PRECIPITATION ON MOUNTAIN SLOPES POCKELS 83 



m/bqn, and if its ordinates are designated by 7), its equation becomes 



or , 





n .„ **' - *" 



sin m«.« 

 m </ 



As long as 77 remains so small that for both the highest and lowest 

 points of the profile of the surface of the earth (q 77/ 2) 2 is negligible 

 in comparison with unity — which is practically always the case 

 for the mountains that come under our consideration — we • can 

 write 



ft 

 = b~~ sin mx.e~ rr> \ 

 m 



i + " 



r = V'm 2 -f ^ 2 /4 



(5') 



In these expressions b and m appear as parameters that can be 

 chosen at will, the first of which determines the altitudes and the 

 second the horizontal distances between the mountain ridges; we 

 have, namely, m = 2iz\ A, if X denotes the wave length, that is to 

 say the distance between two corresponding points, as for example 

 the summits of neighboring mountain ranges. 



It is easy to show that the stream line determined by the velocity 

 potential (3) for the configuration of the ground given by the trans- 

 cendental equation (5') is the only one compatible with the general 

 conditions 1 to 5. Moreover, since a potential current is determined 



single valued, for the interior, by the value of — along the bound- 

 ary of a closed region, therefore, our solution in case it gives hori- 

 zontal velocities that are constant, or slowly diminish with the 

 altitude above the center of the valley, is also applicable to the 

 specially interesting practical case in which only one single moun- 

 tain range rises above an extended plain and is struck perpendicu- 

 larly by a uniform horizontal current of air. To what extent this 

 holds good must be established in each special case. 



The horizontal and the vertically upward velocity components 

 corresponding to our solution are: 



u = a (1 + bm sin mx .e~ ny ) 0) 



v = abn cos mx.e~ ny (7) 



