TABER. — SCALAR FUNCTIONS OF HYPERCOMPLEX NUMBERS. 61 

 to denote the number obtained from the number 



n 



A=^a i e l . 

 1 l 



by replacing, severally, the units of the original system e u e 2 , . . . e n by 

 the units e x , e 2 , . . . e n , respectively, of the reciprocal system. 



Theorem I. Let R denote the domain of rationality of any arbitrary 

 aggregate of scalar s including the constants y# k of multiplication of any 

 given hypercomplex number system e v e 2 , . . . e H . Let 



n 

 1 *' 



be any number of the system ; and let 



1 _ n _" -i n n 



8A = ~ 2. 2. a <yw SA = -% 2 a <yjv 

 i * i 3 1*1* 



Then both SA and SA are invariant to any linear transformation of the 

 units of the system ; and, if p is any scalar and B any second number of 

 the system, 



SpA=pSA, SpA = pSA, 



S(A + B) = SA + SB, S(A + B) = SA + SB, 



SAB=SBA, SAB=SBA. 



If s is a modulus of the system, 



Se=l = Se. 



If A Js rational in the hypercomplex domain ft (R, e,.), then both SA 



and SA are rational in R ; and if, moreover, A is idempotent,* there 



nSA>0 . 7 , 7 , , M 



are - independent hypercomplex numbers, rational in ft (R,e t ), 



idemfaciend f 

 that are . , ,. . with respect to A, in terms of which even/ number 

 idemjacient L J J 



idemfaciend 

 of the system . to A can be linearly expressed, and there are 



idemjacient L 



* If A 2 = A dp. 0, A is idempotent. Benjamin Peirce, Am. Journ. Maths., 4, 

 104. 



t If A B = B, B is idemfaciend with respect to A ; if B A = B, B is idemfacient 

 with respect to A. Peirce, loc. cit., p. 104. 



