136 
ON THE SURFACES THE LOCI OF [503 
where 2 denotes the sum of the three terms obtained by the cyclical interchange of 
a, ß, y; and 
paa = (z a - \w a ) (x - Ay) - (x a - \y a ) (z - Aw), 
pba = (z b - \w b ) (x - Ay) - (x b - \y b ) (z - Aw); 
A here standing for A a ; and similarly for paß, &c. 
60. To obtain the intersection with xw — yz = 0, writing w = —, then 
paa = [z a ~ Aw a - Z -(x a - A y a )] (x - A y), (A = A a ), 
cc 
pba = [z b - \w b -1 (#& - Ay b )] (x - Ay) ; 
or say 
Npaa. pba = VJlf a (a? — A a y); 
also the expression in { } becomes 
= {0 -f^ +c ^ + h \ ( x ~ x ßv) ( x -\v); 
so that the norm in question is 
Norm 2 */M~ a (Aß - A y ) {(a -/) ^ + c ^ + A} (a? - A a y) (® - A^) (a - A y y) ; 
or say 
Norm 2 Vl/ a (A^ — A y ) {hx 2 + (a-f)zx + cz 2 }(x — \ a y)(x — A ß y) (x — A y y) ; 
where M a is now considered to stand for 
{(z a x - zx a ) - A (w a x - y a z)} {(z b x - zx b ) - A (w b x - y h z)\. 
Observing that the norm was originally the product of 8 factors, this breaks up into 
{hx 2 + (a —f) zx + cz 2 } 8 {(x — A a y) (x — Aß y) (x — A y y)} 8 = 0, 
and 
Norm 2 *lM a (Aß — A y ) = 0, 
where the new norm is the product of 4 factors. 
61. Writing for greater convenience A, /i, v in place of A a , Aß, A y , and observing 
that M a is a quadric function of A„, that is of A, the last-mentioned norm is 
Norm VA + PA + 6'A 2 (/X — v), 
which is easily seen to be 
= (440- B 2 ) (fi - v) 2 (v - A) 2 (A - y) 2 ; 
or writing for a moment 
(A + PA + C\ 2 ) = (P - Qx) (P' - Q'A),