102 
ON THE SURFACES THE LOCI OF 
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Special forms of (Quadri-)Cones. 
4. We have to consider the special forms of (quadri-)cones ; these are: 1°. The 
sharp-cone, or plane-pair; that is, a pair of two planes, intersecting in a line called 
the axis, the vertex being in this case an indeterminate point on the axis. Observe 
that a plane-pair passes through a given point when either of its planes passes 
through such point; it touches a given line when its axis meets the given line. 
2°. The flat-cone, or line-pair; viz. this is a pair of intersecting lines, their point of 
intersection being the vertex of the line-pair, and the plane of the two lines being 
the diametral of the line-pair. Observe that the line-pair passes through a given point 
when its diametral passes through such point; it touches a given line when either of 
its lines meets the given line. 3°. There is a third kind, the line-pair-plane ; viz. 
the two planes of the plane-pair may come to coincide, retaining, however, a definite 
line of intersection, or axis: or again, the two lines of a line-pair may come to 
coincide, retaining a definite plane or diametral; that is, in either case we have a 
plane passing through a line; and which is to be considered indifferently as two 
coincident planes intersecting in the line, or as two coincident lines lying in the plane. 
But there is not, in the present Memoir, any occasion to consider this third kind of 
special cone. 
The letters C, P, L in the Table denote that the cone is a (proper) cone, plane- 
pair, or line-pair, as the case may be. 
Singular Lines and Curves on the Surfaces. 
5. We may establish d priori the existence, and even to some extent the multi 
plicity, of the several lines and curves on the surfaces ahcdef... afiySe^. Thus: 
1°. Lines ah: take for the vertex of the cone a point at pleasure on the line ah; 
the cone passing through h will ipso facto pass through a\ and the conditions 
are thus that the cone shall pass through h and satisfy four other conditions— 
in all, five conditions: and there is thus a cone with the point in question as 
vertex; that is, the line ah is situate on the surface. Moreover, for the surfaces 
ahcdef abcdea, abcdafi, ahca/3y, abafiyS respectively, for a given position of the 
vertex on the line ah, the number of cones is 1, 2, 4, 4, 2 respectively: and 
these are the multiplicities of the line ah on the several surfaces respectively. 
2°. Lines a : take for the vertex of the cone a point at pleasure on the line a; then 
the cone ipso facto touches the line a, and there are only five other conditions 
to be satisfied; that is, we have a cone with the vertex in question; or the 
line a is situate on the surface. Moreover, for the surfaces abcdea, ahcda/3, 
abca(3y, aha/3yS, aa/3ySe, a/3y8e% respectively, the number of cones is 1, 2, 4, 4, 2, 1 
respectively: and it may be seen that the multiplicities of the line a are the 
doubles of these numbers, or are = 2, 4, 8, 8, 4, 2 for the several surfaces 
respectively.