Oil Flapping Flight of Aeroplanes. 425 



part of it which extends from W = 2 -5 to W = 3 is shown with the 

 scale of abscissae magnified fifty times, as AB, and corresponding 

 curves for p are given. The case taken is when AY = 60, so that it 

 applies to flight at 60 feet per second, if the wing area is 1 square 

 foot per pound carried, 30 feet per second with an allowance of 2 

 square feet per pound, and so on. 



The principal things to note about the curves are that, in the first 

 place, W has a minimum value, in this case 2 '5 ft.-lbs. per second 

 per pound carried. This occurs when pS cos 6 is infinite, and practi- 

 cally implies that when the rate of flapping is very high, keeping the 

 same stroke, the horse-power required comes very near the minimum, 

 and, remarkably enough, the wing becomes, virtually, exactly equi- 

 valent to a plane of area A, inclined at a constant angle a = m, 

 moving horizontally, as in Langley's experiments. This is not acci- 

 dental, but seems to follow from the circumstance that 2 W = /*pS cos 0, 

 and if p$> cos 8 be infinite and W finite, p must vanish, and, conse- 

 quently, oc = m (1 - p cos 6) becomes a = m simply. 



[Note added March 21, 1899. In the paper as read an erroneous 

 statement here followed, to the effect that the minimum value of W 

 was half that got in Langley's experiments with a plane at angle a, = m. 

 It arose from the left-hand side in equation (A) being, by mistake, 



J'lnjp 

 VP a (s + a)cft, instead of as given above. The author is 

 o 

 indebted to Prof. J. Purser for pointing out the mistake. The least 



value of W is equal to that of Langley's experiments.] 



Of course, as before mentioned, it is not to be taken that the figure 

 2-65 ft.-lbs. per second per pound of bird, is here demonstrated to be 

 that which a real bird, with 2 square feet of wing per Ib. of bird, 

 flying at 60 feet per second, would or must exert. All that is pro- 

 fessed to be shown is, that the bird may be able to fly with the exertion 

 of two or three ft.-lbs. per second per pound, and that 30 to 1300 are 

 quite unnecessary. Further, it appears that, within a very large range 

 of the values of pS cos 0, that is frequency of flapping, stroke, and lag of 

 phase of stroke relatively to phase of bird's body motion and twisting 

 of wing, there may be little comparative variation in the horse-power 

 e.g., between frequency per minute 500 and frequency 250 (suppos- 

 ing S cos 9 = 1), there would only be a change of work per second 

 from 2-51 to 2 -55 scarcely 2 per cent. 



This seems to bear on the explanation of the great range of variation 

 in the manner of birds' use of their wings ; some (as pigeons and 

 ducks) flapping very fast, others (as seagulls and herons) flapping but 

 slowly, without any physiological reason for expecting material 

 differences in the rate at which their muscles can give out energy. 



