386 Lord Rayleigh on the Vibrations of a 



that the motion is strictly in two dimensions. Introducing 

 the further assumption that <£ occos n0, we get in place of (5), 



G^ + ^-J + « 2 V=°> • • • • < 7 > 



of which the solution is 



$■= a n cos pt cos n6 J n (/cr) (8) 



The relation between a n and A n of (1) is readily found by 

 equating the value of d<p/dr 9 when r = a, to d8r/dt, both of 

 which represent the normal velocity at the circumference. 

 We get 



a olA. n / N 



a » C0S ^=«J^M S* < 9 > 



The kinetic energy of the fluid motion is given by 



= \P U ^^<^dO + K 2 (TjV dx dy dzl 



— \irpza I cos 2 pt fa . J n (/ea) J'„(/ea) + k 2 I J 2 n (tcr) r dr\ ., (10) 



For the potential energy of the liquid, if compressible, we 

 have 



= ^nrpz a 2 n s\v? pt K? I J^Kr)rdr (11) 



The sum of the potential and kinetic energies for the solid and 

 liquid together must be independent of the time. The unin- 

 tegrated terms in (10) and (11) cancel, and we find 



W»- 1)» pa S^ca) 



In the application of (12) ica is a small quantity. From the 

 ascending series for J„(/ca) we find 



a J„(«a) _a 2 / kV \ 



«J' n (*a)~nV ^.2n + 2 ^'"r ' * ^ ; 

 so that approximately 



Mg^)! =<7 , (1+n - 2)+B - Va ( 1+ _g_).. (14) 



