324 Jloyal Societi/, 



surface ; but it will still be liable to be moved from its place unless 

 the body of fluid within the level surface have no tendency to change 

 its form or position by all the forces that act on its own particles. 



What has been said not only demonstrates the insufficiency of the 

 principle of equality of pressure for determining the figure of equili- 

 brium of a fluid at liberty, but it points out the conditions which are 

 necessary and sufficient for solving the problem in all cases. The 

 pressure must be a maximum at a central point within the mass : it 

 must be zero at the surface of the fluid : and, these two conditions 

 being fulfilled, there will necessarily exist a series of interior level 

 surfaces, the pressure being the same at all the points of every sur- 

 face, and varying gradually from the maximum quantity to zero. 

 Now all the particles in the same level surface have no tendency to 

 move upon that surface, because the pressure is the same in all di- 

 rections : wherefore if we add the condition that every level surface 

 shall have a determinate figure when one of its points is given, it is 

 evident, both that the figure of the mass will be ascertained, and 

 that the immobility of the particles will be established. 



Maclaurin's demonstration of the equilil)rium of the elliptical 

 spheroid will always be admired, and must be instructive from the 

 accuracy and elegance of the investigation. That geometer was the 

 first who discovered the law of the forces in action at every point of 

 the spheroid ; and it only remained to deduce from the knowai forces 

 the properties on which the equilibrium depends. These properties 

 he states as three in number ; and of these, the two which relate to 

 the action of the forces at the surface and the centre of the spheroid, 

 are the same with the principles of Huyghens and Newton, and co- 

 incide with two of the conditions laid down above. The third pro- 

 perty of equilibrium, according to Maclaurin, consists in this, that 

 every particle is impelled equally by all the rectilineal canals stand- 

 ing upon it and extending to the surface of the spheroid. Now it 

 does not follow from this property that a particle is reduced to a 

 state of rest within the spheroid, by the equal pressures upon it of 

 the surrounding fluid : because these pressures may not be the eft'ect 

 of all the forces that urge the mass of the spheroid, but may be 

 caused by the action of a part only of the mass. Maclaurin de- 

 monstrates that the pressure impelling a particle in any direction is 

 equivalent to the eflfort of the fluid in a canal, the length of which 

 is the difference of the polar semi-axes of the surface of the spheroid 

 and a similar and concentric surface drawn through the particle, 

 which evidently implies both that the pressures upon the particle 

 are caused by the action of the fluid between the two surfaces, and 

 likewise that the pressures are invariably the same upon all the par- 

 ticles in any interior surface, similar and concentric to the surface 

 of the spheroid. Such surfaces are therefore the level surfaces of 

 the spheroid ; and every particle of the fluid is at rest, not because 

 it is pressed equally in all directions, but because it is placed on a 

 determinate curve surface, and has no tendency to move on that sur- 

 face on account of the equal pressures of all the particles in contact 

 with it on the same surface Maclaurin seems ultimately to have 



