Ifj2 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



rectiouof the ^ind makes an angle e with the radial gradient such that 

 tan ^ = '. whereas in the inner region the corresponding- angle £' is given 



A.' 



l)V the equation tan i' = Y-^ — . 

 k—c 



Still less allowable are the consequences that follow when we imagine 

 the inner region bounded by some other curve sueli as an ellipse. In 

 this case by utilizing the special solution it results that at special por- 

 tions of the bo,uudary more air flows inward from without than flows 

 away, but at other s|)ecial portions of the boundary the relation is 

 reversed. One can easily persuade oneself of this by using the known 

 value of the logarithmic potential of an ellipse.* When therefore W 

 can be considered as the logarithmic potential of a stratum of the inner 

 region still it is not to be considered as constant. Its value is to be 

 specially determined for each given region. This computation will now 

 be executed for the case of a circular region. 



YI. CYCLONE WITH A CIRCULAR INNER REGION. 



Let the region of ascending air currents be bounded by a circle of 

 the radius FL Let the center of the circle be the origin of the system 

 of CO ordinates. We put 



r-=.r-+}f. 



First the velocity i)otential is easily computed as follows : 

 For an exterior point 



<p,--^'^R\n'r )■ (23^0 



For an interior jioint 



cp,=—^^EU2]ogR-l)-\-r^\ . . . (236) 



Furthermore for an exterior point we have 



W,^=^Ir log r 



Of the fuuctions :, Wi and P, which are still to be determined, it can 

 certainly be assumed that they depend upon r only. 

 If we further consider that 



}x dr ' r' 

 then equations (13) can be written ; 



dr dr \ dr dr J 



*KirchhofiF, Torlesungcn iiher Mechanik, 187(>, page 217. 



