In the case of the flying bird (apart from the initial difficulty of raising itself into the air, which involves another problem) it may be shewn that the bigger it gets (all its proportions remaining the same) the more difficult it is for it to maintain itself aloft in flight. The argument is as follows:
In order to keep aloft, the bird must communicate to the air a downward momentum equivalent to its own weight, and therefore proportional to the cube of its own linear dimensions. But the momentum so communicated is proportional to the mass of air driven downwards, and to the rate at which it is driven: the mass being proportional to the birdãs wing-area, and also (with any given slope of wing) to the speed of the bird, and the rate being again proportional to the birdãs speed; accordingly the whole momentum varies as the wing-area, i.e. as the square of the linear dimensions, and also as the square of the speed. Therefore, in order that the bird may maintain level flight, its speed must be proportional to the square root of its linear dimensions.
Now the rate at which the bird, in steady flight, has to work in order to drive itself forward, is the rate at which it communicates energy to the air; and this is proportional to mÿ£¢Vÿ£¢2ã₤, i.e. to the mass and to the square of the velocity of the air displaced. But the mass of air displaced per second is proportional to the wing-area and to the speed of the birdãs motion, and therefore to the power 2ô§ of the linear dimensions; and the speed at which it is displaced is proportional to the birdãs speed, and therefore to the square root of the linear dimensions. Therefore the energy communicated per second (being proportional to the mass and to the square of the speed) is jointly proportional to the power 2ô§ of the linear dimensions, as above, and to the first power thereof: {25} that is to say, it increases in proportion to the power 3ô§ of the linear dimensions, and therefore faster than the weight of the bird increases.
Put in mathôÙeôÙmatôÙiôÙcal form, the equations are as follows:
(m =ã₤the mass of air thrust downwards; V its velocity, proportional to that of the bird; M its momentum; l a linear dimension of the bird; w its weight; W the work done in moving itself forward.)
But
Therefore
But, again,
The work requiring to be done, then, varies as the power 3ô§ of the birdãs 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 dimensions16. 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 2ÿ£¢3ã₤:ã₤2ÿ£¢3ô§ã₤, or 8ã₤:ã₤11ôñ3, or, say, 1ã₤:ã₤1ôñ4. If we take the ostrich to exceed the sparrow in linear dimensions as 25ã₤:ã₤1, which seems well within the mark, we have the ratio between 25ÿ£¢3ô§ and 25ÿ£¢3ã₤, or between 5ÿ£¢7ã₤:ã₤5ÿ£¢6ã₤; in other words, flight is just five times more difficult for the larger than for the smaller bird17.
The above inôÙvesôÙtiôÙgaôÙtion 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 =ã₤ãÿ£¢l, a result which happens to be identical with one we had also arrived at in the case of the fish. In the birdãs case it has a deeper significance than in the other; because it implies here