90 Dr. C. Y. Burton on the Scattering and 



and when £=77 (that is, when #=L) this becomes 

 E 77 =-8A2>sin7 7 [(^+l) 2 expi( y a-l)77+( ) a-l) 2 exp{-i^-l)7;} 

 - (ft— l) 2 exp { -i (ji + 1)77} - O + l) 2 expt(/A + 1)17] _1 



= 4A( A 6 1 2 + /^2 2 )"W-"expi(-7 7 -^-' S 7) (37) 



(as we find from (27). after some reductions), where 



W = (/*x 2 + /* 2 2 + 1 + 2/.J 2 exp 2 W + (/V + fi 2 2 + 1 - 2^) 2 exp ( - 2^77) 

 -2( y a 1 4 + ^ 2 4 + l + 2 / x 1 2 / ^ 2 2 -2^ 1 2 -6/. 2 2 ) cos 2^7? 

 + 8/i 2 W + At 2 2 — 1) sin 2 W ; 



C0S*57 = W~*[expylt2?7 . {(/A! 2 — ^ 2 2 + l + 2iU, 1 ) COS (/*!— 1)?7 



4- (2/* ly u, 2 + 2/* 2 ) sin (^ — 1)77} 

 + exp ( — m) . { - (fii 2 —^2 2 + 1 — 2/xi) cos (/^ -f 1)77 



+ (2/A]/a 2 - 2/* 2 ) sin (/*!+ 1)17}] ; 

 sin'ST = W~^[exp/x 2 77 . {(/u 1 2 —/z 2 2 + l + 2/i, 1 ) sin (^ — 1)77 



— (2/^2 -f- 2//, 2 ) cos (^—1)77} 

 + exp ( — ^v){W— /^2 2 + l — 2/x 1 )sin(/^ 1 + 1)77 



+ (2/ai/*2 — 2^2) cos (/Xj + l)^}]; 

 cos &, sin 3 = (/*!, ^X/V + Z^ 2 )" 1 . 



Since yu-!, ^ 2 are given in terms of %i, % 2 by (28), (29), 

 E?/ is known in terms of the complex constant x- 



34. The transmitted regular disturbance is thus 



yjr f/ =:4:A{fj,{ 2 + !i 2 2 )m-*exj)i{pt-vx-d-< sr), . (38) 



which assumes a simpler form when x v ~ 1 * s small. For in 

 that case (28), (29) become 



so that 



W-i^Kl + X^f- 1 )- 1 exp (- Xl L) 

 sin ^ = sin x^~ fa/pi • cos %2^ =f sm (X2L— $), 

 and (38) becomes 



-\Jr" = A exp (— % X L) exp i {pt — vx—x^)- • (39) 



The refractivity, in the ordinary optical sense, is thus, to 

 our order of approximation, % 2 ; though strictly speaking 

 Pi — 1 does not vanish with % 2 v~" 1 when terms in %i 2 v -2 are 

 retained. 



35. When the secondary vibrators are Rayleigh resonators, 

 distributed through the region 0<.ic<L with the complete 

 irregularity of gas-molecules, the values of Xn X2 are given 



