182 Mr PocJclington, On the Configuration of a jMir [Feb. 26, 



The relation between f2 and t by which the boundary in 

 the O-plane is transformed into the real axis in the ^-plane, so 

 that the points D" and F'\ B" and H", A" or I" become the 

 points ^ = + 1, ^= + a, ^=00, is given by 



o_ f dt 1 



= A r di f _i ]_^ 



" 2a j Vf^ U - a ^ + a/ 



fi , f /a^ - 1 + Va'-l Vf-1 



log ( 



2aVa'-l ^ IV ^-a 



-"TT^ J r^- 



Noticing that when t increases through the value a, the imaginary 

 part of ft diminishes by ttl, we put 



^ =1. 



2aVa' - 1 



Giving now to t the values and oo , and to Xl the corresponding 



1 1 



values log -^ and log y., we have the equations 



1 1 , , 1 « + 'v'ti' — 1 



^ a — \fa - 1 



giving finally 



V a — '^a^ — l n V! 1 o /-^ — ^ /^\ 



y-r= = 2a' - 1 -2aVa'^-l (4), 



f^ a + Va-^ - 1 

 and 



^ , 1 /a^ - 1 + Va' - 1 Vf - 1 



12 = log -y^ ; 



^ U\ t-a 



at +1- \^a' -l\/f-L y 

 t + a J 



or, making the same substitution, i = sn u, as before, and putting 

 for fl its value log -^ , 



cZa? 1 a sn ^t — 1 + t Va^ — 1 en ?i / a sn w + 1 — Va^ — 1 



cn'M 



dw V ^nu — a / sn it + a 



_ 1 {2a^ — 1) sn^ u — a^ + Ita'J a' — 1 sn tt en it 



