VERTICAL MOTION IN FREE SPACE COMPLETE KINEMATIC DIAGNOSIS. 151 



The chart of fig. 106 gives the topography of the surface of flow expressed in 

 terms of pressure; qualitatively we can consider it also as a chart giving topography 

 in terms of height. We have given above the approximate height corresponding 



10" 

 to the different integer values of pressure. But if we multiply by - we pass to 



decimal heights. Thus in rough approximation we can interpret the curves 1, 2, 3, 

 ... of the chart as contour-lines which give the heights 1,2,3,. meters or 

 the heights 10. 20, 30, ... or 100, 200, 300, . . . of a surface of flow. 



(C) Vertical component of specific momentum. In order to find vertical specific 

 momentum, we have to draw a chart of divergence of the horizontal motion (see 

 formula (h) of section 187). For this we can use directly the given chart of fig. 102, 

 no special division into bands of flow being required. Divergence of the two- 

 dimensional field of velocity v will according to formula (g) of section 1 70 be given 

 by the equation 



div 2 v = --+z>5 



cS 



s denoting the length of arc along the lines of flow and 8 the divergence of these lines 

 (see section 168). As we here come across the most important construction of kine- 

 matic diagnosis, we will illustrate each of the four separate operations, the last of 

 which gives the result. 



(1) We construct the field of the derivative of the intensity of the vector with 



c S 



respect to its vector-lines. This differentiation is performed in the regular way by 

 use of the differentiating sheet of fig. 81 as illustrated in section 165. The resulting 

 field is given in fig. 107. The numbers added to the curves give the values of the 

 derivative obtained when ds is measured in centimeters on the chart. In order to 

 get the true values per meter we have to multiply by io~ s , as a centimeter on the 

 chart represents 10 5 meters. 



(2) Then we have to draw the field of divergence 5 of the lines of flow. We can 

 determine this field by use of the divided sheet for differentiations of the second 

 order, fig. 90, this sheet being placed with the radii tangential to and the circles 

 normal to the lines of flow. But if the isogons of the lines of flow are given, we get 

 a much better determination by using the ordinary differentiating sheet of fig. 81. 

 We then perform the differentiation of the angle represented by the isogons with 

 respect to the normal curves n to the lines of flow. The resulting field of divergence 

 of the lines of flow is given on the chart of fig. 108. The numbers give the value of 

 the divergence referred to the centimeter as unit of length and to the scale of the 

 chart. Multiplying by io~ 5 we get the true divergence of the lines of flow referred 

 to the meter as unit of length. 



(3) Then we perform the graphical multiplication of this field of divergence 

 by that of the intensity v of the given velocity. The result of this multiplication, 

 which is performed in the regular way (section 150) is given on the chart of fig. 109. 



(4) Finally we perform the graphical addition of the two fields of figures 107 

 and 109, and change the sign in order to pass from divergence to vertical-component 

 of specific momentum. We thus get the chart of fig. 1 10, which contains the result. 



