129
503]
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS.
where, for convenience, I retain the unaccented and accented letters (a,...), (a',...)
instead of these letters with the suffixes a. and /3 respectively. Writing as before
then
L =(af -af)x + ...
M = (ag' — a'g) x + ...
N = {ah' — a'h) x + ...
il = {gh' — g'h ) x + ...
A = fix — Lw,
B = fly — Mw,
G = flz — Nw,
and similarly
A' = fl'x — L'w,
B' = fl'y — M'w,
C' = fl'z - N'w;
where for L', M', N', fl' we have {a",...) and {a'",...). Hence
BC — B'C = w
y
z
w
M, N, fl
M', N', fl'
with like expressions for GA' - C'A and AB' - A'B; and substituting, we have
Pa,
Qa,
r a
= w
Pa,
Qa,
Ta,
A,
B,
G
X ,
y »
Z ,
w
A',
G'
L,
M,
N,
il
L',
M',
B T/ ,
il'
or substituting for p a , q a , r a their values xw a — wx a , yw a — wy a , zw a — wz a , this is
Pa,
Qa,
r a
= — w 2
X ,
y >
z ,
w
A ,
B,
G
x a,
ya,
Z a ,
W a
A',
B\
G'
L,
M,
N,
il
L',
M',
N',
il'
whence, omitting the factors w 2 , the equation is
\p 2 aye ,p 2 8/3e
x ,
y >
z ,
w
— p 2 a/3e. p 2 y8e
x ,
y >
z ,
w
% a,
ya,
Z a ,
W a
<^a ,
Pa ,
Za ,
W a
L,
M,
N,
il
L",
M",
N",
il"
L\
M',
N',
il'
L'",
M"',
N"’,
il'"
+ ^p 2 a.ye . p 2 8fie . p 2 a/3e . p 2 y 8 e
x ,
y »
z ,
w
X ,
y >
z ,
w
,
P a ,
z a ,
Wa
æ a >
ya ,
Z a ,
W a
L,
M,
N,
il
L',
M' ,
N',
il'
L",
M",
N",
il"
L'",
M'",
N"',
il'"
{Surface aafiyde.}
C. Vili.