in reply to M. Poisson. 163 



appears to be no absurdity in supposing that the latter con- 

 dition may be fulfilled, at the same time that no level surfaces 

 exist in the interior, and consequently when there is no equi- 

 librium of the mass of fluid. 



It has been shown that there is a plain distinction between 

 a fluid in equilibrio when the accelerating forces acting upon 

 the particles are explicitly given, and when the same forces 

 depend upon the figure of the mass. Yet, according to the 

 received theory there is no difference in the two cases with 

 respect to the conditions necessary to insure the equilibrium. 

 Supposing that the algebraic expressions of the forces possess 

 the criterion of integrability, nothing further is required in 

 either case, than that the outer surface of the fluid be a level 

 surface cutting the direction of gravity everywhere at right 

 angles. But if the determination of the equilibrium be the 

 same in both cases, we might expect that the demonstration 

 would be different in circumstances so essentially distinguished. 

 It must be confessed, however, that we meet with nothing in 

 the shape of demonstration, except what is vague and unsatis- 

 factory and applies alike to both cases. The principle seems 

 to be this : granting that the exterior surface is a level sur- 

 face, it is always possible to trace the level surfaces in the 

 interior; because, since there is no distinction of density, we 

 may adopt any surfaces we please as level surfaces*. When 

 the accelerating forces are explicitly given, there is no diffi- 

 culty nor ambiguity ; because the equation of all the level sur- 

 faces are nowise different except in the constants introduced 

 by integration, which vary from one surface to another. But 

 the case is not the same when the level surfaces depend upon 

 the figure of the fluid. Taking a point in the interior of the 

 mass, we may indeed conceive a surface to pass through it 

 and to be extended on all sides, so as everywhere to cut the re- 

 sultant of the accelerating forces at right angles, and we may 

 call this a level surface: but we may thus fall into error; be- 

 cause it is not clear that the surface so traced will return into 

 itself and completely inclose a portion of the fluid; and with- 

 out this it would neither be a level surface, nor would there be 

 an e(|uilibrium of the fluid. In order to place the theory on a 

 solid foundation, such gratuitous assumptions must be set 

 aside; and it must be proved, by means of the ec]uations which 

 accurately determine the level surfaces, that they do neces- 

 sarily exist in the interior of the fluid, and produce the equi- 

 librium in question. But this has not been done; and, to say 



* Mecan. Ciksh; Liv. 1", No. 17 ; & Liv. T% No. 22. 



Y 2 the 



