44 ON MAGNITUDE [ch. 



becomes, the more swiftly must the air stream over the wing 

 to give rise to the rarefaction or negative pressure which is more 

 and more required; and the harder must it be to fly, so long as 

 work has to be done by the muscles of the bird. The general 

 principle is the same as before, though the quantitative relation 

 does not work out as easily as it did. As a matter of fact, there 

 is probably little difference in the end; and in aeronautics, the 

 "total resultant force" which the bird employs for its support is 

 said, e^npirically, to vary as the square of the air-speed: which is 

 then a result analogous to Froude's law, and is just what we arrived 

 at before in the simpler and less accurate setting of the case. 



But a comparison between the larger and the smaller bird, like 

 all other comparisons, applies only so long as the other factors in 

 the case remain the same ; and these vary so much in the complicated 

 action of flight that it is hard indeed to compare one bird with 

 another. For not only is the bird continually changing the incidence 

 of its wing, but it alters the lie of every single important feather; 

 and all the ways and means of flight vary so enormously, in big 

 wings and small, and Nature exhibits so many refinements and 

 " improvements" in the mechanism required, that a comparison based 

 on size alone becomes imaginary, and is little worth the making. 



The above considerations are of great practical importance in 

 aeronautics, for they shew how a provision of increasing speed must ac- 

 company every enlargement of our aeroplanes. Speaking generally, 

 the necessary or minimal speed of an aeroplane varies as the square 

 root of its Unear dimensions; if (ceteris paribus) we make it four 

 times as long, it must, in order to remain aloft, fly twice as fast as 

 before*. If a given machine weighing, say, 500 lb. be stable at 

 40 miles an hour, then a geometrically similar one which weighs, 

 say, a couple of tons has its speed determined as follows : 



W:w::L^:l^::S:l. 



Therefore L:l::2:l. 



But V^:v^::L:l 



Therefore V:v::V2:l = 1-414 : 1. 



* G. H. Bryan, Stability in Aviation, 1911; F. W. Lanchester, Aerodynamics, 

 1909; cf. (int. al.) George Greenhill, The Dynamics of Mechanical Flight, 1912; 

 F. W. Headley, The Flight of Birds, and recent works. 



