AND DOUBLE THETA-FUNCTIONS. 
933 
limits as before, and in particular -= co ; it is at once seen that if in the original 
cun 
functions, which I now call A (u, q), B (u, q), C (u, q), D(w, q), we write —. for u, we 
Til 
obtain the same infinite products which present themselves in the expressions of the 
new functions A (u, r), &c., only with a different condition as to the limits ; we have 
for instance 
nn 
au 
7 ri 
a 
m + n — 
7 n 
=nn/14- 
a 
n — m — ; 
7 Tl 
-A=nn/ H 
which, interchanging m, n, and of course also /x, v, is 
1 
a 
m + n—., 
7 Tl' 
with the condition -=0 instead of - = co. Hence disregarding for the moment 
constant factors, and observing that a is replaced by a, we have 
D (u, 7’)-r- qj=[fi-i-v, = co']-i-[jfl-T-V, = 0] 
= ex P =ex P lo g c i)- 
G5. We have equations of this form for the four functions, but with a proper 
constant multiplier in each equation : the equations, in fact, are 
A (u, r)={ A(0, r)-rA(0, q)} exp 
log q) a/^, g) 
B(w, r) = {B(0, ?’)-t-B(0, q)} 
B (S' 4 
C(«, r)={ C(0, r) -4- C(0, q)} 
D(« > r)={D , (0,r)-rD / (0,?)}^ „ 
It is to be observed that r is the same function of k' that q is of k : this would at 
ttK' 
once follow from Jacobi’s equation log q= ——, for then log q log and therefore 
loo 1 r 
7rlv r 
K 
— f only we are not at liberty to use the relation in question log q- 
7tK / 
K 
and assuming it to be true we have 
