FIELDS OF PRESSURE AND MASS IN THE ATMOSPHERE. 121 



measure for the departure from the state of true equilibrium. For this reason it 

 will be useful to develop some simple relations involving the number of unit-tubes. 



Proceeding along an isobaric unit-sheet, we get unit-change of specific volume, 

 and consequently unit-change of thickness of the sheet for every isosteric surface 

 met with. Instead of counting the isosteric surfaces, we may also count the unit- 

 tubes. Introducing the ascendant (section 17) of the specific volume, we see that 

 the projection of this vector on the isobaric surface points in the direction of increas- 

 ing thickness of the sheet. We can therefore count algebraically, reckoning a tube 

 positive when the projection of the ascendant points in the direction in which we 

 proceed, otherwise negative. By this mode of counting we get a measure for the 

 increase of thickness of the unit-sheet. From the unit-sheet we may pass to any 

 sheet composed of any number of unit-sheets; the increase of thickness of the 

 sheet from one vertical to another will be equal to the number of isobaric-isosteric 

 unit-tubes contained between them, counted algebraically in the defined manner. 

 The increase of height comes out in dynamic decimeters if the m. t. s. units be used. 



Counting in the same way the number of equipotential-isopycnic unit-tubes 

 contained in an equipotential sheet, we find the variations in the difference of pres- 

 sure between the upper and the lower limiting surface of the sheet. The rule of 

 signs is formally the same as in the preceding case, the projection of the ascendant 

 of the density pointing in the direction where the difference of pressure increases. 

 Thus, in order to find the increase in the difference of pressure between foot and 

 top of two verticals having their end-points in the same two level surfaces, we have 

 simply to count algebraically the number of equipotential-isopycnic unit-tubes con- 

 tained within the closed curve formed by the two verticals and two level curves 

 joining their end-points. 



73. Relation between Sections and Charts. These rules lead to a new view 

 of the charts representing the mutual topographies or the differences of pressures. 

 The curves of these charts may be considered as the horizontal projections of ver- 

 tical walls, dividing the sheets into a set of tubes. These tubes with vertical walls 

 are easily seen to have a close relation to the unit-tubes with oblique walls. 



To consider first the charts of mutual topography, each vertical wall has a con- 

 stant dynamic height. Two different walls therefore have a constant difference of 

 dynamic height. From the numerical relation developed in the preceding article 

 we therefore conclude that every tube with vertical 'walls must contain a constant 

 number ot isobaric-isosteric unit-tubes. If m. t. s. units be used, this number will be 

 equal to the difference of height between the vertical walls, expressed in dynamic 

 decimeters. Thus every section of the tube, it being plane or curved, normal or 

 oblique, contains this constant number of unit-parallelograms. This does not mean 

 that the course of the unit-tubes with their parallelogram-section is exactly the 

 same as that of the tubes with vertical walls. But the latter give the average course 

 ot the first. Thus, if a unit-tube passes out through the vertical wall, for instance 

 at its base, a corresponding tube will enter through the same wall at its top. We 

 thus arrive at this result: The charts of mutual topography of isobaric surfaces 

 show the average course and the number of the isobaric-isosteric unit-tubes in the 

 sheet between the surfaces. 



