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



to an arbitrary sphere, they will remain each other's inverted 

 functions with respect to the image of the first sphere. 



Or otherwise : When a function is inverted several times 

 in succession the resulting function is independent of the 

 order in which the inversions are effected. 



This theorem may easily be proved with the aid of a simple 

 figure (fig. 2) and a very small amount of calculation. Even 

 without proof it will readily be accepted, being rather obvious. 



We apply it to the potential of the spherical bowl, which 

 we suppose to be a part of the spherical surface, represented 

 (in section) by fig. 2. 



The potential of A consists of two parts, which are each 

 other's inverted functions with regard to the sphere ; they will 

 continue to be so related after an inversion with P for centre, 

 and the tangent C from P to the sphere for radius. The first 

 term is the amplitude of the bowl at A multiplied by the factor 



tt^. Bv the inversion the lines drawn from A to the rim of the 



2tt j 



bowl become circular arcs through P B (the image of A with 

 respect to the sphere), and points of the rim of a new bowl, 

 the image of the first with respect to P : this of course forms 

 a part of the same sphere, the latter remaining unchanged 

 after the inversion- It is this new bowl with which we pre- 

 sently shall have to deal. From the elementary theory of the 

 reciprocal radii, it follows that the angles formed by the cir- 

 cular arcs, referred to above, are the same as those between 

 the lines of which they are the inversion. For shortness' 

 sake I would venture to call the greatest of these the circular 

 amplitude of the new bowl with respect to P and B. I shall 

 designate it by the symbol 2d. 



The first term of the inverted potential now becomes 



*27T PB 



The second term is the inverted function of that represented 

 by the first term with respect to the sphere. Thus it has at 

 the point B the value that the first term has at B', multiplied 



bv -FTPi • We mav write for it 

 J OB 



v 2#' _b a_ 



V ° , 27r'PB / '0B ; 



where 20' represents the circular amplitude of the new bowl 

 with respect to P and B'. In the new system the bowl has 



