ~ ~7d^ Vv) r-y _ \dxO\dy) - \dx dy)\dx)\dy) + Uy V W rf2 dx^ ~ /duY The substitution of these values will accomplish the requisite transformation. For example, the expression for the radius of curvature, art. (78), becomes P — m^m /a-u\ /rf«\3 / U-u \ /du\ /du\ {d-u\ /(/«Y ' \dxO \d^) ~ \d^/) \di) \dy) + \dP) {dr) 132 THE DIFFERENTIAL CALCULUS. which is necessarily symmetrical with respect to the co- ordinates. The corresponding transformation of other formulae is obvious and may be here left to the student. VIII. Multiple Points, (84.) A multiple point is a point in which two or more branches of a curve meet or intersect. If it is common to two branches of the curve it is called a double point; if it is the concourse of three branches it is called a triple point, &c. At a multiple point there will be a tangent to each branch of the curve that passes through it, and therefore the dif- ferential coefficient -^, which determines the position of the dx tangent, must admit of corresponding multiple values. In this case the expression for -^, deduced from the equation dx of the curve, vdll take the indeterminate form -, and its multiple values may be obtained by either of the methods given in arts. (61) and (62). Let u =f{xy y) = be the equation to the curve ; then, art. (61), the conditions for a multiple point will be and if, for the values of x and y which simultaneously fulfil these equations, the second partial diiferential coefficients do not all vanish, the point will be double and the values of Q =; J: will be determined by the quadratic equation dx For the conrenience of abbreviation, let this be denoted by (A) + 2(c)a+(B)a2 = 0; PROPERTIES OF PLANE CURVES. 133 then the two values of a will be + Vc*-^ - AB Diagram 1. Diagram 2. Diagram 3. We may hence, according to the nature of these roots of the quadratic, distinguish three classes of double points : I. If the two roots or values of a be real and unequal, the two branches of the curve will take different directions, and the point will be a point of intersection or real double point as represented in diagrams 1 and 2. These and the following diagrams may be placed in any position with respect to the axes of coordinates. II. If the values of a be equal, the two branches of the curve will have a common tangent, and therefore also have mutual contact at the point under consideration. In this case if the convexities of the two branches be situated on opposite sides, the contact will be external, as shown in diagram 3, and the point is called a point of contact of the Jirst kind or point of emhrassement ; and if the convexities lie in the same direction the contact will be internal, as in diagram 4, and the point is then called a point of contact of the second kind or point of osculation. Diagram 4. If, however, the value of c^ — AB under the radical, which vanishes at the point P, should change its sign and become nega- tive on one side of the point, the cor- responding value of a will be unreal, and therefore the two branches of the curve will be restricted to one side of the point, which is then denominated a cusp. As before, if the convexities of the two branches lie in con- trary directions, the cusp is of the Jirst kind, as shown in J 134 THE DIFFERENTIAL CALCULUS. diagram 5 ; and if the convexities are in the same direction it is of the second kind, as shown in dia- Diagram 5. gram 6. y III. If the values of a be unreal, then no real branch of the curve can pass through or meet the proposed point, which, being thus detached from its associated curve line, is in such case called an isolated or conjugate point, (8.5.) The analytical criteria for discrimi- nating the character of a double point are therefore as follows : Diagram 6. «»(^J-@)(|-:)>". the point is an intersection of two branches of the curve and is a real double point. "• ^^^^ WJ "-fe)(^)=^^ If > Oforpomts immediately preceding and following, it is a contact of two branches ; if of diiferent signs at these points, it is a cusp. The contact or cusp will be of the first or second kind according as — | for the two branches has different signs or the same sign. If — ^ = 0, this will indicate an inflexion, dx" ( d^uY /d'-u\/d^u\ ^^ . . '' ^^^^ \d^j) - v^Avy ^ ' "' '" ^"^ '"'^"''^ or conjugate point. It is easy to extend the process to higher orders of multi- plicity. If, for the values of a: and g which fulfil the equations « = 0,(g) = 0, (I) =0; III. PROPERTIES OF PLANE CURVES. 135 tial differential coefficients do not vanish, then the values of a will be the roots of the cubic equation (S)-"(^,)-'(^)-*($)-''- If the three roots of this equation be real and unequal, the point will be an intersection of three branches or a real triple pointy of which the point P in the annexed diagram. No. 7, is an example. Diagram 7. If two of the roots be equal, it will be a -^ ^^ point of cow^ac^ d^ndi intersection ; if the three \^V/^^^ roots be equal, it will be a point of double /\ contact; but if the equation contain a pair of unreal roots, then only one real branch of the curve passes through the point, and it is therefore in that case not a real triple point. Should the point P be a quadruple point, as in diagram 8, the third partial differential coefficients will also vanish, and the values of a will be deter- J^° ^ mined in like manner by an equation of the fourth degree. Since an algebraic equation of odd dimen- sions must necessarily have at least one real root, it is evident that a conjugate point can only occur when the degree of multiphcity is even. (86.) An examination of the character of multiplicity of any proposed point of a curve may in general be more readily effected by a method analogous to that given in art. (62), for determining multiple values of -^ when of the form -, and dx which we shall here repeat with a slight modification. Let the coordinates of the point V he x = a^ i/ = b ; then if in the equation of the curve x and y be replaced by a -f j/, b -f y'y we shall have an equation in which a/, ?/' are now the coordinates of any other point P' in the curve estimated from the proposed point P as a new origin. In this equation make 136 THE DIFFERENTIAL CALCULUS. if = j3a?' ; then dividing throughout by the power of d/ that may he common to the several terms, we shall obtain an equation in which ^ will denote ^ or the tangent of the angle which X the chord PP' makes with x\ and when x^ is made = the corresponding values of /S^ given by this equation will evidently be those of -^, and the number of such values will, as before, dx determine the multiplicity of the point. Also, by giving to a?' a small positive or a small negative value, we may ascertain the number and situation of the corresponding points P' in the immediate vicinity of P on either side. Since y^—^x' we have, by differentiating with x' as the independent variable, dx' therefore at the point P, where x' = 0, -"^^' dx'^~\dx')o The first of these shows that the values of ^ when a?' = are those of — , as before stated ; the second will determine dx the positions of convexity by art. (81) or the radii of curvature by art. (78) if required, the formula for the latter being The nature of each separate branch of the curve may, however, be easily made known by comparing with /Sq the two values of ^ which correspond to small positive and negative , PROPERTIES OF PLANE CURVES. 