AND DOUBLE THETA-FUNCTIONS. 
Comparison with Jacobi. 
46. The comparison of the formulae with Jacobi gives 
snKn=- -j= Y)u — Cu, 
V ro 
/a fcl—x ( i /—a / x—d\ 
= VgVv I r better Vi V c-d 
cnKw= /y/|-BM4-C«, 
= V- a/—’ 
V g V c—x 
dn K u = y/T' A u -a C u, 
/k /a—x 
" Vf V 
where it will be recollected that 
k 2 = 
k'*= 
ch 
— af 
It may be remarked that we seek to determine everything in terms of a, h, c, d. 
The formula just written down, k 2 = bg-f-af, gives k in terms of these quantities ; 
and k, K being each given in terms of q, we have virtually K as a function of k, that 
is of a, b, c, d : but it would not be easy from the expressions of k, K each in terms 
of q, to deduce the actual expression K 
d<p 
\/'l —Id siu- <p’ 
of K as a function of k. 
The square-set, u±u\ 
47. He verting to the square-set 
A(u+u)A(u-u')= XX' + YY', 
B(u+u')B(u-u') = yx'+xy; 
C (u + u) C (u-u')= XX'-YY', 
D(u+u')V(io-id)=-YX'+XY', 
if we first write herein u — 0, and then u— 0, using the results to determine the values 
of X, Y, X', Y' we find 
«C ~u - /3D z u = («~ - /3~) X, aChi - V = (« 3 - j3-)X', 
/30 2 u —aD%= „ Y, (3C~u — olD : u'= ,, Y, 
and thence 
6 c 
MDCCCLXXX. 
