271 reply to M. Poisson. 165 



equilibrium in the other : for the forces which urge the par- 

 ticles of one body are different in no respect ft'om the forces 

 which urge the particles of the other, except in being all in- 

 creased, or all diminished, in the same given proportion*. 



Prop. 3. — If a homogeneous mass of fluid revolve about an 

 axis which passes through the centre of gravity, and be in 

 equilibrio by the attraction of its particles in the inverse pro- 

 portion of the square of the distance ; any surface in the in- 

 terior, similar to the outer surface and similai'ly posited about 

 the centre of gravity, will be a level surface. 



For, by the hypothesis, the whole mass of fluid and the 

 portion of it bounded by the interior surface are similar in 

 their figures ; and they both revolve in the same time about 

 the common axis which cuts them similarly : wherefore the 

 first of these two bodies being in equilibrio, it follows from the 

 last proposition, that the other would likewise be in equilibrio 

 if it revolved by itself, the exterior matter being taken away 

 or annihilated. Thus the interior fluid body is in equilibrio 

 in two different states : first, when it revolves by itself, in which 

 case the only forces in action are, the attraction of its particles 

 and their centrifugal force ; and secondly, when it is a part of 

 the whole fluid mass, in which case there is superadded to 

 the former forces, the action of all the exterior matter. Now 

 these two states of equilibrium cannot consist with one an- 

 other, if we suppose that the exterior matter has any other 

 effect than to produce an equable pressure upon the surface 

 of the interior body ; that is, unless the same surface be a level 

 surface. 



* In the Annates de P/it/sique et de CJiimie, torn, xxvii. p. 234, M. Pois- 

 son makes nse of a very curious argument with respect to this proposition : 

 — "Proposition dont I'inverse ne serait pas vraie; car on salt que pour 

 une memo densite et une meine vitesse de rotation, qui ne depasse pas une 

 certaine limite, il y a deux ellipsoides dissemblables qui satisfbnt a I'equi- 

 libre d'unc masse fluide." Now all this is quite beside the purpose. There 

 is no question about inverse propositions. The theorem as I have laid it 

 down is undoubtedly true; and it may therefore be legitimately used in 

 deducing from the existence of the c(juilibrium the essential properties 

 that belong to it, and without which it cannot subsist. M. Poisson, not 

 adverting that my reasoning is as much an analysis as if it had been ex- 

 pressed by algebraic equations, goes on to say, that the new condition is 

 not necessary to the equilibrium. Now I have here proved that the e(]ui- 

 librium caruiot take place without it, and that the usual condition is by 

 itbclf insufficient. 



The argument in p. 23,j is equally inapplicable. The case mentioned is 

 one in which the accelerating forces urging the particles is independent of 

 the figure of the fluid : it is one in which the forces are explicitly given, 

 and is therefore attended with no difficulty. 



By means of such loose argumentation any point may be proved or dis- 

 proved at pleasure. 



The 



