108 Applications of BesseVs Functions to Whispering Gallery. 



Similarly,, if we write ka = x' +iy', where x' — a/p, y — ?////,, 



Dn'W + iy') = D/M+VDh'V ) 

 l) n ^' + iy') VnCv'l + iy'D^x') > 



and in virtue of (10) 



D»!V) = -^^T>n'(x f ) + smh 2 /3 D»(V), 

 where cosh @ = njx'. Thus 

 PnV + tyQ _ D»'(J) j\ . • ,/ coshff 



Accordingly with use of (36) 



d«(a- + «y) l v w 



+ S inh^-gf)} . (38) 



Equation (32) asserts the equality o£ the expressions on the 

 two sides of (38) with 



If we neglect the imaginary terms in (38), (37), we fall back 

 on (34). The imaginary terms themselves give a second 

 equation. In forming this we notice that the terms in if 

 vanish in comparison with that in y. For in the coefficient 

 of y' the first part, viz. — m -1 cos1i/3, vanishes when n is 

 made infinite, while the second and third parts compensate 

 one another in virtue of (33) . Accordingly (32) gives with 

 regard to (34) 



ah e 2t ficr g-swos-tanh/s) 



11 = pk smh£ = 7" sinh/3 ' * ' * ^ 39) 



in which cosh/3 = yu, (40) 



In (39) iy is the imaginary increment of Jta, of which 

 the principal real part is n. In the time factor e ikYi , the 

 exponent 



'j,Yf - - 1 —- — l,l ^ t I & (39) Y 

 fjua /ua \ n J 



