ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 151 



Part I., making use of the equation (31) instead of the simpler equation (23) of 

 the previous paper. On putting yj, zero, the equation (31) gives 



^CU-LlCl + ^C^^O (43), 



an equation which must hold for all values of n equal to or greater than .s, it being 

 understood that Cj_ 2 = 0, and C'_, = 0. The series of equations typified by (43) 

 may be divided into two groups, in the former of which the suffixes involved are 

 such that n s is even and in the latter odd. The types of motion resulting from 

 these two groups may be treated independently, the former being characterised by 

 symmetry with respect to the equator and the latter by asymmetry. The treatment 

 of the two groups of equations will be exactly similar, and we shall therefore in the 

 main confine our discussion to the former group. 

 If we introduce the notation 



'/ 1J I 'I I ~l II 



H: < = 'if - t'lT - . . . - "L/ 



(i-O, 



LI J'u+s . . (/ </'/. 



it may be shown as in Part I. that provided L/C*,. = 0, 



'' J "^" . _ TT, y-2*"' _ . T7-. / ( r \ 



~^~ e in,, , <e iv,, (4j), 



and therefore the equation (43) may b written 



whence the period-equation for the free oscillations of symmetrical type is obtainable 

 in the form 



L: - H; t _ 2 - K; +2 = o (46), 



when n s is an even integer. 



The same equation will apply to the asymmetrical types if we suppose that n s is 

 au odd integer, and that the continued fraction H" terminates with the partial 



quotient -'-'-~^ - . 



In particular, putting it, n = 0, we can express the period-equation for the 

 symmetrical types in the form 



L| +2 - !<;+. - ud mj. 

 while, putting n s = 1, that for the asymmetrical types may be written 



L; +8 - 



On the analogy of the problem dealt with in Part I., we may anticipate that, when 



