241-242] METHOD OF NORMAL COORDINATES. 439 



where 88 is an element of the surface r = a. Hence, when the 

 surface oscillates in the form r = a + f n , we find, on substitution 

 from (2) and (4), 



(12). 



To find the potential energy, we may suppose that the external 

 surface is constrained to assume in succession the forms r = a + 0? n , 

 where 6 varies from to 1. At any stage of this process, the 

 gravitation potential at the surface is, by (6), 



fl = const. + ^?n ............... (13). 



Hence the work required to add a film of thickness n &0 is 



W.^2gpfKSdS .................. (14). 



Integrating this from 6 = to 6 = I, we find 



(15). 



The results corresponding to the general deformation (1) are 

 obtained by prefixing the sign 2 of summation with respect to n, 

 in (12) and (15); since the terms involving products of surface- 

 harmonics of different orders vanish, by Art. 88. 



The fact that the general expressions for T and V thus reduce 

 to sums of squares shews that any spherical-harmonic deformation 

 is of a normal type. Also, assuming that f w oc cos (&amp;lt;r n t + e), the 

 consideration that the total energy T + V must be constant leads 

 us again to the result (10). 



In the case of the forced oscillations due to a disturbing 

 potential IT cos (at + e) which satisfies the equation V 2 fy = at 

 all points of the fluid, we must suppose H to be expanded in 

 a series of solid harmonics. If f n be the equilibrium -elevation 

 corresponding to the term of order n, we have, by Art. 165 (12), 

 for the forced oscillation, 



where a- is the imposed speed, and &amp;lt;r n that of the free oscillations 

 of the same type, as given by (10). 



