120 
ON THE SURFACES THE LOCI OF [503 
have, as before, paa = — v. aba and pba = u. abet, with the like formulae with 
13 and y in place of a. The factor thus becomes 
V — uv 
Vaba, Vaba, Vpca 
V abß, V abß, \/pcß 
V aby, V <267, VpC7 
which vanishes, being = 0 1 ; and the norm is thus = 0 4 , viz. the line is 
4-tuple. 
(2) The line a is 8-tuple: in fact, for a point on the line we have paa — 0, 
pba = 0, pea =0, whence each factor vanishes, being = 0^, and the norm 
is therefore 0 8 . 
(3) The line (ab, a, (3, 7) is 4-tuple: in fact, writing z = 0, w — 0 for the 
equations of the line, we have h a = 0, hp = 0, h y = 0, and z a w b — z b w a = 0, 
or say w a = \z a , w b = \z b . Hence, writing 
I =(9- *«)« ~ (/+ X6) y, 
viz. I a = (g a — \a a )x — (f a + \b a ) y, &c., for z = 0, w = 0, we have paa = z a I a> 
pba = z b I a ; and similarly pa(3 = z a Ip, pbfi = z b Ip, and pay = z a I y , pby = z h I y . 
The factor thus is 
^!z a z b 
V/., 
Vi, 
Vpca 
Vi, 
Vi, 
Ipcß 
Vi, 
v/;, 
Vpcy 
which vanishes, being = 0 1 ; there are four such factors, or the norm 
is 0 4 ; whence the line is 4-tuple. 
(8) The line abc. a. ß is a 4-tuple line. To prove it, take as before w = 0 for 
the equation of the plane abc, and (z = 0, w = 0) for the equations of 
the line in question. We have h a — 0, hp = 0, iu a — 0, w b = 0, w c = 0 ; 
whence (if z — 0, w = 0), writing for shortness I — gx—fy (viz. I a =g a x—f a y, 
Ip= g ß x —f ß y), we have paa, pba, pca = I a z a , I a z b , I a z c , and similarly 
paß, pbß, pcß = IßZ a , I ß z b , l ß z c : the factor thus is 
'Jlo.Za , I a Zb, ^ Io.Z c 
^Iß Za , Iß Z b , V/, Z c 
Vpay , \fpby , Vpc7 
which vanishes, being = 0 1 : and 
giving to the radicals the signs 
= 0 4 . 
there are 
+ , — at 
four such factors, obtained by 
pleasure: hence the norm is 
{Surface abcaßy.}