42 Dr. J. N. Kruseman on the Potential of the 



the bowl is electrically connected with the earth and in its 

 centre of curvature a charge — V a is concentrated. 



When we now call to mind the reciprocal property of 

 Green's function, we know at the same time the potential at 

 the centre when the inducing charge is at P, and from that 

 we may derive at once the induced charge of the bowl by 

 multiplying by a. 



Writing E for — Y a, we find thus for that charge 



n Ef. 2c a . Y r 2c 1 * 



0= ^ sin -1 h -sin- 1 -. J- . 



it (, *i'+*s r a 5i4-s 2 J 



§4. As a last application of (2) we proceed now to find the 

 distribution of the electricity on the insulated and charged 

 bowl. 



To this effect we choose two points situated on the same 

 radius, equally distant from the bowl. When their distance 

 from it is taken infinitely small (Br), then these points are 

 each other's images. The outer point we designate by A, 

 the inner one by B. For the amplitudes of the bowl with 

 regard to the point where it is cut by the line AB, we write 

 u and 2it — u, u being the outer and 2it — u the inner value. 



We have thus, 



for the amplitude at A, u + -k-' Sr, 

 ■or 



„ „ „ B,27r-(u-^8rj. 



The inversion gives, 



for the function at A, - — k-< 2ir—u-\- — 8r >, 

 a + br I br J ' 



Thus for the potential, when the bowl is charged to potential 

 2tt, 



at A, 27T—dr< 2-*- f , 



L a br ) 



{a br J 



For the surface-densities on the convex and concave sides 



* Compare this result with Theorem II. Chapter iii. in Maxwell's 

 ' Electricity.' 



