912 
PROFESSOR A. CAYLEY ON THE SINGLE 
exp. ^( —a+/3i), viz. : this is =e - * a {cos ^/3+i sin ^/3}. But usually /3=0, viz., q is a 
real positive quantity less than 1, and cf denotes the real fourth root of q. 
I have given above the three notations, but as already mentioned propose to employ 
for the four functions the notation A u, B u, Cu, D u : it will be observed that E )u is an 
odd function, but that A u, B u, C u are even functions of u. 
29. We have 
The constants of the theory. 
A0= 1 + 2(/ + 2t2 4 +2q y + . . 
B0 = 2^+2^+ 2<p + . . 
.C0=1 — 2q+2f— 2</ 9 + . . 
D0 = 0 
D'0=— 7r{q l — Sq°—5q ¥— . . }. 
If, as definitions of k, k', K, we assume 
B 2 0 C*0 _AO D'O 
k ~ A 2 0’ k ~ A 2 0’ K— “BO’CO’ 
then we have 
}*■ = V2(i-U+i¥+ • ■), 
*'= = l-8 9 +32<f-9S 2 H . . , 
=Wl + ^+^+08*+ . ■ ), 
where I have added the first few terms of the expansions of these quantities. We 
have identically 
P+F 2 = 1. 
It will be convenient to write also as the definition of E, 
we have then 
p«A 
K(K-E) = W : 
E=K—— —■ 
K CO 
AO.BO.CO.D'O 
- {- A 5 0(D'0) 3 + B~0.C0.C ;/ 0}, 
and moreover 
E 1 C"0 27t 2 q— 4A+ 9j 9 — . 
'K = K*"C(r “K r ’l-2 2 +2 ? i + . 
1 
