TILiNSMISSION LINK EQUATIONS 



149 



and PGP~^ may be written as a convergent infinite series, the wth term 

 of which contains the matrix (assuming only three circuits for the sake of 

 simplicity) 



2 



r 2 

 71 



7 







2 

 72 



7 





 



2 



73 - 7 J 



P-' = R", 



(1.57) 



where the equality follows from the definition (1.12) of R and equation 

 (A1.9) of Appendix I. This series for PGP~^ is exactly the same as the 

 series (1.53), and this completes the proof of the first method. 



The equations (1.18) and (1.28) which are incidental to the first method, 

 will now be established for the case in which the matrix F occurring in 

 them is equal to PGP~^. For then we have equation (1.55) and the 

 equations 



cosh xV = P 



cosh X'Yi 



cosh xyi 



cosh xyi_ 



(1.58) 



sinh xV Zo = sinh xY T Z 



where sinh xT T ^ may be expressed in the same fashion as cosh xV, the 



snih v"/)' 

 rth element of the diagonal matrix being '- — . The elements in the 



7r 



diagonal matrices occurring in these expressions may be expanded in series 



by replacing 7r by its representation (1.56), assuming -^ ~ 1 

 using 



< 1, and 



Z^\+T _ 



cosh 2\/l -\- r = 



V^ (rzY 1 /ttz 



it,-iW 



sinh 2\/l + r 



=^(?y^ 



/p-i(s) 



/p+|(2) 



where /p-i(3) and ifp_i(s) are Bessel functions of the first and second kinds, 

 respectively, for imaginary argument. Equations (1.18) and (1.28) are 

 obtained when equation (1.57) and the Bessel function recurrence relations 

 are used. 



^ These are special cases of formulas given in "Theory of Bessel Functions," by G. N. 

 Watson, page 141. 



