Electric Field in the neighbourhood of a Spherical Bowl. 41 



space between the bowl and the plane of its rim, the second 

 is obtuse when the image of the point lies within the same 

 space. In all other cases the angles are acute. 



§ 3. As an application of (2) we may calculate the capacity 

 of the bowl by writing down the potential at the centre of 

 the sphere, which obviously is equal to the charge of the 

 bowl divided by the radius of curvature. 



We find thus 



7r (. a a) 



or, writing 2a for the amplitude of the bowl with regard to 

 the centre, 



Y = ^\a + sma\=-. 

 ir ( > a 



Thus n E a(a + sin a) 



V, 



7T 



the same as the value given in Watson and Burburv's 

 < Electricity ' (p. 142). 



The same authors treat (§ 141) of the effect on the poten- 

 tial of making a small hole in a spherical or infinite plane 

 conductor. 



The result they arrive at needs a correction. From our 

 formula (2) it follows easily that, when c is small compared 

 with a, «i and s 2 the potential is given by the formula, 



v _Vo« Vo cV~a 2 ) 

 V ~ r + 6tt* s 3 a 2 ' 



where s represents the distance of the point from the centre 

 of the hole, while terms of higher order than the third are 

 neglected. 



It follows that the system is not equivalent to a complete 

 sphere charged to potential Y , together with an additional 

 charge on the aperture. 



Our equation gives without difficulty the charge induced 

 on the bowl by a quantity E of electricity concentrated at 

 any point in its neighbourhood. For, if we bring a charge 

 — Y a in the centre of the sphere, the bowl being formerly 

 charged to V and insulated, the potential of the bowl will 

 become null, while the distribution of its electricity remains 

 unchanged. At any point P where the potential had the 



value Y it will now be Y — Y . - ; thus Y represents the 



potential of the induced charge at the point P when 



