342 TIDAL WAVES. [CHAP. VIII 



This is identical with Art. 189 (ii), except that we now have 



<r 2 - 4w 2 4ns 

 gh Q o-a? 



in place of o- 2 /^. The solution can therefore be written down from the 

 results of that Art., viz. if we put 



a 



wehave t=A, ('-) F , 0, y,~ 2 )e" (iv), 



where a = J+Js, 0=l+s-&, y=s + l ; 



and the condition of convergence at the boundary r = a requires that 



=* + 2; ....................................... (v), 



where.;' is some positive integer. The values of a- are then given by (iii). 



The forms of the free surface are therefore the same as in the case of no 

 rotation, but the motion of the water-particles is different. The relative orbits 

 are in fact now ellipses having their principal axes along and perpendicular to 

 the radius vector ; this follows easily from Art. 202 (3). 



In the symmetrical modes (s = 0), the equation (iii) gives 



where <T O denotes the 'speed 3 of the corresponding mode in the case of 

 no rotation, as found in Art. 189. 



For any value of s other than zero, the most important modes are those for 

 which =s + 2. The equation (iii) is then divisible by <r + 2n, but this is an 

 extraneous factor ; discarding it, we have the quadratic 



whence 



This gives two waves rotating round the origin, the relative wave-velocity 

 being greater for the negative than for the positive wave, as in the case 

 of uniform depth (Art. 203). With the help of (vii) the formulae reduce to 



the factor e* (e being of course understood in each case. Since = iR, 



the relative orbits are all circles. The case s 1 is noteworthy ; the free 

 surface is then always plane, and the circular orbits have all the same 

 radius. 



When k > s+2, we have nodal circles. The equation (iii) is then a cubic 

 in <r/2n ; it is easily seen that its roots are all real, lying between oo and 



