FLUID PARTICLES. 79 



librium of A will not be disturbed by the attraction of the 

 stratum $A, if the resultant of that attraction on every particle 

 in the surface of A be directed perpendicularly to it." 



This reasoning satisfactorily shows, that if a fluid mass of 

 attractive matter be increased by a stratum producing equal 

 pressure over the free surface, the equilibrium will be destroyed 

 unless a certain condition is fulfilled, of which the symbolical 

 expression is 



P, Q, R being the attractions, parallel to the axes of co-ordi- 

 nates, of an element of that part of a fluid mass which is external 

 to a given level surface. But the necessity of this condition 

 cannot be proved, unless it is shown to be impossible in any 

 way to increase the mass A, without destroying the equilibrium, 

 supposing it not fulfilled. All that has been shown is, that 

 the mass cannot be increased by a stratum producing equable 

 pressure over the free surface. Now, generally speaking, the 

 mass so increased will not fulfil the condition of having the 

 forces at the new free surface normal to it, those acting at the 

 original free surface being of course so. We cannot, therefore, 

 affirm that we have fallen on a case in which the ordinary con- 

 dition is fulfilled, without producing equilibrium. If, however, 

 we dispense with the limitation, that the stratum added shall 

 produce equable pressure, we lose the simplicity of Clairaufs 

 method, nor can we make any use of his principle, except by 

 setting aside the construction he employs, which confines him 

 to the particular case in which a surface of equal pressure is 

 potentially a free surface. 



This has already been done, and the result is the general 

 equation of equilibrium. It remains to show, that it is in all 

 cases sufficient. It is admitted to be sufficient in the case of a 

 fluid acted on by forces tending to fixed centres. We shall 

 endeavour to reduce the general case to this. Conceive a body 

 acted on by a force directed to a fixed point. It may be so 

 placed, as to remain at rest under the action of the force, that is, 

 the resultant of the force upon it is equal to zero. In this 

 position of the body, the centre of force is some point within it. 

 Let the body, remaining in the same position, diminish sine 

 limite, being always similar to itself, the resultant of the force 

 upon it is always equal to zero ; and ultimately, when the body 



