9 
4.—We consider now the paradox to which reference has 
been made above, returning to the solution of the first two 
sections. If we imagine the fluid to be contained within a 
fixed vessel it is clear that, as the cylinder is moved forward, 
an equal volume of fluid must be displaced backwards. The 
same argument should hold continuously as we suppose the 
containing vessel increased indefinitely, and hence in the 
limit, when we eonsider motion in an infinite fluid subject to 
its being at rest at infinity. Thus there should be a per- 
manent displacement of the fluid backwards on the whole. 
But, according to the paths drawn in Fig. 2, we find that 
every particle comes to rest ultimately at some point D in 
advance of its initial position A; so that there appears to be a 
displacement of the liquid forwards. The interest of this 
paradox lies partly in its recurrence in various writings. 
Lanchester! states the difficulty and leaves it with the 
remark: “it is evident that some subtle error must exist in 
Rankine’s argument, the exact nature of which it is difficult 
to ascertain.” Taylor? points out how with a finite displace- 
ment of the eylinder it can be verified that the fluid is dis- 
placed backwards, but with an inflnite displacement one has 
the curious result of a permanent forward displacement. 
Maxwell? raised the same problem many years ago; he 
admits it as a real difficulty and disposes of it thus: “It 
appears that the final displacement of every particle is in the 
forward direction. It follows from this that the condition 
fulfilled by the fluid at an infinite distance is not that of 
being contained in a fixed vessel; for in that case there 
would have been, on the whole, a displacement backwards 
equal to that of the cylinder forwards. The problem actually 
solved differs from this only by the application of an 
infinitely small forward velocity to the infinite mass of fluid 
such as to generate a finite momentum.”’ 
The difficulty arises chiefly from a loose use of the idea of 
“FW. Lanchester, Aerodynamics, vol. 1, Aerodonetics, p. 20, 1909. 
?D.W. Taylor, Speed and Power of Ships, p. 10, 1910. 
“tio Os Maxwell, Scientific Papers, vol. ii., p. 210, 1870. 
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