TABER. — SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS. 629 



If p IK any scalar, and 



B = IhCi + 62<'2 + . . . + h^e^ 



any second number of the system, we have 



SipA = pSiA, S2pA = pSoA, 



Si{A ^ B) = Si A ± SiB, S2U ± J5) = S2A ± S2B, 



SiAB = SiBA, SiAB^SiBA. 



If e is a modulus of the system, 



(Sje = 1 = §26. 

 // A is nilpotent, 



SiAP = 0, 52.4^ = 0, 



for every positive integer p; and conversely, if either 



SiA^ = (p = 1,2, ... m) 



or 



S2AP = ip= 1,2, ... m), 



A is nilpotent. Moreover, A is nilpotent if 



SiAei = S1AC2 = . . . = SiAe„ = 0, 

 or 



SoAei = SoAe-i = . . . = S^Acn = 0. 



// A is idempotent, there are m SiA > linearly independent numbers 

 of the system satisfying the equation 



AX=X, 



in terms of tvhich every number of the system .satisfying this equation can 

 he expressed linearly, also mS^A > linearly independent numbers 

 satisfying the equation 



XA = X, 



in terms of ivhich every solution of this equation can be expressed 

 linearly.^ 



Let 



(5) A' = a-i(?i + 0:2^2 + . . . + Xn^e^, 



3 Sec paper bv tlie author cited above, pp. 61, 69, and 70, also Trans. Am. 

 Math. Soc, 5, 522, (1904). 



