348 BELL SYSTEM TECHNICAL JOURNAL 



where k is the square matrix [^i , ^« , • • • ^jv] composed of the modal columns 

 kj , and the latent roots are assumed to be distinct. The element in the ith 

 row andy column of hr'^ is, to our order of approximation, equal to —kij 

 when i 7^ j and to 



l-^ll'hshi (A2-19) 



8 = 1 



when i — j. Here, as usual, the prime on S indicates that the term s = j. 

 is to be omitted. 



Another result which is sometimes useful is the approximate solution of 

 UnXi + ^12:^2 + • • • + UipXp + • • • UinXn- = r;i 

 W2i:)Ci + U22X2 + • • • + U2pXp + • • • + U2nXn' = 1)2 (A2-20) 



Up\X\ + Up2X2 + • • • + UppXp + • • • + UpnXn = Vj 



where \ Vj\ "^l Vp\,J ^ P, and the w's are of the type assumed in (A2-14) : 

 Xi = ^' - ^^^^^ , J9^p (A2-21) 



Uj-j- UjjUpp 



N 



tjp . v^/ Upg f V 



Xp ^ — -Y 2^ 2~ ('7p^«p - t]sUpp). 



Upp «=1 ^aa l^pp 



The prime on S indicates that the term s — p (which happens to be zero 

 here) is to be omitted. A somewhat more general result which states that 

 the ith element in the column matrix {D + E)~ t] is 



N 



1 



8=.l 



VidT' +tl Eu d7' d7\Esi Vi - di rjs) (A2-22) 



may be derived from (A2-12). Here the elements of the column matrix 

 need not be restricted hy \ rjj\ <^\ 7jp\yj 9^ p. 



References 



1. Der Einfluss der Kriimmung von rechteckigen Hohlleitern auf das Phasenmass 



ultrakurzer Wellen, E. N. T., 16, 73-85 (1939). 



2. Research on Wave Guides and Electromagnetic Horns (Report III), E. T. /., 4, 64-67 



(1940). 



3. Unpublished work concerned with elbows of small radius of curvature and minimum 



reflection. 



