PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
751 
Again, when k. 2 is uniform, 
dy' 
dx 
1 ! 
( 1 -*) 
2 log 
1 
X 
which is always negative, or y' continually decreases as x increases ; while when x= 1, 
y'= 0, so that y is always positive. 
Hence, keeping the outside boundary the same, if the mass of the core be gradually 
diminished (or the hollow increased) the centre of the hollow moves in, from a point 
(f& 2 ) th of the radius, towards the centre of the outer boundary, coinciding with this 
last in the limit. 
When the added circulation is very large 
y 1=4 !—log 
This is always positive. When x=l, or there is no rotational motion, ^ = 0, as 
x decreases, y increases, and the hollow moves outwards to the boundary. From 
the formula itself we should gather that when the radius of the hollow is decreased 
to a certain small amount, it will be in contact with the outer boundary, and if this 
were possible the hollow would slip out of the inside of the core into the fluid, and 
there would be two rings formed, one with a hollow only and circulation round it, and 
another with rotational core and no added circulation. But this cannot be asserted 
until we have learnt the connexion between k 2 and a, from equation (37), for in the 
actual case it will be impossible to reduce k 1 below a certain limit to be determined. 
It is easy to see that y 1 > y'. 
An idea of the magnitudes of the quantities involved can best be obtained from 
a numerical example. Take for instance the case of a ring 10 cm. radius, and 
radius of cross-section 1 cm. Then k 2 =-^y. Take the three cases where the radii 
of the cross-section of the hollow are respectively f, we find 
x=TS y — '055 y x —-093 
x=i y — '041 y 1= -053 
x=h y -023 V\— '026 
The limit when y x = 1 is found from 
1 
or writing 
O 
i.e. 
a sufficiently small quantity. 
log -=39+a: 
x—e 39 £ 
£==e -39 =10 -16 X‘115 
a;= 10“ 1G X ‘115 
5 d 2 