137 • rallies of j/. Thus, if (/3 — ^^^) x' continues to be positive, the convexity is evidently downwards ; if it continue to be negative, the convexity is upwards ; and if it change sign with a;, the point is one of inflexion. Example 1. — Let x^ — ax^y + hy^ = 0, and determine the nature of the point at the origin where j? = 0, y = 0. Here (d'^i\ ,^ o « ^ / d-u\ ^ „ (cl'^u\ (;^)=12.= _2<.y = 0,(_) = -2«x = 0, (-J=6*y = 0; Therefore the equation for determining the values of a dy dx — Qaa + 6Z»a-^ = 0, or ha^-aa = 0; dy . = -± IS the roots of which are a = 0, and a = + V - , and therefore h the point is a real triple point similar to that shown in diagram 7. Otherwise, the origin being already situated at the pro- posed point P, substitute y = ^x, and x'^ — ax''^(^+ bx^fi^ = 0, which divided by x^ gives a:— a ^ -\- b^^ = 0. Hence, at the origin, - a/3 + i/33 = ; .'. ^ = and /3 = + ^ ?, and the point is a real triple point. Example 2. — The equation being cy- + bx'^ — x'^ = 0, required the nature of the point at the origin. Substitute ^x for y and divide by x'^ ; then, a/S" -h 6 — J? = ; .'. /3- = — , and at the origin, x = and 138 THE DIFFERENTIAL CALCULUS. /3^ = \/ , which being unreal, the point is detached from a its curve, and is a conjugate point. Example 3. — The curve {ay — x'^Y' {a'^ -\- x^) — m^a^x"^ ■= passes through the origin ; it is required to find the nature of this point. Substitute, as before, y = px; then, dividing by x'^, we get, (ap - xy (g3 -f ^3) _ m3«2^2 = ; .'. /3 = - ± a - Va^ -h ^3 At the origin 0^ = 0, and, as the double values of here merge into one, the two branches have mutual contact with the axis of x at this point. Differentiating the value of ^ we have also (//3 _ 1 ma^ . /d^\ _ I ±m dx a ~ (a^ + a?3)t /^\ _ 1 ± ydx/Q a Therefore, if m > 1, the convexities lie in opposite directions and the contact is external; if m < 1, the contact is internal, or a point of osculation, and the two branches have their con- vexities presented downwards ; and in either case the two radii of curvature are p« = — - r-^r— ; — x • 2(1 + m) Example A. — The curve whose equation is cr^-f-a^^^^yS — o has a double point at the origin, and the directions of the branches are determined by jSq = + . /— . Example 5. — The curve (a^— a?^) y 2_ (^3 _j_ ^2)-p3 _- q has a double point at the origin, and /Sq = ± Ij or the branches make equal angles with the axes of coordinates. Example 6.— The Lemniscate {x^ + y'^f — a'^{x^-—y^-) = has a double point at the origin, and the branches make equal angles with the axes. Example 7.— If b (y — xy — x^ = 0, the origin will be PROPERTIES OF PLANE CURVES. 139 a cusp of the first kind, the common tangent making equal angles with the axes. Example 8. — If a:^ -h a'^x'^ — b^y- = 0, the origin will be a cusp of the first kind touching the axis of x. Example 9. — In the Cissoid y'^{2a^x)—x^ = 0, the origin is a cusp of the first kind also touching the axis of x. Example 10. — If {ay — ax— x'^)'^ — x^ = 0, the origin will be a cusp of the second kind, with the two convexities down- wards, and the common tangent making equal angles with the coordinate axes ; also the branches at this point will have the same centre of curvature, the common radius being p^ = — aV 2, so that the contact is of the second order. Example 11. — The evolute to the ellipse, example 1, art. (80), {axY -^ {hyY = {a^--r-)^ has four cusps of the first kind at the points a-' — b'' - ^ a-^ — b-' x=.Oy y = -f — - — , and y =0, .r = + IX. Tracing of Curves, (87.) The equation of a curve being given, it is sometimes required to develop its particular structure, peculiarities of form, and general character. Such an investigation is usually called discussing or tracing a curve from its equation, and only requires the practical application of the preceding for- mulae. It will be sufficient here to indicate the chief points that should engage attention. I. If the equation be in the implicit form, it will be advisable, if practicable, to solve it with respect to one of the variables, pro\ided the result be in a convenient form for calculation. By first making y = and then a: = 0, we shall ascertain if the curve crosses the axes and the positions {xq, 0), (0, y^ of the points of intersection. Also, by assigning to one of the 140 THE DIFFERENTIAL CALCULUS. variables a series of positive values from to oo, and of negative values from to — oo , and calculating the correspond- ing values of the other variable, we shall be enabled to follow the course of the curve, and to discover if it has any infinite branches. In all these calculations both positive and negative results should be carefully included, so as to obtain the com- plete branches of the curve. II. Should the curve possess any infinite branches, ascertain if they have asymptotes and determine their equations, and thence their geometrical positions. III. Determine the value of -—, and from it deduce the maxi- dx mum and minimum values of x and y, and the angles at which the curve cuts the axes, &c. IV. Determine the value of -7-| and thence the relative posi- tions of convexity of the different branches, and the points of inflexion if there be any. dtj V. Should the expression for -^, for particular values of the variables, become of the form —, determine the nature of the corresponding multiple points. Note. — In some cases the character of a curve can be discussed with greater facility when its equation is transformed into polar coordinates. See the following Chapter. X. Envelopes, (88.) Let the equation to a system or family of curves be denoted by U=/(.2:',y, a) =0, where a is a variable parameter which is only constant for each curve. For each specific value of a the equation will be that of a determinate curve ; and when a varies continuously PROPERTIES OF PLANE CURVES. 141 it will determine a continuous succession of curves, the position and character of each of which will differ but little from that which precedes it. Let Uo=yiJ^» y» «) =0, Uj =/(j:, y, a -h da) = 0, \5^ = f(x,y,a + 2da) =0, be three consecutive curves in this series, and suppose P to be a point in which the curves Vq and U^ mutually intersect, and P' the corresponding point in which Uj and Uj intersect. Then, since the two points P, F are both situated in the curve Ui, it is evident that the curve which is the locus of the points P will have the element of its arc, PP' = dsy co- inciding with an equal element of the curve U^. Therefore the curve traced by the intersection P will have contact with the entire family of curves U, and it is hence called the envelope of the system. The envelope to the family of curves U is therefore to be found by determining the locus of the point of intersection of two consecutive curves taken indefinitely near to each other. Let X, y be the coordinates of the point of intersection P ; then these coordinates will falfil both of the equations U = 0, Uj = 0. Hence, in passing from U to Uj, the