Vertical Force on Cylinder Submerged in Uniform Stream. 393 
the moment of an elementary doublet is 2xoca*dp/(p? + 4/7), and its axis 
makes with Ow an angle 
2 tan! (p/2f) — kop + 4x. 
Now at this stage the only additional terms of order a* for the turning moment 
arise from the interaction of the external uniform stream and the semi-circular 
distribution of doublets just described. 
It can easily be seen that the amount due to the uniform stream and an 
internal doublet of moment M, at an angle «, to Ox is 
— 2nceM, sin o,. 
Therefore the additional couple of the specified order is 
— 4rkoec?at |, sin {2 tan (p/2f) = kop-t0/2} 
0 P +4 Wie 
On reduction (25) comes to precisely the expression (24) with a minus sign 
dp (25) 
and therefore the couple on the cylinder is zero to the order specified. 
HARRISON AND Sons, Ltd., Printers in Ordinary to His Majesty, St. Martin’s Lane, 
303 
