112 
ON THE SURFACES THE LOCI OF 
[503 
23. To reduce the foregoing result, we have 
xw a 
-wx a , 
yyJa 
-Wya, 
ZW a 
~ WZ a 
xw b 
— wx b , 
yw b 
- wy b , 
zw b 
-WZ b 
proportional to the three determinants which contain w, of the set 
x , 
y > 
z , 
w 
, viz. 
A = w 
y , z , 
w 
, &C. ; 
x a> 
z a, 
Wa 
ya, Z a , 
Wa 
X b , 
y b , 
z b , 
w b 
y b , z b , 
w b 
and 
similarly A', 
B', 
G' 
are proportional to the three 
determinants which contain w, of 
the 
set 
x , 
y > 
z , 
w 
, viz. 
A'—w 
y , z , 
w 
, (fee. 
x c , 
y c , 
Z c , 
w c 
yc, z c , 
w c 
Xd, 
ya, 
Zd, 
Wd 
yd, z d , 
Wd 
Hence, omitting 
the 
factor w, 
and 
writing (a, 
b, c, f, 
g, h) and (a', 
b', c', f', g', h') for 
the coordinates of the lines ab and cd respectively, we have 
A = hy—gz + SiW, A'— b'y — g'z + &w, 
B = — hx + fz + bw, B' = — h'x + f 'z + b'w, 
C — goc — iy + gw, C' = g'x — i'y + gw ; 
and thence 
BG' — B'G = Clx — Lw , 
CA'-C'A = Cly-Mw, 
AB' —A'B = Clz — Nw, 
where 
L = (af' — a'f) x + (bf' — b'f) y + (cf v — c'f ) z — (be' — b'c) w, 
M= (ag r - a'g)x + (bg' - b'g) y + (eg' - eg) z - (ca' - c'a) w, 
N = (ah' — a'h) x + (bh' — b'h) y + (ch' — c'h) 2 — (ab' — a'b) w, 
fl = (gh' — g'h) x + (hf' — h'f ) y + (fg r — f'g) ^ — (af' - a'f + bg' — b'g + ch' — c'h) w ; 
and consequently 
Pa, 
Qa, 
Ra 
A , 
B, 
G 
A', 
B f , 
C 
— Cl (xP a + yQa + zR a ) — w (LP a + MQ a + Af Bo) 
— — w (LP a + MQ a -(- iV R a + CIS a) ; 
or omitting the factor — w, say it is = LP a + MQ a + NR a + fl$ a , viz. this is =p 2 aab.cd. 
{Surface abedea.)