395] 
INVESTIGATIONS IN CONNEXION WITH CASEY’S EQUATION. 
71 
and substituting for P, Q, R, dU, dV, dW their values, the left-hand side is = - Jdz, 
which is = 0; hence the equations in question are proved, and (f\ g, h) is a point 
on the curve A. 
It is to be noticed, that the two curves X, A are geometrically connected through 
the three arbitrary points as follows: viz. taking as axes the sides of the triangle 
formed by these three points, then starting from any point (f g, h) of A, we take the 
inverse 
inverse the point corre- 
u * 
sponding to the assumed point (/, g, h) of A : and conversely starting with an assumed 
/* h 
point on X, we take the conic - + - + - = 0 which passes through the angles of the 
r x y z 1 ° ° 
triangle and touches X at the assumed point ; the inverse line fx + gg + hz = 0 ; the 
^ of this line; and finally the inverse point (/, g, h), which 
will be on the curve A, the point corresponding to the assumed point on the curve X.