46 Proceedings of the Rotjal Irish Academy. 



The term e-l'^^ 



forms, in general, a rapidly converging series, so that we may write the 

 equation to determine y in the form 



y + Ay/ = 0, (56) 



where A is a numerical coefficient. 



l^ow, integrating over circular disk, we have 



47r X charge = TO j + 2irR%B\,K,{nR), 



jN'ow 1 

 ^B\K,{nR) = ^A,Ti{nR) + B,,K,{nR) - AyB^ — , 



' nH\ I ^ n-l 



47r /2 



The added charge is then 



y being determined as above. 



This will be correct, neglecting smalls of order M — • 



Jt 



This problem has also been discussed by Maxwell, the same general 

 remark applying as in previous. 



H. — Application to fluid Irrotational Motion. 



I. A thin cylindrical disk descends in a vertical cylinder of water, 

 to the axis of which its plane is perpendicular, the centre of the disk 

 lying on the axis. It is further supposed that the height of the 

 cylinder is large compared with the radius of the disk, and the breadth 

 large compared with the height. 



Let <j) be the velocity -potential of the fluid, and let where 2 is 



reckoned from the bottom of the cylinder, take the place of potential 



