332 



MEMO IKS OP THE AMERICAN ACADEMY 



The Sign of Involution. ■ 



I shall take involution in such a sense that xv will denote everything which 

 for every individual of y. 



Thus Z w will be a lover of every woman 

 denote whatever stands to every woman in the relation of ser\ 



Th 



w 



of every 



lover of hers ; and s( lw ) will denote whatever is a servant of everything that 



lover of a woman. So that 



( S W 



s«") 



A servant of every man and woman 

 denote a servant of every man that is 



ill be denoted by s m + w , and s m ,s™ will 



of every woman 



So that 



s 



m 



+ * 



S m ,S w . 



That which is emperor or conqueror of every Frenchman will be denoted by (e -fc 



and e r -fc 2, p e f — P , cP -|r 



denote whatever is emperor of every Frenchman 



emperor of some Frenchmen and conqueror of all the 

 Frenchman. Consequently, 



queror of every 



(«-b-«) 



f 



ef -fe- X p ef~P,cP -fc- cf . 



Indeed, we may write the binomial theorem so as to preserve all its usual coeffi- 

 cients; for we have 



(e + C )f = 4 -fe- [f ] . ef 



tl 



^t-fcr 



U ] • ( [f ] 



I) 



2 



.e 



f 



* 2 ,c 2 t -{^ etc. 



That is to say 

 Frenchman com 



those things each of which is emperor or conqueror of every 

 *t, first, of all those individuals each of which is a conqueror of 



every Frenchman; second, of a number of 

 men, each class consisting of everything w' 



ti is 



qual to the number of French- 

 i emperor of every Frenchman 



but some one 



d 



conqueror of that one ; third, of a number of 



qual 



half the product of the number of Frenchmen by one less than that number, each 

 of these classes consisting of every individual which is an emperor of every French- 

 man except a certain two, and is conqueror of those two, etc. This theorem holds, 

 also, equally well with invertible addition, and either term of the binomial may be 

 negative provided we assume 



x)a 



)Ly].z 7 J. 



In addition to the above equations which are required to hold good by the defini- 

 tion of involution, the following also holds, 



just as it does in arithmetic. 



(s,l) 



W 



sV w , 



