503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
131 
49. Investigation. The projections of the six lines are tangents to a conic: the 
condition for this is (P, Q, P) 2 = 0, where the left-hand side represents the determinant 
obtained by writing successively (P 0 , Q a , R a ), &c. for (P, Q, R). The equation may be 
written 
where 
a/3e . ySe . ay£. 8l3Ç— a(3Ç. yS£\ aye . S/3y = 0, 
a/3e — 
Pa, 
Pp, 
P', 
Qa, Ra 
Qp, Pp 
Qe, R e 
and substituting for P a> &c., their values, we have a(3e = w. p 2 a/3e; whence the fore 
going result. 
50. Singularities. The equation shows that 
(2) The line a is a 2-tuple line: in fact, for each point of the line we have 
p 2 a/3e = 0, p 2 ay£ = 0, p 2 a/3£ = 0, paye = 0. 
(4) The line (a, ¡3, e, £) is a simple line: in fact, for each point of the line 
we have pa/3e = 0, pa(3£ = 0. 
(9) The quadriquadric a/3e. yS£= 0 is a simple curve on the surface: in fact, 
for each point of the curve we have pa/3e = 0, p 2 y8£=0. 
It may be remarked that the surface meets the hyperboloid p 2 a/3e in 
lines (a, /3, e) each twice, 6 
», (a, /3, e, y) „ once, 2 
» (®> S, €> „ „ 2 
» ( a > ft e, f) „ „ 2 
curve a/3e.yS£ „ „ 4 
2 x 8 = 16 
51. It might be thought that there should be on the surface some curve a/3ySe£, 
such as the cubic abcdef on the surface abcdef; but I cannot find that this is so. 
The equation of the surface is satisfied if we have simultaneously (X being arbitrary) 
p 2 a/3e . pay £ — Xpa/3£. paye = 0, 
XpySe .p 2 8/3% — p 2 . p 2 8/3e = 0; 
which equations represent quartic surfaces, the first of them having a. for a double 
line, and passing through the lines ¡3, y, e, £ (13 + 4x5 = 33 conditions, so that the 
equation of such a surface contains only an arbitrary parameter X); and the second 
having 8 for a double line, and passing through the lines /3, 7, € , £ But I see 
no condition by which X can be determined so as to have the same value in the 
two equations respectively. Of course, leaving it arbitrary, the two quartic surfaces 
intersect in the lines /3, 7, e, £ and in a curve of the order 12 depending on the 
arbitrary value of X, which curve lies on the surface a/3y8e£. 
{Surface ajSySef.} 
17—2