point P will remain fixed and only a will vary, so that we must have (S)=«- We have thus the two equations "=»• (S)=». from which the variable parameter a being eliminated we shall obtain an equation involving x and y, the coordinates of the point P, which will be the equation to the envelope of the proposed curves U. (89.) If the equation \] ^=f{x, y, a) be of the first degree 142 THE DIFFERENTIAL CALCULUS. in X and y, it will represent a system of straight lines ; and if, as the parameter a varies continuously, the variahle line be supposed to be in motion, the point P will obviously be the centre of instantaneous rotation ; and its locus will be that curve to which the line is always a tangent. This may be made apparent by conceiving the envelope or the curve which is the locus of P to be represented by a rectilinear polygon of an indefinite number of sides, each of these sides at the same time representing an infinitesimal element ds of the curve. The sides produced will represent tangents to the curve, and the angular points will evidently be the intersections of ' consecutive tangents. This property of a curve being generated by the ultimate intersections of a series of lines determined by a given law may be further instanced in the evolute to a curve. Since, art. (79), the normal drawn to a curve at any point is always a tangent to the evolute, it is evident that the evolute will be the envelope to all the normals, in the same way that a curve is the envelope. to all its tangents. Example 1. — Find the envelope to the system of lines determined by the equation - -j- ^ = 1, where a and /3 are a ^ variable parameters subject to the, condition a/3 = 4m^. By differentiating the equations with respect to the para- meters, we have from which eliminating da, d^, we get _ = ^=:_, ora = 2a:', a iS 2 ^ = 2y. These substituted in aP = 4m^y we have for the envelope the equation xy = m^, which is that of a hyperbola referred to its asymptotes. x'^ v'^ Example 2. — The equation to an ellipse being — + jo = ^> that of the normal drawn through the point x'y' is, example PROPERTIES OF PLANE CURVES. 143 2 7 2 art. (74), ^"^ __ — ^=: «-— ^- ; determine the envelope to J?' y' all these normals. The two variable parameters x\ y' may be reduced to one by making J?' = « cos a, y' = h^\\\a\ then, putting c^:=^a'^ — lr, we shall have COS a Sin a and, differentiating with respect to the variable parameter a, (c^U\ sina . , cosa ^ -— ) = «a? T- + fiy -r-o- = 0- da / cos-a sm-a From the latter equation, tan a = — f -^ ) ; and by sub- stituting the corresponding values of cos a, sin a in U = and reducing we finally obtain (ax)^ + (by)i = (c2)* which is the evolute to the ellipse, and agrees with the result before obtained in art. (80). Example 3. — The envelope to the system of straight lines determined by the equation y — ax -{- - is the parabola a y2 == 4mx, Example 4. — The envelope to the system of circles (x — 7n — a)- -j- y2 — 4^^^ is also the parabola y- = 4mx, Example 5. — If a straight line whose length is c slide with its extremities upon the axes of coordinates, its variable equa- tion will be represented by 1 ^ — = 1 ; and the c cos a c sni a envelope, or curve to which the line is always a tangent, will 2_ i. J. be the hypotrochoid a^^ + y^ = c^. Example 6. — The parabolas described by projectiles dis- charged, in vacuo, from a given point with a given velocity are included in the equation 4my=^4max — (1 -f a^)j:-; and the envelope to these is the parabola x^ = 4m (m — y). 144 THE DIFFERENTIAL CALCULUS. CHAPTER VIII. FORMULiE FOR POLAR EQUATIONS, &;C. (90.) The system of representing positions by means of coordinates relative to fixed axes gives the greatest facihty and the widest range to the appUcations of the analysis. It is on that account much employed in geometry, and almost exclusively in physics, to which in nearly every branch of inquiry it seems to be particularly adapted. In the geometry of curve lines, however, it is sometimes convenient to in- vestigate the properties of certain curves from what is called the polar equation^ and which is especially applicable to curves of the spiral kind. A fixed indefinite right line 0.r, origi- nating at O, is called the polar axis or prime radius; the fixed point O is the pole or origin ; any right line O P drawn from the pole O to a variable point P is called the radius vector to that point, and its angle VOx with the axis i\iQ polar angle. The radius vector OP is denoted by r, and the polar angle PO.r by ^ ; these evidently define the position of the point P, which may be symbolically designated the point rO. The polar equation to a curve expresses a relation between r and ^, and is of the form r(r, ^) = c ; and, in most cases, r may be separated so as to give the explicit form FORMULAE FOR POLAR EQUATIONS. 145 F and/ in most cases involving the polar angle 6 under the form of trigonometrical functions. The quantities r, 6 being thus made subject to an equation, we shall have particular values of r for each successive value of 6 ; and hence the point P becomes restricted to a particular curve determined by the equation. I'he perpendicular OH from the pole upon the tangent being, as before, denoted by j9, the equation to a curve is in some cases advantageously expressed in r and p, (91.) Polar Equivalents. — By taking the axis of a? for the polar axis, and the origin of the rectangular coordinates for the pole, we shall obviously have j7 = r cos 6, y = r sin ^ ; and hence also, by differentiation, dx •= dr cos 6 — rdO sin B, dy =. dr^md -{- rd6 cosd ; d-x = d-rcos6 — 2drdB smS — rd6^ cos6 — rd-S sind, d-y = d^rshid + 2drdecose — rdB^smd + rd-O cos 6. These values substituted in any given formula involving rectangular coordinates, will give the equivalent polar formula in terms of r, 6 and their diiferentials. The following relations are sometimes useful in dynamical investigations : dx cos 6 -f dy sin 6 = dr^ dy cos 9 — (Zr sin^ = rdO, d-jc cosd + d^y smd = d'-r-rd6^ (PycosO - d^xsmd = rd^-e -f 2drde = ^^!::^\ r When B is taken as the independent variable, dB will be constant, and the terms containing d-B will disappear. (92.) Rectification. — Substituting the foregoing values of dx, dy in the equation ds^ == dx^ -f dy^, we get G 146 THE DIFFERENTIAL CALCULUS. .-. ds = ^{dr^ + r'-de'-). and s=fdS^(^r^^ + ^^. (93.) The value of ds may be immediately deduced from the diagram. Thus if OP and OF be the radii vectores, sub- tending the arc W=:ds and containing the angle VO'P' = dd, let P^/^ be a small arc described with the radius O P and meeting O F in w^ ; then, when the elements are infinitesimal, this small arc may be regarded as a right line perpendicular to OP'; also, we shall obviously have mV = dr, and Vm = rde; .