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MR. IVORY ON THE EQUILIBRIUM OF A MASS OF 



coordinates of the point of action *. Now there are cases, and the homogeneous 

 planet is one, in which the forces acting on the interior particles are not deducible, in 

 the manner supposed, from the forces at the surface ; and with respect to such pro- 

 blems, the theory is silent, and has provided no means of solution. 



But it will be satisfactory, and it is not difficult, to acquire just notions respecting 

 Clairaut's theory, by a careful examination of the principles as they are laid down 

 by the author, for whose great abilities and high pretensions as a discoverer in science 

 we entertain the sincerest respect, although we dissent from him on some points. The 

 French geometer assumes for the foundation of his superstructure a mass of fluid, 

 H K I, in equilibrium -{-. If /' represent the force perpendicular to the surface of 

 H K I, at any point K, and k the thickness K* of an additional stratum on I; and if 

 the stratum be so determined that k xy shall have constantly 

 the same value at all the points of the surface ; it will follow 

 that the pressure of the stratum upon the surface on which it / 

 lies, is constant ; and hence the body composed of the stratum 

 and the original mass will be in equilibrium. In like manner, 

 if a second stratum be added to the new body in equilibrium, 

 the thickness being determined by the same condition as 

 before, a third body of fluid in equilibrium will be obtained, 



consisting of two strata and the central mass. By adding more strata indefinitely, 

 the dimensions of the mass of fluid may be enlarged to any extent, at the same time 

 that the conditions of equilibrium are continually preserved. In all this it is evidently 

 supposed that no change in the figure of the successive surfaces is effected by the 

 strata laid upon them ; for without this admission the procedure would be nugatory, 

 and could lead to no determinate conclusion. 



The investigation of Clairaut is very elegant and geometrical, and carries with it 

 the clearest evidence. It is entirely consonant to the theorem in No. 5. When it is 

 not extended beyond its proper assumptions, it leads to a sure, and in truth to the 

 only satisfactory principle of the equilibrium of amass of fluid at liberty. It assumes 

 that the pressure of every new stratum upon the surface on which it is laid, is caused 

 solely by the foi-ces in action at that surface, these forces being supposed to exert 

 the same energy on all the particles of the infinitely small thickness of the stratum, 

 and the thickness being so determined as to make the pressure constant. The pro- 

 cedure is agreeable to the usual rules of mathematical investigation, according to 

 which the forces are conceived, not to flow continuously as the coordinates increase, 

 but to vary from surface to surface by infinitely small gradations. Now this is very 



* When the forces acting upon the interior particles assume singular forms of expression at the surface, 

 Clairaut's theory fails ; aad this makes the distinction in the text necessary. But the whole theory, founded 

 on an assumed principle, or upon an algebraic equation which determines the eiFect of the forces upon a par- 

 ticle taken individually, is so loosely delivered that it is difficult to speak of it with precision. 



t Theorie de la Figure de la Terre, Premiere Partie, § xxi. 



