416 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
a = jq 4 + p 2 4 +6/x, 
b = +^ 2 4 A 2 +3^, 
C = : jq 4 A^ 2 "t "t 2jU-S 2 » 
^ = W+^V + 3 w 25 
e = i5i 4 A x 4 + 2? 2 4 A 2 4 +6 /xs 2 2 , 
and from these by operations which lead to the elimination of p t 4 , p 2 4 
from every consecutive triad of equations 
as 2 -bs 1 + c- /x(8s 2 - 2s 1 2 ) = 0 , 
bs 2 -cs 1 + d - fx(i s 2 - s l 2 )s 1 = 0 , 
cs 2 -ds l + e- /jl(8s 2 - %s-f)s 2 = 0 , 
or, if we put v for — /ul(8s 2 — 2 s x 2 ) , 
as 2 - bs 1 + (c + r) = 0 
bs 2 - (c - + d = 0 
(c + v)s 2 - ds 1 + e — 0 
From the resulting cubic equation 
a b c + v 
b c-\v d =0 
c + v d e 
v can be determined, and thence in backward order s 1 , s 2 ; p) \ , X 2 ; p x , y> 2 ; 
q 1} q 2 ,m. 
In passing, note is taken of the fact that the said cubic when arranged 
according to powers of v is 
ct b c 
(i ae - 4 bcl + 3 c 2 )v + 2 
bed — 0 
c d e 
and that ae — 4<bd + 3c 2 and the determinant here appearing are the two 
invariants * of the quartic under investigation. 
The reduction of the octavic (a 0 , a 1 , . . . , a 8 jj x , y) 8 to the form 
U-f + U 2 8 + U 3 8 + U^ + 7 0 €W 1 2 W 2 2 W 3 2 W 4 2 
where u r =p r x-\-q r y is shown in similar fashion to depend on the solution 
of the quintic equation 
= 0 
% 
a i 
a 2 
a 3 
a 4 -v 
a 4 
a 2 
a z 
a 4 + \v 
a 5 
a 2 
a 3 
a 4 ~ 
a S 
a 6 
CO 
e 
<^4 + 
% 
«6 
a 7 
a 4 -v 
a b 
«6 
a 8 
* The term “invariant” is first used in this paper. 
