529 T. H. Havelock 
(12) back to the form (5); it gives, with the form (11) for f(x), the 
series 
S ca? (ia)"— 2", (13) 
1 
Further, the last term in (12) represents the potential of image sources 
and sinks in the region of the plane for which y >2f, and hence it can be 
expanded in the neighbourhood of the circle |z| =a in a series of 
ascending powers of z. Thus we obtain w in the form 
ee) (ee) 
Wie CONS C2 Cas (ia) a Date Bazi 
1 1 
(__7\n+1 poo ee ' 
3 = Ae | Bay SO as e-2* f* (ie) de, (14) 
n! g k= Ko = 1 
With the potential in the form 
w = const -+ S (C8 EID) 2-8), (15) 
1 
the condition of zero normal velocity on the circle | z | = a is satisfied, 
provided 
1D) == Fe (Ce. (16) 
Hence, from (14) we obtain the equations 
b=1-—2 ie oly or Ue icf (Kk) e *F dic, 
0 
= hey ae Up 
m+1 poo 7 
pres = | ee ees dr. (17) 
0 0 0 
These relations, with (11), may be expressed in the form of an integral 
equation satisfied by the function J («a); it is easily found to be 
a eee MPuty+ipv aban = 
vif (v) = v! I caecata Yuta ut (u) I, (2 Vuv) du, (18) 
where + = x oa, I, is the modified Bessel function, and the limit of the 
integral is to be taken as the positive quantity, » approaches zero. 
For purposes of calculation, we use (17) as a set of linear equations for 
the coefficients bo, b,, .... We write 
Jin (Paste le gm ~2Kofte yh Gf 19 
ee ee ee Uu u. ( ) 
