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Article XXVII. 

 A singular case of the Equilibrium of Incompressihle Fluids; 



by M. OSTROGRADSKY. 



(Read to the Academy of St. Petersburgli, February 19, 1836.) 



From " Mimoires de I'Academie Imperiale des Sciences de St. Petersbourg," 

 vol. iii. part 3. 



XN mechanics there is no other distinction made between different 

 bodies or different systems of bodies, besides that which relates to their 

 masses, their positions, and their possible displacements. These displace- 

 ments, together with the mass or quantity of the inertia of each element 

 being given, we have all that is requisite as well as indispensable to en- 

 able us to treat of the equilibrium and movement of any system. 



That a system subjected to the action of any forces may remain in 

 equilibrio, it is necessary that the forces should be incapable of pro- 

 ducing any of those displacements of which the system is susceptible. 

 Now, as the forces, though capable of producing all the displacements 

 of which the Avhole momentum is positive, are yet unable to produce 

 any of those which correspond to the zero or the negative momentum, 

 the equilibrium of the system consequently requires that the whole mo- 

 mentum should be zero or negative for all the possible displacements. 

 From this leading principle we may in the easiest and simplest manner 

 derive the condition of equilibrium of a system without knowing any- 

 thing more than the masses and the possible displacements. More par- 

 ticularly Avith respect to the equilibrium of liquids, we have, for instance, 

 no need of the experimental principle known by the name of the prin- 

 ciple of equal pressure, which, before the publication of the Mecanique 

 Analytique, mathematicians were accustomed to consider as the basis of 

 the theory of the equilibrium of fluids. It is sufficient to know how a 

 liquid mass can be displaced, and this is the only datum by means of 

 which, in the Mecanique Atudytique, the equations relative to the equi- 

 librium of liquids are deduced. But Lagrange having neglected the 

 consideration of the displacements, accompanied by an augmentation 

 of volume, though such displacements are evidently possible, was un- 

 able to deduce from his analysis the essential condition, that the quan- 

 tity which represents the pressure must necessarily be positive. This 

 condition being added, the theory of Lagrange will be the most satis- 

 factory of all those in which the liquids are considered as continu- 

 ous masses ; and if there is anything further to be remarked, it is that 

 the incompressibility of the differential parallelepipeds is there taken as 

 the condition of the incompressibility of the fluid, though it should be 



