344 REV. T. p. KIRKMAN ON THE THEORY OF 



which have terms and -^— values. 



x+ 1 L 



Nothing is gained by permuting the exponents of the 



term 



If, for example^ we construct ^ on 



we merely form ^ and its values on the derangements by 



— of G and of its derived groups, 



G, AG, BG, &c. 

 In this series of derived groups we find 



^G, A^G, B^^G, &c., 



22 

 whose derangements by ^ are G, AG, BG- = &c, ; that 



23 



is, we have made the functions which are obtained by the 

 first system of exponents, on G and its derivates. 



§ 11. 

 Examples of the construction of m-valued functions. 

 60. The equivalent groups of the eighth order made 



with four elements are 







1234 



1234 



1234 



2341 



3142 



4312 



3412 



4321 



2143 



4123 



2413 



3421 



1432 



1324 



1243 



4321 



2143 



3412 



3214 



4231 



2134 



2143 



3412 



4321 



G, 



G, 



Gts, 



which are maximum groups, each consisting of a group of 

 four powers and of its only derived derangement. 



We have three distinct three- valued functions ^, by the 



