338 
Proceedings of Poyal Society of Edinburgh. [sess. 
Among the remaining six relations we find two that are indepen- 
dent of H but are quadratics in F, and two that are independent 
of F hut are quadratics in H. The elimination <?f F from the first 
pair and the elimination of H from the second pair ought, one 
would think, to lead us to one and the same resultant, and ought, 
indeed, to give us the very resultant obtained previously from the 
set of four relations. On trial this is found to be the case ; con- 
sequently, we shall obtain a new mode of arriving at the eliminant 
of our set of equations 
B x 2 - Dxy + A y 2 — 0" 
C?/ 2 -Eyz + Bz 2 = 0 
Lz 2 - K zw + Civ 2 = 0 
Aw 2 - G wx +,Bx 2 = 0 
if from them we can deduce the equation 
A f L ?/ + B»> 2 \- - B — f + (BG 2 + D 2 L - 4ABL) = 0 
\ yw ) \ yw ) 
the latter being suggested by one of the above-mentioned quad- 
ratics in H, viz., 
1 
2 
2A D 
D 2B 
G H 
G 
H 
2L 
or AH 2 - DGH + BG 2 + D 2 L - 4ABL = 0 . 
Fortunately, the required deduction is perfectly easily made. 
We may either, as before, simply substitute for D, E, G their 
values derived from the given equations, or we may improve upon 
this, as follows : — - 
Starting with the second term we see that 
DG(Ly 2 + B w 2 )yw 
= D y • G^(Lv/ 2 + B tv 2 ) , 
Bx 2 + Ay 2 Aw 2 + Lx 2 
x 
X 
. (By 2 + Biv 2 ) , 
(ABw 2 + + BLx 2 + AL^iy + Bro 2 ) 
X 
= A(L y- + B vflf + ( A 2 ^ + BLx 2 )(L?/ 2 + Bw 2 ) 
X 2 