-. ds'- = PF2 = ^F3 + p^3 = ^,.3 ^ ^2^^2. Several of the subsequent formulae may also be obtained geometrically from the diagram, and the determination of them in this way would form useful exercises for the student. (94.) Perpendicular on the Tangent. — The perpendicular OH from the origin upon the tangent being denoted by^, we have, art. (73), xdy — ydx ^ Js By substituting the preceding polar equivalents, this gives rHe rU6 p = ds " ^/(dr'- -f- r'-dO^Y 1 dr Cor. — If M = - ; then du •=■ ^ -^y and we obtain the neat r H formula 1 -^ 2X ^ (95.) Sectorial Area, — Conceive two consecutive radii vec- tores OP = r, OP' = r -h c?r to be drawn, subtending the element W =^ ds of the curve and containing the angle POP' = de. The sectorial element thus formed by these radii vectores and ds may be considered as a plane triangle, I FORMULAE FOR POLAR EaUATIONS. 14/ A the perpendicular from the origin on the opposite side V produced will obviously be that on the tangent to the » urvc. Therefore, p denoting this perpendicular, the area of the sectorial element ='^-— • That is, denoting by S the iiectorial area of the curve estimated from a given radius pds ^ , ,^,^ xdy—ydx r^dd vector, dS=^' But, art. (94),;? = ^ / - = -j-y JL CIS CLS xdy — ydx r^dO (96.) Inclination of the tangent with the radius vector. — Let the angle OPT included by the tangent and radius vector be denoted by P ; then by the diagram, • p OH ;,. Substituting the value of;?, art. (94), these become rde rde sm P='J^ = ds s/{dr- + r'-de^)' _ rfr _ dr r. rde tanP = -7-. dr Cor. — Hence we obtain, _ f?r _ rdr ^ " ^^ " V(r— yO' Ty» dr ^ „ pdr de^ — tanP = — jfr, -^y -J r"dd _ prdr 148 THE DIFFERENTIAL CALCULUS. which are here expressed in terms of the radius vector and the perpendicular on the tangent. (97.) Tangent and Normal. — Let a straight line NOT be drawn through the origin at right angles to the radius vector OP, and intersecting the tangent and normal in the points T and N. This line we shall here designate the relative axis to the point P. It is evident that the positions of the tangent and normal with respect to this axis will enable us to construct them geometri- cally. The line PT is the polar tangent, PN is the polar normal, OT is t\iQ polar subtangent, and ON is i\iQ polar sub- normal. From the angle P, determined in the last article, the values of these lines are immediately deduced as follows ; r T rds PT = polar tangent = — -j-^ ^ =■ — , ° cosP V V ""i^ ) ^^ r r ds PN = polar normal = -; — :i^ = — =-—, ^ smP p de V r T do OT = polar subtangent = r tan P = — —— 5- = — 7— > v(r — JO") dr V r dr ON = polar subnormal = — - = - \^(r'^ — p^) = — ; tanP JO ^ c?^ UH = » = rsmP= — 7— > ds OK=;?,=:rcosP= V(r2-;?2) ^ !^ . (98.) Asymptotes. — If for any finite value of 6 the value of r becomes infinite, the radius vector does not meet the curve at any finite distance, and therefore it must be parallel to the tangent which belongs to the corresponding point at the infinite r'^dQ distance. The polar subtangent OT = —7- will then become identical with the perpendicular from the pole on the tangent, and if its value be finite, the tangent admits of being con- FORMULAE FOR POLAR EQUATIONS. 14f> structed and is then an asymptote to the curve. If the polar subtangent = 0, the asymptote passes through the pole and coincides with the radius vector : but if the value of the polar subtangent be infinite, the tangent, being at an infinite distance from the pole, is not an asymptote. If the diagram be conceived to be turned round into such a position that the radius vector shall proceed from the pole towards the right hand, the rule of signs to be observed in the construction will be simply as follows : If the value of the polar subtangent OT = —-r- be positive, it must be measured downwards, and if it be negative, it must be measured upwards ; then the right line drawn through the point T parallel to the radius vector, will be the required asymptote. (99.) A polar curve may have a circular asymptote. If, when the value of the polar angle d is supposed to proceed positively or negatively to infinity, the point P recedes from the pole until the radius vector ultimately attains, as a superior limit, the finite value a ; then a circle whose centre is the pole O and radius a will evidently be an exterior asym- ptotic circle. But if the point P approaches the pole, until the radius vector reaches as an inferior limit the finite value a, the circle will be an interior asymptotic circle, (100.) Circle of Curvature. — The value of the radius of curvature obtained by general differentiation, art. {77), is — dyd^x — dxdry But, using the polar equivalents, art. (91), we have dydrx^dxdry = dr{d'^x sine—dy cos^) -f rdd (d-x cos6 -f d-y sm6) = -dr{2drdd + rd-B) + rde{d^r-rde-) = '-ddir'-dS'- -}- 2dr^--rdh)''rdrd^; ds^ P = dd (r-dO- -f- 2dr- - rd'-r) -\- rdrd-O 150 THE DIFFERENTIAL CALCULUS. By taking 6 as the independent variable, __ ds^ (dr^ + r^dd^)i ^ dd{r'-de^ + 2c?r3-r6^r) "" dBir^de^ + 2dr'--rd'-ry which will be positive when the convexity is downwards, and negative when it is upwards. and the expression for p reduces to the convenient form P- ,/ d^u\ „.d'^u (101.) The value of the radius of curvature in terms of r and p may be found as follows : Referring to the diagram, we have the angle OPI = P, POI = 6, and PID = a>; .\ (o = V + 6, and dco = dV + dS. But from the values of sin P, cos P, art. (96), we deduce _ iZsinP^ rdp--pdr cosP "" r\/{r'^'—p^) Also, art. (96), rdr - ,^ pdr ds = —J-— ^; and dB — — jj~^ ^r ; .*. ao) = V('-"-i'-) Hence, art. (78), ds rdr da) dp This neat relation may be verified by substituting for dp the differential of the expression p = — ,, y o . — oiw^x • ^^^ \/\dr^ + r-'dO'^) result will be found to correspond with the value before obtained. FORMULA FOR POLAR EQUATIONS. 151 Examples. 2 1 . In the lemniscate r- = a- cos 2^, p = — - . 3r 3 2. In the spiral of Archimedes r = ady p = ^^ — ^ . Za" + r" Q T .V, • 1-1 ^ r(fl2 + r2)^ 3. In the reciprocal spiral m = ~, p =— ^ — — ^ — ^ . 4. In the cardioid r = a (1 — cos ^), p = - \/2ar. 3 5. In the logarithmic spiral ^j = mr, p = — 2/2 2^ 6. In the epicycloid p'^ = — ^ — g— > 3 2 1 (102.) Chord of Curvature, — The portion of the radius vector, produced if necessary, intercepted hy the circle of curvature, is called the chord of curvature. As this chord evidently subtends an angle, at the centre of the circle, equal to 2 P, its value is Chord of Curvature = 2p sin P = -^ = -^. r dp Example 1. — In the lemniscate r^ = a^ cos 20, the chord of curvature = q ^' Example 2. — In the cardioid r = a(l — cos 6)y the chord of 4 curvature = « ^* o (103.) Evolute and Involute. — The radius of curvature coincides with the normal and touches the evolute, art. (79). Let r^ = O R, /?^ = O K be the radius vector and perpendicular on the tangent which belong to the evolute at the point of contact. By referring to the figure, page 148, it will be seen that p and p^ constitute a rectangle IIOKP with the tangent 152 THE DIFFERENTIAL CALCULUS. and normal to the curve; also that OK3=HP3=OP2-OH2 and 0R2 = RK2 -f OK^, that is Of = {p — p)^ 4- /"^ — j»^ = p^ — 2^/) + r^' The value of p = being previously determined, we can dp usually by means of these two equations and the equation of the curve f{r, p) = 0, eliminate r and p, and so obtain the equation of the evolute in r, andjo^. Example 1. — The evolute to the logarithmic spiral^ ^= mr is a similar logarithmic spiral ^^ = mr^. Example 2. — The evolute to the epicvcloid j)^ = . ? . ^^ ~^J c^ — a^ is another epicycloid w.