SH Mr. Ivory's Remarks on the Theory 



by equalizing the weight of all the columns of fluid drawn 

 from the centre to the surface. There is no doubt that both 

 these conditions are indispensable to the equilibrium of a 

 fluid mass. But Bouguer remarked that they were not al- 

 ways reconcileable in the same figure; and hence he con- 

 cluded that those figures only were in equilibrio in which both 

 the conditions were fulfilled. Clairaut afterwards showed 

 that the equilibrium of a fluid was not always ensured even in 

 those cases when both the principles of Huyghens and New- 

 ton led to the same figure. Maclaurin adopted the more 

 general and undoubted principle, that every particle is in 

 equilibrio when it is pressed equally in all directions. But 

 we are indebted to Clairaut for the discovery of the general 

 equations of the equilibrium of a fluid mass, whether homo- 

 geneous, or composed of parts of different densities. Finally, 

 Euler brought this theory to the more simple form in which 

 it is now taught, by deducing the equations of Clairaut from 

 the principle of an equal pressure in all directions. 



The conditions required by the hydrostatical theory for 

 the equilibrium of a fluid mass are these: 1°. All the par- 

 ticles of the same density must be arranged in distinct strata. 

 2°. The resultant of ail the forces acting upon a particle must 

 be perpendicular to the level stratum, or couche de niveau, in 

 which the particle is placed. These conditions will be ful- 

 filled if all the level strata be defined by the same equation, 

 the arbitrary quantity introduced in the integration alone 

 varying from one stratum to another ; and the same quantity 

 representing always a certain function of the density. 



In the case of a homogeneous fluid, the distinction of the 

 level strata arising from the difference of density is lost; and 

 then the only conditions requisite to the equilibrium are con- 

 tained in this proposition : The resultant of all the forces 

 acting upon a particle in the outer surface must be perpendi- 

 cular to it ; and the differential equation of the same surface 

 must be an exact fluxion. 



Now if, with Newton, we suppose a sphere of a homoge- 

 neous fluid, originally at rest; and inquire what will be the 

 nature of the oblate figures produced by the rotation upon an 

 axis ; it is manifest that we shall only have to fulfill the single 

 condition, that the gravity be every where perpendicular to 

 the meridians. This problem was first solved by Legendre 

 in 1784-, but only upon the supposition of a very small oblate- 

 ness. After the lapse of a century, the conditions of equili- 

 brium assumed ly Newton were thus not only verified, but 

 completely demonstrated ; since it was shown that the equili- 

 brium 



