3 
4. A discussion of the paradox that the fluid appears to 
have a permanent forward displacement ultimately. 
The difficulty is shown to arise from the introduction 
of infinities without precise definition of conditions. 
Analytically the ambiguity occurs in the form of a 
double integral whose value depends upon the order 
of performing the integrations. 
1.—A circular cylinder of radius a and of infinite length 
moves through an infinite fluid with uniform velocity wu at 
right angles to its axis. The fluid is assumed to be perfectly 
continuous, frictionless, and incompressible; and under 
these conditions a certain continuous motion is determined 
in the fluid. Let the diagram in Fig. 1 represent a section 
at right angles to the axis of the cylinder, the circle with 
centre O being a section of the cylinder at any instant. 
zy 
FIGURE I. 
The fluid at any point P(7,6) is moving with velocity 
ua? /r? in a direction making an angle 26 with Oz, that is 
tangentially to a circle through P touching the axis of « at 
O. Thus the fluid at points on a circle such as OPA is 
moving tangentially to the circle at each point at a given 
instant. his solution gives the actual velocity of the fluid 
at any point at any time; it does not follow the motion of a 
given particle of the fluid. If we fix attention upon a fluid 
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