We assumed, for the breadth of the actual water-way in the above Table, a rate of contraction, which is much the same as that observed in the diameter of a jet from an orifice in a thin plate. This may be going too far, but we think it advisable to keep the builder on the safe side of the limits of practicability. Square ended piers, and abrupt projections, are likely to produce as great a degree of contraction, especially when the river runs in floods, the only case that is particularly deserving of attention.
But the discharge through the arches will be material ly improved, by forming he piers with pointed sterlings, and otherwise adapting them to the figure of the stream. In rivers, where the arches are wide in comparison of the depth of water, the contraction does not appear to amount to a fourth of the above, or one twentieth of the whole water-way. And in this we arc confirmed by the experiments of Eytelwcin and Bossut. The former of whom states the contraction, in such a case as this, to be from 8.02 to 7.7, or nearly 2s• We have, therefore, calculated the following Table upon the principle of a contraction of and conceive, that when circumstances are most favourable, allowing for the additional friction caused by the obstruction, Ste. it will be found to come exceedingly near the truth.
By the help of this Table, we may see the effects likely to be produced in rivers by the usual accidents to which they are liable. The velocities above ten feet produce inundations that sweep away every kind of structure. Those in the latter part of the Table are given as fair results of the theory, but, in fact, they are impracticable.
In Westminster Bridge, the piers form about one sixth of the water-way ; the velocity is between 2 and 3 feet, or more accurately 2 feet; the head, therefore, will be between .036 and .082, more accurately .045, or about halfan inch; which is exactly the greatest fall ob served by Labelye.
At London Bridge, the apparent water-way is only one-fourth of the breadth of the river, but is much re duced by the drip shot piles, which have been driven into the bed to protect the foundations. The velocity of the stream above the bridge is 3 feet 2 inches, which, by this Table, would give a head of 2.6 feet, and by the former one 4 feet. We cannot suppose these piles to take off less than one fifth of the water-way, which would make the head 4.31 by this Table. But probably the contraction is greater than this Table supposes, corning nearer that assumed in the former, (which would have given us a head of almost 6 feet,) since a fall of 4 feet 9 inches was observed about the year 1730; and the exca vation had become so very dangerous, as to suggest the measure of cutting out one of the piers of the bridge, and throwing two arches into one.
The fall at Blackfriars will be somewhat less than at Westminster, but will not exceed one inch. In the
same example, as before tried, this Table affords the following results : Now, since the pebbles extend only to three feet, the waters pass to the clay, which bears only one-third of this velocity, and would therefore require a depth of 45.69, or 36.69 below the bed: The bridge the/dare cannot stand in such floods as this. Suppose. then, that it be proposed to make a total change of foundation, as, by paving all across the river, or any s;unlar operation, referring to the Table with an obstruction of and ve locity 5 feet, we find the head is 2.6725, and velocity 14 feet per second, winch would require a bot tom as firm as solid rock With good workmanship, however, the pavement s,ould stand a considerable time, especially if the joints were so carefully closed that wa ter could not readRY penetrate, and work out the finer materials in 10,ich the pavement was bedded : For, al though the water passes through the arch with this great rapidity, yet the general river being in a different train, und running with a much smaller velocity, will not bring along with it much heavier materials than the gravel and pebbles of the bed, and these will not be very injurious to the artificial bed at the bridge : For we are of opinion, that it is by no means the action of the water. hut rather the attrition, or battering and rub bing; of the houldcrstoncs, gravel, and sand, brought down by it, that renders the hardest rock liable to be cut up by the force of a swift running stream. It is nevertheless, extremely difficult so to secure a pave ment, or inverted arch, in a river, that the water will not ultimately carry it away, even when the river does not run foul in its freshes. The great velocity which has been communicated to the river, cannot he sup posed instantaneously to change upon passing the ob struction. Instead of that, we see a swift current shooting along in the line of the arches for a great way below the bridge, while powerful eddies run up in the line of the piers, casting up at length banks or shoals behind them, which tend, in their turn, to strengthen and prolong the original current and eddy. Whatever pains, therefore, we take to secure the pavement or in verted arch, this strong current must cut up and carry away the materials of the bed behind them; an opera tion which, if once begun, must constantly go on with increasing force. The water will have a fall over the lower end of the pavement, and will gradually wash out the foundation of the outer course of stones, which be ing immersed in water, will not be difficult to move. A few stones dropping out will add to the power of the stream, by roughening the bottom. Course will loosen after course, until the whole presents only a loose mass, ready to be torn up and swept away by the first ensuing flood i•the river.