114 Mr. G. B. Jerrarcl on the Resolution of Equations 



any root, x T , continuing fixed during the formation of the same 

 set of groups. 



5. If now we represent by 



D(Vf( W ), Vg(««), Vh(««)) 

 any symmetric and rational function of Vf(«u), Vg(«v)> Vh(*i>)*; 

 on supposing, as is permitted, # a , #3, x y , x$, x e successively to 

 become fixed, we may very readily obtain 



D(Vf(»/J), V G ( a /3), VH(^))=V(ar /J ), 



D(Vf(* 6 ), v G(«e), V H («e))= 5 r(o? 6 ) 



(S) 



V, 2 r, . . b >- being expressive of rational functions. 



6. Hence it appears that the equation (o') may take the form 



(V 3 + V, (#J** + V.&0 V+ V 3 fe) ) x 

 =0. 



(o") 



It only remains therefore to investigate the nature of the 

 rational functions designated by x r l , l r„, . . b r 3 . 



7. Now the first member of the equation just arrived at must, 

 when expanded, be capable of coinciding throughout its whole 

 extent with the function V 15 + C t V 14 + C 2 V 13 -f . . + C 15 . Each 

 therefore of the fifteen coefficients arising from such an expan- 

 sion must be a symmetric function of the roots w a , Xa, . . x e . 



Whence it is manifest that either 



orf 







fa) 



K) 



5 rM = \(0), 

 n being equal to any number in the series 1, 2, 3. 



* D is the Hebrew letter sa-mekh. 



t By r(0) I mean the absolute term of the series for r(x) when reduced 

 to its most simple form a+bx+cx' 2 +dx 3 +ex A . Thus r(0) is here equal to a. 



