338 Mr. R. Hargreaves o?i Radiation 



The scheme is then 



H x \ + HA' =iiK 1 "\", H 2 \ + HA' = H,"X"o 



(H 1 X-H 1 V)N=H/ / \ // F / + kMH 2 // \ // , I . (49) 



(H 2 \ -* H AO N = H a "X/>N" - ^JtuMH/V, J 



and the solution is 



H A'{N + juN" ^N -r fl" + m&VM 2 /V 2 } 



2M^MN 



^H^N-N" N f/iN^^VM'/V 8 }- ^ ~ H a \, 



H,V{1+^ ^JN+JN // + MVM 2 /V 2 } 



= H 2 \{N- / aN 7/ /.N + N"-a*WM 2 /V 2 } + 2 ^ M -H t \. 



If u = 0, we note that the cases are separated, and the 

 solutions differ from (i.), (ii.) in having N, N ;/ for v, v" . The 

 general value of A 2 is 



(H^ + H^V^IH^ + H^X 2 . 



Its dependence on the original polarization is shown by 

 writing 



H x = H cos t/t, H 2 = H sin yjr, 



when A 2 has the form 



a cos 2 i/r + /3 sin 2 yjr -\- 2y sin -ty cos ty. 

 Here a and {3 contain u 2 explicitly, while y shows the first 

 power of n. When the mean of A 2 for all orientations with 

 a given wave-front is taken, the term in y vanishes and the 

 mean is i(« + /3), which contains vr explicitly. Now N N"M 

 contain v implicitly, but not u, since Z™ in U = lu -j- mv + nw. 

 Thus the component of translation which only appears through 

 its square is that along the line of intersection of the wave- 

 front and the reflecting surface. This is the condition stated 

 at the end of § 6, that the tangential forces shall do no work 

 in the aggregate, and it also secures a simplification of the 

 general equations for the collective relations. 



§ 14. In finding the pressure and the relations between 

 p p f and vr we shall be content with first-order work ; and as 

 the influence of u and v only appears in second-order terms, 

 we shall suppose that w only exists. The use of the kine- 

 matical cosine v is more convenient than n because A 2 is 

 expressed in terms of v v", and v has the range to 1, while 

 to the first order 1 — v 2 = yu 2 (l — j/ /2 ). The relations are 



p*^K 2 {n-r)(l-rn)dn=p fi $A*{n f + r)(l+rri)dri, 



(5(T 



