108 
ON THE SURFACES THE LOCI OF 
[503 
{Surface abcdef.) 
or, as this may be written, 
8 
S' 
z 
aL + bM + cN, 
a'L +b'M -|-c'N, 
N 
aL' + bM' + cN', 
a'L' + b'M' + c’N', 
N 
and, for a point on the line aa{3, this 
is 
aL + bM + cN 
a'L + b'M + c'N 
= 0. 
aL' + bM' + cN’, 
a'L’+b'M' + c'N' 
But in the equations — a'L — b'M — c'N = — (af +• b'g + c'h) S', and — aL — bM — cN 
= (af + bg + cti) S, writing S = 0 and S' = 0, we have aL+ bM + cN= 0 and a'L+b'M+c'N = 0, 
and the equation is satisfied; that is, the surface passes through the line aot/3, and 
similarly it passes through the line ay8. 
Surface abcdef. 
16. The equation may be written 
pabe .pcde .pacf.pdbf— pabf. pcdf. pace. pdbe = 0, 
where pabe = 0 is the equation of the plane through the points a, b, e; and the like 
for the other symbols. The form is one out of 45 like forms, depending on the 
partitionment 
( ab . cd 
ac . db 
[ ad. be 
(ef)> 
of the six letters. 
17. Investigation. In the projection, the six points (p a , q a , r a ) are situate on a 
conic ; the condition for this is 
(p, q, rf = 0, 
where the left-hand side represents the determinant obtained by writing successively 
(p a , q a i r a)> & c -> for (p, q, r). The equation in question may be written 
where 
abe. ede. acf. dbf— 
abf. 
cdf. 
abe --- 
Pa, 
Qa, 
r a 
Pb, 
q b , 
n 
Pc . 
q e > 
r e 
, &c.; 
and substituting for p a ,..., their values, we have abe = w 2 .pabe, whence the foregoing 
result.