l6o SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



special case, namely, the central part of the whirl. For a system of 

 circular isobars the equation of continuity can be written 



2n r h v = constant 



Supposing the height to be variable and a function of r, we can write 



v r cos (p — / (r) 



Assuming the following hypothesis: 



/ (r) = c r 2 



where c is a constant, then the equation of continuity takes the form 



v cos (/> = or 



Introducing this value of v in equations (2) and (3) of §10 and by the 

 aid of the formulae of §12, we shall find 



(2) 



(3) 



- G cos d; = v ( k — v sin d> — — c ) 



P \ dr J 



u. „ . . /_ . _ v sin </» ,dd>\ 



- G sin <p = v [2 co sm 6 + + v cos cp — l I 



o \ r dr / 



P 

 Eliminating G we shall have 



d(f> 



= ^ sin <p — 2 oj sin 6 cos <J> — 2 c sin </' — z; — - . . (4) 



dr 



This equation can be written 



cr d{tang ^= (k-2c)tang</j-2cosm8 

 dr 



By integration placing the arbitrary constant equal to zero we find 



tang0 = (5) 



hY k-2c 



The angle of inclination is therefore constant and the trajectory 

 is a logarithmic spiral, but the angle of inclination has a value dif- 

 ferent from the normal value a of § 9, eq. 3. We express this 

 value by /?, and introducing the value 



tan a = — suit/ 

 k 



