RESEARCHES IN THE HIGHER ALGEBRA. Ill 



may be taken to represent such a substitution. Let us 



make 



'Ki = <p{a, h, c, d) 



X3=^(6, d, a, c)=lil 

 Xs = (j){c, a, d, b)=XP 

 X^=cf){d,c, b,a)=XP. 



§38. 



Next, let be any symmetric function of X. The sub- 

 stitution P' or its equivalent Z*"^+'' {m an integer) has^ as we 

 see, no effect upon 6. Therefore 



e=ei=ep=ep. 



§ 39- 

 From 6 = 61 and the transformations afforded by the 

 theory of interchanges we find 



6{"f) = 6{"'?) {"') = ^("^) (*^) =^(*^) («*) 

 6{''.'!) =6{"') {'.'') = 6{"f) i^") = 6["') ("f^) 

 6(^") = 6'(°*) ("^) 3= 6'('"^) (*^) = 6{^^) ("*) 

 6{'"^) = 6{^') (^0 = ^( - ) (• •') = ^(- ) ( -) . 



§ 4°- 

 From 6 = 6P and the transformations afforded by the 

 theory of interchanges, we find 



^("*) =: ^c^^) ("^) = ^C'^') ("^^) = 6{;"^) {'") 

 6{"^) = 6 {"") («") = 6' (*^) ('."') = ^ ("'') (*'') 

 ^("^) = ^(«^) (*«) = ^(*''') (^■^^) = 6{''.') (*^) . 



§41- 

 From 6=6P adjoined to 6=61 we obtain 

 6{"'') = 6{'") = 6{"^) ("0 = ^C") ('^), 

 and all the interchanges are now reduced to binary ones, 

 two of which are equivalent one to the other. Conse- 

 quently 6 may be repi'esented as the root of a sextic 



