44 
DETERMINATION OF THE ORDER OF A SURFACE. 
[806 
and the equations of the normal ON are therefore 
x = y = Z' 
a b c ’ 
we have therefore for the cone 
ax + tn/ + C z _ a . 
+ y 2 + z 2 ) \/(a 2 + b 2 + c 2 ) 
or if, for convenience, cos 2 9 — k, then the equation of the cone is 
(ax + by + cz)' 2 — k (a 2 + b 2 + c 2 ) (x 2 + y- + z 2 ) = 0 ; 
say this is 
{a 2 — k(a 2 + b 2 + c 2 ), b 2 — k (a 2 + b 2 -+ c 2 ), c 2 - k (a 2 + b 2 + c 2 ), be, ca, ab} (x, y, ¿) 2 = 0...(1). 
Taking then (x 0 , y 0 , z 0 ) for the coordinates of the point to, the equation of the 
plane coL is 
Ax + By + Cz + D A'x + B'y 4- G'z + U _ ^ 
Ax 0 + By 0 + Gz 0 + jD A'xq + B'y 0 + (J z 0 + 1) 
viz. it is 
{BC — B'G) (yz 0 — yoz) + ... + (AD' — A'D) (x — x 0 )+ ... = 0, 
that is, 
f (yz 0 - y 0 z) + g (zxo - z 0 x) + h (xy 0 - x 0 y) + a (x - x 0 ) + b (y - y Q ) + c (z - z 0 ) = 0, 
or say 
x (hy 0 - gz 0 + a) + y (- htf 0 + fz 0 + b) + z (gx 0 - fy 0 + c) + (- a« 0 - by 0 - cz 0 ) = 0.. .(2), 
viz. (1) and (2) are the equations of the conic C, and the coordinates (a, b, c, f, g, h) 
satisfy of course the equation 
af + bg + ch = 0 (3). 
Considering now the line L as belonging to a ruled surface, the coordinates (a,...) 
satisfy as before three equations 
F (a, b, c, f, g, h) = 0 (4), 
G( „ ) = 0 (5), 
H{ „ ) = 0 (6), 
of the orders p, q, r respectively, and we can from the six equations eliminate a, b, c, f, g, h. 
The resulting equation V = 0 contains the coefficients of (1) in the order 1.2.p.q.r = 2pqr 
(which is the product of the orders of the other 5 equations), and the coefficients of 
(2) in the order 2.2 .p. q. r = 4>pqr, (which is the product of the orders of the other 
5 equations). But the coefficients of (1) being quadric functions of (x, y, z), and 
those of (2) being linear functions of {x, y, z), the aggregate order in (x, y, z) is 
2.2pqr + 1.4pqr, = 8pqr; 
or, since the order of the ruled surface is n, = 2pqr, the order of the locus is = 4n; 
which is the above-stated theorem.