TABER. — SCALAR FUNCTIONS OF HYPERCOMPLEX NUMBERS. 63 



then the systems e' 1} e' 2 , . . . e' n and e\, e' 2 , . . . e' n are reciprocal pro- 

 vided the determinant of the transformation is not zero.* 



Theorem (3). If p is any scalar, and p A^= C, then pA=0; and 

 conversely. If -4_+ B ==_ 0, then A + B = C; and conversely. If 

 A B = C, then BA = 0;f and conversely. 



Theorem (4). If e is a modulus of the system e l5 e 2 , . . . <?„, then s is 

 a modulus of the reciprocal system e x , e. 2 , . . . e n ; and conversely. If 

 A is idempotent, so also is A, and conversely ; if A is nilpotent, so also 

 is A, and conversely. If B is either idemfaciend, or idemfacient, with 

 respect to A, then B is either idemfacient, or idemfaciend, respectively, 

 with respect to A ; and conversely._ And, if B is either nilfaciend, or 

 nilfacient, with respect to A, tjien B is either nilfacient, or nilfaciend, 

 respectively, with respect to A. % 



* For let 



e 'i e 'j = %k 7',* e 'k (»>i = 1, 2, . . . n), 



n 



e' . e' = £. y' ■ , e', (i, j = 1, 2, . . . n). 



Then 



n n 



2» -, t . t ., 7 = X, t,, 7' ... (t,/, / = 1, 2, . . . n), 



- lh —"k T ih T jk "Yhkl ~ -'k T kl 1 ijk 

 1 1 1 



n ji n 



-* tt r jk T ,/, ym = % k r u y' iik (i,j,l = 1,2, . . . n) ; 



l i l 



therefore, 



n n n 



-k T ki (y'*k - y'jm) = ^a ^ k t m r ik (r*« - W = ° &./» l = h % ■ • • «)■ 



i ii 



Wlience follows 



y',ik= y'nk (*,y,* = i»8,. . .«), 



since, otherwise, the determinant of the transformation is zero, 

 t If 



n n n 



AB=-Zi.a.e.'%b.e.= z,.c e.= C, 

 ill 

 then 



n n n n 



li(i.J. ; y.,=lI(ji.7.., —c. tk = 1,2, . . . n): 



11 11 



and, therefore, 



_ _ n n n 



B A-%.b. ~e. • £ a. 1. - %. c. 1. = C. 

 l l l 



J This theorem follows from the last clause of (3). 



