159 



admitted in this theory. Now, if the matter be considered ac- 

 curately, it will be found that the only point within a mass of fluid 

 in equilibrium which is at rest by the sole action of the surrounding 

 fluid, is the central point of Newton, or the point of maximum- 

 pressure. The reason is that, on account of the maximum, the 

 pressure of all the canals terminating in the central point, increases 

 continually as the depth increases ; so that, besides the pressures of 

 the canals, there is no other cause tending to move the particle. 

 With respect to any other particle, the pressure caused by the action 

 of the forces in some of the canals standing upon the particle, will 

 necessarily increase at first in descending below the surface, and 

 afterwards decrease ; so that the effective pressure transmitted to 

 the particle, is produced by the action of the forces upon a part only 

 of the fluid contained in such canals. If a level surface be drawn 

 through any particle, it is proved in the paper, that the equal press- 

 ures of the surrounding fluid on the particle, are caused solely by 

 the forces which urge the portion of the fluid on the outside of the 

 level surface, the fluid within the surface contributing nothing to the 

 same eff'ect. Thus a particle in a level surface is immoveable by the 

 direct and transmitted action of the fluid on the outside of the level 

 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 equilibrium 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 known 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- 



