752 
PROFESSOR W. M. HICKS OH THE THEORY OF YORTEX RINGS. 
10. Relation between a, k 1} k . 2 .—When there is no hollow the volume of the core 
remains constant. In other words, o?kd 2 is constant. When a hollow exists the relation 
is found from the equation 
{H 2 d'~ (Pi+pT^H 1-x ) + — + 2/r' + — 1 _7 ic / A ^ log " jd (37) 
combined with that of constant volume, viz. : 
7n= 87r 3 a 3 (/% 2 — k*) 
The first gives the ratio kjk 2 , the second then gives k l or k 2 in terms of a, and a 
is determined by the energy of the motion, which is considered below. 
In the case where there is no core, 
and 
or 
— fx —0, 7it —0, k.-) — /c^ 
li\d! 
n= 
327 r 2 ct 2 k 2 c 
2 2t r V 2n 
(39) 
or, the sectional radius of the hollow remains constant. This agrees with the result 
in [I. § 9]. 
It is clear that when there is an added circulation this radius cannot become very 
small. For we may write the equation 
4???n 
M { 5 +v log 1 } =qy - [^d'-^+rfdw-x)+tfd 
w+2^io g ty 
Now as x decreases from 1 to 0, the right hand side continually decreases. Hence 
the greatest value is when x= 1 
Therefore 
d (&. 
Ml q°+2/io g i)< 4 q3+ M q 
a 
4mII 
>- 
a 
■ + {(p i +p')~ fi-py' — 2/W} d 
Hence 
/q 2 <i + 2/qju/xc£ log - 
x > 
Pi 
~d- 
4mII 
> 
*d 
Pi 2 d- 
4mH 
This, therefore, is an inferior limit to 
x. 
