ME. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
1097 
§ 44. Vortex Ring app7vaching a Fixed Plane. 
This question is at once reduced to the case of two vortex rings on the same axis, 
by taking the image of the ring in the plane. 
Let 2 be its distance from the plane at any moment. 
The equation of energy gives 
c' cos (f) d(fj 
+ 4c2 sill' 
where Cg, Wg, are the values of c and a when the ring is at a great distance from the 
plane. 
Put 
Then 
V,s + ,.(F-E) + ^,F = c,(log^"-i 
( 111 ). 
where / v s is the modulus of the elliptic functions. 
VC' +2' ^ 
This equation gives the relation between tlie radius of the ring and its distance 
from the plane. 
Put 
~ 7 === = sin 6. 
\/c' + 2^ 
MDCCCXCIII.—A 
7 A 
