EEV. E. HAELEY ON THE :METH0D OF STMMETEIC PEODUCTS. 341 
+ 120S8M22 
+ 
19258*3*2 
+ 
20587*1* 
4-236587621 
-t-120S8754 
+ 
288587531 
+ 
38058752* 
4-2405874*1 
+486^87432 
+ 
3485873* 
+ 
372586*4 
-[-304586*31 
+200^86^2^ 
+ 
364586541 
+ 
702586532 
4-6465864*2 
-f 804^8643^ 
4-1188585*1 
+ 
684585*42 
4-844585*3* 
4-4885854^3 
4-1920584^ 
+ 
36057*3 
4-40057*631 
4-55257^62^ 
+ 
26457*5* 
-f 
77657*541 
4-56557*532 
4-48057^^2 
4-102457*43* 
+ 
264576*5 
4-564576*41 
4-707576^32 
+ 
3325765*1 
+ 
942576542 
4-74057653* 
4-6605764^3 
+ 
504575*2 
+ 
936575*43 
4-4205754* 
4-74456*51 
+ 
44456*42 
4-138056*3* 
4-76856*5*2 
4-47656*543 
4-109256*4* 
+ 
444565*3 
4-228565*4* 
4-16825M 
+ 
{60583^ 
+ 
3057*2* 
4- 65753*2 
4- 6574*1 
+ 
14574*32 
+ 
85743* 
-f 2056*1* 
4-4856*42^ 
4- 656*3*2 
+ 
6565*31 
+ 
12565*2*4-122654*1 
-f 10565432 
4- 465653* 
+ 
20564*2 
+ 
32564*3* 
4- 1855*41 
4-5855*32 
4- 2655*4*2 
+ 
1455*43* 
+ 
24554*3 }5x 
4- {2582^ 
4- 4563*2* 
+ 
1255*2* 
+ 
65543*1 
-f 655432* 
4-6553*2 
4- 454*31 
+ 
1254*2* 
+ 
854*3*2 
4- 4543^}5x*4-{8542* 
4- 453*21 
+ 
253*2* }5x* 4- 
2{54r 
+ 52*1*} 5x^ 
The bar placed over a number consisting of two digits indicates that it is a single expo- 
nent. Ecc. gr. 2126^ represents 211751 represents and so on. 
14. AYe may regard \ as the root of a sextic equation; and since the coefficients of 
any equation may be considered as known, when the sums of the powers of its roots are 
known, therefore the formulte in the last article may be considered as giving implicitly 
the coefficients of the sextic in X. The equation in X being once obtained, we may, by a 
transformation linear in X, deduce the sextic in & or in rr'. But the high dimensions of 
the symmetric functions in x present practical (almost insuperable) difficulties in passing 
to the corresponding functions of the coefficients of the quintic. Mr. Cayley has 
suggested to me a more convenient method, which, however, I must leave its author to 
expound himself. 
15. But if, in place of the complete quintic, we take any one of the trinomial forms 
af-^ex +/= 0 , 
x^-\- bx*-\-f=0, 
