190 REV. R. HARLEY ON SYMMETRIC FRODUCTS, AND 
p =PQ(P°Q?+ 12P°S? — 2PQ’S + 8%), and 
q =— P(110P°Q?S — 16P!Q* — 11 P*S*— 29 P°Q*S* — 2P°Q'S 
— 13P2S!+ PQ®— PQ’S' — Q'S?- 8°). 
The quantities w and v being now known, Bi, Ba and BG 
are given by 
/p4 oe 
B= / zy = V Pv, and B,= a 
and the roots of the quintic 
a2 —5Pe-—5Q2*?-5Rer+E=0 . . (10), 
where 
R=-ic=S8-P’, 
are the values of the expression 
A wd 3 
arya )* Pv + (1) = ee - 
22. Substituting for w its value in (8), we obtain 
A’(pE+q)?+B'(pE +9) (mE +n) + C’(mE + )?=0, 
a relation which admits of simplification. In fact, if we 
restore the values of A’, B’; C’, m, n, p and gq in terms of 
P, Q and §, that relation takes the form 
A’m(PQSE?’ + «hk? + AE + p) =0, 
where x, X and yp are rational and integral functions of 
PQ and 8. 
; +, PQSE’+ «E?+r’E+p=0; 
or, in effect, 
PQSES + § P!‘Q?+ P’S?+ 4P?Q*S + PQ! + PS’ + Q’S*} FE” 
+ §121P°QS + 10P°Q? + 175 P!QS? — 88P*Q'S + 12 P*Q° 
+27P°QS? — 7PQ’S? + 2Q°S + 2QS*2E + 121 P°Q? 
4+ 121 P8S?— 561P7Q?S + 102P°Q! + 286 PS + 227 P°Q’S? 
— 137P!Q!S + 191 P*S* + 17P°Q* — 30P2Q’S* + 8P*Q#S? 
+ 26P°S' — 7PQ?S! — 7PQ'S + Q’+ 2Q'S?+S°=0 . (11), 
a relation due to Mr. Cockle, who first announced it in 
the Appendix to the Lady’s and Gentleman’s Diary for 
last year, p. 82. Of quintics, whose roots have, as yet, 
been exhibited,* this is the most comprehensive form. It 
* Mr. Cockle, in the Paper above referred to, does not ewhibit the roots 
