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ON THE SURFACES THE LOCI OF 
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Mode of obtaining the several Equations: Notations and Formulae. 
6. The equations of the several surfaces are obtained by taking as centre of 
projection an assumed position of the vertex, and projecting everything upon an 
arbitrary plane; the projections of the given points and lines are points and lines in 
the arbitrary plane, and the section of the cone by this plane is a conic; the equation 
of the surface is thus obtained as the condition that there shall be a conic passing 
through m given points and touching 6 — m given lines. 
7. We take as current coordinates (X, Y, Z, W), or when plane-coordinates are 
employed (£, y, £, w): the coordinates of the vertex are throughout represented by 
(x, y, z, w); but in explanations &c., these are also used as current coordinates. The 
plane of projection is taken to be W = 0. The coordinates of the given points a, &c., 
are taken to be (x a , y a , z a , w a ), &c. There is no confusion occasioned by so doing, 
and I retain the ordinary letters (a, b, c, f, g, h) for the six coordinates of a line, it 
being understood that these letters so used have no reference whatever to the given 
points a, b, &c.; viz. the coordinates of the given lines a, &c., are (a a , b a , c a , f a> g a , h a ), 
&c.; there is sometimes occasion to consider the coordinates of other lines ab, &c., but 
the notation will always be explained. 
8. I write l, m, n, p, q, r for the coordinates of the line joining the vertex 
(x, y, z, w) with a point (x, y', z, w); viz. 
I =yz' —y'z, p — xu/ - x'w, 
m = zx' — z'x, q=yw' — y'w, 
n = xy' — x'y, r = zw' — z'w, 
(l a = yz a — y a z, &c., this being explained when necessary) ; and also 
P = . hy — gz + aw, 
Q — — hx . + fz -f- bw, 
F = goc-fy . +cw, 
S = — ax — by — cz . , 
(P a = h a y — g a z + a a w, &c., this being explained when necessary). 
This being so, then projecting from the vertex (x, y, z, w), say on the plane W = 0, 
the x, y, z coordinates of the projection of a point a are as : q a : r a (p a = xw a — x a w, &c.); 
and the equation of the projection of a line a is 
P a X + QoY-\- R*Z — 0, 
(P a = h a y — g a z + a a w, &c.). We thus have, in the projection on the plane W = 0, the 
m points and 6 —m lines situate in and touched by the conic.