814 MR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
The bracket in the first term = 0 oo (y) 
„ second =_A%M 
7 r dy 
third 
fourth 
w 
W tfc/ 3 
/A\ w ft0 (y) 
W 
, /AY ^o.o(y) 
* 
the sign being + if n=Ap or 4p-j-l, and — if w=4p+2 or 4p-f 3. 
Since 
lo g» ,= 5|i o gp' 
logp'=^.| iogr 
Hence 
o _zi r r \a i \ 2KAlog r d0 oo (x) d0 Oi( £y) /2KAlogr\ 3 l d~0 Q ^(x) <P0 oo (y) 
% 
I /_t y/2KA log rVl d s 6 0 0 {x) d s dQ Q{y) ^ 
A '\ tt 3 Jsl dbd dy . W 
which may be expressed in the symbolical form 
2KA log r fP 
^y0 0 , 0 (x)e M {y) . (93); 
and it may be proved by an exactly similar process to hold for all functions, so that 
generally 
K X o\ 1 2ICA log r P 
^h y r r •* "A4*Ap(2/).( 94 ) 
the parameters of 6^ k (x) being p, K ; and of 0 VtP (y) q, A. 
25. The functions <4> have already been distinguished by the oddness or evenness of 
p\-\-vp; this formula (94) enables us to verify the division of the even and uneven 
functions given in the table. The latter will be obtained by taking one of the single 
0’s even and the other uneven; and since there are three even and one uneven func¬ 
tion 6 for each of the variables, there will be six uneven functions <E>, obtained by 
taking the uneven function of each variable with the three even, functions of the other 
variable : hence there will be ten even functions, since there are 16 (=4 2 ) in all. 
