GKOUPS AND MANY-VALUED FUNCTIONS, 393 



order : that is, 



iV-3'^= 23 1564897 



312645978 



123456789. 



The result of operation with {k^) is to add to the model 



H the two derivates 



564789123 



645897231 

 456978312 



897123564 



978231645 



789312456, 



which complete a group of nine containing three cube 



roots of 231564897, and three cube roots of 312645987, 



all six of the ninth order. 



If we operate with each of 



123 123 123 123 123 

 2 3^1 23 1^ 2^3 1 2 3''i 231^ 

 3^ 1^2, 3 1^2^, 3^1 2^, 3^ 1^2, 3 1^2^, 

 which are all alike groups, we add to H five more sets of 

 six substitutions of the ninth order ; and we thus solve the 

 curious problem in evolution, to find the eighteen cube roots 

 0/231564897 and the eighteen cube roots 0/312645978. 



It will be found impossible to modify the vertical rows 

 of these operative groups, as we can those of Art. 39, so 

 that the same element shall everywhere shew the same 

 exponent. 



98. Let the model be 



12345678 

 21436587 



K. 



and the auxiliary 



1 234 



2^1 4 3' 



SEE. III. VOL. I. 3e 



3^4^ 1 2 



4 3^2 1^ 



