Vertical Force on Cylinder Submerged in Uniform Stream. 389 
doublet making an angle « with the axis of 7, we have with the same notation 
as before, 
y Met 1» Met (6) 
Forming (dw/dz)? we see that again the only terms which give any contribution 
to the integral (1) are the product terms in 7 and s, and for a typical term we 
have 
dz drt 
| (z— 2,)? (z aad z,)° OR , a ye @ 
Thus we obtain 
Ppt (ar+as) 
BNC ee MMei("* 8 
X—1 Te Dy Gar (8) 
and the contribution to the total force due to M, and M, is 
xX, = — 470MM, cos (a, + a, — 36,,) /Rre?, 
Woy = 4roM,M, sin (a, ap = 30,.) /B,.3, (9) 
0,, being the angle between Oz and the vector R,, drawn from the doublet 
(r) to the doublet (s). Further, calculating the total moment M from (2), 
the product terms M,M, are the only terms which give any value, and the 
corresponding contour integral is 
er ea ee (10) 
G—aFe—aP  &—*) 
Hence we obtain 
M = — 2notuM,Me “*™) (z, + 2,) (%, — 2), (11) 
the real part to be taken. 
On reduction it is seen that this consists of the sum of the moments of the 
forces given by (9) acting at the internal doublets, together with a couple for 
each pair of internal and external doublets of amount 
2neM,M, sin (a, + «, — 26,,) . /R,.”. (12) 
The contribution to the forces and moment on the body when the external 
field includes also a uniform flow can easily be obtained in the same manner. 
3. We now apply these expressions to a circular cylinder of radius a sub- 
mergedinauniformstream. Take Oz in the undisturbed surface of the stream. 
Oy vertically upwards ; and let the stream velocity be cin the negative direction 
of Ox. Let the centre C of the circle be at the point (0, —f). Then the image 
of the stream in the circle is a horizontal doublet at C of moment ca. The 
