n 



322 Mr. C. G. Darwin on the 



7, Reflexion of Monochromatic Radiation. 



Returning to the reflexion we get as the whole reflected 

 amplitude 



— ig ( 1+ exp { — pa cosec 6 + ik(p 1 —p 2 ) — 2iq } 



pl \ 



+ exp { — 2 pa cosec 9 + ik(pi — p z ) — 4iq } + k 



where the slowly varying quantities have been replaced by 

 their values for the first plane. As has been indicated, the 

 whole radiation at a finite distance is the same as at an 

 infinite, so that we may take the p's as large as we please* 

 Thus p g =p 1 + 2ass'm and the whole expression is 



/ 1 + § exp —s.fia cosec 6 — sika2 sin 6 — s2iq J 



I <1— exp — (jm cosec 6 + ika2 sin 6 + 2iq ) > * 



If lea sin 6 is near nir this has a strong maximum. Let 

 Jca sin (j> = mr. Then ha sin 6 = nir + ka cos $(d — <£) and the 

 amplitude is 



e ik(Ct-p) I r- -i 



— iq < /jLacosec(f) + i[2kacos<fi(6 — (f)) + 2q ] > .. 



Corresponding to this we have an intensity 



-I/{(^cosecc/)) 2 +[2^cos^)((9-^)4-2( ?0 ] 2 |. . (3) 



This has its maximum at kacos(f>(0 — (j)) + q = 0, If q is 

 replaced by its value in terms of the refractive index, [the 

 expression at the end of § 3 can be recovered. I mojbao j 



Suppose now that we measure the ionization in an electro- 

 scope of length / and sufficient breadth to include the whole 

 beam. The effect then is 



e i7c(Ct-p) 



P 

 e i7c{Ct-p) 



P { J (/*« 



dO 



cosec <f>)' 2 + 4:[ka cos <j> (# — </>) 4- q ] 2 ' 

 where I is the intensity of the incident beam 



t ? 2 7r 



p- pa cosec (/) . 2ka cos <£ ' 

 which reduces to 



I- /2(2 * , * ) N 2 .^, 3 coscc2<f> U\ 



pp. 