^ = - — — . (104.) The value of the radius of curvature maybe simply deduced from the equation rf' = p2 — 2pp -f r^. Since, when we proceed to a consecutive point in the curve, OR = Ty and PR = p, which have reference to the pole O and the intersection R of consecutive normals, do not change, we may differentiate with respect to r and p only, which gives vdr — 2pdp 4- 2rd^r = 0, .-. p = —-. dp (105.) Let /, y be the radius vector and perpendicular on the tangent which belong to an involute of the curve. As the curve is its evolute, we have from the foregoing equations, substituting — - for p , dp FORMULAE FOR POLAR EQUATIONS. 153 The values of p and r given by these equations being substituted in the equation of the curve, we shall find an equation involving r'y p' and their diiferentials. If it can be integrated, the equation of the involutes of the curve will thence be found. (106.) With respect to the evolute, let p, be the radius of curvature at the point R, ds^ the element of the arc, and u>^ the inclination of the tangent RP with the polar axis. Then o)^ = 0) -f - and ds^ = dp ; ds __ ds^ ^ dp d^s d(o, d(o d(i) '" the differentiations being with respect to co as the independent variable. * Tliese formulae are useful if 5 or p can be expressed as a function of <», or when a curve can be reduced to an equation of the form F(5, o)) = 0, or/(p, o)) = 0. Thus in the example of the cycloid, page 1 24, we have dx A / '^ cos CO = ~r-= \/ -—y ds ^ 2a p = 2\/2a(2a — j7) = 4asino); .'. p. = -7^ = 4flCosa) = — 4a sin 0). ; dco and the two equations p = 4asin, and p^=^ — 4 « sin o)^ which determine the respective curves, show that the evolute to the cycloid is an equal cycloid placed in an inverted position. (IO7.) Positions of Convexity and Points of Inflexion. — When p is constant or dp = 0, the curve becomes a straight * It may here be suggested that a curve may be determined by an equation between any two, or more, of the quantities r, 6,p, w, p, s, and that in particular cases the investigation of the properties of a curve may be greatly simplified by an approi)riate selection of variables. 154 THE DIFFERENTIAL CALCtJLUS. line and therefore has no convexity. On examining the diagram it is evident that if a curve is concave towards the pole, r and j) will either both increase or both decrease, and therefore -^ will be" positive ; and if the curve is convex dr towards the pole, r and 'p will one of them decrease when the other increases, so that -^ will be negative. dr Hence, we have this rule : If dp . r positive 1 ^, . f concave 1 ^ j ^i ^ -i- is < ^ ^. > the curve is < > towards the pole. dr I negative J [ convex J ^ When -^ changes sign by passing through or - the direction of curvature will become reversed, and this will indicate a point of inflexion, (108.) Locus of the point where the perpendicular meets the tangent. — Let it be required to find the equation to the curve which is the locus of the point H, where the perpendicular from the pole intersects the tangent. Denote the radius vector OH of this curve by r^^, and the corresponding polar angle and perpendicular upon the tangent by 6^, and^^^. Then we shall havej9 = r^^, and, since O H is perpendicular to PH, the angle between two consecutive positions of OH will be equal to that between corresponding positions of the tangent PH ; that is, d6,,=^ d(o. But, art. (101), .-. J = Sli and r=!jL.. \/(^^^y/) ^//V(r,/-jp,/) p^^ Hence, if the polar equation to the given curve be/(^, r) = 0, that of the locus of H will be/( r^^, -^ ) = 0, being ob- EQUATIONS, &C. OF KNOWN CURVES. 155 r 2 tained by simply substituting the values p = r^^, r = -^ in the Pii given equation. Example 1 . — In the case of the logarithmic spiral, the locus of the point H is an equal and similar logarithmic spiral. Example 2. — In the case of the rectangular hyperbola, the locus is a lemniscate. The preceding articles present a complete digest of the [' most useful formulae which relate to curves referred to polar coordinates, and by them we are enabled to trace and discuss all the peculiarities and properties of curves from their polar equations. (109.) For convenience of reference, we shall here collect together the equations of the principal known curves ; and we shall then conclude with some general theorems, which have been deferred for insertion at the end of the volume. 1. The Parabola; referred to its vertex and axis, y2= 4mj7; the focus 2m 1 + COS0' 2. The Ellipse ; referred to its centre and principal axes, the equation is "i" "^ 73 ~ ^ ' ^^^^ the centre is the pole, the polar equation is y.2 = a2 I j . and, when the focus is the pole, it is r = . -, ovp = h \/ , where e = — ^ -- 1 + ecos0 V 2a — r a 3. The Hyperbola. — Referred to its centre and principal axes, the a^ y^ e^ — 1 equation is — — - - = 1 ; when the centre is the pole, r^ = a^ — . and when the focus is the pole, r = — 5^ or p = b \/ , ^ 1 + ecos0' ^ V 2a + r* where e = • The hyperbola has two asymptotes. 4. The Equilateral Hyperbola^ when referred to its asymptotes, has for its equation 2xy = a^; and the polar equation is r^ = -, ,or» = — sin20 -^ r 156 THE DIFFERENTIAL CALCULUS. 5. The Cycloid. — Referred to its vertex and axis, the equation is y = ^{2ax—oc^) + avers -, a which may be otherwise stated x = a(l— cosc^), y — a{

13. The Reciprocal Spiral. — Its polar equation is a ar 14. The Logarithmic Spiral. — Its polar equation is r = a^; ox p = mr\ the curve intersects its radius vector at a constant angle P ; and its evolute and involute are spirals equal to the original one. 15. The Cardioid. — Its polar equation is r = a (1 — cos 0) or y^ = 2«j»2. the origin is a cusp of the first kind, and its evolute is another cardioid ; also the lines drawn through the pole, and intercepted by the curve, are all of the same length 2a. irla^x) 16. Quadratrix of Dinostratus.— Its equation is y = x tan — ' GENERAL THEOREMS. 157 and it has an infinite number of asymptotes perpendicular to the axis 2a of x. When :c = 0, y = xoo = — . TT 17. Quadratrix of Tsehirnhausen. — Its equation is y = a sin — -, and it lias inflexions at the points where y = 0. 18. Companion to the Cycloid: x — a(l — cosc^), y = acp. 19. Trochoid; x = a(l— n cos + hcosl^-^^ \ (pt y = (a — b)sin y)' ^y differentiating first with respect to x and then with respect to j, we have or ^ = »y/^ . dx 1 - x'yf^ ' or ^/= f'^ ,. dz 1 — x(t>'yf'^ ' dx dz This equation heing independent of the form of the function y z=zf^ must evidently he true if y be replaced by any function of iS or by any function of y. Substituting therefore u = Fy, we get du du , / 1 >, ^ = ^*^ (^)- Again, since w is a function of y, which is a function of two variables x and r, we have, art. (37) and this equation (1), d^u __ d du(j)y __ d ducfyy __ d ^ du ., .oX (o\ dx^ dx dz dz dx dz Idz i d^u _ d d du((t)y)^ __ d^ du((l)yy __ d^ f du / , .3! dx^ dx dz dz dz^ dx dz'^ Idz J .... (3), &c. &c. &c. d^'u _ d d''-^ du{yY-^ dx^ ~~ dxdz^-^ dz "~ dz^-^ dx ^"-1 f du ,. .