754 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
So also if r 0 be the radius of the hollow when a= co and r for any other value of x 
0 Q , , N0 , „ /O l-x + xlogx , „ , log 1/x /4oN 
—^\p=n\p—(hi+p) + ~+ 2 / x I-* • ' ( 43 ) 
Where also 87r 2 IIr 2 0 =/x, 2 2 c^ / . 
Hence r 2 0 (l + p) = r 1 2 p, or when the hollow first forms the sectional radius is 
! + p\ : 
its ultimate value. 
It will be important to know how the section of the core alters as the aperture 
increases. If r be the radius of the section, r=2ak z . Also m=87r 2 a- 3 (& 2 2 —& 2 X ) 
= 2n~ar z (l—x). 
Hence 
da , dr dx 
—+2 ---= 0 . 
a r I —x 
Differentiating (37) and substituting for dx/( 1—x) it will be found that 
= — 2 
2 (l —«) 3 ,l-x , 3 ,/1 + Jr , \, 11 da 
tf-jr-+ 2 w ^^ '+ 2 i* (w? -1*1) lo S~j- 
{tfp-ki+nV 
Here the left hand factor is always positive, and the right hand factor is clearly 
positive when x is small whatever the relations between the circulations and densities. 
Hence the section always decreases as the aperture increases, when thcrinterior hollow is 
small compared with the outside. 
When x is nearly = 1 = 1 — z say, the equation becomes 
V 3 )~= +i/V * 
Hence, when 2 is small, so also is dr/da, and it vanishes with 2 . In other words, 
when the hollow is large compared with the outer boundary, the section of the ring 
decreases very slowly as the aperture increases. When there is no core at all we have 
seen that it remains absolutely constant. In all cases where them is a hollow the 
section decreases more slowly than if the core were continuous. 
It remains to see how the radius of the hollow itself changes with increasing- 
aperture. Here r 2 =xr' 2 . Hence 
