Electric Field in the neighbourhood of a Spherical Bowl. 47 



parameters naturally belonging to the figure in question. 

 We adopt as such : — 



The perpendicular p from B on the plane of the circle. 



The perpendicular p x from P on the same plane. 



The perpendicular q from the centre of the sphere (5) on 



this plane. 

 The perpendicular q^ on this same plane from the centre of 



a sphere through P and the circumference. 

 The radius c of the circle. 

 The distance R from B to P. 



It is easily seen that these six independent quantities are 

 sufficient to define the figure. 



The following relations exist between the old and the new 

 parameters :— pi =p _ A/ p> ^ 



q =p-F, 

 c*=Y 2 -q 2 , 

 R(2w-R) 



With the aid of these we transform the formulae (6), (7), and 

 (8) into the following : — 



W+? 2 )' ' 



Vpi 2 (?-?i) 2 



c 2 R 4 - 4/?V(<7 - <7i V 2 + 4R 2 #pi Wi + ^RV 



(6a) 



&(* + ?) ' (8A) 



And, finally, writing k 2 for the common factor, 4p 2 (c 2 + £ 2 ), 



/ca=R 2 c, 

 fch=2pp 1 (q — q l ), 

 K 2 (b 2 -a 2 + A 2 ) =4,R 2 pp l (qq 1 + c 2 ). 

 When we now return to fig. 1, and suppose it to be the 

 inverted figure,- we see that 



/i = PF, Z>=OF, a = 00; 

 and j as before, we set 



PA = 5 1? PC = 5 2 . 



Thus s *=(b + a) 2 + h 2 , s 2 2 =(b-a) 2 + h 2 , . 

 s* + s 2 = 2(a 2 + b 2 + h 2 ), 

 5l 2 + S2 2_ 4a 2 = 2 (6 2 -a 2 + A 2 ), 

 2^2 = 2 V(a 2 + 6 2 + A 2 f-4a 2 6 2 = 2 V(6 2 -a 2 + A 2 ) 2 + 4a 2 A 2 . 



