189-191] SPHERICAL SHEET OF WATER. 315 



space standing on the element of area a sin 6$co . a$6, whilst the 

 right-hand member expresses the rate of diminution of the volume 

 of the contained fluid, owing to fall of the surface. Hence 



d_ 1 _ ( d (hu sin 0) d(hv)} 

 dt~ a sin I d6 da&amp;gt; }&quot; 



If we neglect terms of the second order in u, v, the dynamical 

 equations are, on the same principles as in Arts. 166, 185, 



du d dl dv d c?H ,. 



dt~ add~ad6 dt ~ asinOda) ~ a sin 0da&amp;gt; 

 where U denotes the potential of the extraneous forces. 

 If we put 



(3), 



these may be written 



du_ gd - dv_ g d 

 dt~~ad@ ( ^~ Qj K-^iam?*^ V ri 



Between (1) and (4) we can eliminate u, v, and so obtain an equa 

 tion in f only. 



In the case of simple-harmonic motion, the time-factor being 

 e i(&amp;lt;rt+ e ) } the equations take the forms 



,,_ i (d(husin0) d(hv) 

 ^~ ~ 



191. We will now consider more particularly the case of 

 uniform depth. To find the free oscillations we put ?=0; the 

 equations (5) arid (6) of the preceding Art. then lead to 



1 d . dfr 1 d* 



n x 



This is identical in form with the general equation of spherical 

 surface-harmonics (Art. 84 (2)). Hence, if we put 



o--a 2 /gh = n (n + 1) (2), 



a solution of (1) will be 



?=# (3), 



where S n is the general surface-harmonic of order n. 



