376. The water-way of a bridge should be sufficient to give free passage to the highest floods, and particular regard must be had to this circumstance in fixing the height and width between the piers.

The form and area of the water-way is often so much altered by the bulk of the piers, as to cause an increase of velocity in the current under the bridge; and when the bottom is of such a nature that it will yield to this increased action of the current, there is much danger of the foundation of the piers being undermined; also in navigable rivers it renders the navigation difficult and often dangerous. Whereas, if the forms and dimensions of the piers be so contrived that there shall be only a very small increase of velocity under the bridge, those evils will be avoided, and the floods will pass through without doing any material injury.

Whenever the velocity is increased by contracting the width of the stream, the bottom tends to wear deeper, unless it be so hard as to resist the increased action of the current. In the latter case the chief evil will be the fall of water under the arch.

The velocity of rivers is extremely variable; it depends chiefly on the declivity of the bed, and is most considerable in mountainous countries. In level districts there is little to be apprehended from the effect of the velocity; nevertheless it would not be prudent, even in level situations, to contract the water-way so as to produce a rapid fall under the bridge, particularly if the bed of the river be not sufficiently firm to withstand it.

The danger of a considerable fall under a bridge is well known, as in the case of old London Bridge, where the fall during the ebb was generally about 4 feet; and many lives had been lost in attempting to pass it. The want of a sufficient water-way appears to have been one of the causes, if not the chief cause, of the failure of Hexham Bridge, in which the fall was not less than 5 feet at the time the bridge fell,* and the bottom not of a nature to withstand such an increase of velocity.

* Smeaton's 'Reports,' vol. iii., p. 338. It appears that the bottom was sufficient to withstand a fall of 3 feet 9 inches, but failed in the flood, which rose to 5 feet (p. 313).

Want of water-way is a fruitful source of danger to a bridge; but care should be taken not to run into the opposite extreme, as a certain amount of velocity in the current is required to prevent deposits of sand and gravel, the movement of which, in a sluggish stream, is checked by any obstacle, however slight, and which, in process of time, reduces the water-way so much as to impede the passage of floods.

377. The following Table will enable the reader to compare the firmness of bottoms of different materials. The experiments were made by Du Buat.* The second column gives the greatest velocity the material in the third column is capable of resisting; and the fourth column contains the specific gravity of the material. In the first column the popular stages of accumulation are stated.

It is to be observed, however, that the scouring action of rivers depends also upon the depth or weight of water resting upon their beds.†

378. It appears then that the velocity of the water which should be permitted under a bridge is determined either by the nature of the bed of the river, or by the fall that would be hurtful to navigation.

* ' Principes d'Hydraulique,' tom ii., art. 399. † Law's ' Rudiments of Civil Engineering.'

If 6 represent the breadth of the natural water-way, and c, the breadth as reduced by the construction of the bridge; also V the velocity in feet per second of the river in its natural state; then the velocity v under the bridge will be expressed by the equation v = m. V b/c , and c = m.b V/v. Where m is a constant quantity which expresses the contraction a fluid suffers in passing through a narrow passage. According to Sir Isaac Newton's experiments, the value of m is 25/24;* this value of m should be used when the ends of the piers are square. They are, however, generally made of a form better adapted for dividing the stream; some experiments were made by Du Buat with models of piers having the end facing the stream in the form of an equilateral triangle, according to which we may take m = l.09.† Adopting this yalue, v = 1.09 V b/c, and c = 1.09 6 V/v.

Example. - Let the bottom of the river be fine sand, and the breadth of the natural water-way 36 feet, and the velocity V = 0.25 foot per second. Then for a fine sandy bottom, v should not exceed 0.351 foot; hence c = 1.09 b V/v = 1.09x36x0.25/0.351 =27.7 nearly, which is the breadth of the contracted water-way; and 8.3 feet may be occupied with piers without endangering the bottom.

379. Retaining the same notation, the amount of fall, h,

* ' Principles of Natural Philosophy.'

† Du Buat's lowest number is 1.097, but in wide rivers perhaps it will be less; therefore 1.09 is assumed as near the truth. See Du Buat, ' Principes d'Hydraulique.' will be found by the equation m2b2 -c2/ 64 c2xV2 = h.* And taking the value of m = 1.09, then m2 = 1.1881; or near enough for practice, m2 = 1.2; consequently, 1.2b2-c2/64 c2 x

V2 =h, the fall.

Example. - The breadth of the Thames above London Bridge is about 936 feet, according to the observations of Labelye in 1746; and the sum of the water-ways of the old bridge at the time of low water was about 200 feet; the mean velocity of the stream just above the bridge was 3 1/6 feet per second. Therefore 1.2b2-c2/64 c2xV2 =1.2x876996-40000/64x40000 x

361/36 = 1011315.2/2560000 x 361/36 = 3.96 feet, or 4 feet nearley; which rendered the passage extremely dangerous.

The velocity of the current and the sectional area of the water-way should be ascertained at the time of the highest floods if practicable, otherwise we must be satisfied with an approximate value of the velocity at that period; which may be obtained by taking the velocity and depth at the time of observation, and assuming that the velocity during floods is increased in proportion to the square root of the depth. In a river the surface velocity in feet per second is nearly as the square root of the hydraulic mean depth multiplied by the fall in two miles, both in feet, and the mean velocity is nearly 9/10 ths of this quantity.

The fall under the bridge is directly as the square of the velocity, therefore there is much danger in contracting the water-way of a rapid river, and the fall will also be nearly as

* An investigation of this formula is given by Dr. Button in Lis ' Tracts;' also in his ' Course of Mathematics.' the depth of the river; which shows how necessary it is to ascertain the height of the highest floods.*