508 MR. IVORY OF THE EQUILIBRIUM OF A MASS OF 



from which it follows, that every interior level surface is similar to the upper surface, 

 and similarly posited about the centre of gravity. 



The expression of the integral in equation (3.) is evidently true in all similar sphe- 

 roids, without any change in the function F ; for F, being of no dimensions, contains 

 only the proportions of the linear dimensions of the geometrical figures, and these 

 proportions are the same when the figures are similar. And, since all the level sur- 

 faces are similar to the upper surface, it is obvious that the equation of a level sur- 

 face may be thus expressed, 



because the integral between the brackets, which stands for the sum of all the mole- 

 cules within the level surface divided by their respective distances from a point {xy z) 

 in that surface, is equal to the part of the equation of the level surface which contains 

 the function F. Now the equilibrium of the mass of fluid will be impossible, unless 

 the pressure determined by the equation (1.) be constant at all the points of the same 

 level surface ; which requires that the equation 



'^iff + ^iy' + -')} = '^■{Uf\+'^'->' + ^'^} 



be verified, the coordinates of the attracted point varying in any level surface*. This 

 differential equation will be fulfilled if the equation 



rdm r rdm~\ 

 constant = / -j I / — f-J 



/ L7 / 



hold at all the points of every level surface. And as the integral without brackets 

 is the sum of all the molecules of the whole mass of fluid, divided by their respective 

 distances from the attracted point in the level surface ; and the integral with brackets 

 is the like sum relatively to all the molecules within the level surface ; the last equa- 

 tion may be expressed more simply thus, 



constant = / —?^, 



the integral being extended to all the molecules of the stratum between the level 

 surface and the upper surface of the fluid. In the figures which verify this equation 

 there will exist in the interior parts no surfaces of constant pressure except the level 

 surfaces, which is a necessary condition of equilibrium ; and the intensity of pressure 

 in every level surface will be determined by the equation (1.), as required in the 

 problem. 



We have next to investigate the figures which verify this last equation. Let s re- 

 present the distance oi dm from the centre of gravity; and, r being drawn to an 



* 'ITiat is, of every surface determined by varying the constant in the equation of the upper surface, accord- 

 ing to the definition in No. 6, 



