819] TWO CASES OF THE QUADRIC TRANSFORMATION BETWEEN TWO PLANES. 101 
Writing for convenience (a, b, c) for the coordinates of the fixed point, and (x 2 , y u z?), 
(x. 2 , y 2 , Zo) for those of the other two points, the formulae with A, B, G give thus the 
correspondence 
x 2 : y 2 : z 2 — bcx? — a 2 y 1 z 1 : cay; 1 — b' 1 z [ x 1 : abz? — &x x y 2 , 
which is the first of the two cases in question. These equations give reciprocally 
x x : y x : z x — bcx? — a % y 2 z. 2 : cay? — b 2 z 2 x 2 : abz? — c 2 x. 2 y 2 , 
or the correspondence is a (1, 1) quadric correspondence. 
The formulae with P, Q, R give in like manner 
x 2 : y > : z. 2 = a {ax? + by? + cz?) — x x {a?x 1 4- b 2 y l + c 2 z 1 ), &c., 
or if for shortness 
0, = ax? + by? + cz?, ©j = a?x x + b 2 y 1 + c%, 
then 
x 2 : y-2 ■ z. 2 = aCli — «i©i : b£l x — y 2 ®i : cil 1 — z 1 ® 1 , 
which is the second of the two cases. We have reciprocally 
x x : y 2 : z 1 = a£lo — x 2 © 3 : bfl 2 — yM< 2 : cil 2 — z 2 ® 2) 
where 
il 2 = ax? + by? + cz?, ©a = a 2 x 2 + b-y., + crz 2 , 
and the coi’respondence is thus in this case also a (1, 1) quadric correspondence.