163 



The law of the square of the velocity does not hold except in a 

 few instances, and in peculiar circumstances, as regards resistance 

 of fluids to the motion of floating bodies. If the water be not very 

 deep, and if the velocity be more than three or four miles an hour, 

 the deviation will become exceedingly striking. The following are 

 causes of the deviation : 



I. An Emergence of the floating body from the fluid takes 

 place at all velocities. This emergence is produced by the velocity 

 of the bod3\ Wlien at rest the body, in virtue of the force of gra- 

 vitation, displaces a portion of fluid equal to its own weight; but if 

 it be moved it will require to displace a greater and greater por- 

 tion of fluid in a given time. Now the force of gravity acting 

 downwards on the body is constant force, and the force with 

 which the fluid must be displaced is an increasing one ; by comparing 

 these two quantities together the following result is obtained — 



Where R is the resistance, s the transverse section of statical dis- 

 placement, V the velocity of the body, g the measure of gravity, and 

 g the density of the fluid, and by differentiating we find R a maxi- 



4 o 

 mum, when r ^ -^ and the immersion =: when v =: 2g. 



II. The Wave of the fluid interferes with the resistance. Those 

 portions of a fluid which are thrown aside by the prow of a vessel 

 form Waves. These waves are propagated in the direction of the 

 motion of the vessel. 



The wave of a fluid moves with a velocity which is nearly uni- 

 form. 



The velocity of the wave is dependent solely upon the depth of 

 the fluid, reckoning from the summit of the wave. It is quite inde- 

 pendent of the width of the channel. It is quite independent of 

 the velocity of the vessel which creates the waves. The velocity of 

 the wave is equal to that which would be acquired by a body fall- 

 ing freely by gravity through a space equal to half the depth of the 

 fluid. 



The resistance of a fluid at a given velocity is intimately connect- 

 ed with the velocity of the wave in the fluid. 



When the velocity of the floating body is less than that of the 

 wave, the effect of the wave is to increase the resistance of the 

 fluid. 



