CHAP. XDC] NtJMEBICAL INTEGRALS AND DEEIVATIVBS. 451 Bougaief60a noted that, for an arbitrary function ^, i = S 2 MO, S Z iMeO=n S S » J t*=»l n»l tt-1 n-1 u-ln-1 N. V. Bervi61 treated numerical integrals extended over solutions of indeterminate equations, in particular for n=a+b(x+y)+cxy, b2=b+ac. Bougaief62 considered definite numerical integrals, viz., sums over all divisors, between a and &, of n. He expressed sums of [x], the greatest integer ^x, as sums of values of f (ti, m), viz., the number of divisors ^*n of m. Also sums of f's expressed as ?»(l)+&(2) + . . .+ft(n), where f i(ri) is the number of the divisors of n which are ith powers. 1. 1. Cistiakov62a (Tschistiakow) treated the second numerical derivative. Bougaief626 gave 13 general formulas on numerical integrals. Bougaief63 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,64 using distinct primes a, 6, . . . , called the derivative of N=aabP.... Similar definitions are given for derivatives of fractions and for the case of fractional exponents a, 0,___ The primes are the only integers whose derivatives are unity. 6WComptes Rendus Paris, 120, 1895, 432-4. "Mat. Sbornik (Math. Soc. Moscow), 18, 1896, 519; 19, 1897,182. M/6id., 18, 1896, 1-54 (Russian); see Jahrb. Fortschritte Math., 27, 1896, p. 158. ***Il)id., 20, 1899, 595; see Fortschritte, 1899,194. **blbid., 549-595. Two of the formulas are given in Fortschritte, 1899, 194. w/6id., 21, 1900, 335, 499; see Fortschritte, 31, 1900, 197. MABOciaci<5n espafiola, Granada, 1911, 1-12.