TABER. — SCAL.\R FUNCTIONS OF HYPEK COMPLEX NUMBERS. 639 



then, for any pair of integers u, v from 1 to )\ 



muv 



/ . (luhcJuhv ^ iuJ -Iv ^ Uj 

 h=l 



and, since by supposition Juw, Juzv, etc., are linearly independent, we 

 have 



guhv = {u, V = \,2, ... r\ h = 1,2,... ?n„r). 



Whence it follows that 



r VI po r Wop 7ngo 



J = Yi JL 9phoJpho-{- Y. Y. (JohpJohp-\- Y 9ohoJoho= 0; 

 p=l h = \ P=l h = \ /t=l 



and, therefore, for l^u = r, 



Y duhoJuho = luJ ~ 0, Y ffohuJohu = J lu = 0. 



h=l h=l 



From these equations we derive 



Quho =0 (u = 1,2, . .. r; h = 1,2, ... muo), 

 Qohu =0 (u = 1,2, ... r; h = 1, 2, ... Mou). 



Thus, ultimately, we have 



moo 



J — Y QohoJoho = 0; 



h = l 



whence follows 



goho =0 (// = 1, 2, . .. viqo). 

 Since 



r r 



(28) ei= Y Y luCiU 



u=l r=l 



r r T r 



+ Z iJuCi-Y JueiU)+Y {eJu-Y J'>eilu) 



U=l »=1 «=1 1=1 



r r r r 



+ {ei-Y hci-Y Cih+Y Y luCiU) 



u=l u=l «=1 »=1 



(i = 1,2, ... w), 



it follows that each unit of (ei, €2, ... e„), and thus that any numl^er 

 of this system, can be expressed linearly in terms of lunnbcrs in the 

 (r + 1)- aggregates V^,{u, t; = 0, 1, 2, . . . r), and, therolore, linearly 



