CHAP. V] FAEEY SERIES. 155 K be the area of the region bounded by this curve, and N the number of points (x, y) within it or on its boundary such that # is a multiple of k and is prime to y. Then ,. „ where K increases by uniform stretching of the figure from the origin. In particular, consider the number N of irreducible fractions x/y^l whose denominators are ^n. Since x^y, the area K of the triangular region is n2/2. Hence N— (n2/2)(6/7r2), approximately (Sylvester55). Again, the number of irreducible fractions whose numerators lie between I and Z+ra, and denominators between I' and Z'+m', is 6ram'/7r2, approximately. There is a similar theorem in which the points are such that y is divisible by k', while three new constants obey conditions of relative primality to each other or to x} y, k, k'. Extensions are stated for w-dimensionai space. E. Cahen219 called /&(n) the indicateur of kth order of n. G. A. Miller220 evaluated Jk(m) by noting that it is the number of operators of period m in the abelian group with k independent generators of period m. G. A. Miller221 proved (10) and (11) by using the same abelian group. E. Busche222 indicated a proof of (10) and (12) by an extension to space of k+l dimensions of Kronecker's223 plane, in which every point whose rectangular coordinates x} y are integers is associated with the g. c. d. of x, y. A. P. Minin224 proved (14) and some results due to Gegenbauer.204 R. D. Carmichael225 gave a simple proof of Zsigmondy's216 formula for $. G. Me*trod226 stated that the number of incongruent sets of solutions of xy'— x'y^a (mod m) is 2dmJ2(m/d), where d ranges over the common divisors of m and a. When a takes its m values, the total number of sets of solutions is It is asked if like relations hold for Jk, /c>2. Cordone91 and Sanderson115 (of Ch. VIII) used Jordan's function in giving a generalization of Fermat's theorem to a double modulus. FAREY SERIES. Flitcon248 gave the number of irreducible fractions <1 with each denominator <100, stating in effect the value of Euler's <j>(n) when n is a product of four or fewer primes. 219Th6orie des nombres, 1900, p. 36; I, 1914, 396-400. 220Amcr. Math. Monthly, 11, 1904, 129-130. 221Amer. Jour. Math., 27, 1905, 321-2. 222Math. Annalen, 60, 1905, 292.