HOMOGENEOUS FLUID AT LIBERTY. 519 



of the attracting- mass, and are brought to depend entirely upon the situation of the 

 particle with respect to the equator and the axis of rotation. This second investiga- 

 tion, therefore, concurs with the first, in proving that the Newtonian property is 

 necessary to the equilibrium of the spheroid, and not merely accidental. 



Maclaurin's demonstration is different in some respects from either of the two 

 investigations that have been mentioned. He requires three separate conditions for 

 the equilibrium : first, the resultant of the centrifugal force and the attraction of the 

 mass, must be perpendicular to the surface of the spheroid ; secondly, every particle 

 must be pressed equally in all directions ; thirdly, all the columns reaching from the 

 centre to the upper surface must balance and sustain one another. Now if the first 

 of these conditions be fulfilled, and that too whether the mass of fluid be an elliptical 

 spheroid or have any other figure, the other two will follow as necessary consequences. 

 It maybe observed further, that a demonstration proceeding on an arbitrary enumera- 

 tion of properties, which may not be complete, makes a vague impression, and falls 

 short of the conviction produced by a proof that rests on determinate principles bear- 

 ing directly upon the point to be investigated. The conditions essential to Maclau- 

 rin's demonstration are only these two : first, the attraction upon a particle propor- 

 tional to its distances from the equator and the axis of rotation, which is peculiar to 

 ellipsoids, and necessarily connected with the Newtonian property ; secondly, the per- 

 pendicularity to the upper surface of the resultant of the forces acting upon a particle 

 contained in that surface : and notwithstanding the beautiful train of reasoning em- 

 ployed by the author, his demonstration would gain in precision and clearness by 

 omitting all that relates to the superfluous properties. 



Clairaut's Theory. 



To Clairaut belongs an important part of the theory of the figure of the earth. 

 He was the first that entertained correct notions respecting the eff'ect to alter the form 

 of the terraqueous globe, produced by heterogeneity in its structure. At present we 

 confine our attention to his general equations of the equilibrium of fluids, and their 

 application to the case of a homogeneous planet. His theory is constructed with 

 great analytical skill, and is seducing by its conciseness and neatness. From the 

 single expression of the hydrostatic pressure are derived the equations of all the level 

 surfaces, and of the upper surface of the fluid. But these equations are not sufficient 

 in all cases to solve the problem. They are sufficient to solve it when the forces are 

 known algebraic expressions of the coordinates of the point of action : they are not 

 sufficient when the forces are not explicitly given, but depend, as in a homogeneous 

 planet, on the assumed figure of the fluid. In this latter case, the solution of the pro- 

 blem requires, further, that the equations be brought to a determinate form by elimi- 

 nating all that varies with the unknown figure of the fluid. 



In the theory of Clairaut it is tacitly assumed that the forces urging the interior 

 particles are derived from the forces at the upper surface merely by changing the 



3x2 



