756 
PROFESSOR W. M. HICKS OK THE THEORY OF VORTEX RINGS. 
In the simplest case where there is no added circulation at all /a 2 =/x'. 
/a /. 5 Id 
V =8^ 2L + 4 -iS-' 
— ^ when d=d'. 
In general, when x is small, the most important term is that involving log - . 
It is given by 
L fh + A 6 
1 r 
jl ! + -f—, lid E+ (1 
L 
4 
jd _i_\ 
fl^+fJb' 1 —xj * ‘ \ yU-j+ft 7 1- X) 
When there is no added circulation at all, and d=d' 
v =8^jE+E-Wh#(L-L 2 ) 
(1-af 
. . . ( 48 ) 
As an example of the use of the formulae let us find the alteration in V, in this 
simplest case, when the radius of the ring is doubled, supposing originally that the 
radius of the ring =10 cm., outside radius =1 cm., and inside cm. 
Since the volume is unaltered, if k. 2 , k y be the new values of Jc z , k x 
also since 
or 
where 
therefore 
Q/V2 __ ^ 2\_7. 2 7, 2 — _ 1 _— __3_ 
°\ K 3 K \ ) — ^3 /t l — 400 1600 — 160 0 
4 mII „ - / X , 1 \ , 
XT = \ l ~Y^x log xr 
1—2 l og "j = l('i—2—V—] 0D . h 
K- 2 _ „ 2 21 7, 2 h 2 » h 
A-O TV y y A 2 /tj, /L| 
—i l°g e 2 = i+Hog^ 2 
z =(xJk i) 3 
log 10 2=731(2-l)log 10 e 
= -3174(2:— 1) 
