TABER. — SCALAR FUNCTIONS OP HYPERCOMPLEX NUMBERS. 65 

 Whence follows 



n n n n 



2«- 2 ; a>i y'jv - 2»' 2 a < 7*i ; 

 ii ii 



that is, S A is invariant to any linear transformation of the units of the 

 system e 1} e 2 , . . . e n . 



Let p be any scalar, and A and B any two numbers of the system 

 e x , e<_, . . . e n . Then, by (1), (3), and Theorem I, 



S( P A) = S(pl) = P SA = pSA, 

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

 S(AB) = S(BA) = S(AB) = S (B A)J 



If e is a modulus of the system e lt c 2 , . . . e n) then by (4), £ is a modu- 

 lus of the system e 1} e 2 , . . . e H ; and therefore, by (1) and Theorem I, 



Se= Sl= 1. 



Let A be rational in 3& (B, e,). Then by (6), A is rational 

 in H£ (B, e { ) ; therefore by Theorem I, S A, and thus, by (1), 

 S A = SAis rational in B. Let, moreover, A be idempotent. Then, 

 by (4), A is idempotent. Therefore, by Theorem I, there are 



* The equations S pA = p S A and S (A + B) — S A + SB are immediate 

 consequences of the definition of S A. 

 t Since e. • e . e . — e, e . • e . we have 



n i j hi j 



n 71 



2* y.jk Inki = 2* y m 7 kjl (iJ, h,l = l,2, . . .n); 



i l 



and, therefore, 



1111 



-.n n n n 



= - !£■ E.L Si. a by^-i y, •*» 



„ **i ~*j ^*k ~h i ) 'hik ' kjh J 

 " 1 1 1 1 



h n n n n 



*-^ „ *»i -^ **& ^n j i ';jfc 'Aift 



1111 

 1 n n n n 



= »^^/^*^ a i 6 .-7 ftjfc 7^- 

 ' i l l l 



"Whence follows S A B = S B A, since the two last members of these two equa- 

 tions are equal, as may be seen by the interchange in either of i and j, and of h 



and /.-. 



VOL. XLI. — 5 



