304 ELECTROSTATICS. [FT. II. CH. VII. 



account of the change of ^-coordinate we must take X as the root 

 of the equation 



and if X' be the same function of x', y', z that X is of x, y, z we 

 must have 





v ~ 4 ' 



We will now define our function W by two different analytic 

 expressions. 



In the space T we take 



and in the space T' and T" 



This makes TF continuous at 2 as required, since on the disc 

 X = 0, and the change of sign in the second term makes the 

 normal derivative continuous in crossing the disc 2. By the 

 definition of W we have in T and T" (since the inverse of T is T', 

 and of T" is itself) 



and in T' 



Accordingly we have for the values of F in 



(9) r, F'=|-tan-.f H-fgn-tan-f), 



