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.}