1080 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Section IV. 
§ 29. The motion of two or more vortex rings on the same axis is most easily dis¬ 
cussed by means of the stream-line function. The determination of this function is 
easily reduced to the determination of the potential of a distribution of gravitating 
matter, so that the preceding methods apply to this problem. 
The Steady Motion of a Vortex ring of Finite Cross-section. 
As before, let the figure represent a section through the axis of the ring. 
Let 0 be the centre of the ring, Oz the axis, C the centre of gravity of the cross- 
section. 
z 
Let OC = c : the cross-section is nearly circular; let its equation be 
L = « (1 -p cos X + /3o cos 2 ;)( -f &c.) . . . 
Avhere &c., are small quantities. 
Since C is the centre of gravity of the cross-section 
• (56), 
r'ZTT rR 
Pt cos X Pi- H c?Y = 0 . 
Jo Jo 
Therefore 
or 
(1 -f /3i cos X + / 8 o cos 2 y -f . . .cos x dx = 0 , 
0 
(^1 + A + A + /Sg +... = 0 . . 
Now it will be shown that 
A is of the second order in ajc ; 
A of the third, &c. 
(57). 
