42 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



Let us determine the elementary force component dF s , which, as a consequence 

 of the pressure, tends to move a volume element dr of the fluid in the direction s. 

 To consider the simplest case, let the volume element have the form of a straight 

 cylinder with its axis in the direction s and with its bases normal to this direction. 

 As the pressure in a perfect fluid acts normally to the surface, the pressure against 

 the lateral surface can be disregarded, as giving no addition to the component of 

 force along the axis s. We have thus only to consider the pressure against the 

 two bases of the cylinder. Let the value of the pressure at the first base be p. 



At the other it will then be p -\- -~ds, ds being the height of the cylinder. The 



area of each base being dcr, we find that the exterior fluid exerts the force pda 



d-t> 

 against the first and the oppositely directed force {p-\-f- ds) dcr against the sec- 

 ond base. From these two oppositely directed forces will therefore result the force 



dp 

 dF s = - dsda. Now dsda is the volume dr of the element. Further, dp/ds is 



the component G, of the gradient in the direction s (section 33, b). We therefore 

 get 



(a) dF, = G,dT 



Thus the elementary force tending to move a volume element of the fluid in any 

 direction is equal to the component of the gradient in this direction, multiplied by 

 the volume of the element. Or, in other words: The gradient represents the 

 force per unit-volume uue to the field of 'pressure in the fluid. 



By this we see that there is a close relation between potential gradient and 

 pressure gradient. For both gradients represent moving forces. But there is this 

 important difference, that the potential gradient represents the force of gravity per 

 unit-mass, while the pressure gradient represents the force of pressure per unit- 

 volume (section 3). To get the force of pressure per unit-mass we have to multiply 

 the gradient by the specific volume, exactly as we get the force of gravity per 

 unit-volume by multiplying the acceleration of gravity by the density of the body 

 considered. Force of gravity and force of pressure, both referred to unit of mass, 

 are therefore, respectively, 



(b) g and aG 



while the same two forces, both referred to unit-volume, are, respectively, 



(c) pg and G 



The consistent use either of forces per unit-mass or of forces per unit-volume 

 leads to mutually equivalent but formally different methods of formulating the 

 principles and of treating the problems of hydrostatics. We shall develop both of 

 them in parallel, as they are complements of each other from a practical point of 

 view. 



34. Condition of Equilibrium in Terms of Forces per Unit-Mass. The con- 

 dition of internal equilibrium of a fluid is fulfilled if the force of gravity and the 



