162] STREAM LINES OF A VORTEX-RING. 259 



nearly, if g denote the distance between two infinitely near points 

 (x, CT), (V, CT ) in the same meridian plane. Hence at points 

 within the substance of the vortex the value of ty is of the order 

 m vr log OT/, where e is a small linear magnitude comparable with 

 the dimensions of the section. The velocity at the same point, 

 depending (Art. 93) on the differential coefficients of i/r, will 

 be of the order m /e. 



We can now estimate the magnitude of the velocity dx /dt of 

 translation of the vortex-ring. By Art. 161 (3) T is of the order 

 prn ^ ur log -cr/e, and u is, as we have seen, of the order m /e ; whilst 

 x XQ is of course of the order e. Hence the second term on the 

 right-hand side of the formula (8) of Art. 160 is, in this case, small 

 compared with the first, and the velocity of translation of the 

 ring is of the order m /^ . log cr/e, and approximately constant. 



An isolated vortex-ring moves then, without sensible change 

 of size, parallel to its rectilinear axis with nearly constant 

 velocity. This velocity is small compared with that of the fluid 

 in the immediate neighbourhood of the circular axis, but may be 

 large compared with m /w , the velocity of the fluid at the centre 

 of the ring, with which it agrees in direction. 



For the case of a circular section more definite results can be obtained 

 as follows. If we neglect the variations of OT and o&amp;gt; over the section, 

 the formulae (7) and (10) of Art. 161 give 



or, if we introduce polar coordinates (s, %) in the plane of the section, 



T- 1 ) ** ..................... &amp;lt; 



where a is the radius of the section. Now 



JO 



and this definite integral is known to be equal to 277 logs , or 27rlogs, 

 according as s ^s. Hence, for points within the section, 



&quot; s ds 



(ii). 



172 



= - 2o&amp;gt; wof* flog 0-2^ s ds -Mw, [ a (\o S *2*- 



J Q\ s / J S\ S 



