I horizontal to i vertical, and 2 to 1: being made flatter according as the soil has less tenacity. In some cases even 2 to 1 has been adopted; the half regular hexagon has slopes of 0.58 to 1; in channels for temporary use we may have 1 to 1.

And so also must the velocity be given; and, for the same reason, some kinds of earth being worn away, and the form of channel destroyed, by a rate which carries down the particles of the soil through which it is excavated, a velocity must therefore be assigned within this rate of motion. It has been determined experimentally for many kinds of earth.

The effect of the velocity of the water, in carrying down the particles of the ground through which the channel is excavated, depends jointly upon their tenacity and size. As to the size, we know that the cubical quantities or weights of any similar bodies decrease faster than their superficial areas; and the pressure or force urging a body down stream being, ceteris paribus, proportional to the surface, is relatively greater the less the volume; the smaller the particles, therefore, the less is the velocity required to move them. Mr. Beardmore* in Table 3 gives the following statement of the limit of bottom velocities in different materials in feet per minute:

30 ft. will not disturb clay with sand and stones.

180

angular stones, about 1 in. do.

The beds of rivers, protected by aquatic plants, however, bear higher velocities than this Table would assign.

*Hydraulic Tables, by N. Beardmore.

Such being the natural limitations in the choice of any particular rectangle or trapezium, the engineer must proceed to determine the figure of the transverse area without violating the conditions they impose.

115. When it is desired to convey the greatest possible quantity of water in an open channel with a given area of transverse section, then the volume discharged being directly proportional to the area, and inversely as the wetted border, we must select the figure which for a given area has the least border, and for a given border has the greatest area.

Geometry informs us that the circle has this property: the semicircle, and therefore the semicircular channel, has the same property; the ratio between the area of the semicircle and semi-circumference being the same as that between the circle and the entire circumference. Then follow the regular demi-polygons, with less and less advantage as the number of their sides is less; and among the more practible forms are the demi-hexagon, and finally the half-square.

As the transverse sections of artificial open channels are, when without masonry, trapezoidal, the question as to the form of greatest discharge is reduced to taking among all the trapeziums with sides of a determinate slope, that which gives the greatest section for a given wetted border; or, in other words, which has the greatest hydraulic mean depth; and every different area and ratio of slopes has its particular maximum trapezium. Let p be the depth of the trapezium BF (Fig. 75),

and b the bottom width BC, and n: 1 the ratio of the

slopes, or AF: FB; then the general values of S and C

S

Since, then, S in the expression, with slopes of n : 1,

is a maximum, its differential will be zero, and we have

and as the border is constant, its general value being differentiated, gives db + 2dp √ n2 + 1 = 0. Hence 2dp √n2 + 1; this being substituted in (18), gives b = 2p ( √ n2 +

db

==

(19)

I

n); with which value of b we have

Therefore in all trapezoidal channels of the best form, with certain given slopes and area, the hydraulic mean depth is half the depth of the water: and hence we derive a construction for the cross section of a maximum

discharging channel; remarking that as

s_p
C

=

we have

S = C. Let the trapezium ABCD (Fig. 76) be the

2

E

D

B

Fig. 76.

channel sought; from the middle point E of the top

width draw lines EB and EC dividing the figure into three triangles, of which AEB and CED are identical; let EP be the perpendicular from E upon AB; then

EP

AB + BC + CD × 2 = AB + CD × + BC ×

and therefore

p EP

==

2

2

Hence from E as centre, and with

pas radius describing a circle, it will touch the two sides AB and CD. If, therefore, conversely, we describe a circle (Fig. 77) with any radius, and draw a tangent, parallel to a horizontal diameter, produced on each side indefinitely, and then between these lines draw tangents having the given inclination, we obtain a figure similar to that re

quired, from which, by proportion, we find the transverse section of the channel sought: a construction given by Mr. Neville in his Hydraulic Tables.

Other properties of the trapezium of greatest discharge, Figs. 76 and 77, are, First, that the line of surface of water AD or A'D', Fig. 77, is equal to the sum of the slopes AB and CD, or A'B' and C'D', and consequently the wetted border is equal to the sum of the top and bottom widths, or the mean breadth equal half the border. Secondly, the triangle BEC, Fig. 76, is similar to the triangles EAB and EDC; the vertical

angle BEC being equal to the angle of inclination of the sides to the horizon. Thirdly, the angle between the perpendiculars from E upon the sides AB and CD is double the angle of inclination of the sides, and the angle PEB half of the same, that is, of the angle BEC. This gives another construction when p and the angle of inclination of sides are given. On a vertical line lay off p, and from the upper point E (Fig. 76), on each side, lay off the angle of inclination, bisect each of them by EB and EC, and through the lower point of p draw a perpendicular to intersect EB and EC, which gives the base BC, then from B draw BA perpendicular to EP, to intersect the horizontal line through E at A, and in like manner on the opposite side, giving the required trapezium ABCD. From the second property we obtain an expression for the area of the trapezium of greatest discharge in terms of the depth and angle of inclination ẞ of the side slopes with the horizon, for the area of the triangle BEC is equal to p × þ tan 1ẞ, as half BC is the tangent of half the vertical angle to radius p; also the sum of the areas of the triangles EAB and EDC is equal to EP × AB, but EP is equal to p, and AB is the cosecant of ẞ to radius p, as is evident if from B, Fig. 75, we draw the perpendicular BF, the angle ABF being the complement of BAF, that is ẞ; thus the area of the trapezium is

p (tan + cosec. B).

We may deduce from the Table that very large channels formed in any kind of earth cannot be designed so as to be of the best discharging form, as the depth of excavation would be too great; the ratio of the depth to the mean width must rather resemble that observed in large rivers.