ON THE VIBRATION OF PENDULUMS IN FLUID MEDIA. 317 



When the fluid is supposed to move over a fixed solid 

 ellipsoid, the principal difficulty will be so to satisfy the equa- 

 tion (2), that the particles at the surface of this solid may move 

 along this surface, which may always be effected by making 



supposing that the origin of the co-ordinates is at the centre 

 of the ellipsoid ; X and //, being two arbitrary quantities constant 

 witli regard to the variables x, y, z : and a, b, c being functions 

 of these same variables, determined by the equations 



= -+/ b* = b"+f, c> = c'*+f, and ^ + |! + J=l ....(4), 



in which a', >', c are the axes of the given ellipsoid. 



To prove that the expression (3) satisfies the equation (2), 

 it may be remarked, that we readily get, by differentiating (3), 



P x /ay dy dy\ 



<?bc \dx* "*" d * dz') 



dx* df - dz* a 3 bc dx <?bc \dx* " df 



JL _ 



' * 2b* * 2cV \dx \dy \dz 



* In my memoir on the Determination of the exterior and interior Attrac- 

 tions of Ellipsoids of Variable Densities 1 , recently communicated to the Cam- 

 bridge Philosophical Society by Sir EDWARD FFRENCH BROMHEAD, Baronet, 

 I have given a method by which the general integral of the partial differential 

 equation 



_<PV &V d?V d*V n-s dV 



-^ + d^ + --' + dtf + dtf + u du 



may be expanded in a series of peculiar form, and have thus rendered the deter- 

 mination of these attractions a matter of comparative facility. The same method 

 applied to the equation (2) of the present paper has the advantage of giving an 

 expansion of its general integral, every term of which, besides satisfying this 

 equation, may likewise be made to satisfy the condition (6). The formula (3) 

 is only an individual term of the expansion in question. But in order to render 

 the present communication independent of every other, it was thought advisable 

 to introduce into the test a demonstration of this particular case. 



1 [Vid. supra, p.IiSs.] 



