638 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



of the system, we have, for some positive integer p, 



SW^'' = (h = 0,1,2, ...r-1), 

 or 



SiAP-"^ = (A = 0,1, 2, ...r-1), 



A is nilpotent. Conversely, if A is nilpotent, these equations are all 

 satisfied for any positive integer p.^ 



With respect to the idempotent numbers /i, I2, ... It, hnear in 

 powers of any non-nilpotent number A, the number system may be 

 regularized as follows. Let Too denote the aggregate of numbers 



T T T T 



u=\ u=l u=\ v=l 



uo 



for i = 1,2, ... m. For any assigned integer u from 1 to r, let V 

 and Tou denote, respectively, the aggregates 



T r 



hei— Y. luCilv and eilu — Y IvCih 



v=l v=l 



for i = 1, 2, . . . m; and, for any assigned pair of integers u, v from 

 1 to r, let r„5 denote the aggregate of numbers I^Ci /„ for i= 1,2, . . . m. 

 Further, for u and v any two integers from to r, let m^v denote the 

 greatest number of linearly independent numbers of the aggregate T^v', 

 and, if m^v 5^ 0, let Juhvt for h = 1, 2, ... Muv, denote any system of 

 rriuv linearly independent numbers of r«„. We then have, by (20), 



(ZDJ J-uJuhv ^ J uht ^^ Juhv-l-vt J-wJuh'o ^ "uh'o Joh"v-lv ^ J oh"v 



{u,v= 1,2, . .. r; h= 1, 2, . . . w^; h' = 1, 2, . . . m«(,; h" = 1,2, .. . mov), 



i^l) lu'Juhv = = Juhviv' 



(w, V — 0, 1, 2, . . . r; h= 1, 2, . . . viuv; u'v' = 1, 2, ... r; u' 9^ u, v' 7^ v) . 

 We ma}' now show that the J's are linearly independent. For, if 



r r npq r '"po 



J = Jl Y H gphqJphq + H Z! (JphoJpho 

 p=l fl=l h = X p = l fe = l 



r ^op '"too 



~^ JL Jl gohpJohp + Y 3 oho J oho = 0, 



6 Cf. paper by the author in the Trans. Am. Math. Soc, 5, 545, note. 



