TABER. — SCATAR FUNCTIONS OF HYPER COMPLEX NUMBERS. 049 



Let 



US) -juv^^' = -l—. Ju,mu^^^ 



{p = \,2, ... v; u, V = 1,2, ... fxp). 

 Then 



(49) J.<'') = ~V) Ju.^"^ J^^'" = - '^""^''!,?^'' /./^^ 



(p = 1, 2, ... I/; M, V = 1, 2, ... Up) 



by (46) ; and, therefore, we may take the J's as new units. We now 

 have 



(50) JJPQJ^'^ = . ^^ / ^^ ^ Jui<^Vi/P^-J.i<^Viu,^^> 



Vp,ui(^)pW^> 



Vpi„i(P)p,M(^)-pi«'l(^Wl^^^ 



(p = 1, 2, . . . j^; w, 1J, «', tv = 1,2, ... Up] v 9^ v), 



(51) rj^U~'/''^ = , ^^ / ^^ ^/ ui^^Vi/P)-Jua^'')Jx/^^ = 



Vpiu/P^in^^'-pW^Pi.'!^'^ 



(p, g = 1, 2, . . . J/; 5 ?^ p; M, y = 1, 2, . . . /zp; u, v = 1, 2, . . . n,). 



For 1 =p~v, the units Ju/''^ for u, p = 1,2, . . . )Up constitute a quad- 

 rate of order Hp-, and, therefore, in the present case, the number sys- 

 tem is constituted by v mutually nilfactorial quadrates. ^^ For the 

 modulus e of the svstem we now have 





(52) e= z L h^'^ = Z L '/-'^^ 



15 For 



= Pin^'^^/./P'/u.(P>. 



16 A quadrate i.s a hyper complex number system with m = m} units 

 €ur (w, i^ = 1, 2, . . . Wi) which can be so chosen that 



tuxttw = Cuu-. Cut-e/u' = (m, i', «'', «• = 1, 2, . . . m; v' ^ v). 



B. Peirce, Am. Journ. Maths., 4, 217. 



