704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
495 
or finally between the proper limits the value is 
id 
log 
¡X — idv 
fX 4- idv 
where the logarithms are 
6v 
log (fx — idv) = log Vyur + v 2 — i tan -1 —, &c., 
/ 1 
and the tan -1 denotes always an arc between the limits — + ^tt. Hence, if 
- = oo , —j = 0, the value is 
v v 
27f 1 
Tp (- 0i — 0i + + £7ri) = = — ; or K = J —. 
id 
Hence finally 
[tx + v, = 00 ] -r [fx -r v, = 0] = exp. ( l — u 2 ). 
64. We have a, =logq, negative; hence taking r such that log q log r = we 
have a/ = logr, also negative; and r, like q, is positive and less than 1. We consider 
the theta-functions which depend on r in the same manner that the original functions 
did on q, say these are 
' u I 
A (u, r) = A (0, r) nn 
1 + 
_ _ a , 
m + n —-. 
7nl 
B (u, r) = B (0, r) nn 
c (u, r) = G (0, r) nn 
II + 
_ a 
m + n —. 
7Tl) 
1 + 
_ a t 
m + n —. 
7Tl) 
D (u, r) = D'(0, r)wnn 
II + 
m + n 
7TII 
limits as before, and in particular - = go ; it is at once seen that if in the original 
functions, which I now call A (u, q), B (u, q), C (u, q), D (u, q), we write for u, we 
7Tl 
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 1 + 
au 
rri 
a 
m + n —-. 
TCI, 
= nn/1 + 
t\> — nn / 1 + 
u 
n — m 
TCI, 
a ) 
n + m—: / 
7Tl /