124 HISTORY or THE THEORY OF NUMBERS. [CHAP, v P. Mansion42 proved (6), showing that <£(w, k) equals <£(w) or 0, according as m is or is not a divisor of k. [Cf . Bachmann, Niedere Zahlentheorie, I, 1902, 97-8.] He repeated his41 "generalization." He stated that if a and 6 are relatively prime, the products of the <£(a) numbers <a and prime to a by the numbers <b and prime to b give the numbers <db and prime to ab [false for a = 4, 5=3; cf. Mansion44]. His proof of (4) should have been credited to Catalan.11 E. 'Catalan43 gave a condensation and slight modification of Mansion's42 paper. C. Le Paige (ibid., pp. 176-8) proved Mansion's44 theorem that every product equals a determinant formed from the factors. P. Mansion44 proved that the determinant |c#| of order n equals rc^. . .xn if Cij =S#P, where p ranges over the divisors of the g. c. d. of i} j. To obtain a "generalization" of Smith's theorem, set Zi=xif Z2=Xi+x2, . . ., Zi=2xd, where d ranges over all the divisors of i. Solving, we get a^=*m~V+~2/3--. • •> where the Ts are defined above.39 Thus each c# is a z. For example, if n = 4, For Zi= i, Xi becomes <f>(i) and we get (6) . [As explained in connection with Mansion's41 first paper, the generalization is due to Smith.] J. J. Sylvester45 called <£(n) the totient r(ri) of n, and defined the totitives of n to be the integers <n and prime to n. F. de Rocquigny46 stated that, if <f>2(N) denotes <fr\<l>(N)\, etc., <P(N») = <j>v-2(Nm-2) •4>p-1 1 (N - 1)2 } , if N is a prime and m>2, p>2. He stated incorrectly (ibid., 50, 1879, 604) that the number of integers ^ P which are prime to N = a*W . . . is P(l — I/a) A. Minine47 noted that the last result is correct for the case in which P is divisible by each prime factor a, 6, ... of N. He wrote symbolically nE— for [n/x], the greatest integer ^n/x. By deleting from 1, . . . , P the x [P/a] numbers divisible by a, then the multiples of b} etc., we obtain for the number of integers ^P which are prime to N the expression [equivalent to (5)]. If 2V, N', N", ... are relatively prime by twos, «Annales de la Soc. Sc., Bruxelles, 2, II, 1877-8, 211-224. Reprinted in Mansion's Sur la the"orie des nombrea, Gand, 1878, §3, pp. 3-16. "Nouv. Corresp. Math., 4, 1878, 103-112. "Bull. Acad. R. Sc. de Belgique, (2), 46, 1878, 892-9. «Amer. Jour. Math., 2, 1879, 361, 378; Coll. Papers, 3, 321, 337. Nature, 37, 1888, 152-3. ^Les Mondes, Revue Hebdom. des Sciences, 48, 1879, 327. "Ibid., 51, 1880, 333. Math. Soc. of Moscow, 1880. Jour, de math. Stem, et spe"c., 1880, 278.