CHAP. V] EULEK'S ^-FUNCTION. 119 G. L. Dirichlet21 added equations (4) for N=n,..., 2, 1, noting that, if s^n, <f>(s) occurs in the new left member as often as there are multiples ^ n of s. Hence The left member is proved equal to 2J^[n/s], where It is then shown that \l/(ri) — S^2/71"2 is of an order of magnitude not exceeding that of n*j where 2>5>y>l,7 being such that P. L. Tchebychef22 evaluated #(n) by showing that, if p is a prime not dividing A, the ratio of the number of integers ^pAN which are prime to A to the number which are prime to both A and p is p:p — 1. A. Guilmin23 gave Crelle's17 argument leading to <j>(Z). F. Landry24 proved (3). First, reject from 1, . . ., N the N/p multiples of p ; there remain N(l — l/p) numbers prime to p. Next, to find how many of the multiples q, 2q, . . . , N of q are prime to p, note that the coefficients 1, 2,. . ., N/q contain N/q-(l — l/p) integers prime to p by the first result, applied to the multiple N/q of p in place of N. Daniel Augusto da Silva25 considered any set S of numbers and denoted by S(a) the subset possessing the property a, by S(ab) the subset with the properties a and b simultaneously, by (a)S the subset of numbers in S not having property a; etc. Then symbolically. Hence (ba)S=(b)\(a)S\=S\l-(a)\\l-(b)\, (. . .cba)S = S\I-(a)\ \l-(b)\ \l-(c)\ . . .. A proof of the latter symbolic formula was given by F. Horta.25a With Silva, let S be the set 1, 2, . . ., n, and let A, B, . . . be the distinct prime factors of n. Let properties a, b, ... be divisibility by A, 5, .... Then there are n/A terms in S(a), n/(AB) terms in S(ab), . . ., and <£(n) terms in (. . .cba)S. Hence our symbolic formula gives "Abhand. Ak. Wiss. Berlin (Math.), 1849, 78-81; Wcrke, 2, 60-64. 22Theorie der Congruenzcn, 1889, §7; in Russian, 1849. "Nouv. Ann. Math., 10, 1851, 23. "Troisifcme m6moire sur la th<5orie des nombres, 1854, 23-24. "Proprietades geraes et resolu<jao directa das Congruencias binomias, Lisbon, 1854. Report on same by C. Alasia, Rivista di Fisica, Mat. e Sc. Nat., Pa via, 4, 1903, 13-17; reprinted in Annaes Scientificos Acad. Polyt. do Porto, Coimbra, 4, 1909, 166-192. MaAnnaes de Sciencias e Lettras, Lisbon, 1, 1857, 705.