102 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



rectionof the wind makes an angle s with the radial gradient such that 



tan e = -, whereas in the inner region the corresponding angles' is given 

 ft 



by the equation tan £ — r 



Still less allowable are the consequences that follow when we imagine 

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

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

 tions of the boundary more air flows inward from without than flows 

 away, but at other special 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. 



VI. CYCLONE WITH A CIRCULAR INNER REGION. 



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

 the radius R. Let the center of the circle be the origin of the system 

 of coordinates. We put 



r 2 =.x 2 -\-y 2 . 



First the velocity potential is easily computed as follows : 

 For an exterior point 



^-^itJlogr (23a) 



For an interior point 



<Pt=-l{lP{2logR-l) +r 2 } . . . (236) 

 Furthermore for an exterior point we have 



W *=2k R l0g r 



Of the functions C, W { and P, which are still to be determined, it can 

 certainly be assumed that they depend upon r only. 

 If we further consider that 



df(r) = df(r) x 



dx <lr ' r 7 



then equations (13) can be written ; 



X d f^ + Jh = _ r f\dcp 



dr \ / dr dr J 



yp-x^=+z(x d ^ +y d JT\ 



dr dr ^ ' \ dr T J dr J 



"Kirehhoff, Vorlpsungen tiler Mechanik, 187H, page 217. 



