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



groups are obtained equivalent to that given when all the 

 exponents are unity. 



Thus, for example, if we operate on the model 

 123456789 



231564897 (H) 

 312645937 



by the auxiliary 



1 2^3^ 

 2^3^-1 {k) 



3^1 2^, 



which adds to (H) its derivates by 



Q ^ 645 -897 -123 ^2^1 



' 1230645.897 12^ 



and by 



Q ^ 897-i23-645 ^ 3^i 2^ 

 ' 123.645.897 12Y' 

 we form a group of nine, equivalent to that obtained by 

 operation with the auxiliary 



123 



231 (^"0) 

 312. 

 But there is a cm^ious extension of this theory, whereby 

 we obtain groups of nine not so equivalent. 

 Let the operative group be 

 1 23 

 2^3 1 {k^) 



with the definitions, 2*= 2, 3^ = 3, i*=i. 

 We find that [k^ is a group by the test 



2^31 .3212^=1^2^3^ = 3^12-. 2^31 ; 

 for first, the operation 2^31 on the subject 3^12^ puts 2^ for 

 1, 3 for 2, that is 3^ for 2^, and 1 for 3, that is i^ for f" ; 

 and the operation 3^1 2^ on the subject 2^31 puts 3^ for 1, 

 1 for 2, that is 1^ for 2^, and 2^ for 3 ; and secondly, 



i''2"3°= 123 

 is the model, whatever a may be, written in a diflPerent 



