PARTICULARLY THOSE OF TWO VARIABLES. 
791 
*{C; £) a+sK +^ io gff+^ io g r > ?+*i A +~ io g r +^ iQ gg 
$ {0>')* +sK+ ^ lo g^+^ lo g r > y+*'i A +^log»-+^]og S } 
/A H-s, p f 
\p -| -t, V +2 ) 
ky} 
*{ 
-s', p'-^s'i) 
\p'+t, v'+z j 
K y\ 
• (14) 
where, in. the last formula, s, s 1} s', s\, t, z are integers, and the functions on the right 
hand side may be reduced by formulae (2), (3) to functions in which the numbers of the 
characteristics differ from \, p, p, v, A/, p, p, v\ respectively by less than 2. This 
combination of periods in quotients is similar to the combination of real and imaginary 
periods in elliptic functions. 
The product theorem . 
6. We multiply four theta-functions 
$ 
M 
j*i* v J 
<i> 
p 2 
2 > v 2 
y s 
[ As> ps\ 
1W vJ 
(r 3 , 
such that each sum of the four corresponding numbers in the characteristics is even. 
Let such a product be denoted by 
5 I 
MDCCCLXXXII. 
(15) 
