distance from the orifice which is in the middle of the side, and to equal distances from orifices equally distant from the middle.

Let C be the orifice in the middle of the side, and E, e, equal orifices at equal distances Plate 5. from C.

The distance to which the fluid will spout at C (by Prop. XVI.) is twice CH, and at E twice ED. But CH (El. III. 15.) is greater than DE, any line drawn from the diameter parallel to the radius: therefore twice CH is greater than twice ED.

Also, since the horizontal distances to which the fluid will spout at E and e, are twice ED, or ed; and that ED, ed, being equally distant from the centre, and parallel to the radius, (El. III. 14.) are equal; the horizontal distances from E, e, are equal.

Hence if in the plane of the horizon, GB be drawn perpendicular to the side AB, and GB be double of CH, and FB double of DE, or de, the fluid spouting from C will fall upon G, and from E and e, upon F.

COR. If the side of the jet be inclined, in any angle to the horizon, and the direction, and velocity of the spouting fluid be known, the amplitude, altitude, and time of flight, may be discovered by the rules investigated in Book II. on Projectiles.

EXP. Let water spout from the middle orifice, and from orifices equally distant from the middle, the truth of the proposition will be manifest.

REMARK. In all the propositions respecting the times in which vessels empty themselves, the orifice is supposed to be very small in respect to the bottom of the vessel, otherwise the experiments do not agree with the theory.

DEF. II. A river is a stream of water which runs by its own weight down the inclined bottom of an open channel.

DEF. III. A section of a river is an imaginary plane, cutting the stream, which is perpendicular to the bottom.

DEF. IV. A river is said to flow uniformly when it runs in such a manthat the depth of the water in any one part continues always the same.

ner,

PROP. XVIII. If a river flows uniformly, the same quantity of water passes in an equal time through every section.

Fig. 11.

Let AB be the reservoir, BC the bottom of the river, and ZX, QR, sections of the plate. 5 river. Because the river flows uniformly, the same quantity of water which passes through Fig. 15 ZX in a given time must pass through QR in the same time: otherwise the quantity of

water

Plate 5.

Fig. 15.

water in the space ZQXR, must in that time be increased or diminished, and consequently the depth of the water in that space altered; contrary to the supposition.

COR. Hence if V, B, D, be the velocity, breadth, and depth respectively, VxBxD will be a given quantity, and V will vary as

I

BxD

PROP. XIX. The breadth of the channel being given, the water in rivers is accelerated in the same manner with any body moving down an inclined plane.

For each drop of the water moves down upon the inclined plane of the bottom, or upon the inclined plane of the sheet of water, next below it, parallel to the bottom.

PROP. XX. The breadth of the channel being given, the velocity of each drop of water in a river is the same that a body would acquire in falling from the level of the surface of the water in the reservoir, to the place of the drop.

Let AB be the depth of the reservoir, AP the level of its surface, and BC the bottom of the channel. Any drop at E, after it comes out of the reservoir at K, (by Prop. XIX.) rolls down the inclined plane KE, parallel to the bottom. And this drop, when it comes out of the reservoir AB at K (by Prop. XIII.) has the same velocity which a heavy body would acquire in falling from A to K: and, in rolling down the inclined plane KE, it acquires (by Book II. Prop. XXXIV.) the same velocity which any heavy body would acquire in falling down GE the perpendicular height of the plane. At E the drop will therefore have acquired a velocity equal to that which a body would acquire by falling through AK and GE, that is, through MGE, the perpendicular drawn from the level of the reservoir to the place of the drop.

COR. I. Hence the breadth of the channel being given, the velocity of each drop of water in a river is as the square root of its distance from the level of the surface of the reservoir. For, if E and R be two drops in different parts of the river, and AP the level, the velocity of the drop E is the same that a body would acquire by falling down ME, and that of R the same which a body would acquire by falling down HR. Therefore (by Book II. Prop. XXVI. Cor. 2.) the velocity of the drop E is to the velocity of R, as the square root of ME to the square root of HR.

COR. 2. Hence the breadth of the channel being given, the water at the bottom of a river will run faster than the water at the surface.

PROP.

SPROP. XXI: The breadth of the channel being given, the depth of the river continually decreases as ît runs.

The same quantity of water (by Prop. XVIII.) passes through each of the sections ZX, QR, in the same time. But (by Prop. XX. Cor. 2.) the water runs faster at the lower section QR, than at the upper ZX. Therefore the area of the section QR must be as much less than the area of the section ZX, as the velocity at QR is greater than the velocity at ZX. But the breadth of the sections are by supposition equal; therefore their areas are (El. VI. 1.) as their heights. Consequently the heights of the section QR, ZX, will be inversely as the velocities at those sections; that is, the depth of the water at QR will be as much less than the depth at ZX, as the velocity at QR is greater than the velocity at ZX.

PROP. XXII. At a given distance from the reservoir, if the river flows uniformly, the velocity of the water will be inversely as the breadth of the channel.

Because the river flows uniformly, the depth at any given section ZX is always the same: and in any given time, the same quantity of water must flow through the different sections ZX, QR, as was shewn in Prop. XVIII. But a given quantity of water cannot flow in a given time through any section, unless as much as the area is increased, so much the velocity' is diminished, and the reverse; that is, the velocity must be inversely as the area of the section, or the depth being given, as its breadth.

