156, 157] ELECTRICAL IMAGES. 303 



us now form a function W which behaves at infinity like a 

 potential function, and is harmonic in all space, except that it is 

 discontinuous everywhere on the spherical segment S. (If it were 

 not for the discontinuity, such a function would vanish everywhere.) 

 Let us also form a function W defined by 



Multiplying by l/R = E/l, 

 ( W(af v' A-2 W '(x V ,)-*W 



r (% , y , % ) y ' \%, y> Z) ~ y ' 



If the values of the functions for points internal and external 

 to the sphere S be distinguished by the suffixes i and e, on the 

 surface S, since x = x' } y = y', z = z, 



(3) W t '=W t , W.'-Wt. 



W' (x, y, z) vanishes for / = oc and is finite for I = since 



(4) W (x, y,z) = j^W (V, y', z'} and 



lim W f =~ lim I' W (of, y', /) = const. 



Let us now put 



(5) V(x, y, z) = W(x, y, z) + W (x, y, z), 



and we shall show that the function W may be so defined that V 

 will be the potential of an equilibrium distribution on the spherical 

 segment. 



We have seen in 141 that the potential at any point due to 

 an equilibrium distribution on a circular disc of radius a is 



e (TT J\] 



-15- tan" 1 } , 



a (2 a } 



where X is the greater root of the quadratic 



The derivative of this function according to the normal of 

 course has a discontinuity by changing sign on crossing the disc. 

 If we consider the disc placed in the mouth of the bowl, on 



