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 =