382 
ME. W. SPOTTISWOODE ON THE FORTY-EIGHT 
§ 2. The identical relations between the forty-eight coordinates. 
On inspecting the expressions (13) and (14) it will be seen at once that the following 
relations subsist, viz.: 
. N,+M,+F,=0, . N/+M/+F/=0, .... (16) 
N* + . +L- + (f = 0, N, + . + L, + z — 0, 
M*+L^ + . +H^=0, M/+L y + . +H*=0, 
F.* +G y +H z + . =0 ; F* +G^ +H/+ • —0. 
This is a set of 4 + 4 = 8 identical relations between the forty-eight coordinates. 
Added February 19, 1880. 
The next set of identical relations is to be sought among the first minors of the 
discriminants of U, V, . . . , bordered respectively with the coefficients of P, Q, . . . 
Thus, selecting T, we have to examine the coefficients of A, B, . . . (say the quantities 
St, 33, . . .) in the development of 
A, H, G, D, (A, B, . . . , suffix t). 
H, B, F, 
G, F, C, D- 
D*, D ;/ , D, 
We then have, changing the sign, 
St = BD, 2 —2FD,D y +CD/ 
= 
p, P', 8 | 
X 1 
8, S', y | 2 
-{1 
P, y, 8 | 
X I 
y, P', 8 1 } 
8, S', y | X | 8 
S', iS 1 
+ 1 
7; y , S | 
X I 
8, 8 ',p | 2 
= { 1 
P, P', 8 
X I 
8, S', y | — 
I y, P', 8 | x | 
8, S', £ | } 
8, 8 
7 
-{1 
P, y, 8 
X i 
8, S', y ] — 
1 7^ /, S 1 X | 
8, S', /3 | } 
1 8,8 
',P 
=P, 
P', 8, . 
7 X I 8, S', 
7 t Xy, y', 8 
• • P 
Pi, 
Pi, §1; 
7i 
7n 7m S n 
. . ft 
P,, 
Pff, . 
• 
7a 
7a> 7a) 8 a , 
. . & 
P, 
. . 8, 
S', 
7 
7) • • 
8, S', £ 
Pi, 
• • s. 
s/, 
7i 
7n • • 
s t , S/, Pi 
Pi, 
§, 
s/, 
7a 
Y-2, ■ • 
S., 87, & 
