512 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



this supposition, exactly compensated by the decrease. But this requires that U^ 

 be exterminated, because this function varies its sign when & and ct' are changed into 



§ -\- ^ and w^ + T. Wlierefore, leaving out U^ , if we assume 



—F- after the first, both 



those in which i is even and those in which it is odd, will vanish, so that we shall 

 have 



which is constantly of the same value at all the points of the same level surface. 

 Taking the most general expression of U^^ , and observing that the constant 



may be blended with U^^ , we shall have 



^3 = A a'2 -I- B i'2 + C c'^ + D a' ^>' + E a' c' + F ^' c' : 

 but a/, y, z' being the coordinates of R' in the surface of the fluid, we have 



R' — ^ ' R' — ^ ' R' ~ ^ • 



and these values being substituted, the result will be 



1 = A ^'2 + B 3/'2 + C ^'2 4_ D x' y< J^Ex %' + F?/' t!, 



which is the equation of an ellipsoid, the coordinates <r', y, z' being parallel to three 

 diameters intersecting at right angles. It is therefore demonstrated, that the ellipsoid 



/d m 

 -J-, taken between the 



assigned limits, of the same value at all the points of the same level surface, that is, 

 at all the points of any interior surface similar to the upper surface, and similarly 

 posited about the centre. 



The foregoing reasoning is independent of the centrifugal force ; but by attending 

 to the rotatory motion which causes that force, it is easy to prove that the axis about 

 which the fluid revolves, or the diameter parallel to the coordinate x\ must coincide 

 with one of the axes of the geometrical figure. For, there being no centrifugal force 

 at the poles of the axis of rotation in the surface of the fluid, the only force in action 

 at these points is the attraction of the mass. But the resultant of the forces urging 

 every particle in the surface of a fluid in equilibrium must be perpendicular to the 

 surface : and as there are no points on the surface of an ellipsoid at which the attrac- 

 tion of the mass is perpendicular to the surface, except the extremities of the three 

 axes, it follows that, with one or other of these, the axis of rotation of the fluid in 



