A RESULT IN THE THEORY OF WEIERSTRASS ELLIPTIC FUNCTIONS By ANDREW D. McGETTRICK [Received 13 November 1970—Revised 25 August 1971] 1. In his paper 'On Kummer sums' (see [1]) Professor J. W. S. Cassels enunciated a conjecture which suggests that there is a close relationship between a certain product of values of a Weierstrass p-function and the Kummer sum. In verifying this conjecture he made use of a result which yields a value for the argument of this product. The main purpose of this paper is to show how this result can be proved. The work contained here was carried out at Cambridge under the guidance of Professor Cassels (see [4]). I would like to take this oppor- tunity of expressing my thanks for his constant help and encouragement. My thanks are due also to Dr B. J. Birch and Dr J. Hunter for their helpful comments on the style of this paper. 2. We begin by introducing some notation. Let p be a rational prime such that p = 1 (mod 3) and let o> be a primitive cube root of unity. Then it is well known (Eisenstein ([2], p. 4)) that there exists -m e Z[o»] with the following properties: (i) m = a + bo), a, b e Z and a = - 1 (mod 3), 6 = 0 (mod 3), (ii) p = a 2 -ab+b*. It is worth noting that m is not unique; for the complex conjugate of -m would also satisfy these conditions. Further, let p denote the Weierstrass p-function with periods VJ,(DVT and let p 0 be the corresponding function with periods \,io. In this paper we start by denning logp in such a way that it too is periodic with the same periods as p. The main theorem, namely, Theorem 3.1, then yields a value for d where Im(z) denotes the imaginary part of z e C and the sum is taken over a set of p — 1 non-zero -nr-division points of p. 3. Before giving our definition of \ogp we construct a fundamental region H for p. Proc. London Math. Sac. (3) 25 (1972) 41-54