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.