503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
119 
{Surface abcaßy.} 
32. The equation is 
Surface abcaßy. 
Norm 
paa, fpba, Vpca 
\/paß, fpbß, fpcß 
^pay, fpby, \!pcy 
where the norm is a product of 16 factors, each of the order f. As before, paa = 0 
is the equation of the plane through the point a and the line a; viz. paa has the 
value already mentioned. 
33. Investigation. In the projection, the equation of the conic touching the pro 
jections of the lines a, /3, y is 
A VR a X+Q a Y + R a Z+ B V P ß X + Q ß Y+ RfZ + C*JP y X + Q y Y + RfZ = 0; 
and to make this pass through the projection of the point a, we must write herein 
X : Y : Z—p a : q a \ r a . As before, we have 
PaX + Q a Y+ R a Z = w a (x l\ + y Q a + z R a ) 
W (x a 1 a + y a Qa + ZgRß, 
= -w (x a P a + y a Qa + z a R a + w a s a ), 
= — w .paa ; 
and so for the other terms; the equation thus is 
A fpaa + B fpaß + C fpay = 0; 
or forming the like equations in regard to the points b, c respectively, and eliminating 
we have a determinant = 0, and then, taking the norm, we obtain the above-written 
equation of the surface. 
34. Singularities. The equation of the surface shows that 
(0) The point a is 8-conical: in fact, for the point in question we have 
paa = 0, paß = 0, pay = 0 ; each factor is 0*, and the norm is 0 s . 
(1) The line ab is 4-tuple. To prove this, observe that the sixteen factors are 
obtained by attributing at pleasure the signs +, — to the radicals 
Vpbß, fpcß, fpby, fpcy; hence there are four factors in which fpbß, Ppby 
have determinate signs, but in which we attribute to the radicals 
Ppcß, fpcy the signs + or — at pleasure. It is to be shown that the 
four factors each vanish for a point on the line ab; that is, on writing 
therein for x, y, z, w the values ux a + vx b , uy a + vy b , &c. But we thus