388 T. H. Havelock. 
couple on the cylinder. This should, of course, be zero for a complete solution ; 
it is verified that the method used here gives zero moment up to the stage of 
approximation in terms of the ratio of the radius of the cylinder to the depth 
of its centre. 
2. Consider steady two-dimensional flow of a liquid of density p past a 
solid body, the motion being irrotational and there being no field of force. The 
motion being specified in the usual manner by a function w of the complex 
variable x + iy, the resultant force (X, Y) on the solid and the moment M 
about the origin are given by 
AY ce Sad () ih, (1) 
a 
C 2 
M = —1e| (2) ih, (2) 
where in (2) the real part is to be taken. In each case the integration is taken 
round the contour of the rigid body, or indeed round any contour enclosing 
the body but excluding any external sources and sinks. 
Now suppose the motion to be given by 
w = — Xm, log (z — z,) — um, log (z — z,), (3) 
where the suffix s refers to the given distribution in the liquid, and r to the 
image system within the surface of the body, m, and m, being real. 
Forming (dw/dz)*, we see that this quantity has simple poles at the points 
z, within the contour of integration ; and we obtain at once from the theory 
of residues 
ON ee aso ee) (4) 
Z—Z 
T <8 
the summation extending over the external and internal sources taken in 
pairs. Hence we obtain 
X = 2r0 Lm,m, (x, — 2,)/R,.”, 
Y = 29 m,n, (Ys — y;)[R,<?. (5) 
It follows that the resultant force is the same as if each pair of external and 
internal sources attracted each other with a force 2mpm,m,/R,,, where R,, 
is the distance between them. 
It may easily be verified in the same way from (2) that the moment M is 
accounted for by the same forces acting at the internal sources. It is con- 
yenient to have a similar analysis for doublets, If M is the moment of a 
