EEY. E. IIAELET ON THE METHOD OF STMMETEIC PEODHCTS. 
347 
19. We have thus obtained the symmetric product for the quintic wanting in its 
second term ; but it seems desmable on many grounds to calculate the product for the 
perfect form. A Tariety of methods of performing this calculation might be suggested. 
The reversal of the problem of transformation appears, at first sight, the most easy and 
practicable, and if in the foregoing expression for 11 the following substitutions be 
made, viz. — 
2b^ 
— for c. 
4b^ 3bc 
^ + diord. 
36 -* , 352 2bd . „ 
“53 + 52 
4^ 
■5^ 
+/for/, 
the result ufill he the symmetric product for the complete quintic 
(1, b, c, d, e,fjjv, l/=0. 
But it will be found on trial that this process, though apparently simple, does in point 
of fact involve prodigious labour. Mr. SAiiUEL Bills of Hawton, who kindly undertook 
to assist me in the calculation, communicated to me in the early part of the present year 
that portion of the expression into which enters. But the difficulties of the calcula- 
tion and the want of means of verifying successive results have led him to abandon the 
work as impracticable. An equally effective and a much more expeditious process is 
supplied by the following considerations. 
20. It occun*ed to Mr. Cockle that the symmetric product for the perfect quintic 
— 5Mar^ — 5P.r^ — 5Qa;^ — 51\.r +E= 0 
corresponding expression in P, Q, E and E is easily effected, and I find that tlie result (as yet unpublished) 
of the transformation is — 
n = 5''x 
+ 
625 
P‘- 
+ 325 
P^Q 
E + 2 
P' 
+ 2000 
P'^R 
+ 
560 
P'^QR 
+ 5 
P^R 
— 
250 
— 
85 
P>Q3 
+ 6 
P^Q- 
+ 2400 
P^R" 
+ 
268 
P'QR" 
+ 4 
P'R2 
— 
225 
PRPR 
— 
102 
P'Q^R 
+ 6 
P-Q'R 
— 
25 
P'^Q^ 
+ 
14 
P-Q® 
+ 1 
PQ" 
+ 
1330 
P'R^* 
+ 
35 
p:QR3 
+ 1 
PID 
+ 
95 
P’Qqi^ 
— 
7 
PQ3R2 
+ 1 
Q=R2 
— 
115 
FQRt 
+ 
2 
Q+l 
+ 
336 
P'lV 
+ 
0 
QR" 
+ 
10 
P^QS 
— 
58 
P3Q2R,-, 
+ 
14 
P^Q^R^* 
+ 
32 
P-R5 
— 
7 
PQ'R 
— 
7 
PQ^R" 
+ 
1 
Qt 
+ 
2 
Q'R* 
+ 
1 
R« 
