Vertical Force on Cylinder Submerged in Uniform Stream. 391 
The part arising from the interaction of the line distribution of doublets and 
the doublet at C is 
Empxyeat |S a Faye i) dp, (18) 
0 
which reduces to 
— 12f?) cos cop — 2f (3p? — 4 f?) sin kyp 
Brora [ pea Meio eg eS Ee ESI eG as 91 (19) 
: (Par ary 
This. integral may be evaluated by differentiating twice with respect to f the 
integral 
fs Pp COS Kop — 2f sin Kop dp = — eS Ii (e? * °F), (20) 
Jo jiP ap ee 
where Ji is the logarithmic integral. 
Collecting the terms from (17) and (19) we obtain finally 
Zt. 
Y= — {1 ++ QKof | 4x2 f? ram Srp em li (Gey, (21) 
This vertical force changes from upwards to downwards at a certain velocity. 
For when c is small, that is cof large, using the asymptotic expansion of the 
logarithmic integral we find that Y approximates to mec?a*/2f% ; on the other 
hand, when c is large, Y is approximately —pca*/2f%. The value of (21) 
can be calculated readily from tables and it is of interest to compare the 
relative values of R and Y and their variation with velocity. 
The figure shows R and Y graphed on the same scale on a base of c/V (gf). 
R is very small at low velocities and then increases rapidly to its maximum at 
301 
