PRECIPITATION ON MOUNTAIN SLOPES POCKELS 8l 



vertical plane, and consequently depend only on the vertical co- 

 ordinate (y), and one horizontal coordinate (x) ; 4, the internal fric- 

 tion, as well as the external (or that due to the earth's surface), 

 may be neglected; 5, at great heights there must prevail a purely 

 horizontal current of constant velocity (a). As to the configura- 

 tion of the ground, we must, corresponding to proposition 3, assume 

 that the profile curves are identical in all vertical planes that are 

 parallel to the plane of xy; 6, and further, we assume the surface 

 profile to be periodic, that is to say, the surface of the earth is formed 

 of similar parallel waves of mountains without, however, deter- 

 mining in advance the special equation of the profile curves. 



If we designate by u and v the horizontal and vertical components 

 of velocity and by e the density, then, in consequence of assumptions 

 1 and 3, there follows the condition 



9 (e u) d (e v) = Q 

 dx dy 



and in consequence of 2 there must exist a velocity potential, <p, 

 which, according to 3, can only depend upon x and y, so that 



u=% z, = ^and ±( 9 *\ + ±( .?*) -0. 



dx dy dx \ dx J dy \ dy / 



If we consider that the density of the air (excluding large differ- 

 ences of temperature at the same level) changes much more slowly 

 in a horizontal than in a vertical direction, then we can regard e as 

 a function of y only, and obtain for <p the differential equation — 



eJ<p=- --L- (1) 



dy dy 



The law of the diminution of density with altitude will, strictly 

 speaking, be different for each particular case, because the vertical 

 diminution of temperature in a rising current of air, which deter- 

 mines the rate of diminution of density, depends upon the conden- 

 sation. But it is allowable, as a close approximation and as is 

 usually done in barometric hypsometry, to assume the law of dimi- 

 nution of pressure which obtains, strictly speaking, for a constant 

 temperature only, and which, as is well known, reads as follows: 



P 



nat log = q y, 