^1 , . In deducing the values of the differential coefficients when j: = we may obviously make x = before differentiating ; that is, we may at once use Uq = Fy^ = Y/z, and (/)yo= (p/z. Thus we find, 160 THE DIFFERENTIAL CALCULUS. \dx/Q dz yacT-'/o dz V dz J &c. &c. /J%\ d^-^ j d. Ffz . , ^ x^ 1 and by substituting these values in we obtain the theorem stated. Lagrang^ s Theorem, (112.) If y •=. z •\- x^y, where ^y denotes a given func- tion ; then the development of another function Fy in ascend- ing powers of x will be This is a case of Laplace's more general theorem, from which it immediately follows on making fz ■=■ z\ and when f^z =1, it becomes Taylor's theorem. Hughes, Printer, King's Head Court, Gough Square. Jxiitrimtntari) Scientific WBov'k^. MR. WEALE'S SERIES OF RUDIMENTARY WORKS FOR THE USE OF BEGINNERS. NEW LIST FOR 1852. The whole Senes, comprising 105 volumes, will be succeeded by other interesting and useful works more especially intended for Public Instruction, written by learned and efficient masters in the several branches of Education. 1. Rudimentary Chemistry, by Professor Fownes, F.ll.S.. &c. 3rd edition, and on Agricultural Chemistry, for the use of Farmers . Is. Natural Philosophy, by Charles Tomlinson, 2nd edition . 1*. Geology, by Lieut.- Col. Portlock, F.R.S., F.G.S., &c. 2nd edit, '\s.6d. . Mineralogy, by D. Varley, vol. i. 2nd edition . . , Is. . vol. ii. ,, ... 1*. Mechanics, by Charles Tomlinson, 2nd edition . . .1*. Electricity, by Sir WiUiam Snow Harris, F.R.S., &c. 2nd edit., with the important addition of the Cavendish Papers . Is.Sd. Magnetism : an Exposition of the General Principles of Magnetical Science, by Sir W. Snow Harris, vol. i. . .Is. vol. ii. . Is. vol. iii. . ls.6d. History, Progress, and Present State of the Electric Telegraph in its several applications, by Edward Highton, C. E. . 1*. Pneumatics, by Charles Tomlir^^on, 2nd edition . . .Is. Civil Engineering, by Henry Law, C.E., vol. i. 2nd edition Is. — _ vol. ii. Is. vol. iii. Is. Architecture (Orders), by W. H. Leeds, 2nd edition . Is. Ditto, (Styles — their several examples,) by T. Bury, Architect Is. Principles of Design in Architecture, by E.L.Garbett, Arc*, v. i. Is. vol. ii. Is. Perspective, by G. Pyne, Artist, vol. i. 3rd edition . . Is. vol. ii. ., . . . Is. Art of Building, by E. Dobson, C.E., Assoc. Inst. C.E. . Is. Brick-making, Tile-making, by the same, vol. i. .Is. vol. ii. . Is. Masonry and Stone-cutting, by the same . . Is. — Illustrations of the preceding, in 16 4to atlas plates . . Is. — Art of Painting, or a Grammar of Colouring, by George Field, Esq., vol. i. . ... Is. vol. ii. . . .Is. Draining Districts and Lands, by G. D. Dempsey, C.E. Is. Draining and Sewage of Towns and Buildings, by the same ......... Is. Well-sinking and Boring, by J. G. Swindell, Archi- tect, 2nd edition, revised by G. R. Burnell, C.E. . . Is. 2 RUDIMENTARY SCIENTIFIC WORKS 32. Rudimentary Art of Use of Instruments (generally), by J. F. Heather, M.A., of the Royal Mil. Acad., Woolwich, 2nd edit. . I*. 33. ■ Constructing Cranes for the Erection of Buildings and for Hoisting Goods, by J. Glynn, F.R.S., C.E. . . 1*. 34. Treatise on the Steam Engine, by Dr. Lardner. {Written specially/ for this Rudimentary Series.) . . . . 1*. 35. Art of Blasting Rocks and Quarrying, and on Stone, by Lieut.-Gen. Sir John Burgoyne, K.C.B., R.E., &c. &c. . Ls-. 36. Dictionary of Terms used by Architects, Builders, Civil and Mechanical Engineers, Surveyors, Artists, Ship-builders, &c. vol. i * ... Is. 37. — — vol. ii Is. 38. vol. iii 1*. 39. vol. iv . . Is. 40. : — Art of Painting on Glass, or Glass-Staining, by Dr. M. A. Gessert, v^ith an Appendix on the Art of Enamelling, &c. l.ercritical could be dissatisfied. In going over so large a field, and the vast amount of pains taken, the insignificance of a few •lips of the pen render them venial. We cannot but feel the superiority of a work of this kind to some more ambitious hand-books, which are made up by a paste-and-scissors process, with an aT)nn dance of quotations from old books, containing mere nominal allusions to places and things, ' I all interest but that which the philosophical inquirer may need in noting the misdirected iity of the compiler. IVIr. Wcale's book takes a higher position than these, and he is justly c..-.;. d to higher reward. Hiu volume is a sensible and useful guide." — Art- Union Journal, Sept. 1851. I. London. — Section i. The Physical Geography of the Basin of the Thames — II. Chmate — in. Geology — iv. Natural History — v. Statistics — Spirit of the Public Journals — 'Times' Printing-press — vi. Legislation and Government, Municipal Arrangements, Police, Postal Arrange- ments — Banking — Assurance Offices — Export and Import Duties. Il Wood-cuts of * Times* Machine 1*. NEW SERIES OF ' LONDON^ Church — Westminster Abbey — St. Stephen's Chapel — St. Paul's — Churches, including those by Sir C. Wren, Inigo Jones, Sir W. Cham- bers, &c. 30 wood-cuts, interior and exterior of Churches 3. London. — Somerset House — St. Paul's before the fire — Almshouses — Arts, Manufactures, and Trades — Tables of Life Assurance Companies, with the Rates of Premiums — Asylums — the Bank of England — Baths and Washhouses — Buildings for the Labouring Classes — Breweries — Bridges — Canals — Cemetery Companies — Club-Houses. 27 wood-cuts 4. Club-Houses — Churches — Colleges — an elaborate account of the Privileges and Constitution of the City of London, a special article — Customs, Custom House, Docks, and Port of London — Royal Dock- yards, with plans— Ducal Residences — the Electric Telegraph — Educa- tion — Engineering Workshops — the Royal Exchanges, Coal and Corn Exchanges — Coffee Houses, &c. 30 wood-cuts of Club Houses, the Docks, and the three Royal Exchanges, plans and elevations 5. Galleries of Pictures. — Succinct account of all the Pictures, with the names of4he Masters, in the Galleries and Collections of Lord Ash- burton — Barbers' Hall, City — Bridewell Hospital — Thomas Baring, Esq., M.P. — the Society of British Artists — British Institution — British Museum — the Duke of Buccleuch — Chelsea Hospital — the Duke of Devonshire — G. TomUne, Esq., M.P. — Dulwich College — the Earl of Ellesmere — the Foundling Hospital — School of Design — Greenwich Hospital — Vernon Gallery — Grosvenor Gallery — Guildhall — Hampton Court — T. Holford, Esq. — H. T. Hope, Esq., M.P. — St. James's Palace — H. A. J. Munro, Esq. — Kensington Palace — the Marquis of Lans- downe — the National Gallery — National Institution — the Duke of Northumberland — Lord Overstone — Mr. Sheepshanks — Lord Garvagh — Earl de Grey — Lord Normanton— Sir Robert Peel — the Queen's Gallery, Buckingham Palace — Samuel Rogers, Esq. — Royal Academy — Society of Arts — the Duke of Sutherland — Lord Ward — the Marquis of Hertford — the Duke of Wellington — Whitehall Chapel — Windsor Castle, &c. 13 wood-cuts Gas Works and Gas-lighting in London — Gardens, Conser- vatories, Parks, &c. around London, with an account of their for- mation and contents. 