80 EQUILIBRIUM OF MUTUALLY ATTRACTIVE, &c. 



becomes a physical point, it coincides in position with the centre 

 of force, and is in the same state with respect to the action of 

 other forces upon it, as if this force did not exist. 



This being granted, conceive a homogeneous mass of fluid 

 composed of mutually attractive particles, the free surface of 

 which fulfils the required equation 



Let the attractive power of each particle be conceived trans- 

 ferred to a fixed centre of force coinciding with it. Then the 

 action of all the other particles on one particle is precisely re- 

 placed by that of the fixed centres ; and it has been shown, that 

 the resultant of the action of the centre coinciding with a par- 

 ticle on that particle, equals zero. Hence, the supposition we 

 have made does not change, in any way, the forces acting on 

 any particle of the mass. Were the system in its present and 

 former state respectively to move, the motions would be widely 

 different; but in the arbitrary position we have placed it in, 

 the action on it is precisely the same in the two cases. Now, 

 the single equation given above assures its equilibrium, when 

 we regard it as a system acted on by forces directed to fixed 

 centres; and as the hypothesis by which we are enabled to 

 look upon it in this way nowise affects the forces acting on it, 

 it follows, that the system considered as acted on by mutual 

 attraction must be in equilibrium. Consequently, a mass of 

 homogeneous fluid, the particles of which are mutually attrac- 

 tive, will always be in equilibrium when the free surface fulfils 

 the single condition implied in the general equation obtained 

 above. The same reasoning applies to the case of any fluid, 

 elastic or incompressible. 



If this demonstration be thought satisfactory, the question 

 raised by Mr Ivory, as to the sufficiency of the general equation, 

 must be looked upon as settled. The suggestions here made 

 with respect to the new condition tacitly introduced in Clairaut's 

 reasoning, will, it is thought, enable us to trace the source of 

 the difference of the view taken by Mr Ivory, and that gene- 

 rally entertained. In one form or other, it seems to recur in 

 eveiy way in which that distinguished mathematician has treated 

 the subject. 



