161 



ure is restricted, so the expressions of the forces must be functions 

 that fulfil the conditions of integrability, without which limitation 

 an equilibrium of the fluid is impossible. Thus, when the forces 

 are given, the pressure may be found by an integration, which is 

 always possible when an equilibrium is possible : and as the pressure 

 is constant at all the points of the same level surface, an equation is 

 hence obtained that must be verified by every level surface, the 

 upper surface of the mass being included. But although one equa- 

 tion applicable to all the level surfaces may be found in every case 

 in which an equilibrium is possible, yet that equation alone is not 

 sufficient to give a determinate form to these surfaces, except in one 

 very simple supposition respecting the forces in action. When the 

 forces that urge the particles of the fluid, are derived from independ- 

 ent sources, the figure of the level surfaces requires for its determi- 

 nation as many independent equations as there cu^e different forces. 



In the latter part of the paper the principles that have been laid 

 down are illustrated by some problems. In the first problem, which 

 is the simplest case that can be proposed, the forces are supposed to 

 be such functions as are independent of the figure of the fluid, and 

 are completely ascertained when three co-ordinates of a point are 

 given. On these suppositions all the level surfaces are determined, 

 and the problem is solved, by the equation which expresses the 

 equahty of pressure at all the points of the same level surface. 



As a particular example of the first problem, the figure of equili- 

 brium of a homogeneous fluid is determined on the supposition that 

 it revolves about an axis and that its particles attract one another 

 proportionally to their distance. This example is deserving of at- 

 tention on its own account ; but it is chiefly remarkable, because it 

 would seem at first, from the mutual attraction of the particles, that 

 peculiar artifices of investigation were required to solve it. But in 

 the proposed law of attraction, the mutual action of the particles 

 upon one another is reducible to an attractive force tending to the 

 centre of gravity of the mass of fluid, and proportional to the di- 

 stance from that centre : which brings the forces under the condi- 

 tions of the first problem. 



The second problem investigates the equilibrium of a homogene- 

 ous planet in a fluid state, the mass revolving about an axis, and the 

 particles attracting in the inverse proportion of the square of the 

 distance. The equations for the figure of equilibrium are two ; one 

 deduced from the equal pressure at all the points of the same level 

 surface ; and the other expressing that the stratum of matter be- 

 tween a level surface and the upper surface of the mass, attracts every 

 particle in the level surface in a direction perpendicular to that surface. 

 No point can be proved in a more satisfactory manner than that the 

 second equation is contained in the hypothesis of the problem, and 

 that it is an indispensable condition of the equilibrium. Yet, in all 

 the analytical investigations of this problem, the second equation is 

 neglected, or disappears in the processes used for simplifying the 

 calculation and making it more manageable : which is a remarkable 



