140 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



transport T v . Let dn be a horizontal element of line which is normal to the lines 

 of flow. The expression 



(a) T = Vdndz 



will then give the horizontal transport through the area dndz, which extends from 

 the bottom to the top of the sheet. Thus we have to draw a chart representing 

 the expression (a). 



In order to do this we shall first consider the expression 



(b) r, = Vdn 



which represents the transport in a sheet of the thickness of dz = i . The curves 

 Ti = const, will be the curves of equal transport for the two-dimensional vector V. 

 In order to draw these curves we may proceed as we have developed already (section 

 167) : On the chart which represents V we first draw an arbitrary initial curve C 

 and divide it into elements which give equal values of the two-dimensional transport ; 

 i. e. , for each element we shall have 



(c) Vdn' = c' 



dn' denoting the projection of the element of the curve C upon the normal to the 

 lines of flow, c' is an arbitrarily chosen constant, equal either to the unit of transport 

 used in practice or equal to a simple multiple or fraction of this unit. The essential 

 point is to choose the constant so that we get bands of flow of suitable breadth 

 for the construction. Through the points of division we draw lines of flow which 

 will then define the bands of flow to which the transport T is to be referred. Using 

 the divided sheet of fig. 86, we then draw curves for equal values of the breadth dn 

 of these bands of flow. Finally we perform the graphical multiplication of this 

 field by that of V. The field resulting will be that of Ti , which represents the hori- 

 zontal transport in a sheet of unit thickness, dz =1. 



In order to get a chart of T we have finally to perform the multiplication by 

 the thickness dz of the sheet. If dz is constant this will lead to a simple change of 

 the intervals between the curves T Y = const. In the general case, where the thick- 

 ness of the sheet is variable from place to place, dz will be represented by a chart 

 which gives the topography of the upper limiting surface of the sheet relatively 

 to the lower. We have then to perform the graphical multiplication of this field 

 by that of T t . The result will be the field of T represented by curves for integer 

 values 



r = ... ii, 10, 9,8, ... . 



This field directly represents the average horizontal transport in the sheet, 

 but indirectly it will also represent the correlated free vertical transport. Let us 

 suppose, for the sake of simplicity, that the lower limiting surface of the sheet 

 is a surface of flow. The bands of flow in the two-dimensional drawing will then 

 represent tubes, the bottom and the two lateral walls of which are surfaces of flow, 

 while a transport goes through the top. The curves T = const, will represent 

 vertical walls which are sections of these tubes. When we proceed along a tube 



