44 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



therefore, is constant. But this ratio, taken with the negative sign, is, by equation 

 (c), section 34, equal to the specific volume of the fluid in this sheet. We there- 

 fore conclude that the specific volume is constant all along the sheet. This con- 

 dition being fulfilled for every infinitesimal sheet, it follows that the surfaces of 

 equal specific volume must have the same course as those of equal pressure and 

 of equal potential. Hence: 



In the state of equilibrium there is coincidence between the isobaric, the iso- 

 sleric, and the equipotential surfaces. 



This remarkable coincidence of the equiscalar surfaces of three different fields 

 is a necessary but not sufficient condition for equilibrium. 



(II) Principle of the Unit- Sheets. From equation (r), section 34, we further 

 conclude that in every direction the variation of potential is a times more rapid than 

 that of pressure. In reference to infinitesimal unit-sheets this means that every 

 isobaric unit-sheet contains a equipotential unit-sheets. For practical reasons it 

 will be important to have this principle formulated not only for infinitely thin 

 sheets, but also for sheets of finite thickness. Integrating, therefore, equation (c), 

 section 34, and denoting by a m the mean value of the specific volume in the interval 

 between the pressures p x and p 2 , we g et this relation between finite differences of 

 potential and the corresponding finite differences of pressure: 



( a ) 4> 2 - 4>, = - ,(A -A) 



Applying this to an isobaric unit-sheet, we get p 2 pi = T > an d thus 



(*) fc - +1 - - . 



Here <p 2 $, is the number of equipotential unit-sheets contained within the con- 

 sidered isobaric unit-sheet. Disregarding the sign, we thus get this numerical law: 



In the state of equilibrium the number re-presenting the mean specific volume 

 of the fluid in an isobaric unit-sheet also represents the number of equipoten- 

 tial unit-sheets contained in the isobaric unit-sheet. 



These two principles, taken in connection with the rule of signs that increasing 

 potential gives decreasing pressure and vice versa, give the full equilibrium relation 

 between the three fields that of mass, that of pressure, and that of potential. 



36. Determination of Heights or Depths of Given Pressures. In the m.t.s. 

 system of units the thickness of an equipotential unit-sheet is 1 dynamic decimeter. 

 The number of equipotential unit-sheets contained in an isobaric unit-sheet is there- 

 fore the number of dynamic decimeters giving the thickness of the sheet. The 

 principle of the unit-sheets, therefore, enables us to find the thickness of any isobaric 

 sheet, adding the thicknesses of the successive unit-sheets and to determine thus 

 the height or depth where the pressure has any given value. This is the dynamic 

 principle of the barometric measurements of heights or of manometric measure- 

 ments of depths. 



Performing this operation practically, it will generally be convenient to pass 

 from the m.t.s. units, dynamic decimeter and centibar, to the greater units, dynamic 



