HOMOGENEOUS FLUID AT LIBERTY. 525 



forces, and the effect of these to vary the direction of the forces in action at the suc- 

 cessive upper surfaces from exact perpendicularity, will continually become less and 

 less, and may be ultimately neglected. No objection can therefore be made to sub- 

 stituting-, for the equilibrium of the central mass, the supposition that it is infinitely 

 small, in so far at least as it is purposed to construct a body of fluid such that the 

 forces in action at the upper surface shall be perpendicular to that surface. 



If we suppose that the forces urging- the particles of the fluid are expressed by 

 known and explicit functions of the coordinates of their point of action, the body of 

 fluid, as it acquires finite dimensions, will likewise approach continually to a known 

 figure ; for the equation of the surface, deduced from the perpendicularity of the 

 forces, has a determinate form, which ascertains the fig-ure of the mass when its vo- 

 lume is given. In this case, too, all the forces acting upon every individual stratum 

 being taken into account, and the strata exerting a constant pressure upon one an- 

 other, the equilibrium of a mass of fluid will be fulfilled simultaneously with the con- 

 dition of the perpendicularity of the forces to the upper surface. 



It remains to examine what will be the result when the central body H K I, sup- 

 posed infinitely small and of any figure, consists of attracting particles. In this case 

 there is no question about an equilibrium ; because, although the forces at the suc- 

 cessive upper surfaces are exactly estimated, Clairaut has neglected the attraction 

 of every stratum upon the body of fluid to which it is added, an omission which is 

 fatal to an equilibrium of the mass. But as the procedure of that geometer always 

 induces a figure which fulfills the condition of the perpendicularity of the forces to 

 the upper surface, it is interesting to inquire whether, in the case of an attraction 

 between the particles, the resulting figure is determinate and invariable, or indeter- 

 minate and varying with the figure of the small central body. Assume any body of 

 finite dimensions similar to the small central mass H K I, and consisting of the same 

 fluid ; and supposing, for the sake of simplicity, that the law of attraction is that of 

 nature, it is easy to prove, that the attractive forces acting in similar directions at 

 similar points of the surfaces of the two bodies have constantly the same proportions 

 as the linear dimensions of the bodies : and if the two bodies revolve with the same 

 rotatory velocity about axes similarly placed, the centrifugal forces acting in similar 

 directions at similar points of the surfaces, will likewise be to one another as the linear 

 dimensions of the bodies. It appears, therefore, that the forces perpendicular to the 

 surface of the central body H K I, although they are infinitely small, yet being pro- 

 portional to the like forces at the surface of the finite body, they have given and finite 

 proportions. Now upon the proportion of these forces depend the relative thicknes* 

 and figure of the first additional strata at least ; and as no limit can be assigned 

 when this influence will cease, the conclusion undoubtedly is, that the ultimate sur- 

 face will vary with the figure of the central mass. And thus the form induced by the 

 procedure of Clairaut upon a mass of fluid consisting of attracting particles is inde- 

 terminate, and susceptible of being varied indefinitely. 



MDCCCXXXIV. 3 Y 



