ON THE SCALAR FUNCTIONS OF HYPER COMPLEX 



NUMBERS. 



SECOND PAPER. 



By Henry Taber. 

 §L 



In this paper I shall denote by 7,jji,., for i, j, I: = 1, 2, . . .m, the 

 constants of multiplication of a given non-nilpotent hyper complex 

 number system (cu e-i,. .e^} We then have 



m 



(1) eiCj = Y. yijkCk (i,j = 1, 2, . . . m). 



k=l 



In These Proceedings, vol. 41 (1905), p. 59, I have shown that there 

 are two functions of the coefficients of any number 



(2) A = aiei -\- o-: (■+ ... + a^p„ 



of the system (^i, e-i,... O constituting generalizations of the scalar 

 function of quaternions, to which they reduce, becoming identical 

 when m = 4, and, at the same time, the system (ci, e-2, e^, 6^) is equiva- 

 lent to the system constituted by the four units of quaternions. These 

 functions, in designation the first and second scalar of A, are defined 

 as follows: 



(3) SiA = ~~Y. Z "tTw» 



^ 1=1 ;=1 

 1 mm 



(4) S2A= ^ y Z aiyjih 



and conform to theorem I given below. In this paper I shall employ 

 these functions to establish a simple criterion for the existence of an 



m 

 1 A number A = 2 Oi^i of anj' hvpcr coniple.x system (ci, e-z, ... em) is 

 i=l 

 idempotenl if A- = A 9^0; A is nilpotenl, if A ^ but Ap = for some positive 

 integer p > 1. A system is iiilixih'nt, if it contains no idempotent number; 

 otherwise, iton-nilpotent. Every nunilK-r of a nilpotent system is nilpotent. 

 See B. Peirce, Am. Journ. Maths., 4, 113, (ISSlj; ef. H. E. Hawkes, Trans. 

 Am. Math. Soc, 3. .321 (1902). 



