442 Lord Rayleigh on the Light emitted from a 



R to R/ with introduction of a factor decreasing from unity 

 (the value from to R), as we pass outwards from R to R'. 

 The form of the factor is largely a matter of mathematical 

 convenience. 



As an example we may take e~ h '( r ~ n \ or e~ hk( - r ~" B, \ which 

 is equal to unity when r = R and diminishes from R to R'. 

 The complete integral (31) is now 



4<7rf E . 4:7rC w 



-t-I sin kr . r dr + — I e' 71 ^-^ sin kr .r dr. (34) 



From the second integral we may extract the constant 

 factor e hkB , and if we then treat sin Tcr as the imaginary part 

 of eJ kr , we have to evaluate 



At—njkr^ Jy, 

 'E 



We thus obtain for (34) 



-T5- (sin &R — kR cos frR) 

 k 6 v 



I 



- %1 + W [eos*R'{(A» + l)*R'+2A} 



+ sinffi'{(A 2 + l)MR' + A 2 -l}] 

 [cosm{(li 2 + l)m + 2h\ 



P(l + A 2 ) 5 



+ sinm{(A 2 + l)MR-f A 2 -l}]. . (35) 



When we combine the first and third parts, in which R' 

 does not appear, we get 



[cosffi{2A-A 2 (A 2 +l)^R[ 



kHl + h 2 )< 



+ sin&Rj7i 4 + 3A 2 + 7<A 2 + l)&R}]. . (36) 



The first part of (35), representing the effect due to the 

 sphere R suddenly terminated, is of order AR ; and our 

 object is to ascertain whether by suitable choice of h and R' we 

 can secure the relative annulment of (35). As regards (36), 

 it suffices to suppose h small enough. In the second part of 

 (35) the principal term is of relative order^R'/R)*?-**^'-^ 

 and can be annulled by sufficiently increasing R', however 

 small h may be. 1 



Suppose, to take a numerical example, that h = I ^ )i and 

 that e-W-V is also JL, Then 



R'-R= *^=^ = 1100A. 



