20 INFINITESIMAL CALCULUS in which Bj, B 2 , B 3 , &c., are constants. These constants are called Bernoulli s numbers, and it can be shown without much difficulty that The complete investigation of the method of their determination is due to Euler. See his Calc. Diff. , lib. ii. cap. 5. In order to develop tan by aid of them, we write it in the form tan = Hence we find tan0 = 2 2 (2 2 -l observing that + 2 6 (26-l)^ , In like manner we get V-l e 2 *-! 2 4 B,0 3 2 6 B,0 5 Also, since cosec = cot 6 + tan , we get s , c Q i ~ 2(2"-l)B a O L IA For the completion of this investigation it would be necessary to consider the convergence or divergence of these series. This ques tion would occupy too much space for treatment here. 54. The numbers B 1? B 2 , . . . were arrived at by James Bernoulli (Ars conjectandi, 1713, p. 97) in studying the summation of series of powers of the natural numbers 1, 2, 3. . . . Thus, if Sn? represent the sum of the series Bernoulli proved that The numbers, Bj, B 2 , B 3 . . . were defined by Bernoulli as being the coefficients of the first power of n in the expressions for S?i 2 , S>i 4 , Su 6 , &c., respectively. This series of Bernoulli may be established as follows If each side of the identical equation be differentiated p times with respect to x, and we make x in the result, we get f f nx_~ 1P + 2P + 3" . . . + (n-l]P = DP[ -I, when 2 = 0, e x -l J where D stands for . ax This may be written Again c nx -l x e x -l x e x -l we get by Leibnitz s theorem 27, , . 2 1.2 1 . Now it is easily seen that and hence Bernoulli s series follows immediately. From the preceding we have 7n{l-l + 2-l + 3-l . . . + (-_!) 1.2 The function at the right hand side of this equation has been re presented by <p (z, m), and called Bernoulli s function of the with order, by Professor Raabe (Crclle, xlii.). Raabe has arrived at many remarkable properties of these func tions, of which a few of the most elementary are here added. ) = 2n(f>(z, In - 1) , M here n > 1. where z>Q and <1, and n>. For their demonstration the reader is referred to Raabe s memoir, as also to Schlomileh s Comjxndium dcr Hohern Analysis. It may be noted that the first fifteen of Bernoulli s numbers were given by Euler in his Inst. Calc. Dif. P. 2, ch. 5. The next sixteen were calculated by Professor Rothe of Erlangen, and published by Ohm in Crelle, vol. xxii. ; and thirty-one additional numbers have been recently calculated by Professor Adams, and published in the Proceedings of the British Association for 1877. " The fractional part in each of these numbers was calculated by Professor Adams, by aid of Von Staudt s theorem (Crclle, xxi.). This remarkable theorem is as follows. If 1, 2, a, a . . . In, be all divisors of In, and if unity be added to each so as to form the series 2, 3, a+l, . . . 2n + l, and of these the prime numbers 2, 3, p, 2 }> be selected, the fractional part of B, t will be 1 11 -TT+ <* P P 55. Several methods have been given for facilitating expansions by series, of which one of the most general and remarkable is that given by Arbogast in his Calcul dcs Derivations (1800). This is a method for expanding a function of a + b + c ^ + d - - + &c. 1 . t J. . 2*. O iii a series of ascending powers of x. Let and suppose Also, let then we have A=/(0) = ^>(a). Also, writing u , u", u ", &c. instead of d represents the required function. e) q (*) = A + B y + C * " 2 + D j- + &c. we obtain, by successive differentiation of the equation /(^) = < f (x) = <p ( f" (x) = ( Now, it, u , u", u" , spectively, when x = 0. Accordingly B =/ (0) = &0 () D =/"(0) = d<t> (a obviously become a, b, c, d, . . . re "(a), &c. From the mode of formation of these terms, they are seen to be each deduced from the preceding by an analogous law to that by which derived functions are deduced one from the other ; and, as/ (x), f"(x) . . . are deduced from f(x) by successive differenti ation, so in like manner B, C, D, . . . are deduced from <f>(u) by successive derivation ; where, after differentiation, a, b, c, &c. , are substituted for ctu ct u u, -j- , -7-5 , . . . c. Ct/SC WX/ If this process of derivation be denoted by the letter 8, then B = S.A, C = 5.B, D = S.C, &c. From the preceding, we see that in forming the term 5 . 4>(a) we take the derived function <t> (a), and multiply it by the next letter b, and similarly in other cases. Thus 8 . & = c , 8 . c = d , . . . 5 , l, m = mb m ~*c , 8 . c m = mc m ~ l d. . . . Also 8 . <j> (a)b = 4> (d)c + <j>"(a)b*. This gives the same value for C as that found before ; D is de rived from C in accordance with the same law ; and so on. As an
