TABER. — SCALAR FINCTIONS OF HYPF.R COMPLEX NUMBERS 641 



Therefore, if in the s(jiiare array, 



I'll. 112) • • ■ 1 ir, Tio 

 I'2i> rac, . . . r2r> V20 



Toi) ro2j • • • Tor, Too 



we strike out any p rows or any p columns, the units of the aggregates 

 in the resulting array constitute a sub system of (ci, €2, ... €„). In 

 particular, for ^ u = r, the units of r„u constitute a sub system. 

 Since, by (32), (w, i') is nilpotent if u 9^ i\ we have 



(33) Siiv, r) = 0, 82(11, v) = (u, i' = 0, 1, 2, ... r; v 9^ w). 



Let now A be so chosen that r = r, where, as above, r is the greatest 

 value of r for any number A of the system. The units of Too then 

 constitute a nilpotent sub sj'stem; and, since every number of a 

 nilpotent system or sub system is nilpotent, we now have 



(34) Si(0,0) = 0, 82(00) = 0. 



For, otherwise, if Too contains an idempotent number Iq, we have 



/o/„ = 0= 7„/o {u= 1,2, ...r) 



by (27) ; and thus the number system (ci, e^ . . . Cm) contains r -\- 1 

 idempotent numbers mutually nilfactorial, which is impossible, as 

 shown above p. 19. Moreover, for 1 £ w ^ r, there is now but one 

 idempotent number in the aggregate Tuu- For, if possible, let r„„ 





contain a second idempotent number /'„ = J^ ChJuhu other than /, 



in which case we have /V = I'u', let 



I" =1—1' 



when we have, by (20) and (26), 



f" 2 = T 2 _ J J' _ J' J _L A' 2 _ r _ 9 7' -1-7' = T" 



i'J"u = I'uiL-i'u) = 0= ih-r:)!', = I" J' J 



and, by (32), 



I J", = iMu - I'u) - = (/u - i'u)U = l".U (t- ^ u). 



9 Cf. B. Peirce, loc. cit., p. 112. 



