PRINCIPLES OF HYDROSTATICS. 43 



force of pressure are everywhere directed oppositely to each other, and if their 



amounts per unit-mass are equal, 



() g=-aG 



Other forms for this condition are easily deduced. Remembering that the 

 negative derivatives of <j> and p along the direction s are equal to the components 

 of g and G in this direction, we get 



(*) Ts*% 



Writing equations of this form for each of the three rectangular axes x, y, z, we 

 get the hydrostatic equations in their traditional form, referred to rectangular 

 coordinates. For us, however, the introduction of artificial systems of coordinates, 

 having no relation to the intrinsic geometry of our problems, will only cause 

 complication. It will, on the contrary, be most convenient for us to have the condi- 

 tion of equilibrium referred as closely as possible to the natural coordinate surfaces, 

 the level or equipotential surfaces. This is obtained if we multiply equation () 

 by the line element ds, and use the differential formula? 



dcj) dp 



&*-<** J~ s ds = dp 



Between the total increases d<f> and dp of pressure and of potential along the line 

 element ds, we thus get the relation 



(c) d<f> = adp 



This equation gives in its simplest form the intrinsic relation which, in the case of 

 equilibrium, exists between pressure, specific volume, and gravity potential. 



35. Equilibrium Relation between the Fields of Potential, of Pressure, and 

 of Specific Volume. We have considered, independently of each other, the fields 

 of potential, of pressure, and of mass, and the description of each field by means 

 of its proper equiscalar surfaces and sheets. The condition of equilibrium which 

 we have formulated gives a relation between these three fields which can be 

 expressed as a relation between their surfaces and sheets. Expressed in this wa} r 

 the equilibrium relation will contain two distinct principles, the first of which is 

 purely descriptive, dealing with the course of the surfaces, while the other is of 

 metric nature, giving a numerical relation between the unit-sheets. 



(I) Principle of Coincidence of Surfaces. The gradients of potential and 

 of pressure being oppositely directed, while the first of them is normal to the equi- 

 potential and the second to the isobaric surfaces, we at once conclude that isobaric 

 and equipotential surfaces must coincide. 



From this it follows that every isobaric sheet must coincide with an equipoten- 

 tial sheet. Let the two coinciding sheets be infinitely thin. The passage from 

 the one limiting surface of the sheet to the other, then, gives a certain increase of 

 potential d(f>, and a corresponding increase of pressure dp; all along the sheet d<f> 

 has the same value, and the same will be the case with dp. Their ratio d<f>/dp, 



