REFLECTIONS FROM CIRCULAR BENDS 325 



Likewise, the element in the i*^ row and j^^ column of the square matrix 

 F~^ To is — Viibj when i 9^ j and is 



(1 - VU + S' Virr, V„^i) di (3.3-4) 



TO 



when i = j. 



By combining the approximate expressions for the elements of T tanh cV 

 (given by (3.2-5) and (3.2-6)) and F'^ To we find that if Ua denotes the i'^ 

 row andy*^ column of T tanh cT + V~^ Tq then 



Uij = Dij - Vij8j, i 9^ j 



Uii = yih + Z' Dmikirr. + 6^ (l - Vu + E' Cm^ (3.3-5) 



= (Tt — SilVt + 2.^ {Dmikim + 5iC;„i) 

 m 



In these equations we have set 



Oi = 5i + 7i/i, Cmx = VimVmi 



Dmi = (jiti — ymim)kmi = {yiti " ymtm)Fmi/{8i — dm) 



(3.3-6) 



where 7i and k^i are given by (3.2-2) and (3.2-3). 



We are now in a position to identify the matrix equation (3.3-1) for x 

 with the set of equations (A2-20). The quantity rji which appears on the 

 right hand side of the i*^ equation in (A2-20) is given by (3.3-3). The 

 coefficients which appear on the left hand side are the m's defined by (3.3-5). 

 Therefore, from (A2-21), when i 9^ p, 



Xi = — — [—Vip(ap - dp) — Dip] 



(Ti(Tp 



where we have neglected higher order terms and in so doing have replaced 

 7,- by the simpler 5/ . . 



W^en i = p (A2-21) yields * 



Xp = U^plVp + z2 ('i^pmUmpVp ~ UpmUppTIm) / (UmmUpp)] (3.3-8) 



TO 



In order to combine the second order terms in \/upp with those in the rest 

 of the expression for Xp w^e assume that dp is the major portion of Upp . 

 Then, approximately, 



\/Upp = a^[\ + bpVpp/<Tp — zL (Dmpkpm + 8pCmp)/a-p] (3.3-9) 



