GROUPS AND MANY-VALUED FUNCTIONS, 315 



N = A« + B6 + Cc+ • • -I- jy, 

 A>B, B>C, &c., 

 k being the least common multiple of ABC • • J^ and one at 

 least of abC' -j being > i . 



Let G be any one of the W groups of powers of a sub- 

 stitution constructible on this partition of N by theorem C 



(12). 



Let 



Pi Pi ' ° Pa Qa+\ Qa+2 ' ' Qa+b '^a+b+1 ' *^a+6+c " ' 



be the circular factors of G of the orders ABC • • • 



Let pi.=p^'^^ be the i"" cyclical permutation of p^. 



Let w equivalent groups g be constructible of I substi- 

 tutions made with 



a + b + c+-'+j 

 elements, of which the a elements are consecutive in unity, 

 the b elements consecutive &c. ; and such groups g that 

 every circular factor in them made with the a elements is 

 a divisor of A, every circular factor made with the b ele- 

 ments is a divisor of B, &c. ; and also such groups g that 

 every vertical row of the group written in a column shall 

 comprise no elements but of the a or of the b &c. ele- 

 ments. 



Let M be the least common multiple of 

 AAA B B C C 

 Ai A2 A3 sji Bg Cx L/2 

 where AiAgAg- • are the orders of the circular factors of ^ 

 made with any of the a elements, BiBj- • those of factors 

 made with the b elements &c. 



Let the substitutions following be formed : 



Q^ PJPpPy - p¥ g^ •• yf - ^ 



P1P2P3 " Pa la+lQa+l " ^a+b^a+b+l " 



where a^^- '6rj- -e- • is every one in turn of the /- i sub- 

 stitutions of the group g, a/37 • * ^ being the a elements, 

 r) • • being the b elements, e • • being the c elements, &c. ; 

 and where the terms of the numerator are those of the 



