142 
which both remain invariable by any even number, and are 
changed the one into the other by any odd number, of trans- 
positions. 
Hi in the above expression for x 0 is invariable by the cyclical 
permutation of x 0 x\ x 2 x z x 4 , and therefore receives all its possible 
values by permutations of x 1 x 2 x z x 4 . H 2 is formed by effecting 
on Hi the Substitution 2143, H 3 by 3412, and H 4 by 4321, per- 
formed on the subindices of x l x. 2 x z x A in H x . 
Our object is to construct the values of H 4 H 2 H 3 H 4 in terms 
of the coefficients of Fr=0. 
The functions H 4 H| HJ can all be formed for any integer 
index i. Let 
H; + H- + H‘ + H‘ = (A) i 
Hj+H- + H* + H- =(B), 
Hh + H 22 4* H.23 + Hg 4 — (F) f - 
where the four functions (B)* are formed by effecting on Hi the 
four substitutions written over B, &c.; or, which is the same 
thing, by raising to the ^th power H 5 H 6 H 7 and H 8 , made from 
Hj by the four substitutions over B, &c. 
It is plain that 
(A) i +(B) i -f(C) 4 -f(D) i -f(E) i +(F), =M, 
and 
((A) i+ (B) j+ (C) j )((D) i +(E) i+ (F) j j=N ( 
are both given symmetrical functions of the roots, and rational in 
the coefficients of the quintic F=0; for the former is the sum of 
all the 24 values of Hf, and the latter is the product of two fac- 
tors, either of which becomes the other by any odd number of 
transpositions of x x x. 2 x z x 4 , and neither of which is changed by 
any even number of such transpositions. 
Hence (A) i -F(B) i -f-(C) i is the root of a given quadratic, whose 
coefficients are rational functions of those of the given quintic. 
