608 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



The differential equation (lb') for the tangential component 

 V t for the inner region now becomes 



1 d(rV.) k rV. 



The general integral of this equation is the sum of a function that 

 satisfies the complete differential equation, added to the product of 

 an arbitrary constant multiplied by a second function (V' t ) that 

 satisfies the same differential equation after omitting the term X. 

 For this latter function we find 



Since this expression becomes infinite when r is positive and r = o 

 therefore we must have C = o. On the other hand this constant 

 must be retained in the case of the cyclone where y has a negative 

 value whose absolute value however can not exceed 2k. 



Hence our solution for V t is only attained through the integral 

 of the complete differential equation for which the following expres- 

 sion is found 



V = —( H +2 R2 - 1 Y' /r f - r2k/r d < >2 > 



2~r\~2~ 7* ~ I J /n + 2 R2 _ r2 yr 



which does not allow of presentation in definitive terms for all 

 values of . A simplification is possible when - is a simple rational 



r r 



fraction. I give the following result for the special case when 

 r = k 



2r\ 2 r 2 / 



X [ r 2 + — ■ — R 2 log nat < 1 - > • 



\ 2 I n + 2 R 2 j J 



The last factor in this expression (and therefore also V t itself) is 

 negative and for r = o becomes infinitely small in such a way that 

 V t also itself becomes zero. 



The preceding relatively simple result was attained without 

 special limitation as to the exponent n in the law for the vertical 



