604 



SMITHSONIAN MISCELLANEOUS COLLECTIONS 



VOL. 5 I 



but at a moderate distance from the boundary circle it assumes 

 the following simpler very approximate form 



r R 2 k 



p r * 2 



2 ' r 3 



where the second negative term is the important one as it ought 

 to be since the atmospheric pressure diminishes outwardly. This 

 last given law for the gradients differs only by the sign of y from 

 that which *\vas deduced by Oberbeck for the outer region of a 

 cyclone. 



Finally, in order to express the barometric pressure as a function 

 of the distance r from the center we have to integrate the differential 

 equations (6) and (6') which only requires simple quadrature. If 

 P represents the pressure on the boundary circle whose radius is R 

 we obtain 

 for r < R 



X 2 



r(2k+r) 

 P+P ---b I 1 



(fc + Tf 



(R 2 - r 2 ) 



(7) 



On the other hand for the exterior region, we find the following 

 complex expression 

 for r>R 



p=p-^( r 4 2 ) 2 ( i+ ^)(^-^ 



-p'-kR?[l +-jlognat- + 





k/ r 



]{ 



r 2*/rl 

 2(k r)/ k [ e \ 



2k(k + r ) 



r XR \ 2 2 r 



+ B J - 

 A' - B' 



(70 



where 



e -■ 



I - 



x 



A' = 



I 



e ~ x 

 x 



x = fc r>/x R- 



x = k/ r 



x = 2 k r-/y R- 

 J x = 2 k/y 



B 



J k/ r x 



r2kr*/ r & e -x di 



J 2 k/r x 



The integral 



J 



: dx 



