342 
EEV. E. HAELET ON THE METHOD OE STMMETEIC PEODHCTS. 
to which, by the method of Mr. Jeeeaed, Mr. Cockle, or Professor Stlvestee, the 
complete quintic may be reduced, then the foregoing formulae may be made available. 
For, any symmetric function of the roots may be resolved into a function of the sums of 
the powers of the roots, and these sums, in any of the above cases, can of course be easily 
calculated. Thus, taking the second form, 
which in many respects is the most convenient, we readily find 
21=0, 22 = 0, 23=-3tZ, 24=0, 
and therefore 
2A=222r=(21)^22-22123-(22)^+224=0. 
In like manner we obtain 
2X^=4. 5V/, 
2x^=6cZ^ 
2X‘‘=4.5^^^y’^ 
2x^ = 158.5<Z/+5yb 
2A®=6(^«+4.5«^^®/k 
Therefore the equation in X is 
A®-2.5VyX^-26Z^X^+5^c?y’V-(58cj!^+5y‘^)/X+(?«=0; 
and since, in this case (art. 11), 
■r4(ir)=— 5?i, 
the corresponding equation for the resolvent product is 
^-2 . 5V(/’^‘+2 . 5W+5«6Zy^^H5=(58(^®+5®/)/^+5®(Z®=0, 
where I have written 6 for This equation was first given by Mr. Cockle. See 
his paper entitled “ Pesearches in the Higher Algebra,” printed in the second part of 
the fifteenth volume of the ‘Manchester Memoirs.’ Some interesting and curious trans- 
formations of the equation may be found in the same paper. 
Section III. 
The Symmetric Product for Quintics. 
16. We know that, for the perfect form (art. 11), 
= — 15a^e 5aV — 5al>^c 5a*X. 
And by definition (art. 1), 
n(x)^0A¥A^6‘ 
It hence appears that the symmetric product H is of twenty-four dimensions with respect 
to X. The partitions of twenty-four, for the quintic, are as follows : — 
