1906-7.] Periods of the Elliptic Functions of Weierstrass. 357 
XXXV. — On the Periods of the Elliptic Functions of Weierstrass. 
By R. T. A. Innes, Director of the Government Observatory, 
Johannesburg, F.R.A.S. 
(MS. received June 19, 1907. Read July 15, 1907.) 
I. 
The elliptic functions introduced by Professor K. Weierstrass of Berlin are 
best known through the admirable pages of Professor H. A. Schwarz’s 
Formeln und Lehrsatze zum Gebrauche der elliptischen Functionen, 
Gottingen, 1885. These pages have been translated literally into French 
by Dr Henri Padb, Paris, 1894. Halphen’s well-known three-volumed 
treatise on elliptic functions is devoted almost entirely to the Weierstrassian 
forms. In the English language, the following references may be useful : — 
Whittaker, Modern Analysis ; Greenhill, Elliptic Functions ; Harkness and 
Morley, Theory of Functions] Forsyth, Theory of Functions. 
II. 
The Weierstrassian forms are intimately connected with a cubic equation 
which may be written 
4(s-ei)(s-e 2 )(s-e 3 ), 
or 
4=s*-g 2 s-g 3 , 
yielding the well-known relations between the roots and the invariants 
e i + e 2 + e 3 = 0 > e i e 2 + e i e 3 + e 2 e 3 = ~ Iff 2’ e i e 2 e 3 = iff 3 > 
with the discriminant 
G = f w (gi-27gi). 
As the object of this paper is mainly a numerical one, we limit our 
attention to real values of the roots ; e x is the largest positive root, e s the 
largest negative root, and e 2 is intermediate in value. 
III. 
The problem usually presented in physical applications is : — Given the 
roots or the invariants of the cubic, to find the periods of the elliptic 
functions. 
If the roots are given, the formulae required have been given with all 
necessary fulness by Schwarz and Halphen. If, instead of the roots, the 
invariants are given, it would seem, at first glance, that the cubic must be 
solved before the periods can be found, but it has been shown that this is 
