AND DOUBLE THETA-FUNCTIONS. 
995 
Differential relations of the tlieta-functions. 
137. In “The second set of 16,” selecting the eight equations which contain Y x and 
W l5 these are 
u+u 1 
u—u 
U + U 
u—u 
(Suffixes 1 .) 
3 
. 3 
3 
. 3 
Y W 
,- A -> 
U 4 
0 
— 
0 
4} 
Y' + W', 
12 
8 
— 
8 
12 
— 
Y' -W', 
6 
2 
— 
2 
6 
— 
W' + Y', 
14 
10 
— 
10 
14 
= 
W' - Y', 
11 5 
1 
+ 
1 
5 } 
x' + z\ 
13 
9 
+ 
9 
13 
— 
X' -Z', 
7 
3 
+ 
3 
7 
— 
Z' +x # , 
15 
11 
+ 
11 
15 
— 
Z' -X', 
and then, considering any line in the upper half and any two lines in the lower half, we 
can from the three equations eliminate Y x and W x , thus obtaining an equation such as 
S3 
S3 
o 
1 
S= 
S3 
Y', 
w 
=o, 
*5 ^1 + ^5* 
X' 
Z' 
di3^o + ^9'^13> 
X',- 
Z' 
2X'Z' 
-V.) 
+ ( X'W'+Y'Z')^ ^i+Vs) 
+ (—• X' W' -f Y'Z') (S 13 S 9 + 3 g 3 ls ) = 0, 
where the arguments of the theta-functions are as above, u-j-u', u — u', u-\-u', u — u'; 
and the suffixes of the X', Y', Z', W' are all =1. 
138. Suppose in this equation u! becomes indefinitely small; if u were =0 the 
values of X', Y', Z!, W' would be a, 0, y, 0: and hence u being indefinitely small we 
take them to be a, 6/3, y, 68, where 
=( u '£ +v '£) Y ’ anJ kS ’ = ( u £u +v '£) w ’ («= ! ’= o )- 
are in fact linear functions of u and v. 
6 M 2 
