30 BELL SYSTEM TECHNICAL JOURNAL 



Equation (11) : z = \l — — -.- \ 



\ a-\-ir 



X y 2{a 2 + F 2 ) 



(l-a)F 



2(a 2 + F 2 ).r 

 agz — — | tan 



, (l-a)F 



a + F 2 



Equation (12): ^=y/\-i/F\ 



" r= V2 ^+VT+i7^. 



i = 



1 2 



8^ 2 (.r + l).r 2 



Gg Z = — 2 COt -1 /^. 



The relation y = — V x 2 — 1 can be written in the form 



x — 1 I v— 1 



= \ - — -i 



-y \x+\ 



which shows that x—1 is always smaller than — y; and is very much 

 smaller except at small values of F, where the two approach equality 

 as F approaches zero. 



The locus of z in the xy-plane is the hyperbola x 2 — y 2 = l. For any 

 preassigned value of F the corresponding values of x and y on this 

 locus can be accurately calculated by means of the above equations 

 for x and y. For any pair of values of x and y situated on this locus 

 the corresponding value of F is given by F= — 1 2xy. 



Equation (19): K = \]^^' 



,, J & + £ 2 £ + V(l+o 2 ) (1+frg) 

 M= V" 2(l+6 2 )£ 



\-bk 2 E 

 N = 



2{l+b 2 )EM' 



\-bk 2 E 



««*=-* tan ^ 2£ 



