496 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
which, interchanging m, n, and of course also /x, v, is 
with the condition - = 0 instead of - = oc. Hence disregarding for the moment 
v v 
constant factors, and observing that a is replaced by a', we have 
D (u, r) 4- D , q\ = [fj,~v, = oo ] 4- |> tv, = 0] 
= exp. (i % it 2 ) , = exp. {\u 2 log q). 
65. We have equations of this form for the four functions, but with a proper 
constant multiplier in each equation: the equations, in fact, are 
m 
A (u, r) = {A (0, r) -r- A (0, q)} exp. (\u 2 log q) A , q), 
B (u, r) = {B (0, r) -r B (0, 5)} 
G (u, r) = {C (0, r) + C (0, q)} 
D (u, r) = [D (0, r) -r• U (0, §)} ^ 
m 
au 
iri 
au 
7TÎ 
au 
It is to be observed that r is the same function of k' that q is of k. This 
ttK' 
would at once follow from Jacobi’s equation log q = —, for then log q tog r = nf- 
ancl 
7tK' 
therefore logr = — j r (only we are not at liberty to use the relation in question 
lo£ 
7TK 
g q = —j£~) * assum i n g if f° b e true, we have 
, £ 2 (0, q) V _C*(0, q) A (0, q) D' (0, q) 
A*(0,qy K A 2 (0, q) ’ B (0, q) G (0, q) ’ 
7 C 2 (0, r) 7/ B 2 (0, r) A (0, r) jy (0, r) 
"“^*(0, ry "¿ 2 (0, r)’ ~ 5(0, r) G(0, r) ’ 
. 7rK' . 7tK 
logq=-~K-’ lo § r = --j?> 
where, if the identity of the two values of k or of the two values of k' were proved 
independently (as might doubtless be done), the required theorem, viz. that r is the 
same function of k' that q is of k, would follow conversely: and thence the other 
equations of the system.