320 



MEMOIRS OP THE AMERICAN ACADEMY 



Involution, also, follows the indexical principle. 



a#-frz 



Jb*? j dj • 



Involution, also, satisfies the binomial theorem 



(x -k- y) z = a* -fe- 2 P x*-P,yP -fc- p , 



where 



denotes that j9 is to have every 



than z. and is to be taken 



of z in all possible ways, and that the sum of all the terms so obtained of the form 

 x 2 — P,yP is to be taken. 



Subtraction* is the operation inverse to 



subtraction with a comma below the usual sign. Then we shall have tthat 



addition. We may write indeterminative 



( x — y) -fc y 

 x — y) + y 



X 



X 



(* + y) 



y 



X 



D 



is the 



oper 



to multiplication. Since multiplicat 



not 



generally commutative it is necessary to have 



signs for division. I shall take 



{x-y)y 



X 



X 



y 



X 



y 



D 



indicated by a sem 



that multiplication which is indicated by 



comma may 



be 



So that 



(*#)# 



x . 



Evolution and 



the 



the operations 



to involution. 



X 



W9) 



x io 9x y 



y 

 y 



These 



d 



are to be 





garded as imperat 



But in addition to them 



there are certain other characters which it is highly desirable that relations and op 



should possess, if the ordinary signs of algebra 



These I will here endeavor 



to be applied 



th 



enumerate 



1. It is an additional motive for using a mathematical sign to signify a certain 

 operation or relation that the general conception of this operation or relation should 



