146 
H36"h 
Hs 6 + .... +H6o= 3 M l5 
I mean that xJ* is the sum of the &th powers of the functions 
obtained by effecting, on the subindices of x 1 x 2 x 3 x i x 5 in H : , the 
substitutions written under j\, 2 J 7 being the sum of the five 
functions made by operating on Hf by the substitutions undergo, 
&c. We form thus 60 values of Hj with the positive, and 60 
more with the negative, substitutions. 
By + x M£=X 
2 J?+ 2 K?+ 2 U+ 2 Mf= 2 X 7ii 
8 J?-f 3 K?+ 3 L?+ 3 M^A 
I mean to say that is the sum of the hth. powers of the four 
functions ^ &c. 
It is evident that Hi, H|, &c., can be formed for any integer 
index i, and as well as x Jj, &c., for any integer h. 
I denote by HLx, H!_ 2 , &c., and by ^JJ, on _iK 7 , &c., and by 
_xX w , &c., the results of operating on Hi, H 2 , &c., on X J^, &c., and 
on jX^-j &c., by the substitution 2134, effected on the sub- 
indices of x x x 2 x 3 x 4 x 5 throughout the function considered. 
It is evident that 
(lXfti+^Xfti+.sXa*) + (-iX Ai 4 -_ 2 X 7ii + - 3 X W ) 
and (1X7^+ 2 X hi + 3 X w ) (_.iX /u . -f- _ 2 X 7 , -f _ 3 X M ) 
are both symmetrical functions of the roots, and rational func- 
tions of the coefficients, of G=0 ; as may be readily seen by 
making h—i— -1, and observing that the involutions cannot destroy 
the symmetry in the roots, or the rationality in the coefficients 
of G=x0 ; for the former is the sum, and the latter the product 
of two functions, either of which becomes the other by any odd 
number of transpositions of the subindices of x 1 x 2 x 3 x 4 x 5} while 
neither is altered by any even number of such transpositions 
Wherefore 
Ai+ 2X7^4” A* 
is the root of a given quadratic, whose coefficients are rational 
functions of those of G=0, 
