241-242] METHOD OF NORMAL COORDINATES. 439 



where SS 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), 



***5//fe?0 ..................... (12). 



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

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

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

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



11 = const. + 21 y0Cn ............... (13). 



Hence the work required to add a film of thickness n S0 is 



W.^j-^ffpfftJW .................. (14). 



Integrating this from 6 to 6 = 1, we find 



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 n oc cos (<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 1' cos (at -f- e) which satisfies the equation V a n' = 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 cr is the imposed speed, and <r n that of the free oscillations 

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



