t' 206 ] 

 XXXIX. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



Jan. 13 & 20, A PAPER was read, On the Equilibrium of Fluids 5 

 18M I. * and the figure of a homogeneous Planet in a 

 fluid stale ; by James Ivory, Esq. A.M. F.R.S. 



The author considers the essential property of a fluid, and that on 

 which its definition should be founded, as consisting in the perfect 

 mobility of its particles among one another. If abstraction be made 

 of the force of gravity, or other accelerating force, when a conti- 

 nuous fluid is at rest, and consequently in a state of equilibrium, 

 all its particles are equally pressed in every direction, are equally 

 distant from one another, and are similarly arranged about every in- 

 terior point. No fluid is absolutely incompressible ; but the degree 

 of compressibility may be conceived to be so small as not to affect the 

 results j and it is accordingly disregarded in the investigations which 

 occupy the present paper. 



These investigations are built on the assumption that the hydro- 

 static pressure at every point of the fluid is the same function of 

 the three rectangular co-ordinates of the point drawn to three planes 

 intersecting one another at right angles. The author shows that the 

 algebraical expressions of the accelerating forces producing the pres- 

 sure are not entirely arbitrary ; because they must necessarily be 

 equal to the partial differential co-efficients of a function of three in- 

 dependent variables, and therefore they are likewise the same func- 

 tions of the co-ordinates of their point of action in every part of the 

 mass. This is one of the conditions required for the equilibrium of 

 a mass of homogeneous fluid; and a second necessary condition is, 

 that these functions of the ordinates are capable of being integrated. 

 When these two conditions are fulfilled, the determination of the 

 figure of equilibrium is reduced to a question purely mathematical. 

 For we can form an equation expressive of an equilibrium between 

 the accelerating forces and the variation of pressure, and by integra- 

 ting this equation we may obtain the hydrostatic pressure ; whence 

 maybe deduced the equation of all those points at which there is no 

 pressure, that is, of the outer surface of the fluid. All that is then 

 requisite for securing the permanence of the figure of the fluid, is 

 that the pressures propagated through the mass be either supported, 

 or mutually balance one another. The upper surface, which is at 

 liberty, and where there is no pressure, and all interior surfaces, 

 where the pressure is constant, have the same differential equation ; 

 and from this the author infers that such surfaces are perpendicular to 

 the resultant of the accelerating forces acting upon the particles con- 

 tained in them. These interior surfaces were denominated by Clai- 

 raut level surfaces ; and they are distinguished by the two proper- 

 ties of being equally pressed at all their points, and of cutting the re- 

 sultant of the forces at right angles. 



The author next extends the investigation to heterogeneous fluids, 

 the different parts of which vary in their density, and deduces a si- 

 milar 



