II] OF THE SPEED OF A SHIP 31 



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

 speed. 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 the linear 

 dimensions*: whence, 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 velocity (F) 

 which the fish attains depends on the work (W) it can do and the 

 resistance (R) it must overcome. Now we^ have seen that the 

 dimensions of W are l^, and of R are l'^ ; and by elementary mechanics 



WocRV^ or F^oc^. 



73 



Therefore V^oCr-=l, and F oc Vl. 



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

 of speeds — a simple and most elegant instance of "dimensional 

 theory!." 



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 vary as 

 the cubes of their linear dimensions, for it varies on the one hand 

 with the area of the piston, and on the other with the length of the 

 stroke; so is it likewise in the animal, where the corresponding 

 ratio depends on the cross-section of the muscle, and on the distance 

 through which it contracts. But in two similar engines, the available 

 horse-power varies as the square of the linear dimensions, and not 

 as the cube ; and this for the reason that the actual energy developed 

 depends on the heating-surface of the boiler J. So likewise must 



* We are neglecting "drag" or "head-resistance," which, increasing as the cube 

 of the speed, is a formidable obstacle to an unstreamlined body. But the perfect 

 streamlining of whale or fish or bird lets the surrounding air or water behave like 

 a perfect fluid, gives rise to no "surface of discontinuity," and the creature passes 

 through it without recoil or turbulence. Froude reckoned skin-friction, or surface- 

 resistance, as equal to that of a plane as long as the vessel's water-line, and of area 

 equal to that of the wetted surface of the vessel. 



t Though, as Lanchester says, the great designer "was not hampered by a 

 knowledge of the theory of dimensions." 



X The analogy is not a very strict or complete one. We are not taking account, 

 for instance, of the thickness of the boiler-plates. 



