428 
on MR Wilkinson’s rectangular transformation. 
[783 
we have 
A — q'z 2 + r'y 2 — p'x 2 + q'x 2 + r'x 2 
= — p'x 2 + q' (x 2 + z 2 ) + r' (x 2 + y 2 ) 
= —p'x 2 + (q 1 + r') (x 2 + y 2 + z 2 ) — q'y 2 — r'z 2 . 
By what precedes, we have an identity of the form 
x 2 + y 2 + z 2 — a (p'x 2 + q'y 2 + r'z 2 ) + /3 (px 2 + qy 2 + rz 2 ), 
where, determining a from the equations 1 = q'a + q/3, 1 = r'a + r/3, we find 
a = (q — r) -7- (qr' — q'r); 
but px 2 + qy 2 + rz 2 = 0, and the relation thus is 
x 2 + y 2 + z 2 = a (px 2 + qy 2 + r'z 2 ) ; 
hence 
A = {(q +r') a — 1} (p'x 2 + qy 2 + r'z 2 ), 
or, substituting for a its value, this is 
and, forming the like values of B and G, the relations to be verified become 
which are, in fact, the values of the ratios p" : q" : r" obtained above; and the 
theorem is thus seen to be true. It may be remarked that, if the first and second 
cones, instead of intersecting in four lines on the absolute cone, had been arbitrary 
cones; then, taking in the first cone a line (1) and in the second cone a line (2), 
the reciprocal of (1) in regard to the absolute, the envelope of the plane (12) would 
have been (instead of a quadric cone) a cone of the class 8.