503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
11.7 
factor operated upon by A =XB x +7B y + ZS z +WB w , reduces itself for z=0, w = 0 to 
a multiple of W. Considering the factor in the form of a determinant, the result of 
the operation is 
X, 
Y, 
Z, 
w 
+ 
x , 
y > 
'dpa a . pa/3, 
Xa, 
Va, 
z a, 
• 
A g paa. pa/3, 
x a, 
ya, 
z a 
Vpba .pb/3, 
Xb, 
yb, 
z b, 
A Vpba .pb/3, 
x b, 
yb, 
z b 
'dpea . pc/3, 
x C) 
ye, 
z c, 
• 
A dpea .pc/3, 
X c , 
ye, 
z c 
Vpda .pd/3, 
Xd, 
yd, 
z d, 
w d 
A dpda .pd/3, 
x d, 
yd, 
z d 
the first term is 
2(1 ^1 a 7/3 > 
z h V/./p, 
X, Y, Z, W 
x a, y<i> z a, 
Xb , Vbi z b, 
x c, Vc> z e, 
Z C ^ I a Ip , 
'/pda. pd/3, x d , y d , z d , w d 
where the first column may be replaced by 
-Z-Jljf, 
*Jpda. pdp — Za ^ I.Ip, 
and the term in question thus becomes 
{w i Z4TJ f +W(- Za'Jlalp + 'J,pda.pd/3)}. abc, 
— abc. 
if for shortness 
x a , y<t> ^a 
x b> yb> z b 
X c , y<i> 
As regards the second term, we have 
which is 
But 
, ^ paa. Apa/3 + pa/3. Apaa 
A \/paa.pap = - ^dpaaTpap ' 
I a Apa/3 + Ipàpaa 
~ 2 dljp 
pcia. = X (-g«z a ) + y (f a Z a ) + £ (g a X a -f«ya) + w ( a * æ a b «y* GaZ(tb 
= Xa (g« z - a«w) + y a (~f* z - M + z * (" 9* x + f ay ~ CaW ) ’ 
and thence 
Apaa = x a (g.Z- a. W) + y a (~U Z ~ MV) + *. (“3^ +f‘ Y ~ c “ W) ’ 
{Surface cibcdap.}