503] 
THE VERTICES OE CONES WHICH SATISFY SIX CONDITIONS. 
121 
Surface aba/3y8. 
35. The equation is 
Norm {Ipaa .pba. p 2 /3y8 — *Jpa/3 .pb/3 .p 2 y8a + fpay .pby ,p 2 8a/3 - Vpa8 . pb8 .p 2 a/3y] = 0, 
where the norm is the product of 8 factors each of the order 3. As before, paa = 0 
is the equation of the plane through the point a and the line a; viz. paa has the 
value previously mentioned: and p 2 {3y8 = 0 is the equation of the quadric surface through 
the lines ¡3, y, 8. 
36. Investigation. In the projection, taking £, rj, £ as current line-coordinates, the 
equation of the conic passing through the projections of the points a, b is 
^(pa^ + q a v + r a £) (pbZ + qiv + r b 0 + + Br) + (7£ = 0, 
where A, B, G are arbitrary coefficients. To make this touch the projection of the 
line a, we must write f : tj : £ = P a : Q a : Ra', and then 
and similarly 
we 
PaZ + q a V + r a f= p a P a + q a Q a + r a R a , 
= W a (x Pa + y Qa + Z R a ) 
- w (<v a P a + y a Qa + Z a R a ), 
= -w (x a P a + y a Qa + z a R a + w a S a ), 
= — w. paa, 
PbZ + qbV + r b £ = - w.pba. 
Hence the equation is 
w Vpaa .pba + A P a + BQ a + CR a = 0; 
and forming the like equations for the lines ß, y, 8 respectively, and eliminating, 
have 
Vpaa. pba, P a , Q a , R a 1 = 0 
fpaß.pbß, Pß, Qß, Rß 
'Ipay. pby, P y , Q y , R y 
\/pa8.pb8, P s , Qs, R& 
which, throwing out a factor w, becomes 
fpaa. pba. p-ßy8 — Ipaß. pbß ,p 2 y8a + I pa y .pby .p 2 8aß — fpa8 .pb8. p-aßy = 0 ; 
or, taking the norm, we have the above written equation. 
37. Singularities. The equation shows that 
(0) The point a is a 4-conical point; in fact, for the point in question we have 
paa = 0, paß = 0, pay = 0, pa8 = 0 ; each factor is =0*, and the norm 
is =0 4 . 
{Surface abaßyd.} 
C. VIII. 
16