AND DOUBLE THETA-FUNCTIONS. 
999 
144. But on the right hand side we have the term ^\/ defa / b / c / multiplied into 
(a — b) c(X+/xy) + (b- c)a(X'4- fiy) + (c — «)b(X"+/P 'y), 
and the term — -^v/d^Xabc multiplied into 
(a —6)c / (Xd- fix) + (b —c)a / (X / +/ i'x )+(c—G^b/X" +/x"y), 
and it is clear that the whole can only vanish if these two coefficients separately 
vanish. This will be the case if we have for X, X', X” the equations 
(a—b)\-\-(b—c)\.'-\-(c—a)\"=0, 
o ,, -{-« ,, ~\~b „ =0, 
and the like equations for y, y, fx". The equations written down give 
( a—b)\ : (b-c)X : (c—a)\"—a—b : b—c : c —a 
that is — and similarly p,=p,'=p/'. 
145. But this being so, the three equations in P, Q give 
that is 
P+^(k+/xy)\/X—0, Q l -\--^(X-\-fxx)\/Y — 0, 
.dr jCl/ih 
u 7v +v i = 
dir dv 
: ~(\+lxx)\/Y. 
x u 
In these equations u and v are arbitrary ; hence X and /x must be linear functions of 
u and v ; say their values are = am' -\-pv ', ctu' + tv' respectively. We have therefore 
%,=-fo+*yW x • 
| = -^+o-</)VY . 
:!=-^+v)vx 
r-jip+r^v'Y, 
or, what is the same thing, 
—j>6-^=(7x+ay)dii+(p+nj)dv, 
—\ = (sr+ c rx)du + (p + tx) clv > 
