116 
ON THE SURFACES THE LOCI OF 
[503 
viz. 7 a = ((jo. Tut*)x — (f a + X6 a ) y, &c., and writing z = 0, w — 0, we have paa .paß = z a 2 I a I ß , 
pboL .pbß = z b 2 I a I ß , pea. pcß = z 2 J a J ß , pda.pdß = z d 2 J a Jß ] and the factor of the norm 
(reverting to the expression thereof as a determinant) is 
X , 
y 
z a ^ Ialß, 
x a, 
Va, 
Za, 
^Z a 
Zb 'll «Iß, 
x b, 
Vb, 
Zb, 
\z b 
Zc'/JaJß, 
X c , 
Vc, 
Ze, 
fXZ c 
Zd'ljo.jß, 
'Zd, 
dd, 
z d , 
pz d 
which vanishes. In fact, resolving the determinant into a set of products of the form 
+ 2.13.45, where the single symbol denotes a term of the top line, and the binary 
symbols refer to the second and third lines, and the fourth and fifth lines respectively 
(denoting minors composed with the terms in these pairs of lines respectively); then 
each product will contain a term 14, 15, or 45, and the minor so designated (to which 
ever of the two pairs of lines it belongs) is =0. The factor is thus evanescent, being, 
as it is easy to see, = 0 1 . There are two factors which vanish; viz. taking the first 
radical to be +, the second radical must be also +, but the third and fourth radicals 
may be either both + or both - ; the norm is thus = 0 2 , viz. the line (ab, cd, a, /3) is 
a double line. 
30. (8) The line cibc, a, (3 is a double line. To prove this, take w = 0 for the 
equation of the plane abc, and (z — 0, w = 0) for those of the line in question; we have 
K = 0, hp= 0, w a = 0, w b = 0, w c = 0; and writing I a = -g a x +f a y, Ip = - gpx+f a y, then 
for z = 0, w = 0, the factor expressed as a determinant is 
which is 
X , 
y 
Za'llJß, 
^a, 
y a , 
Za, 
(Z b , 
yb, 
Zb, 
z e 'lljß, 
x c , 
y c , 
Z c , 
'Jpda. 
pdß, 
(Zd, 
yd, 
Zd, 
'JTJt 
X , 
y 
Za, 
(Za, 
ya, 
Z a 
z b , 
(Zb, 
yb, 
z b 
Z c , 
(Zc, 
yc 
Zc 
w d 
and consequently vanishes, the form being 0 1 . There are two such factors, viz. the 
radical Vpda. pd/3 may be either + or —, hence the norm is = 0 2 . 
31. But it is to be further shown that the line is tacnodal, each sheet of the 
surface being touched along the line by the plane w = 0: we have to show that the 
{Surface abedap.}