Mr. W. M. Buchanan's Theory of the Reaction Water- Whetl 119 



cicnt to determine the quantity of action expended by the jot. Those forces 

 we know to be equal and contrary; and therefore, by ascertaining the power 

 expended in giving motion to the water ejected, we arrive at the true 

 measure of the reflex action produced upon the vessel. Now, the force 

 expended must obviously depend upon the velocity and volume of the 

 jet, and will therefore be known when those elements are found. If ^ be 

 the mass of a particle of the fluid, (moaning by the mass the weight 

 divided by gravity,) and V the velocity with which it is ejected at the 

 orifice, its vis viva is expressed by ^V 2 , and therefore 2^V 2 will bo the 

 sum of the vires viva? of all the particles ejected with that velocity referred 

 to a unit of time. But the number of particles which flow through the 

 orifice will obviously be as their velocity, and that velocity as the square 

 root of the height of the fluid-level above the orifice, representing the 

 pressure by which the particles are urged. Assuming, for simplicity, that 

 the orifice is formed in the bottom of the vessel, and that some means are 

 contrived for maintaining the water-level constant ; if we suppose that 

 under these circumstances, a lamina of water immediately over the orifice 

 is put in motion, at every indefinitely small instant of time, by the pressure 

 of the whole column of fluid standing above it, the entire gravitation of 

 the column, being employed in generating the velocity of the lamina, will 

 urge it forward by a force as much greater than its own weight as the 

 column' exceeds it in height, and through a space as much less, in the 

 same proportion. But when the forces are inversely as the spaces described, 

 the final velocities are equal, and, therefore, the velocity with which the 

 laminae of the water issue by the orifice must be equal to that which they 

 would acquire by falling in vacuo from the height of the surface of the 

 water to the orifice. Denoting this height by H, we shall then have the 

 relation V 2 = 2 </H, and consequently 



2^Y 3 = 2^ x 2H, 

 which is the sum of the weights of all the particles of a column of the 

 fluid of a height = 2H, and expresses the measure of the pressure which 

 is constantly being expended during the efflux of the water. But agreeably 

 to the principle of reaction equal and contrary to action, the orifice being 

 vertical, an equal amount of weight will be deducted from the entire pres- 

 sure of the fluid upon the bottom of the vessel. Now, what is true with 

 respect to the effect of a vertical jet must be equally true when the efflux 

 is lateral, since the vertical and horizontal components of prossure at equal 

 depths, and referred to the same unit of surface, are equal; and, there- 

 fore, the jet being projected horizontally, 2^ x 2H will represent the 

 weight which must be applied to the suspended vessel in the line and direc- 

 tion of the efflux, to prevent it from being deflected from its vertical posi- 

 tion. 



This conclusion may be arrived at simply by reflecting, that when part 

 of the weight of a body is expended in producing motion in any direction, 

 :ui rqual weight must necessarily be deducted from its pressure in the oppo- 

 site direction, since the gravitation employed in generating velocity can- 



