84 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



It would now be desirable, in order to be able to handle the cases 

 actually occurring in nature, to adapt our solution to some form 

 of the earth's surface arbitrarily chosen. The first thought would 

 be to attempt this by the superposition of a series of velocity poten- 

 tials of the form of equation (3) having different constants m and 

 b, or in other words to write 



(p =^<f h = a \x -^\b h cos m h x.e- n hy\ . . . . (8) 



but we find that this solution only corresponds to a superposition 

 of the profile curves, that is to say, it gives 



V = V Vh = V K — sin m h x.e~ r M (9) 



only when we can put the exponential functions e^ffland c^hVboth. 

 equal to unity. In this case 77 is at once transformed into the 

 simple trigonometrical series 



v =^\b h 



m h 



sin m h x (9') 



and therefore, by putting m h = h m l we can develop any arbitrary 

 function, r) = f (x), into a series, proceeding for any value of x 

 greater than zero and less than XI 2. But the condition that e± hmr l 

 is equal to unity for any large value of the quantity h will not be 

 fulfilled for any arbitrary form of the profile curve if its maximum 

 altitude is assumed to be very small in comparison with the wave 

 length X. Therefore, we must limit ourselves to an approximate 

 representation of the desired profile curve by a definite number of 

 terms of the series that enters equations (9) or (9'). Especially can 

 we in this way never attain the rigid solution for a ground profile 

 that has sharp angles. However, the neglected higher terms of the 

 series have a proportionately smaller influence on the vertical vel- 

 ocity at great altitudes and, therefore, on the resulting precipitation, 

 in proportion as their serial number h is larger. 



II 



As a first example, we choose a form of profile to correspond as 

 closely as possible to a plane, broad, valley and a plateau-like moun- 

 tain range, because, in this case, we may expect nearly the same 

 conditions on the slope of the mountain as if it were struck by a 



