Chap. XXV.

ON STONE BRIDGES.

479

ment will measure a velocity of 6 feet 4 inches in a second ; but great velocities are toomuch for the instrument, and very slight are not sensible enough to be perceived ; but inrapid currents the pallet may be reduced, and in slow, enlarged ; the same graduation willthen serve. We know that this velocity depends in a great measure on the fall, and asthis commonly diminishes in proportion to its distance from the source, it follows thatthe same mass of water which would flow' very rapidly amidst the mountains where theriver takes its rise, and where it is only a torrent, would flow considerably diminished invelocity as it approached near the sea, or the channel into which it emptied itself: thus,admitting that it receives no considerable increase, and that the bed has always an equalconsistencv, if two bridges were to be constructed over the same river, the greatest spanmust be given to that nearest the source, although a passage is to be afforded to the sameflood, but which will flow' through the first in two days, whilst it will take eight to flowthrough the second. To value the quantity of water, we must multiply the surface of thesection by the mean velocity: the first of these tw'o elements of the calculation it is easy toascertain, but not so with the second, which must be deduced by more or less approximationfrom the measure of the velocity of the threads of water of which the river is composed,and particularly from those at the surface and middle of the current.

Dubuat, in his experiments upon the motion of fluids, had for his object the determiningthe relations betw'een the velocity of the surface and that at the bottom of the river, as wellas to discover its mean velocity. lie also deduced from his observations the followingformula: calling V the velocity of the surface, and U the mean velocity, he found,

U = (V>-JW)* + >\V,

W being a constant number =0 02707 metres : and to express the relation between thevelocity of the surface and that of the bottom, calling this last \V, we have

w=(v s -\v y

On these two formulas he calculated a table, which gives the value of the velocities of thebottom and the mean velocity, relatively to that of the surface, from V = 0-027 metres toV = 2-707 metres.

On this subject M. Prony has remarked that the above formula? do not correspond withexperience, for supposing the second V = o, we have W = tc, whence it follows that thevelocity of the surface being null, that of the bottom would be equal to 0-027 metres, aresult which could not be admitted. If in the first formula, we make V = o, we mustdeduct for U a finite value, which does not agree with what we find in nature : the relationestablished between U and V must necessarily give at the same time U = o and V = o, or

U = co and V =

AI. Prony therefore adopted another equation, U =

V(V + a)V + b 5

and the

value of the constant numbers a and b having been determined, after seventeen experimentsby Dubuat upon the values of A V, which varied from V = 0*15 metres to V= 1 -30 metres :he found also a = 2-37187, and 6 = 3-15312, the metre being the unit of the measure, whichgives for the preceding formula

U =

V(V + 2 37187)V+ 3-15312

This formula has the advantage of representing more faithfully than that of Dubuat theresults of his own experiments, and of agreeing likewise with the phenomena in question,confined within their proper limits. AI. de Prony remarks that from V = o to V = 3V + 9-37187

metres the relation-,—_— is evidently equal to 0*82, and as practical cases are\ -4" 3 -1 o312

commonly within these limits, this formula will be sufficiently exact:

U = 0*82 V, or even U = |V.

Neither practice nor theory has hitherto afforded any closer method of deducing thevalue of the mean velocitv of a current of water from that at the surface and middle ; butsome observations may be made upon the manner of applying the results just explained.The experiments of Dubuat were made in artificial canals, in which the section was a rect-angle or trapezium, and where the depth of the water varied from 54 millimetres to27 centimetres : some reservations must therefore be made in considering the results de-duced, and the nature of the motions of fluids is not yet sufficiently well-known to allow ofan absolute decision, by comparing great things with small : moreover, the experimentsupon which the preceding formula is established have appeared to indicate that therelations existing between the three velocities, V, LT, and W, are independent of the sizeand figure of the bed of the river, and these relations did not appear to change sensiblywhen the breadth of the bed was six or seven times greater than the depth. “ It is thus,”observes M. de Pronv “ difficult to persuade oneself that the different elements have no

5 b 4