GROUPS AND MANY-VALUED FUNCTIONS. 301 



be the circular factors of the orders ABC • ■ , of a group 

 G made on the partition of N elements (Art. 1 2, theorem 



N=Aa + B6 + Cc+ • • + J;' 



(A>B, B>C, &c.), (A>2). 

 Let 



Pm = «m ^mCm- ■ Ki , (A elements) 



qm = a'm b\n c'm ' 'fm ' ' (B elcmcnts) 



P—m — "m "m 9m. ' ' ^m 

 ?— TO — " mj m^ m' ' " in) 



and let 



Pvn P—m 'im ^—m ' 



be the i*'* cyclical permutations of the factors 



PmP — m' 'y[m3 'J—m') {Pro. — Pm, /• 



The substitution 



Pi P2 Ps •• qa+\ Qa+'i • • 



__ P-\P-i P-^ g-o+i y-(+2) • 



P\ Pi Ps ' ' qa+l qa+i • • 



is a square root of unity ; for, let 



p'{=abcdefg', 

 we have 



p'^^_agfedcb 

 Pi ~ abcdefg 

 p"iX^ _gfedcba 

 p'{ ~ abcdefg 

 p^ _fedcbag 

 pY ~ abcdefg 

 &c. 



We see that all the factors in these substitutions are of 

 the second order. 



25. Further: EG is a derived derangement of G, of 

 which all the substitutions are square roots of unity. For 

 the vertical circles of EG are those of G read in contrary 

 order; and since, if 



