THE FORCED VERTICAL MOTION AT THE BOUNDING SURFACES. 1 35 



v being the resultant velocity, the vertical component v v will then be given by the 

 formula 



(b) v v = v d 



In accordance with this expression we can construct the field of v v . In the case of 

 motion along the bottom of the sea we should have to use the depth below sea-level 

 instead of the height above it. But we shall henceforth consider exclusively the 

 case of the atmosphere. As soon as the observations are at hand, it will be easy 

 to adapt the same methods to the investigation of sea-motions. 



Formula (b) reduces the drawing of a chart of vertical velocity to a simple 

 problem of graphical differentiation and of graphical algebra. 



A rough sketch of the field (b) can easily be made by the discontinuous method. 

 Evidently the field (b) will contain a zero-line v v = o, which separates from each 

 other the windward and the leeward sides of the mountains. The general course of 

 this line is seen at once and can be drawn by eye-measure in those parts of the 

 country where the slope is strong enough to produce a vertical motion of any impor- 

 tance. By use of the differentiating sheet of fig. 81 , we can then make a few determi- 

 nations of v v in the places where it is seen to have its greatest positive and negative 

 values. Afterwards the curves v v = const, can be drawn by eye-measure. It will 

 not be difficult in this way to draw such charts in the daily meteorological service. 



For more detailed investigations we can bring the continuous graphical methods 

 into application. The method of proceeding will be this : 



We construct first the chart of the angle of inclination (a) . The construction 

 is that which has been exemplified in fig. 83. In this figure we can interpret the 

 lines a = const, as contour-lines, and the lines s as the lines of flow of the wind. 

 The stippled curves will then be curves for equal values of the angle of inclination i. 

 Of these curves we first draw that for the angle of inclination zero. This curve will 

 pass through all the points of tangency of the lines of flow and the contour-lines. 

 A zero-curve must therefore pass the summit of every mountain as well as the 

 highest point in every pass. Inasmuch as the wind does not travel precisely along 

 the chain, but has a component across it, the zero-line will follow near the highest 

 ridge of the chain, passing all the summits and the highest point of the passes. 

 In the same manner, when the wind does not travel precisely along a valley, but 

 has a component across it, a zero-line will run along it, near its bottom. 



As soon as the zero-line is drawn, we determine the course of the curves for 

 integer values of the angle of inclination by making continuous use of the differen- 

 tiating sheet of fig. 81 as described in section 165. 



Finally we perform the graphical multiplication (section 150) of the field of 

 the angle of inclination i with that of the scalar value v of the velocity of the wind. 

 The chart resulting will then represent the field of the vertical velocity v v . 



182. Ascendant-Charts. From a theoretical point of view the drawing of the 

 charts of vertical velocity is exceedingly simple. But still, when it is to be done 



