and Electromagnetic Theory. 323 



is the equation for flux of energy. [The formula for ts or 

 ct" may be used, r. infra.] The terms representing a rate of 

 mechanical working are 



p\\\ nJJ dn d</>-p' 4 jjAVU' dn' d<j>, 



and the pressure depends on the term multiplying w contained 

 in U and U' or I'u 4- m'c — n'ic. If we are only concerned with 

 its main term, that independent of u vw, we may write ft' = ra, 

 p' =p. use statical limits for n and a statical value of A 2 , and 

 then with x f° r tne eu ergy-content of the incident radiation, 

 the pressure is 



^ . jj (1 + A> 2 dn d<f> (20) 



The terms containing the other section of U, viz. lu + mv 

 will vanish in the integration with regard to <£, if A 2 is a 

 rational integral function of (lu + mv)* and of (lv — mu) y 

 i. e. of (ucos <p + v sin cj>) 2 and of (y cos <j>>— u sin <j>) . This is 

 readily seen by taking v = 0, i. e. by measuring (f> from the 

 component of translation in the reflecting surface, in which 

 case the quantities named vary as cos 2 <j> and sin cf>. The 

 component which must only enter through its square is for 

 each wave-front the component in the reflecting surface 

 along its line of intersection with the wave-front. On the 

 present electromagnetic scheme this will be shown to be true 

 in an exact manner, when A 2 stands as above for the mean 

 of all polarizations taken for that particular wave-front. 

 Thus there is no work expended in connexion with tangential 

 forces (in the collective result) ; a further consequence is 

 that in (19) nw may be written for U and —riw for U', and 

 that all first-order results are independent of u and v. 



§ 7. For perfect reflexion A 2 =l, and the vanishing of the 

 above integrals is obvious, but in that case u and v do not 

 intervene at all, as the variables of the reflected wave are 

 determined independently of u and v. In fact 



J:J' = m:m' = V-U: V-U' 



is equivalent to 



Z : r = V— wn: V + wn 



or to vA — n 2 : x /L~n' 2 = Y~~ wn: V + wn' ; 



i. e. n is determined in terms of n precisely as in (4) with 

 only w present. The omission of u and v in the account of 

 perfect reflexion is therefore justified, and that not merely 

 as a preliminary course to secure simplicity in the opening 

 statement. 



Y2 



