AND DOUBLE THETA-FUNCTIONS. 
1001 
1 . 1 
and hence substituting for the foregoing value, and introducing an indeterminate 
x y 
multiplier M, we obtain 
c n 'u + c n "v = M£ n { mi -fi pv + (era + tv) + (cru . + tv) } , 
which breaks up into the two equations 
c ii /= M£h{et+ (iS+ia)cr}, 
and thence also 
c/ = M& 7 { 
55 
b 
h 
c 7 " =ML { 
55 
b 
■> 
1 5 
o-: =m h { 
55 
c 
}> 
c s " =M£ 5 { 
55 
C 
}, 
C 13 ~ M7t 13 { 
5 5 
d 
■> 
s> 
Cig // =M^ig{ 
55 
d 
}’ 
c i4 = M^ u { 
55 
e 
}> 
C 14 — MX' 14 { 
5 5 
e 
s> 
o 
if 
o 
55 
f 
}> 
c io = M^i 0 { 
55 
f 
s> 
which twelve equations determine the coefficients m, c r, p, r in terms of the c', c" of the 
odd functions 5, 7, 10, 11, 13, 14 ; and moreover give rise to relations connecting these 
c', c" with each other and with the constants a, b, c, d, e,f 
148. It is observed that if as before 
b=u 
'A 
cLu 
then, substituting for P, Q their values, we have 
k=-| ( ™' +/ «o(v^|+^|)-|(«'+ro(yv^ 
= [zsU + pv) b 1 -f- (c Til ' -f- Tv)b. z , 
if for shortness 
.=rd 
y=d 
>*=-^K x 7 ,+' /y ,W> -e{y^+ x ^,} 
d\ 
dyj’ 
6 N 
MDCCCLXXX. 
