Chap. XXV.

ON STONE BRIDGES.

1509

V*

should have flowed over a space equal to j > if we suppose that it. continues in the

same manner to accelerate its movement, it must, to require the rapidity v, flow over aA 2 V 2

space equal to —j. -—The length of the narrowing will then be represented byV 2 /A 8

r a /A 8 \

a distance — 1 J from the point where that width is equal to A.

I% /A 8 \ ...

I —— i|; and to have the intermediate widths between A and a , it will be sufficient

U 2 / . , ^ „

to remark that the point where the width of the section is equal to x should be situated at

The preceding analysis gives the means of enabling a current to pass from a greatersection into a smaller, without the inclination being altered, or -without any fall or contrac-tion being formed; and although no account is here taken of the resistance produced byfriction, this want of exactness cannot produce any sensible errors, when the section isconsiderable: but these results cannot be applied to the form of the starlings of piers,because in the more ordinary cases it is necessary to give them an excessive length,and to render their angle very acute; the projection of a starling being limited, wecannot avoid the change of the section being made more rapidly than should be for thefall to remain the same, and all that can be done is to give to the faces of the starlings theleast disadvantageous forms possible. To obtain this it will be sufficient to bringtogether the ordinates of the curve, previously determined for the form of the narrowing,V 8 /A 8 \

so as that the distance-1 —^ 1 } between the points, where the width of the sections

2p I V a 2 }

is A and a, is equal to the portion of the length of the pier to which a curvilinear form isto be given. The principle of the solution is not satisfactory, as we may remark,according to the construction which results from it, that the curve of the starling willnot be tangent to the direction of the flanks of the pier; nevertheless it appears that toprevent as much as possible all contraction in the current to the passage of the arches, itis necessary that the surface of the starling .should not form an angle with the lateral faceof the pier, but should be, on the contrary, traced in the prolongation of that face. Thecondition here proposed is that the velocity of the water should only augment in insensibledegrees, and that from the effect of the figure of the starlings, the small threads ofthe stream should be conducted, so that they enter under the arches, in directionsparallel to the flanks of the pier, and not flow from the flanks in such a manner asto increase that contraction of the current which results from the pier. This conditionwould be satisfied by giving to the sides of the starling a convex curve tangent to the sidesof the pier, by which the starling will be prolonged, and the sides rendered more acute;the nature of the curve does not appear to be of great importance.

When determining the best form to be given to piers, we must always bear in mind thatwhen they are totally immersed in water they will lose § of their weight; and when we areobliged to place the bridge with a considerable obliquity to the current of the stream, weproduce an obstruction which varies with the cosine of the angle of obliquity, and, conse-quently, the additional head is as the versed sine of that angle; but if the banks of theriver are parallel lines, the water-way under the bridge will increase as the secant of theangle of obliquity, or inversely as the cosine.

The effect of the gyration at the shoulders of piers deserves also some notice, as it oftencauses their entire destruction; and the beds of rivers being all porous, the springs workthrough them, with a force equal to a pressure of the whole depth of the river : whereverthere is a void or means for the escape of such a spring or source, its force or pressureupwards will act against any construction that comes within its influence, and whereversuch a void extends to the depth of 4 or 5 feet, it is capable of lifting up an enormousweight: such a force was exerted upon the floor of the dock constructed by Smeaton atRamsgate harbour: that celebrated engineer found that the natural springs rose in the chalkbed, and issued with such violence as to break the paving-stone with which the area wascovered. Stones weighing a ton and a half, laid down at Plymouth, with a wall 100 feetin length, were hoisted up and overthrown by a similar force.

However necessary it may be to give to the piers the most proper form to prevent toogreat a contraction, this condition is not the only one to which regard must be paid : wemust endeavour to protect them from the shocks of ice, from floats of timber and boats,and this can only be effected by giving great solidity to the starlings, taking care tomake their projecting angle not too acute, which would subject it to wearing away, orblunt the angle too much, which would render it less capable of opposing the masses ofice : to avoid the contraction, it is indispensable to make the starlings very long and ter-minating in a point; it is thus impossible to satisfy at the same time the two con-ditions ; and as, according to peculiar circumstances, it may be essential to have regard

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