18 Prof. J. EL Jeans on the Interaction 



suppose the origin from which t is measured to remain 

 indefinite. We readily obtain 



iv 2 /c 2 mV 



and hence 



, -* 7 ^ x , eAv ~) 



X={lO—\ T )t-\ T 7 COS KWt 



ek I, . . (3) 



y— vt —v—^j. COS KICt i 



tVK~m\ 



z— w t J 



in which, strictly speaking, constants of integration must be 

 added to x, y, 0, and £. 



It now appears that the motion of the electron may be 

 regarded as compounded of 



(i.) a uniform velocity of translation, u 0f r , w ; 

 (ii.) oscillations parallel to the axes of x and y, each of 

 a purely harmonic nature, and of frequency tew. 



Since w = u + V, the result is such as might have been anti- 

 cipated from the Doppler theory, but it is not easy to give a 

 rigorous proof without a detailed examination of the equations 

 of motion. 



3. The electron will, according to the classical dynamics, 

 absorb light of frequency /cY ; it will emit light Avhose 

 frequency will vary according to the direction of emission, 

 the frequency in any direction being obtained by modifying 

 the frequency of oscillation kid in accordance with Doppler's 

 principle. 



Let polar coordinates r, 6, ^ be taken, the axis of x being 

 taken for = 0, and the plane of xy for i/r = 0. The velocity 

 of the electron has a component in the direction 0, yfr equal 

 to 



u cos 6 + v sin 6 cos yjr + w sin 6 sin yjr, 



so that the frequency, say q, of the radiation emitted in this 

 direction will be given by 



q= -y- (Y—u cos — Vq sin 6 cos ty — iu sin sin yfr). (4) 



Let us assume the distribution in different directions of 

 the radiation emitted by the electron to be 



1(0, y/r) sin 0d0dyfr. 



