372 



Lord Ray lei gh on the Dispersal of 



The next step is the calculation of the series included in 

 the square brackets o£ (22) and (26) for various values of 

 6 from # = in the direction backwards along the primary 

 ray to = 180° in the direction of the primary ray produced. 

 If we add the terms due to even and odd values of m sepa- 

 rately, we may include in one calculation the results for 

 and for 180 — 0, since (— l) TO cosm(180 — 0) = cos??i0 simply. 



In illustration we may take the numerically simple case 

 where = and = 180, choosing as an example z = 2*4 

 in (22). Thus 



m. 





m. 





 2 

 4 

 6 



•26905 - i X '48842 



•57166 - i X -91676 



• 830 - i X • 6 



2 



1 

 3 

 5 



•39906 - i X 113268 

 •13448 - i X '01436 

 • 38 - i 



2(even)= -84903 - i x 1 "40524 j 2( dd) == '53392 - i x 114704 



Accordingly for = 0, we have 



'odd 



'31511-z x -25820, 



and for = 180° 



^even+^odd = 1*38295 -zx 2'55228. 

 These are the multipliers of 



J gi(nt—kr) 

 I 



For most purposes we need only the modulus. 



(•3151) 2 +('2582) 2 =(-4074) 2 , 

 (l-383) 2 + (2-552) 2 = (2-903) 2 . 



\2ikr> 



in (22). 

 We find 



and 



As might have been expected, the modulus, representing the 

 amplitude of vibration, is greater in the second case, that is 

 in the direction of the primary ray produced. 



For other angles, except 90°, the calculation is longer on 

 account of the factor cos m6. The angles chosen as about 

 sufficient are 0, 30°, 60°, 90° and their supplements. For 

 2 or 3 of the larger z's the angles 45° and its supplement 

 were added. The results are embodied in Table II., and a 



