1906-7.] Periods of the Elliptic Functions of Weierstrass. 359 
The relation of n x and w x to q x is the same as that of n 3 and w 3 to v 3 . 
If the roots e 1} e 2 , and e 3 are given, q ± has to be computed by known 
formulae, and then w 1 and n x by the above relations. 
IV. 
When the roots are not given explicitly, it is either necessary to find 
them by solving the cubic or to use the invariants. To make use of the 
invariants, put 
cos q = Jyfgjgl , 0° < q < 180° 
(N.B . — All radicals are to take the positive sign ; thus cos t is of the same sign as g s ), 
and to abbreviate put 
then 
.. 1 1 1 ) 
— i#2 5 
12 Af „q , 
1 COS^-l- { 
o 
j ,F ( - ¥ ■ ¥ ■ 2 • “’I ) + 5 ''K ¥ ■ ¥ • 2 • | • 
'-i A! 4 7r [l».W-¥'¥- 2 ’""^) + 5r (¥’f 2 ’“T)[' 
*»-n r, ^{ 7F (4’¥- 2 '~^)- 5 “‘- f (¥’¥- s -“y)}- 
n, = ± At -*_<! -7cos. 1 F(-l > l,2 > cos»i) + 5F(l J A ) 2 > cos>i.')l 
si„2i| V 6 6 2 J V6 6 2j 
9 
36 
As the hypergeometrical series concerned are always convergent, these 
four formulae constitute a formal solution of the problem. 
y. 
Although the two series converge for all values of the argument, the 
convergence is slow when the variable sin 2 -h- or cos 2 -h- exceeds J. It is 
A A 
therefore convenient to compute w 1 and n x when q is under 90°, and w 3 
and n 3 when q exceeds 90°. If we suppose q less than 90° and we require 
w 3 and n 3 , we compute w 1 and n ± and also v 1 . The latter quantity is 
found as follows : — 
