Regular Reflexion of Light by Gas Molecules. 101 



the values of p which are effective belong to a very narrow 

 spectrum-line in the middle of a much broader incident line, 

 we may simplify by treating /(p) as a constant. After a 

 thickness (L say) of the vapour has been traversed, the 

 •distribution of energy in the primary beam will be different. 

 For any assigned value of p, the energy of the primary 

 beam has been changed iu the proportion exp(—^ 1 L), and 

 X\ as we know is proportional to sin 2 7 ; so that in place of 

 * x p( — XrL) we ma y P u ^ ex P( — K srn2 %)> where K is a 

 constant multiple of L, and is equal to % t L cosec 2 7. At the 

 same time it has to be remembered that, corresponding to 

 any definite value of p (which determines 7), the diffuse 

 secondary radiation emitted per molecule* has an intensity 

 proportional jointly to the intensity of the exciting radiation 

 and to sin 2 7- It follows that the intensity of the diffuse 

 secondary radiation at a depth L within the mass of vapour 

 will be less than that where the primary beam first enters in 

 the ratio 



J sin 2 7 . exp ( — K sin 2 7) d{p 2 ) : j* sin 2 7 d(p 2 ). 



Making use of (61), and at the same time putting M in 

 place of {p 2 — u 2 )I'ik, we can write for the ratio 



(tJ^, exp(--^W: fl— rfM. . (62) 

 JM 2 +1 *\ M 2 + l/ Jl\P + l 



'Only a very small range of values (negative and positive) 

 -of M contributes sensibly to the integrals, and the limits of 

 integration must be wide enough to include the whole of this 

 range. 



55. Using a rough graphic method, and ranging from 

 M= — 10 to M = +10 (or from to 10, which amounts to 

 the same thing), I find that the ratio (02) would be as small 

 as 0-5 probably for K = 1'5 and certainly for K = 1'6. The 

 latter figure may be compared with Wood's experimental 

 result, referred to in § 52 above. Thus 



l*0 = K = j£ 1 cosoc 2 7 x 0*5 cm. 



'On reference to (DO) the values to be attributed to v (the 

 number of resonators per c.cm.) are seen to be 



v = Sttx l'6/3\ 2 = 6*7 X 10° for type I. molecules \ . 



v=8irxl'G/\ 2 =201xl0 9 „ II. „ J ' ' 



* The energies of the diffuse secondary radiations from the various 

 molecules being simply additive in an attenuated vapour sensibly obeying 

 the gas-laws (cf. § 7, part I.). 



