224 LECTURE XXIV. 



alone, would not produce any material retardation of its motion, if the par- 

 ticles of the fluid themselves were capable of moving on each other without 

 the least resistance ; for in this case a small portion of the fluid, in imme- 

 diate contact with the solid, might remain at rest, and the remaining mass 

 of the fluid might slide over this portion without any retardation. It ap- 

 pears, however, that the water in contact with the bottom of a river moves 

 with a very considerable velocity, and the water next above this only a 

 little faster, so that the velocity increases almost uniformly as we ascend 

 towards the surface. It follows, therefore, that the resistance must be much 

 greater where the particles of water slide on each other, than where they 

 glide along the surface of a solid. This internal friction operates gradually 

 throughout the water ; the surface being retarded by the particles immedi- 

 ately below it, those particles by the next inferior stratum, and each stra- 

 tum being actuated, besides its own relative weight, by the friction of the 

 water above, tending to draw it forwards, and by that of the water below 

 tending still more to retard it ; the retardation being communicated from 

 below upwards, in such a manner as to be every where equivalent to the 

 relative weight of the water above the part considered. It appears from 

 observation, that when we have determined the mean velocity in English 

 inches, we may find the superficial velocity, very nearly, by adding to it 

 its square root, and the velocity at the bottom, by subtracting from it the 

 same number : thus the square root of 48f being nearly 7, the superficial 

 velocity of the Ganges will be about 55 inches, or 4 feet 7 inches in a 

 second, and the velocity at the bottom 41 f. There are, however, frequent 

 irregularities in the proportions of the velocities at different depths, and it 

 has sometimes been observed, perhaps on account of the resistance of the 

 air, that the velocity is a little less immediately at the surface, than a few 

 inches below it. 



For similar reasons, the velocity of a river is also generally greater in 

 the middle than at the sides ; and the motion of the particles in the middle 

 must be retarded, not only by those which are below them, but also by 

 those on each side, while these, on the contrary, are dragged on by the 

 water in the middle : the middle parts tend, therefore, to draw the sides 

 towards them, which they cannot do, without lowering the surface of the 

 fluid next to the banks in such a degree as to make the difference of level 

 an equivalent to this tendency to approach the middle. This appears to be 

 the reason that the surface of a large river may generally be observed to 

 be slightly convex, or a little elevated in the middle. 



The course of a river is sometimes interrupted by a were or a fall, natu- 

 ral or artificial ; in such cases the velocity may be calculated in the same 

 manner as when a fluid is discharged from a reservoir through an aperture 

 of considerable height : supposing the whole section of the were to be such 

 an aperture, in a vessel so much higher, that the velocity of a fluid issuing 

 from it at the upper part of the aperture would be precisely equal to the 

 actual velocity of the river. The extent of the swell caused by a were, or 

 by any partial elevation thrown across the bed of a river, may also 'be 

 found by first determining the height at which the surface must stand 

 immediately above the were, and then calculating the inclination of the 



