1480

THEORY AND PRACTICE OF ENGINEERING.

Book II.

Influence on the relative values of V, U, and W : and experiments made on a larger scalewould possibly prove that too much deduction has already been made.

In the sixth volume of the Memoirs of the Academy of Sciences are some researches onthe motion of fluids, applied principally to what is called linear motion, that is to say, tothe case in which the molecules of the fluid move in a rectilinear bed, following right linesparallel to the axis of this bed; and the results relative to the question now under conside-ration are, that, whatever be the figure of the transverse section, the mean and greatest velo-cities at the surface and middle of the current tend to become equal as the dimensions ofthe bed decrease : the relation between these two velocities is independent of their absolutevalue : if in a rectangular bed the horizontal breadth be supposed very great, and thevertical depth of the water very small, the mean velocity will be about U*64 of the greatestvelocity. If the two dimensions of the rectangular bed are supposed to be very large, therelation between them will be about 0*41, which formula* give the means of calculating therelation for different beds, of which the transverse sections are semicircles or rectangles.

Experience proves that the true laws of the motion of waters in the beds of rivers or inpipes differ from those of the linear motion : the only case in which nature realises theselatter laws is that in which fluids flow in rectilinear pipes of very small dimensions, andthe results deduced immediately from this observation are those alone to which we can payany consideration in the case before us.

At the time of great floods, which is that for taking the elements of a calculation on thequantity of water, it is generally overflowing the banks and extending on both sides over alarge surface, where it flows slowly, w’hilst there is a considerable velocity in the midst of thecurrent, and it is very probable that gross errors would occur if the rules for calculationjust cited were applied to these cases. If possible, a part of the river should be selectedwhere the water is so confined that during floods it docs not overflow to any great extent;and the section of the velocity should be taken at a bridge, if any exist near the site of thatprojected.

It must be acknowledged that the means we at present possess for obtaining the directvalue of the mean velocity of a river are very limited, and subject to more or less error.As a certain relation must exist between the section, the fall, and the velocity of the cur-rent, and as it is always possible to measure these, the value of the velocity would naturallybe deduced from it, if its relation were well known ; M. de Prony in the work before citedhas deduced it from the best experiments he could collect, and has arrived at the equation,

U= -0 0719+ VO 005163 + 3232 : 96 II I,

in which U being always the mean velocity of the current, R represents the mean radius,that is to say, the area of the section divided by the part of the perimeter of that sectionwhich belongs to the solid side in which the fluid flows, and I, the fall in a metre.

Since the researches of M. de Prony, M. Eytelwein has published in the memoirs ofthe Academy of Berlin some experiments, on the means by which he has established aformula similar to the preceding, but of which the coefficients have somewhat differentvalues : this formula, taking the metre for the linear unity, is,

U = —*003319+ V0-0011016 + 2735 66 II I.

The experiments which have determined the coefficients comprise the values from two tothree metres for the velocity, whilst the experiments which M. de Prony has made donot exceed 0 m. 88 c.. the formula of 31. de Prony agrees, as well as that of 31. Eytelwein,with experience, as regards small velocities, but they appear to give very feeble results forlarge ones. 31. de Prony has given for the motion of water in pipes this expression,

U = — 0 02488 + VO-OOOG192 + 2871-43 R I,

which agrees better than that mentioned above with those experiments in which the valueof the velocity is above one metre. The first equation will give the value of the meanvelocity with sufficient practical exactness; but it must be observed, that it supposes thesize of the section of the river, and the value of the fall, to be the same in a sufficient lengthfor the mean velocity to he regarded as constant; and it is necessary to take this into consi-deration when the formula is used.

The dimensions of the beds of rivers are generally sufficiently constant, with the ex-ception of particular cases in which their course is through sandy lands, which yield withso much facility to the action of water that the current may at each increase he carriedinto different parts: at each point of the course of the river a certain equilibrium isestablished between the action exercised by the waters on the cavities, and the tenacitiesof the matters of which the bed is composed ; and in virtue of this equilibrium remarkablechanges seldom occur either in the size or form of the bed, which nearly always preservesthe same rule. But if, from any cause whatever, the force with which the water tendsto corrode and destroy the bottom and shores of its lied be augmented, the bed will beenlarged until a new equilibrium is established, unless the side walls are so composed