157, 158] ELECTRICAL IMAGES. 307 



which being partially differentiated gives 



[a 2 + 2X - O 2 + if) -(z- c) 2 ] ^ = 2X#, 



(22) [a 2 + 2X - O 2 4- f) -(z- c) 2 ] 7:- = 2Xy, 



[a 2 + 2X - O 2 + y 2 ) - - c) 2 ] ^- = 2 (a 2 + X) (5 - c). 



Multiplying these respectively by at/R, y/R, zjR and adding 

 gives 



Pr\ 9~X IY<r> 2 _t- tft -L- f z \ rv\ -4- 9/^ 2 <?Y y r\ 

 . . C/A< ^iA/ I oC "t" 7 i ^ / v* ~r *** I* O I 

 /2^J\ = 1 , 



Putting now 2 + ^/ 2 + z* = R* = a 2 + c 2 and using the quadratic 

 (21), the numerator 2 {X (R* cz) + aV cPcz], becomes equal to 

 the denominator, 2R (X + cz c 2 ), multiplied by (a 2 + \)/R so that 

 finally 



o\ 9X tt 2 + X 



(24) = = ^ , 



dn e dn{ R 



and for the density within and without we obtain 



1 ^ tan" 1 - ^= 



lj 6^ 



(25) 



1 8F' 1 (IT 



^-tan-^l. 



Vx Vxj 



The difference of the densities within and without is 

 The smaller the opening of the bowl, the smaller is the density 

 within. At the edges of the bowl, where X becomes zero, the 

 density is infinite, as in the case of the circular disc, but the 

 capacity in either case remains finite. 



15S. Application of the Conformal Representation to 

 two-dimensional problems. In cases where the densities of a 

 distribution are the same at all points situated on the same line 

 parallel to a given direction, as for instance, in the case of the 



202 



