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



or P would be in G, contrary to hypothesis. 



If Q, be neither in G nor in VG, no arrangement in 

 QG can be in G or in PQ,; for if 



(qI=PA^A^ =:PAs, 



contrary to hypothesis. 



If then we add to G + PQ + QG • • the derivate RG by 

 E,, which is not in the preceding groups, we shall partition 



the UN permutations of N elements into — — groups, which 

 are G and its — — - i derived groups. 



K 



We easily demonstrate the known theorem. The derivate 

 of G by V is the derivate of G by PA„j, A„^ being any sub- 

 stitution of G. 



Cauchy has also proved in the Memoir above quoted 

 that the substitutions 



AP and PAP-i 

 are similar substitutions ; that is, that they differ neither in 

 the number nor in the orders of their circular factors. 

 We shall say that the model groups 

 G and PGP-^ 

 are equivalent groups, when they are not the same group. 

 All that precedes is well known. 

 8. The derived groups of G 



PG, QG, RG, • • 

 are derangements by P, Q, R, • • of the equivalent or iden- 

 tical model groups 



PGP-i, QGQ-\ RGR-i • • 

 Let us suppose that one of them is identical with G ; or 



that 



G=PGP-\ 



then GP=PG, 



and consequently GP'^=PGP = P'^G, 



