503] 
113 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
We have similarly 
Pa, Qa, Pa 
A", B", C" 
A'", B'", O'" 
taken to be =p 2 aac.db. 
24. We have in like manner the other two determinants 
Pa, 
Qa, 
Ra 
and 
Pa, 
Qa, 
R a 
A , 
B, 
c 
A', 
B' , 
G' 
A", 
B", 
C" 
A'", 
B"\ 
C" 
taken to be — p 2 a ab • ас and p 2 a cd . db respectively. 
But we have 
p 2 a ab. ас =paa. pabc, 
(viz. geometrically the hyperboloid through the lines a, ab, ас breaks up into the plane 
paa through the line a and point a, and the plane pabc through the points a, b, c). 
And similarly 
p 2 a cd .db = —p 2 a dc ,db = +p 2 a db . dc = pad .pdbc; 
whence, substituting for the several determinants, we have the foregoing equation of the 
surface. 
25. Singularities. The form of the equation shows that 
(0) The point a is a 4-conical point: in fact, for this point we have pabe = 0, 
p 2 a ac ,db= 0, pace = 0, p 2 a ab. cd = 0. 
(1) The line ab is a double line: in fact, for any point of the line we have 
pabe = 0, p 2 a ab .cd = 0, pabc = 0. 
(2) The line a is a double line: in fact, for any point of the line we have 
p 1 a ac.db = 0, p 2 a ab .cd = 0, pact = 0, pda. = 0. 
(7) The line abe.cd.a is a simple line: in fact, for any point of the line we 
have pabe = 0, p 2 aab.cd = Q. Observe that, on writing in the equation 
pabe = 0 the equation becomes (p 2 a ab. cd)' 2 = 0; so that the surface along 
the line in question touches the plane pabe. 
Surface abcdaß. 
26. The equation of the surface is 
Norm {*\/paa . paß. pbcd — 'fpba.pbß .pcda + ^pea .pcß .pdab — Vpda . pdß. pabc] = 0, 
where the norm is the product of 8 factors. 
As before, paa — 0 is the equation of the plane through the point a and the 
line a; and pbcd = 0 the equation of the plane through the points b, c, d. The form 
is unique. 
{Surface abcdaß.} 
C. VIII. 
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