68 Prof. Oliver Lodge on the Stream-lines 



translated through the rest, forms an ovoid mass with dimples 

 before and behind — the dimples, however, becoming less and 

 disappearing when the translatory motion is made still slower. 

 The lines near the core are in this case rather displaced away 

 from the axis. The dottedness of the portion of the line which 

 crosses the axis of the ring is purely subjective, and only 

 indicates uncertainty on my part as to its exact course, from 

 want of knowledge. It is probable that the same defect ex- 

 hibits itself in my terminology, which is probably incorrect, 

 or at least unusual. Thus I cannot help calling the actual 

 circular axis of the ring its "core," instead of the whole of 

 the rotational portion, as is usual in dealing with rings of very 

 small cross section in proportion to area of ring itself. The 

 rings drawn are not of small cross section, and so one wants a 

 name for their innermost axis or core. 



We can try to apply Sir William Thomson's rule* for the 

 velocity of translation of very thin or high-speed rings, to the 

 case of fig. 2 ; though this ring is not nearly thin enough for 

 the formula to be properly applicable. 



Using the symbol X for the ratio of radius of ring itself to 

 its cross-section radius,, the rule may be written: — 



2ir X velocity of translation 

 vortex velocity at centre of ring 

 __ 2\ x velocity of translation , /Q\ _ 1 \ 



vortex velocity at surface of rotational portion 



In fig. 2 the value of X is about 3 ; and accordingly each 

 of the above terms is about 3 also ; or the two vortex velo- 

 cities specified in the formula are nearly equal, and about 

 double that of the translational velocity. 



This does not agree with what I said before, about the ratio 

 of uniform field to ring-field at centre being as 64 : 5; hence 

 there is something wrong, but I don't know what. The lines 

 of uniform velocity in fig. 2 were taken 8 times as close 

 together as in fig. 1; and this surely represents a velocity 

 64 times as great. I can only suppose that the ring is much 

 too fat for the formula. 



Plate III. is an attempt to represent a vortex-ring advan- 

 cing in a very imperfect or viscous fluid, showing its gradual 

 increase in size, and decrease in forward velocity. It is easily 

 drawn by superposing a diverging equiangular pencil on the 

 stationary vortex which forms the basis of all three diagrams ; 

 but that it really represents the effect of viscosity does not 

 seem very probable. No slip, due to inertia of displaced fluid, 

 is shown in any of the diagrams. This figure better repre- 

 sents a ring moving towards a large distant obstacle. As 

 * Phil. Mag. June 1867, xxxiii. p. 511. 



