Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/44

is expanded in ascending powers of $\mathrm {R} /c$, 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 applications.

In Section II. 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 ${\tfrac {1}{5}},{\tfrac {2}{5}},{\tfrac {3}{5}},{\tfrac {4}{5}},1$ 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 annul us is small compared with its mean radius.

It is shown that the form of the cross section may be represented by $\mathrm {R} =a\left(1+\beta _{2}\cos 2\chi +\beta _{3}\cos 3\chi \,+,\,etc.\right)$, where $\beta _{2},\beta _{3},etc.,$ are of the second, third, &c., order in $a/c$. Their values are found as far as $\left(a/c\right)^{4}$.

To the second order

${\frac {\omega ^{2}}{\pi \rho }}=\left({\frac {a}{c}}\right)^{2}\left(\log {\frac {8c}{a}}-{\frac {5}{4}}\right),$

$\beta _{2}={\frac {5}{8}}\left({\frac {a}{c}}\right)^{2}\left(\log {\frac {8c}{a}}-{\frac {17}{12}}\right).$

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 Ii and the quantities 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.