Manchester Memoirs, Vol. xlii. (1898), No. 3. 19 



and thence, from (94), 



We have a formula of this type for each interpolated 

 system. Distinguishing the successive systems by suffixes, 

 we have, from (6) and (7), 



^' ^ sml^(^^+' ~ 24cos^^ + ^,_i) . (97), 

 and therefore 



|,+i - 2{cos^« -/{k)smka}^, + 4_i = o . (98), 

 where 



/W = ^{.V^P-<2-.<^^7 . (.9). 



The difference-equation (98) is to be solved as before. 

 The critical values of k are determined by 



cos ka -f{lc) sin ^« = + I . . (100) 



and are therefore given graphically by the intersections of 

 the curves 



>' = cotJ^, j=-tanja; . . (loi) 

 with the curve 



y=/{x/a) . . . (102). 



The latter curve has asymptotes parallel to r correspon- 

 ding to 



kc={b,la,)\ [r=i,2,.,.n\ . . (103). 



It is not difficult to establish that these special values of 

 k will occur always in intervals of total reflection. 



It may be shewn also that the terminal conditions, in 

 the problems of §§ 2, 3, reduce to the forms given in 

 equations (24), (27), and (41)... (44), and thence that the 

 expressions obtained in § 3 for the coefficients of reflection 

 and transmission in terms of ka and B are perfectly general. 



