GROUPS AND MANY-VALUED FUNCTIONS. 339 



is identically the derangement 



47. Let r be any model group of the /''* order made 



with 



a + b + c-\- ' ■ +j 



elements^ such that every vertical row of T is composed 



only of the a or of the b elements^ &c. 



Let 



a^'y- ', 6cK- •, Trp- •, • • 



be any substitution of P, ajB^y- • being the a elements, 



6lic- ', the h eletneuts, Trp- • the c elements, &c. 



Form the substitution 



PlPiP^ ' 'Qa+\ 9a+Z' ''^a+b+1 '^a+b+2' ' 



where g^+i qa+% ° • are the circular factors of the b equiva- 

 lent groups /^l7^2• -hj, made with the B6 elements of N &c. 



The effect of the factor — * of Q in the derived group 



Pn 



QJ is to change the group 



n -P-itn (P^\^ 



^'-p.^'y¥J 



into 



Pm „ (Pn\ 



which is the derangement of 



PjiLqjPjB\ by '^iP^\ . 

 i?o ^P^' Pa ^PJ 



It is then evident that QJ is a derived derangement of 

 J. If we form 



J + QiJ + Q2J + • • + Q^_i J, 

 employing all the substitutions of P, we shall have by the 

 reasoning of (29) a model group of krl substitutions. In 

 fact the effect of the derivant 0,^ is to put for any elemen- 

 tary group of J a certain derangement of some other 

 equivalent elementary group of J. 



SER. III. VOL. I. UTJ 



