3G2 MR. E. W. BARNES ON THE THEORY OF THE 
We evidently obtain 
-log 2 - z)-- yn( - oA + - • 
(O^ 2^ A 
l > ’ (Z/2 
' 4:V* 
a)i Wj 
\ l> <1 
^ 
log 2 y^.j (wj, ajo)?- y. 2 i ^ 3 )^ ,,^3 + . . . 
H“ H" + • • • + 27rt/i,^ 
OJj w., 1 
d ’ i 
0) 
+ Y 2^0 ( 0 
ft), W.1 
R <1 
where we put symbolically 
+ 
00 GO 
1 
mi = 0 »,2=o(''^P«i + 
And now, equating coefficients of the various powers of 
’ TJ ~ ^ '/b ~7 + , - 27n/r^,y.2bo 0 
\ 1 ' 1 ' »-0 s=^0 \ P 'J I \ 
/W, a).-,\ , . Sw.A 
\R ?/ / r=0 s=0 \ 7/ \ 
Wj W.i 
2 d 7 
a)| o)., 
id (/ 
the transformation equations for the first and second gamma modular forms. 
Note that we also have, where m > 2 , 
\(i‘o)\ sco,i\ , , , t \ I r 1 11 
..r..rA“(i7 ’ 7;j = (-)''0-i)'i„-:-s4- 
If T = we may put or yoj (ojj, Wo) = now, putting p = I, (J — 
■\ve liave 
,1-1 
fh 
And putting^; = n, q = 1, 
f /21 {'^) = — wi® N xjj. 
'll 
'll 
'^^"t /21 - 9ii (^) = - " 1 ^ - 27rt^;,,i - 
We get analogous formulae by writing wj y^o (ojj, Wo) = 
The analogy between tliese results and those obtained in the analysis of elliptic 
functions is obvious. We cannot obtain, however, results which will connect such 
expressions as 
f/ 2 i(^+ 1 ) and goi(T). 
