329] Conjugate Functions 273 



At B, we now have f = - a, u = oo , and therefore 

 z 7T.A A/ - iirA. 



Thus the distances between the pairs of arms are TT\ / - A and TrA 



v a 



respectively. 



Let P be any point in EF which is at a distance from E great compared 

 with EB. Let the value of f at P be >, so that fp is positive and greater 

 than 6. 



We have TF= Z7 + iF=log?, so that along the conductor FED, F= 

 and 7 = log. 



The total charge per unit width on the strip EP is, by formula (233), 

 dS^^ (U P - U s ) = ~ (\ogb-\ogb) ......... (247). 



If P is far removed from E, the value of p is very great, and since 



the value of w 2 will be nearly equal to unity at P. 

 From equation (246), 



z = - 2 A y^tan- 1 ^ u + 2A log (I+u)-A log (1 - u*), 

 so that log (1 - <) = 2 log (1 + u) - 2 ^ tan- 1 ^ u - ^ . . .(249), 



in which the terms log(l -u>\ -'Z/A, are large at P in comparison with the 

 others. Again, from equation (248), we have 



(250), 



in which log log (1 - u 2 ) are large at P, in comparison with the term 

 log (an? + b). Combining equations (249) and (250), 



log f = log (aw 2 + 6) - 2 log (1 + w) + 2 ^/ - tan- 1 y^ u-^-~ 



(251), 



in which the terms log ? and -| are large at P in comparison with the other 

 terms. At P we may put u = 1 in all terms except log f and z/A, and obtain 

 as an approximation 



log ?> = log (a + 6) - 2 log 2 + 2 / y/-tan- 1 y/ + J. 



18 



