832 
MR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
• _ r' (ApK-A^V) x (A^KWV) x (A 3 .K f A 
' 3 ~7r 3 (A^EM 4 ) x (A x .KA ! 7 } ) x (A x .K*AV^ 
. _ r' (A,.ICA¥) x (A v K 4 AV 4 ) x(A^K^cVW 1 ) 
' 3— 7T 3 (A lt K*A i ) x (Aj.I^AV) x (A^KM 4 /*) 
_ r' (A^KAV^) X ( A 1 .K i AVV i ) x (A 3 .K l A i c*c' i 7V 4 ) 
1— 7T 3 ( A V K 4 A 4 ) x (Api^AV) x (Aj.KAV 7 ) 
(198) 
(199) 
( 200 ) . 
Section III. 
The Addition Theorem . 
38. There is no addition theorem proper for the theta-functions, but the product of 
a theta-function of the sum of two variables by a theta-function of the difference of 
those variables can be expressed in terms of the functions of those variables. Thus 
with the previous notation for the single theta-functions 
0oa(°)0&i( u ‘+ v ) 0 oat u v ) — d o, i( u )K ii v ) ~ i{ u ) ^ i , i( v )> • • • • 
00,1 2 (°)0], i( U + V )01, i(“—®)—01, i a (“) 00, i a (») - 01, i*fp) 00. i a (“). 
0o,o(°)0o,i( o )0i,i(“-t-i')0o,i(“-' y )=0o,i(“)0i,i(“)0i,o( w )0o,o( 1 ’) 
"l" 00 , l( v ) 01 , l( v ) 0i,o(»)0o,o( M ) • 
0lA5)_ k and 
0o,o s (O) _ ’ 0o,o s (O) _ 
( 201 ), 
( 202 ), 
(203). 
Dividing the third of these by the first and substituting for the O' s, there results 
the ordinary expression for sn(u-\-v) ; and the division of the second by the first gives 
sn(u-\-r)sn{u — v). The object of the present section is to obtain, by means of the 
theorem (23), the complete expression of the sum-and- difference theorem for the 
double theta-functions ; it will be given by 256 formulae similar to (201), (202), (203). 
39. Some abbreviations in the notation are desirable ; in the subsequent formulae 
© denotes r9-(*+f, y-\-r]), 
©' „ &(x—£,y—y)), 
3- ,, 3-(x, y ), 
0 >, <$(£ y), 
and, in order to simplify the first forms which are obtained from (23), some subsidiary 
equations are necessary. For example, writing down equations similar to (i)-(x) in 
Section I., but involving 0 2 3 2 instead of c 2 3- z , the following simpler relations are found 
to be their equivalent and include (31), (32), (33) as particular cases. 
