ON PADDLE WHEELS. 



123 



much less the same as if the body were at rest. We have not yet met with any explanation 

 of the law according to which this pressure decreases, but we suggest the following. 



Let A be a reservoir of fluid, which we will suppose to be water, but the reasoning will 

 apply as well to any other fluid, and let B C be a bent tube 

 communicating with the reservoir by means of the aperture a, 

 at a depth d below the level of the water in the reservoir, 

 which is supposed to be maintained constant, the part B 

 being horizontal, and the part C vertical and open at the 

 extremity c, which is situated at a height h above the level 

 of the aperture a. It is evident that, neglecting the effect of 

 friction, the water will issue from the orifice c with the 

 velocity due to the head d h, or, calling v the velocity of 

 the water, v= ^ 2 g (d h) ; and, if the sectional area of the tube B be equal to the area of the 

 orifice c, the water in it will have the same velocity ; so that, if a solid piston were fitted into the 

 tube B, and made to move forward towards the branch C with the above velocity, the pressure 

 on both sides of the piston would be equal, or if the branch C were removed, the pressure of 

 the water in the reservoir on the after surface of the piston would be equal to the weight of a 



v 3 

 column of water, whose altitude is equal to h or d . Or it may be explained other- 



i/ 



wise, thus : Let a solid body, terminated by a plane at its stern end, move through water in 

 a direction perpendicular to that plane with a given velocity v. The water in immediate 

 contact with the surface must evidently have the same velocity if it remains in contact with 

 it ; to generate which velocity in the water a pressure is required equal to the weight of a 



v 2 

 column, whose altitude is ; this pressure cannot produce any effect on the surface, its 



*9 



force being already expended in producing the motion of the water, and must therefore be 

 deducted from the weight of the total column of water pressing upon the surface when 

 at rest. 



Applying this to the case of a prismatic body terminated at both ends by planes perpen- 

 dicular to its axis, moving through water in the direction of its axis, which is supposed to be 

 horizontal, with a certain velocity v, we find for the pressure on an indefinitely small portion 



v 2 



e of its after surface, situated at a depth d below the level of the water, e \d ] ; so that 



* ff 



if we suppose the pressure on the head end of the body to be the same as if the body were at 

 rest, i. e. e d on an element of the front surface equal and opposite to that on the after surface, 



v* v 2 



the excess of pressure on the former will be e [d (d 5)] or e ., which is equal to the 



y * & 



whole resistance as generally assumed. There seems to be no reason to suppose the head 

 pressure to increase with the velocity ; so that the only addition to be made to the above 

 resistance would be that produced by the cohesion of the particles of water. 



Another theorem, founded on the preceding, and applicable to paddle wheels, is the 

 following : 



Prop. 2. If a plane surface move with a given velocity through a fluid in a direction not 

 perpendicular to itself, the resistance is equal to the weight of a column of the fluid, whose 



