BY SMALL OBSTACLES OF CYLINDKICAL AND SPHERICAL FORM. 247 



and using the approximations for D 2 () and D () given in (8), we have 



D (Aw) =-| 



77 



D (/(-a) = - 



TT 



Substituting these expressions for D (ka) and D 2 (&) in the above formula for A lf 

 we obtain 



A = -TT -'' 



77 ' 



_ _ 



XV ' log ^\a + y + ^ITT + t|XV (log |Xa + y + \ LIT- -) ' 



In general, it will tend to sufficient accuracy if we take 



AI = -^7r{logiX + y +ir}-' ....... (15) 



in the case when \a is small. 



Similarly we obtain from (11) the approximation 



B^cTT^OoglXa + y + ltTr)- 1 ....... (1G), 



when \a is small. 



Let us next consider the case when X is large. 

 Since | ka is great, we may write 



' te+1 4 



D (ka) = 



Substituting these expressions in the formula given for A, in (13), we obtain 

 approximately 



A > = ^ Ml ~ 



When \ae~ l ''" is written for ka, this reduces to 



A! = itTT/iV [1 + v/2 . (\a)- 1 -1/iV (log %ha + y + f) 



-i{ v /2.(X)- 1 +|(\a)- a + i7r/iV}] . . (17). 



As a first approximation to the value of B 1; we find in the case when X is large 



B 1 = 2ifca(--Y / V ta+1 - to) ........ (18). 



The approximate value for A , obtained from equation (4) with the help of (8) and 

 (9), is easily seen to be 



A = -l7r/tV{l + ^V(log|-Aa+y+^7r-f)} ..... (19). 



