34G 
EEV. E. HAELET ON THE METHOD OE ST]\OIETEIC PEODHCTS. 
SO that the eyanescence of (3 involves the evanescence of and consequently of 11. 
Making (3^ vanish, the equations at the foot of the last article become 
c — 5l3.^f33, 
— d — 5 (f3^f3s + /3i/3y, 
—e = b((3l(3,-\- (3,(31)— 
—f — (3'l-3-(3l-{-(3l-{- 2(3,(31 g + jcd. 
And the elimmation of (3,, (3^, (3^ gives 
n= 
+ 
1 
c'^ 

65 
c^d 
/- 10 
C7 
— 
16 
+ 
560 
c^de 
+ 125 
c^e 
+ 
2 
c^d^ 
— 
85 
c^d^ 
+ 150 
c^d" 
+ 
96 
c^e" 
— 
1340 
c''de' 
— 500 
— 
9 
c^d^e 
+ 
510 
(?d^e 
— 750 
c'^d-e 
— 
1 
chi' 
— 
70 
ed^ 
— 125 
cd‘' 
266 
+ 
875 
c‘de' 
+ 625 
ce^ 
— 
19 
c^d‘e^ 
— 
175 
cd^e^ 
+ 625 
d"^ 
+ 
23 
d'd^e 
+ 
50 
dh 
+ 
336 
c^d' 
— 
250 
dd' 
— 
2 
cH^ 
— 
58 
c'^dre' 
+ 
14 
c-d'e^ 
— 
160 
c‘e® 
— 
7 
cd^e 
+ 
35 
cd'^e' 
+ 
1 
d^ 
— 
10 
d'e' 
+ 
25 
e® 
/=+ e25c^d 
— 3l25cde 
/^=0. 
In order to determine Ic (a constant numerical factor, dropped in the course of calcula- 
tion), let us take the particular equation 
for which we have (art. 11) 
and consequently 
^=5c^ 
n=5V^ 
Then since, in this case, the 
gives 
and therefore 
coefficients d, e, f severally vanish, the foregoing formula 
* The symmetric product for the quintic, deprived of its second term, 
5Pa;^— — 5Ea;+E=0, 
■was first calculated by Mr. Cockle. See his paper “ On Equations of the Eifth Degree,” published iu the 
Appendix to the Lady’s and Gentleman’s Diary for 1858. In the second section of my original memoir I 
have verified his result by an independent calculation and supplied the constant numerical multiplier. 
Mr. Cockle presented the product in the form of a function of P, Q, S and E, S being given by 
S=PHE. 
EoUowing the notation of my friend, I gave the product in the same form. But the passage to the 
