188 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
—c=56,63+5f3 Ps, 
— d=5 6} 8, +58, B3- 36", 
and — e=f}+ 83+ B3— 6(8, B3— BBs). 
Or,* if we put 
—b=5P, —c=5Q, —d+10°=5S, 
e=H, Bz 8;=u, and B;=Puv, 
we shall have 
0 =O abe « ates 
w+ Pyu— PSv=0......». . 2) 
—10P? v2 w+ P® v3 + P?(5PQ + E)v?+ Pov= 0. <8); 
ae elimination of v between (1) and (2) gives 
u’—Quv?+ P(P?+S)u-P?Q=0 . . (A), 
and between (1) and (3), 
u® — 8Qu?> — (1LP?— 3Q?)ut + (PE + 15P°Q — Q*)u3 
— P?(QE + P+ 5PQ*)u?+ 2P°Qu—P°Q’?=0 . (5). 
Proceeding witb (4) and (5) by division, we obtain 
awv+bhut+e=O0 ... . (6), 
where 
a=—11P*+13P"S — 2PQ’+8’, 
b= — P(P?+S)E- Q(16P*+8P*S — PQ’+S"), 
and c= P?Q(PE+5P°Q+4+ QS). 
We might now carry on the process of division until we 
arrived at a linear equation in uw, but we shall be con- 
ducted to a more simple and elegant result by proceeding 
thus: — Arrange the terms in (4) and (5) according to 
ascending powers of wu, divide the latter equation by w’, 
and the result by the former, until there arises a biquad- 
ratic in wu. Arrange the terms of this equation according 
to descending powers of uw, and divide it by (4); there will 
result a quadratic in uv. In effect, we shall be conducted 
to 
* Mr. Cockle’s notation is here adopted, so far as my own objects will 
allow, in order to facilitate the comparison of our results. See a series of 
papers by that gentleman “On Equations of the Fifth Degree,” published 
in the appendices of the Lady’s and Gentleman’s Diary for 1848, 1851, 
1856, 1857 and 1858. 
