Let the function be represented by y, and the variable included in it by x. Supposing at pleasure the series it is proposed to determine the co-efficients A, B, C. series itself. that is to say, the First, to determine A, x must be made equal to zero in the two members of the equation; the value to which y is reduced when a is made = 0, is represented by Yx=0; we then have Ayxo The other coefficients may be determined, by making = 0 in the successive differential coefficients of the equation in which the notation x = 0 joined to the differential coefficients (which might be expressed by the phrase, if x is made equal to zero) indicates that a ought then to be made = 0. These values found, and substituted in the series developed above, We may obtain the same result by determining the indeterminate coefficients in The reader may easily also obtain the same result by dividing a by b + x. prove this for himself by actual calculation. Example 2. Suppose the function y = (a + x)", to be developed according to the integral powers of x. We have 2) (a + x)-3, . . . &c. Yx=0=a", Px=0 = maTM ̄1, qa = m (m 0 1) am-2, &c. These values substituted in the general series above, we have It is evident that the expression of this series retains its value, whatever number may be designated by m, whether a positive, negative, integral, or fractional number, because the differential coefficients already developed are the same for all cases (art. 12, 13). Thus the binomial theorem is demonstrated both generally and according to the affection of any exponent. As, in an expression of many quantities, each of them may be regarded either as arbitrary or variable, unless there is something which determines it otherwise, the proposition enunciated in this article shows in fact how to develop an expression according to either of the quantities included in it. It is often differently expressed; but the same in effect: and is called, from its inventor, Maclaurin's theorem. Thus, in the first example, we may develop the function y = integral powers of b, taking the values of α b + x according to the In the second example, if the function y = (a + )" is developed with respect to a, we obtain 17. Given a function developed according to the integral powers of X, then " 1.2 If instead of a we put x + Ax = x + h, y becomes y + Ay = y + k (80 that Ah here designates the difference of the independent, Ay = k that of the dependent or of the function). From this substitution we have (A, B, C.. not containing a), y + k A+B (x + h) + C (x + h)2 + D (x + h)3 + + h (B+ 2 Cx + 3 Dx2 + .) h3 (D + But the function proposed being y = A + B + Сx2 + .) we deduce 2 C + 6 Dx+ = 2 (C + 3 Dx + . .), &c., !! as we have already obtained in the preceding article. Thus we have, finally, ̧† Taking away y from the two members of the equation, we obtain Thus, not only is the function changed, but the change of the function is expressed in a series according to the integral powers of the difference of the independent. This series is called, from its author, Taylor's theorem. We can, therefore, by means of this article, and the preceding, develop a function according to the integral powers of the independent, and likewise the change of the function according to the powers of the variation of the independent. The coefficients of these series are the differential coefficients of the proposed function, which are determined in the first series by a particular supposition of the independent, namely, by x = 0. These two methods of development may be used for every function whose successive differential coefficients can be found. This having been shown hitherto only for all the algebraic functions, these two methods of development can thus far be applied only to algebraic functions. ON THE DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS. 18. Let y hea*. Substituting x + Ax for x, we have y + ▲y = a*‡A, whence Ay = aa‡a* · a* = a*. (aa*— 1). This expression being arranged according to the powers of Ax, we have Ax2 1.2 Expressing the coefficient of Ar by k, (that is to say •) + (62 b + .) + Ax2 1 [(a 1)2 1.2 we have Ay=a* . ▲x . k +· a* . Whence (art. 6) dy =d. a* =k. a*. dx. From thence, according to art. 15, we may deduce the differentials of the superior orders; we have, in fact, By means of these differentials, we may develop (according to art. 16) the function y =a* in a series arranged according to the integral powers of x. Taking here k = 1, and making the value of a, changed by this supposition, equal to x, we have, by developing a few terms, It is to be understood that in this expression we must take both logarithms in the same system. and thence, if y is considered as the independent (according to art. 9) dy = from y = a*, x. log. a= log.y; (log. y and log. e must here be considered as in the same system.) If we take the system whose base is equal to e = 2·71828 . . (art. 18), we have log. e = 1, and designating the logarithm of y in this system by log.' y, we have This system, whose base is e, is called the system of natural logarithms, or Neperian or hyperbolic. The proposition contained in this article, on the differential of the logarithm of a number, is then that is to say, that the differential of the logarithm of a number taken in any system is equal to the differential of the number, divided by the number itself, and multiplied by the logarithm of e271828 . . . taken in the system adopted. For the Neperian system, the differential of the logarithm of a number is equal to the differential of this number divided by the number. If we take the logarithms in the Neperian system, (according to art. 18) we have also the following series: 20. We shall now develop (by the method in art. 16) the function y = (1 + x) in a series according to the integral powers of x. We have log. Substituting these values in Maclaurin's series in art. 16, we have It is observable that the terms of this series become smaller in proportion as they are distant from the commencement, supposing a to be a true fraction (whose numerator is less than the denominator); while they become greater when x is a fraction greater than unity, because the value of the successive powers of a proper fraction diminishes as they proceed, while that of the successive powers of a fraction greater than unity, increases. The first terms then are not sufficient for finding the logarithm by approximation, unless x is a true fraction. This, as is well known, has been thus expressed: "The above series is convergent only when x is a proper fraction." Another inconvenience of this series consists in the change of the signs by which the respective terms are affected. The following is a mode of finding a series, more convenient for calculation. In the above series, putting x instead of +x, we have log. (1 − x) = log. e ( — x − 22 - 23 = 2 3 Taking away this series from the first, we obtain ..). By means of this series, the logarithm of a number may, with tolerable exactness, be found by a few of the first terms, because the series converges sufficiently, and the successive terms diminish sufficiently quickly. If n is greater than unity, the convergence of the series will increase still more. To find the logarithm of the number n + z, we must evidently first know that of the number n. For the Neperian system, whose base is e, the above series is changed, according to the notation adopted in art. 19, into 1 3 log."(n + z) = log'n + 2 [2n + 2 + 3 (2n 2+ z )2 + z The first terms of this series will be sufficient to find the Neperian logarithm |