TABER. — SCALAR FUXCTIONS OF HYPER COMPLEX NUMBERS. O.')? 



we have 

 (SO) 



v'= r-v, 



where T is the determinant of the transformation. Therefore, the 

 equation V = is invariant to any transformation of the units of the 

 system (t'l e^, ... e„). 



We have now the following theorem: 



Theorem V. Let {ci, ci, ... Cm) he any given number system consti- 

 tuting a sub system of the quadrate e^v («, ^ = 1> 2, . . . u): thus let 



Ci= T. Z ^uv(u^ (1= 1,2, ... m). 



u=l v = l 



For any given number 



n n 



.rl = ^ ^ Ouvfuv 

 u=l v=l 



of the quadrate, hi 



1 



SA = - Yi "uu, 

 n , 



wheri, for any given number 



m 



m n n 



X= Z Xiet = L I L ^idu.^'^^u. 



i = l t = l u=l v=l 



of the system, (ci, e^, . . . em), we have 



1 



SX = ' I L Xiduu^"^. 



n 



Let 



4=1 U=\ 



V 



S ei Cj 



{i,j = 1,2, ... m) 



denote the resultant of the system of equations 



SXci = £ XjHejei =0. (i = 1, 2, . . . m). 



