862 
MR. A. 31. FORSYTH ON THE THETA-FUNCTIONS. 
^V,,„=2 lo giV+ |(lo g P.,,f d ^+ 3 kg P, gg S (logft <)^} 
+ • • 
\dpidpj 
/ d 3 
3asri slnl12 i log M^7 + lK lo « 
d 3 
+sinh 2{log ft ^) - S S log ft/^J 
(249) 
where in the summation s, t take all the values 1, 2, 3,.. ., r, except l and m together, 
then 
C 
l/M’ Xg, . . . , 1, . . . , 1, . . . , 
^2’ • • * 5 lj • * • 5 1? • • * * V t 
= ( VVi, m.cdc'fcJc'jK^Kj n cdc'd Kd . ( 250 ) 
in which t has all values except l, m. This formula gives the constant terms for all 
those S r ~' 2 even functions in the characteristics of which \ l =\ m =l=v i =v m , but no 
other corresponding numbers are unity at the same time; and since l, m may be any 
whatever of the suffixes, this formula comprises \r{r — l)3 r ~ 2 constants. 
The above will suffice to indicate how all the constants may be obtained. 
Note. —Since the above memoir was written I have seen a paper by Professor H. J. 
S. Smith (in vol. x. of Lond. Math. Soc. Proc., 1879) in which the results of §§ 6, 62, 
and 63 are given in an equivalent but somewhat different and more concise form. 
[Sept. 29, 1882.] 
