528 MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



by their respective distances from any point in the surface abc, has the same inva- 

 riable quantity. It follows from what is now proved, that the exterior fluid presses 

 with the same intensity at every point of the interior surface ah c. 



The least attention to the internal pressures at the surface ah c, and to the forces 

 by which these pressures are caused, will show that the equilibrium of the mass ABC 

 is secured by these two conditions : first, the resultant of the forces in action at the 

 exterior surface must be directed perpendicularly towards that surface ; and secondly, 

 the level surfaces, that is, the interior surfaces, which are perpendicular to the re- 

 sultant of all the forces acting- upon the particles contained in them, must be similar 

 to the outer surface, and similarly posited about the centre of gravity. These con- 

 ditions of equilibrium, although enunciated in different terms, it will readily appear 

 are not inconsistent with those before laid down, but are equivalent to them, and 

 must necessarily bring- out the same result. 



The same things tliat have just been proved were investigated in the paper on this 

 subject published in the Philosophical Transactions for 1824. There is no inaccuracy 

 in that paper in deducing the conditions which the equilibrium requires to be fulfilled. 

 These are, the perpendicularity to the upper surface of the resultant of the forces in 

 action at that surface, and the immobility of a particle by the attraction of a stratum 

 within which it is placed, and which is bounded by two surfaces similar and similarly 

 posited to the upper surface. What is really exceptionable in that paper consists in 

 the manner in which the second of the true conditions of equilibrium is conceived 

 to be fulfilled. It is supposed in the paper that every individual particle within the 

 stratum is attracted by the matter of the stratum so as to be drawn in all directions 

 with equal intensity, which no doubt fulfills what is required, and is exact in par- 

 ticular figures ; but being deficient in generality, it is an improper foundation on 

 which to place the determination of the figure of equilibrium. To correct this mis- 

 conception, it must be observed that the stratum, by attracting the particles within 

 it, produces pressures in every part of the interior mass ; and the immobility of a 

 particle requires that it be pushed by the surrounding fluid with equal force in all 

 directions. The difference between the two modes of action will be stated with most 

 precision in mathematical language. 



Assume a particle within the stratum, y being its distance from dm, n molecule 

 of the stratum ; the condition that the particle be attracted by the stratum equally 



in all directions, requires that the integraly — ?, extended to all the molecules of the 



stratum, have constantly the same value at all the points within the stratum ; and 

 the condition that the particle be at rest by the equal pressure of the surrounding 



fluid, requires that the same integral^ -^ have a constant value at all the points of 



the lower surface of the stratum. The second determination, which admits the in- 

 tegral, although it must be constant in any one surface, to vary in any manner in 



