Numerical Solutions 



Polar coordinates in the i plane have been designated as (x,^) in (18). Arc 

 length along the contour will be denoted by s , with s = at A'" and increasing 

 in the counterclockwise sense of traversing the contour. We shall also require 

 the polar coordinates of the chord directed from a point P at arc length s to a 

 point Q at arc length t along the contour, 



PQ - r^, ei'P^' (23) 



in complex notation. In particular, the chord A"'p would have the polar coordi- 

 nates (r„^, (P^J. 



The Gershgorin integral equation [3j may be written in the form 



= J K (0,0') d (0') dst' + 2q)^^ , (24) 



,2-n 



(0) 



"0 



where o'" P = \e''^ , o"'Q = \'e''^ ; 



K (0,0') = - - ^ 



K' [\' - X. cos (0 - 0')] - ^TT^ sin (0 - ) 

 k^ + k' ^ - 2kk' cos (0 - 0') 



(25) 



arcs in | sin j , (26) 



^os = f^o' +^' - 2^0 ^ cos 0]l/^ \o = ^ (0) . (27) 



When 0' = 0, the expression for k(0,0') in (25) is indeterminate. Although the 

 limit can be readily obtained, it is preferable to avoid this difficulty by writing 

 (24) as 



61(0) 



^2-n 



I K(0,0') [5(0') - 6'(0)] d0' + qp^^ , (28) 



which is equivalent to (24), since, by (25), 



r K(0,0')0(0)d0' = - ^^ r -^ d0' = -5(0) . (29) 







The integral equation (28) can be solved approximately by reducing it to a 

 set of linear equations in a discrete number of values of and 6. Because of 

 the double symmetry, (28) can be collected so that the range of integration ex- 

 tends from zero to 77/2. Mappings were computed for =: 1°, 2°, 3°, • • •, 89° 

 and the resulting values of e also range between and 90°. It is assumed that 

 6 = when = and = 90° when v = 90°. 



1624 