PROP. XXIII. The depth of a river being given, the pressure upon any part of the bank will be the same, whatever is the breadth of the river.

The pressure upon any given part in the bank (by Prop. I. and III) will be as the distance of that part from the surface; which remains the same whilst the depth is the same, whatever be the breadth of the river: therefore the pressure will remain the same.

PROP. XXIV. If the breadth of a river be given, the pressure on any part of the bank will be as the depth of the river.

For the pressure on any part of the bank is (by Prop I. and III.) as the depth of that part below the surface, which depth will increase with the depth of the river.

PROP. XXV. The pressure against any given surface in the bank of a river, if that surface reaches from the bottom to the top of the stream,

97

Plate 5.
Fig. 15.

Plate 5.
Fig. 11.

is equal to the weight of a column of water whose base is the surface, and whose height is half the depth of the stream,

Let ZQXR be a given surface in the bank, reaching from the bottom BC of the river to its top AD. The pressure upon this is (from what was shewn in Prop. IX) half the pressure on an equal surface at the bottom XR; which pressure (by Prop I and III) is equal to the weight of a column of water whose base is the surface ZQ, and whose height is the depth of the stream. Therefore the pressure against the surface ZQXR is equal to the weight of a column whose base is the surface ZQ, and its height half the depth of the

stream.

PROP. XXVI. When a stream which moves with the same velocity in every part strikes perpendicularly upon any obstacle, the force with which it strikes is equal to the weight of a column of the same fluid, whose base is the obstacle, and whose height is the space through which a body must fall to acquire the velocity of the stream.

Let a stream of water flow horizontally out of the orifice e. If this stream were to strike upon an obstacle of the same breadth every way as the orifice or stream, placed perpendicular to the horizon, the stream must strike upon the obstacle with its whole force. But this force is equal to the weight of a column of water whose base is e, and height Ae. And (by Prop. XIII.) Ae is the height from which a body must fall to acquire the velocity with which the stream spouts from e. Therefore the force with which this stream would strike such an obstacle is equal to the weight of a column of water whose base is e, and height that from which a body must fall to acquire the velocity of the stream. And because no part of the stream, however broad; can strike the obstacle except so much as is contained within a section equal to the surface of the obstacle, no other part of the stream is to be considered in estimating this force. It is also manifest, that if the stream flow horizontally with the same velocity, in any other manner than through an orifice, as in the current of a stream, it will strike an obstacle with the same force.

PROP. XXVII. When the obstacle is given, the force with which a stream strikes upon it, will be as the square of the velocity with which the

stream moves.

If any stream strikes upon a given obstacle, the force will (by Prop. XXVI.) be equal to the weight of a column of water whose base is the obstacle, and whose height is equal to the space through which a body must fall to acquire the velocity of the stream Since then. the base is given, the weight will be as the height of such a column. But the spaces through which bodies fall to acquire different velocities are (by Book II. Prop. XXVI.) as the squares of those velocities. Therefore the height of this column, and its weight, and consequently the force of the stream, which is equal to this weight, will be as the square of the velocity with which the stream moves.

CHAP.

CHAP. III.

Of the RESISTANCE of Fluids.

PROP. XXVIII. If a spherical body is moving in a given fluid, the resistance which arises from the re-action of the particles of the fluid is, within certain limits of the velocity, as the square of the velocity with which the body moves.

A spherical body moving in a given fluid, the number of particles which it will meet with in a given time will be as its velocity; for the space through which it will pass will be as its velocity, and the number of particles it will meet with will be as the space through which it passes. But the re-action of the particles of the fluid, and consequently the resistance, is as the number of particles or quantity of matter by which the the resistance is made. Again, if a given quantity of matter is to be moved, the moving force is (by Book II. Prop. IX) as the velocity communicated; and the resistance of that given quantity of matter is as the moving force. Therefore the resistance arising from re-action in a given number of particles of fluid is as the respective velocities with which they are moved; that is, as the velocities with which the bodies which pass through the fluid move. The resistance of the fluid being then as the velocity on a double account, first, because the number of particles moved are as the velocity of the moving body, and secondly, because the resistance of a given number of particles is as the velocity of the moving body; the resistance will be in the duplicate ratio, or as the square of this velocity.

SCHOL. In very swift motions, the resistance of the air increases in a greater ratio; (see Remark to Prop. LVIII. Book H) and in other fluids, the same consequence would follow for the same reason, with respect to projected bodies. Besides, the greater the velocity is. the less will be the pressure against the back of the body which will cause a deviation in the law of resistance.

PROP. XXIX. When a spherical body moves with a given velocity in any fluid, the resistance of the fluid arising from its re-action, will be as the squares of the diameter of the spherical body.

A spherical body, in moving through a fluid, displaces a cylindrical column of that fluid, the height of which is the space which the sphere describes, and its base a great circle of the spherical body. Because the velocity is given, the space described in a given time, that is, the length of the column is given: whence, the quantity of fluid in the column, that is, the column will be as its base, a great circle of the sphere. And the resistance which the column of fluid makes by re-action to the motion of the sphere will be as its quantity of matter: it will therefore be as the base of the column, or as the great circle of the sphere, or (El. XII. 2.) as the square of its diameter.

PROP.

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