174 PROFESSOR K. PEAESON AND MISS A. LEE ON THE VIBRATIONS 



us write r /r, = q*, where q is generally a small quantity. Then the denominator of 

 the expression for V, v in terms of r (see xxx) may be written 



r (r - r - qr } ) {(r r f + (r - r ) yr, + q^}. 



The second factor is negative for r < r + qr t) then vanishes for r = r -f- qr t , and 

 is ever afterwards positive. 



The third factor would vanish for imaginary values only, and since it is positive for 

 r = r , it remains positive always. This supposes that r'{ is positive. Similarly the 

 numerator in the value of <f> t , or (r r )~ -\- ; will always be positive, if r'f be positive. 

 Now by (xxix), r$ cannot be negative unless tan x > \/% m ' X > ^0. This corre- 

 sponds to a degree of damping in which the amplitude would be reduced to "000019 

 of itself every period, or a degree immensely higher than that of the usual Hertzian 

 oscillator. Hence both the numerator and the third factor of the denominator are 

 always positive. 



Accordingly Vj v is negatively infinite for r = 0, becomes negatively finite, but 

 very large, until r = r -j- qr^ when it again becomes negatively infinite, after this it 

 Ixjcomes positively infinite, and rapidly decreases to zero as r increases. Thus, as in 

 the case of the other waves, there is a sphere round the oscillator within which the 

 waves move inwards for transverse electric vibrations. This sphere is of somewhat 

 larger radius (r + qi\ as compared with r ) than in the case of the magnetic wave. 

 In terms of x the radius of this sphere is given by 



^(sin2 x ){(fcosec' x -l) 13 +l}. 



For example, when tan x = '2/77 or % = 3 38' 33"'5, we find for the radius of this 

 sphere, 6'69604 ?, or between six and seven times the radius of the sphere within 

 which the velocity of the magnetic wave is negative. Substituting the value of ?, 

 the radius = '135X. Hence, for a small oscillator, say $ of a wave-length long, 

 this sphere would be well outside the sphere '05 of a wave-length circumscribing 

 the oscillator, and thus practically within the field to which our theory might be 

 approximately applied. The inward moving wave of transverse electric vibrations 

 ought thus to be capable of experimental demonstration. 



We have, in the next place, to find the minimum negative velocity, and its distance 

 from the centre. 



Writing r r = u and r = q*i\, we have to find the maxima or minima of 



Differentiating, we find for the required values 



