158 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



The component velocities are : 



M -^_ A W v=^+-^ (18) 



jx fc ^ n fc ^ 



-^^-{(SKD'K 1 ^ • • • (19) 



Finally, from equation (17) we obtain 



^eo U s taU t-(l4 2 ){^[(f)' + (f) 2 ]}-- « 



All these expressions still contain the as yet undetermined function <p, 

 which is only limited by the condition Jcp=(). Such functions can be 

 easily found in various ways. Thus if we bring the function of a com- 

 plex variable x+iy into the form 



F {x+iy)=<p+iip, 



then both cp and also >/< satisfy the above given differential equations. 

 Moreover, both functions stand in the following relations to each other. 



dq>_dtp. d<P__d$ 



dx cV dy dx 



With the assistance of these equations one can easily find the general 



equation for the path of the wind. We obtain this from the differential 



equatious 



u dy=v dx, 



1 1 Wdx+^dy \ =^dy-^-dx. 

 k \ dx dy f dx dy 



If we introduce >/• into the right-hand side of this equation we obtain 

 as the equation for the path described by the wind 



$—j- (p— constant (21) 



The path of the wind intersects the system of lines defined by the 

 condition q>= constant at an angle that is everywhere the same. 

 If we designate by s the angle that the direction of the wind makes 

 with the normal to the curves cp= constant then we have 



tan € =-=- . 



K 



For currents of air of moderate velocity the term 



in equation (20) can be neglected in comparison with cp. In this case 

 the isobars, for which p equals a constant, are identical with the curves 

 f= constant and we obtain the following general theorem ; 



