154 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



In these we have put 



X = 2ff sin /i, 



wherein a is the angular velocity of the earth (0.00007292) and ft the 

 mean geographical latitude of the region in question. Herein the sys- 

 tem of coordinates is to be so taken that the resultant produces in the 

 northern hemisphere a deviation of the path toward the right. There- 

 fore the axis of A r is positive toward the east and the axis of Y positive 

 toward the south. For the resistance of friction, according to the 

 adopted assumption (c), we put 



X 2 — — ku, Y, = — k v. 



The factor k is dependent on the nature of the earth's surface. It is 

 smaller for the surface of the ocean than for that of the land and is of 

 the same order of magnitude as A. 



By the introduction of these forces in the equations of motion (2) we 

 obtain 



Jv a» . dv 7 , 1 dp 



(3). 



If after the addition of ±« {dv / ' d$) to the first equation and of ±m 

 pM/ ity) to the second we introduce the double angular velocity £ in 

 reference to the axis of Z, so that 



C = du - dv (4). 



" dy dx v ' 



then equations (3) can be written in the form 



y p +*(*■+ v*)i+j t +ku^ -(a+o« 



ij\$+i(* + *l\+% + **= (*+0« 



. (5). 



III. DEDUCTIONS FROM THE FUNDAMENTAL EQUATIONS AND THEIR 



TRANSFORMATION. 



From equations (5) we can deduce without further special assump- 

 tion a theorem that expresses a general relation between the gradient, 

 the wind velocity and the wind direction. As is well known in mete- 

 orology, the term gradient indicates the difference in the atmospheric 

 pressure at two localities that lie at a definite distance apart in the di- 

 rection of the most rapid change of pressure. According to this we can 

 consider the differential quotient 



1 dp 



(j dn 



