1 54 Proceedings of Boyal Society of Edinburgh. [sess. 
and eliminating x, y from this by ( /S ) we get 
4B'C 'm 2 n 2 = ( acn 2 + abm 2 + 2 ghmn) 2 , 
and therefore 
abm 2 + acri 2 + 2(gh± JVC')m?i = 0 . . (e) 
By eliminating m, n from (8), (e) we get 
(5c -/ 2 ){a 2 5c - (gh ± JVC') 2 } = {abc -f(gh ± JWC)} 2 
or, after a simple reduction 
5c{F± n /B 7 C'} 2 = 0, 
and on rationalizing we get 
a 2 b 2 c 2 A 2 = 0 . 
9. In the third place, it may be shown by the geometrical 
process of § 4 that a 2 b 2 c 2 is a factor of the eliminant. In that 
section it was shown that either PQR, or P'QR is a straight line. 
As we have already seen, if zero values of x or y or z are excluded 
and PQK he straight, the conic (5) must break up. But if zero 
values are not excluded, PQR is straight if any two of the three 
points P, Q, R coincide. How this may happen when a = 0 or 
5 = 0 or c.= 0. Thus, if PQR he straight, it follows that abc A = 0. 
Exactly the same result holds if P'QR is straight. Hence 
a 2 b 2 c 2 A 2 is a factor of the eliminant, and since the eliminant must 
he of order 12 in the coefficients there can he no other factor. 
In a similar way it can he shown that the eliminant of (2) is 
abcfgh A 2 . 
10. The dialytic method as exhibited in §§ 1, 3 does not give 
the true eliminant, because, first, the system (2) does not follow 
from (1) when any one of the three, x , y, z, is zero; and, second, 
the system (1) is satisfied by x = 0, bz 2 + cy 2 - 2xyz = 0 provided the 
single relation a = 0 is satisfied. But the dialytic process as given 
in § (7) does give the true eliminant, because the differential co- 
efficients of the Jacobian of A, B, C vanish when (1) are satisfied, 
provided x , y, z are not all zero. The elementary algebraic 
methods given by Tait, Lord M £ Laren, and Muir do not give the 
complete result of elimination, because they assume that x, y, z are 
all different from zero. 
