1896 - 97 .] 
Dr Muir on Quaternary Quadrics. 337 
efficients, but from which the two not desired may be eliminated. 
These are 
2A 
D 
F 
D 
2B 
E 
or 
DFH - 2BGF - 2 A EH + 4 ABK + DEG - D 2 K = 0 , 
G 
H 
K 
D 
2B 
E 
F 
E 
2C 
or 
EFH - 2BKF - 2CDH + 4BCG + DEK - GE 2 = 0 , 
G 
H 
K 
D 
E 
H 
F 
2C 
K 
or 
KFH - 2ELF - 2CGH + 4CDL 4- GEK - DK 2 = 0 , 
G 
K 
2L 
2 A 
D 
G 
F 
E 
K 
or 
GFH - 2DLF - 2AKH + 4AEL + DGK - EG 2 - 0 , 
G 
H 
2L 
where it 
is clear 
that we have the means of eliminating F and H 
and obtaining a relation connecting A, B, C, L, D, E, K, G. The 
thought therefore occurs to one, that if it were possible to deduce 
four equations like these from the four given equations, our difficulty 
would be overcome. Of course, the deduced equations could not be 
exactly like these, for they could not possibly contain the letters F 
and H. But any quantities whatever that might occur as F and H 
do occur would suit our purpose equally well, because we only want 
them in order to eliminate them. And here another suggestion 
comes in, viz., that as the equivalents of F and H in the set of ten 
equations, a 1 a 2 = A, etc., are 
a i72 + a 2ri 
and 
+ AA » 
or a r — + 
7i a i 
and 
0 L BS, 
Pv 8, + ft ’ 
CaG + Ay-, 2 
or 1 ' ' 
and 
Lft 2 + BV 
a i7i 
A 8 1 
it is almost certain that these will suit. As a matter of fact, on 
turning to the solution which was obtained (§ 6), it will be found 
that they are exactly those which were taken and found to suit. 
8. These suggestions, however, are not all that are obtainable 
from the vanishing of the ten primary minors of the discriminant. 
