Chap. XIX] NUMERICAL INTEGRALS AND DERIVATIVES. 451 
Bougaief^"" noted that, for an arbitrary function \p, 
<* J M=l n = l M=l n=l u = ln=l 
N. V. Bervi" treated numerical integrals extended over solutions of 
indeterminate equations, in particular for n = a-\-h{x-{-y)-\-cxy, h^ = b+ac. 
Bougaief^" considered definite numerical integrals, viz., sums over all 
divisors, between a and b, oi n. He expressed sums of [x], the greatest 
integer ^x, as sums of values of f (n, m), viz., the number of divisors ^n 
of m. Also sums of ^s expressed as ^i{l)+^i{2)-\- . . .-{-^i{n), where 
fi(n) is the number of the divisors of n which are ith. powers. 
1. 1. Cistiakov^^" (Tschistiakow) treated the second numerical derivative. 
Bougaief^^^ gave 13 general formulas on numerical integrals. 
Bougaief^^ gave a method of transforming a sum taken over 1, 2, . . ., n 
into a sum taken over all the divisors of n. He obtains various identities 
between functions. 
D. J. M. Shelly, ^^ using distinct primes a,h,. . ., called 
^'=K^^•••) 
the derivative of iV = a^b^ .... Similar definitions are given for derivatives 
of fractions and for the case of fractional exponents a, j8, . . . . The primes 
are the only integers whose derivatives are unity. 
ec^Comptes Rendus Paris, 120, 1895, 432-4. 
«iMat. Sbornik (Math. Soc. Moscow), 18, 1896, 519; 19, 1897, 182. 
^mid., 18, 1896, 1-54 (Russian); see Jahrb. Fortschritte Math., 27, 1896, p. 158. 
«2o/6id., 20, 1899, 595; see Fortschritte, 1899, 194. 
^^^Ibid., 549-595. Two of the formulas are given in Fortschritte, 1899, 194. 
«/6id., 21, 1900, 335, 499; see Fortschritte, 31, 1900, 197. 
"Asociaci6n espanola, Granada, 1911, 1-12. 
