56 Mr Green on the Vibration of Pendulums 
co-ordinates x, y, 2 of any particle of the fluid mass, and of the time ¢ that 
the velocities of this particle in the directions of and tending to increase the 
co-ordinates 2, y and 2 shall always be represented by 5 ae ats and ae re- 
spectively, Moreover, 9 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 
supposing that the origin of the co-ordinates is at the centre of the ellip- 
soid: A and p being two arbitrary quantities constant with regard to the 
variables w, y, =; and a, b, c, f being functions of these same variables, de- 
termined by the equations 
ae—a°+f, =U? +f, C=? +f, and eae sees Sate eaers (4.) 
in which a’, b’, c’ 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 Epwarp Frrencu Bromueap, Baronet, I have given a method by which 
the general integral of the partial differential equation 
GV, a2Ni av Lies n—s dV 
= ae Pees ai az ban oa ae 
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. 
