AND DOUBLE THETA-FUNCTIONS. 
931 
A u— (l -\-2q cos TTU-\-<f)(l + 2 q 3 cos 7 tu + 2 °) • • , 
Blt = COS 2(f COS TTU-\-q*)(l -f-2 ^ 4 COS 7T?^ + 'Z 8 ) • • J 
C« = (1—2(7 C0S TTUfqfjf — 2^ COS TTW + ( 7 6 ), 
Dm= sin ^7 tw( 1 — 2g 3 cos , n^+g ,4 )(l — 2q 4 cos 7^^+< 7 8 ), 
ct 
and writing for a moment a= — and therefore q^-\-q~ }s —e in!ll -\-e~ iv,t \= 2 cos lira, &c., 
° 7TI 
each of these expressions is readily converted into a singly infinite product of sines 
or cosines 
Aw=11. cos \tt(u-\-iicl), 
B(( = n. cos \Tr(u-\-na), 
Cw=II. sin \Tr(ii-\-na), 
Dw=II. sin \Tr(ii-\-ncL) , 
where n is written to denote n-j- 1, and n has all positive or negative even values 
(zero included) from — oo to + 00 , or more accurately from — v to v, if v is ultimately 
infinite. 
6L. The sines and cosines are converted into infinite products by the ordinary 
formulae, which neglecting constant factors may conveniently be written 
sin ^7rM=n(w-f-m), cos \Tru=H[u-\-m), 
where m is written to denote m+1, and m has all positive or negative even values 
(zero included) from — go to + co , or more accurately from — y to y, if y be ultimately 
infinite. But in applying these formulae to the case of a function such as sin Ipr(w+wa), 
it is assumed that the limiting values y, — y of m are indefinitely large in regard to 
M-j-na ; and therefore, since n may approach to its limiting value Tv, it is necessary 
that y should be indefinitely large in comparison with v, or that =0. 
y 
62. It is on account of this unsymmetry as to the limits - = 0,- = co , that we 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— tT. 
63. In general, if we consider the ratio of two such infinite products where for the 
first the limits are (iy, Tv), and for the second they are (Try", ±k), and where for 
convenience we take y>y', v>v, then the quotient, say [y, y]-i-[~y\ is = exp. (Mm 2 ) 
where 
6 r> 2 
