394 llEV. T. P, KIRKMAN ON THE THEORY OF 



This {k') is a group, as appears by the tests, 



2H4.f{k') = {k')2h^3', &c. 

 We have in the case of this model, if i = , 2= ft, &c, 



i'=i. 2=*= 2, 3=^ = 3, 4='=4. 



The result of operation by [k') on K is 



12345678 



21436587 



43127865 



34218756 



65871234 



56782143 



78653421 



87564312; 



a group first divined and defined by Mr. Cayley in an 



elegant little paper in the Philosophical Magazine for 1859. 



The difficulty of the step from the analytical definition of 



a group to its actual construction, is shewn by the fact, 



that Mr. Cayley did not succeed in constructing this group 



till long after he had published its definition. 



I hope that one great use of this Memoir will be to 

 facilitate the tactical construction of groups, as well as the 

 enumeration of their equivalents. Hereby the fonctions 

 Men definies (68) of the Paris Prize Question for i860 will 

 be accurately formed and exhausted. 



This group of Mr. Cayley's has six substitutions of the 

 fourth order all square roots of 21436587. 

 If we operate on the same model with 

 1234 



2'2l 423 



324 1 22 



[k") 



we obtain the group 



42322 1^ 



12345678 

 21436587 

 43128756 



