240 On the Potential of a Symmetrical System. 



It follows that the expansion 



{cq-Ci) + {c 1 -c 2 )T 2 -\- . . . -±-(c n -c n+l )F 2)i + . . . 



has the value + jn if ^>0, and the value — fM if /a<0. 

 Consequently Y = V 1 at r = a, by what has been explained 

 before. 



Consider next the value of ( -^— - — ■-— ° ) at r — a\ this 



should also vanish, since there is no surface-density on the 

 sphere. We find that its value is 



2'7t/d[— /* + c 1 + ; (2c 1 '-f-'3c s }P 2 + . . . 



+ {2nc n +(2n + l)c n+l }F 2n i- . . .] (n>0), 

 or 



27r / o[ + ^ + c i +(2c 1 + 3c 2 )P 2 + . . . 



+ {2n<?„+(2w + l)c„+i}P 2re + . . .]. (^<0) 



Each of these expressions vanishes, according to the value 

 found above for A 2?J . 

 Hence 



Br d** 



Now Vj, V satisfy the same differential equation of the 

 second order (Laplace's), 



and, at r = a. V = Vi and -^r^ = "^ - f° r a U values of 9 



between and 7r. It follows that V must be the analytical 

 continuation of Vg beyond the sphere r=a; and the dis- 

 continuity at r = a is only apparent, not real. A similar 

 point occurs in connexion with the magnetic potential of 

 a circular coil, carrying an electric current; the expressions 

 for this are given in the same article of Thomson and Tait. 

 The numerical details are slightly different, but the principle 

 involved is exactly the same as in the above work. 



S. John's College, Cambridge. 

 26th Jane, 1901. 



