156 
ON THE DOUBLE ^-FUNCTIONS. 
[661 
venient form, we assume i2S' 2 il — (SH) 2 = Gl 2 M (St^) 2 , where M is a function of x. We 
thus obtain an equation A8 2 A — (SA) 2 = il 2 2l (8u) 2 , where the value of 21 depends upon 
that of M. The value of M has to be taken so as to simplify as much as may be 
the expression of 21, but so that M shall be a symmetrical function of the constants 
a, b, c, d: this last condition is assigned in order that the like simplification may 
present itself in the analogous relations B8 2 B — (8B) 2 = il 2 23 (8u) 2 , &c. The proper 
expression of M is found to be 
M = — 2x 2 + x(a + b + c + d) + a 2 + b 2 + c 2 + d 2 — 2 be — 2 ca — 2 ab — 2 ad — 2bd — 2 cd, 
viz. M having this value, the one other equation above referred to is 
i2S 2 H - (bny = &M(Suy; 
and we then have a system of four equations such as 
¿8 2 A-(SJ.) 2 = n 2 2l(&0 2 , 
where 21 is a linear function of x, and where consequently il 2 2l can be expressed as 
a linear function of any two of the four squares A 2 , B 2 , C 2 , D 2 . 
To establish the connexion with the Jacobian H and ® functions, the differential 
relation between u, x may be taken to be * 
a Sx 
OU = ; 
V x . 1 — x . 1 — k 2 x 
viz. we have here d = oo, and to adapt the formulse to this value it is necessary to 
write instead of u, and make other easy changes. The result is that il differs 
from D by a constant factor only, so that, instead of the five functions A, B, G, D, f2, 
we have only the four functions A, B, C, D. The equation ilS 2 n — (Si!) 2 = D?M(hu) 2 
is replaced by an equation D8 2 D — (SD) 2 = D 2 2) (8u) 2 , or say 8 2 (log D) = 2) (8u) 2 , which 
gives a result of the form 
j) = e w +*i Sw fa£ 
showing that D differs from Jacobi’s 0 (u) only by an exponential factor of the form 
Ce Xu \ And it then further appears that A, B, G differ only by factors of the like 
form from the three numerator functions H (u), H (u + K), © (u + K), so that, neglecting 
constant factors, the functions 
A B G 
D ’ D ’ D equal 
to 
H (u) 
H (u + K) 
0(u) 5 
0(w+ K) 
that is, to the elliptic functions sn u, cn u, dn u.