R I V E R. 
height through which a heavy body would fall in a second) 
is the parameter, the velocity of this particle will be repre¬ 
sented by the ordinate P D of this parabola; that is, P I) is 
the space which it would uniformly describe in a second. 
From this principle is derived the following theory of run¬ 
ning waters. 
Let D C (fig. 2.) be the horizontal bottom of a reservoir, 
to which is joined a sloping channel C K of uniform breadth, 
and let A B be the surface of the standing water in the reser¬ 
voir. Suppose the vertical plane B C pierced with an infinity 
of holes, through each' of which the water issues. The 
velocity of each filament will be that which is acquired by 
falling from the surface A B. The filament C, issuing with' 
this velocity, will then glide down the inclined plane like 
any other heavy body ; and (by the common doctrine of the 
motion down an inclined plane) when it has arrived at F, it 
will have the same velocity which it would have acquired by 
falling through the height O F, the point O beiug in the' 
horizontal plane A B produced. The same may be said of its 
velocity when it arrives at H or K. The filament immediately 
above C will also issue with a velocity which is in the subdupli¬ 
cate ratio of its depth, and will then glide down above thefirst 
filament. The same may be affirmed of all the filaments; 
and of the superficial filament, which will occupy the sur¬ 
face of the descending stream. 
From this account of the genesis of a running stream of 
water, we may fairly draw the following consequences. 
1. The velocity of any particle R, in any part of the 
stream, is that acquired by falling from the horizontal plane 
AN. 
2. The velocity at the bottom of the stream is every where 
greater than any where above it, and is least of all at the 
surface. 
3. The velocity of the stream increases continually as the 
stream recedes from its source. » 
■ 4. The depths E F, G H, &c. in different parts of the 
stream, will be nearly in the inverse subduplicate ratio of the 
depths under the surface AN: for since the same quantity of 
water is running through every section E F and G H, and 
the channel is supposed of uniform breadth, the depth of 
each section must be inversely as the velocity of the water 
passing through it. This velocity is indeed different in diffe¬ 
rent filaments of the section; but the mean velocity in each 
section is in the subduplicate ratio of the depth of the fila¬ 
ment under the surface A B. ' Therefore the stream becomes 
more shallow as it recedes from the source; and in conse¬ 
quence of this the difference between L H and M G continu¬ 
ally diminishes, and the velocities at the bottom and surface 
of the stream continually approach to equality, and at a 
great distance from the source they differ insensibly. 
5. If the breadth of the stream be contracted in any part, 
the depth of the running water will be increased in that part, 
because the same quantity must still pass through •, but the 
velocity at the bottom will remain the same, and that at the 
surface will be less than it was before; and the area of the 
section will be increased on the whole. 
6. Should a sluice be put across the stream, dipping a 
little into the water, the water must immediately rise on the 
upper side of the sluice till it rises above the level of the re¬ 
servoir, and the smallest immersion of the sluice will produce 
this effect. For by lowering the sluice, the area of the sec¬ 
tion is diminished, and the velocity cannot be increased till 
the water heap up to a greater height than the surface of the 
reservoir, and this acquires a pressure which will produce a 
greater velocity of efflux th rough the orifice left below the sluice. 
7. An additional quantity of water coming into this chan¬ 
nel will increase the depth of the stream, and the quantity 
of water which it conveys; but it will not increase the velo¬ 
city of the bottom filaments, unless it comes from a higher 
source. 
All these consequences are contrary to experience, and 
show the imperfection, at least, of the explanation. 
The third consequence is of all the most contrary to expe¬ 
rience. If any one will but take the trouble of following a 
single brook from its source to the sea, he will find it most 
123 
rapid in its beginnings • among the mountains, • gradually 
slackening its pace as it winds among the hills and gentler 
declivities, and at last creeping slowly along through the flat 
grounds, till it is checked and brought to rest by the tides of 
the ocean. 
Nor is the second consequence more agreeable to observa¬ 
tion. It is universally found, that the velocity of the surface 
in the middle of the stream is the greatest of all, and that it 
gradually diminishes from thence to the bottom and sides. 
And the first consequence, if true, would render the run¬ 
ning waters on the surface of this earth the instruments of 
immediate ruin and devastation. If the waters of our rivers, 
in the cultivated parts of a country, which are two, three, 
and four hundred feet lower than their sources, ran with the 
velocity due to that height, they would in a few minutes 
tear up the most tenacious earth. 
The velocities of ora rivers, brooks, and rills, being so 
greatly inferior to what this theory assigns to them, the other 
consequences are equally contrary to experience. When a 
stream has its section diminished by narrowing the channel, 
the current increases in depth, and this is always accompa¬ 
nied by an increase of velocity through the whole of the 
section, and most of all at the surface ; and the area of the 
section does not increase, but diminishes, all the phenomena, 
thus contradicting, in every circumstance, the deduction from 
the theory; and when the section has been diminished by a 
sluice letdown into the stream, the water gradually heaps up 
on the upper side of the sluice, and, by its pressure, pro¬ 
duces an acceleration of the stream below the sluice, in the 
same way as if it were the beginning of a stream, as ex¬ 
plained in the theory. The velocity now is composed of 
the velocity preserved from the source and the velocity pro¬ 
duced by this subordinate accumulation; and this accumu¬ 
lation and velocity continually increase, till they become 
such that the whole supply is again discharged through this 
contracted section: any additional water not only increases 
the quantity carried along the stream, but also increases the 
velocity, and therefore the section does not increase in the 
proportion of the quantity. 
The next authors whose labours we shall notice, are Pro¬ 
fessor Michelotti at Turin, and Abbe Bossut at Paris. The 
first made a prodigious number of experiments both on the 
motion of water through pipes and in open canals. They 
were performed at the expense of the sovereign, and no 
expense was spared. A tower was built of the finest 
masonry, to serve as a vessel from which the water was to 
issue through holes of various sizes, under pressures from 
5 to 22 feet. The water was received into basons constructed 
of masonry and nicely lined with stucco, from whence it 
was conveyed in canals of brick-work lined with stucco, and 
of various forms and declivities. The experiments on the 
expence of water through pipes are of all that have yet been 
made the most numerous and exact, and may be appealed to 
on every occasion. Those made in open canals are still 
more numerous, and are no doubt equally accurate ; but they 
have not been so contrived as to be so generally useful, being 
in general very unlike the important cases which will occur 
in practice, and they seem to have been contrived chiefly 
with the view of establishing or overturning certain points 
of hydraulic doctrine which were probably prevalent at the 
time among the practical hydraulists. 
The experiments of Bossut are also of both kinds; and, 
though on a much smaller scale than those of Michelotti, 
seem to deserve equal confidence. As far as they follow the 
same track, they perfectly coincide in their results, which 
should procure confidence in the other; and they are made in 
situations much more analogous to the usual practical cases. 
This makes them doubly valuable. They are to be found in 
his two volumes entitled Hydrodynamiqne. He has opened 
this path of procedure in a manner so new and so judicious, 
that he has in some measure the merit of such as shall follow 
him in the same path. 
This has been most candidly and liberally allowed him by 
the Chevalier de Buat, who took up this matter where the 
Abbe Bossut left it, and has prosecuted his experiments with 
great 
