NEWTON S PRTNOTPIA. 317 



an equation which we shall have frequent occasion to use 

 in illustrating Newton s propositions. 



Let us now consider some of the consequences of this 

 proposition. 



Consequence L In order that there may be equilibrium, 

 the forces and density of the fluid must be such that 



p Xdx + pYdy + pZdz 

 is a perfect differential, that is, we must have 



dpX. d pY dp^K. _ dpT* /Wftt 



~d^~ ~J^&amp;gt; ~d7 dx 



If we eliminate p from these equations, we get 



X + Y t Z- = 0. 



\dz dyJ \dx dx) \dy dx ) 



Hence unless this equation be satisfied there is no fluid 

 which will be in equilibrium under the action of the forces. 

 But if the equation be satisfied, the law of density must 

 still be such as to satisfy one of the equations (5) (6). 

 The only forces we meet with in nature are those which 

 tend to fixed centres, and vary as some function of the 

 distance from those centres. In all such cases the quantity 



Xd*+ Ydy + Zdz 



is a complete differential. For calling r the distance of 

 any point from one of the centres of force, and &amp;lt;p(r) one of 

 the laws of attraction, we have 



where a is the abscissa of the centre of force, and similar 

 expressions hold for Y and Z. Hence 



