Minimum Energy in Vortex Motion. 539 



for the case of a circular enclosure and a small part of its 

 whole volume occupied by vortex, to which exigency of time 

 limits the present communication, 



24. Commence with the vortex lining uniformly the en- 

 closing cylinder, and the stirrer in the centre of the still 

 water within the vortex. The velocity of the water in the 

 vortex increases from zero at the inside to ^b 2 /a at the out- 

 side, in contact with the boundary ; according to the notation 

 of §§ 14 and 15. Now move the stirrer very slowly from its 

 central position and carry it round with any uniform angular 

 velocity Kfyja and >\%bja. A dimple, as shown in fig. 6, 

 will be produced, running round a little in advance of the 

 stirrer, but ultimately falling back to be more and more nearly 

 abreast of it if the stirrer is carried uniformly. If now the 

 stirrer is gradually slowed till the dimple gets again in advance 

 of it as in fig. 6, and is then carried round in a similar relative 

 station, or always a little behind the radius through the middle 

 of the dimple, the angular velocity of the dimple will decrease 

 gradually and its depth and its concave curvature will increase; 

 till, when the angular velocity is J $>/a, the dimple reaches the 

 bottom (that is, the enclosing wall) with its concavity aright 

 angle, as shown in fig. 7, and the angular velocity of propa- 

 gation becomes \ZJbja. 



25. The primitively endless vortex belt now becomes divided 

 at the right angle, and the two acquired ends become rounded ; 

 provided the stirrer be carried round always a little rearward, 

 or considerably rearward, of abreast the middle of the gap. 

 Figs. 8 and 9 show the result of continuing the process till 

 ultimately the vortex becomes central and circular (with only 

 the infinitesimal disturbance due to the presence of the stirrer, 

 with which we need not trouble ourselves at present). 



26. Suppose, now, at any stage of the process, after the for- 

 mation of the gap, the stirrer to be carried forward to a station 

 somewhat in advance of abreast of the middle of the gap ; or 

 somewhat rearward of the rear of the vortex (instead of some- 

 what in advance of the front as shown in fig. 8). The velocity 

 of propagation will be augmented (by rearward pull I), the 

 moment of momentum will be diminished : the vortex train 

 will be elongated till its front reaches round to its rear, each 

 then sharpened to 45° and brought into absolute contact with 

 the enclosing wall : the front and rear unite in a dimple 

 gradually becoming less ; and the process may be continued 

 till we end as we began, with the vortex lining the inside of 

 the wall uniformly, and the stirrer at rest in the middle of 

 the central still- water. 



[To be continued.] 



