516 Prof. J. J. Thomson on the Relation between the 



with irrotat ion ally moving liquid, the pressure over a cross 

 section of the cylinder will depend upon the distribution of 

 the vortex filaments in the cylinder. Let the cylinder be a 

 right circular one. Let v be the velocity, and p the pressure 

 at a distance r from its axis, p the density of the liquid ; then 

 we have 



v 



_ dp 



n=d? •••••• (o 



The pressure over the cross section of the cylinder is equal to 



>b 



tirvp dr, 

 o 



where b is the radius of the cylinder. Integrating this by 

 parts, we find that the pressure II over the cross section is 

 given by the equation 



1. 



Jo dr 



= Ptt& 2 - \\pv 2 2irrdr .... (2) 



Jo 



by equation (1), P is the pressure at the surface of the cylinder. 

 The tension A exerted by the cylinder on the solid is given 

 by the equation 



A = P7rZ/ 2 -II 



C b 



= I ^pv 2 2irrdr (3) 



= kinetic energy per unit length of the cylinder. (4) 



Since 



27m; = vorti city inside a circle of radius r, 



we can, if we know the distribution of vorticity, easily cal- 

 culate by means of equation (3) the value of A. 



Let us suppose that if all the vortex filaments were collected 

 round the axis to the exclusion of the irrotationallv moving 

 liquid, they would occupy a cylinder of radius a. Let f be 

 the rotation in the vortex filament, and let 



J 2wr£dr=m. 



Jo 



Then we find that when the vortex filaments are as close to 

 the axis of the cylinder as possible, 



A pm 2 , pm 2 , b 



A= T77- + ^l— log-. 



107T 4tt °a 



