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.}