378 
MR. W. SPOTTISWOODE OR THE FORTY-EIGHT 
we shall have, for [as 3 ], . . 
respectively, 
. 3 [as 2 ?/], . . . , i.e., 
the coefficients 
of x 3 , . . . 3as 2 ?/, . . . 
[a; 3 ] —a, a', 8, 8', . 
. (suffix 0) . 
.(9) 
a, a', 8, S', . 
■ ( » 1) 
a, a , 8, S', . 
• ( .. 2) 
. . a, a , 8, 8 
G ( „ 0) 
a, a', 8, 8 
■>( i) 
a, a , 8, 8 
S( .. 2) 
3[ary]=a, a', 8, S', . 
. +a, /3 , 8, 8 , . 
• +A « , S, 8 , 
. . (suffix 0) 
cl, a', 8, S', . 
cl, /S', 8, S', . 
A a', S, S', 
■•(»!) 
a, a', 8, S', . 
• a, A, 8, S', . 
A a ', 8, 8', 
• • ( „ 2) 
A A, S, S' . . A A, 8, 8' . . A A, 8, 8' ( „ o) 
A A, 8, S' • • A A, 8, 8' . . A A, 8, S' ( „ l) 
A A, 8, S' . . A A, 8, S' . . A A, 8, S' ( „ 2) 
But there is another and very useful form which may be given to the equations (8) 
and (8'). In fact, writing down only the first lines of the determinants, and putting 
I M, 
It, 
a | 
=u, 
I u. 
w', 
a' ] 
=U', 
(r, 
w'. 
suffix 
as) 
1 w, 
u', 
0 i 
=v. 
| u, 
u, 
A I 
=V', 
K 
w', 
?5 
2/) 
I u. 
It, 
y 1 
=W, 
I u, 
< 
y 1 
=W', 
(w. 
?d, 
5 ? 
*) 
| w, 
It, 
8 1 
= T, 
1 tt, 
?d, 
8' | 
= T', 
(?q 
w', 
? > 
0 
i w. 
a, 
CL j 
=P, 
I U, 
a, 
/ | 
a 
= F, 
{u, 
?d, 
J > 
as) 
1 u, 
A 
A| 
=Q, 
I u', 
A 
A I 
G 3 
II 
(«» 
u, 
2/) 
! 
y> 
y 1 
=R, 
1 u \ 
y> 
y' i 
II 
(«, 
u , 
>> 
*) 
| u, 
s, 
S' 
1 =s, 
1 tt'. 
8, 
S' | 
= S', 
U, 
0 
We may write the results of eliminating x, y, z, t respectively in the following 
forms : 
UP' -U'P = 0 ..(11) 
VQ' — YQ =0, 
WR'-W'K=0, 
TS' — T'S =0; 
and these may be regarded either as the equations of the projections of the curve on 
the four coordinate planes respectively, or as equations of conical surfaces passing 
through the curve, and having their axes parallel to the coordinate axes respectively. 
