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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    nl                 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.