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



Wherefore Gg gives a function of two terms only, which 



is 



{aa^^= iioo) ^6=12 + 43; 



and Gj gives a function of four terms which is a value of 

 the function given by G3, viz., 



^7=12 + 23 + 34 + 41. 

 61 • The first halves of the groups Gj Gj Gg are groups 

 of the fourth order, and will give six-valued functions. 

 They give by the system of exponents {a/3'yS = 22io) the 

 halves of the functions ^i^g^a* 

 The group 



1234=5' 

 4312 

 2143 

 3421 

 gives no six-valued function by the system of a^jy. 



because 



P=^«^^44 = i243P 

 where 



12435' 

 is a derived derangement of ^. 



But the groups 



1234 and 1234 



2341 3142 



3412 4321 



4123 2413 



give each a value of the six- valued function 



#6= i»2^3T4'>' + 2"3^4'>'i'>' + 3«4'^i>'2v + 4«i^2V3v, 

 of which one form is 



1^2 + 2^3 + 3^4 + 4^1. 

 62. The six groups each of the 20th order, made with 

 five elements (18) give each a six- valued function by any 

 system a^iySa of five exponents. 



The system of four exponents a/3<y8S gives only three 

 functions; for each group G^^ gives the same algebraic 



