808 
MR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
20. Let 
Then 
And 
and 
so that 
K 
7 T% 
(VA + Ba? 
A 
TTI 
- log r= f 3 
i s J i -v/X 
, r k?a'+ B'x 
• lo g 2=J j 
(Ac 
IA a/X 
dx 
r 1 dx „ 
J 0 7x _k<)i 
[ l xHx „ 
Jo7x- K2! 
C 1 a?efcc T7 , 
J„7x =Ku 
f 1 fiAfe T;r 
J 0 7x = K31 
J 
dKpi _ 
C^/q 1 cZ/q 
- 1 dK-Q cfKcjj 
/q <i/q 1 d /q 
1 ^K 31 dK gl ^ 
/q~ri&q “ Ki ^ 7 _ 
li 
21 
31 
dK n . -T7- _f 1 _ dx _ 
Kl <Z/q ^ 01 “J ov / X(l-/qV) 
Pi 
A. a/x __ 
fte 1— /q 3 $ /y/X.(l —/c/a?) 
R t + Sps -f- T r ® 2 -T TJfr 2 
-s/X 
dIC 
1 c£/q 
■K 01 =PK 31 +QK sl +RK n +SK 01 
(G2). 
(03) 
P, Q, B, S being determinate functions of /q, k 2 , k 3 . By eliminating K n , K ai , K 31 
from these equations it is obvious that K 01 satisfies a differential equation of the 
fourth order in «q as the independent variable. K u will satisfy a similar equation ; 
and hence also K, equal to 
ak 01 +bk 113 
(A, B being functions of /q, /q, k 3 ,) will satisfy a linear differential equation in /q of 
the fourth order. 
