the Fifteen Young Ladies. 201 



we have 



G^G^', andG 3 =G 1 0"; 



which merely affirms that G 2 is formed by writing in order the 

 1st, 2nd, 3rd, 4th, ( 5 th = )12th, ( 2 nd=)9th, &c. vertical rows of 

 Gj, &c. Or, if we please, we may interpret G 2 = G 1 ^ / as express- 

 ing the result of operating on the substitution 6 1 (with unity, 

 that is, the first line of G supposed written under it) by every 

 arrangement of G : (with unity so subscribed). 



The three systems G v G 2 , G 3 cannot, as Mr. Woolhouse justly 

 observes, be mutually elicited by any direct substitution of the 

 form 



Gi = $G 2 =%G 3 ; 



but this does not prevent one being obtained from another by a 

 right-handed operation with a substitution, as 



G 2 =G,0', G^G^'- 1 , 

 where 



6> / - 1 = 1234i57463256 7 0. 



It will not be found difficult to demonstrate that the three 

 systems given by Mr. Woolhouse are all the simple derangements 

 of G which give different solutions of the problem, if we define 

 that two solutions are different when one G ! cannot be obtained from 

 the other G" by a direct substitution, as G' = PG ,; . 



But we need not begin with the model group Gj, whose two 

 vertical circular factors are 1234567 and 1334567. 



The late Mr. Anstice has shown (Camb. and Dub. Journ. 

 vol. vii. p. 285) that we may employ the two vertical circles 



1234567 1235674, which determine 



the group 



H, 







1 



2 



3 4 



5 6 



7 



12 3 4 5 6 7 











2 



3 



4 5 



6 7 



1 



2 3 5 16 7 4 











3 



4 



5 6 



7 1 



2 



3 5 6 2 7 4 1 











4 



5 



6 7 



1 2 



3 



5 6 7 3 4 12 











5 



6 



7 1 



2 3 



4 



6 7 4 5 12 3 











6 



7 



1 2 



3 4 



5 



7 4 16 2 3 5 











7 



1 



2 3 



4 5 



6 



4 12 7 3 5 6 







This 



group is 



one 



of the 













nis 





15 



.13. 



12.: 



11 



.10.9.8 



.5.4.3.2.1 



6 . 2 . 7 2 



equivalents of G : vide art. 12 of my memoir " On the Theory 

 of Groups and Many-valued Functions," Manchester Memoirs, 

 1861. 



One derangement of the group H given as a solution by Mr. 

 Anstice is 



