26 



THE INTERNAL WORK OF THE WIND. 



value, as the variable x is not one of the unknowns important in reaching the 

 desired result. 



In order that the trajectory described by the aeroplane may be definite, the 

 law of variation for the angle a between the aeroplane and the tangent of the 

 trajectory must be stated, for it is really by varying this angle at will that a bird 

 chances its course in the air. 



To realize the most favorable conditions, this angle should be diminished as 

 the velocity increases, for instance it can be made to vary inversely as the square 

 of the velocity ; this voluntary change of orientation can be accomplished, more- 

 over, without a perceptible expense of energy. 



Fig. 4. 



It is probable that this law of variation which we have attributed to the 

 angle a is not that which offers the most favorable conditions for flight, but if our 

 object can be attained in this special case, the proof will hold a fortiori under all 

 more favorable circumstances. 



Suppose R is the resistance of the air pressing normally on the aeroplane at 

 its center of gravity, p the radius of curvature of the trajectory, and t the time 

 (Fig. 4). The equations for the movement of the center of gravity are: 



!/ sin />'- 



in 



! di 



[ p =0CO3/J--COS« 



Furthermore: - - — ll ! s 



t) as 



as the arc s increases. 



the minus sign indicating that the angle ft decreases 



The resistance of the air R = — Vf( a ) '. * being the density of the air, A 

 the surface of the aeroplane, and /(«) a function of the angle a, experimentally 

 determined. As the angle a varies inversely as the square of the velocity, we 



