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VII. On the Steady Motion and Small Vibrations of a Hollow Vortex. 
By W. M. Hicks, M.A., Felloiv of St. John’s College , Cambridge. 
Communicated by J. W. L. Glaisher, M.A., F.Ii.S. 
Received May 31,—Read June 21, 1883. 
Contents. 
PAGE. 
Introduction. 161 
Section I.—The stream and velocity functions. 164 
§ 1.—Equation of conjugate toroidal functions . 164 
2. —The cyclic constant... 166 
3. —Velocity potential for given normal motion . 167 
4. —Expansions of P, Q, R, T . 171 
Section II.—Motion of rigid tore. 172 
§ 5.—Stream function for the cyclic motion... 172 
6. —Stream function for translation. 173 
7. -—Amount of fluid carried forward . 175 
8. —Energy of motion .... 177 
Section III.—Steady motion of hollow vortex... 179 
§ 9.—First approximation to velocity of translation, and 
surface velocity . 179 
10. —Stream functions for a tore whose section deviates 
slightly from a circle... 183 
11. —Second approximation to the form of the hollow .... 185 
12. —Energy of vortex. 190 
13. —Waves round hollow. Stability of hollow. 191 
14. —Pulsations of hollow . 195 
The following pages form a continuation of some researches commenced about 
three years ago, but which the author was compelled by other engagements to lay 
aside until the beginning of the present year. The general theory of the functions 
employed was published in the Transactions of this Society (Part III., 1881), under 
the title of “ Toroidal Functions.” These and analogous functions are employed in the 
present communication, and references in square brackets, with the letters T.F., refer 
to this paper. Since it was written I have found that Carl Neumann had already 
given the general transformation [T.F. §1] by means of conjugate functions, in a 
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