496 MR. IVORY ON THE EQUILIBRIUM OF A MASS OP 



canal presses at the point (.r y z) will be equal to the sum of all the variations of the 

 function j9 in the whole length of the canal, that is, to the difference between the 

 value of jt at the point {x y z) and at the surface of the fluid. Now the value of 

 j» at the surface of the fluid is equal to zero ; wherefore, the intensity with which all 

 the fluid in the canal presses at the point {x y %) is equal to the value of p at that 

 point. 



It follows from what has been proved, that every narrow canal drawn from any 

 point {x y z), and terminating in the surface of the fluid, will press at that point with 

 equal intensity. Hence, if an infinitely small mass of the fluid, such as a sphere, or 

 a cube, &c., be situated at the point {x y z), it will have no tendency to move by the 

 action of the surrounding fluid ; for it will be equally pressed by all the narrow canals 

 standing upon different portions of its surface, and extending to the surface of the 

 fluid. This property is perfectly general and necessary ; and it may become a ques- 

 tion, whether it be not alone sufficient to secure an equilibrium. Without entering 

 upon the discussion of this question, we here confine our attention strictly to what 

 has been demonstrated, namely, that in a fluid in equilibrium, every infinitely small 

 portion is pressed with equal intensity by all the narrow canals issuing from it, and 

 terminating in the surface of the fluid *. 



4. According to what has been shown, the forces which urge the particles of a fluid 

 in equilibrium, and the consequent pressures, depend upon one function (p' {x,y, z,), 

 varying in its value as the coordinates change their place from the centre of gravity 

 to the surface of the fluid. The same function likewise determines the figure of the 

 mass ; for, the fluid being at liberty, the surface will contain all the points at which 

 there is no pressure. If jo denote the pressure at any interior point {x y %), this equa- 

 tion has been investigated, viz. 



Q — 'p — (p'{x,y,z)i 



and if we make /? = 0, the result, viz. 



C=r.(p^{x,y,z) 



must be verified at all the points of the surface. But it is to be observed, that instances 

 may occur in which the function (p' (x, y, z) in passing from a point within the fluid 

 to a point in the surface, undergoes a modification in the form of its expression. It 

 may happen that the quantities which it contains acquire particular relations at the 

 surface; and on this account the function may put on a singular form, distinguished 



* If the mathematical principle of the property respecting the canals be stated abstractly, it will be found 

 to lie in the nature of the function p, which must be a maximum at the centre of gravity, the point of greatest 

 pressure ; and continually decreasing in receding from that point, it must have the same value at all points of 

 the surface of the fluid. Now it is not impossible but, in some problems, there may be more than one function 

 that will satisfy the two conditions ; and, should this be the case, the figure of the fluid remaining the same, 

 the property respecting the canals would be verified in more than one supposition respecting the pressure and 

 the forces in action. 



