314 Mr. R. Hargreaves on Radiation 



The special feature of the transformation is the relation of 

 the formula for Doppler's effect to the mode in which allow- 

 ance is made for this conversion of energy. 



The translation in the first section is not general, hut it will 

 appear in § 7 that when reflexion only is in question it is not 

 necessary to consider other components. 



§ 1. First take the case of reflexion at a perfectly re- 

 flecting plane surface, which is retreating directly from 

 the free aather space with uniform velocity w. If 

 p\t — (1% + my + nz)/Y \ is the argument of an original, 

 and p r {t— (Vx + m'y — n'z)jY} that of the reflected wave, the 

 kinematical part of the law of reflexion consists in the iden- 

 tification of arguments at the moving surface, i. e., for a 

 value z = z + wt : the identification to hold for all values of 

 t, #, and y. The conditions are 



pl=p } l\ pm—p'm 1 , p(l — wn/Y)=p'(l-rwn , /Y). (1) 



The former pair maybe replaced by p 2 (l— n 2 )=p /2 (l — ?i /2 ), 

 and the law of reflexion is 



(l-n f )/(l-ii7fi/V) 8 =(l-n /1 )/(l+iwi / /V) f f . (2) 



with a common value <f> for an azimuthal angle involved in 



I— \/l — n 2 cosc£, m— \/l — n 2 sin<£, l'= V 1 — n' 2 cos </>, 



m' — Vl — tt /2 sinc/>. 

 The differential of (2) gives 



(Yn-w)dn/(l- wn/Yf = (Yn f + w)dn'/(l 4- twi'/V) 3 , 



or p z (Yn — iv)dn=p /z (Yn' + w)dn', 



and multiplication by the last of (1) gives 



p ±(Yn-w){l-wnlY)dn=p' A (Vn' + w){l + wn f /Y)dn f , (3) 



a formula of fundamental importance. The law of reflexion 

 may be put in linear form, viz. with r=w/Y, 



Yn—w Yn f 4-w n—r n' + r /A . 



— or = (4) 



Y — wn Y + wn n 1 — rn l + rn n 



which follows from (2) by using 



/ n-r >£ (l-r 2 )(l-n 2 ) 

 VI — rnj ' (1—rn) 2 



We may note also the relations 



p(Yn— w)=p r (Yn f + w), and p 2 dn=p' 2 d?i l . . . (5) 



