503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
107 
and 
hM - g N + ail = (af + bg' + cb!) P, 
— hL . +fN + b£l = ( „ ) Q, 
gL-fM . +cil = ( „ )R, 
-aL-bM-cN . = ( „ )S. 
14. p 3 a. a/3.78 = 0 is the equation of the cubic surface through the lines a, /3, y, 8 
and aa/3, ay8 (viz. cia/3 is the line from a to meet a, /3, and so ay8 is the line from 
a to meet y, 8). Observe that the conditions which determine this cubic surface thus 
are that the cubic shall pass through 
a; the points of aa/3 on a and $ respectively, 3 other points on a, 3 on /3, and 
1 on aa/3 ; 
also the points of ay8 on y and 8 respectively, 3 other points on y, 3 on 8, and 
1 on ay8 ; in all, 1 + 9+9=19 points ; 
viz. the conditions completely determine the surface. 
15. We have 
p 3 a .a/3.7 8 = 
x 
V 
w 
> Va > ¿a > ^a 
L a ß , LI a ß , -A aß > Haß 
Lyü, 
viz. this determinant, equated to zero, gives the equation of the surface. 
To prove this, take as before the unaccented letters (a, b, c, f g, h) to refer to 
the line a, and the letters with one, two, and three accents to refer to the lines 
/3, 7, 8 respectively; write also L, M, N, fl and L', M', N 1 , IT for L a ß, &c„ and L y s, &c., 
respectively. Referring to the foregoing expressions for L, M, i\ r , i), and observing that 
for a point on the line a, the values of P, Q, R, S are each = 0, then for such a 
point we have L + (af+b'g + c'h) x— 0, &c., that is, L : M : N : H = x : y : z : w, and 
these values satisfy the equation of the surface, which is thus a surface passing through 
the line a ; and similarly it passes through the lines /3, y, 8. 
To show that the surface passes through the line aa/3, take the coordinates of 
the point a to be 0, 0, 0, 1; then the line aa/3 is given as the intersection of the 
planes ax + by + cz = 0 and ax + b'y + c'z = 0, that is, >3 = 0 and >3=0. And the 
equation of the surface, writing therein x a , y a , z a , w a = 0, 0, 0, 1, becomes 
x , y , z 
L, M, W 
L\ M\ N' 
14—2