GROUPS AND MANY-VALUED FUNCTIONS. 331 



for 1, 123 for 1, 456 for 3. 789 



231 564 897 



312; 645; 978; 



for 1^, 231 for 1?, 564 for 3^, 897 



312 645 978 



123; 456; . 789; 



for i^j 312 for 2^, 645 for 3^, 978 

 123 456 789 



231; 564; 897; 



we obtain a group equivalent to J above written. And if 

 for tbe elements of ^" we substitute 



for 1, 1234 for 2j 5678 for 3, 90a6 

 2341 6785 oah^ 



3412 7856 ah(^o 



4123; 8567; hc^oa; 



for 1^, 2341 for i^j 3412 



3412 4123 



4123 1234 



1234; 2341; 



&c. 



we shall complete a grouped group of 24 substitutions 

 made with 12 elements. 



Ifj instead of transforming g' into g" , we make the 

 transformation 



1 2^3^ 123 



2^3^ 1 2^3^ 1'^ 



3^1 2^ 3^ 1^2^ 



1 3^2^ 1 3^2^ 



3^2^i 321^ 



2^1 3^ 2 1^3 

 by the addition of two to every exponent in the second 

 row, and of one to every exponent in the third row, and 

 then make in the new auxiliary thus formed the substi- 

 tutions above-mentioned, we shall have grouped groups 

 equivalent to those before formed. 



The auxiliary chosen may be anij group whatever. If 



SER. III. VOL, I. TT 



