THE SCALAR FUNCTIONS OF HYPERCOMPLEX 

 NUMBERS. 



By Henry Taber. 



Presented March 8, 1905. Received March 16, 1905. 



In a recent number of the Transactions of the American Mathematical 

 Society * I have extended the quaternion scalar function to hypercomplex 

 number systems in general, establishing by the aid of this function 

 certain important theorems in the theory of hypercomplex numbers, t 

 I fiud that there are two generalizations of the quaternion scalar function. 

 Thus, denoting (as, throughout this paper) the constants of multiplica- 

 tion of any given hypercomplex number system e u e.,, . . . e u , by 

 y iJk , for i,j t k = 1, 2, . . . n, — when we have 



n 



e.ej = 2 lak e k (t t j = 1,2, ... w), 



i* 



and, by 



n 



-4=2 a ' g «» 



1 » 



any number of the system, I employ SA and SA to denote those 

 functions of the coefficients a and of the constants of multiplication 

 defined as follows : 



1 n n 1 n n 



SA = n 2. 2 # *y«r» ^^ = -2.2 a ^- 



n 1 > i J n ! » ! * 



When n = 4, and e u e 2 , e 3 , e 4 are the units of quaternions, these functions 

 both coincide with the quaternion scalar function as customarily defined. 

 Thus, if e 4 = 1, and e 1( c 2 , «3 are three mutually normal unit vectors, 

 we have 



"1m = = ypj , yijj = 1 = 7jij (i = 1, 2, 3 ; j = 1, 2, 3, 4) ; 



* Vol. 5 (Oct., 1904), p. 514. 



t In the Proc. Lond. Math. Soc. for Dec, 1890, vol. 22, p. 67, I had previously 

 extended the scalar function of quaternions to matrices in general, employing this 

 generalization to prove certain theorems of Sylvester's. 



