• Manchester Memoirs, Vol. xlii. (1898), No.'^, 9 



The dynamical equations, analogous to (26), are found to 

 give 



i^-co's>ka , j^e~^ - cos ka 



^_A=C'-;^^^' + D'^-^-.^^^^ . (43), 

 ismka tsmka 



and 



^^-^nka_ _ ,nO e ' '' ' COS ka _ ^ ^,^, /' - COS ka 



From (41) and (43) we obtain 



C(/Q _ ^-ika^ ^ j^^,-i^ _ ,-ika^ ^ 2,-sin ka . (45), 

 C(e'^'' - /) + i9(/'^^ - e - ') = 21 A sin ka . (46). 

 Again, from (42) and (44) we find 



—ika -i^ 



nr^ J . ■ ' Be-'''^'' . . (48). 



2t Sin d 



The last four equations may also be written 

 Ce^^^-^^^ sin^(0 + ka) -De' ^^'^^^^^^ sin^(0 - ka) = sin ka (49), 



- Ce^'^^'^^''hm\{Q-ka)-^ne-^'^^-^''hm~{Q+ ka) = Asmka (50) 



sin-(0 + >^rt) ., ,^,. ^ ^ 

 sinti 



sin-(0->^<a;) ., ,,/a , x 



sin Q 

 Substituting from (51) and (52) in (49), we find 



sin sin ^^/^^'-^^^^^ 



^~sin4(0 + ^^)/^''^^)^-sin^^(0-.^«)r'^"^^^' 



sin0sin.^^/<'^ + ')^^ , , 



sin0sin/^«cos(;z+ i)0 + i{i - cos cos >^«)sin {n+ i)B ^ 



