503] THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 123 
where 
A = {— bcghx 4- cbhfy — bcfgz +fabcw}, 
B = {— caghx + cahfy + acfgz 4- gabcw}, 
C = \+ baghx — abhfy 4- abfgz + habcw), 
D = {— afghx — bfghy — cfghz 4- (bcgh 4- cahf+ abfg) w\, 
P = ( . hy - gz + aw), 
Q = (— hx . + fz + b w), 
R=( gx — fy . + gw), 
S = ( ax + by + gz . ) = 0, 
the right-hand factors vanishing for the values ux a + vx b of the coordinates. 
38. It thus appears so far that the factor is = 0 1 ; it is, in fact, = 0 2 , viz. we 
can show that, operating upon it with 
A = Xd x ”1* 1 dy + Zd z + Wdj w , 
the value (for any point of the line ab) is =0. We have 
A Vpaa .pba ,p 2 /3y8 =B au • p 2 (3yS + Vjpaa .pba . A .p 2 /3y8, 
2vpaa. pba. 
where Iba. (= Apba) is what pba becomes on writing therein (X, F, Z, W) in place of 
(x, y, z, w). Writing, as before, for x, y, z, w the values ux a + vx b , &c., we have 
paa = — v. aba, pba = u . aba; and putting for shortness 
- v. Iba + u. la a = Ika, &c., 
the expression in question, divided by V — uv, is 
= — 2vu [aba . Ap 2 /3yS — &c.} 
+ [Ika . p 2 fiyh — &c.}, 
where, denoting the determinants 
X Y Z W 
ux a -vx b , uy a -vy b , uz a vz bf uw a — vw b 
by (a', b', c', f, g', h'), we have 
Ika = a'f a + b'f a 4- c'g^ 4 fa a 4- g'b a 4- h'c«. 
But aba. Ap 2 l3y8 = A aba. p 2 /3y8, since aba is independent of (x, y, z, w); and the 
expression is 
= - 2wA (AP +BQ +CR+ PS) 
+ AP' + BQ' + CR' + DS', 
{Surface abcdap.} 
16—2