348 REV. T. p. KIKKMAN ON THE THEORY OF 



The systems of exponents a/S'yyy and aa^^jB are ex- 

 cludedj because each of tlie six groups gives the same 

 symmetrical function by these systems. 



The six groups of 20 (18) comprise each a group of 10. 

 By these we can form twelve-valued functionSj wbicb are 

 the halves of the six-valued already constructed. 



64. The system a/Syyy is not excluded here, but gives 

 by any of the groups a value of the twelve-valued function 

 (ayS777= 21000) 



v=iH + 2'^3 + 3^4 + 4^5 + 5~i + 2^1 + 1^5 + 5^4 + 4^3 + 3^2 ; 

 nor is the system aa/3/3y8 excluded, for it gives by any of 

 the groups a value of the twelve-valued function 

 ^'=12 + 23 + 34 + 51. 



The functions ViVaVsVjV comprise each a twenty-four- 

 valued function and one of its values. 



There are six twenty-four-valued functions made with 

 the system of exponents {a^ySe) ; three with the system 

 {a/3<y88), one with the system {a/S^jy), and one with the 

 system (a/3777), upon the models with which G'lG'g" '^'& 

 begin, (Art. 18). 



65. The six groups each of 120 substitutions made with 

 six elements are thus formed, theorem F, (23). 



We first write the group made with five elements 

 G' = S{pi + c) (mod 5) 

 of Art. 18, with the addition of six final, thus; 



123456 241356 314256 432156 

 234516 413526 142536 321546 



345126 135246 425316 215436 (H) 

 451236 352416 253146 154326 

 512346 524136 531426 543216. 

 We next form the derivants (23), {(3=2), (20), 



W^= (2=^(i-i)HiUd.5 +^ + - = 645231, 



2 6 



F,= (2^(l-2)'-(?-2)^ + 2)^,a.5 + ^ + -=465132, 



