BY SMALL OBSTACLES OF CYLINDKICAL AND SPHERICAL FORM. 263 



Now, at the external boundary r = R we may write approximately 



= 2 [A n /T ( " +1) . i-P^/t)^"-*^] 



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

 enclosed is to be taken into the account. 

 Hence we obtain 



- I [A a /r<" +1 V +1 P n (fi) . e^-^], Rgr, = h I [A n r<" + V +1 . P n (ft e^'^]. 



n=0 n=0 



Combining these results we find 



K( Ml +p 1 g D ) = (-^a^ +^)S[A,,7r< +1 >. l +I P 1 ,( /t )e'<''- AK >]. . . (2). 



B=0 



Again we have from (3) 7, since /iR is large, 



= 2 {(2u+ l)h~ l P n (n) sin (Mi-^iTr) cos ( 



n = 



Hence we have 



1) h~ l P n (/A) sin (/*R-^H7r) sin 



= - h {(2/1+ 1) /rTn (/A) cos (/iR-f HTT) cos 



n=0 



Combining the two last results we find 



o) = p<r 2 {(-)(2n+ !)?(/*) cos (o-e- 



=0 



Substituting this result in (2) and integrating over the surface of the sphere r = R, 

 we obtain 



ff (PM+Ptfi) <*S = 477^0- i [(-) A sl - +1 A-<- +1 > cos (o-t-hll) e-<"-*>], 



JJ n=0 



of which the mean value is 



2[(-)*A, lt " + Vr<'' +1 >] ........ (3). 



This last expression then represents the loss ot energy to the primary waves in 

 consequence of the presence of the obstacle. From the value of A M obtained in (15), 7, 

 we see that the summation (3) consists of a series of terms arranged in descending 

 order of magnitude. Consequently, in determining its value we may limit our 

 attention to the first two terms. Hence the total loss of energy to the primary 

 waves is given very approximately by 



(4). 



