The integrals in these equations can be evaluated either for the entire contour of a 

 stationary body or for any part of it, or even for part of a streamline in the fluid. In any case 

 X and y stand for the components of the total force transmitted toward the left across the 

 chosen path of integration, and L for its moment about an axis through the origin. 



The new formulas are especially advantageous when the path of integration is closed. 

 Then, if dw/dz is given by a mathematical function that is analytic on the path, and also 

 throughout its interior except for a few singular points, the integrals are given at once by 

 the sum of the residues of the integrand at the singular points. It does not matter if these 

 points actually lie in a region devoid of fluid. 



As an example, the Blasius theorem may be used to prove the Kutta-Joukowski theorem. 

 Under the conditions specified in Section 72, with the cylinder stationary, dw/dz is a regular 

 function everywhere outside of the cylinder; hence, by the Cauchy integral theorem, the path 

 of integration may be displaced toward infinity in all directions. As before, let the motion be 

 steady; and at infinity let the fluid approach at speed V from a direction making an angle y 

 with the positive a;-axis, so that its components of velocity are -V cos y, -U sin y. Then, 

 in view of the results in Sections 35 and 72, w can be written for large z in the form 



ir ^ *2 



w ^ He 'y z + — In 2 + a. + — + [74i] 



2n- ° 3 ,2 



where F is the circulation about the cylinder. The origin may be located at any finite point. 

 Then, in powers of 1/z, 



dw . ir h "*2 ^ ^ 



— = Ve-'y+ — - — - [74j] 



dz 2nz ^2 ^3 



— =t/2e-2'y+ e-'y-[ +2 6, f/e-^y — 



dz ^z \4^2 y ^2 



Upon substituting this series for dw/dz in Equation [74g] and noting that ^ dz/ z'^ = 2ni 

 if 71 = 1 but = for other integral values of n, as shown in Section 30, it is found that 



? I ? \ dz 



2 \n I z 



Hence 



X = - pTV sin y, y = pTV cos y. [74k, 1] 



The magnitude of the force per unit length is thus {X^ + Y")'''^ = pVV, 



166 



