The slope remaining the same, the velocities are as ✔RThe velocities of two great rivers that have the same declivity, are as the square roots of the radii of their sections. If R is so small, that R= 0, or R, the velocity will be nothing; which is agreeable to experience; for in a cylindric tube R the radius; the radius, therefore, equals 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 declivity becoming so small, that mile, the water will have sensible motion. 4. In a river, the greatest velocity is at the surface, and in the middle of the stream, from which it diminishes toward the bottom and the sides, where it is least. It has been found by 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 at the bottoin. Hence, if the former velocity bev, the velocity at the bot2 √ v + 1. tom =v 5. The mean velocity, or that with which, were the whole stream to move, the discharge would be the same with the real discharge, is equal to half the sum of the greatest and least velocities, as computed in the last proposition. The mean velocity is, therefore, =VI Vu+1. This is also proved by the experiments of Du Buat. 6. Suppose that a river having a rectangular bed, is increased 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, v and v' the mean breadth of the river to be such, that we may reject the small quantity subtracted from R, in art. 3; and, in like manner, Then, substituting for R and R', we have bd v and of the sections b d and b x, we have the discharges, viz. b d v = 307 b d ✓ b d 307 b x ✓ b x √b + 2 d √6+2x Now the last of these is double of the former; therefore, 2 b d ✔ b d ✓ b + 2 ď or ways be resolved by Cardan's rule, or by the approximating method given at page 92. = = 1, then x3 - 3 x As an example, let b = 10 feet, and d x = 10, and 1-4882, which is the depth of the increased river. Hence we have 1.488 × v' = 2 v, and 1·488: 2 :: v : v', or v : v' :: 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 things that are augmented. The increase of the velocity increases 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 The bed of the river then resistance to the force of the water. changes only by insensible degrees, and, in the ordinary language of hydraulics, is said to be permanent, though in strictness this epithet is not applicable to the course of any river. 7. When the sections of a river vary, the quantity of water remaining the same, the mean velocities are inversely as the areas of the sections. This must happen, in order to preserve the same quantity of discharge. (Playfair's Outlines.) 8. The following table, abridged from Du Buat, serves at once to compare the surface, bottom, and mean velocities in rivers, according to the principles of art. 4, 5. 9. The knowledge of the velocity at the bottom is of the greatest use for enabling us to judge of the action of the stream on its bed. Every kind of soil has a certain velocity consistent with the stability of the channel. A greater velocity would enable the waters to tear it up, and a smaller velocity would permit the deposition of more moveable materials from above. It is not enough, then, for the stability of a river, that the accelerating forces are so adjusted to the size and figure of its channel that the current may be in train: it must also be in equilibrio with the tenacity of the channel. We learn from the observations of Du Buat and others, that a velocity of three inches per second at the bottom will just begin to work upon the fine clay fit for pottery, and however firm and compact it may be, it will tear it up. Yet no beds are more stable than clay when the velocities do not exceed this for the water soon takes away the impalpable particles of the superficial clay, leaving the particles of sand sticking by their lower half in the rest of the clay, which they now protect, making a very permanent bottom, if the stream does not bring down gravel or coarse sand, which will rub off this thin crust, and allow another layer to be worn off; very a velocity of six inches will lift fine sand; eight inches will lift sand as coarse as linseed; twelve inches will sweep along fine gravel; twenty-four inches will roll along rounded pebbles an inch in diameter; and it requires three feet per second at the bottom to sweep along shivery angular stones of the size of an egg. (Robison on Rivers.) 10. Mr. Eytelwein, a German mathematician, has devoted much time to inquiries in hydrodynamics. In his investigations he has paid attention to the mutual cohesion of the liquid moleculæ, their adherence to the sides of the vessel in which the water moves, and to the contraction experienced by the liquid vein B when it issues from the vessel under certain circumstances. He obtains formulæ of the utmost generality, and then applies them to the motion of water; 1st, in a cylindric tube; 2dly, in an open canal. D A 11. Let d be the diameter of the cylindric tube E F, h the total height F G of the head of water in the reservoir above the orifice F, and the length E F of the tube, all in inches: then the velocity in inches with which the fluid will issue from the orifice F will be v= 233 √ 57 h d :(English measure.) this multiplied into the area of the orifice will give the quantity per second. 12. For open canals. Let v be the mean velocity of the current in feet (English), a area of the vertical section of the stream, p perimeter of the section, or sum of the bottom and two sides, length of the bed of the canal corresponding to the fall h, all in feet then The experiments of M. Bidone, of Turin, on the motion of water in canals, agree within the 80th part of the results of computations from the preceding formulæ. 13. The following table also exhibits Eytelwein's coefficients for orifices of different kinds; their accuracy has, in many cases, been amply confirmed. 41 |