The Forces on a Circular Cylinder 530 
‘Substituting the power series (11) for f(«) on the right of (17), we obtain 
the infinite set of equations 
4 5 
bol + y*) + Gev%b1 + a Bs ar a by +... = | 
G2¥° ur) Gav? = 
eV by + (14+ BY) by + AO b, + JO bg +... = 0 iit 
qv" Gav q5¥ WX ai 
Be bo + LY oy + (1+ A) bs + BE B+. =0 
Peary srg eal ORIYA OID rie Sl gaet when LN h <u itisy =0 
From the integral expression for q, given in (19), and also the fact 
that a/f< 1, it can readily be shown that the infinite determinant formed 
by the coefficients of bo, b,, ..., on the left of (20) is convergent. 
Evaluating the expression (19) and putting 
Os, = Fi, = US, (21) 
we find 
— ip! Ni 
r= A {CD Ge tte tie}, 
a 
Lae) (22) 
where « = 2«,f, and /i denotes the logarithmic integral. 
For any given values of a, f, and c, we have in (20) a set of equations for 
the b’s with complex numerical coefficients. 
Although expansions in terms of other parameters may be more suit- 
able for special ranges, it is convenient to assume that the coefficients b 
can be expanded in power series of the quantity y, that is kya. These 
expansions will be of the form 
by = 1+ Doay® + Boay* + Dosy® + --- 
b, = bygy? + Bysy° + bry" = 000 
Dey = = saat + bog y® + bogy® + O06 
° (23) 
Substituting in (21) and collecting the various powers of y, the new 
coefficients may be found to any required stage. For the calculations 
which follow, it was found sufficient to obtain the results: 
boo = —- hh 
bo = 4% 
bog = 292 — G0 
Diy = Che = CRO AP Say Oe 
424 
