Potential of the Electric Field near a Spherical Bowl. 39 



designate it by the more or less appropriate term of amplitude 

 of the disk with regard to P. 



Accepting this term, the following verbal expression may 

 be given to formula (1) : — 



When a thin circular disk is charged to potential 7r ? the 

 potential at every point of space is numerically equal to the 

 amplitude of the disk with regard to that point. 



Of course it is supposed that no other electrical charges are 

 present in the field. 



Now by Thomson's theory of electrical images, from every 

 function V, which satisfies Laplace's equation (A 2 V = 0) ? 

 another function which satisfies the same equation may be 

 derived by the following process of inversion. A point 

 within the space where A 2 V = is connected by a straight 

 line with the centre of any spherical surface of radius a ; let 

 the distance be p ; on the same radius another point is chosen 

 at a distance r from the centre, both distances being con- 

 nected by the relation pr = a 2 . This point is the image of the 



former. A function having the value -/at the image, while 



the particular value of the function Y at the original point 

 is/, satisfies Laplace's equation. 



§ 2. We shall apply this theorem to a function analogous to 

 the potential of a charged disk, but distinct from it in certain 

 particulars. Instead of the amplitude of a disk we shall 

 consider the amplitude of a segment of a spherical surface, 

 i. e. the amplitude of a bowl. This amplitude may be defined 

 thus : For nearly all points of space it is identical with that 

 of a disk having the same boundary ; there exists only a 

 difference for points situated in the space between the bowl 

 and the plane of its rim. For those points the amplitude of 

 the bowl is 2ir minus the amplitude of the rim ; thus it is 

 greater than 7r, much in the same way as the solid angle sub- 

 tended at the same points by the bowl exceeds 2ir. 



In the case of a disk the potential attains a maximum value 

 for points on the disk ; the function now under consideration 

 has its maximum on the bowl ; the differential-quotients of 

 both functions are identical as to their absolute value, but they 

 differ in sign for points within the space between the bowl 

 and the plane of its rim. Thus our new function satisfies 

 Laplace's equation throughout space ; it is discontinuous 

 with its differential-quotients only on the bowl, where it has 

 two values, those for outer and inner points supplementing 

 each other to 2ir. The equipotential surfaces are still sphe- 

 roids ; but to those which cut the segment belong two distinct 

 potentials. 



