Manchester Memoirs, Vol. xlii. (1898), No.%. 11 



and the mean value of /' is then 



r=i-I . . . (60). 

 We may notice that if q denote the coefficient of reflection, 

 as given by formula (30) of § 2, we have 



--=^:- r-^ ■ ■ (-)• 



in accordance with known optical results. 



So far, we have supposed to be real. When 6 is 

 imaginary, the formula (40) must be replaced by 



or by 



^^ = Ce^{^^^ + •^«') - -^^ ^ Z)e^^^^^ +S'r)+su ^ /^^^ 



according as the upper or the lower sign is taken in (14). 

 The rest of the notation being as above, we find in the 

 case of (62) 



^^ sinh^.sin^^/^+^)^^ 



sinh u sin ka cosh {n + i)u ■{• i{i - cosh z^ cos ^^)sinh(;2 + i)u 



...(64), 



. /(cosh ut- cos ka) sinh (n+ i)u e ^ 



sinh u sin ka cosh {n •\- \)u -v i{\ - cosh u cos ka) sinh \ji-\-\)u 



...(65), 

 these results being obtained at once from (53) and (55) by 

 writing = — iu. When nu is great, we may put 



cosh (« + \)u = sinh {n + \)u = 00, 

 which gives 



„ ^ /(cosh u - cos ka) eika 



B = o, A = -^-r — V— r 7 '-r — i-x- (66). 



sinh2<!sm/ea + z(i - cosh?<!Cos/e«) ^ ^ 



The modulus of the latter expression, which is, moreover, 

 easily indentified with (35), is unity. The incident wave 

 is now reflected with unchanged amplitude, and there 

 is no transmitted wave. 



4. It remains to examine for what ranges of ka the 

 incident wave in the problem of § 2 is partially transmitted, 

 or wholly reflected, respectively. For this purpose we 



