503] THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
127 
The whole coefficient of Z thus contains the factor (ofg . x + bfg. y) ; and similarly it 
would appear that the whole coefficient of W contains the factor (abg. x — abf. y), the 
other factor being the same in each case ; viz. the two terms together are 
- (z b x a + z a x b ) h s y 
+ 0& y a + z a yb) h s x 
+ 2 \z a z h (a s x + b s y) 
[Z (ofg . x + bfg .y)+W {abg. ¿c - abf. y)} ; 
+ 2z a z b (-g b x+fy), 
where the second factor is Ap 2 a/3y, which is the required result. See post, Nos. 59 
et seq. 
42. (4) The line [a, /3, y, S] is an 8-tuple line; in fact, for any point of the 
line in question we have p 2 (3y8 = 0, p-yba = 0, p 2 8a/3 = 0, p 2 a(3y = 0; whence each factor 
is 0 1 , or the norm is 0 8 . 
I notice that the surface meets the quadric p 2 a/3y in 
lines a, (3, y each 8 times 24 
„ («, 7> S) » » 16 
„ (ab, a, /3, y) „ 4 8 
24 x 2=48 
Surface aafiybe. 
43. The equation is 
(p 2 a/3e ,p 2 y8e. p 3 aay . S/3 + p 2 aye. p 2 8f3e. p 3 aa/3. yS) 2 
- 4p 2 a/3e . p 2 y8e. p 2 aye .p 2 8/3e .p 2 a[3y ,p 2 8/3y. paa. pSa = 0 ; 
or, what is the same thing, 
(p 2 cc/3e. p 2 ySe. p 3 aay . 8(3 -p 2 aye .p 2 8fie. p 3 aa/3. y8) 2 
- 4tp 2 a/3e. p 2 y8e. p 2 aye. p 2 8/3e . p 2 /3aS. p 2 ya8. p(3a. pya = 0 ; 
the equivalence of the two depending on the identity 
p 3 aa(3. y 8. p 3 aay. 8/3 
- p 2 a/3y. p 2 8(3y. paa. p8a 
+ p 2 /3a8. p 2 ya8 . p/3a .pya = 0; 
where, as before, p 2 a(3e = 0 is the equation of the quadiic through the lines a, /3, e, 
and paa = 0 is the equation of the plane through the line a and the point a\ viz. 
p 2 a(3e, &c., and paa, &c., have the values already mentioned: p 3 aa/3.yS = 0 as already 
mentioned is the cubic surface through the lines a, ¡3, y, 8 and aa/3, ay8. 
{Surface aapySe.}