[704 
704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
559 
tions, 
= 0; 
The 
136. Observe that, dividing the first equation by + 4 2 (it), or say by + 4 2 , the left- 
hand side is a mere constant multiple of d 2 log + 4 ; and the right-hand side depends 
only on the quotient-functions + 7 4- + 4 , + 8 -^+ 4 , + u -^+ 4 ; each side is a quadric function 
of u, v'. Equating the terms in u 2 , u'v, v 2 respectively, we have 
d 2 
du 2 
l0g +4, 
d 2 
du dv 
log + 4 , 
df 
dv 2 
log+4, 
each of them expressed as a linear function of the squares of the quotient-functions 
+ 7 -=- + 4 , + 8 -r-+ 4 , + u -r-+ 4 . The formula is thus a second-derivative formula serving for 
the expression of a double theta-function by means of three quotient-functions. 
Differential relations of the theta-functions. 
137. In “The second set of 16,” selecting the eight equations which contain Fj 
and W 1} these are 
u+u' 
U—u’ u+u 
M _ M ' (Suffixes 1.) 
it of 
+ . 
+ + 
. + Y 
w 
third 
2 { 4 
0-0 
4} = T + W', 
12 
8-8 
12 = Y' - 
W', 
— 
6 
2-2 
6 = W' + T, 
oY )> 
14 
10-10 
14 = W - 
Y', 
o-a? ), 
oV), 
H 5 
1+1 
5} = X' +Z\ 
13 
9+9 
13 = X - 
- Z', 
№ ), 
7 
3+ 3 
1 =--Z r + X', 
4 (0), 
15 
11 + 11 
15 = £' - 
-X'. 
Then, considering 
any line 
in the 
upper 
half and any two lines in the 
lower half, 
we can from the 
three equations 
eliminate Y i and 
W lt thus obtaining 
an equation 
such as 
+4 +0 
Y', W 
= 0, 
) J 
+ 5 +i + +i+ 6 , 
X', Z' 
c 2 . +'- 
+13+9 
+ +9+13, 
X\ - Z' 
8 7)» 
viz. this is 
8 4), 
- 2X'Z' 
(+ 4 +0 - +o+ 4 ) 
8 11), + ( X'W + Y'Z’) (+ 5 % + + 4 + 5 ) 
8 8), +(- X'W'+Y'Z') !3 ) = 0, 
urse where the arguments of the theta-functions are as above, u + u', u — u', u + u', u — u'; 
and the suffixes of the X', Y', Z', W are all =1.