21 wood-cuts of the principal Conservatories, Gardens, &c Iv. Halls, Hospitals, Inns of Court — Jewish Synagogues — Schools, Learned Societies, Museums, and Public Libraries — Lunatic Asylums — Markets — Mercantile Marine — the Mint — Music, Opera, Oratorio — Musical Societies, &c. 17 wood-cuts Observatories in London and its Vicinity — Observatories and Astronomical Instruments in use at Cambridge and Oxford, with 20 wood-cuts of interior and exterior of Observatories, and of Astronomical Instruments ........... I*. Patent Inventions in England — PubHc and Private Buildings of London, criticisms on the taste and construction of them — Houses of Parliament — Prisons, &c. 16 wood-cuts Is, 10. Railway Stations in London — Sewers — Statuary — Steam Navi- _Tlio V¥rivVc nf flip TVmmPS Tnnnpl WMf^r. NEW SERIES OF EDUCATIONAL WORKS. / Supply to the Metropolis — Excursion to Windsor, with views and plans ; plans of the Stables, &c. — The Two Universities of Cambridge and Oxford, with views and plans of the Colleges ; and an Index and Directory. 25 wood-cuts Is. *^* The following gentlemen were contributors to the preceding : P. P. Baly, Esq. C.E. George Hatcher, Esq. C.E. William Pole, Esq. G. R. Bunnell, Esq. C.E. Edward Kemp, Esq. (Jeorge Pyne, Esq. M. H. Breslau. Esq. Henry Law, Esq. C.E. Charles Tomlinson, Esn. Hyde Clarke, Esq. C.E. W. H. Leeds, Esq. W. S. B. Woolhouse, Esq. E. h. Gurbett, Esq. Architect. Rev. Robert Main, LL.D. Actuary. J. Harris, Esq. C.E. H. Mogford, Esq. NBW SERIES or EDUCATIONAL WORKS; OH Volumes intended for Public Instruction and for Reference : To be published in the course of 1852. The public favour with which the Rudimentary Works on scientific subjects have been received induces the Publisher to commence a New Series, somewhat different in character, but which, it is hoped, may be found equally serviceable. The Dictionaries of the Modern Languages are arranged for facility of reference, so that the English traveller on the Continent and the Foreigner in England may find in them an easy means of communication, although possessing but a slight acquaintance with the respective languages. They will also be found of essential service for the desk in the merchant's oflace and the counting-house, and more particularly to a numerous class who are anxious to acquire a knowledge of languages so generally used in mercantile and commercial transactions. The want of small and concise Greek and Latin Dictionaries has long been felt by the younger students in schools, and by the classical scholar who requires a book that may be carried in the pocket ; and it is believed that the present is the first attempt which has been made to offer a complete Lexicon of the Greek Language in so small a compass. In the volumes on England, Greece and Rome, it is intended to treat of History as a Science, and to present in a connected view an analysis of the large and expensive works of the most highly valued historical writers. The extensive circulation of the preceding Series on the pure and applied Sciences amongst students, practical mechanics, and others, affords conclusive evidence of the desire of our industrious classes to acquire substantial knowledge when placed within their reach ; and this has induced the hope that the volumes on History will be found profitable not only in an intellectual point of view, but, which is of still higher importance, in the social improvement of the people; for without a knowledge of the principles of the English constitution, and of those events which have more especially tended to promote our commercial prosperity and political freedom, it is impossible that a correct judgment can be formed by the mass of the people of the measures best calculated to increase the national welfare, or of the character of men best qualified to represent them in Parliament; and this knowledge becomes indispensable in exact proportion as the elective franchise I may be extended and the system of government become more under the influence of public opinion. 8 NEW SERIES OF EDUCATIONAL WORKS. comparison of the text with the examinations for degrees, given at the end of the second volume of the History, will show their applicability to the course of historic study pursued in the Universities of Cambridge and London. 1. Outlines of the History of England, with special reference to the origin and progress of the English Constitution, by Wm. Douglas Hamilton, of University College, with illustrations . . . , Is. 2. , Continuation, bringing the His- tory down to a recent period 1^. *;,c* This history is designed to communicate, in an accessible form, a knowledge of the most essential portions of the great works on the English Constitution, and to form a text-book for the use of Colleges and the higher classes in Schools. 3. View of the History of Greece, in connection with the rise of the arts and civilization in Europe, by W. D. Hamilton, of University College . l^. " To Greece we owe the Arts and Sciences, but to Rome our knowledge of them." 4. History of Rome, considered in relation to its social and political changes, and their influence on the civilization of Modern Europe, designed for the use of Colleges and Schools, by the same . . Is. 5. A Chronology of Civil and Ecclesiastical History, Literature, Science, and Art, from the earliest time to 1850, by Edward Law, vol. i. G. _____ yol. ji. 7. Grammar of the English Language, for use in Schools and for Private Instruction 8. Dictionary of the English Language, comprehensive and concise . 9. Grammar of the Greek Language, by H. C. Hamilton 10. Dictionary of the Greek and English Languages, vol. i. by H. R. Hamilton II. , vol. ii. by the same 12. English and Greek Languages, vol. iii. by the same 13. Grammar of the Latin Language, by H. C. Hamilton 14. Dictionary of the Latin and English Languages, vol. i. by H. R. Hamilton 15. ■ , vol. ii. by the same 16. English and Latin Languages, vol. iii. by the same 1 7. Grammar of the French Language 18. Dictionary of the French and English Languages, vol. i. by D. Varley 19. English and French Languages, vol. ii. by the same 20. Grammar of the Italian Language, by Alfred Elwes, Professor of Languages 21. Dictionary of the Italian, English, and French Languages, v. i. by the same 22. English, Italian, and French Languages, v. ii. by the same 23. French, Italian, and English Languages, v. iii.by the same 24. Grammar of the Spanish Language, by the same 25. Dictionary of the Spanish and English Languages, vol. i. by the same 28. Enghsh and Spanish Languages, vol. ii. by the same . 27. Grp.mraar of the German Language, by G. L. Strausz, (Ph. Dr.) 28. Dictionary of the English, German, and French Languages, vol. i. by Nicolas Esterhazy S. A. Hamilton Is. 29. German, English, and French Languages, vol. ii. by the same Is. 30. French, English, and German Languages, vol. iii. by T In one Volume large 8ro, with 13 Plates, Price One Guinea, ' ill half-morocco binding, MATHEMATICS PRACTICAL MEN: A COMMON-PLACE BOOK PURE AND MIXED MATHEMATICS, DESIGNED CHIEFLY FOR THE USE OP CIVIL ENGINEERS, ARCHITECTS, AND SURVEYORS. BY OLINTHUS GREGORY, LL.D., F.R.A.S. THIRD EDITION, REVISED AND ENLARGED. BY HENRY LAW, CIVIL ENOINEEB. 10 MATHEMATtCS FOR PRACTJCAL MEN CONTENTS. PART I.— PURE MATHEMATICS. CHAPTEE I.— Arithmetic. Skct. 1. Definitions and Notation. 2. /addition of Whole Numbers. 3. Subtraction of Whole Numbers. 