978 Dr. G. Breit on the Effective Capacity of 



at AB, !~ -= — ;— . Hence at CD the part of a involving 

 On ay 



2a 4tt d? 



and at AB this part of cr is 



/c e L di 

 + 2a&rdt 

 But the potential of DC exceeds the potential of AB by 

 T di 



Therefore the effect of this part of <r is precisely the same 

 as would be caused by a condenser of capacity 



K e l 



47T ' 



This part of the effective capacity can be interpreted as the 

 capacity between two conducting sheets separated by a 

 medium of specific inductive capacity K e at a distance 2a, 

 the area of each sheet being 2al. The contribution of the 

 above quantity to C will be called 



C '=r- Z = 0-0796 *.Z (20) 



Now V must be found. For this purpose it is necessary 

 to solve the potential problem in two dimensions for the 

 case of a square. This can be done b}^ means of elliptic 

 functions in various ways. The one given here is con- 

 venient on account of the fact that rapidly converging 

 series are obtained for the results. Also it can be understood 

 without any knowledge of elliptic functions. 



Let z^x+jy (j =v /—l), 



and consider the transformation defined by the differential 

 equation 



J=Ae%-T-4)*, .... (21) 



where A is a constant and e is the natural base. It will be 

 shown presently that this transformation has the property of 

 transforming the unit circle in the r plane into a square in 

 the z plane. Also it transforms the region outside the unit 

 circle into the region outside the square. 



Before proving the above-mentioned properties, a few 

 words must be said as to the meaning of the square root 



