562 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
the coefficients of P and Q are at once found to be 
1 (a — b) Va^ 1 (a, — b,) Vab 
Vab ’ " 2 Va^T~’ 
respectively, or observing that a —b, = a, —b y , =ci — b, the equation becomes 
P "vtb + ^ \)t! = ~ I + w) ^ cdefa ' b/ + + /**) ^ c / d / e / f / a M; 
or multiplying by Vaba,!», and writing for shortness abcdef = X, a / b / c / d / e / f / = Y, this 
becomes 
a / b / [P + -q (A + ¡Ay) X] + ab {Q + -q (A + fix) V T"} = 0. 
143. There are, it is clear, the like equations 
b A (P+|(v +/i '2/)VZ) + bc{e + |(x' +AW7) = 0, 
c,a, 1 P+- 0 (\"+ y!'y) VZ) + ca {Q + |(X" + p"x) V7} = 0, 
and it is to be shown that A = A' = A" and /a = ¡a = ,0/'. For this purpose, recurring 
to the forms 
Vaa y 0Vbb, — Vbb, 0 Vaa y = {(A + fAy) Vcdefa / b / + (A + g,«) Vc/^e^ab}, 
Vbb / 0Vcc, — Vcc / 9 Vbb / = —{(A' + ¡Ay) Vadefb / c / + (A' + /ax) Va/be^bc}, 
Vcc / 0 Vaa / — Vaa / 0 Vcc, 
{(A" + fA"y) VbdefcA + (A" + y!'x) Vb/i^ca}, 
multiply the first equation by Vcc,, the second by Vaa /} and the third by Vbb,, and 
add: the left-hand side vanishes, and therefore the right-hand side must also vanish 
identically. 
144. 
But on the right-hand side we have the term 
1 
0 
Vdefa / b / c / multiplied by 
(a — b) c (A + ¡Ay) + (b — c) a (A' + yly) + (c - a) b (A" + yi'y), 
and the term — ^ Vd^fabc multiplied by 
(a — b) c y (A + fAX) + (b — c) a, (A' + y!x) + (c — a) b, (A" + ¡a' x), 
and it is clear that the whole can vanish only if these two coefficients separately 
vanish. This will be the case if we have for A, A', A" the equations 
(a — b) A + (b — c) A' + (c — a) X' = 0, 
c „ + a „ +6 „ =0,