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II. Notes on Hydrodynamics. 

 By Professor A. Gray, F.R.S.* 



II. Determination of Translational Velocity of a Vortex Ring 

 of Small Cross-section. 



1. A VALUE for the velocity of displacement of a vortex 

 J\. ring in an unlimited fluid was given without proof 

 in a note by Lord Kelvin appended to Professor Tait's trans- 

 lation of Helmholtz's celebrated paper (Phil. Mag. 1867). 

 A demonstration of the result was promised in the note, but, 

 so far as I know, Lord Kelvin never published it. Several 

 investigations have since been published with results differing 

 slightly from Lord Kelvin's value, which, however, has been 

 confirmed by Hicks (Phil. Trans, 176) and by Lamb 

 (Hydrodynamics, 3rd ed. p. 227). The following direct and 

 elementary proof (in which no use is made of elliptic in- 

 tegrals as such) may possibly be of interest. 



It is well known that the velocity at any point P in a 

 frictionless incompressible fluid, in which exists a single 

 vortex filament of any form, may be found in the following 

 manner. Let k be the strength of the vortex, that is double 

 the (constant) product of the elemental angular velocity 

 in the filament at any point by the area of cross-section 

 there, r the distance of the point P considered from 

 an element E, of the filament, of length ds, and 6 the 

 complement of the angle between the directions of the 

 element E and the line EP. The velocity Bq at P due to 

 the element E may then be taken as given by the equation 



~ x ds . cos 



s *=i^ — " d) 



The direction of Bq is at right angles to the plane determined 

 by the element E and the line EP, and the flow is towards 

 the side of the plane specified by the rule given below. 



This rule applied to all the elements of the filament, gives 

 the velocity at P as the resultant of all the vectors Sq given 

 by the elements composing the complete filament. Of 

 course there might be added to the right hand side of (1) 

 any term of: proper dimensions for which the integration 

 round the closed filament gives a zero vector. 



The theorem here stated is the analogue of that by which 

 in electromagnetism the magnetic force at any point due to 

 a linear circuit may be found. There, if y be the current in 



* Communicated bv the Author. 



