v = {r2 — 2r (a sin 0 cos <f> + b sin 8 sin <f> + c cos 6) + a2 + 62 + ca}-i; (203) so that (202) is a property of this last expression. A further reduction of the equation (202) has been made by Mr. Hargreave in a paper in the Philosophical Transactions for 1841; but as it introduces considerations beside our present purpose, I must omit it. Also a discussion of some properties of it by Mr. Boole will be found in the first volume, page 10, of the Cambridge and Dublin Mathematical Journal. These memoirs however for the most part are with the view of inquiring whether (202) is or is not a differential equation derivable from some finite expression more general than (203). CHAPTER IV. THE RELATIONS BETWEEN FUNCTIONS AND DERIVED-FUNCTIONS, ON WHICH CERTAIN APPLICATIONS OF THE CALCULUS DEPEND. 109.] In the preceding part of the work, with the single exception of Art. 71, 72, we have considered the changes of the variables to be infinitesimal, and have estimated the changes of the functions and their derivatives as they are due to such infinitesimal increments; but many subsequent applications require a knowledge of the properties of functions, when the variables are increased by finite augments. ODe object of this Chapter is therefore to connect such finite changes with differentials and derived-functions; a relation of the kind has been established in equation (81) of Art. 72, the second member of which consists of a finite number of terms; and an accurate equality exists between the sum of them and the left-hand member, except so far as 6 is an undetermined proper fraction. But the proof is insufficient, inasmuch as it does not afford answers to such questions as follow: Are all functions of the form f(x + h) capable of expansion in a series of the form of the second member of equation (81); and if all are not, what are the characteristics of those which are? and supposing the function to be capable of expansion, can it be so expanded for all values of x and A, or is it possible for some and impossible for others? and what are the characteristics of x and A for which it is possible? and is the series true when continued to any number of terms, or must it cease at a certain term, because the requisite conditions are not satisfied by subsequent terms? Hence arises the necessity of proving certain theorems which establish relations between functions and their derivatives, and which involve certain conditions subject to which they are, and in failure of which they are not, within the general range of the Calculus; and by means of these, the infinitesimal calculus will be extended to changes of functions due to the finite changes of the variables. 110.] Theorem I.—Given that y=f(x) is a continuous function of x, for a given value of it, viz. x = x0, and for values a little greater and a little less than x0, that is, for x0 + dx, and x0—dx: then, if f'(x0) is positive, x and f(x) are, for that particular value, increasing or decreasing simultaneously; and if f\x0) is negative, as x increases and passes through x0, f(x) is decreasing, or vice versd. Let Ay and Ax he, as before, the simultaneous and finite changes in the values of y and x; then it is plain, that according as — is positive or negative when the increments are less du than any assignable quantity, so is — or f'(x) which is its limit. Since y=f(x), y + Ay =/(x + Ax), Ay _ f(x + Ax) -f(x) AX (X + AX) — X On the supposition that ^ is positive, the numerator and denominator of the fraction must have the same signs; and therefore, if x + Ax is > x, that is, if x increases, /(x + Ax) is > than f(x): but if X + Ax is < x, that is, if x decreases, then /(x + Ax) is < than f(x); and the same is true of the limiting value when Ax and Ay become dx and dy. So again if — is negative, the numerator and denominator must have different signs; and therefore, if x increases, f(x) must decrease: and if x decreases, f{x) increases, and the same will be true in the limit. Hence we conclude that, if = fix) is positive for x = x0, at that particular value, x and f(x) are increasing or decreasing simultaneously; and if f'(x) is negative when x = #o» as x increases f(x) decreases, or as x decreases f(x) increases. Thus, if /(a?) = a?3-6tf2 + ll#-6, f\x) = 3#2-12.r + ll. Now f\x) is positive when x — 1; therefore / (a?) and x are simultaneously increasing at that value of x; but f\x) is negative when x = 2; therefore as x increases, /(x) is decreasing at that value. Corollary I.—Hence if f(x) is continuous for every value of x between x0 and xn, x„ being greater than x0, x and f(x) are increasing or decreasing simultaneously through all values for which fix) is positive; and through all values for which f'(x) is negative, as x increases, f(x) decreases, and vice versd. Cor. II.—Hence also if, up to a certain value, x = a, as a? increases f{x) increases, and after that value f{x) decreases, f'(x) will be positive until x = a, and then will become negative; and as the sign of such a quantity changes only by the quantity passing through zero or infinity, according as the factor, to the change of sign of which the function's change of sign is due, is in the numerator or denominator, so, when x=a, will f'(x) be equal to either zero or infinity. In further illustration of this theorem and its corollaries, let us consider the following examples: In fig. 10, let ObA be a semicircle, the radius of which is equal to a; let o be the origin and Oca the axis of x; then the equation to it is y2 = 2ax—x2; .-. f(x) = y = {2ax-x>}t, g =/'(*) = — ■ ax (2ax—x2p Now for all values of x between 0 and a, f\x) is positive, and therefore as x increases, f(x) increases also; and for all values of x between a and 2 a, f\x) is negative, and therefore as x increases, f{x) decreases. And when x=a, f\x) = 0, and changes sign from + to —. Or again, let y =f(x) = sin x; So that for all values of x in the first quadrant, cos x is positive, and x and sin x are simultaneously increasing or decreasing. But for all values of x in the second and third quadrants, f'(x) or cos x is negative, and sin x decreases as x increases; and similarly for all other values of the arc. It is also to be observed, that not only by its sign does or f'(x) indicate whether an increase of the variable is accompanied contemporaneously by an increase or decrease of the function, but it also by its value denotes the rate of such increase or decrease; the greater f'(x) is, if it is positive, the faster does f(.v) increase as x increases; and the less f'(x) is, if it is negative, the slower does /(x) decrease as x increases. Thus if y = x, the equation to a straight line passing through the origin, dy = dx, and the simultaneous increments of x and y are equal; but if y = 2x, dy = 2dx, and the increment of y is always twice that of x. 111.] Theorem II.—If x„ and x0 are two definite values of x, xn being greater than x0, and xn— x0 being a finite quantity; and if T(x) is a function of x, which, as also its first derivedfunction, is finite and continuous for all values of x between x„ and x0, then F(,r„)-F(d?0) = (xn — Xo)¥{x0 + 6(xn — x0)}, 6 being some proper positive fraction. Let the difference x„ — x0 be divided into n parts, and let xu ■r2,... xn_i be the values of x corresponding to the n — 1 points of division; and let us moreover suppose n to be so large, that each of the divided elements, xi—x0) x2—Xi, ... xn — x„_i, is an infinitesimal. From the definition of a derived-function given in (6), Article 18, we have the following series of equations, whence, adding all the first and second members of the series of equations, the sum of the first is F (xh) — F(.r0); and the sum of the second is, by Preliminary Theorem III, the product of the sum of the first factors, viz. x„ — x0, and some mean value of the second factors; that is, r(x„) — r(x0) = (xn—xo)T{xo + 0(xn-xo)}, (2) in which 6 is some positive proper fraction: and therefore the last factor is a correct symbol of the required quantity; for putting 0 — 0, we have only the first term of the series, and which would therefore be correct if all the derived-functions in equations (1) were equal; and if 0 = 1, we have v'(xu), which is a term just beyond that which the series reaches; hence 8 must be greater than zero and less than unity. Let the finite difference xn— x0 be represented by h, so that |