503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
133 
or, as the second equation may also be written, 
(QoS - QS 0 ) (P xy R zw - P zw R xy ) — (P 0 R — PR 0 ) (QxySw, - Q Z wS xy ) = 0 ; 
viz. the second equation represents a cubic surface having upon it the lines (P = 0, R = 0) 
and (Q = 0, S = 0) : it therefore intersects the quadric PS — QR = 0 in these two lines, 
and besides in an excuboquartic curve, which is the required locus. 
55. Representing the determinants 
P , Q , 
R > 
S 
by (a', V, 
c', {', g', W), viz. a' = QR n - Q a R,... 
Po , 
Qo > 
K , 
$o 
f = PS 0 - P 0 S,...; 
¿V 
Qxy) 
Rxy, 
S X y 
by (a, b, 
c, f, g, h ), viz. a = Q X yR zw QzwPxyy • • • > 
PZW> QzW) 
Pzw y 
S Z w 
(a', ...) are 
linear 
functions, (a,.. 
) quadric functions, of the coordinates; the 
equation of the cubic surface is gb' — bg' = 0, viz. the excuboquartic arising from the 
generatrices is the partial intersection of the quadric PS — QR = 0 and the cubic 
gb' — g'b = 0 ; the two surfaces besides intersecting in the lines (P = 0, R = 0) and 
(Q = 0, S = 0). 
It appears, in the same manner, that the excuboquartic arising from the directrices 
is the partial intersection of the quadric PS — QR = 0 and the cubic he' — ch' = 0 ; the 
two surfaces besides intersecting in the lines (P = 0, Q = 0) and (R = 0, S = 0). 
56. But the elimination may be performed in a different manner, as follows: 
from the first two equations in 0, multiplying by P zw , - P xy and adding, and so with 
Qziv, - Qxy, &C., we obtain 
(Qo-0S o )( 
(Qo-es 0 )( 
(Qo-OS 0 )( 
(Qo-OS 0 )( 
-0b) - (P 0 — 0R O ) (— c + 0f ) = 0, 
c + f?a) - (P 0 - 0R O ) ( %) = 0, 
b ) - (P o -0Ro)( a + 0h) = 0, 
{-eh) - (P o -0Ro)( g )=o. 
We then have 
0 = 
— G+0Î c + 0a 
or, what is the same thing, 
a + 0b f - 0h ’ 
h0 2 + (a - f ) 0 4- c = 0. 
Using this equation, written in the form (a + dh) 0 = — c + 0f, to transform the first or 
third of the four equations in 0, we obtain 
— aP 0 — hQ 0 — cP 0 + 0 (— hP 0 • +fP 0 +b£ 0 ) = 0; 
and using the same equation, written in the form (f— #h) 0 = c + #a, to transform the 
second or fourth equation, we obtain 
gPo — fQo + c$ 0 + 0 ( 
hQ 0 - gPo + a$ 0 ) = 0 ;