TABER. — SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS. 047 



I shall now assume that the number system (^i, ei, ... e^) contains 

 no invariant nilpotent sub system, in which case, by what has just 

 been proved, we have 



(40) m^o = »iou = vioo = {u = 1,2, . . . r), 



that is, no number of the system is contained in V^o nor in either of the 

 aggregates F^o, F^u for u = 1,2, . . . r. Further, 



(41) w„„=l (/A = l,2,...r), 



that is, 7u is the only unit in Fuu for 1 ^ it ^ r. Finally, for u and v 

 any two distinct integers from 1 to r, if 7„ and /„ are connected, 



wi„, = m^ = 1 ; 

 whereas, if /„ and 7, are not connected, 



w„, = m^ = 0. 

 In the present case, the number system contains a modulus, viz., 



(42) e = 7i + 72 + . . . + 7„ 



since, for w, v any two integers from 1 to r, if F^, contains a unit J„i„ 

 we have 



by (26) and (27). 



It is, with the present assumption, convenient to modify our nota- 

 tion to indicate the connection which may exist between certain of 

 the idempotent numbers, I\, I2, ... 7?. I shall, therefore, suppose 

 these numbers arranged in v aggregates, 1 = j* = r, containing respec- 



tively fj.1, M2. • ■ • l^v of the 7's, where X Mp — >', any two idem- 



p=i 



potent numbers in the same aggregate being connected, but no 

 pair of idempotent num])ers in different aggregates being connected; 

 and, for l=p = v,l .shall denote by Ij^^ {u = 1,2, . . . fip) the idem- 

 potent numbers in the j^^'' aggregate. The r- aggregates of numbers, 

 formerly denoted by Fu,, for u, v = 1, 2, . . . r, into, one or other of 

 which the units fall when the system is regularized as above and 

 contains no invariant nilpotent sub system, will now be denoted by 

 Fu,/^''^ for p,q=l,2,...v, and for w = 1,2, . . . Hp and u = 1, 2, . . Hq\ 



