fl«k XULbTUKI U*' THIS XHLBUKY OF IN UMBERS. [CHAP. XXIV

AN EQUIVALENT PROBLEM IN THE THEORY OF LOGARITHMS.

The system of equations £a*=2&; (fc = l, • • •, n) which we have been considering is equivalent to the system Sai = 2&i, Saia2 = 2&i&2, •••, • • a» = S&i&2 • • • &n» Consider the equation having the roots d, a2, and that having the roots 61, 62, • • •• Thus our problem is equivalent to the following: Find two equations of the same degree each having all its roots integral and the first n coefficients of the one equal to the corresponding coefficients in the other.

The latter problem occurs in the investigation of rapidly converging series convenient for the computation of logarithms. In the familiar series

•, ,

m-\-n

take, for example, m=#2, n-(x—l}(x+l}. Then log ($+1) differs from 21ogx— log (x— 1) by a series in k = l/(2x2— 1). In general, we desire that m and n shall be polynomials in x whose roots are all integers such that k becomes a fraction whose numerator is a constant. We may remove the second terms of the polynomials by a linear substitution.

J. B. J. Delambre56 took m~x3-\-px+q, n=x3+px~-q, and assumed that m=0 has the roots a, 6, —a— 6, and n=0 the roots —a, —6, a+&, whence p = — a2 — ab — 62, q = a26 + a&2. For a = 6 = 1 , we have the formulas ra, n-x3— 3z±2, ascribed to Borda.

J. E. T. Lavern£de57 gave an extensive treatment of such polynomials, chiefly of degrees 3 and 4, and noted the examples

m, n= (xd=2) (xdb4) (x±10) (x=T=7) (s=F9) =x5 - 125x3+3004x±5040.

S. F. Lacroix58 quoted the preceding results and the following, attributed toHaros:

John Muller59 had made only the following contribution to our subject: log (d+l)2=log d+log

log (d4-3)2=log (d+l)2+log (d+4) -log d-log q, g

The latter is applied when d= 14 to find log 17, knowing log 15, log 18 and log 14. Then q = 2025/2023. Taking a = 2024, z=l, we have g=(a+x)/ (a— x), a series for the logarithm of which is found by subtracting the

w J. C. de Borda's Tables trigonom£triques d£cimales ou Tables des logarithmes . . . revues, augmente'es et publi4es par Delambre, Paris, an IX (1800-1). Introduction.

" Notice des travaux de 1'Acad. du Card, 1807, 179-192; Annales de Math, (ed., Gergonne), 1, 1810-11, 18-51, 78-100. See AUman.60

" Trait6 du Calcul Diff. . . . Int., ed. 2, 1, 1810, 49-52.

" Traite" analytique des sections coniques, fluxions et fluentes . . ., Paris, 1760, 112. This topic does not occur in the earlier English edition, A Math. Treatise: containing a System of Conic Sections; with the Doctrine of Fluxions and Fluents . . ., London, 1736.