[ 38 ] 



III. On the Potential of the Electric Field in the neighbour- 

 hood of a Spherical JBowl, charged or under influence. By 

 Dr. J. NlEUWENHUYZEN KllUSEMAN*. 



[Plate I.] 



THE contents of this paper were suggested by Sir W. 

 Thomson's memoir, " Determination of the Distribution 

 of Electricity on a circular segment of plane or spherical 

 conducting surface, under any given influence.'" 



It has been my aim to iind the potential at every point of 

 space in those cases in which Sir W. Thomson has given the 

 distribution of electricity. I found it possible to expand the 

 desired function as a series of spherical harmonics, and after- 

 wards to bring the expansion into a finite form. The result, 

 however, may be stated and proved by a very simple method 

 which I shall now proceed to describe. 



§ 1. When a very thin circular plate is charged with elec- 

 tricity to a given potential V , the equi potential surfaces are 

 spheroids, and the potential at every point of the surround- 

 ing space is given by the equation 



V= ?Io sin - 1 ^L ) (1) 



where c represents the radius of the plate, and s 1 and s 2 the 

 longest and shortest lines drawn from the point under con- 

 sideration to its circumference. Now the angle figuring in 

 this formula has a simple geometrical meaning. For, let in 

 fig. 1 the plane of the plate represent a plane through the 

 point P and the centre of the disk normal to the latter. 

 This plane cuts the circumference of the disk at the points 

 A and C. Thus we have 



AC = 2c, PA= 5l , PC = s 2 . 



Bisecting the angle APC by AD, drawing the perpendicular 

 DE on AC in the plane of the disk, it is easily seen that 



sinEPD=-^. 



The double of this angle EPD is the greatest angle which 

 any chord of the plate subtends as seen from P. Since it 

 frequently occurs in the following considerations, I venture to 



* Abstract from a paper in Verslagen en Mededcelingen der Koninhlyke 

 Academic van Wettenschajopen. Communicated by the Author. 



