TABFR. — SCAL.VK FINITIONS OF lIVI'Fli COMPLKX NTMUKUS. (565 



necessary condition, is tlu> existence of a iiiodiilus, and, a fortiori, 

 that Ai 9^ 0. In this case, we may put /; = m and assign to the d's 

 the values given hy ec(uations (88), antl to the 77's the vahies gi\en hy 

 equations (90). W(^ then have 



m m m m 



^■1 = L C'Ci = Z Z Z "iTiiuem, 

 , , 1=1 1=1 ii = l t = l 



(92) 



m m m m 



'^' = JL (lic'i = L Z Z "iT.iueui.; 



i=l »=1 W=l 1=1 



and. therefore, 



, m m 



SiA = - Y. Z (Hjiuu = SA, 

 m , , 



(93) 



-.mm _ 



S-iA = Z Z (niuiu = '^'--l'. 



I = 1 w = 1 



On the other hand, if we assign to the 0's the values given l)y (90) and 

 to the Tj's the \alues given hy (88), which is now possible, we shall have 



m m m ?« 



A = Y. ('i(^i = Z Z Z «i7uir€ut, 



,_ , 1=1 »=1 M=l V=l 



(94) 



m m m m 



A' = Y. (^ic'i = Z Z Z <''i7uaei<r; 

 1=1 »=1 u=l 1=1 



whence follows 



1 m m 



S1.4 = ~ 23 Z "iTiua =^ ^ A' , 



(95) . 



1=1 ii = 1 



, m m 



S2A = - Y Z "iTmu = SA. 

 t=l u=l 



When either the representation of the number system (ci ('2, ■ . ■ 0^) 

 and its reciprocal system given by equations (88) or by equations (90) 

 fails, and indeed in any case, we may proceed as follows. Let // = 

 m -\- 1, and let 



(96 a) dj^> = Ji,^, ^m+l,/'^ = em.i,m+/'^ = 



{i, u,v = 1, 2, ... ?/i),. 



(96 6) d«.n.+/'^= 0, ^.>-i''' =\ 



{i, u = 1,2, ... m\ 119^1); 



