II] THE PRINCIPLE OF SIMILITUDE 25 



that is to say, it increases in proportion to the power 3| of the 

 linear dimensions, and therefore faster than the weight of the 

 bird increases. 



Put in mathematical form, the equations are as follows : 



{m = the mass of air thrust downwards ; F its velocity, 

 proportional to that of the bird; M its momentum; I a Hnear 

 dimension of the bird ; w its weight ; W the work done in moving 

 itself forward.) 



M = w = l^. 



But M^mV, and 7n = PV. 



Therefore M ^ l^ V^ 



and ?2 y2 _ jz^ 



or 7 = V?- 



But, again, W = mV^ ^PV x V^ 



= PxVl^l 



The work requiring to be done, then, varies as the power 3| 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 hnear dimensions*. 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 hnear dimensions 

 of the bird, the difficulty of flight is increased in the ratio of 

 2^ : 2^-, 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^* and 25^, or 

 between 5'^: 5^; in other words, flight is just five times more 

 difficult for the larger than for the smaller birdf. 



The above investigation includes, besides the final result, a 

 number of others, explicit or implied, which are of not less im- 

 portance. Of these the simplest and also the most important is 



* This is the result ari'ived at by Helmholtz, Ueber ein Theorem geometrisch 

 ahnliche Bewegungen fliissiger Korper betreffend, nebst Anwendung auf das 

 Problem Luftballons zu lenken, Monatsber. Akad. Berlin, 1873, pp. 501-14. It 

 was criticised and challenged (somewhat rashly) by K. Miillenhof, Die Grosse 

 der Flugflachen, etc., Pfluger''s Archiv, xxxv, p. 407, xxxvi, p. 548, 1885. 



t Cf. also Chabrier, Vol des Insectes, 3Iem. 3Ius. Hist. Nat. Paris, vi-viii, 

 1820-22. 



