350 
EEY. E. HAELEY ON THE METHOD OE STMMETEIC PEODHCTS. 
22. But it is noteworthy that the complete value of a seminvariant or critical function 
may always be deduced from the result obtained wheii any coefficient (excepting only a 
the first) vanishes. Thus, for the equation 
(«, 5, c, c?, 1)®=0, 
the operator is 
and if ex. cjr. we Avrite 
and suppose given, all the other terms can be fomid. For, in this case, let 
V'=V— (3cBrf+2fZ^g)=5aB6+4Z•Bc^-eBJ; 
then 
and therefore 
V Iq -j-cZy/Ti-j-fZ^Vlad- . . . 
+ 3c(Ii -}-2(Zl2 4-3(ZT3 -j- .. .) 
+2 +•••) 
3cIi=-vTo, 
6cl2=-VT,-2SJo, 
9 cl 3 =— V'lg— 2 BJi, &c., 
Avhich give Ii, Ig, I 3 , &c. And it is easy to see how to obtain the corresponding formulse 
for the other cases, viz. when the given function is free from either c, ory. The 
extension of the process to equations of other degrees than the fifth presents no difficulty. 
23. In applying the formulse in art. 21, it will be convenient to omit the constant 
numerical factor 5®. The calculation may be thus conducted : — 
Terms in Do. 
Ao 
Bo 
( 
Co 
Do 
+ 
1 
dd 
65 
c" 
10 
dd 
+ 
625 
c^°e 
16 
dde 
+ 
560 
de 
+ 
125 
cde 
— 
3125 
cV- 
+ 
2 
dd^ 
— 
85 
dd“ 
+ 
150 
c®e" 
1 
"T 
96 
ddd 
— 
1340 
dd 
— 
500 
c‘d-6 
9 
dd^e 
+ 
510 
dd’^e 
— 
750 
d^d^- 
— 
1 
dd^ 
— 
70 
cd'^ 
— 
125 
c®e’ 
— 
266 
ddd 
+ 
875 
cd 
+ 
625 
dd-e 
19 
cd?d 
— 
175 
(Pd 
+ 
625 
dd’’e 
+ 
23 
d^e 
+ 
50 
de" 
+ 
336 
dd 
— 
250 
- 2 
c^d'e " — 58 
+ 14 
cV — 160 
cd'^e - 7 
cd'e* 4- 35 
d^ + 1 
JV - 10 
e® + 25 
