Multilayer Coils with Square and Circular Section. 981 



Thus as t moves from j to — 1, z moves in a straight line 

 parallel to the axis of pure imaginaries and in the negative 

 sense. 



Similarly it may be shown that as t moves from — 1 

 to — j i z moves in a straight line parallel to the axis of reals 

 and in the positive sense. Finally, as t moves from — j 

 to + 1, z moves in a straight line parallel to the axis of pure 

 imaginaries in the positive sense. 



It will now be shown that as t returns to the value 1, 

 z returns to its original value. In fact by (21), if 



T 



' = e 



n r, 



dz 



= 



dz 

 dr 



Therefore, the length of the straight line corresponding to 

 the values of t between 1 and j is the same as that of the 

 straight line corresponding to the values of r between j and 

 — 1. Similarly the remaining two straight lines are equal 

 to the ones mentioned above. Thus the figure is a convex 

 rectangle with all sides equal, and hence is closed and is a 

 square. 



From the finiteness and continuity of the functions on 



dz 

 the right-hand side of (21), it follows that -=- is finite, 



single-valued and continuous for t>-1. Also if |t| is 



large, ~ approaches e 4 . Therefore, changes in t call for 



equal (in absolute value) changes in z. Hence parts of 

 the t plane lying at an infinite distance from the origin 

 correspond to parts of the z plane also lying at an infinite 

 distance from the origin. 



If |t]>1, the right-hand side of (21) may be expanded 

 by the binomial theorem, viz., 



(22) 



■=- = Ae J ± 1— Z 

 dr L s = 



^t" 4 *], 



where p s = 



1.3. 5... (25-3) 

 •2. 4. 6... 2s 



Hence by integration a particular solution for z is 



■(4s-l)- 



7T f 00 _(4*-l)-| 



