Chap. XXV.

ON STONE BRIDGES.

1483

increase, a part of the .waters flow slowly over inundated banks, the exact figure of thesurface of the fluid cannot be obtained, but it may by a sufficient approximation whenthe bed is uniform and the movement of the waters permanently established. 31. Belanger,engineer of the Ponts et Chaussecs , has given for this purpose the following calculation :—Letus call s a length taken on the bottom of the bed, in the direction of the stream ; h, thevertical height of the transverse section of the current; w, the area of this sectioncorresponding to the height h ; x y the width of the section w taken on the surface of thewater ; It, the mean radius ; the fall of the bottom of the bed in metres, which is supposedto be constant; Q, the volume of water which the river discharges in a second; we have theequation

Q 2

S

gw*

,-1

(o,

Q Q*

00002427— + 0*0003655—r-

in which the metre and the second sexagesimal are taken as units of length and time,g representing the velocity 9*809 metres, which the gravity may impress on the body in a

second ; all the quantities comprised under the sign j may be valued as the function of the

height h of the section. In calculating the value of the integral indicated by this sign,setting off from the greatest value of h , which takes place at the point of elevation, that isto say, immediately above bridge, to a second value intermediate between the precedingand natural value of the section, the result will give the length comprised between abovebridge and the section to which belongs this second value of h. We may thus ascertainthe relations between the given heights of the sections and the distances from the bridge towhich these heights extend, and consequently the figure of the surface of the fluid throughthe whole extent of the eddy, which is generally prolonged to an infinite distance, but theincreased or greatest height at a limited distance ceases to be evident. The formation ofan eddy above bridge producing great inconveniences, both in relation to the solidity of anedifice and the difficulties to which it subjects navigation, every effort should be made todiminish it as much as possible.

It has been observed that it was dangerous to give too great an opening to the water-way, for the reason that under some arches alluvial deposits might be formed, which,acquiring by time sufficient hardness to resist the action of the current, would, in a flood,oblige the waters to flow under the arches which had remained free, and thus expose thepiers to injury by attrition. For a similar reason building a bridge of two parts, separatedby an island, should be avoided : it might happen that one portion becoming encumbered,the whole current would be obliged to pass under the other, and occasion its destruction :the bridges of Chazey and Roarine were carried away from this cause, and the fall of mostbridges may be attributed to- some error in the water-way producing too great adiminution of the section, from too little or too great a length having been given. All theelements for the calculation of the area of the water-way should be taken at the time ofthe greatest floods; and according to the quantity which flows at this period, the areamust be determined. It is essential in rivers which will admit of them, so to dispose thearches that at low water there should be under some at least a metre in depth, in order thatthe navigation be not interrupted : it will always be possible to combine the differentconditions between the size and the form of the arches used, and in each particular case,by making different experiments, to arrive at the best solution of the problem.

Gauthey applies the preceding ideas to the bridge over the Durance at Bonpas, which isconstructed in wood between two banks, distant from each other 534 metres. The widthof the river at low water is only 110 metres, and the mean depth 1 *30 metres : but theflow of the river rises 3 metres, and the surface of the water-way is then 1530 square metres.The bed of the river not being confined, and the width being considerable in reference toits depth, it was difficult to ascertain the mean velocity with exactness; but at about 4 my*riametres above Bonpas, near to 3Iirabeau, the Durance passes between two precipitousrocks, distant from each other only 1 80 metres. At this point, the height and velocity of thewater were ascertained during the floods, more or less considerable, and the result was,

1st, That the river being 2*44 metres reduced depth, the mean velocity was 1*95 metresfor a second :

2nd, That when the river was 2*92 metres reduced depth, the mean velocity was 2 44metres per second :

3rd, That at the moment of the greatest floods, the depth of the waters being 4*87 metres,the mean velocity was 4*12 metres. Thus, in this latter case, which must be particularlyconsidered, the expenditure of the river is 180x4*87 metres x4T2 metres «363 2 cubemetres per second.

The distance from 3Iirabeau to Bonpas not being considerable, and the river only re-ceiving between these two points some rivulets or unimportant streams, there cannot be a