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.)