114 ON THE SURFACES THE LOCI OF [503 
27. Investigation. In the projection, the equation of the conic touching the pro 
jections of the lines a, /3 is 
V(P„X + Q.Y+ R.Z) (P ß X + QßY + R ß Z) + AX + BY+CZ= 0, 
where A, B, G are arbitrary coefficients. To make this pass through the projection of 
the point a, we must write X : Y : Z=p a : q a : r a \ viz. we thus have 
P a X + Q a Y + R a Z = w a (x P a + y Q a + z R a ) 
- w (sc a P a + y a Q a + z a R a ), 
~ w (x a P a 4- yaQa z a R a I- w a SI), 
= — w . paa.; 
and similarly 
PßX + QßY+ R ß Z = — w .paß. 
We thus have 
w si pact. pa/3 + Ap a + Bq a + Gr a = 0. 
Or, forming the like equations for the points b, c, d respectively and eliminating, the 
■equation is 
which, substituting for (p a , 
converted into 
si paa. paß, 
Pa, 
q a , 
r a 
= 0; 
s/pbcL .pbß, 
Pb, 
qb, 
n 
sJpea .pcß, 
Pc, 
q c , 
r c 
s!pda. pdß, 
Pd, 
qa, 
r d 
r a ), &c., their values, 
viz. 
Pa = 
x a w, &c., is readily 
x , 
y > 
2 , 
W 
slpaa .paß, 
x a, 
P a, 
&a, 
Wa 
s/pba .pbß, 
Xb, 
yb, 
Zb, 
W b 
si pea .pcß, 
x c , 
y c , 
Zc, 
w c 
si pda. pdß, 
Xd, 
y d > 
z d , 
w d 
or, what is the same thing, 
\Ipaa . paß. pbcd — s/'pba. pbß . pcda + si pea.. pcß. pdab — si pda. pdß. pabc = 0; 
viz. taking the norm, we have the form mentioned above. 
28. Singularities. The equation shows that 
(0) The point a is an 8-conical point; in fact, for the point in question 
paa = 0, paß = 0, pcda = 0, pdab = 0, pabc = 0 ; each factor is of the form 
0 1 , and the norm is 0 8 . 
{Surface abedaß.}