506 MR. j. J. THOMSON ON THE VIBRATIONS OF A VORTEX RING, 
Substituting in the expressions for A 0 and A x , we find on neglecting small quantities 
of an order higher than ci 2 /f 2 -\-h 2 
1 
A.= 
« 2 
_ a ^ ■ i - a~(h sin e+/cos e ) 2 q c/ 2 (3 / 2 — 2A 2 ) 
0 (f- + W)* 4 (/ 3 + A 3 ) i + 4 <J* + Wf 
„ oa(h sin e+/co s e) a-(hmi e+/oos e) s 
C0S& (7™ + COS 20- -^5-^5- 
p'U= 
oct p 
O^+r) 1 
_ _oa/f 2 15 aV/ 2 1 n ^a'a?f*(h sin e +/cos e) 2 3aV(l— £sin 3 e) 
(/ 3 +Ay 2 ff 3 +A¥ i ~ 4 f/ 2 +/, 2 v* ■* 
(/ 3 +A 2 ) 
15 a'a 2 f(1i sin e +/cos e) 
(/ 2 + A 2 )’ 
(/ 2 + A 2 )’ 
§aa'f cos 
Tf + Wf 
, a [ Qaa'f cos e 15 aa'f 2 (h sin e -F/cos e) 
T" 008 “ 1 / /-O VlOvi-, *, . ,nvl 
(/ 2 +^) 1 
A sin e + 
(/ 2 +A 2 )* 
f 10 s aV/ 2 (A sine -f/cos e) 2 
15aA 2 /cos e(A sin e +/cos e) 
/ o • o ^ 
3 war sin? e ] 
[ 4 (/ 2 + A 2 ) f 
(/ 2 + A 2 )* 
2 (/ 2 +^)‘j 
+ cos 2# 
Although for reference we give the complete values of A 0 and A x to the order of 
approximation we are working to, yet when we have in the expressions for the velocities 
a coefficient consisting of terms of differerent orders, we shall only retain the largest 
term. 
If we do this we find 
y=i 
mV 2 
(/ 2 + A 2 ) 
T^-P) 
11V Ci! ^ 
+t cos ^ (/8+ ^I (/~(/ cos e +3/i sin e) — 2A 3 (2/‘cos e+A sin e)) 
+f cos 2 9 
m'a'' 2 ci? 
TF+if) 
jj-K/oos e+h sin € )3--3 a/ ( /^ y+^ me ) +5 y cos e (/ cos e +A s i n 
i sin 3 e(/ 3 +A 3 )j 
+ 
From the formulae ( 1 ) and (3) we find the velocity along p 
=JmV£ / A 1 
_ 3 m'a'Zpfc' 
~V a +£'*)* 
Hence a, the velocity along af at the vortex C D due to the vortex A B, 
m'a’~£'( f+ci cos e cos 6) 
3 mV 2 /A 
2 (/ 2 + A 2 ) 
m'o! 2 a 
+ cos (/2 + ^ r j ( h cos e +/sin e) 
5/A(A sin €+■/ cos e)' 
p + h~ 
-j- cos 20.§ 
f /-7 O 
(/ 2 + A 2 )U 4 
3 5 (A sin €+/cos e)~fli 5 (A cos e +/sin e)(A sin e +/ cos e) } . 
(P+WY 
C/ 8 + A 3 ) 
+A sin e cos e 
