On Flapping Flight of Aeroplanes. 427 



unity imply (on reference to the equation a = m (1 - /x cos ft) ) that 

 there is a reversed pressure on the wing during part of the stroke, as 

 shown in fig. 1 at the third position, where P 3 is directed partly down- 

 wards. This condition is however unfavourable, probably in all cases, 

 to economy of labour, though it may be favourable to forward pro- 

 pulsion. At all economical rates of working p is quite small, that 

 is, the inclination of wing plane to wing path varies but little. 



It may also be observed that, for any value of pS cos 0, there are 

 two of W, one of which lies close to its least possible value, and the 

 other is very large e.g., for pS cos = 15, W = 2'65 or about 40, 

 so that a person who was going on observed values of pS and took 

 cos to be nearly unity, might easily overlook the small value of W, 

 and base his estimate of work on the large one. This is the more- 

 likely as the curve of pS cos is of the third degree, and the solving 

 of a cubic by a direct process, especially when there are three real 

 roots, is troublesome, and avoided in consequence. Whether anything 

 of this kind occurred in Navier's or other earlier investigations, the 

 present writer is unable to say. Langley says that Navier added the 

 work done, here expressed by the right-hand side of equation (A) to 

 that on the other to find the whole. Why it should have seemed 

 necessary to do this is not clear, for if the wing has no sensible mass, 

 as here assumed, the pressure of the engine downwards cannot exceed 

 F, and no work can be done by a vertically moving engine, except 

 that expressed on the right-hand side of equation (A). The way in 

 which forward motion is maintained bears some analogy to the process 

 of so-called " invisible " skating, and to screw propulsion, wherein 

 it would be manifestly absurd to add the work done against the ship's 

 resistance to that given by the turning moment applied to the screw, 

 to find the horse-power of the engine. In fact, the construction of 

 Langley's whirling table virtually made the aeroplane a portion of a 

 screw propeller with axis vertical, and his coefficient k has, bound up 

 in it, the inefficiency of the propeller in converting horse-power of 

 turning couple into thrust. 



With regard to the absolute magnitude of the figure 2 '5 as the least 

 value possible for W in the case illustrated by fig. 2, it is of impor- 

 tance to note that this depends on m, and m has been made = 

 on the strength of the equation 



which again depends on the assumption that the downward accelera- 

 tion of the body attached to the aeroplane would be g, irrespective of 

 the horizontal velocity of the plane, if the angle of inclination of the 

 plane to its path were zero. Langley's experiments on the " Plane 



