488 Lord Rayleigli on the Influence of Obstacles 



in which 



A = — sin imir. 

 Thus 



log -jeos f — eot*'m7rsin f j- = log ( 1— - — j 



4-log/l- r - f ,■■)+.... 

 ° \ innr + tt/ 



If we change the sign of ra, and add the two equations, we 

 get 



Io s { x - iiSL} = log { * _ isfe 5 } 



+1 °4 1 -^^ 2 } + log { 1 -(li^} + -" ; 



whence, on expansion of the logarithms, 



sin 2 g sin 4 g sin 6 g 



sin'tmir 2 sin 4 /m7r 3 sinHmir 



= p r i , i | i '+ ...a 



5 \(wra7r) 2 (innr + 7ry (imir—iry J 



2i \(im7r) 4 (iw27r + 7r) 4 {imir— 7r) 4 J 



+ W\{SS7 + (^TT + TT) 6 + J + 



By expanding the sines on the left and equating the corre- 

 sponding powers of £, we find 



1 _1_ _1_ _J_ 7T 2 



(?'ra) 2 (iw + 1) 2 (em- 1) 2 (e'm + 2) - snAViwr 



? + re: ±iw + • • • = "" a o^2^,^ + ai n *; m ~ > ■ v^>) 



(im) 4 (zra + l) 4 3 smi)N7r sin ira7r 



i i i , . . . = ^ ^ | ^ 6 . (29) 



(im) 6 (era + 1) 6 15 sin 2 tm7T BinSffur sin b m7r v ' 



In the summation with respect to m required in (19) we 

 are to take all positive and negative integral values. But in 

 the case of m = Q we are to leave out the first term, corre- 

 sponding to m! = 0. When m — 0, 



7T 



sin 8 i»i7r (tm) 2 3 r 



