13 
difference between the two cases, the latter being included 
in the former. But we have seen that it is necessary to 
define conditions more precisely in order to avoid ambiguity. 
We may illustrate this by a definite problem in initial 
motion for which the solution is known. 
Let the fluid be contained within a fixed concentric 
cylinder of radius c, and let the inner cylinder be suddenly 
started with velocity wu. If ¢ is the velocity potential of 
the initial fluid motion, the boundary condition at the outer 
cylinder is that d¢/dr should be zero. The value of ¢ is 
The second part of ¢ represents the fluid motion already 
studied, with an additional factor c¢?/(c2—a?). Superposed 
on this there is a uniform flow backwards of amount 
ua? (c2—a?). The total momentum can be found by integ- 
rating throughout the liquid as before. In this case there is 
no ambiguity and it is easily shown that the second term in ¢ 
contributes nothing to the momentum. Adding the part due 
to the uniform flow, we find the total fluid momentum to be 
asa’u backwards; this result is independent of the radius 
of the outer cylinder, and, of course, agrees with elementary 
considerations. 
Now suppose the radius ¢ to become infinite. The fluid 
motion then differs from that studied in the previous sections 
only by a superposed uniform flow backwards of infinitesimal 
magnitude; but when integrated through the infinite extent 
of liquid it gives use to a finite momentum sau backwards. 
Further, in any finite time the additional term makes no 
more than an infinitesimal difference to the paths of the 
particles; but if we attempted to extend the solution to 
“infinite ” time we should be faced with various ambiguities 
in making any allowance for the extra term. 
The velocity potential ¢ for a finite extent of fluid is 
determinate when the values of ¢ or 5¢/dn are given over 
all the boundaries. If the outer boundary becomes infinite 
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