Field of Cylindrical Currents. 433 



Hence the total kinetic energy of the system 'J^+T^ equals 



W 



2pm 2 a I log — 1*75 J . 



This is the formula given by Lamb*. By means o£ (22) 

 we could have written it clown at once. The formula,, 

 however, Lamb gives for the velocity of translation of a 

 vortex ring is 



m /, 8a 1\ /7 \ . 



This formula is due to Lord Kelvin J. In Basset's 

 ' Hydrodynamics ' § both formulae are given. The author 

 thinks that when r/a is small formula (a) must be the more 

 accurate. If this be not the case, then the electrical analogy 

 must break down at points contiguous to the filament. 

 Chree ||, in his paper 'Vortex Rings in a Compressible 

 Fluid,' gives formula (a).. He also points out reasons for 

 the slight divergences in the formulae for the motion of vortex 

 rings given by various physicists. In another important 

 paper H by the same author, the similarity of the equations of 

 vorticity to those of electrodynamics is shown very clearly. 

 When the compressibility of the fluid is considered, the 

 hydrodynamical problem is much the more difficult. 



6. The magnetic force due to a cylindrical current sheet. 



We shall suppose that there are n bands of current per 

 unit length. We shall first find the magnetic force at a point 

 P in a plane through the base of the cylinder and at a 

 distance r from the axis. Let the length of the axis of the 

 cylinder be I and its radius a, Then, by (10), 



i=2taj j 



r cos d> ndz , , 



-r-.#, 



'o Jo (~ 2 + a 2 + r 2 — 2arcoS(/>) 3 ' 2 ' I 

 where i is the current in a filament. 

 Hence ~ 2ni C" a(a — r cos 6) r z "I l 7 , 



z= ~Jo — * — Lf+a^v* 



, .f'a(a— r cos <f>) 



where A 2 = a 2 + r 2 — 2ar cos (/>. 



* ' Hydrodynamics,' p. 227, 3rd edition, 

 t ' Hydrodynamics,' p. 227. [Note that K=2m.] 

 X Phil. Mag. [4] vol. xxxiii. Supp. p. 511 (1867). 

 § Vol. ii., (5) p. 62, and (86) p. 87. 

 || Proc. Edin. Math. Soc. vol. vi. p. 59 (1887). 

 II Proc. Edin. Math. Soc. vol. viii. p. 43 (1889). 

 Phil. Mag. S. 6. Vol. 13. No. 76. April 1907. 2 H 



