122 
ON THE SUBFACES THE LOCI OF 
[503 
{Surface abcdaß.} 
(1) The line ab is a 2-tuple line. To prove this, we have for the coordinates 
of a point on the line in question ux a -\-vx b , uy a + vyi, &c.; the values 
of paa, pba become as before — v. aba, + u . aba, and similarly for paß, pbß, 
&c.; so that, omitting the constant factor V — uv, the value of the 
factor is 
aba. p 2 ßy8 — abß. p 2 y8a + aby. p 2 8af3 — ab8. p 2 aßy. 
Taking (a, b, c, f, g, h) for the coordinates of the line ab, we have 
aba = af a + hg a + ch a + fa a + gb a + hc a , 
with the like expressions for abß, &c.; and substituting for p 2 ßy8, &c., their values, 
the factor is 
X 2 
y 2 
z 2 
w 2 
xw 
yw 
zw 
yz 
zx 
xy 
a 
fagh 
fabc 
fabg — fetch 
fbch 
-fbcg 
fegh 
fbgh 
b 
gbhf 
gobe 
- gach 
gbch - gabf 
geaf 
gehf 
hafg 
gahf 
c 
hcfg 
habe 
habg 
— httbf 
heaf - hbcg 
hbfg 
f 
abhf 
acfg 
abch 
- abcg 
abfg + achf 
aegh 
abgh 
g 
bagh 
bçfg 
— bach 
bcaf 
behf 
bcgh + bafg 
bahf 
h 
cagh 
cblif 
cabg 
— cabf 
ch fg 
cafg 
cahf + cbgh 
viz. the value of the factor is {a (fagh) + g(bagh)+ h (cagh)} x 2 + &c., where fagh =f a apg y h s 
is the determinant 
f a, g, h , 
the suffixes in the four lines being a, ¡3, y, 8 respectively. 
Collecting, this is 
( . cbhfy — bcfgz + fabcw) ( . hy — gz + aw) 
(— caghx . + acfgz + gabcw) (— hx . + f z + b w) 
(+ baghx — abhfy . + habcw) ( gx — fy . + c w) 
(- afghx - bfgliy - cfghz . ) ( &x + by + cz . ) 
+ bcgh \w (ax +by + cz)-x( . hy — gz + aw)] 
+ cahf[u> (ax + by + cz) — y (— hx . + fz + bw)] 
+ obfg \w (ax + by + cz) — z ( gx — iy . + cw)] = 0; 
or, what is the same thing, 
AP + BQ + CE + DS = 0,