340 Proceedings of Royal Society of Edinburgh. [sess. 
9. These modes of solving our problem suggest further to us 
that, by ringing the changes in selecting eight of the ten equations 
cqa 2 = A, etc., we may obtain a variety of new problems in elimina- 
tion, the eliminants of which may be foretold, so to speak, by 
keeping an eye on the ten vanishing minors of the related dis- 
criminant. Thus we may choose the eight equations 
= 5 a i/^2 ~b a 2^l = » Pi7* + ^ 2 ”/] = > 
yi72 = C, a i72 + a 2 7i ==F » 7 i 8 2 + 72 8 i = K > 
Sj 8 2 = L , cq 8 2 + a 2 = G , 
and seek to eliminate cq, /? x , y T , 8 X a 2 , /3 2 , y 2 , 8 2 ; or, what is the 
same, to eliminate cq, /3 V y 15 8 1? a 2 from 
c A 2 + By, 2 = e /3|7i . 
LV + CV-KyA; 
or, what is still the same, to eliminate /?, y, 8, from the triad 
FL/3 2 y - DLy 2 /3 + GBy 2 S - FBS 2 y + DC^/3 - GC/3 2 S = 0 \ 
C/3 2 — E/3y + By 2 = 0> 
Ly 2 -KyS +CS 2 = 0). 
It is not readily apparent how this would be accomplished in a 
direct manner, yet we know that by the elimination of H from 
the two equations 
2B 
E 
H 
E 
2C 
K 
H 
K 
2L 
D 
2B 
H 
F 
E 
K 
G 
H 
2L 
a l^ + “ iPl = D > 
c 
a i~ + a 27i = * > 
7i 
a iy + a 2^i = & » 
the same resultant will be 'obtained; and the latter equations, 
being quadratics in H, the resultant is immediately found to be 
I C LE 2 + EK 2 
F 2EGK + 2DEL 
C EK 
F GE + DK 
EK LE 2 + EK 2 - 4BCL 
GE + DK 2BGK + 2DEL-4BFL 
