238 Mr. D. L. Webster on the 1 heory of the 



Of the n(l — e~ nY ) charges per unit volume not radiating 

 alone, to which expression (5) cannot be applied, a fraction 

 e -nv ma y -fo e S11 pp 0se( ] to radiate in pairs only, each pair 

 giving an electric vector in the — y direction, of intensity, 



^[/(^X^l 



(«) 



where the origin is on one of the electrons of the pair. 

 Squaring and integrating with respect to t, we find for the 

 energy scattered per unit area from each pair 



2E?T(1 + F), (7) 



where F is the mean product of the functions f for pairs of 

 which one electron is at the origin, while the other may be 

 in any part of the volume V with no preference for any part 

 over any other. Evidently F is constant for large values of #, 

 but variable, increasing as 6 decreases, at smaller angles, but 

 always remaining less than (1). 



The case of three electrons in the volume V presents no 

 additional difficulties, except the introduction of the mean 

 product F' of the functions /for pairs of which both members 

 are scattered indiscriminately over the volume. 



We may now discard the assumption of a single, polarized, 

 primary pulse, and assume that there are N primary pulses 

 per unit time, and that the mean square of the z component 

 of the primary electric vector is the same as that of the 

 y component, where the mean is over a long* time, rather 

 than over a single pulse. This change in our assumptions 

 must, of course, result in the introduction of a factor 

 N(l + eos 2 #) into the expression for the scattered radiation, 

 to obtain the total energy scattered per unit time, per unit 

 area of the receiving screen at a distance r, and per unit 

 volume of the radiator. The resulting quantity, which takes 

 account of all re-enforcements of radiations, is 



E?TN(l + cos 2 6>)ne-" Y [l + (l4-F)(l-^- ?iV ) 



+ (l + F + F')(l-^- ?iV ) 2 +(l + F-f2F / )(l-^ nY ) 3 +....]-W 



To obtain what Mr. Crowther terms the "excess radia- 

 tion," we may subtract from expression (8) the value, 

 EiTwN(l + cos 2 #), of the same quantity calculated without 

 reference to re-enforcements of the radiations. And since, 

 for large values of 6, nV is very small, and the excess 

 radiation extremely small, the total excess radiation between 

 the angles 6 and 6 + d6 may be obtained by multiplying this 



