SMALL VIBRATIONS OF A HOLLOW VORTEX. 
195 
The time of vibration may also be written in the forms 
UP 
2n 
+ 1 “t P 1l-\ 
V —P M _ X 
or 
lip 
‘m^/n 
which shows that the time is independent of the velocity of translation, a result which 
has important bearing on the theory that atoms of matter are hollow vortices. For 
the different orders of vibration, the time of vibration varies inversely as the square 
root of the number of crests running round the hollow. 
14. Pulsation of hollow .-—In the preceding case, n — 0 would correspond to pulsa¬ 
tions of the hollow, in which therefore the whole motion is a change of volume, and 
the use of the stream function is not allowable. But as it happens, the application of 
the velocity potential is here very easy. Let, as in Art. 13, the displacement be 
given by 
8k=£= 
Then the velocity potential is 
with 
Therefore 
C-c\* 
M 
<f>=\/ C—c2A„— cos nv 
* n 
£ a C — c b(b , 
t*——=-- — when u= 
lc L — c a ou 
u 
l —1^- M= - ' /f< ’ C) 2 iSP„+(C-c)^ [ cos nv 
whence the principal term is 
9«2 
A n- 
P, 
fi p 
SP 0 + 20 -f- - 
au 
M 
4P n 
Hence 
as before 
Therefore 
_ 2«V2& ^- 2 p o44/ .-i E 
= -4«V2^LM 
M 
<l>=-4a*y/2tfU/L^/(C-c)=? 
J- n 
■ , TT rfU . 
4 «V (2C)i*LM + = 0 
U 2 
M- 
Therefore time of pulsation 
4« 2 FL 
M = 0 
47 TClk flp / , 4\* 
~JL=%A log 7 
and therefore varies slowly with the energy. 
2 c 2 
