82 
[730 
730. 
[ADDITION TO MR SPOTTISWOODE’S PAPER “ ON THE TWENTY- 
ONE COORDINATES OF A CONIC IN SPACE.”] 
[From the Proceedings of the London Mathematical Society, vol. x. (1879), 
pp. 194—196.] 
Write 
U = (a, h, c, d, f g, h, l, m, n\x, y, z, t)\ 
U 0 = ( „ ££ y, £ to) 2 , 
TF=( „ \x, y, z, ty, £ «), 
P = (a, /3, 7, BQx, y, z, t), 
P 0 = (a, (3, 7, y, £, w). 
Then the equation of the cone, having for its vertex the arbitrary point (£, y, £, <w), and 
passing through the conic U = 0, P — 0, is 
UP 0 2 — 2 WPP 0 + U 0 P 2 = 0. 
Or if, to put the coefficients (f, y, Ç, co in evidence, we write for a moment 
and therefore 
then the equation is 
A = (a, h, g, l \x, y, z, t), 
B = (h, b, f „ ), 
G = {g, f, c, n $ „ ), 
D = (l, m, n, d\ „ ), 
W = AÇ + Br] + CÇ+Do) ; 
U (<*% + /3y + 7^ + Bar)' 2 — 2P (ai; -f- ¡3y + 7^ + Ba>) (A£ + By + CÇ + I)a>) 
+ P 2 (a, b, c, d, f, g, h, l, m, , y, £ w) 2 = 0.