632] on aronhold’s integration-formula. 
13 
then y is determined as a function of x by the equation U= 0, that is, 
(a, b, c, / g, K$x, y, l) 2 = 0; 
or, what is the same thing, 
by — — {hx + f »/(— Rx 2 + 2Gx — A)}; 
the constants a, /5, £, y are such that 
{a, b, c, f g, y, 1) 2 = 0, 
«£ + fiv +y = 0, 
that is, 
i2 0 = 0; 
and the value of A is given by 
A 2 = -(A, B, C, F, G, HI«, ¡3, 7 ) 2 . 
The theorem may therefore be written 
[ dx . Tf 
A JnQ~ log n > 
ilQ 
where the several symbols have the significations explained above. 
The verification is as follows. We ought to have 
A dx_P 0 dx+Q 0 dy adx+fidy 
ilQ 
W 
n 
when dx, dy satisfy the relation P dx + Q dy = 0, viz. substituting for dy the value 
jP d x 
-Q-, the equation becomes 
A _ P 0 Q — PQ 0 aQ — f3P 
n W il ’ 
that is, substituting for il its value, 
AF= (P 0 <2 - PQo) (ax + /3y + ry)- (ocQ - /3P) W. 
On the right-hand side, substituting for W its value, 
coeff. a = x(P 0 Q — PQ 0 ) - Q (P 0 x + Q 0 y + P 0 ), =Q 0 R- QR 0 , 
coeff. ¡3 = y (P 0 Q — PQo) + P (P 0 x + Q 0 y + P 0 ), = R 0 P - RP 0> 
(as at once appears by aid of the relation U = Px + Qy + R = 0), 
coeff. 7 = P 0 Q-PQ 0 . 
The equation to be verified thus is 
a ,/3, 7 
Po, Qo> Ro 
P, Q, R 
A W =