ON THE FRICTION OF FLUIDS. 223 



the same, whatever the velocity may be : nor does the friction increase with 

 the pressure, as is demonstrated by an experiment of Professor Robison 

 on the oscillations of a fluid through a bent tube, terminated by two bulbs, 

 which were performed in the same time, whether the tube was in a hori- 

 zontal or in a vertical position. Mr. Coulomb has also proved the same 

 fact by experiments on the vibrations of bodies immersed in fluids, and 

 suspended by twisted wires ; he finds that precisely at the surface, the 

 friction is somewhat greater than at any depth below it : he also considers 

 a certain part of the friction as simply proportional to the velocity, and 

 a very small portion only, in common fluids, as perfectly independent 

 of it.* 



It is obvious that wherever the friction varies as the square of the velo- 

 city, or even when it increases in any degree with the velocity, there must 

 always be a limit, which the velocity can never exceed by means of any 

 constant force, and this limit must be the velocity at which the resistance 

 would become equal to the force. It is for this reason that a light body, 

 descending through the air, soon acquires a velocity nearly uniform ; and 

 if it be caused, by any external force, to move for a time more rapidly, it 

 will again be speedily retarded, until its velocity be restored very nearly to 

 its original state. In the same manner the weight of the water in a river, 

 which has once acquired a stationary velocity, is wholly employed in over- 

 coming the friction produced by the bottom and the banks. 



From considering the effect of the magnitude of the surface exposed to 

 the friction of the water, in comparison with the whole quantity contained 

 in the river, together with the degree in which the river is inclined to the 

 horizon, we may determine, by following the methods adopted by Mr. 

 Buat, the velocity of any river of which we know the dimensions and the 

 inclination. Supposing the whole quantity of water to be spread on a hori- 

 zontal surface, equal in extent to the bottom and sides of the river, the 

 height at which it would stand, is called the hydraulic mean depth ; and 

 it may be shown that the square of the velocity must be jointly propor- 

 tional to the hydraulic mean depth, and to the fall in a given length. If 

 we measure the inclination by the fall in 2800 yards, the square of the 

 velocity in a second will be nearly equal to the product of this fall multi- 

 plied by the hydraulic mean depth. For example, in the Ganges, and in 

 some other great rivers, the mean depth being about 30 feet, and the fall 4 

 inches in a mile, the fall in 2800 yards will be about 6 inches, which, 

 multiplied by 360 inches, gives 2340 inches for the square of the mean ve- 

 locity, and 48 inches, or about four feet, for the mean velocity in a second, 

 that is, not quite three miles an hour, which is the usual velocity of rivers 

 moderately rapid. If, however, great precision were required in the deter- 

 mination, some further corrections would be necessary, on account of the 

 deviation of the resistance from the exact proportion of the squares of the 

 velocities ; since the friction, as we have already seen, does not increase 

 quite so fast as this. 

 Jit is obvious that the friction of a fluid, moving on the surface of a solid 



* Hist, et Mem. de Paris, 1784, p, 229. Mem. de 1'Institut, vol. Hi. Phil. 

 Mag. vii. 183. 



