Of the film of water in the immediate neighbourhood of the pier, every part is urged laterally with the same force, viz. the excess of pressure produced by the accumulation. And, upon the supposition that all the water of the passing stream proceeds with the same velocity, which, by the way, is not strictly true, we might conclude that the deflection produced at the shoulder of the pier is the same at any depth. Now this Dec tion will have some proportion to the velocity of he stream. It evidently depends on the rapidity of the cur rent. In the language of mathematicians, it will Le a function of that velocity. Suppose, for a moment, that the velocity of the lateral discharge at the shoulder of the pier be equal to that of the current, it is plain that these waters, with those which are immediately conti ous in the stream, moving parallel to the side of the pier, would be projected in the diagonal of a square which had the side of the pier as its side. In that case we must suppose a certain space immediately behind the shoulder of the pier to be void of water ; and at the same time, this sort of repulsion will produce a head of water, or accumulation in the stream, immediately be yond that void space. This accumulation will be pro pagated front the shoulder of the pier, as a centre across the arch, at the same time it will be carried down the stream; and accordingly, we do always see a wave, which proceeds from each sterling or abutment of an arch, and which meet perhaps a considerable way be low the bridge : but, at the same time, and what is of more importance, the waters, which are, as it were, projected from the shoulder of the pier, are not at li berty to proceed in that direction ; having on one side the void space above mentioned, they are repelled on the other, by the lateral and hydrostatic pressure of the general mass. This will, of course, act perpendicularly to their direction, and produce ultimately a sort of gyra tion or revolution. The superficial waters will like wise tend to run over and deseend into the void ; as they descend, they arc exposed with the lower waters, to the lateral pressure of a greater depth ; the gyration will become more rapid in descendinz: and its radius of curvature shorter. The void will assume the appear ante of a hollow cone, the apex pointing downwards. The descending titter mill at length strike the bottom, and be reflected upwards ; the motion of the apex will still be the most rapid, and will thereby produce that boiling appearance which we generally see towards the tail of the pier, and for a good way below the bridge. The general current is no sort of obstruction to this re flection front the bottom ; it is perpendicular to its di rection, and therefore neither helps nor hinders it, hut it is the cause why the vortex is at first elongated in the direction of the stream, and why, after reflection from the bottom, it reappears considerably below the place of its original formation.
We have as yet taken into consideration the effect only of that film which lies in the immediate neighbour hood of the pier, and assuming it to pass laterally with g velocity equal to that of the general current, we have supposed that the primary deflection will be at an angle of 45°, but the second, third, &c. films, which are in the front of that, will have a similar lateral discharge, and will therefore have deflected a part of the waters of the general current, before they have reached the pier and the last film ; but the deflection cannot be so great, because the accumulation is not so great. In fact, the first deflection is in a manner nothing. It is a nascent quantity ; but each succeeding film having some room made for it by the deflection produced by the former, will be broader than it, measuring across the current, and will in its turn add a little to the former deflection; yet so, that the ultimate breadth can hardly be much greater than that of the pier, let the deflection be what it may.
All this while we are speaking of a square ended pier ; and it is now clear, that the water which lies in a manner stagnated before it, is bounded on the plan by two curve lines, which have their convexity turned to wards the axis of the pier, and are of course concave on the outside. If this, therefore, be the most advisable form of the sterlings of piers, it is, in all probability, different from any that have ever yet been constructed. Before, however, going further, we may observe, that the water in the front of the pier is by no means stag nant ; passing in the direction of the stream, every film has a greater velocity than the succeeding ; it has to supply not only its own waste by the lateral discharge, but that of all the succeeding, or rather, perhaps, its own waste, in passing through all the succeeding stages.
At the commencement of the accumulation and de flection, the direction, as well as the velocity, is, in fact, that of the stream at the pier ; the velocity in the direction of the stream vanishes, for the whole is de flected.
We conceive, therefore, although with great diffi dence, that it is with impropriety, Newton has said, that the motions are the same as if a certain part of the wa ter in front of the pier or obstruction were frozen, and he conceives this part to end in a point. No part of the water which is before the pier is perfectly stagnant, and it can therefore by no means be considered as frozen, neither can that portion of the water be supposed to come to a point ; for, if we take, as the quantity of cur rent intercepted by any pier, to the lateral discharge at the shoulder, so the breadth of the pier to a fourth proportional, it is plain that, roundly speaking, we have the distance on each side of the axis, to which this wa ter extends, at that part of the stream where it may be supposed to be confounded with the general current ; and this breadth must be something, since the discharge is something.
Giving up then the idea of making a sterling, or point to our pier, which shall be the same as the water that is supposed to stagnate before it, we think the best thing that can be clone is to offer some maxims of construc tion, which, though they do not constitute a complete rule for the perfect formation of starlings, will yet serve to improve the practice of the bridge builder, and preserve him from falling into gross and danger ous errors.
It is evident that all abrupt angles at the junction of the sterling with the pier are to be avoided : this part should be ngatly and regularly rounded away, so as to prevent the gyration above alluded to, by giving the fi gure of theIncipient p t•t of the curve to the shoulder of the pier. Neither do we approve of the parallelism of the sides of the pier. A small convexity might be gi ven them with much advantage, and without adding to the trouble of erection.
It were perhaps to be wished, that in the horizontal section of the curved part of the sterling, it should have a contrary flexure, so as to make the point of it, by be ing as sharp as possible, turn aside the waters with the greater ease ; but this is attended with a manifest dis advantage. The point will then be liable to damage, and the most trivial variation in the thread of the stream, would produce in an increased degree all those gyra tions we wanted to avoid ; such points too, in navigable rivers, would be fatal to loaded craft. But while we do not attempt to give this acuteness to the pier in the ho rizontal section, we may acquire it with great facility in the longitudinal or vertical section. For that purpose, the lower courses should be made gradually to project before the upper, and if the formation of the pier ad mits of it, they may also be made to project on the sides ; but care should be taken in so cluing to attend to the rules we have formerly given for its stability, and not uselessly to increase the mass of the pier. As we de scend, however, the courses of the sterling should be come more acute, and project further up the stream, and thus the pier may be supposed ultimately to end in a mere point. In short, the figure of a pier, or sterling of this kind, will have a considerable resemblance to the sock of a plough,--an instrument which, in its applica tion, has a considerable analogy with the sterling. It may be asked, whether these projecting courses should be left by the mason in steps, or trimmed away to a re gular curve surface ? We conceive, that there are good reasons for choosing to leave them in steps. For, in dependent of the saving of labour, when the current has any obliquity to the direction of the pier, it will flow over the acute point of the sterlingoand being reflected by these steps, will be in a great measure prevented from injuring the bottom. The safety, however, of craft would seem to make it proper to round away the fore corners of the courses.