Electric Field in the neighbourhood of a Spherical Bowl. 43 

 of the bowl we find then, 



C or a J 



Now we have 



2c 



u— 2 sin -1 , ds, ~ ^- Sr. ds.> = 'J- &r. 



Si + s 2 ? * 2a ' 2 2a 



Thus 



Su 2c 



8r a x /( Sl ^-s 2 y 2 -Ac 2 ' 



Vo 

 2lT 



And, finally, multiplying by — - to make the potential on the 



A/7T 



bowl Y instead of 2ir : 



V n V r 2c . - 2c 1 



4tt(j= -^ + — ^ , — snr 1 \ , 



ira ira (_ ^(s^^) 2 — 4c 2 s 1 -\-s 2 ) 



Ynf 2c . , 2c 



7™ I */(»! + . 



Assuming polar coordinates, taking the centre of the bowl 

 for pole and its centre of curvature for origin, then 



a I v'f^+so) 2 — 4c 2 Si + s 2 J 



. a + r) . a— r) 



Sj = 2asm— -~ } s 2 = 2a sm— r , 



• o 2c /l + cos a 



sm zw = 



Si ■+ S 2 V 1 + COS q 



tan2u= 1+cosa 



\/(.si + s 2 ) 2 — 4c 2 v cos?;— cos a 

 Substituting these values, we have 



W= -"(a/ -1+cojjr ^tan-i A / 1+cm. 1 



7Ta L V cos?? — cos a V cos 77— cos a J 



and for 47rcr the same value with the addition of — -. 



ira 



These are Thomson's expressions for the density (Re- 

 print, p. 185). 



§ 5. To solve the second part of our problem, we shall make 

 use of the following theorem : — 



When two functions, being so related that by inversion 

 their values interchange, are inverted once more with regard 



