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503] THE 'S ERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
3 . Lines [ab, a, /3, 7]: taking the vertex in one of these tractors, the cone cannot 
be a piopei cone, but (if it exist) it must be either a line-pair having the 
tractor for one of its lines, or else a plane-pair having the tractor for its 
axis. The two cases are: 
Surface abcaf3y. Cone is a plane-pair, the two planes intersecting in the tractor, 
and passing, the one of them through the points a, b, the other through the 
point c. The vertex being an indeterminate point on the tractor, the tractor is 
situate on the surface. 
Surface aba/3y8. Cone is a line-pair, one line being the tractor, the other a line 
drawn in the plane of the tractor and ab to meet 8, and which meets the 
tractor in an arbitrary point thereof: the tractor is thus a line on the surface. 
4°. Lines [a, /3, 7, 3]: taking the vertex in one of these tractors, then, as in the last 
case, the cone is either a line-pair having the tractor for one of its lines or 
a plane-pair having the tractor for its axis. The three cases are: 
Surface aba./3y8. Cone is a plane-pair, the two planes intersecting in the tractor 
and passing through the points a, b respectively. 
Surface aa/3y8e. Cone is a line-pair, one line being the tractor, the other a line 
in the plane of the tractor and a, meeting the line e and meeting the tractor 
in an indeterminate point. 
Surface cc/3y8e£. Cone is a line-pair, one line being the tractor, the other a line 
drawn from an indeterminate point of the tractor to meet the lines e and £. 
5°. Lines [ab, cd, a, /3]. Cone is a plane-pair, the two planes intersecting in the 
tractor, and passing through the points a, b and the points c, d respectively. 
6°. Line abc, def. Cone is a plane-pair, consisting of the two planes abc and def. 
7°. Line abc, de, a. Cone is a plane-pair, the two planes intersecting in the line; 
one plane being abc, the other a plane through the line de. 
8°. Line abc, a, /3. There are two cases: 
Surface abcda/3. Cone is a plane-pair, the two planes intersecting in the line ; the 
one being abc, and the other passing through the point d. 
Surface abca(3y. Cone is a line-pair; one line being abc, a, ¡3, the other a line 
in the plane abc meeting the line 8, and meeting the line abc, a, ¡3 in an 
indeterminate point. 
9°. Cubic abcdef. Each point of the cubic is the vertex of a proper cone passing 
through the cubic, and therefore through the six points; that is, the cubic is 
a line on the surface abcdef. 
10°. Quadriquadric oc(3y, Se£. Cone is a line-pair; viz. it is composed of the lines 
drawn from any point of the curve, one of them to meet the lines a, /3, 7, 
and the other to meet the lines 8, e, £. 
11°. Excuboquartic a/3y, Be, a. Cone is a line-pair; the two lines being, one of them 
a line at pleasure meeting a, /3, y, the other the line which, in t e p ane 0 
the other line and the point a, meets the lines 8, e.