144 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



for equal values of the breadths dn of these bands of flow and perform the graphical 

 multiplication of this field by that of the scalar value v of the velocity. This gives 

 the field of 7\. 



The field of T x will represent the final result if the thickness of the sheet is 

 defined by unit decrease of pressure. If it has a thickness defined by any variable 

 decrease of pressure, a chart of this decrease of pressure dp must be given. 



This chart will give in terms of pressure the topography of the upper limiting 

 surface of the sheet relatively to the lower one. If we perform the graphical multi- 

 plication of this field of pressure dp by that of T t , we get the field of T. 



The direct interpretation of the chart of T is this : it gives the horizontal mass- 

 transport in the sheet the thickness of which is defined by the decrease of pressure 

 dp from bottom to top. But at the same time it represents the vertical mass- 

 transport through the top of this sheet in an indirect way: The curves T= const, 

 divide the bands of flow into elementary areas ; for each of these areas we have unit 

 mass- transport through the upper limiting surface of the sheet. 



(B) Topographic method. We retain that division of the given velocity-chart 

 into bands of flow which we have performed by drawing the curve C and dividing 

 it into elements. The curve C will now represent a vertical wall the height of which 

 is given by the condition that there shall be constant decrease of pressure dp' 

 from bottom to top. At this wall the tubes will then have the given mass-transport 

 T' = v'dn'(-dp'). We propose to draw a chart of that decrease of pressure 



v'dn' 



which must define the thickness of the sheet if the tubes are to have everywhere the 

 same mass-transport as they have at the wall C. 



We perform the construction first for the case in which the wall C has the 

 height which is defined by unit decrease of pressure from bottom to top, dp' = i. 

 This is done according to the formula 



c' 

 ^ ~ d ^= vdn 



where c' is the value of the two-dimensional transport v'dn' at the curve C. In 



c > 

 order to find the field of dp l we first draw the field of by use of the differen- 

 ce 



tiating sheet of fig. 81. Then we perform the graphical division by the field of the 

 scalar value of the velocity v. The resulting field will be a chart which gives in terms 

 of pressure the topography of the upper limiting surface of the sheet relatively to 

 the lower one in the case dp,= i. If the wall C has a height defined by another 

 constant decrease of pressure dp' , we have finally to perform the multiplication of 

 the field of dp l by this constant dp'. The field resulting (d) represents in terms 

 of pressure the topography of the upper limiting surface of the sheet relatively to 

 the lower one. If the lower is a surface of flow, the upper will also be a surface of 



