316 NEWTON S PKINCIPIA. 



But the moving force 



p X dx dy dz 



tends to do the same, and since the element is in equili 

 brium, the sum of these two must be zero. 



,.g = ,X - - - (1). 



By Law II. the pressure referred to a unit of area on all 

 the faces meeting at the corner (xyz) are equal, and as 

 similar reasoning applies to all the sides of the element, we 

 shall have 



These three equations are necessary and sufficient for 

 equilibrium. 



These equations may be put under a form which is very 

 useful, and may sometimes be used independently of any 

 axes of coordinates. For draw any curve in the fluid and 

 consider an element ds of its arc. Since no assumption 

 has been made as to the axis of #, take it as tangent to the 

 arc, and if S be the resolved part of the force along the 

 tangent, the equation (1) shows that 



\ dp _ 



-= io. 

 p d s 



Reverting to the old position of the axes, the resolved parts 

 of X Y Z along the tangent is 



hence 



dx ^ r dy dz 



Jv -, + JL - t + A -j ; 

 as d s d s 



1 dp = Jidx -f Ydy -f r Ldz 9 



