118 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



of these functions are continuous. We assume that xpi = \l/, D\pi = D\l/, 

 DVi = D~\p, D% = D^yp and D% = D*xl/ at the ends of the intervals. 

 From (2.20), Wi'D^i ^ -H as y ^ ijo . 

 Since 



2 8iy - 2m + lyo) = ^ + - £ (-1)" cos Ky (2.21) 



we obtain from (2.20) 

 / 1 



2?/oi/'i 



,/32ff(i22 - Wp2) 



+ 2i;(-l)" ^"^'"^ 



Since 



^ + ^F = (Z) + /3)(t.x^Z)^ + fi^)V, 



using (2.4), the condition for matching to the external field, 



dV 



^ + /37 = 0, 



dy 



yields, using D\p = DV = and Ui*D^\f/ = — i^, the relation 



(ui'D' + fi')Vi = 3^/3 at 2/ = 2/0 . 

 Applying this to (2.22), we then obtain, finally, 

 yo ^ 1 



+ 2Z 



r (^2 4- X„2)[(i22 - Ml2X„2)2 - cOp2(Q2 + ,,^2X„2)] 



(2.22) 



(2.23) 



For the odd solution we use a function, yp2(y), equal to ;/'(?/) in — //o < 

 y < yo and representable by a sine series. To ensure the vanishing of D^p 

 and 7)V at ?/ = ±?/o it is appropriate to use the functions, sin n„y, where 

 Mn = (n -\- l'2)ir/yo . The period is now iyo and we define \p2(y) in /yo < 

 y < 32/0 by the relation i;'2(2/) = ^{2yo — y) and in — 32/o < 2/ < — 2/o by 

 ^2(2/) = ^{ — '^Uo — y)- Thus, we write 



00 



1^2(2/) = 2 C?n sin UnV Hn = (w + 3^)^7/0 







^2(2/) ^^i" ho supposed to satisfy 



