189-191] SPHERICAL SHEET OF WATER. 315 



space standing on the element of area a sin #&> . aW, 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 6) d (hv)} 



dt a sin \ 



.(i). 



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

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



<lM = _d^_dQ L dv = _ d dQ, 



dt $ add ad6 ' dt ~ ^ a sin 0dco a sin Qdco' ' '' '' 



where fl denotes the potential of the extraneous forces. 



If we put 



? = -/</ .............................. (3), 



these may be written 



dt add l dt~ 



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 

 ) t t ne equations take the forms 



d(hv)} , . 



j ' ' W> 





Uss i(-f) t ,. f '__ .(f-g) ...... (6). 



^ b -^ *' 



191. We will now consider more particularly the case of 

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

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



1 d / . *d\ 1 d 2 ? (7 2 a 2 ^ 



This is identical in form with the general equation of spherical 

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



<T 2 a 2 /gh = n(n + l) (2), 



a solution of (1) will be 



?= (3), 



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



