1133

THEORY AND PRACTICE OF ENGINEERING.

Book II.

vertical surface will be to that sustained by the inclined surface reciprocally as the breadthN O to X Q.

When an immovable surface is directly opposed to a current, the power sustaining itwill be equivalent to the weight of a column of water having this surface for its base, and themean height of the reservoir for its height. Now if in the interval K Y (fig. 1727.) a roller Zbe placed, which traversing the canal can turn freely on gudgeons, having two pulleys above, itis evident that if a cord be attached to the centre Y, passing over the roller, and thence overpulleys, with a weight P equal to the impulse of the current, this weight will take the place ofthe power R, and sustain the surface N () M Q in equilibrio, as before ; but if it be diminished,the surface will be driven forwards with a velocity equal to that of the weight in ascending,since there is no difference between the arms of the levers ; the impulse of the current willbe diminished precisely by the same quantity as the weight was, since these two forces willalways be equal. When the weight P is diminished by acertain quantity, the surface of the fluid acquires the greatestvelocity of which it is capable, and preserves it constantlyuniform while the weight mounts, and the impulse whichpresses it will be expressed by the square of the excess of thevelocity of the current over that of the surface.

Since among all the elements of a rectangle there is alwaysone which can be divided by the diagonal A C, in such a man-ner that the square of the greater part H K multiplied by thelesser K I gives the greatest of all the products that can beformed, it follows that the entire velocity of a current may bedivided into two parts, the smaller of which becoming that ofthe surface, and the other that with which it is struck, this

surface will have the greatest possible quantity of motion. It

is ascertained that the velocity of the surface should be one-third that of the current for the greatest effect; that is to say,that it may at the same time receive the greatest velocity and impression possible, the com-bination of which answers to the greatest quantity of motion, the force of the shock beingequal to | the weight of the column of water which measures the absolute force of thecurrent against the surface when it is immovable. In a state of equilibrium the quantityof motion of the mover is always equal to that of the weight, and for the greatest effectwe can only count on £ of the moving power ; it follows that it can only raise £ of theweight with which it was in equilibrio when acting in full force.

If the aperture F G II D be closed, in fig. 1727., and the water comprised in the spaceF G M Q be at rest, leaving the weight P out of the question, the power which will drive

the surface N O M Q with a uniform velocity in the direction RY will be the same as

that which would be necessary to support this surface in equilibrio against the impact ofa current which would have the same velocity; for whether the water meets the surface,or the surface the water, the impact will always be expressed by the square of theirrespective velocities.

If the surface meet a current issuing from an aperture, the power having to support notonly the impulse of which the velocity of the current may be capable, but further thatwhich arises from its own velocity, the resistance resulting from their concussion must beexpressed by the square of the sum of the velocities of the surface and current; that is, ifthe current have a velocity of 3 feet per second, and the surface in ascending passes over 2feet in the same time, it would be the same as if it supported in equilibrio the impressionof a current whose velocity was 5 feet per second, or as if it were driven with this velocityin still water; for if, when a surface retreats from a current, its velocity must be subtractedfrom that of the current, to obtain the velocity with which it is struck, it is natural, whenthe surface meets the current, to add the velocity of the former to that of the current: onthe other hand, when the same surface is moved in the natural direction of the current, witha velocity greater than the impulse which the power sustains, it should be expressed by thesquare of the difference between the velocity of the surface and that of the current, becausethe surface is then relatively to the water which retires, what the current is when the surfaceretires from its impact.

M. de la Hire first started the idea that the uniform velocity of running water may beregarded as acquired by a fall, consequently as the mean velocity which the water from areservoir would acquire whose height would be equal to this fall ; whence he concludesthat the direct impression of a current against a vertical surface should be measured by theweight of a column of water having for its base the surface struck, and for its height thefall relative to the velocity of the current.

If the surface retreat from the current, we shall, in like manner, find the impulse whichit sustains, by subtracting its velocity from that of the current, if the first is less than thesecond, or by subtracting the velocity of the current from that of the surface, if it be thecontrary, and by dividing the square of the difference by 60, to have the height of the prism

Fig. 1731.