VERTICAL MOTION IN FREE SPACE COMPLETE KINEMATIC DIAGNOSIS. 141 



from one section to the next, we have unit change of horizontal transport. By the 

 solenoidal condition we must therefore have unit vertical transport through that 

 area of the top which is contained between these two sections. Thus the curves 

 T = const, will divide the bands of flow into areas for each of which we have unit 

 vertical transport through the upper limiting surface of the sheet. In the case of 

 decreasing horizontal transport the vertical transport will go up, and in case of 

 increasing vertical transport it will go down through the top of the sheet. 



If there is a vertical transport through the lower limiting surface of the sheet, 

 the areas will represent that addition to the vertical transport which arises on 

 account of the horizontal motion in the sheet. 



We thus see that we have a method of arriving at a representation of vertical 

 motion like that illustrated by figs. 43 c and 45 c. 



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

 V into bands of flow which we have performed as an introduction to the construc- 

 tion of areas of equal vertical transport. The curve C represents a vertical wall 

 of the given constant height dz' . The bands of flow on the chart represent tubes of 

 flow in space, which at this wall have the given transport T' = V'dn'dz' . In case 

 (A) we have examined the change of transport T as we proceeded along tubes, 

 which were limited below and above by given surfaces. Now only the lower limit- 

 ing surface will be given. The upper will be subject to this condition, that it shall 

 pass through the upper edge of the wall C. We will determine its height dz above 

 the lower surface so that the tubes retain in all sections the transport T' which they 

 have in the section formed by the wall C . 



For this we have to introduce into (a) the value V'dn'dz' for T, and to solve 

 with respect to dz, 

 (d) dz = * dz > 



and construct a chart of this height dz. This will be a topographic chart which 

 gives the height of the upper limiting surface relatively to the given lower surface. 



The construction will be very like the preceding one. We first perform the 

 construction for the case of a wall C of unit height. Setting dz' = 1 and remem- 

 bering that V'dn' has been determined to be equal to the number c' , we have 



{e) dz < = Vdn 



As c' is equal either to unity or to a simple multiple or decimal fraction of the unity, 



we can determine the field of the quantity ^ in one operation, using the divided 



sheet of fig. 81. Then we perform the graphical division of this field by that of V. 

 The field resulting will be a topographic chart representing the upper limiting 

 surface when the initial wall C has unit height. 



Performing the multiplication by the constant height dz' we get the field of dz, 

 i. e., the topographic chart representing the upper limiting surface for any given 

 constant height of the initial wall C . 



