1896-97.] Dr T. Muir on Ternary Quadrics . 229 
(13) Another mode of arranging the expansion is suggested by 
the mode just employed for arranging that part of it which in the 
preceding is independent of o\ Doing this we have 
o 
(1) all the terms free of ]$, viz. 
(AMR 4- amr) 3 + (A 2 M 2 R 2 + a 2 m 2 r 2 - 4 AMR. amr) ( bn p + clq) 
- 4(AMR + amr)bnp.dq 
- bnp.clq(bnp 4- clq) (F r ) 
(2) all the terms having an expression of the 3rd degree 
o 
following 2}, viz. 
{AMR + amr)%(amr - 2AMR)3A.j?.Z 4- ( bnq> + clq)%{amr — AMR)^A.^.£ 
+ bnpxlq^A.p.l (F 2 ) 
(3) all the terms having an expression of the 6th degree 
o 
following 2, viz. 
(AMR + amr)QZA.a.np.lq - 2J3A .m.p 2 .cT) 
+ ( bnp + clq + 3 AMR + 3amr)(%'ZAM.. r pb.lq - HAdsi.am.p.q) 
- 'Zamr^AM.pb.lq + 2 AMR2A V.Z 2 
4- 3(2AMR - amr ~ clq)%A.am.np 2 
4- 2(2AMR - amr - bnp)%A.ra.l 2 q 
- %bnp'2 l A.m.p 2 .el (F 3 )^ 
(4) all the terms having an expression of the 9th degree 
following % 
The full expansion is thus 
Fi 4* F 2 4- F 3 4- E 4 . r ■ ... 
(14) There is, however, another form of the eliminant which is 
of far greater interest than any of the preceding. It not only bears 
on its face the peculiarities of structure which we know it ought 
to possess at least inherently, but it is also — and mainly by reason 
of this — much better suited for the calculation of the final ex- 
pansion. 
It is got by obtaining seven equations in the compound 
variables 
x 2 y, y\ z\ yd, zx 1 , xy\ xyz 
and eliminating dialytically. The first six equations are readily 
