56 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



When the section of the vortex thread is infinitely small and € is an 

 infinitely small quantity of the same order as (I— A) and the remain- 

 ing linear dimensions of the section, but a dp d\ is finite, then ip and 

 also E are ot the same order of infinitely large quantities as log s. 

 For very small values of the distance v from the vortex ring we have 



K 2 = l- 



V 2 





= Wl loo- A. 



7T °8g 



In the value of K, ip is multiplied by p or g. If g is finite, and v of 

 the same order as s, then K is of the same order as log s . Only when 



g is infinitely large of the order - will K be infinitely large of the order 



— log £. But in this case the circle becomes a straight line. On the 



other hand, if p , which is equal to-', is of the order -, then the sec- 



dt dz * 



ond integral will be finite, and for a finite value of p will be infinitely 

 small with respect to K. In this case we can, in the first integral, substi- 

 tute the constant I in place of X and obtain 



2 d(mRH) _ K 

 dt 2rch 



or 



2mm=c-J^t 



2,7th 



Since 8ft and R are constant, I can only vary proportionally to the 

 time. When §Jt is positive the motion of the liquid particles on the 

 outer side of the ring is directed toward the side of positive £, but on 

 the inner side of the ring toward the negative z. K, h, and R are by 

 their nature always positive. 



Hence it follows that for a circular vortex filament of very small 

 cross-section in an infinitely extended mass of liquid the center of grav- 

 ity of a cross-section has a motion parallel to the axis of the vortex 

 ring, which is of approximately constant and very large velocity, aud 

 which is directed toward the same side as that toward which the liquid 

 flows through the ring. Infiuitely slender vortex filaments of a finite 

 radius will have infinitely large velocities of propagation. But if the 



radius of the vortex ring is infiuitely large of the order _ , then will 



( £ ) 

 R 2 be infinitely large with respect to Zl, aud I will be constant. The 



vortex filament which has thus transformed itself into a straight line 

 will be stationary, as we had already previously found for rectilinear 

 vortex filaments. 



