44 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
is expanded in ascending powers of R/c, and the expansions of the integrals, obtained 
by differentiating this with respect to c, are deduced. Section I. is devoted to the 
discussion of these functions, and some similar ones, needed in hydrodynamic 
apj)lication3. 
In Section IL the potential of an anchor ring at all external points is found in a 
very convergent series of integrals. The expansions of Section I, are not needed ; but 
the first few terms are reduced to elliptic integrals. The equipotential surfaces are 
drawn for the ratios y, f, |, f, 1 of the thickness of the ring to its mean diameter. 
In Section III. the potential of a conductor, in the form of an anchor ring, is found 
at external points ; the surface density at any point of the ring and the charge are 
also determined. Section IV. consists of a discussion of the motion of an anchor 
ring in an infinite fluid ; the velocity potential or stream line function for motion 
parallel to the axis, perpendicular to the axis, &c., being first determined. The 
kinetic energy is determined in the several cases; and in the case of the cyclic motion 
through the ring, the linear momentum. In this last case, the solid angle subtended 
by a circle at a point near a circumference, is incidentally found. 
In Section V. the annular form of rotating fluid is discussed, when the thickness of 
the annulus is small compared with its mean radius. 
It is shown that the form of the cross section may be represented by 
R = a (1+^2 cos 2^ -f ySg cos 3y +, &c.), where ySg, /3^, &c., are of the second, third, 
&c., order in ajc. Their values are found as far as (a/c)h 
To the second order 
The method employed throughout the paper has not the analytical elegance of 
Mr. Hicks’ Toroidal Functions, but it has many advantages. The potential of an 
attracting ring takes a very simple form. The boundary conditions to be satisfied in 
hydrodynamical applications are very simple. The results are obtained in terms of 
R and the c|uantities most obviously connected with a ring. 
I am greatly indebted to Mr. Herman, Fellow and Assistant Tutor of Trinity 
College, Cambridge, for the careful manner in which he has read over much of the 
work, and the many errors he has corrected. In consequence, the paper will, I think, 
be found free from any serious mistakes. 