4. Multiplication of Whole Numbers. 5. Division of Whole Numbers. — Proof of the first Four Rules of Arithmetic. 6. Vulgar Fractions. — Reduction of Vul- gar Fractions. — Addition and Sub- traction of Vulgar Fractions. — Mul- tiplication and Division of Vulgar Fractions. 7. Decimal Fractions. — Reduction of Decimals. — Addition and Subtrac- tion of Decimals. — Multiplication and Division of Decimals. 8. Complex Fractions used in the Arts and Commerce. — Reduction. — Addi- tion. — Subtraction and Multiplica- tion. — Division. — Duodecimals. 9. Powers and Roots. — Evolution. 10. Proportion. — Rule of Three. — Deter- mination of Ratios. 11. Logarithmic Arithmetic. — Use of the Tables. — Multiplication and Division by Logarithms. — Proportion, or the Rule of Three, by Logarithms. — Evolution and Involution by Log- arithms. 12. Properties of Numbers. CHAPTER II.— Algebra. 1. Definitions and Notation. 2. Addition and Subtraction. 3. Multiplication. 4. Division. 6. Involution. 6. Evolution. 7. Surds. — Reduction. — Addition, Sub- traction, and Multiplication. — Di- vision, Involution, and Evolution. 8. Simple Equations. — Extermination. — Solution of General Problems. 9. Quadratic Equations. 10. Equations in General. 11. Progression. — Arithmetical Progres- sion. — Geometrical Progression. 12. Fractional and Negative Exponents. 13. Logarithms. 14. Computation of Formulae. CHAPTER III.— Geometry. 1. Definitions. 2. Of Angles, and Right Lines, and their Rectangles. 3. Of Triangles. 4. Of Quadrilaterals and Polygons. 5. Of the Circle, and Inscribed and Cir- cumscribed Figures. 6. Of Planes and Solids. 7. Practical Geometry. CHAPTER IV.— Mensuration. 1. Weights and Measures. — 1. Measures of Length. — 2. Measures of Surface. — 3. Measures of Solidity and Ca- pacity. — 4. Measures of Weight. — 5. Angular Measuie. — 6. Measure of Time. — Comparison of English and French Weights and Measures. 2. Mensuration of Superficies. 3. Mensuration of Solids. CHAPTER v.— Trigonometry. 1. Definitions and Trigonometrical For- mulae. 2. Trigonometrical Tables. 3. General Propositions. 4. Solution of the Cases of Plane Trian- gles. — Right-angled Plane Triangles. 5. On the application of Trigonometrj^ to Measuring Heights and Distances, • — Determination of Heights and Distances by Approximate Mechani- cal Methods. MATHEMATICS FOR PRACTICAL MEN. 11 CHAPTER VI.— CoNio Sections. Sect. 1. Definitions. 2. Properties of the Ellipse. — Problems relating to the Ellipse. 3. Properties of the Hyperbola. — Pro- blems rclatins^ to the Hyperbola. 4. Properties of the Parabola. — Problems relating to the Parabola. CHAPTER VII. — Properties OP Curves. Sect. 1. Definitions. 2. The Conchoid. 3. The Cissoid. 4. The Cycloid and Epicycloid. 5. The Quadratrix. 6. The Catenary.— Tables of Relations of Catenarian Curves. PART II.— MIXED MATHEMATICS. CHAPTER I.— Mechanics in General. CHAPTER II.— Statics. 1. Statical Equilibrium. 2. Center of Gravity. 3. General application of the Principles of Statics to the Equilibrium of Structures. — Equilibrium of Piers or Abutments. — Pressure of Earth against Walls. — Thickness of Walls. — Equilibrium of Polygons. — Sta- bility of Arches. — Equilibrium of Suspension Bridges. CHAPTER III.— Dynamics. 1. General Definitions, '? On the General Laws of Uniform and Variable Motion. — Motion uniformly Accelerated. — Motion of Bodies un- der the Action of Gravity. — Motion over a fixed Pulley. — Motion on Inclined Planes. 3. Motions about a fixed Center, or Axis. — Centers of Oscillation and Per- cussion. — Simple and Compound Pendulums. — Center of Gyration, and the Principles of Rotation. — Central Forces. — Inquiries connected with Rotation and Central Forces. 4. Percussion or Collision of Bodies in Motion. 5. On the Mechanical Powers. — Levers. — Wheel and Axle. — Pulley. — In- clined Plane. — Wedge and Scriw. CHAPTER IV.— Hydrostatics. 1. General Definitions. 2. Pressure and Equilibrium of Non- elastic Fluids. 3. Floating Bodies. 4. Specific Gravities. 5. On Capillary Attraction. CHAPTER v.— Hydrodynamics. 1. Motion and Effluence of Liquids. 2. Motion of Water in Conduit Pipes and Open Canals, over Weirs, &,c. — Velocities of Rivers. 3. Contrivances to Measure the Velocity of Running Waters. CHAPTER VI.— Pneumatics. 1. Weight and Equilibrium of Air and Elastic Fluids. 2. Machines for Raising Water by tlie Pressure of the Atmosphere. 3. Force of the Wind. CHAPTER VII.— Mechanical Agents. 1. Water as a Mechanical Agent. 2. Air as a Mechanical Agent. — Cou- lomb's Experiments. 3. Mechanical Agents depending upon Heat. The Steam Engine. — Table of Pressure and Temperature of Steam. — General Description of the Mode of Action of the Steam Engine. — Theory of the Steam Engine. — Description of the various kinds of 12 MATHEMATICS FOB PEACTICAL MEN. Engines, and the Formulae for calcu- lating their Power. — Practical appli- cation of the foregoing Formulae. 4. Animal Strength as a Mechanical Agent. CHAPTER YIIT. — Strength op Materials. 1. Results of Experiments, and Principles upon which they should be practically applied. 2. Strength of Materials to Resist Tensile and Crushing Strains. — Strength of Columns. I Sect. 3. Elasticity and Elongation of Bodie subjected to a Crushing or Tensil Strain. 4. On the Strength of Materials subjecte( to a Transverse Strain. — Longi tudinal form of Beam of uniforr Strength. — Transverse Strength c other Materials than Cast Iron.— The Strength of Beams according t the manner in which the Load i distributed. 5. Elasticity of Bodies subjected to Transverse Strain. 6. Strength of Materials to resist Torsioi APPENDIX I. II. III. IV. V. VI. VII. VIII. IX. X. XL XIL XIIL XIV. XV. Table of Logarithmic Differences. Table of Logarithms of Numbers, from 1 to 100. Table of Logarithms of Numbers, from 100 to 10,000. Table of Logarithmic Sines, Tangents, Secants, &c. Table of Useful Factors, extending to several places of Decimals. Table of various Useful Numbers, with their Logarithms. A Table of the Diameters, Areas, and Circumferences of Circles and also tl sides of Equal Squares. Table of the Relations of the Arc, Abscissa, Ordinate and Subnormal, in tl Catenary. Tables of the Lengths and Vibrations of Pendulums. Table of Specific Gravities. S Table of Weight of Materials frequently employed in Construction. Principles of Chronometers. Select Mechanical Expedients. Observations on the Effect of Old London Bridge on the Tides, &c. Professor Parish on Isometrical Perspective. Supplementary to the Rudimentary Series of 105 Volumes. Mr. "Weale has to announce a very important addition to his useful and practic series of volumes ; viz., '' The Practice of Embanking Lands from the Se treated as a means of profitable Employment of Capital; with Examples and Particula of actual Embankments, and also practical Remarks on the Repair of Old Sea Wall.' by John Wiggins, F.G.S.— Double Volume, Price 25. JOHN WEALE, 59, HIGH HOLBORN. m UNIVERSITY OF CALIFORNIA LIBRARY This book is D UE on t he last date stamped below. Fine^feifediileT4'^'^!R#ai«.on,first day oyerdue^,^ NOV 13 IS47 I^ECD LP MAR 9 ^9^' 21-100m-12,'46(A2012sl6)4120 I ';^,, THE UNIVERSITY OF CAUFORNIA UBRARY