HOMOGENEOUS FLUID AT LIBERTY. 517 



In the second problem, the forces acting upon a particle within the surface are the 

 same functions of the coordinates as the like forces acting upon a particle in the sur- 

 face ; because the forces which urge a particle in any situation depend only on the 

 whole mass of fluid, and the distance of the particle from the centre of gravity. But 

 in the first problem, if we except the particular class of figures susceptible of an equi- 

 librium, the finding of which is an additional condition to be investigated, the forces 

 urging a particle within the surface are not deducible from the forces at the surface 

 merely by changing the coordinates of the point of action. 



9. To complete the theory in this paper, it would be necessary to determine the 

 figure of equilibrium of a revolving mass of homogeneous fluid, on the supposition 

 that the particles attract one another with a force varying as any power of the distance. 

 The solving of this problem would enable us to decide whether the equilibrium be 

 possible in any other law of attraction but the direct proportion of the distance, or 

 the inverse proportion of the square of the distance. The principles that have been 

 laid down are suflicient to solve the problem enunciated in this general manner ; but 

 the application of them would require mathematical discussions too extensive to be 

 entered upon at present. To conclude this paper, some observations will be made 

 that seem to be called for by the notions that prevail on the subject of which it 

 treats. 



On Maclaurin's Demonstration of the EquiUhriuin of the oblate elliptical 



Spheroid. 



In treating of the figure of the earth, Newton begins with observing that a homo- 

 geneous mass of fluid, supposing its particles urged only by their mutual attraction, 

 would arrange itself in a form perfectly spherical. If this sphere acquire a revolving 

 motion about one of its diameters, a new force will be impressed on its particles, 

 causing them to recede from the axis of rotation ; and, in obedience to this force, the 

 fluid will subside at the poles and dilate itself in the direction parallel to the equator. 

 Newton assumes, without alleging any reason in support of his assumption, that the 

 revolving fluid will permanently settle in an oblate elliptical spheroid. Admitting 

 tacitly that this is the figure of equilibrium, he proves that the relative dimensions 

 of the spheroid depend upon the proportion of the centrifugal force to gravity at the 

 equator ; and this proportion being ascertained by experiment in the case of the 

 earth, he finds that the equatorial diameter is to the polar axis as 230 to 229. The 

 whole of this speculation, when published in the Principia, was entirely new ; it in- 

 volves many points of difficult investigation ; and the ability has always been ad- 

 mired by which the difficulties are either overcome or evaded by ingenious approxi- 

 mations sufficiently exact and requiring the least possible calculation. But this 

 splendid theory was incomplete till it should be proved that a fluid sphere turning 

 upon an axis must assume the form of an elliptical spheroid. The attention of 

 geometers was therefore turned to this point. The subject was treated by Mr. James 



MDCCCXXXIV. 3 x 



