166 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



//I ,,2\TT !f H W*0*M#> I liJ I ^ "~ 8 . P' I p I _>i p I 



V I L A* / *~> 7 ^n t; J* 1 * * ' T' "~ 2 T^ " ' r< "+ 2 ' ' ' f 



ft L i L u J 



r i )* iy 



J i _Ll'irJ. ps i +' p i 



_ J _i_ ^'"-i ps I -^"+1 p 



L ^ ^ 



f". . . + ~ -^ 9*-* P'- 2 + 9n P?, + -^T- </,, +8 P rt+ 

 L n 



and we shall obtain expressions for the height of the surface-waves and the velocity- 

 components. 



I have not thought it worth while to compute any of these series in detail, as the 

 general character of them may be inferred from the series computed in Part I. for 

 the special case where s = 0. For large values of n, and even for comparatively 

 small values of n when ligj^ora^ is large, the quantities C will rapidly diminish as 

 we pass away in either direction from Q r Hence the most important term in the 

 series for will be that involving P', and this term will in general sufficiently 

 predominate to decide the number and approximate position of the nodal parallels of 

 latitude. 



If we neglect C'_i in comparison with C*. + J and suppose that w is small in com- 

 parison with \, the formula (49) gives 



r (r + s+ 1) CJ +1 = (r + 1) (27- + 3) Dj 



and therefore Dj will be of the same order of magnitude as C* +1 . Hence, when r is 

 less than n, D* will be of the same order of magnitude of C' +) , and similarly it may 

 be seen that when ? is greater than n, D'. will be of the same order of magnitude as 

 Q._I. Thus the predominant terms in the expressions for the velocity-components 

 will be those involving C*, D' ( _,, D' l+1 . 



The exponential factors indicate that the type of motion involved will consist of 

 waves propagated round the sphere with uniform angular velocity X/s about the 

 polar axis, and that there will be s crests or troughs on each parallel. Positive 

 values of A. will correspond with waves propagated in the opposite direction to the 

 rotation, that is westwards, while negative values will correspond with easterly 

 waves. The paths of the fluid particles will be ellipses with their axes directed 

 along the meridians and parallels. 



11. Oscillations of the Second Class. 

 In dealing with the oscillations of the second class we proceed in the same manner 



