Multilayer Coils with Square and Circular Section. 979 



In (21), because the square root is a two-valued function, 

 and confusion will be caused unless it is known definitely 

 what branch of the function is used. In the following, the 

 only case of interest is that of 



|t| > 1. 



This excludes the possibility of negative values of the real 

 part of 



The square root will then be chosen so as to have its real 

 part always positive or zero. This involves a cut along the 

 negative of the axis of reals in the plane of 1— t"" 4 . Since, 

 however, the real part of 1 — t ~ 4 is never negative, no dis- 

 continuity in (1 — t~ 4 )* is introduced by the cut. 



Fig. 10. -Illustration of proof of conformal transformation of 

 the outside of a square. 



Consider now the value of ~ given by (21) in the case 



of t moving on the unit circle (see fig. 10) in the counter- 



dz 

 •clockwise direction starting at the point (1). At 1, — =0, 



dr 



which shows that 1 is a branch of the transformation (21). 



The velocity of motion of z is therefore zero as t approaches 1 



with a constant velocity. But if t is slightly different from 1 



and lies on the unit circle, say if r occupies the position B 



(fig. 10), then the point t~ 4 has the position Q on the same 



circle such that 4ZB01=Z10Q. 



The stroke Ql represents then in magnitude and direction 



the quantity 1 — r -4 . Now the angle which this stroke makes 



with the positive direction of the axis of: a? is readily shown 



to be 



J-2ZB01. 



3 H2 



