REFLECTIONS FROM CIRCULAR BENDS 



335 



The elements of the diagonal matrix a^Vl are given by (5.1-1) and those of 

 P and 5 by the equations and tables of Appendix I. 



a?T\ 



(5.1-6) 



The next step is to use (5.1-6) to evaluate the coefficients of x and y in 

 (2.3-3). The square matrices VaC tanh r«c and VaC coth VaC cause most of 

 the computational difficulties. We shall deal with these matrices by using 

 Sylvester's theorem (an account of this theorem is given in Section 3.9 of 

 Reference®). This requires the determination of the latent roots and modal 

 rows of aT\. However, it is interesting to note that the matrices in ques- 

 tion may also be computed from cVa (which is easily obtained from a^r^) 

 by processes which employ only matrix multiplication, addition, and in- 

 version. 

 Thus, setting A for c Fa, 



A sinh A 



cosh A 



v4' A^ 



3! ^ 5! ^ 



a"" A^ 



2! 4' 



A coth A = (cosh A){A-'^ sinh A)' ' 

 A tanh A = A\A coth A)-"^. 



Although the series always converge, they do so too slowly to be of use in 

 our computations. The same is true of the series 



A tanh A 



E 8^' [(2m 



1)'// + 44^'. 



For the matrices we shall encounter it appears best to use Sylvester's 

 theorem even though this requires the determination of the latent roots and 

 modal rows of a r« . The square matrix formed from the modal rows* 

 will be denoted by k. 



* As has already been mentioned in the footnote associated with equation (1.3-9), we 

 shall use the notation and theory set forth in Sections 3.5 and 3.6 of Reference'. 



