The forces on the cylinders in any of the preceding cases, when the motion is steady, 

 are most easily found from the Blasius theorem, which will be proved in Section 74. When 

 only the two original vortices are present, from [42a] 



\dz' \2-C 2 + C/ 



Substitution for (dw/dz)^ in Equation [74g] gives, after a slight algebraic change, 



where A", and Y^ represent x and y components of the force per unit length on any cylinder due 

 to fluid outside it, and the integral is to be taken around the circle representing the cylinder. 

 The integral is easily evaluated by the method of residues as explained in Section 30. If the 

 cylinder encloses the singular point (-c,0) but not (c,0), 



2+ C 



whereas all other terms of the integrand give zero. Thus 



A^ pT^ 



X, -iY, = 

 ^ ^ c 47rC 



where F = 2rtA and represents the circulation around the cylinder. Since pr^/4:iTC is real, 

 y = 0, and the total force per unit length on the cylinder is 



X,=^ [42d'] 



^ 477C 



If the cylinder encloses the point (c,0) instead, 



dz 



(z-c) 



2ni 



and the sign of A'j is reversed. 



If another cylinder or a wall is present, as in Cases A, C, E, an equal and opposite 

 force acts on it. The force on a wall is easily verified by direct integration of the Bernoulli 

 term in the pressure. 



If there is only an ideal line vortex at the point (c,0), the reactive force may be imagined 

 to act on the vortex, but the formula for the force is correct only if the vortex is assumed to be 

 stationary. If the fluid and stationary vortex are inside a cylindrical shell, the force on the 

 shell is the same as if an inner cylinder were present with a circulation F around it equal to 

 that around the vortex. 



The direction of the force on a cylinder or on the wall is in all cases such as to draw 

 it toward the other cylinder, or toward the wall or vortex, along the shortest path between them. 



97 



