408 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvii J. B. Fourier's111" error in applying recurring series to the solution of numerical equations was pointed out by R. Murphy.1116 P. Frisianilllc applied recurring series to the solution of equations. E. Bettillld employed doubly recurring series to solve equations in two unknowns, by extending the method of Bernoulli.102 W. Scheibner112 considered a series with a three-term recursion formula, deduced the linear relation between any three terms, not necessarily consecutive, and applied his results to continued fractions and Gauss' hyper-geometric series. D. Andre*113 deduced the generating equation of a recurring series F, from that of a recurring series 17,-, given a linear homogeneous relation between the terms 7* multiplied by constants and the terms Un, Un_ly . . ., multiplied by polynomials in n. D. Andre*114 considered a series Ult U2, . . . , with where un, Xn are given functions of n, X* being an integer rgnó 1, while Ain) is a given function of ky n. It is proved that Un = 2 *(n, p)up, *(n, p) * p-i where the second summation extends over all sets of integral solutions 'of Application is made to eight special types of series. D. Andre*116 discussed the sums of the series whose general terms are unxn l)' (an+0)! where un is the general term of any recurring series. G. de Longchamps1160 proved the first result by Lagrange108 and expressed yz as a symmetric function of the distinct roots a, /3,.... He1166 reduced Un=AiUn^l+. .+AgUn-g+f(ri), where / is a polynomial of degree p, to the case /(n)ssO by making a substitution Un= Vn+\Qnp+ .. +AP, - C. A. Laisant115c studied the ratios of consecutive terms of recurring series, in particular for Pisano's series. 1110Analyse des Equations, Paris, 1831. lu6Phil. Mag., (3), 11, 1837, 38-40. lllcEffemeridi Astronomiche di Milano, 1850, 3. llldAnnali di Sc. Mat. Fis., 8, 1857, 48-61. 112Berichte GeseU. Wiss. Leipzig (Math.), 16, 1864, 44-68. 113Bull. Soc. Math. France, 6, 1877-8, 166-170. U4Ann. sc. l'6cole norm, sup., (2), 7, 1878, 375-408; 9, 1880, 209-226. Summary in Bull, des Sc. Math., (2), 1,1, 1877, 350-5. 118Comptes Rendus Paris, 86, 1878, 1017-9; 87, 1878, 973-5. 115aAssoc. franc., 9, 1880, 91-6. "8b/Wd., 1885, II, 94-100. 116cBuU. des Sc. Math., (2), 5, I, 1881, 218-249.