316 History of the Theory of Numbers. [Chap, x i>^ 
M. Lerch"° proved relations of the type 
The number of solutions of [n/ x] = [n/ (x-\-l)], x<n, is 
2i/^(n-r, r)+2 x{n-p, p) (A:= -§ + Vn + 1/4 ). 
F. NachtikaP^^ gave an elementary proof of (15). 
M. Lerch^^' proved that 
2 \yp{vi—aa,-)-]r'4^{m—(Ta,ra)\ 
remains unaltered if we interchange r and s. He proved (18) and showed 
that it also follows from the special case (17) . From (17) f orn = 2 he derived 
L. Gegenbauer^^^" proved a formula which includes as special cases four 
of the five general formulas by Bougaief .^^^ When x ranges over a given set S 
of n positive integers, the sum2/(x)[x(a:)] is expressed as sums of expressions 
$(p) and <J>i(p), where p takes values depending upon x, while $(2) is the sum 
of the values of /(x) for x in *S and x^z, and $1(2) is the analogous sum with 
X'^Z. 
F. RogeP^^ differentiated repeatedly the relation 
|x|<l, 
00 
n(i- 
-xr''-'=e-^, 
then set 
a: = and found that 
22 - 
(-1)'' 
\<T^{2)\^ Jcr 
.(r)l 
r'=ss( 
-!)■/ 
the summations extending over all sets of a's for which 
CI1+CI2+ . . . +ar = i, ai+2a2+ . . . -\-ra.r = r. 
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 
.„(r)=."+|22{nf^+^;^-i)-(-i)'np} 
ai!. . .a,_3!;=2l j J 
""Casopis, Prag, 24, 1895, 25-34, 118-124; 25, 1896, 228-30. 
"i/Wd., 25, 1896, 344-6. 
»«Jornal de Scienciaa Math, e Astr. (TeLxeira), 12, 1896, 129-136. 
"^'Monatshefte Math. Phys., 7, 1896, 26. 
i»Sitzung8ber. Geaell. Wiss. (Math.), Prag, 1897, No. 7, 9 pp. 
