124 
R I V E H. 
great assiduity; arid we must now add with singular success. 
By a very judicious consideration of the subject, he hit on a 
particular view of it, which saved him the trouble Of a minute 
consideration of the small internal motions, and enabled him 
to proceed from a very general and evident proposition, 
which may be received as the key to a complete system of 
practical hydraulics. 
M. [Buat assumes this leading and self-evident proposi¬ 
tion, viz.: 
1. “ When water flows uniformly in any channel or bed, 
the accelerating force, which obliges it to move, is equal 
to the sum of all the resistances which it meets with, whether 
arising from its own viscidity, or from the friction of its 
bed.” 
From this proposition, ingeniously combined with the 
result of his own and Bossut’s experiments, he then draws 
these fundamental propositions, viz.: 
2. “ The motion of rivers depends entirely on the slope of 
their Surfaces. 
3. “ Since the velocity of the water depends wholly upon 
the slope of the surface, or of the pipe through which it is 
conveyed, it follows that the same pipe will be susceptible of 
different velocities, which it will preserve uniform to any 
distance, according as it has different degrees of inclination ; 
and each inclination of a pipe, of given diameter, has a cer¬ 
tain velocity peculiar to itself, which will be maintained 
uniform to any distance whatever. But this velocity 
changes continually, according to a certain function of its 
inclination for all degrees between its vertical and horizontal 
positions.” 
It is obvious that, considering the number of causes that 
may give rise to inequalities in the motion of water, whether 
in pipes or canals, it would have been vain to attempt the 
determination of the function above-mentioned from theory 
only: the results of the several experiments were, therefore, 
examined with the most scrupulous attention, and penetrating 
ingenuity, and from which at length the author derived the 
following theorems, viz.: 
Let V be the velocity of the stream, measured by the 
inches it moves over in a second; R a constant quantity, 
viz.: the quotient obtained by dividing the area of the trans¬ 
verse section of the stream, expressed in square inches, by the 
boundary or periphery of that section, minus the breadth of 
the stream, expressed also in inches, viz.: R= i 
where w is the mean width of the section, h the mean 
height or depth, and b the breadth at bottom. 
The line R is called by Du Buat the radius, and by Dr. 
Robison the hydraulic mean depth. 
Lastly, let S be the denominator of a fraction, which ex¬ 
presses the slope, the numerator being unity ; that is, let it be 
the quotient obtained by dividing the length of the stream, 
supposing it extended in a straight line, by the difference of 
level of its two extremities; or, which is nearly the same, let 
it be the co-tangent of the inclination or slope. 
Then the general formula expressing the velocity V, sup¬ 
posed uniform, is, 
307 V R 
V = 
VS—|h. log. (S + 
VR- 
V = VR 
-TO x ( 
307 
VS —|h.log. (S + j|) 
or 
— to)’ 
But when R and S are both great, then 
( QQ7 
vs--h.iog.s -* )near| y- 
Hence it follows, that the slope remaining the same, the 
velocities are as R, or as the area of the section divided by 
its perimeter, minus the breadth of the river at the surface, 
z-ry nearly; for they are as V R —; and when the river 
is large, the V R may be used without any sensible error. 
Again, if R is so small, that VR — ^ = 0, or R = to> 
the velocity will be nothing, which agrees very well with 
experiments; for in a cylindric tube R ~ | the radius: th 
radius, therefore, is only two-tenths, so that the tube is 
nearly capillary, and the fluid will not flow through it. 
The velocity may also become nothing, by the slope be¬ 
coming so small, that 
307 
VS —fir. log. (S + j§) 
but if -g- is less than or than ^th of an inch to an 
English mile, the water will have sensible motion^ 
In a river, the greatest velocity is at the surface, and in the 
middle of the stream ; from which it diminishes towards the 
bottom, and the sides, where it is the least. It has been 
found, from experiment, that if, from the square root of the 
velocity in the middle of the stream, expressed in inches per 
second, unity be subtracted, the square of the remainder is 
the velocity of the bottom. 
Hence, if v be the velocity in the middle of the stream, 
the velocity of the bottom will be expressed by ( V» — l) 2 
= o —*2 Vc + 1. 
The mean velocity, or that with which, were the whole 
stream to move, the discharge would be the same as the real 
discharge, is equal to half the sum of the greatest and least 
velocities, as computed in the last proposition. Therefore, if 
v. represents the greatest velocity, then will the mean velocity 
— v — Vv + 
Suppose that a river, having a rectangular bed, is in¬ 
creased by the junction of another river equal to itself, the 
declivity remaining the same; required the increase of depth 
and velocity. Let the breadth of the river = b, the depth 
before the junction d, and after it x ; and in like manner, 
f ..>W ... , _ 3 -l*-_ t> d 
v and v' the mean velocities before and after; then 
b x 
R before, and 
b + 2 x 
— R' after, so v 
b + 2 d 
307 VR 
VS 
supposing the breadth of the river to be such, that we may 
reject the small quantity subtracted from R; and, in like 
307 V R/ 
manner, » f = -; then substituting for R and R', 
we have 
VS 
307 
VS 
307 
VS 
307 / 
vs x V 
b d 
2 d + b’ 
b x 
and 
b + 2 x 
Multiplying these into the area of the sections b d, b x, we 
have the discharges, viz. 
307 b d V b d 
V S V [b 4- 2 d) 
307 bx V bx 
VS 
of 
b d v = 
b x v' = — 
and 
V b d 
And since the last 
obtain 
b x V b x 2b d 
V (b + 2 d) ’ 
_ / 8 d 3 
U + 2d 
V {b + 2 x)' 
these is double the former, we 
V (b + 2 x) 
whence 
or 
b + 2 x 
4 b d 3 
b + 2d’ 
4 d 3 
b + 2 d 1 
a cubic equation solvable by the formula of Cardan. As 
an example, let b — 10 feet, d = 1, then .r 3 — ? s x = tf, 
where x = 1.4882, which is the depth of the increased river. 
Hence we have 1.4882 x »' = 2 s, and 1.4882 : 2 :: v : v r ; 
or » is to v 1 as 37 to 50 nearly. 
When the water in a river receives a permanent increase, 
the depth and the velocity, as in the example above, are the 
first that are augmented. The increase of the velocity in¬ 
creases the action on the sides and bottom, in consequence of 
which the width is augmented, and sometimes also, but 
more rarely, the depth. The velocity is thus diminished, till 
the tenacity of the soil, or the hardness of the rock, affords 
a sufficient resistance to the force of the water ; the bed of 
the 
