704 
this 
ring 
and 
mish 
rtely 
704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
and the like equations for ¡x, p!, p!'. The equations written down give 
(a — b)\ : (b — c) X' : (c — a) X" = a — b : b — c : c — a, 
that is, X = X' = X": and similarly p = p! = /x". 
145. But this being so, the three equations in P, Q give 
P + 1 (X+yay) VX = 0, Q + i (X +/lub) VF:::!), 
that is 
dv dv ® — y 
u’ijL + v'^- = - (X + fix) V Y. 
du dv x—y 
In these equations u and v' are arbitrary; hence X and /x must be linear 
functions of u and v'; say their values are = -atv! 4- pv', av! 4- tv' respectively. We 
have therefore 
■£~S(- + <*)vX * —l(#.+n-)VT, 
$ — §(- + «),/* | = -2 (p + ra) VT 
or, what is the same thing, 
whence also 
- 7T = + + ^ + T2/ ^ ^ 
— = (ot 4- ax) du 4- (p + rx) dv, 
v F 
adu + rdv= i 
which are the required relations, depending on the square roots of the sextic functions 
X = abcdef, and Y = a / b / c / d / e / f / of x and y respectively; but containing the constants 
ar, p, a, t, the values of which are not as yet ascertained. 
146. I commence the integration of these equations on the assumption that the 
values v = 0, v = 0 correspond to indefinitely large values of x and y. We have 
X = afi(l-- + .. 
V x 
F=s ’( 1- f + - 1 ' 
where S=a+b+c+d+e+f; and thence the equations are 
dy 
dx 
adu 4- rdv = ^ ^ ( 1 4- 2 
dx 
hS 
4 ... 
_1 
2 yS 
-, P 
1 4- — + 
-asdu 4- pdv = — I 
1+ iJ+\ +i ài( 1 + ^ + 
x J y 
71—2