11 
denies that the limit of the dotted curve in Fig. 5 can ever be 
attained. ‘These two arguments can, of course, be stated 
simply in terms of the flow of liquid at any instant across a 
line behind the cylinder. If we draw lines OQ, OR at 45° to 
the line OO!, then at any time the flow across AO'B is 
forwards in the range QO!R and backwards beyond Q and 
R. According to (A), the range QO!R can be made infinite 
by taking w large enough; while the argument (B) points to 
the region within lines at 45° and 90° to the axis OO!. 
Analytically, the matter reduces to the evaluation of a 
double integral which gives different values according to the 
order in which the integrations are performed. We can see 
this by writing down an expression for the total momentum 
of the whole liquid in the direction of motion of the cylinder. 
Referring to Fig. 1, we have wa?r-? cos 26 for the component 
fluid velocity at any point, or wa?(a? — y?)/(z? + y?)? in terms 
of rectangular coordinates. Thus with s for the density of 
the fluid, the total momentum forward is given by 
z2 — y2 
M = ua? 2s [, @ ae ;drdy, 
where the integration extends throughout the fluid. 
We divide the integration into two parts, writing f for 
(v? —y*)/(@? + y*)*. First, the region abreast of the cylinder, 
extending to infinity in both directions, gives without 
ambiguity 
M,= suars fide f fay =—7sau. 
Var — 
For the rest of the fluid, fore and aft of the cylinder, we 
have 
Why = 4ua?s ff fdedy, 
where 7 ranges from a to o0, and y from 0 to 0. 
The integral M, has difterent values according to the 
order in which the integrations are performed. We have 
2» =4ua* Siy J fle = 2Qrsa?U. 
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