REFLECTIONS FROM CIRCULAR BENDS 



347 



We shall now state several results related to the above formulas. Let u 

 denote the matrix Z) + E so that the typical element iiij = En , i 9^ j, 

 and Uii = di-{- Eii . Then the latent roots Xi , As , • • • Xa- of u are the roots 

 of the equation obtained by setting the determinant of X/ — u to zero: 



X — Wii — Wi2 



— W21 X — U22 



\\I 



= 0. 



(A2-14) 



The modal column kj corresponding to the^th root X/ satisfies the matrix 

 equation 



(Xy/ - u)kj = 0. (A2-15) 



Since the non-diagonal elements of u are small, we see from (A2-14) that 

 we may label the roots so as to make Xy nearly equal to w/y , and this together 

 with (A2-15) shows that all the elements of ^yare nearly zero except the 7th 

 which we may choose to be unity. When these approximate values are 

 taken as a first approximation in the process of solving (A2-15) by successive 

 approximations, the second approximation is found to be 



Xy = Ujj + YJ 



^ij "T" / J Ujs Ksj 



ki 



(A2-16) 



k ■ = 



s^j 



where the 1 in the column for kj occurs as the^th element. This expression 

 for Xy occurs in the perturbation method often used in wave mechanics. 

 For the modal row kj corresponding to Xy we have in much the same way 



Kj{\jl — U) = 

 Kj = [Kij , K2i , ■ ' • Kjj , " ' KNj] (A2-17) 



wjV Us 



where the last expression is an approximation and where /cyy may be chosen 

 at our convenience. 



The results (A2-10) and (A2-11) may also be obtained from (A2-2), 

 (A2-16) and the relation* 



~f{Xi) • 

 /(X2) • 



f(u) = k 











/(^i^). 



k-' 



(A2-18) 



* This is equation (12) in Section 3.6 of Reference^, .\lthough proved only for poly- 

 nomials it may be verified to be true for the applications which we shall make. 



