EMERSION AN ELEMENT OF RESISTANCE. 57 



Extensive series of experiments with the tube of Pitot, conducted by the 

 most eminent experimentalists, and confirmed by the accordance of collateral 

 phenomena, have established the doctrine as an axiom in hydrodynamics, That 

 the resistance of a small unit of surface to a fluid, when either the fluid is in mo- 

 tion, or the surface itself, is equal to the statical pressure of a column of fluid 

 having for its height the height due by gravity to that velocity. Had not this 

 been satisfactorily established by previous experiments, and universally received 

 as an unquestionable truth, my own experiments with the tube of Pitot would 

 have been sufficient to shew the truth of the doctrine, which is merely the con- 

 verse of the theorem, That the statical pressure of a column of fluid generates a 

 velocity in the effluent jet equal to that which is required by a heavy body fall- 

 ing freely by gravity through a height equal to the depth of the fluid. This sta- 

 tical quantity being the measure of the pressure of the fluid upon the anterior 

 surface of the immersed solid, will also be the measure of the quaquaversus 

 pressure of the fluid in every direction, and therefore will measure the pressure 

 of the water upon the vessel causing its emersion. Opposed to this we have 

 the downward pressure arising from the gravity of the solid. Now, the measure 

 of this pressure is the weight of the column of water displaced by the body, the 

 depth of which is equal to the depth of the statical immersion of the solid, and 

 each of these pressures is at every velocity equal to the other, and in the opposite 

 direction to it. Whence, 



Let s = Transverse section of Statical Immersion. 

 V = Velocity of Motion. 

 g = Measure of Gravitation. 

 s' = Section of Dynamical Immersion. 

 .'.vs z= Volume of Fluid displaced by Statical Section ; and 

 Ts' = Volume of Fluid displaced by D3niamical Section ; and 



— = Height due to the velocity v. 

 If P be the density of the fluid. 



^vp -. 



= svp- 



2^ 



'. S'V : 



= s (v- 



-?>^d 



S' = 



-.s{^- 



2gi 



Proceeding from this equation of the dynamical section, to determine the 

 variation of total resistance, on the condition of proportionality to the law of the 

 squares of the velocities, as regards that portion of the section of the solid which 

 remains immersed, from the general equation 



VOL. XIV. PART I. H 



