714 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxiv 



AN EQUIVALENT PROBLEM IN THE THEORY OF LOGARITHMS. 



The system of equations Za* = Z&; (k = l, , ri) which we have been 

 considering is equivalent to the system Sai = Z&i, Zaia 2 = 2&i& 2 , , 

 2ai2 n = 2&i& 2 - - b n . Consider the equation having the roots i, a 2 , 

 and that having the roots 61, 6 2 , 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 corre- 

 sponding 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 



take, for example, m = x*, n=(x l)(x+l). Then log (:c+l) differs from 

 2\ogx log (x 1) by a series in k = l/(2x 2 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. Delambre 56 took m = x 3 '+px+q, n = x 3 -\-px q, and assumed 

 that m = has the roots a, b, a b, and n = the roots a, b, a-\-b, 

 whence p= a? ab 6 2 , q = a?b+ab 2 . For a = b = l, we have the formulas 

 m, n=x* 3zd=2, ascribed to Borda. 



J. E. T. Lavernede 57 gave an extensive treatment of such polynomials, 

 chiefly of degrees 3 and 4, and noted the examples 



m, n= 



S. F. Lacroix 58 quoted the preceding results and the following, attributed 

 to Haros: 



John Muller 59 had made only the following contribution to our subject: 

 log (d+l) 2 = log d+log 



log (d+3) 2 = log (<Z+l) 2 +log (d+^-logd-logq, 



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, x = l, we have q=(a+x)l 

 (a x), a series for the logarithm of which is found by subtracting the 



66 J. C. de Borda's Tables trigonometriques d^cimales ou Tables des logarithmes . . . revues, 



augment6es et publi^es par Delambre, Paris, an IX (1SOO-1). Introduction. 



67 Notice des travaux de 1'Acad. du Card, 1807, 179-192; Annales de Math, (ed., Gergonne), 



1, 1810-11, 18-51, 78-100. See Allman. 60 



68 Trait6 du Calcul Diff. . . . Int., ed. 2, I, 1810, 49-52. 



69 Trait6 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. 



