PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
753 
Now we saw above that y x =^i 1— cc-flog Up 
In the case where x is very small, the value of y depending on the added 
circulation is 
Pi + P + fi) ° & :c 
and ,x> 
P\d 
/xpd + 4mII 
a 
therefore 
wi <. **» bg/i+ngy 
Pi + p' Kpi + p) & \ ' fJr^daj 
When fx^da is large compared with Amli 
y< -tt*. 
Hence, the centre of the hollow never approaches near the outer boundary. 
Two or three cases further may be usefully considered. When there are no added 
circulations, ^ = 0, /r 2 =/x', and 
~ =/**{(1 -x)d’+(l+x)d-- l d x log ! 
• ■ (40) 
This gives the condition when a continuous core begins to develop a hollow. In 
this case £c=0, and 
4?/iII o/7 | y\ 
or 
4mII 
a = 
p\d+d') ’ 
• (41) 
Now ct depends on the energy and increases with it. Hence as the energy increases 
a point is at last reached at which a continuous ring begins to develop a hollow. The 
sectional radius r x is then given by 
8 tt 2 I1 rp= y\(d +d') 
If a x be the value of a when a hollow is just formed, then in general 
2 xd 
\d+d') = (\-x)d' + (1 +x)d-f- log 1 /x : 
7 ’ X ““ Jj 
or if d'/d~p 
J(l+p)=(l-a:)p + l+ai—— log 1/a 
• • (42) 
