982 Dr. Gr. Breit on the Effective Capacity of 



This particular solution for z makes the centre of the 

 square coincident with the origin. Other solutions may 

 be obtained by adding a constant to the above. 



In the applications to the coil, it is desirable to transform r 

 in such a way as to make the real part of the new variable 

 (w = u+jv) have a constant value at the surface of the square. 

 This is accomplished by writing 



r = e u+ J l \ (24) 



so that (23) becomes 



it r- oo -(4a-l)(u+jv)-i 



z = A.'f [<■** + 2 ii ,._ i _ 1 _]. . . (25) 



If u = 0, |t|= 1 in virtue of (24) and, therefore, z is 

 on the unit circle. Thus (u, v) will now be curvilinear 

 co-ordinates in the plane (%,y). Only positive values of u 

 are considered because only positive values of \r\ — 1 

 are required. A curve corresponding to u = constant is 

 a closed curve. If w = 0, this curve is the square. If 

 w = co, r = co and therefore, as was previously shown, the 

 infinitely distant part of the z plane is attained. Further, 

 there is no difficulty in showing from (25) that larger 

 values of u correspond to curves enclosing those corre- 

 sponding to smaller values of u. If u is large, (25) becomes 



: = Ae t(+; v + ^ and for varying v represents a circle with 

 a radius Ae w . The value v = corresponds to a point 



subtending at the origin an angle j with the axis of 



reals. Similarly on the square u=0, v — gives a point 



subtending an angle - with the axis of reals at the origin, 



and the points v = ^, 7r, -~- are as indicated in fig. 11. 

 The value of A in terms of a is clearly 



ay/2 



1+2 



Ps 



(26) 



since for v = 0, ~=1+; ; 

 and other expressions may also be given. 



