AND THE ACTION OF TWO VORTICES IN A PERFECT FLUID. 
503 
or neglecting terms on the left-hand side which are not multiplied by the large 
quantity log = log A we find 
or 
! 8 O 3 —1) T 2 a - n 
pnaa» - -— -log —=& 
(19) 
Differentiating equation (15) with respect to the time and substituting for fS n from 
(16) we get 
/ />9 9/-A 2 
.( 20 ) 
— (i“ 7 l°g ~) (m s -!)«*«»=a" 
or 
a, 
,=A cos |^log y n.\/n 2 -~ l£+/3j.(21) 
/3,,=A sin |~log ^•ns/'rP—l.t+P 
( 22 ) 
where A and ft are arbitrary constants. 
These equations show that the circular vortex ring of indefinitely small section is 
stable for all displacements of its circular axis, and that the time of vibration for a 
displacement expressed by 
p=a-\-ot n cos nd 
is 
2 '7 t /~7 log ~.n. \/n 2 —1 
/ 2<x 3 & e 
2a 
(23) 
2a / 
If V be the velocity of translation of the vortex, viz.: we 2 log — 2a, the time of 
_ G / 
vibration is l 27ra/V.n\/n 2 —l. 
Sir W. Thomson has proved that the circular vortex ring is stable for all alterations 
in the shape of the cross section. If we combine this with the result just obtained we 
see that the circular vortex ring is stable for all possible displacements. Sir W. 
Thomson has also proved that for a displacement of the n th order in the shape of the 
cross section of the vortex arc the time of vibration =27 r/(n —l)w ; hence these vibra¬ 
tions begin by being much quicker than those we have been considering, but since for 
large values of n the latter are proportional to n 2 whilst the former are only propor¬ 
tional to n, the vibrations of a higher order will be quicker for the circular axis than 
for the core. When n is very great ^/n 2 — l=n, thus the amplitude of a n is equal to 
the amplitude of /?» and a } 2 -\-/3 n 2 = A 2 a constant quantity ; thus each point on the arc 
describes a circle about its mean position with an angular velocity (oe 2 n 2 log 2 a/eja 2 . 
3 T 
MDCCCLXXXII, 
