358 TIDAL WAVES. [CHAP. VIII 



where /z- is written for cos Q. In this form the equation is some- 

 what intractable, since it contains terms of four different dimensions 

 in fju. It simplifies a little, however, if we transform to 



as independent variable ; viz. we find 



1/2 (1 ~ " 5) 5' - " ' - (S- 2 "*-/ 3 " 4 )?' = -&*? = - /8JETV. . .(9), 

 which is of three different dimensions in v. 



To obtain a solution for the case of an ocean covering the 

 globe, we assume 



Z' = B + B.^ + B^+...+Bij V y+ ............ (10). 



Substituting in (9), and equating coefficients, we find 



#0 = 0, B 2 = 0, O. 4 = ............... (11), 



=0 .................. (12), 



and thenceforward 



2j (2j + 6) B 2j+4 - 2j (2j + 3) B. 2j+2 + 0B aj = ...... (13). 



These equations give B 6 , B 8) ... B 2 j, ... in succession, in terms of 

 B 4 , which is so far undetermined. It is obvious, however, from the 

 nature of the problem, that, except for certain special values of h 

 (and therefore of @), which are such that there is a free oscil- 

 lation of corresponding type (s = 2) having the speed 2n, the 

 solution must be unique. We shall see, in fact, that unless B^ 

 have a certain definite value the solution above indicated will 

 make the meridian component (u) of the velocity discontinuous 

 at the equator*. 



The argument is in some respects similar to that of Art. 210. 

 If we denote by Nj the ratio B. 2 j +2 /B. 2 j of consecutive coefficients, 

 we have, from (13), 



N . -9 + 8 ft 1 (U} 



~ ' 



from which it appears that, with increasing j, Nj must tend to 

 one or other of the limits and 1. More precisely, unless the 

 limit of NJ be zero, the limiting form of ^V j+1 will be 

 (2j+3)/(2j + 6), or 1 - 3/2J, 



* In the case of a polar sea bounded by a small circle of latitude whose angular 

 radius is <*-, the value of B 4 is determined by the condition that w=0, or d'ldv=Q, 

 at the boundary. 



