354 



MEMOIRS OF THE AMERICAN ACADEMY. 



involution we have, 



in backward involution we have 

 (126.) 



(*) 



W 



K* w) ; 



I 



M 



( ls )w ; 



that is, the things which are lovers to nothing but things that are servants to nothin 

 but women are the things which are lovers of servants to nothing but women. 

 Th,e other fundamental formulae of backward involution are as follows : 



(127.) 



J-MW 



I 



W, S W , 



or, the things which are lovers or servants to nothing but women are the thing 

 which are lovers to nothing but women and servants to nothing but women. 



(128.) 



/ 



(f,n) 



1 , 'u , 



o 



which are lovers to nothing but French violinists are the things that 



nothing but Frenchmen and lovers to nothing but 



This is perhap 



quite axiomatic. It is proved as follows. By (125) and (30) 



i 



(f» 



6-'0 



f,u) 



6-(*< 



i 



f) + m 



u)) 



By (125), (13), and (7), 



'f,'u 



e -l(l-f) } Q-l(l 



«) 



6 



(i { i 



f ) + 1(1 



«)). 



Finally, the binomial theorem holds with backward involution. For those persons who 

 are lovers of nothing but Frenchmen and violinists consist first of those who are lovers 

 of nothing but Frenchmen ; second, of those who in some ways are lovers of nothing 

 but Frenchmen and in all other ways of nothing but violinists, and finally of those 



* 



who are lovers only of violinists. That is, 



(129.) 



<(u -fe-f) = «u -fc- V 



PU,V{-4rll. 



In order to retain the numerical coefficients, we must let {/} be the number of per- 

 sons that one person is lover of. We can then write 



'(u + f) = Ju+ {^-tiu/if-f- 



\ll\l 



!?l 



2 



*2 U ; 2 f _|_ etc. 



We have also the following formula which combines the two involutions : 



(130.) 



i 



w 



C*) w ; 



that is, the things which are lovers of nothing but what are servants of all women 



