564 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
Hence integrating, we have 
<TU + TV = -^ 
\x 2 y-1 
and thence 
,u + pv- + 
‘U + pv + (<7W + TV) = ^ ^ 
1 1 
where the omitted terms depend on —, — &c. 
x 3 ' y° 
Hence, neglecting these terms, we have 
cru + tv 
)• 
HTU + pV + ^S (<TIL + TV) \X y 
an equation connecting the indefinitely small values of u, v, with the indefinitely 
large values of x, y. 
147. From the equations A = k u vj Va, B = k 7 tx Vb, taking (u, v) indefinitely small 
and therefore (x, y) indefinitely large, we deduce 
kA- ia (l + l) 
Cn'u + Cn 'V _ fc n “ \x yj 
c 7 'u + c 7 "v ~ h 7 1 _ ^ /1 iy 
2 \x + y) 
hence substituting for - + - the foregoing value, and introducing an indeterminate 
x y 
multiplier M, we obtain 
c u 'w + c n "v = MIc n {urn + pv + (au + tv) + \a (au + tv)), 
which breaks up into the two equations 
Cn = Mk n {trr + + ^a) cr}, Cn" = Mk u {p + + j^ct) t}. 
Similarly 
c 7 = Mk 7 { „ 
b 
}, c/' =Mk 7 { 
99 
b 
}, 
c 5 = Mk 5 { ,, 
c 
}, c 5 " =Mk- 0 { 
99 
c 
}> 
Gvi = Mk l9 { >) 
d 
}, Cl3 " = m 13 { 
99 
d 
}> 
C14 = Mku { » 
e 
}, c u " = Mk u { 
99 
e 
}> 
Gw — Mku { » 
f 
}, c w " = Mk w { 
99 
f 
L 
which twelve equations determine 
the 
coefficients ot, 
<r, p, 
T in 
terms of 
of the odd 
functions 5, 7, 10, 11, 13, 
14; 
and moreover 
give 
rise to 
relations 
these c\ c" 
with each other and with 
the 
constants a, b, 
c, d, 
e,f 
148. It is observed that if, as before, 
d = u'i+rfi, =p£+q£, 
du av ax ay