600 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



The first of these equations is satisfied by the function 



) 



r 



2 (* + y) 



the second equation is satisfied by 



1 r I A' 2 

 ~ 2 ~k" r 



as is easily seen without further comment. In order to obtain the 

 general integral, we have still to add the integral, multiplied by an 

 arbitrary constant, of the differential equation that arises from the 

 omission of the terms that do not contain V t in equations (Ha) or 

 (lib) ; this homogenous differential equation reads 



d(rVA2k rV t n . ^ __ 



— - + — . — - = for r < R 



dr r r 



whose integral ir, 

 and again 



r V t = C r 



d (rV t ) 2 k . T/ , _ , . . 



— - — - = . r . (r V,) = for t > / 



dr r R 3 



whose integral is 



rV t = &9\r») 



Hence the complete solution for V t reads as follows: 



2 k + j 



(2) 



(20 



The integration constants C and C" are now to be determined 

 in harmony with the conditions of continuity. In the case of a 

 cyclone for which we have negative values of r (that is to say, an 

 ascending current in the interior region) we must put C" = o, 

 since otherwise V t would become infinitely large at an infinite dis- 



