TABER. — SCAL.VR FUNCTIONS OF HYPER COMPLEX NUMBERS. 645 



and, therefore, 



(m, v) = (m, v) U = {u, v)-{v,uy (u, v)' 



k 



= {Colu + L CAiVu(')) {U, V)' 

 h=l 



k 

 h=l 



Whence it follows that m^, cannot exceed 7/i„u = k -\- I; and, there- 

 fore, 7)1 uv = '"uu- Similarly, we may show next that {v, u)' and the 

 product (r, w)'AV\ for h = 1,2, . . : k, are linearly independent, 

 and that in terms of these numbers every number of the aggregate 

 r,u can be expressed linearly. Finally, that /„ and the k products 

 (r, ii)' Xj-^^{u, v)', for A = 1, 2, ... A", are linearly independent, and 

 that in terms of these numbers every number of the aggregate r,j can 

 be expressed linearly. Therefore, in particular, 7/ /„ and /, are con- 

 nected, 



'"uu "'ur '"iiu "'iif 



For 1 = i = ni and u, v any two integers from to r, let {u, v)i 

 denote the component of c,- in Tu^. We then have 



(38) ei= Z L (P>q)i (^■= 1.2, ...m). 



p=0 q=0 



Whence, from (32), we derive 



r r 



(39) S,iu,v)ei=Z Z^ Sx{u,v)ip,q)i 



p=0 q=0 



f 



= T. Si (w, v) {v, q)i 



r 



= T. Si (»» Q)i (w, v) = Si (v, u)i {u, v) = Si {u, v) {v, u)i 



q=0 



{u, V = 0, 1,2 . . . r; l = 1, 2, ... w). 



« 



We may now show first that if, for = w = r, the aggregate Vqu 

 contains any unit, that is, if Muq > 0, the number system (ci, 62, . . .^m) 

 contains an invariant nilpotent sub system. For, let {u, 6) 7^ 0, 

 and let 



(0, u)i{u, 0) = (0, o)/ (i = 1, 2, . . . m); 



