514 MR. J. J. THOMSON ON THE VIBRATIONS ON A VORTEX RING, 
The vortex C D itself contributes to this coefficient the term 
o m , 2 a 
lo s 7-“* 
The vortex A B contributes the term 
(AV+BV+OT+DT+E') 
/ 0 /o 
mara!* 
2 i 
say 
Thus 
(c 3 + 4v 3 sin 3 
F» 
f =»5 h Vn 
Eliminating /V we find 
Oil 
or 
^ 1 
L 3m V 
dt 3 
^ n 4 \ 
logTt“ S =/'0-7 lo gT- F 0 
say 
or writing n 3 for 
the equation takes the form 
=x(0 
^(log 
« 4 \ & <3 / 
d> 
CLci 
A+ B A=x(0 
The solution of this differential equation is 
a . 2 =A cos w£-j-B sin n£-f- C °A^ f x(0 S4n 
sin nt 
n 
fx(0 
n 
cos nt'dt' 
cl/CCcy 
or choosing the arbitrary constants so that a 2 and ~ both vanish when t = 
find 
cos nt[ 1 , ,\ . , 7 , sin nt[ l . n , , , 
—— I x\t ) sin nt at —- Xv ) cos ni dt 
co we 
n 
n 
The complete value of y(£) is given by the equation 
x(0= 
x mW°-( 3 Ff‘ + 2 G't + H') 
4 (c 3 + 4v 2 sin 3 -^ei 3 ) 1 
21 mV 3 a?; 3 sin 3 ^e(E'£ 4 + G-T 3 + HT 3 + K't) 
(c 3 + 4v 2 sin 3 \e.t 2 )^ 
2 m , 2 a g m'a' 2 a 2 (A't i + B^ 3 + CT 3 + ~D't + E') 
a 3 ° & e ’ 2 (c 3 + 4v 2 sin 3 ^ei 3 )“ 
