TABER. — SCALAR FUNCTIOXS OF HYPER COMPLEX NUMBERS. 633 



Let now Ai ?^ 0, in which case Ao 5^ 0, and there is no number of the 

 system satisfying equations (C^), or equations (8); and, therefore, the 

 system contains no invariant nilpotent sub system. In this case, 

 therefore, if 



SiAci = SiBci {i = 1, 2, .. . m), 



we have A = B; otherwise, A — B 9^0 is a. sohition of equations (6). 

 Similarly, if 



S-iAci = SoBci (i = 1, 2, . . . m), 



then A =^ B. 



We have now the following theorem. 



Theorem II. Let (ei, eo. . .em) he any non-nil potent hyper complex 

 number system; let 



X = .Tifi + -c-iei + . . . + x^em, 

 and let 



Ai = 



{i,j = 1, 2, . . . m) 



Ao = 



S2 ei Cj 



{i, j = 1,2, .. . m) 



be the resultants, respectiiely, of the two systems of equations 



(a) SiXei = XiSieiei + x-iSie-iei -h . . . -\- XmSiemSi = 



(i = 1, 2, . . . m), 

 and 



(/3) S'iXci = xi 8261^4 + .T2S2(?2^i + • ■ ■ -\- XmS^emCi = 



{i= 1,2, ... m). 



Then, if the number system is transformed by the substitution 



e'i = TiiCi + T{ieo + . . . + TimCm [i = 1, 2, . . . m), 

 and if 



A'i = 

 we have 



Sie'ic'j 



{i,j= 1,2, ...m) 



A'2 = r-Ai, 



, A'2 = 



Sie'ie'j 



{i, j = 1, 2, . . . m) 



A'2 = 7'-A2, 



ivhere T is the determinant of the substitution. Further, the condition, 

 necessary and sufficient, that the number system shall contain no invariant 

 nilpotent sub system is that Ai j^ 0, or A2 p^ 0. In this case, if either 



