MR. J. J. THOMSON ON THE VIBRATIONS ON A VORTEX RING, 
A reference to equation (17) will show that the vortex ring C D itself contributes 
nothing to this term, therefore 
y) — the part independent of 6 in the expression for the velocity due to the 
vortex ring A B along the radius vector of C D. 
Hence from equation (28) we have 
h=i ^Bm {W+&+m+K] 
If we integrate this equation, substitute for F, G, H, K their values as given in 
equation (28) and determine the arbitrary constant introduced by the integration, so 
that a 0 =0 when t — — co , we find 
a 0 =§mV 3 ct' 
—c 2 
2t>(e 3 +4r 2 sin 2 £ei 3 ) T 6r(c 2 + 4v 3 sin 2 1 (c 3 -f 4r 2 sin 2 -^ei 3 )® 
(1 + sin 2 |e) 
c cos \e.t 
(cos fe + 3 cos \e)t 3 cos + cos fe 
12c(c 2 + 4v 3 sin 2 ieb) Tj 
6 c 3 
Jc 3 + 4v~ sin 2 ^ei 2 )® ~2v sin 
This expression vanishes when t= — co ; it begins by being negative, so that the 
radius of C D is diminished at first when ^=0, the sign of a Q depends upon the value 
of e, if e be less than 60° it is certainly positive when t= 0 ; when t= <x> a 0 is positive, 
and its value is 
/ /o 
ma-a 
8rc 3 sin -^e 
(3 cos ^e+ cos fe) 
m'a'-a 
vc 3 sin 4e 
COS 8 -Jo. 
As this is positive, the vortex ring C D is bigger after the collision. The effect of 
the vortex ring C D on the vortex A B can be got as we saw before by writing 2n—e 
for e in the formula given above. We have thus the ultimate increase a 0 ' in the 
radius of A B given by 
, -j marcd cos 3 ^(2rr — e) 
a ° 2 c 3 sin ^(2-77-— e) 
or since a—a, m—m 
— _ i 
m'a' 3 a 
Oi..\ Q t? 1 
u * <? sm 
cos 3 -Je 
Hence the radius of the vortex ring A B is diminished by the collision. The effect 
of the collision on the size of the vortices is thus to increase the radius of the one 
which is in front when the vortex rings are nearest together, and decrease that of the 
one in the rear by ma 3 cos 3 -^e/2yc 3 sin -|e. Hence the alteration in the radius is, 
cceteris 'paribus , inversely proportional to the cube of the shortest distance between the 
vortices. 
