228 Proceedings of Royal Society of Edinburgh. [sess. 
and that 
o o 
^ a iftP 3 = bnp.clq%$A.p.l 
- %amriAM.pb.lg+ %AMK%A 2 .p 2 .l 2 - Zbnp^AM.a.lq 2 
- %bnp^A.ra.l 2 q - o-(%'£<A.mp 2 .d - 2A .a.np.lq) 
+ %’%A 2 'hLam.]o 2 .lq + 225 AM 2 .am.pb.q 2 - 22AM 2 .pb 2 .lq 2 
+ 222AM .r.pb 2 .l 2 q - 22A 2 M .p 2 b.l 2 q. 
On account of the four terms in o- which here make their appear- 
ance the full coefficient of <r becomes 
— 4 bnp.clq 
+ 2 (amr - 2AMR)2A.^>.Z 
o o 
- 32AM.am.j9.g + 2A .a.np.lq 
+ 322AM.^?AZ^ - 22A .m.p 2 .cl. (D) 
The remaining terms, independent of a, may be classified in four 
o 
groups, viz. (1) those without the sign 2 
(bnp + cZ<?)(A 2 M 2 R 2 + a 2 m 2 r 2 - 4 AMR.emr — bnp.clq) ; (E^) 
o 
(2) those having an expression of the 3rd degree following 2 
bnp.clq'%%A.p.l + ( bnp + clq)%(amr - AMR)2A.p.Z ; (E 2 ) 
o 
(3) those having an expression of the 6th degree following 2 
O o 
- (bnp + clq)%AAl.am.p.q + (bnp + clq^^AAL.pb.lq 
- 2 amr%KAL.pb.lq + 2AMR2A 2 .p 2 .Z 2 
+ 2(2AMR - amr - clq)^A.am.np 2 
+ 2(2AMR - amr - bnp)^,A.ra.l 2 q 
- %bnp%A.m.p 2 .cl ; (E 3 ) 
o 
(4) those having an expression of the 9th degree following 2 
- 22A 2 M.am 2 . j p 3 - 22AM 2 .pb 2 .lq 2 — 22A 2 M p 2 b.l 2 q 
+ 22A 2 M .m.p z b.l + 22 AM 2 . a. b.lq 5 
+ '%%A 2 AL.amp 2 .lq + %'%AAi 2 .am.pb.q 2 
+ 22AM.r.pZ> 2 .Z 2 g 
- 222AM.rar.p 2 & 2 .Z. (E 4 ) 
The full expansion of the eliminant is thus 
cr 3 + cD + E 4 + E 2 + E 3 + E 4 . 
