69] CYLINDER WITH CIRCULATION. 89 



The figure shews the lines of flow. At a distance from the origin they 

 approximate to the form of concentric circles, the disturbance due to the 

 cylinder becoming small in comparison with the cyclic motion. When, as in 

 the case represented, u>K/2na ) there is a point of zero velocity in the fluid. 

 The stream-line system has the same configuration in all cases, the only effect 

 of a change in the value of u being to alter the scale, relative to the diameter 

 of the cylinder. 



To calculate the effect of the fluid pressures on the cylinder when moving 

 in any manner we write 



where % is ^ ne angle which the direction of motion makes with the axis of x. 

 In the formula for the pressure [Art. 68 (i)] we must put, for r=a, 



and il? 2== iu 2 -f - 2 2 + ~ usin (0~x) (iii). 



The resultant force on the cylinder is found to be made up of a component 



in the direction of motion, and a component 



.................................... (v), 



at right angles, where m'=7rpa 2 as before. Hence if P, Q denote the 

 components of the extraneous forces, if any, in the directions of the tangent 

 and the normal to the path, respectively, the equations of motion of the 

 cylinder are 



(m+m-) = P, 



If there be no extraneous forces, u is constant, and writing 

 where R is the radius of curvature of the path, we find 



(vii). 

 The path is therefore a circle, described in the direction of the cyclic motion*. 



* Lord Rayleigh, "On the Irregular Flight of a Tennis Ball," Megs, of Math., 

 t. vii. (1878); Greenhill, ibid., t. ix., p. 113 (1880). 



