376 MR. IVORY ON THE THEORY OF THE ELLIPTIC TRANSCENDENTS. 
A 4 sin 4 X 2 
_ sin 2 a 2 
i 0 sin 2 Af 
1£L 
sin 2 A 2 ’ 
Now, observing that 
A 4 sin 4 X 2 , 
(sin 2 
3 - 3 
sufXo ^ sin 2 A t ? we °btain by equating the values of 
, 1 ~ / 3 2 \ 2 _ 2(1 -< 3 ) 
‘ 2 / sin 4 Aj "l" sin 2 a l ' 
from which we deduce 
f3 = sl 
In order to simplify the formulas I shall put = 1 + £ : then (3=1+2 £, 
1 + 2 e 
and sin 2 = |8 sin 2 Tq = ^ + e - j a : and, having substituted these values in the 
first of the foregoing equations, we shall get, 
£ 4 + 2 £ 3 — 2 A' 2 £ — • A' 2 = 0. 
This equation may be resolved by the usual method into the two following 
quadratic factors, 
g 3 = 4 A 2 A' 2 , 
= 2 + (1 + e - i */? 2 + 4/<’ 2 - i f = 0 , 
£ 2 + (1 - + kJFTiTJi- i f = 0 . 
The second of these equations has two impossible roots : the first has two real 
roots, one being negative and foreign to the question, and the other positive, 
which solves the problem. Thus £ has only one value, which may be con- 
structed geometrically, but the algebraic expression of it need not be written 
down. 
We have next to derive the amplitude + from <p. We readily obtain from 
the foregoing biquadratic equation, 
(i+.)»(i-.) 
