Mr. Ivory on the Equilibrium of an Incompressible Fluid. 325 



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 all depths." It thus appears that the conditions 

 laid down above as necessary and sufficient for an equilibrium, agree 

 exactly with the demonstration of Maclaurin, 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 eflfort of the 

 fluid contained in it will be invariably the same, and equal to the 

 difl^erence 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 every fluid at rest. If a, b, c, and a, V , c , de- 

 note the co-ordinates of the two orifices of a canal ; and («, b, c) 

 and (f) (a, b', c) represent the pressures at the same points ; the 

 function <p (a, b, c) must have such a form as will be changed into 

 <p (a', b', d), 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- 

 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 are diff'erent 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 



* Fluxions, § 640. 



