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



The shortest way to satisfy one^s self that the 120 sub- 

 stitutions form a group, is to write the five derangements 



ll{W, + -¥,+ W,+ W,+ W,), 

 which will be found to contain exactly the substitutions of 

 the five derived groups above written. 



The five equivalent groups to the one (J) above formed 

 of 120 substitutions are 



123564J123645 =3\ 



I23645J123564 =J3 



I23465J123465 =J3 



I23654J123654 =J4 



I23546J123546 =J5. 



All these may be readily formed by writing out the 

 derived groups 123564J, 123645 J, &c., and then effecting 

 a simple derangement of the three final vertical rows of 

 the derived groups, whereby what follows is easily veri- 

 fied. 



66. The '7r3 = 6 groups each of 120th order, made with 

 six elements, give six functions by the system of exponents 

 (a^ySe^), each function of 120 terms. 



The functions are reduced to three by the system of 

 exponents a^ySee, since each of the six groups G^ gives 

 the same function with its derangement (Artt. 57, 58), 



^<ii 23465, 

 which is the derived of another group. 



There are then three distinct functions of the tenth 

 degree, made with the exponents (a;S7See=432ioo), each 

 function of 120 terms. 



The system of exponents a^yySS gives the same alge- 

 braic function on (Artt. 57, 58) 



(^d, 0^123465, 0^124356, and 6^124365. 



None of the six groups has either 123465 or 124356; 

 for none has four letters undisturbed in any substitution. 



Two of the six groups have 124365; wherefore these 

 give values of the same six-valued function of 60 terms. 



