Chap. XVII.

THEORY OF THE MOTION OF FLUIDS.

11ST

Fig. 1729.

current whose mean and uniform velocity per second will be equal to that which a bodywould acquire in falling from the height L K.

As the threads of water of the current increase in velocity in approaching the bottom ofthe canal, their impression on the surface will diminish as they approach the summit O Min an arithmetical progression, since these impressions will he the same as those of thepressure sustained by the sluice of the aperture when closed: if the surface in questionbe represented by the float-board of a wheel, it would be necessary to know its centre ofimpression, because the arm of the lever which corresponds to the mover must always beexpressed by the distance from this centre to the axis of the wheel.

A body rolling or sliding on ahighly polished inclined plane BCDEacquires in descending from the sum-mit to the base the same velocitywhich it would have acquired in fallingfrom the height BA of the sameplane. Hence, water at rest allowedto run down an inclined plane wouldacquire a velocity which may be ex-pressed by the root of the height ofthe plane. If a reservoir be placed atthe summit of an inclined plane, thewater which issues from F G II I,having already a velocity expressedby the root of the mean height L K, orits equal M N, will acquire a greater,expressed by a/MA, after havinggone over the inclined plane; but as

the volume of water which issues from the aperture each instant will always be expressedby 0\/MN, whatsoever be the height of this plane, calling 0 the aperture, the quantityof motion, or the shock of the water, will be expressed by ©a/MNx a/MA =© x a/M F x M A, which show’s that if there be at the foot of the inclined plane a surfaceequal to the aperture which receives the impression of the water directly, the power R,which sustains this surface in equilibrio, will be equal to the weight of a prism of waterhaving a plane equal to the aperture for its base, and the mean proportional between MNand M A for its height.

Whatever may be the dimensions of a surface, to measure the impact or shock, the partwhich receives the impression must alone be taken into the calculation : if the water inissuing from the bottom or one of the faces of a reservoir flows over several contiguousinclined planes, without meeting any other obstacle than the planes themselves, its directimpulse against a surface, relatively to the height of the water in the reservoir, will be thesame as that of a solid body under similar circumstances.

When the direction of a current is not perpen- I Q

dicular to the opposing surface, it does not actwith its absolute force. Suppose, for example, thatthe line N O represents the base of a vertical sur-face placed obliquely to a current at the bottom ofa canal, and that PV expresses the velocity anddirection of the current; it is evident that if thissurface be impressed by a solid body, the im-pression will be expressed by the perpendicularPT. But as the object under consideration is afluid whose impression must be measured by thesquare of the velocity, its absolute weight will hear thesame proportion to its respective force, as the squareof PV to the square of PT, that is, as the square

of the total sine, to the square of the sine of the Fig. 1730 .

angle of incidence.

Suppose the line N Q to represent the base of a surface directly opposed to the current,as it will be perpendicular to the side I X of the canal, the parallels QO and P V will givethe similar triangles QNO, TPV; then calling N Q,a, NO,6, PV, m, PT, «, weshall have aa, bb l nm, mm, which gives aamm = bbnn, and amm, bnn l b, a. As thefirst term of this proportion expresses the product of the surface N Q. multiplied by thesquare of the entire velocity of the current, and the second that of the surface N O by thesquare of this modified velocity, the impulse which the direct surface sustains, is to theimpulse which the oblique surface sustains, reciprocally as the length N O is to the lengthNQ. If the surface N O be inclined, and have the same base as QN directly opposed tothe current, the line II V representing the level of the water, the impulse sustained by the

4 D

Fig. 1729.