On Flapping Flight of Aeroplanes. 429 



as before, remembering, however, that though a may be assumed 

 small, the angle of slope, s, of the wing path, is not, we shall find, if the 

 orbit be taken to be circular, m = 1/#V 2 where V is the velocity of 

 the motion along the orbit. In order that m may have values similar 

 to those in the case of progressive flight, the frequency of flapping 

 must be much higher, twice as fast, or thereabouts. But there are 

 several considerations which render results of this kind much more 

 unsatisfactory than those previously obtained. In the first place 

 the motion assumed is very much less like that of a bird's wing 

 than in the former case, inasmuch as it involves complete revolution 

 of the plane of the wing about a horizontal axis in its own plane, 

 and, in the second place, the edges of the plane are continually cutting 

 across the eddies, created by their previous motion, in a very different 

 way from that in which they interfere with those created in progressive 

 flight, so that the numerical value of k, and the value of the function 

 of V which is put down as 1 in the formula ffiZ/dP = g(F - 1) are 

 really unknown, and cannot be inferred from experiments with soaring 

 planes, or even propellers. We can only expect that the actual num- 

 bers will be tolerably like those given by Langley's and other such 

 experiments. 



All that can be inferred, therefore, is that, provided a sufficiently 

 high speed of flapping is attainable, we may reasonably anticipate that 

 the horse-power for hovering need not differ very materially from that 

 for progressive flight. Internal stresses in the wings or machinery, 

 due to their inertia, as well as physiological difficulties connected with 

 high velocities of reciprocation, may also come in to limit the rate of 

 flapping, and prevent an equally economical rate of working being 

 attainable in hovering, as in progressive flight. A considerable 

 simplification of the mere mathematics could have been effected by at 

 once assuming that the arrangement for altering the angle a was so 

 contrived as to make the pressure Pa, or its vertical component, F, an 

 assigned function of the time; but this has the objection that, there being 

 then no expressly formulated relation between a and P or F, experimental 

 evidence would be wanting to settle the values of the numerical con- 

 stants involved. In fact, the more the mathematical work is 

 generalized, the less definite do the numerical results become, in the 

 present state of our experimental knowledge, and this must form the 

 writer's apology for working out, in a mathematically clumsy way, so 

 limited a case of a problem which has been treated already, in a much 

 more powerful and general way, by far abler hands. For example, 

 Thomson and Tait mention, as an example of motion of a solid in a 

 liquid, the vibratory and irregular movements of such an object as an 

 oyster shell sinking pretty slowly with a swaying motion when thrown 

 flatwise into water. The rate of expenditure of energy necessary to 

 prevent its sinking would, if these motions were forcibly produced, 



