324 . Mr. C. G. Darwin on the 



The intensity corresponding to this will be proportional to 



1 f °° 1 f °° 



jJ2 1 (cj> 2 + y}r 2 )dk or ^ j w*4£. 



The reflected amplitude is 



f" r i j*(&-p) m 



/a 1 — exp— (yita cosec # + i . 2Arasin0 + 2/^ o )' 

 corresponding to which there is an intensity 



ttf: 



If" o 



5\L Ukq \i 



P 2 Jo \l— e^~p —{pa cosec0 + i . 2ka sin -t 2 iq )\ 2% 



We suppose that w* only varies slowly. As /j varies the 

 integrand has strong maxima when ka sin 9 -\-g = mr. Denote 

 this value of k by k n and near k n put k = k n (l + ic). 



Then the expression is approximately 



-ptu n (q")n\ 



k n dx 



p 2 n n K ^ ' J _ oo (/^acosec 0) 2 + (2mTX~) 2 ' 



where the w subscript denotes that the quantity has reference 

 to k n . Performing the integration and putting in the value 

 of q, we have 



P" n Pn *? 



or in terms of the more usual E\ where 'E k d\=uj c dk, 



^^i/(2 W )(^) ; . . . (5) 



By virtue of the assumption that u k only changes slowly, 

 the correction depending on the refractive index has been 

 neglected. The expression would require a little modifica- 

 tion for approximately monochromatic radiation. 



It was by the above processes that the result quoted on 

 pp. 230-231 of Moseley aud Darwin was reached. Unfor- 

 tunately there was made in that paper an assumption which 

 cannot be maintained, viz. that the scattering and absorption 

 are proportional. It would seem better to suppose them 

 independent. The abbreviated proof there attempted cannot 

 be maintained. In the first place, the area assigned for a 

 Fresnel zone is incorrect and also the argument should be 

 carried out with amplitude and not with energy. Amended 

 in this way it gives the right result. 



