60 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and, therefore, 



S («! e x + a 2 e 2 + a 3 e 3 -f a A e 4 ) — a± = S (a t e 1 + a 2 e 2 + a 3 e 3 + a A e t ) 

 by the above definition. For hypercomplex number systems in general, 



n n n n 



2 2 fl, ^+ 2 2 ^v«/» 

 i * i j i ; i j 



that is, £.4 =j^ £.4 f° r every number A of the system.* The chief 

 properties of the two scalar functions, S A and S A, are enumerated in 

 Theorems /and //below; and in Theorem III, I give certain properties 

 of these functions relating to nilpotent numbers. 



Throughout this paper I shall denote by R the domain of rationality 

 of any arbitrary aggregate of scalars including the constants y, jk of 

 multiplication of the number system e x , e 2 , . . . e„, and by 3& (R, e t ) the 

 hypercomplex domain of rationality constituted by the totality of numbers 



A = 2 a > e ' 

 i 



of the 6ystem for which the a's are rational in R.f Any such number 

 A will be termed rational in this hypercomplex domain. Further, I 

 shall denote the units of the system reciprocal to e x ,e 2 , . . . e„ by 

 €i, e 2 , . . . e„, — when, if y ijk (i,j,k = 1, 2, . . . n) are the constants of 

 multiplication of the latter system, that is, if 



n 



e^j = 2 y*«* (hj> = 1,2, ... w), 



i *' 

 we have 



7m = 7j,k (*,/, ft, = 1, 2, ...'»); $ 



and I shall write 



n 



-4 = 2 a,e '' 



1 ' 



* Thus, let 7i — 3, and let e z be a modulus of the system, 



e{- = e x e 2 — 0, e 2 e x = e 1} eo 2 = e 2 . 



We then have 



SA = a s + J « 2 ; 

 whereas, 



5i = « 3 + Jo 2 . 



t See Trans. Am. Math. Soc, 5, 513. 

 J Encycl. d. Math. Wissensch., 1, 163. 



