El \Ti\ES. 319 



Multiplication is an operation which is doiibly distributive with rcfermee t ddit >n 



That is, 



My + *) = •'•// Hr *z • 



(x-^y)z = xz^yz. 



Multiplication is almost invariably an associative operation. 



(xy)z = x(yz). 

 Multiplication is not generally commutative. If we write commutative multipl 



cation with a comma, we have 



x,y = y,x 



Invertible multiplication, is multiplication who--' corresponding inverse operation 

 (division) is determinative. We may indicate this by a dot; and then the conse- 

 quence holds that 



j 



If x.y = z 



and x .y = z , 



then y = y • 



Functional multiplication is the application of an operation to a function. It may 

 be written like ordinary multiplication; but then there will generally he certain 

 points where the associative principle does not hold. Th. , if we write (sin abe)dej 

 there is one such point. If we write (log {h ^abc)def)<j hi, there are two h 

 points. The number of such points depends on the nature of the ij mbol of oper- 

 ation, and is necessarily finite. If there were many such point, in any ca . it 

 would be necessary to adopt a different mode of writing such functions from th 



lly employed. We might, for example, give to "log" such a meaning that 



In 



what followed it up to a certain point indicated by a t should denote the base o 

 the system, what followed that to the point indicated by a J should be the function 

 operated on, and what followed that should be beyond the influence of the s.gn 



"log." Thus log ale t deflghi would be (log abc)ghu the be* being dtf. 



this paper I shall adopt a notation very similar to this, which will be more cor 



iently described further on. 



The operation of Involution obeys the formula * 



:: 



* In the notation of quaternions, Hamilton has assumed 



(x y)z = *<»»> , instead of (*)« = *<** ■ 



assume 



(XV) 





former 



