BRIDGE. 



517 



by the accumulation. And, upon the supposition 

 *V that all the water <>f the pis-, ing stream pro> 



with ttie same wl.u-ity, 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 deflection will have some pro- 

 portion to the velocity of the stream. It evidently 

 dfepradt on the rapidity of the current. In the lan- 

 .ige of mathematicians, it will be 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 contiguous 

 in the stream, moving parallel to the side of the pier, 

 would be projected in .he 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 

 thoulder 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 tlu str-am, immediate- 

 ly bey .ind that void space. This accumulation will 

 be propagated from the shoulder of the pier, as a 

 centt;- 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 ster- 

 ling or .-.hutment of an arch, and which meet perhaps 

 a considerable w:iy below the bridge : but, at the 

 same time, and what is of more importance, the wa- 

 ters, which are, as it were, projected from the shoul 

 der of the pier, are not at liberty to proceed in that 

 direction ; having on one side the void space above 

 mentioned, they are repelled on the other, by the la- 

 teral and hydrostatic pressure of the general mass. 

 This will, of cuurse, act perpendicularly to their 

 direction, and produce ultimately a sort of gyration 

 or revolution. The superficial waters will likewise 

 tend tn run over and descend into the void; as they 

 descend, they are exposed with the lower waters, to 

 the lateral pressure of a greater depth ; the gyration 

 will become more rapid in descending; and its radius of 

 curvature shorter. The void will assume the appear- 

 ance of a hollow cone, the apex pointing downwards. 

 The descending water will at length strike the bot- 

 tom, 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 reflection from the bottom ; it is 

 perpendicular to its direction, and therefore neither 

 helps nor hinders it, but it is the cause why the vor- 

 tex is at first elongated in the direction of the stream, 

 and why, after reflection from the bottom, it reap- 

 pears 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 neigh- 

 bourhood of the pier, and assuming it to pass la- 

 terally with a velocity equal to that of the general 

 current ; we have supposed that the primary deflec- 

 tion 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 there- 

 fore 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 IK- so 'i 



great, her .vumulati' 't. In ~v~ 



fact, the first is in a n..i Ii 



M a nascent quantity ; but eaeli >g film having 



some room in.ide tor it by the deflection produced 



the former, will be broader than it, measuring acros4 



the current, and will in its turn add a little to th- 



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 towards 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 i 

 yet been constructed. Before, however, going fur- 

 ther, we may observe, that the water in the front of 

 the pier is by no means stagnant ; passing in the di- 

 rection of the stream, every film has a greater velo- 

 city 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 deflection, 

 the direction, as well as the velocity, is, in fact, that 

 of the stream at the pier ; the velocity in the direc- 

 tion of the stream vanishes, for the whole is de- 

 flected. 



We conceive, therefore, although with great dif- 

 fidence, that it is with impropriety, Newton has said, 

 that the motions are the same as if a certain part of 

 the water 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 per- 

 f'Ctly 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 current 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 water 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 done is to offer some max- 

 ims of construction, which, though they do not con- 

 stitute a complete rule for the perfect formation of 

 sterlings, will yet serve to improve the practice of 

 the bridge builder, and preserve him from falling into 

 gross and dangerous 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 neatly and regularly rounded away, so 

 as to prevent the gyration above alluded to, by giving 

 the figure of the incipient part 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 



