TABER. — SCALAR FUXCTIONS OF HYPER COMPLEX NUMBERS. 655 



and lot tho miinbcr system {cu e'^, . . . Cm) contain at least one number 

 satisfying the system of equations 



(75) SXci = .\\Si\ei-\- x-iSe-yCi + . . . + XmSemCi = 



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



(76) V— Seiei, Sc-id, ... ScmCi 



SeiCi, SeiCi, . . . lSemi'2 



Seiem, Seiem, ■■■ i^emOm 



we, therefore, now have V = 0, Let A' = B be any number of 

 (ei, eo, • • • Cm) satisfying equations (75). Then B is nilpotent; 

 moreover, the product, in either order, of B and any number 



A = YL (tk^h of the system (^i, <?2, ... Cm) is also a solution of equa- 



k=l 



tions (75). For, for any number 



y = yit\ + yoC2 + . . . + l/mCm 



of the system (^i, e-z, ... em), we now have 



SB Y = y.SBei + y.SBe. + . . . + ymSBcm = 0: 

 wherefore, in particular, 



SB'-*'' = SBB''*' = {h= 1,2, ...n -1), 



and thus, by the theorem given on p. 654, B is nilpotent; further, 



S.{BA-ei) = ^{B-Aed = 0, 



S(AB-ei) = S{ei-AB) = S{eiA-B) = S{B-eiA) = 



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



Since l)Oth B A and A B are .solutions of equations (75), it follows 

 by what has just been proved that both B A and .1 B are nilpotent. 

 In particular, for 1 = i ~ m, Bci is nilpotent; and, therefore, by 

 theorem I, S\Bei = 0. Whence it follows that .B is a solution of the 

 system of equations 



(77) SiXci = XiSiCiCi + XiSie^ei + . . . + .imSiCm^i = 



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



