The work requiring to be done, then, varies as the power 3½ of the bird’s linear dimensions, while the work of which the bird is capable depends on the mass of its muscles, and therefore varies as the cube of its linear dimensions[39]. The disproportion does not seem at first sight very great, but it is quite enough to tell. It is as much as to say that, every time we double the linear dimensions of the bird, the difficulty of flight is increased in the ratio of 2^3 : 2^{3½}, or 8 : 11·3, or, say, 1 : 1·4. If we take the ostrich to exceed the sparrow in linear dimensions as 25 : 1, which seems well within the mark, we have the ratio between 25^{3½} and 25^3, or between 5^7 : 5^6; in other words, flight is just five times more difficult for the larger than for the smaller bird[40]. The above investigation includes, besides the final result, a number of others, explicit or implied, which are of not less importance. Of these the simplest and also the most important is {26} contained in the equation _V_ = √_l_, a result which happens to be identical with one we had also arrived at in the case of the fish. In the bird’s case it has a deeper significance than in the other; because it implies here not merely that the velocity will tend to increase in a certain ratio with the length, but that it _must_ do so as an essential and primary condition of the bird’s remaining aloft. It is accordingly of great practical importance in aeronautics, for it shews how a provision of increasing speed must accompany every enlargement of our aeroplanes. If a given machine weighing, say, 500 lbs. be stable at 40 miles an hour, then one geometrically similar which weighs, say, a couple of tons must have its speed determined as follows: _W_ : _w_ :: _L_^3 : _l_^3 :: 8 : 1. Therefore _L_ : _l_ :: 2 : 1. But _V_^2 : _v_^2 :: _L_ : _l_. Therefore _V_ : _v_ :: √2 : 1 = 1·414 : 1. That is to say, the larger machine must be capable of a speed equal to 1·414 × 40, or about 56½ miles per hour.