On Flapping Flight of Aeroplanes. 

 FIG. 1. 



423 



By fig. 1 we see that if F be the vertical supporting force, 

 F = P a cos (s + a) = P a sensibly, 



since (5 + a) is supposed a small angle ; and if H be the horizontal 

 component of P a , 



H = -Pas 



= - Po(s + a) nearly. 



This is the resistance to forward motion horizontally, and the work 

 done against it, taking the average rate of working, is 



U = ~" 



ft.-lbs. per second. 



Again, if the bird's body be rising at the rate dZ/dt, and the wings 

 at a lesser rate, dz/dt, the engine attached to the body, which exerts a 

 downward force, F, on the wings, is doing work, so that, when F is 



positive, and (Z - z) positive, the engine does work, and this is 



Clt 



to be the same work, on the average, as that done to overcome the 

 resistance of the air to tjie plane taken along its actual path, and is 



Consequently we get, as a condition for determining some relations 

 among the constants, 



.(A) 



Another condition arises as follows. As F, the vertical force, is 

 sensibly equal to P a , we have, for the motion of the bird's body, 



