780 
PROFESSOR W. M. HICKS OH THE THEORY OF VORTEX RINGS. 
In general 
2 2 log x 
all these terms are negative. Hence the core is always stable for pulsations. 
Substituting for u, v, w the time of pulsation is given by 
I677 
/ 1 />L 3 + i log 1 lx 
IHP—P 1 
For a coreless ring this is 
2 2 log £ 
1 ()7r~crk 1 2 3 4 5 / , 4 
- r V log J 
(81) 
which agrees with the value obtained in [I. § 14]. 
[Added April 3, 1886.—The cubic giving the times of vibration of a hollow ring 
of the same density as the fluid, and with no extra circulations, can be solved. The 
equation is 
1 _ r 11 
Q A ,(y o 
f-jzjr- 
n~ 
l-x ^ (1 -xf 
1 —X n 
1 —X 
= 0 . 
The roots are 
y=-n, i(»+Y^)±l \Z{( M +TA)’-r^( n “lA’ 
47r 2 r 2 
The corresponding times of vibration are -, where r is the radius of the section.] 
Pl' 
Errata in Paper on “ Steady Motion and Small Vibrations or a Hollow 
Vortex,” Phil. Trans., Vol. 175. 
1. Page 188, line 6 from bottom, the coefficient of Td in U is wrong, since the full 
value of \p 0 was not substituted. 
2. Page 191, line 4, for 1—2 Jd read \-\-2ld. 
3. „ „ 6, for (L— f) read L— 
4. ,, ,, 7, the coefficient of Id is f(L —^) 3 . 
5. In § 13 the effect of the surface velocity in modifiying the normal motion of the 
wave motion has been neglected. The time of vibration there given is 
therefore wrong. The correct value is given in the foregoing discussion in 
§17- 
