322 Mr. R. Hargreaves on Radiation 



Thus if the formula in A" is used it is clear that 





\"- i dn" not 



<**' 



/* 



must be proportional to energy-content, otherwise there 

 would be nothing to mark the moving standpoint. This 

 agrees with a formula used above (8) for the energy- 

 content in space opened out by the motion. The formula 

 uses wave variables for direction. 



y 

 If energy-content were taken proportional to — A" -4 dn", 



then the term in U" would necessarily be interpreted as on 

 the other side of the equation, and would imply a back- 

 pressure on the aether caused by the refracted ray {v. infra). 



The formula in vr has no term which could be interpreted 

 in this manner, and a special feature is that (u v w) only occur 

 in second order terms. The use of -sr and ray-cosines both 

 imply a new standpoint; the fact that tsr and -sr" are equal 

 makes the change of standpoint rather than that of medium 

 the important fact in (17). 



§ 6. The collective relations when reflexion and refraction 

 are combined must take account of the proportion in which 

 the radiation is divided into two streams, a proportion varying 

 with the angle of incidence.and with the plane of polarization. 

 For the present we denote by A* and B 2 the fractions (sum 

 unity) of the original energy which undergo the respective 

 transformations; the value of A 2 is given by electromagnetic 

 theory, and will be found to depend on (u v w) as well as on 

 the angular variables. It is proposed to weight the equations 

 of transformation with A 2 for the reflected and B 2 for the 

 refracted section ; and then to integrate them in a manner 

 which will clear the results of all reference to particular 

 waves. The relations involve n and (/> (through U), and A 2 

 also involves an angle defining the plane of polarization. 

 But if we suppose A 2 to stand for the mean with regard to 

 this last angle, the plane of polarization taking all possible 

 positions for each wave-argument, then only n and </> need 

 appear, i. e. 



jp 4 Jf'A^Vn-w) (1-U/V)in # 



=p" jY A 2 (W + w) (l-U'/V) dn' d<j> 

 p<§B*{Vn-w)(l-U/V)dnd(l> 



= w-JJ B 2 Vrc r dn r # r /(l -2tt 2 /V 2 + U r 2 /V 2 ) 2 

 and the sum 



p* jj (Vn-to)(l-U/V) dn dcj> 



=p' i jfiA 2 (Vn' + w){l-TJ'/Y)dn'd<l> + i* i §B 2 Vnr ... 



y (i9) 



