NEWTON S PBINCIPIA. 173 



p dy dz and (p -\--r-.dx) dy dz 

 acting along the axis of a:. And the weight will be 



and since there is equilibrium the sum of these must be 

 zero. The equation of fluid equilibrium is then 



dp=-f. -..** --- (1). 



To solve the problem we require the relation between p 

 and p. In fluids generally we have 



p = x P ----- (2). 



This is the law which Newton takes for granted in the 

 two cases which he has worked at length. He also states 

 the results that would be arrived at if we had assumed 

 other laws ; and, as we shall see, Laplace has been led to 

 believe that the above is far from being true within the 

 earth. 



Substituting from the second equation the value ofp in 

 the first, 



*&amp;lt;lp=-p ^ dx. 

 Hence dividing by p and integrating, 



- - - (3), .- 



71 1 * 



where C is some unknown constant. Hence 



u. 1 



p = D s t&quot;^ 7^ 



where D is the density at the centre or at an infinite dis 

 tance, according as n is less or greater than unity, and can 

 only be determined by some of the given conditions of the 

 fluid. 



Generally, we conclude from the above, that when the 



