Page:Proceedings of the Royal Society of London Vol 4.djvu/184

Rh 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 effect 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. Alaclaurin de- monstrates that the pressure impelling a particle in any direction is equivalent to the effort 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 aU 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 aU the particles in contact with it on the same surface. Alaclaurin seems ultimately to have taken the same view of the matter, when he says that* " the sur- faces similar and concentric to the surface of the spheroid, are the level surfaces at aU depths," It thus appears that the conditions laid down above as necessary and sufficient for an equilibrium, agree exactly with the demonstration of iMaclaurin, when the true import of what is proved by that geometer is correctly understood.

The general conditions for the equilibrium of a fluid at liberty being explained, the attention is next directed to another property, which is important, as it furnishes an equation that must be verified by every level surface. If we take any two points in a fluid at rest, and open a communication between them by a narrow canal, it is obvious that, whatever be the figure of the canal, the effort of the fluid contained in it will be invariably the same, and equal to the difference of the pressures at the two orifices. As the pressure in a fluid in equilibrium by the action of accelerating forces, varies from one point to another, it can be represented mathematically only by a function of three co-ordinates that determine the position of a point : but this function must be such as is consistent with the pro- perty that obtains in ever}- fluid at rest. If a, h, c, and a, b', c, de- note the co-ordinates of the t«'o orifices of a canal ; and (a, 1, c) and {d, h' , c) represent the pressures at the same points ; the function 9 (a, I, c) must have such a form as will be changed into (p (a, b', c'), through whatever variations the figure of a canal re- quires that a, b, c must pass to be finally equal to a, b', c. From this it is easy to prove that the co-ordinates in the expression of the pressure must be unrelated and independent quantities. The forces in action are deducible from the pressure ; for the forces produce the variations of the pressure. As the function that stands for the press-


 * Fluxions, § 640.