503] 
109 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
18. Singularities. The form of the equation shows at once that 0) 
(0)0 r ^ 10 P oint a * s a 2-conical point; in fact, for this point we have pabe = 0, 
pacf= 0, pabf= 0, pace = 0. 
(1) The line ab a simple line; in fact, for any point of this line we have 
pabe = 0, pabf= 0. 
(2) Ihe line abe.cdf a simple line; in fact, for any point of this line we have 
pabe = 0, pcd/= 0. 
(9) To show analytically that the cubic curve abcdef is a line on the surface, 
observe that the equation of the surface is satisfied if we have simul 
taneously (X being arbitrary) 
pabe. pacf— X. pabf. pace = 0, 
X. pcde. pdbf — pcdf. pdbe = 0. 
The first of these equations is a cone, vertex a, which passes through the points 
b, e c, f and which, if X is properly determined, will pass through the point d; the 
second is a cone, vertex d, which passes through the points b, e, c, f and which, if 
X is properly determined, will pass through the point a; the two determinations of X are 
dabe . dacf- X . dabf. dace = 0, 
X . acde . adbf— acdf. adbe = 0; 
giving the same value of X; and the equations then represent cones, the first having 
a for its vertex, and passing through d, b, e, c, /; the second having d for its vertex, 
and passing through a, b, e, c, /; the two intersect in the line ad, and in the cubic 
curve abcdef, which is thus a curve on the surface. 
Surface abcdea. 
19. The equation may be written 
(pabe.pcde. p 2 aac. db —pace.pdbe. p 2 aab. cd) 2 
+ 4pabe. pcde.pace. pdbe .pabe .pdbe. paa. pda = 0, 
or, what is the same thing, 
(pabe. pcde. p 2 aac. db + pace . pdbe. p 2 aab. cd)- 
+ 4pabe. pcde .pace. pdbe. pbad. pcad. pba. pea — 0, 
(the equivalence of the two depending on the identity 
— p 2 aab . cd. p 2 aac. db + pabe . pdbe. paa. pda — pbad . pcad. pba. pea = 0) 
1 Or course, as regards the present surface and the other surfaces for which the equation is given in 
an unsymmetrical form, the conclusion obtained in regard to any point or line of the suiface app les to 
every point or line of the same kind. Thus ab being a simple line, we have also ad a simple line, althoug i 
the equation, as written down, does not put this in evidence. 
2 The bracketed numbers refer to the lines of the Table. 
{Surface abedeu.}