494 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
where n is written to denote n +1, and n has all positive or negative even values 
(zero included) from — oo to 4- oo , or more accurately from — v to v, if v is ultimately 
infinite. 
61. The sines and cosines are converted into infinite products by the ordinary 
formulae, which neglecting constant factors may conveniently be written 
sin ^7ru = II (u 4- m), cos \nru = II (u + m), 
where m is written to denote in + 1, and m has all positive or negative even values 
(zero included) from — oo to + oo , or more accurately from — /u to /x, if /i be ultimately 
infinite. But in applying these formulae to the case of a function such as 
sin ^7r (u + na), 
it is assumed that the limiting values g, — g of in are indefinitely large in regard to 
u + na.\ and therefore, since n may approach to its limiting value 4 v, it is necessary 
that /x should be indefinitely large in comparison with v, or that - = 0. 
t" 
62. It is on account of this unsymmetry as to the limits - = 0, - = oo, that 
fM v 
Ave have 1 as a quarter-period, —. only as a quarter-quasi-period of the single theta- 
functions. 
The transformation q to r, log q log r = ir 1 2 . 
63. In general, if we consider the ratio of two such infinite products, where for 
the first the limits are (4 /x, ± v), and for the second they are (± f, ± v), and 
where for convenience we take /x > g, v> v, then the quotient, say [/x, v] -r [g, v'] is 
= exp. (Mu 2 ), Avhere 
taken over the area included between the two rectangles. We have 
(m, 11) = m-f —-. 11, = m + idn 
m 
suppose, where (a being negative) 6 = — - is positive: the integral is 
dm dn f 7 1 
;> = dm • — 72 
J J (m + idn) 2 ’ J ’id \m + idn) 
=rei dm 
1 1 
m — idv m + idvj ’ 
1 . m — idv 
id m + idv ’