168 PROFESSOR K. PEARSON AND MISS A. LEE ON THE VIBRATIONS 



From these we deduce 



E/(27r/X)_ _ B(2ir/xyco8(ft-2 x ) 



P ( r ) ~ (2wr/xyco"s~ X sin (ft - X ) ~ cos** sin 8 (ft - x ) 



From(xix) i <*ft = MX ;7 



sin* (ft - X ) rfr ~(27rr/X) 2 ' 

 or, 



rfft _ 2WX 1 



rfr == (27rr/X)*/_J__ ta V+l 



To find the wave-speed, we have from (xvii) to find dr/dt from 



t T \ 



' ) + {} = constant, 



IT X / 



i.e., 



X dr_( i dPt>\ o 



~2r"~di\ ~ 2ir ~(fr) ~ 



Let X/2r = v as before, and dr/dt = V . Then 



1 



(xix), 



1 



1 + (f - tan ; 



Wheref =2^' HenCe 



?L=I? = _ _H_ X-x- __ (xxi). 



v 1 + tan 2 * - 2f tan x 2 tan x cosec 2 X f 



Now this result is quite independent of the direction of propagation, or the wave 

 moves outwards with the same velocity at the same distance in all directions. 



V v 

 For x = 0, - - = f. This is the value given by GRAY for propagation in the 



equatorial plane,* but it is clearly independent of direction. At considerable distances 

 f is very small, and therefore V = v. Thus v is the limit of wave velocity as we 

 recede from the source of disturbance. 



Now a remarkable result flows from (xxi), which could not in any way be predicted 

 from HERTZ'S theory, neglecting the damping. HERTZ tells us that the velocity of 

 propagation at the source is infinite, and GRAY draws the same conclusion, but (xxi) 

 shows us that near the source the velocity of propagation, although very great, is 



* ' Absolute Measurements," vol. 2, p. 781 



