818 
MR. A. R. FORSYTH OH THE THETA-FUHCTIONS, 
/i ^ 3 K , 1-WOK Tr A 
( 1 - k2 )^+—^- k=0 
k die 
(1 -«’)J|-^Y ^f+K(M=0, 
k \dK/ ' 
so that k may be considered known, and likewise k, E, given by 
k 3 +/c' 3 =1, E= f 2 \/1 “K 3 sin 3 0 d6. 
J o 
x , 3 +x 3 =i, 
Similarly if 
F= ( 2 a/ 1 —X 3 sin 3 6 d9, 
Jn 
X', 1.' may be considered known. 
31. It is proved in Cayley’s ‘Elliptic Functions/ § 310, that the general single 
theta-function satisfies the differential equation 
E \d0 , n 
\ _LOr/ J% -I 
did l ’( K ' K )dx^ 2KKi dfc ° 
(100) 
and 
dK E IC 
dk 
-=™- =-*J K '-l 
V KK K 
K 
so that (100) may he written in the form 
d?0 , 2™ /3 dK dO , „ ,M „ 
^+ur T K x d*+ %KK d* =0 - 
Differentiating 5 times with respect to x 
d*Q_ 2iae^ dK d_ &0 dK d?6_ , 3 _d 
did daf K die dx dx s K die dx s K die dx s 
Now the general term in in (94) is a numerical multiple of 
. . ( 101 ) 
or 
of 
, _ /2KA log rvd s 6(x) d s 6(y) 
V ~\ t ? ) dx s dtf 
r dx s 
so far as x, k are concerned. Then 
d z yjr 2 kk' 2 dK dyjr^^, . , 1V1 
-£*di =K t first tw0 terms in ( 101 )1 
