824 
and therefore 
that is 
Now 
MR. A R. FORSYTH ON THE THETA-FUNCTIONS, 
1 d*«+s0 ui 
4 cc' dx 0 2n+ * 
u l*l 
dx 0 2n+s 
2n + l dK d 2n+ W hl 
K dc dx 2 ^ ~ 
= -4cc'K 3 |k 2 “ +1 
dc 
d ip* +1 0 I<1 
dc dx 0 2n+1 
d 2 ^e lA 
dx 0 2n+ * 
K2 « +1 ^1U_ 
dx 0 2n+1 
-4cc'k4Y K^ 1 ’ 1 
(115). 
1 /2KA\ 
0 h J -j = 2 vp l sin x—2p l sin 2>x-\- 2ipr sin 5a? — . . . 
= 2p*(l —y> 3 )(l — p 3 )(l— y> 6 ). . .sincc(l — 2p 3 cos2a?-|-p 2 )(l—- 2y> 4 cos 2cc-|-p 8 ). .. 
Hence 
K d0 hl 
iri dx o 
Thus (115) gives 
Similarly 
=p 4 |(l —y> 2 )(l — y> 4 )(l — p 6 . ..) 3 
=(-YWk' 
\7 T] 7r 
K 2 “ +1 ?^ 1 =(-l)' +4 0Y4cc / K 3 £) c s c ,i K‘ . . . 
^ 0 2 » +1 
A*“«^ 1 =(-l)'‘«g) 4 (4 r y'A 2 0VV*A i . • • 
*V* +1 
. . (116). 
. . (117). 
35. Another form may be given to several of these formula?. Let 
logy>=p' 
log q=q 
2 log r=/ - 
Then 
and therefore 
and 
(118). 
dp' 1 dp tt s 
dx p dx 2/c/e' 3 K 3 
4cc ' K2 I=^7.( 119 )> 
4 yv'A 3 —=7r 2 — 
77 dy dq r 
( 120 ) 
These formulae practically contain the expansions in powers of x of the single theta- 
functions ; restoring k, k, these are 
