ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. 319 



and consequently at the surface in question, where f= 0, 



Q-\-JL r^LjiJ^L^. < fy-J*lL. ( tf d<l> _ jiz df 

 dx~ ^^J^bc^a'tVc'dx' dy~ a*b'c r dy> dz " a' 3 b'c' dz ' 



These values substituted in (6) give, when we replace -f , 



7/ 7/r 



-f- and J- with their values at the ellipsoidal surface, 

 dy dz 



, 



abc 



(8), 

 v ;> 



which may always be satisfied by a proper determination of 

 one of the constants X and /*, the other remaining entirely 

 arbitrary. 



From what precedes, it is clear that the equation (2) and 

 condition to which the fluid is subject may equally well be 

 satisfied by making 



provided we determine the constant quantities therein contained 

 by means of the equations 



v 



=x 



c abc J ^abc abc 



respectively. The same may likewise be said of the sum of the 

 three values of </> before given. However, in what follows, we 

 shall consider the value (3) only, since, from the results thus 

 obtained similar ones relative to the cases just enumerated may 

 be found without the least difficulty. 



Instead now of supposing the solid at rest, let every part of 

 the whole system be animated with an additional common velo- 

 city X in the direction of the co-ordinate x. Then it is clear 

 that the equation (2) and condition to which the fluid is subject, 



