GROUPS AND MANY-VALUED FUNCTIONS. 363 



in which the first vertical circle is given by 57683241, the 

 second is the same circle reversed, and 

 ^^ 18472635 

 ' 15362748 

 We next complete the vertical circles thus : 

 12345678 12438765 «! 



(f) 57683241 86754231 a^ 



4>' 34216785 43217658 «3 



4>^ 68752413 75682314 a^ 



(f>* 21437856 2134.6587 a, 



<j>' 75864132 68573142 ae 



<^« 43128567 34125876 a, 



</)' 86571324 57861423 «8- 

 We see that 



^ ( = )«1«2( = )«2«3( = )«3«4 ( = )a7«8( = )«8«l 



j)'^{ = )a^a^{ = )a^ai[ = )a^ar^, &c. 



</>^( = )«l«4( = )«2«5( = )«3«6, &c. 



<^^(=:)a2ai( = )a3«2( = )«4«3, &c. 

 And there are sixteen equations of the form 



«2«1«2( = )«3. «3«2«3( = )«4, 



or of the form a^a^a^-^^a^, where ]pqr are consecutive 

 radicals of the system. 



We shall speak of «i, a^, ag- • as of the didymous radi- 

 cals of (j), that is, of the group of powers of (j), and any 

 pair a^a,i, giving a,„a,j=0' are didymous factors of <p\ 



The whole group both of powers and radicals is given 

 with any two consecutive radicals. Thus, if ai= 143256, 

 ^3= 2 13546, we have the skeletons 



123456 143256 

 413526 213546 



5 5 



2 4 



where 



