126 
ON THE SURFACES THE LOCI OF 
[503 
{Surface abcdap.} 
where we have 
z a Pba + z b Paa 
= z b [x a (Zg a - Wa a ) + y a (- Zf - Wb a ) + z a [X (- g a + Xa a ) +Y(f a + Xb a ) + (XZ - W) c a ]} 
+ z a [x b (Zg a - Wa a ) + y b (- Zf a - Wb a ) + z b [X (- g a + Xa a ) + F(/ a + Xb a ) + (XZ - W)c a ]}, 
= (z b x a + z a x b ) ( Zg a - Wa a ) 
+ (ZbVa + z a y b ) (- Zf a - Wba) 
+ 2z a z b {X (— g a + \a 0 ) + l 7 " (f a + A6 a ) + — W) c a }. 
Also 
^¿>4 = {(- + Xa a ) x + (fa + \b a ) y\, 
2?ft y 8 = x 2 . agh + xy (ahf + bgli) + y 2 . hbf 
Ap 2 fty8 = X. 2x. agh + y {ahf + bgh) 
+ F. x(ahf+ bgh) + 2y . hbf 
+ Z .x (cgh + afg) + y (bfg + chf) 
+ W. x {abg — cah) + y (bch — abf). 
41. The whole expression is a linear function of X, Y, Z, W, and it is easy to 
see à priori, or to verify, that the coefficients of X, Y, each of them vanish. The 
coefficient of Z is 
= t {(z b x a + z a x b ) g a - {z b y a + z a y b )f a + 2Xz a z b c a ] p 2 fty8 
+ ^Z a z b [(- g a + Xcia) x + (fa + Xba) y] [x (cgh + afg) + y (bfg + chf)], 
with a like expression for the coefficient of W. 
The foregoing expression may be written 
(z b x a + z a x b ) X g [agh. x 2 + (ahf + bgh) xy + bhf. y 2 ] 
- (z b y a + z a y b ) Xf[agh. x 2 + (ahf+ bgh) xy + bhf. y 2 ] 
+ 2Xz a z b X {c [agh . x 2 + (ahf+ bgh) xy 4- bhf. y 2 ] 
+ (ax + by) [(cgh + afg) x + (bfg + chf ) y]} 
+ 2 z a z b X (- gx +fy) [(cgh + afg) x + (bfg + chf) y\ 
The first sum is 
¿• 2 . gagh + xy (gahf+ gbgh) + y 2 . gbhf 
= ~ xy . afgh - y 2 . bfgh, 
= — h$y (afg . x + bfg. y) ; 
where afg, bfg denote determinants with the suffixes a, ft, y. Similarly the second 
sum is 
= — h$x (afg . x + bfg. y) ; 
the third sum is 
(a s x + b s y)(afg .x + bfg .y), 
and the fourth sum is 
(- 9& x +fs y) (of g .x + bfg. y).