PROFESSOR W. M. KICKS ON THE THEORY OF VORTEX RINGS. 
755 
2 * 1= i f rfr 
r x r 
1—xfda 2 clr\ 2 dr 
TV + V 
x \ a r 
da . dr 
it r 
Substituting for d> 
r r 
dr , ] {PiP ~ Oi + /0 2 } (! -x) s -2/i 1 /i , ( 1 -«) + VH +x) + 2/Y^ - j^^log * 
2-7-=- p — 
x da. 
a 
Where D is the coefficient of —2 dr Jr in the equation found above. 
The maximum radius of the hollow will be when x has the value given by 
{ P'ip —(/H2/x 1 /x'( 1 — x) + 2 f' 2 (l-\-x) +2 //U 1 - ] ' J log - = 0 
2x 
This is satisfied by x— 1 . 
It is easy to show that the expression, on the left increases as x decreases, and 
therefore that dr/da is always positive. Hence 
The radius of the hollow continually increases, as the aperture groivs, up to 
coincidence ivith the outer boundary. 
11. Velocity of translation. The velocity is given by the formula in ( 36 ). If the 
value of fijk.2 be substituted in the second term it becomes 
1 fff-TV/f-filf 1 log 1 
.... ( 44 ) 
Case I. Where there is no core cl— 0, ix x = f = 0, ce=l, ^ = 0. 
V= 
th 
87 TO, 
(2L—1) 
(45) 
which agrees with the result obtained in [I. 21J, 
9 
Case II. Continuous core. — 0,^ = 0, x=0, /V—“’7 
4 V= W<t ( 2 L + 4 ) - b” d 
( 46 ) 
