IN THE FIELD ROUND A THEORETICAL HEKT/IAN OSCILLATOR. 181 



We now proceed to trace the velocity of the electrical disturbance in the equatorial 

 plane. 



We have 



,, 2 tan Y - 



tan a. = - * - * - 

 1 - tan 1 x + tan * - ' 



whence, by differentiation, we find 

 ^ dfr _ *{! 



2ir rfr ~ (1 + tan* %f 2 tan x (I + tan 1 x ) - * (1 - "> tan' x ) - _'* tan x -f ' 

 Hence, if V 4 be the velocity and < = X/(2r), as before 

 V 4 



_ 

 r " " 1 - 



, 



h (1 + tan'x) 1 - 2tanx(l -f tan*x)~ 2 t (l - tan*x) + 2f tan x 

 Now put 



:=+ 



-'tanx V L^Jan^l 

 1 + tan'^ Nl + tan'x) 1 ! 



' 1 + tan 1 x " + tan* x)* (1 + tan* %f 

 Again put 



f- _ 1 Stan'x _ 2 , 2 3 sin* ) = r* 

 Then 



tl + II = coe J x, C. (5 4- {I) = 2 tan x j( 1+ tan 2 x )% 



2 + i * = cos " X ( cosl X ~~ 8m * X) = ( ^ ~" * an * 

 Thus we may write the above result 

 4 = 1 + 



nor- r,) 1 <?+&)- 2a (?-&)- 



_ ! , _ x- _ ( Hi) 



h ( J/X)F r {(r - r.)F (r + r.) - 2rj(r - r.) - rjr.} 



This is a form directly comparable with (xxx). 



If the wave be not damped, Co = . .> = l which gives 







r : h 



a value identical with that given by GRAY.* 



* ' Absolut* Measurements,' vol. 2, P;irt II., 



