II] OF RATE OF WALKING 41 



of the leg permits us to use it as the shortest possible lever while 

 it is swinging, and as the longest possible lever when it is exerting 

 its propulsive force. 



The bird's case is of peculiar interest. In running, walking or 

 swimming, we consider the speed which an animal can attain, and 

 the increase of speed which increasing size permits of. But in flight 

 there is a certain necessary speed— a speed (relative to the air) which 

 the bird must attain in order to maintain itself aloft, and which nfiust 

 increase as its size increases. It is highly probable, as Lanchester 

 remarks, that Lilienthal met his untimely death (in August 1896) 

 not so much from any intrinsic fault in the design or construction 

 of his machine, but simply because his engine fell somewhat 

 short of the power required to give the speed necessary for its 

 stability. 



Twenty-five years ago, when this book was written, the bird, or 

 the aeroplane, was thought of as a machine whose sloping wings, 

 held at a given angle and driven horizontally forward, deflect the 

 air downwards and derive support from the upward reaction. In 

 other words, the bird was supposed to communicate to a mass of 

 air a downward momentum equivalent (in unit time) to its own 

 weight, and to do so by direct and continuous impact. The down- 

 ward momentum is then proportional to the mass of air thrust 

 downwards, and to the rate at which it is so thrust or driven: the 

 mass being proportional to the wing-area and to the speed of the 

 bird, and the rate being again proportional to the flying speed; so 

 that the momentum varies as the square of the bird's linear dimen- 

 sions and also as the square of its speed. But in order to balance 

 its weight, this momentum must also be proportional to the 

 cube of the bird's linear dimensions; therefore the bird's necessary 

 speed, such as enables it to maintain level flight, must be pro- 

 portional to the square root of its linear dimensions, and the whole 

 work done must be proportional to the power 3 J of the said linear 

 dimensions. 



The case stands, so far, as follows : m, the mass of air deflected 

 downwards; M, the momentum so communicated; W, the work 

 done — all in unit time; w, the weight, and F, the velocity of the 



we know that the pendulum theory is not the whole story, but only an important 

 first approximation to a complex phenomenon. 



