200 The Rev. T. 1\ Kirkman on the Puzzle of 



school-girl problem given by Mr. Woolhouse, if we write his files 

 of young ladies horizontally instead of vertically. 

 The derangements 



GP, GQ, GR, &c. 

 of a group G are in general of two kinds — those which are, and 

 those which are not derived derangements. The first are such that 



GP = PG, GQ=QG, &c.; 

 in the other, 



GP is not PG, GQ is not QG, &c. 



A derived derangement GP of G is obtained either by the 

 operation 



PG, 



which denotes the result of effecting on every substitution of G 

 the substitution P, or by the operation 



GP, 



which denotes the result of effecting upon the substitution P in 

 turn every substitution of G. 



A derived derangement GP, of G by P, is both the derivate 

 PG of G by P, and the derangement GP of G by P. 



Any other derangement Gil of G, which it will be convenient 

 to call a simple derangement of G by R, is no derivate RG of G 

 by R, nor is it a derivate of G by any substitution. 



Thus the simple derangement Gj above written is obtained by 

 effecting upon the substitution 



e _ 12432 5 5166-34770 



~~ 1234567i2345670 



in turn every one of the substitutions of the model group G, 

 namely by adding to 6 the products 



234567123456710 



123456712345670 ' 

 345671234567120 



ft 



ft &c. 



123456712345670 



Or Gj may be formed, by a known property of all groups and 

 their derangements, by effecting upon any arrangement of Gj in 

 turn all the substitutions of G. 



The same account can be given of the generation of G 2 and G 3 . 

 Further, G 2 can be obtained, and any derangement whatever of 

 G can be obtained, as a certain derangement of the derangement 

 G r That is, if 



#'=1234 5 2 75 431 6 67 0, 



6"= 1234 1 675 235 4ct0 ; 



