118 BELL SYSTEM TECHNICAL JOURNAL 



an alternative expression. Now when "/" is very small the functions of the 

 angles become equal to the angles and we write, for the dielectric power fac- 

 tor itself , 



2at 



"'"■ = wr:- <"' 



Dividing this expression by eq. (10) 



P.F. = pi 



X 2at 



47r'\/e i sinh 2ai 



X 



and as the last term is always very nearly unity we have, if we put 

 4X, 



4xV'e ^ 



sinh ^ (d — d') . , ^ 

 P.F. = —^ '^. (12) 



sin ~ {AC + t) 



A 



Ordinarily the "sinh" is very closely equal to the angle. 



The reactance of the dielectric segment of line is necessarily equal to the 

 reactance of the part of the original line which it displaces, since space 

 resonance occurs in both cases. Hence, 



AC + t /-^ VTt f.^. 



tan TT — - — = Ve tan tt ^ — [16) 



A X 



which we can rewrite to 



wt A( + 1 \/7t ^ VTt 



— • tan TT — - — = X ^ — • tan x -— — . 

 X X X X 



Putting 



f TT/ A^ + / 



' y = — tan tt — - — 

 X X 



_ we have y = X tan X, (14) 



"y" is directly determinable by measurement and this gives X from the X 



^~ 

 tan X table suppHed.^ The value of e = — follows and P.F. is immediately 



irt 



_\ 



calculable. This completes the reduction of the observation. 



1 As no X tan X table to the necessary subdivision was available, one was calculated 

 from the Hayashi tan A' tables. 



