exactly the same as the number of the Sylow subgroups of order pm in the subgroup G1 composed of all the substitutions which omit one letter of the given transitive group G of degree n whenever a <m. This theorem follows directly from the facts that the number of the Sylow subgroups of G cannot be less than the number of the corresponding Sylow subgroups in a subgroup of G, and that all the subgroups of order pm in G which involve a subgroup of order pa contained in G1 are conjugate under G1. Each of these subgroups of order pm involves only one of the given Sylow subgroup of order pm-a, and is transformed into itself under G by a group whose order is på times the group which transforms it into itself under G1. As the order of G is also p° times the order of G1, it results that each of the Sylow subgroups of order pm in G involves one and only one Sylow subgroup of order pm in G1. In other words, the number of Sylow subgroups of order pm-a in G1 is the same as the number of the Sylow subgroups of order pm in G. From the preceding paragraph it results that if a Sylow group of order 2m is transformed into itself by only its own substitutions in the alternating and the symmetric group of degree 2o 1, it is transformed into itself by only its own substitutions in the alternating and the symmetric group of degree 24. The Sylow subgroup of order 2m in a symmetric group, or in an alternating group, is transitive only when the degree of this group is either 2a or 2a + 1*. From the structure of this group it follows therefore that the Sylow subgroups whose orders are of the form 2m are transformed into themselves only by their own substitutions in every symmetric group and in every alternating group whose degree exceeds 5. Hence the theorem: The number of the Sylow subgroups of order 2m in the symmetric group of degree n > 5, is exactly the same as the number of the Sylow subgroups of order 2m-1 in the alternating group of this degree, and these Sylow subgroups are transformed into themselves by only their own substitutions under each of these groups. From this theorem it results that the group of order 4 is the only Sylow subgroup, whose order is of the form 2m, which is transformed into itself by more than its own operators under * Amer. Jour. of Mathematics, vol. 23 (1901), p. 176. a symmetric or under an alternating group. It may also be observed that if G is an imprimitive group of transformations of Sylow subgroups of order pm, then G cannot involve an invariant subgroup whose order is of the form på, since its degree is of the form 1+kp. If the systems of imprimitivity of G are transformed according to a group involving smaller Sylow subgroups of the form pa than those contained in G, it results from the theorem proved above that G contains other systems of imprimitivity which are transformed according to Sylow subgroups whose orders of the form på are equal to the orders of the corresponding Sylow subgroups of G. Hence the theorem: If G is an imprimitive group of transformations of Sylow subgroups of order pm and involves Sylow subgroups of order pa, then G must have systems of imprimitivity which are transformed according to a group involving Sylow subgroups of order på. UNIVERSITY OF CHICAGO. THEOREMS ON FUNCTIONAL EQUATIONS. BY MR. A. R. SCHWEITZER. (Read before the American Mathematical Society, April 27, 1912.) 1. IN the BULLETIN, volume 18 (1912), page 300, we have referred to Abel, Crelle's Journal, volume 2 (1827), page 389, in relation to the equation as a correlative of the functional equation* discussed by Abel, 1. c., namely, (2') f(x)+(y) = 1{xp(y) + yo(x)}. Further special cases of the equation (1) are obtained by considering the generalizations of equation (2') by Lottner, * Cf. Cayley, Mathematical Papers, vol. IV, pp. 5-6. Crelle's Journal, volume 46 (1854), pages 367, 368, etc. For example, we obtain In equations (2), (3), (4) let 0(xy) 1(x) = (x); then the resulting equations are particular instances of the equation which we have derived in the BULLETIN, 1. c. Following the suggestions of Abel, 1. c., page 386, and Lottner, 1. c., page 368, we remark that equation (2) for 1(x) = (x) contains a trigonometric subtraction formula, and equation (3) for '(x) = (x) contains an elliptic subtraction formula. Abel, 1. c., pages 388-389, has pointed out that the relation (2') leads to an equation in ø(x) which in general is incapable of solution. The relation (2), on the contrary, gives In connection with the discussion of Lottner, 1. c., and the preceding equation (4) the functional equation of Lelieuvre* is possibly of interest. 2. Abel, in Crelle's Journal, volume 1 (1826), pages 11-15, has shown that if * Bulletin des Sciences Mathématiques, ser. (2), vol. 27 (1903), p. 31. and in the BULLETIN, 1. c., page 302, we have shown that if The following theorems have a bearing on these two remarks. THEOREM 1. If p{x, þ(y, z)} of (y, z), z} = y and ƒ {(y, z), y} Χ = {y, p(x, z)} and =z, then there exists a function x(x) such that p(x, y) = x¿ ̄1{x6(x) + x$(y)} and f(x, y) = x {x。(x) — x$(y)}. -1 In this relation we write (y, z) for z; then Combining (6) and (7), we have by Abel's theorem, Crelle's Journal, volume 1, page 13, that there exists a function X(x) such that |