1100 MR. P. W. DYSON OX THE POTEXTIAL OF AX AXCHOR RING. 
§ 45. Vortex ring passing over a Fixed Spherical Obstacle. 
When a vortex ring lies on a sphere concentric with a sphere in the fluid, the 
method of images may be applied, as is shown by Mr. Lewis, ‘ vol. 16, p. 338. 
Let m .2 be the strength of the external ring 
C 3 its radius, 
the radius of its cross-section, 
23 the distance of its centre along OZ. 
Let nil, corresponding quantities for the image 
Then 
w.j = — ni2 
fi 
a 
c 
2 
a 
2 
At any external point P (coor. s, cr, </>), 
= WI 2 1 
Jo 
c^us cos ^ 
0v/{(2 — 22)" -fro' — 2 c 5 -C 2 cos ( f ) + Cj-} 
mo 
q Jo 
CjOT cos (f) 
v/ {(^ ~ -fro® — 2ti7q cos ^ + q-} * 
The equation of energy becomes 
