1142

THEORY AND PRACTICE OK ENGINEERING.

Rook II.

slow,) but because, when a body is moving in lines of a given curvature, the deflectingforces are as the squares of the velocities, and the particles of water in contact with thesides and bottom must be deflected in consequence of the minute irregularities of thesurfaces on winch they slide, nearly into the same curvilinear path, whatever their velocitymay be. The principal part of the friction is as the square of the velocity, and the frictionis nearly the same at all depths ; it varies, however, according to the surface of the fluid incontact with the solid, in proportion to the whole quantity of the fluid, viz. the friction forany given quantity of water will be as the surface of the bottom and sides of a river directly,and as the whole quantity of water in the river inversely ; or, suppose the whole quantity ofwater to be spread on a horizontal surface equal to the bottom and sides, the friction isinversely as the height at which the river would then stand, which is called the hydraulicmean depth.

When the river flows uniformly, and is neither accelerated nor retarded by the action ofgravitation, the whole weight of the water is employed in overcoming this friction, and ifthe inclination vary, the relative weight, or the force which urges the particles along theinclined plane, will vary as the height of the plane when the length is given, or as the fallin a given distance : consequently the friction, equal to the relative weight, must vary asthe fall, and the velocity, which is as the square root of the friction, must be as the squareroot of the fall; and supposing the hydraulic mean depth to be increased or diminished,the inclination remaining the same, the friction would be diminished or increased in thesame ratio ; therefore, in order to preserve its equality with the relative weight, it must beproportionally increased or diminished, by increasing the square of the velocity, in the ratioof the hydraulic mean depth, or the velocity in the ratio of the square root. 'Hie velocitieswill therefore, be conjointly as the square root of the hydraulic mean depth, and of the fallin a given distance, or as a mean proportional between these two lines.

Taking two English miles for a length, we must find the mean proportional between thehydraulic mean depth and the fall in two miles, and inquire what relation this bears to thevelocity in a particular case, and thence we may determine it in any other : according toEvtelwein’s formula, this mean proportional is ^ of the velocity in a second.

Mr. Watt found that in a canal 18 feet wide above, 7 feet below, and 4 feet deep, havinga fall of 4 inches in a mile, the velocity was 17 inches in a second at the surface, 14 inchesin the middle, and 10 inches at the bottom, the mean velocity being 14 inches in a secondor thereabouts ; therefore, to find the hydraulic mean depth, we must divide the area of thesection 2 (18 + 7) = 50, by the breadth of the bottom and length of the sloping sides added

together, whence we have or 29-13 inches ; and the fall in two miles being 8 inches,

we have */ 8 x 29‘13 = 15-26, for the mean proportional, of which, is 13*9, agreeingexactly with Mr. Watt’s observation.

The Po falls 1 foot in two miles, where its mean depth is 29 feet, and its velocity is about55 inches in a second ; our rule gives 58, which is a sufficient approximation to show itsaccuracy.

To find the velocity of a river from its fall, or the fall from its velocity, we have only torecollect that the velocity in a second is of a mean proportional between the hydraulicmean depth and the fall in two English miles.

For the Slope of the Banks of a River or Canal, Eytelwein recommends that the breadth atthe bottom should he two-thirds of the depth, and at the surface ten thirds ; the banks willthen in general retain their form ; the slope of the bank within the water making an angleof 37 degrees with the horizon, or of 4 to 3. The area of such a section is twice thesquare of the depth, and the hydraulic mean depth two-thirds the actual depth.

The superficial velocity of a river is nearly a mean proportional between the hydraulicmean depth and the fall in two miles, and the mean velocity of the whole water is stillmore, or nearly nine-tenths of this mean proportional.

The Ganges at 500 miles from the sea has its channel 30 feet deep when the river is atthe lowest, and it continues at least this depth to the sea : a section of the ground wastaken parallel to one of its branches, in length 60 miles, where the windings of the riverwere so great as to reduce the declivity on which the water ran to less than 4 inches permile, its medium rate of motion being less than three miles an hour in the dry months.Allowing a little for the banks, we may take 30 feet as the hydraulic mean depth ; then ifthe fall in two miles were precisely two-thirds, we should have £ x 30 = 20, and V20 = 4 ’47for the velocity of a second, or 3 05 miles in the hour.

Hi is river when full has thrice the volume of water, and its motion is accelerated in theproportion of 5 to 3 ; the hydraulic mean depth is then doubled, whence the velocity isincreased in the ratio of 7 to 5 : the inclination of the surface is somewhat increased,which increases the velocity from 1 *4 to 1 *7.

Smeatons Experiments relative to undershot Water-ichreh. — ARC!) is the cistern for sup-plying the water, I) E the upper cistern or head, F G a small rod divided into inches, andmoving up and down as the water rises and falls; II I is a rod by which the sluice is drawn,and stopped at any height required by the pin K, which is placed as a diagonal scale upon