316 HISTORY OF THE THEORY OF NUMBERS. M. Lerch130 proved relations of the type The number of solutions of [n/x] = [n/(x+l)], x<n, is p<* F. Nachtikal131 gave an elementary proof of (15). M. Lerch132 proved that i— <ra, ra) }• remains unaltered it we interchange r and s. He proved (18) and showed that it also follows from the special case (17). From (17) f or n = 2 he derived L. Gegenbauer132a proved a formula which includes as special cases four of the five general formulas by Bougaief.129 When x ranges over a given set S of n positive integers, the sum S/(rc) [x(z)l is expressed as sums of expressions 3>(p) and $i(p), where p takes values depending upon x> while <&(«) is the sum of the values of f(x) for x in S and a^z, and $1(2) is the analogous sum with F. Rogel133 differentiated repeatedly the relation n (i-a-r^W*, «-l then set x = 0 and found that ,i the summations extending over all sets of a's for which Starting with the reciprocals of the members of the initial relation, he obtained similarly a second formula; subtracting it from the former result, he obtained ^ aj / y-i\ aJ /J a1!...oP_3!y-2 "°Casopis, Prag, 24, 1895, 25-34, 118-124; 25, 1896, 228-30. ™Ibid., 25, 1896, 344-6. 132Jomal de Sciencias Math, e Astr. (Teixeira), 12, 1896, 129-136. 13MMonatshefte Math. Phya., 7, 1896, 26. 133Sitzungsber. Geaell. Wiss. (Math.), Prag, 1897, No. 7, 9 pp.