320 TIDAL WAVES. [CHAP. VIII 



193. The oscillations of a sea bounded by meridians, or 

 parallels of latitude, or both, can also be treated by the same 

 method*. The spherical harmonics involved are however, as a 

 rule, no longer of integral order, and it is accordingly difficult to 

 deduce numerical results. 



In the case of a zonal sea bounded by two parallels of latitude, we assume 



* (i), 



where /z = cos#, and p(p), q(fi) are the two functions of //, containing 

 (1 /n 2 )* s as a factor, which are given by the formula (2) of Art. 87. It will 

 be noticed that p (/z) is an even, and q (/z) an odd function of /*. 



If we distinguish the limiting parallels by suffixes, the boundary conditions 

 are that u=0 for /M = /X I and/u = fi 2 . For the free oscillations this gives, by 

 Art. 190 (6), 



3 (ii), 



(iii), 



whence 



which is the equation to determine the admissible values of n. The speeds 

 (o-) corresponding to the various roots are given as before by Art. 191 (5). 



If the two boundaries are equidistant from the equator, we have fi 2 = ^ 

 The above solutions then break up into two groups ; viz. for one of these we 

 have 



5 = 0, p M = ................................ (v), 



and for the other 



.............................. (vi). 



In the former case has the same value at two points symmetrically 

 situated on opposite sides of the equation ; in the latter the values at these 

 points are numerically equal, but opposite in sign. 



If we imagine one of the boundaries to be contracted to a point (say 

 /z 2 = 1), we pass to the case of a circular basin. The values of p (1) and q (1) 

 are infinite, but their ratio can be evaluated by means of formulae given in 

 Art. 85. This gives, by (iii), the ratio A : B, and substituting in (ii) we get 

 the equation to determine n. A more interesting method of treating this 

 case consists, however, in obtaining, directly from the differential equation of 

 surface-harmonics, a solution which shall be finite at the pole /i = l. This 

 involves a change of variable, as to which there is some latitude of choice. 

 Perhaps the simplest plan is to write, for a moment, 



2 = J(l- /x ) = sin 2 ^ .............................. (vii). 



* Ct . Lord Eayleigh, I. c. ante p. 314. 



