576 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



We shall now say no more about this pole. 



Putting (25) into (24) we get at once the form 



-{BU, + k),j, + /,•) + {G-'h + G'GBij, + k), 2h + 2GB(j, + k)) 



+ (2Fh + 2FGB(j, + A-) - Fj, - Fk, 2h + 2GB(j, + k) - ./i - k). 



Here we cannot at once put G'"'G' = 1, because this is only true in U. 

 We expand in the following way: The first product is left intact, the 

 second is expanded by distributivity into four terms, and in the third 

 we use (22) and expand into five terms by distributivity. The ten re- 

 sulting terms are: 



- (5(ii + /.■),7i + /0 + 2{G-%h) 



+ 2(G-'GB(jy + k), h) 4- 2(G-'h, GB(j, + A-)) 

 + 2{G-'GB{j, + A-), GB{j, + A-)) 

 + 4(F/i, h) - 2{Bi_n + /■■), h) 

 + 2(F/i, 2GB{j, + A-) - h - /'■) 



- 2(5(ii + /.•), GB{j, + A-)) + (5(ji + A-), Ji + A-). 



Enumerate these terms 1,2, • • • , 10 in the order written. We shall show 

 by combining that only 2 and (3 remain. 



Clearly 1 and 10 cancel. 



Consider the operator G~^G as we have defined it. If v e V, we can put 



V = U -{- Vi 



where u e U, wi e Vi . Then 



Gv = Gu + Gvi = Gu, 



because of the matrix form for G in the coordinate system chosen in 4.5. 

 By definition of G~^ (in 4.56), since u e U, 



G~'Gu = u. 



Hence, combining the last three relations, 



G ''Gv = V - vi (26) 



for any v e V, where i'l is a suitable element of Vi (depending on v of 

 course) . 



Using (26) in term 3, we get for this term 



2(5(ji + A-), h) - 2(ri, h) 



