546 NOTE ON DR MCJIR’s PAPER ON ELIMINATION. [963 
But L, M, N are sums of mere numerical multiples of U, V, W, viz. we have 
2 L = — aU + bV + cW, 
2M — aU-bV+cW, 
2 N= aU+bV-cW, 
so that the original equations U=0, V=0, TT=0 are equivalent to and may be 
replaced by L = 0, M= 0, N = 0. 
Muir shows that we have identically 
L — fP = x {Ax + Hy + Gz), 
M -gQ = y {Hx + By + Fz), 
N — JiR = z (Gx +Fy + Gz), 
where observe that the first of these equations is 
(fx 2 - ayz) {bz 2 - 2fyz + cy 2 )\ 
— (fy 2 — byz) {ex 2 — 2gzx + az 2 ) r = 2xyz (Ax + Hy + Gz); 
— (fz 2 — cyz) (ay 2 — 2hxy + bx 2 )) 
and similarly for the second and third equations. 
He thence infers that the elimination may be performed by eliminating x, y, z 
from the equations 
Ax + Hy + Gz = 0, 
viz. that the result is 
Hx + By + Fz = 0, 
Gx + Fy + Gz = 0, 
A, H, G 
H, B, F 
G, F, G 
= 0, 
that is, K 2 — 0 as before. 
The natural inference is that K being =0, the three linear equations in (x, y, z) 
are equivalent to two equations giving for the ratios x : y : z rational values which 
should satisfy the original equations U=0, V=0, W =0: the fact is that there are 
no such values, but that, K being = 0, the three equations are equivalent to a single 
equation: for observe that, combining for instance the first and second equations, 
these will be equivalent to each other if only 
A_H_G 
H~ B~F’ 
that is, 
AB-H 2 = 0, GH-AF = 0, HF-BG = 0,