160 



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. 



