422 Prof. M. F. FitzGeralcl. 



force is, on an average, equal to that required, on an average, to propel 

 the aeroplane at its average inclination. Observe that the force P OT 

 depends for its magnitude, not on the actual inclination of the wing 

 path at all, but on the angle between that path and the plane of the 

 wing, while its horizontal component depends on both angles, so that 

 although, as long as there is any supporting force, there is a resistance 

 to forward motion along the wing path, there may nevertheless be a 

 forward force acting horizontally on the whole machine, wings and 

 body taken together, as in the first position shown, for instance. 



The thing to do, then, is to formulate this in symbols, and for the 

 present purpose it will suffice to make some simplifying approxima- 

 tions to facilitate the work. We shall then take it that the inclination 

 of the wing path to the horizontal, and the inclination of the plane of 

 the wing to the wing path, are both small enough to assume that for 

 either of them, as well as their sum, the circular measure, the sine, and 

 the tangent do not sensibly differ, and the cosine is unity, for example, 

 for 20 circular measure = 0*349, sine = O342, tangent = 0'363, and 

 cosine = 0*94 ; so that, up to this at any rate, we shall not be incur- 

 ring errors exceeding 5 or 6 per cent, by this assumption. 



We shall also assume that for a small angle (2) of inclination of an 

 aeroplane to its path, the resultant air pressure is given by the formula 

 Pa = 2&V 2 sin a in pounds per square foot, where V is velocity in feet 

 per second, and k a coefficient, which, according to Langley, is about 

 0-0017, and P tt is directed normally to the plane. This agrees nearly 

 with Du Chemins's formula at small angles, as pointed out by Langley. 



Now let the angle a be varied according to the law 



a = m(l - \L cospt), 



m and p being arbitrary constants, p being 2irf where / is the fre- 

 quency or number of flaps per second, t being time in seconds. 



Then the normal pressure, P a , on a 'plane of area A, at forward 

 velocity V, inclined at a to its direction of motion is 



P = 



For convenience we shall take the bird's weight as 1 lb., so that, 

 in general, P a will be, for any other weight, normal pressure in Ibs. 

 per lb. of bird, and A is wing area in square feet per lb. of bird. 



Let, at any moment, Z be the height above some datum level of the 

 bird's centre of gravity, and z that of centre of pressure of wings, 

 and suppose that 



z = Ssin(p*+0). 



Then the slope, s, of the wing path is at any moment sensibly 



supposed to be always a small angle. 



