430 On Flapping Flight of Aeroplanes. 



evidently practically coincide with the rate at which gravity works, 

 when the rate of sinking is, on an average, uniform, and should be 

 quite small. The case of hovering flight of a bird is manifestly closely 

 analogous, and that of progressive flight is similarly closely analogous 

 to that of the same oyster shell, or a piece of slate, projected under 

 water and forcibly maintained in a state of rhythmic motion. The 

 only use of the investigation given in the present paper is to reduce 

 the matter to figures, in a case sufficiently simple to enable us to use 

 numbers supplied by existing experimental results. 



[Note added March 21, 1899. The error alluded to in the previous 

 note was detected by checking equation (A) and that for H to see if 

 the horizontal component of the acceleration was exactly periodic. 

 Equation (C), as now given correctly, is the condition in question, 

 with the additional information that each side is the work done. Mr. 

 F. Purser, F.T.C.D., suggested that the wings as well as the body 

 might be supposed heavy. The author has examined this point, and 

 finds that, supposing, as before, that the engine, say a cylinder and 

 piston, to whose rod the aeroplane is attached, works vertically, no 

 difference is made in equation (C). The force Pa being inclined, has, 

 or may have, a moment about the centre of inertia of the whole 

 machine ; the effects of this should be wholly periodic, but inasmuch 

 as it appears from experiment and otherwise that P does not pass 

 through the centre of figure of the plane, and moves about in some 

 way depending on the angle a, it would be impossible to test this con- 

 dition without assuming details of dimensions, &c. Any actual bird 

 or flying machine, must have some steering apparatus, capable of 

 correcting disturbances of a rotational kind in both the vertical and 

 the horizontal plane, but its management plainly cannot involve any 

 material alteration in the work to be done. The author conceives it 

 to be possible, if not highly probable, that the motion in flying is, in 

 respect of direction, unstable. Small periodic terms may be assumed 

 also to be added to V, but lead to nothing up to the order of quantities 

 involved in neglecting the difference between cosine and radius.] 



