282 Sir William Thomson on the " Oscillations of the 



in this order the equations so found, we can calculate successively 

 by means of them, B 8 , B 10 , B, 2 , . . . each in terms of B ; and we 

 thus have a solution of (2), with one arbitrary constant, B , which 

 may be written thus, 



«'=H./[ ft «)+B .F(/*,(f), .... (8) 



where /(/a, e) denotes the function of /jl and e expressed by the 

 series (4), with the coefficients calculated for the case B = and 

 H = 1 ; and F (/*, e) tbe function similarly found by taking H = 

 and B =l. 



The constant B , as Airy has pointed out [Tides and Waves, 

 art.' (113)] with reference to a corresponding question in the solu- 

 tion for semidiurnal tides, may be assigned so as to make the 

 north and south component motion of the water zero in a given 

 latitude. In the present case (that is, the case of symmetry round 

 the axis of rotation) we have [Airy, art. (95), or Laplace, Liv. iv. 

 chap. i. art. (3)] 



northward displacement of water =- — —- • . (9) 



1 4w?V(l— /a 2 ) dp v ' 



and therefore to make the north and south motion zero we must 

 have 



g*°> do) 



whence, by (8), 



if\r, e) 



d/jb 



If, then, we find B by this equation for any given value of //,, 

 we have a solution of the determinate problem of finding the 

 motion of the water under the given tide-generating influence 

 when, instead of covering the whole earth, the sea covers only an 

 equatorial belt between two equal circular polar islands. 



The solution thus obtained is in a series essentially conver- 

 gent, except in the extreme case of the polar islands vanishing. 

 For, taking the equations (6) in the order indicated above, and 

 so calculating B 2 successively from smaller to greater values of * 

 by the formula 



_2 + 2 4eB, 



J, '+*-J+4 J5 '+»"( l - + 4 ) (,-+!)• ' ■ • ( 12 ) 



we inevitably find for greater and greater values of i, 

 B l+4 i + 2 



B; +2 z + 4 



, more and more nearly the greater is i, . (13) 



