134 
ON THE SURFACES THE LOCI OF 
[503 
and hence, eliminating 0, we obtain 
(hQo — gPo + a$„) (— aPo bQ 0 cP 0 ) — (— hi 3 0 + f-Po + b>S^ 0 ) (gPo — fQo • + c$ 0 ) = 0, 
which, as being of the second order in (a, ...), represents a quartic surface. The 
equation remains unaltered by the interchange of Q, R, and the consequent interchanges 
among (a, b, c, f, g, h): hence the quartic surface contains not only the excuboquartic 
arising from the generatrices, but also that arising from the directrices; and these 
two curves are the complete intersection of the quartic by the quadric PS — QR = 0. 
57. I obtain this same result also as follows. Consider a point (P u Q u R 1} S x ) 
on the quadric surface; P 1 S 1 — Q 1 R 1 = 0; the tangent plane at the point is 
PS.-QR,-RQ, +SP l =0; 
and if this passes through the point a, then 
- QoPi — RoQi + S 0 Pi = (). 
The line which in the tangent-plane meets the lines 8, e is given, as before, by the 
equations 
PxySi Qxy P\ Pxy Qi d - S X yP 1 = 0, 
Pzw^i QzwRi PzwQi d~ S Z wPi = 0. 
Remembering the significations of (a,...), the last three equations give 
& : Ri : - Qi : —P x = . hQ 0 -gR 0 + &S 0 
: — hP o . + f P 0 + bS 0 
: gPo-fQo • d-c# 0 
• a Po bQ 0 cPq . ; 
and substituting these values in SiP 1 — Q 1 R 1 = 0, we have the above equation of the 
quadric surface. 
58. Or again, changing the notation, I take the equation of the quadric surface 
to be 
(a , b , c, d , /, g, h , l, m , n \x, y, z, w) 2 = 0. 
A tangent-plane hereof is 
%X + 7)y + tz + cow = 0, 
where £, y, w are any quantities satisfying the relation 
(A, B, G, D, F, G, H, L, M, y, £ ®)» = 0 > 
the capitals denoting the inverse coefficients. 
Supposing that the tangent-plane passes through a fixed point a, coordinates 
(a, /3, y, 8), we have 
af + /3y + y£+ 8o) = 0;