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



The thus defined function we are to invert, taking for 

 centre and radius of the inversion the centre and radius of 

 the sphere, of which the bowl forms a part, so that this latter 

 becomes its own image. 



The function obtained by the inversion will be discontinuous 

 for points on the bowl, since the original function is discon- 

 tinuous for the same points ; the inner and outer values have 

 only interchanged. 



And now- finally we may imagine a function the value of 

 which at any point is the algebraical sum of the values of 

 both the original and inverted functions. It is evident that 

 this new function has the following properties ; — 



1st. It is continuous for points on the bowl, as well as for 

 all other points. 



2nd. On the bowl it has the constant value 2ir. 



3rd. Its differential-quotients of the first order are con- 

 tinuous for all points of space, those situated on the bowl 

 making the only exception. 



4th. It satisfies Laplace's equation. 



5th. It vanishes at infinity. 



Now a function having these properties fulfils all the con- 

 ditions necessary and sufficient for the potential of the bowl. 



Thus the first part of the problem mentioned at the head of 

 this article is solved for the case that the bowl is charged to 

 potential 2tt. Had it been charged to any other potential V , 

 we should have had of course to multiply our function by 



the factor tt^* 



The algebraic expression of the function is then 



V=— -j sin * h -sin * y, 



where t x and t 2 are related to the image of the considered 

 point as s x and s % are to the point itself. 

 But from a figure it is seen at once that 



s i _ Sc , 

 ti ti 



Thus we have finally 



V f • _x 2c , a . r 2c \ ... 



— -< sin -+ -sin x > . . (2) 



7r I $i + s 2 r a s 1 -\-s 2 J v y 



With respect to the angles that appear in this formula it 

 follows from the foregoing considerations that, since the first 

 must be taken obtuse when the considered point lies in the 



