56 Mr Green on the Vibration of Pendulums 



co-ordinates oc, y, z of any particle of the fluid mass, and of the time t that 

 the velocities of this particle in the directions of and tending to increase the 



co-ordinates oc, y and z shall always be represented by -p> ~^-, and -^- re- 

 spectively. Moreover, g represents the fluid's density, p its pressure, and V 

 a function dependent upon the various exterior forces which act upon the 

 fluid mass. 



When the fluid is supposed to move over a fixed solid ellipsoid, the prin- 

 cipal difficulty will be so to satisfy the equation (2.) that the particles at the 

 surface of this solid may move along this surface, which may always be ef- 

 fected by making 



* = ( x +^fc)^* ; <N 



00 



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

 soid : A and /jl being two arbitrary quantities constant with regard to the 

 variables oc, y, z; and a, b, c, /"being functions of these same variables, de- 

 termined by the equations 



a* = a'*+f, b* = b>*+f, c* = <<*+f, and^ + |!+^ = l; (4.) 



in which a', b', (f are the axes of the given ellipsoid. 



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

 lipsoids of Variable Densities, recently communicated to the Cambridge Philosophical So- 

 ciety by Sir Edward Ffrench Bromhead, Baronet, I have given a method by which 

 the general integral of the partial differential equation 



rf 2 V ^V rf 2 V rf 2 _V n—sdV 



~dtf + dot? + + drf + du? + ~V~ du 



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

 tion 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 text a demonstration of this 

 particular case. 



