[503 
503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
101 
Tangent plane along line is plane abc. 
Tacnodal line, each sheet touched along line by plane abc. 
Tacnodal line, each sheet of surface touched along line by hyperboloid a(3y. 
Surface touched along line by hyperboloid afty. 
2. In the Table, the upper margin refers to the surfaces, and the left-hand margin 
to the points, lines, and curves situate on these surfaces respectively; the body of the 
Table showing the number, and in ( ) the multiplicity, of these points, lines, and 
curves in regard to the several surfaces respectively. Thus, points a; for the surface 
abcdef, 6 x (2), there are 6 such points, each of them a 2-conical (ordinary conical) 
point on the surface: so abcdea, 5 x (4), there are 5 such points, each a 4-conical point 
on the surface (viz. instead of the tangent plane there is a quartic cone); and so on. 
Similarly, lines ab (viz. these are the lines joining two points a, b); for the surface 
abcdef, 15 x (1), there are 15 such lines, each a simple line on the surface; surface 
abcdea, 10 x (2), there are 10 such lines, each a double (ordinary nodal) line on the 
surface; and so on. We have in two places the multiplicity (2 + 2), which refers to 
a tacnodal line, as presently explained. The corner letters C, P, L denote respectively 
proper cone, plane-pair, and line-pair, as afterwards explained. 
3. The lines and curves referred to in the left-hand margin are : 
(1) ab, line joining the points a and b. 
(2) a, line a. 
(3) [ab, a, /3, 7], pair of lines meeting each of the four lines, or say the 
tractors of the four lines ab, a, /3, 7. As regards the surface aba/3y8, 
the multiplicity is given as (2 + 2), viz. the line is (not an ordinary 
nodal, but) a tacnodal line, each sheet touching along the whole line the 
hyperboloid a(3y. 
(4) [a, /3, 7, S], tractors of the four lines a, /3, 7, 8. 
(5) [ab, cd, a, /3] tractors of the four lines ab, cd, a, /3. 
(6) abc, def line of intersection of the planes abc and def. 
(7) abc, de, a, line in the plane abc joining the intersections of this plane by 
the lines de and a respectively. 
(8) abc, a, /3, line in the plane abc joining the intersections of this plane by 
the lines a and /3 respectively. As regards the surface abcda/3, the 
multiplicity is given as (2 + 2), viz. each line is (not an ordinary nodal, 
but) a tacnodal line, each sheet touching along the whole line the plane 
abc. 
(9) Cubic abcdef, cubic curve through the six points a, b, c, d, e, f, common 
intersection of the cones each having its vertex at one of the points 
and passing through the other five. 
(10) Quadriquadric «£7, Se£, intersection of the quadric surfaces a/3y and 8e£ that 
is, the quadric surfaces through the lines a, /3, 7 and 8, e, £ lespectively. 
(11) Excuboquartic a/3y, 8e, a, quartic curve generated as follows: viz. taking any 
line whatever which meets the lines a, /3, 7 (or say any geneiating line 
of the quadric a/3y), the plane through this line and the point a meets 
the lines 8, e in two points respectively; and the line joining these 
meets the generating line in a point having for its locus the excubo 
quartic curve in question (theory further considered in the sequel).