22 ON MAGNITUDE [ch. 



in the case of the aquatic animal there is, as Spencer pointed out, 

 a distinct advantage, in that the larger it grows the greater is 

 its velocity. For its available energy depends on the mass of 

 its muscles ; while its motion through the water is opposed, not 

 by gravity, but by "skin-friction," which increases only as the 

 square of its dimensions ; all other things being equal, the bigger 

 the ship, or the bigger the fish, the faster it tends to go, but only 

 in the ratio of the square root of the increasing length. For the 

 mechanical work {W) of which the fish is capable being pro- 

 portional to the mass of its muscles, or the cube of its linear 

 dimensions : and again this work being wholly done in producing 

 a velocity (F) against a resistance {R) which increases as the 

 square of the said linear dimensions ; we have at once 



W = l\ 



and also W = RV'- = l^VK 



Therefore P = IW\ and V = y/l. 



This is what is known as Fronde's Law of the correspondence of 

 speeds. 



But there is often another side to these questions, which makes 

 them too complicated to answer in a word. For instance, the 

 work (per stroke) of which two similar engines are capable should 

 obviously vary as the cubes of their linear dimensions, for it 

 varies on the one hand with the surface of the piston, and on the 

 other, with the length of the stroke ; so is it likewise in the animal, 

 where the corresponding variation depends on the cross-section of 

 the muscle, and on the space through which it contracts. But 

 in two precisely similar engines, the actual available horse-power 

 varies as the square of the linear dimensions, and not as the 

 cube; and this for the obvious reason that the actual energy 

 developed depends upon the heating-surface of the boiler*. So 

 likewise must there be a similar tendency, among animals, for the 

 rate of supply of kinetic energy to vary with the surface of the 



Forme et la Vie, 1900, p. 815). The effect of gravity on outward form is 

 illustrated, for instance, by the contrast between the uniformly upward branching 

 of a sea-weed and the drooping curves of a shrub or tree. 



* The analogy is not a very strict one. We are not taking account, for instance, 

 of a proportionate increase in thickness of the boiler-plates. 



