1894,] of equal and opposite hollow straight vortices. 



185 



The two hollows of our original problem are therefore in this 

 case of small breadth, while the length, since the perimeter is -yr , 



is the finite quantity ^ . They are very close, their distance apart 



being, in fact, of the second order of small quantities if their 

 breadths are of the first. The velocity of translation is 



V = {^a^ - 1 - 2a^aF^) U=U, 



which is the greatest value it can have. 



It has been suggested to me that it would be of interest 

 to discuss the shape of the sharp extremity of the vortex in this 

 case. If we confine our attention to what takes place near the 

 extremity, we may consider the case as that of a jet, proceeding 

 from infinity with velocity U, which meets an infinitely broad 



Fig. 4. Plane of z. 



stream of fluid moving in the opposite direction with the same 

 velocity, both the jet and the stream being bounded by the axis of 

 X. The diagram in the plane oi z = x + ly is that given in fig. 4, 

 where the arrows indicate the direction of motion of the fluid. 

 Since FE and GHI are flow lines, and ^ increases from — oo 



Fig. 5. Plane of lo. 



F' 



E< 



I' 



-ir 



E" 



Q" 



0=0 



H" 



H" 



-00 



Fig. 6. Plane of 0. 



