twice the? momentum in the first layer or pFUdt, in the downward direction in Figure 111. By 

 the law of action and reaction, this must equal the opposite momentum given to the cylinder, 

 or LSt. Equation [73a] follows; and the direction of L is easily seen to be as stated. 



74. THE BLASIUS THEOREM. 



This theorem provides useful formulas for the force on a cylinder of any shape, and 

 also for the torque or moment of force, in the case of steady two-dimensional motion. 



Consider an element of the surface of the cylinder which has a width ds and unit length 

 in a direction perpendicular to the xy-plane, or to the planes of motion. Let the tangent to 

 ds, drawn in the counterclockwise direction around the cylinder, make an angle 6 with the 

 ^-axis, as shown in Figure 112. Then the force on the element due to the pressure p, taken 

 positive when directed toward the interior of the cylinder, has a magnitude pds and 

 Cartesian components 



Figure 112 — Force on element ds of the 

 surface of a cylinder due to a pressure p. 



dX = - p ds sin d, dY = p ds cos 



But ds cos 6 = dx, ds sin d = dy, where dx and dy are the components of ds. Hence 



X = -\pdy, Y = \pdx [74a, b] 



Also, by multiplying each component of force by its lever arm, the total moment L 

 about an axis passing through the origin is similarly found to be 



N = \p{xdx+y dy) = \ prdr ' [74c] 



164 



