318 NEWTON S PRINCIPTA. 



since x | 2 -\-y b 2 -fz c 2 =r 2 , and hence the above 

 quantity is a perfect differential. The criterion is there 

 fore satisfied, and it is therefore always possible to find 

 a fluid that can be kept in equilibrium by the forces in 

 nature. 



Consequence II. Def. A surface along which the pres 

 sure is the same is called a level surface. It is also some 

 times called a surface of equal pressure. A surface along 

 which the density is the same is called a surface of equal 

 density. Let us call the perfect differential 



then we have 



and this cannot exist, by a known theorem of the Dif 

 ferential Calculus, unless the three quantities p, p, P are 

 functions, each of each. Hence the density is always a 

 function of the pressure, and therefore 



All level surfaces are also surfaces of equal density. 



The differential equation to all level surfaces is 



or, which is the same thing, 



P = constant. 



Def. The bounding surface of any liquid exposed to 

 any constant pressure, as for instance that exposed to the 

 atmosphere, is called a free surface. All free surfaces are 

 clearly level surfaces. 



The resultant of the forces at any point is perpendicular 

 to the surface of the level surface passing through that 

 point. 



Take the point as origin, and the tangent plane to the 

 surface as the plane of xy, so that dz = 0, and the axis of 

 z is normal. Along the surface we have 



