BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 249 



we see that the i/ terms in the expression for the secondary waves are inappreciable 

 at the external boundary owing to the exponential factor exp. ( XR/y/2). 



We shall suppose then that R is great compared with the wave-length of the 

 incident sound and yet such that o-VR/c 3 is small. 



Let us now return to the consideration of (3). At the external boundary, r = R, 

 we may write approximately 



\\/a a. 



2 [A n i" cos 



where square brackets are used to denote that only the real part of the expression so 

 enclosed is to be considered. 



Hence we obtain 



o \ i/a < 



V S TA i." 4 



cos 



- n 



cr \TrhIi / n = o 



Using these expressions for p l and q l we obtain 



(/Wo-fy>o) 2 [A nt " +1 cos n^---W]. . . (4). 



cos 



sin (h'. 



sin (o-t + ^UTT) cos 

 and 



Again, since ^R is great, we have from (1), 3, approximately 



/ 2 V' 2 . / 2 V' 2 



= ( I sin (liR + ^ir) cos crt+ 2 S ( J-^T ) sin (hH + ^Tr ^n-rr} cos 

 \irriL\J i-i \ir/4Xv/ 



Hence we obtain 



2 \ 1/2 / 2 X 1 ' 2 



- 



2 \ 1/2 



j-=?) COS (flR + ^TT) COS i 



a, i 2 \va 



2 ( TTT) cos(/iR + l 

 n=i \TrMv 



cos (o-t + fy 

 Combining these expressions for p and q we find without difficulty 



/ 2 Y /2 / 2 V'" 



(-To) cos(or Ml i77-) + 2/ip o- ppr 



\7T^R/ \7r/tR/ 



CD 



cos 



cos (o-< AR |TT) cos 



D 



2 ( )" cos (o-< AR 



n = l 



Substituting in (4) and integrating with respect to & we obtain 



r(jMktW&)B<** = ^^ ^ [(-)"Ai B+1 cos(o--/iR-i7r)e' 



=0 



VOL. OCX. A. 2 K 



