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XV. Researches on the Theory of Vortex Rings. —Part II. 
By W. M. Hicks, M.A., F.R.S., Principal and Professor of Mathematics and 
Physics in Firth College , Sheffield. 
Received June 13,—Read June 18, 1885. 
Contents. 
n t t-. • PAGE 
section 1.—Preliminary 
§ 1.—Stream function .... . 729 
2. —Equation to boundary of core. .. 730 
3. —The circulations . 732 
4. —Pressure inside a vortex mass in steady motion. 733 
Section II.—Steady motion 
§ 5.—Number of conditions and constants at disposal...... 733 
6. —Stream function and velocities of external motion .... 734 
7. —Stream function and velocities of internal motion .... 737 
8. —Pressure conditions and determination of constants of 
motion . 741 
9. —Relative position of hollow to outer boundary of core.. 750 
10. —Connexion between sizes of aperture and cross-sections 
of the core and hollow.. 752 
11. —Velocity of translation. 755 
12. —Energy.. 75 7 
Section III.—Small vibrations and stability 
§ 13.—Statement of conditions... 759 
14. —Stream or flux function for vibration . 760 
15. —Flow function . 762 
16. —Pressure conditions. 764 
17. —Cases of no rotational core and no hollow . 767 
18. —Case of no inner additional circulation. 768 
19. —General case... 773 
20. —Pulsations. 775 
The present communication forms a continuation of some researches the first part of 
which was published in Part I. of the Transactions for 1884. # In that paper was 
considered the case of a circular hollow with cyclic motion round it. In the 
following pages the more general case is investigated where the core is of different 
density from that of the surrounding fluid, has a hollow inside it, and circulations 
additional to that due to the rotational filaments actually present. The investigation 
* References to this are in square brackets, thus [I. 5]. [T. F.] refers to a paper on “ Toroidal 
Functions” in Phil. Trans., Vol. 172 (1881), containing the theory of the functions used. 
MDCCCLXXXV. 5 A 
