horse-power of the engine; and, with a vessel whose engine has a power equal to 200 horses, the diameter of the wheel, between the outer extremities of the paddleboards, is about 20 feet; the lengths of the paddles are rather less than half, and the breadths between one-ninth and one-tenth of the diameter. Hence, if the circumference of a wheel 20 feet in diameter be furnished with 20 paddle-boards 2 feet broad, when the upper edge of the lowest vertical board is a-wash, there will be three boards wholly or partly immersed, one will be nearly entering the surface of the water, and a fifth will have just emerged from it. 20. If a vessel be retained at rest in still water while the wheels revolve, the reaction of the water against a paddle-board will be the greatest when the tio board is in a vertical position in the water, but this will not always be the case when the vessel is free to move by the rotation of the wheel. In order to explain this subject, let S be the centre of the wheel's rotation, and AB the momentary position of a paddle making, with the vertical line SZ, an angle ZSB represented by 0. Let V be the velocity of the point C (supposed to be the centre of pressure on A B) in a direction perpendicular to the surface of the paddle A B, and V' the velocity of the vessel in the water, in a horizontal direction; then, by the Resolution of Forces or Velocities, V' cos. O is that velocity in a direction perpendicular to the surface of the paddle A B; therefore V – V cos. O will express the relative velocity of the paddle and vessel in the same direction. But the resistance of a fluid against a body moving in it varies with the square of the velocity ;a therefore (V – V' cos. 0)2 may denote the force of resistance, or pressure, against the paddle : this being multiplied by V, the product is the efficient W * Experiments have shown that this rule is very nearly correct notwithstanding the perturbation of the water by the wheel's rotation. momentum of that resistance in a direction perpendicular to the surface of the paddle ; and consequently (V - V' cos. 6). V cos. O is the efficient force by which that resistance impels the vessel forward horizontally : which, for the vertical paddle, where 0 = 0, becomes (V - V)V. 21. But in these expressions it is supposed that the paddles are wholly immersed : this is evidently not the case with the oblique paddles when the upper edge of the lowest vertical paddle is on a level with the surface of the water, for then the immersed part of an oblique paddle is expressed by SB - SA sec. 0; or, r being the radius of the wheel to the outer extremity of a paddle-board, and a the difference between r and the breadth (6) of the paddle, it is expressed by r — a sec. 0 : consequently the ratio between the efficient resistances against a vertical and an oblique paddle will be as (V – V9% V : (V – V' cos. O)V (r cos. 0 – a). These expressions being put in numbers according to the data, for different values of 0, it will be found that the first will be less than the second till the part of the paddle's breadth which is out of the water causes a diminution of power which more than compensates for the superiority which is due to the obliquity. Making the differential of this last expression equal to zero, we may obtain the value of 0 which makes the resistance a maximum. Assuming V'=#V, r = 10 feet, a = 8 feet, whence b = 2 feet, the greatest resistance takes place when 0 = 18°; and the force on the vertical paddle is, to that on the oblique paddle in this position, as 10 to 10.865. The resistance against a vertical paddle being thus proved to be less than the resistance against an oblique paddle, in the most effective part of the motion of the latter, it follows that to obtain equal speed for two vessels, one of which is furnished with paddles of the ordinary kind, and the other with such as are kept by machinery always in a vertical position, the wheels being of equal dimensions, that which has the vertical paddles must revolve with greater velocity than the other, and consequently it must cause a greater consumption of steam and fuel. 22. If a vessel were at rest, every point in the arms or radii of a paddle-wheel would, during a revolution, describe a circle; but when the vessel is in motion, each point describes a trochoidal curve, which is the common cycloid when the forward rectilinear motion of the vessel, during the time of a revolution of the wheel, is equal to the circumference of the circle which would be described by the point if the vessel were at rest. Every point farther from the centre of the wheel than that which describes a common cycloid must describe what is called a curtate or contracted cycloid, and every point nearer the centre a prolate or extended cycloid. 23. The curves described by points on the opposite edges of a paddle-board, and the various positions assumed by a paddle-board during a revolution of the wheel, are exhibited in the annexed figure : Fig. 2. Let A be the centre of a wheel having twenty-four paddle-boards, and let T be a point on the exterior edge of one of the boards when in a vertical position ; also, let the wheel turn about A in the direction Ta b c, &c., and, at the same time, let the centre A be carried towards B by the movement of the vessel, the straight line, A B, being supposed equal to the circumference of the circle, described about A by some point as U, if the vessel were at rest. Then, if A B be divided into twenty-four equal parts in the points 1, 2, 3, &c., the board at T will take, successively, the positions indicated by 1 E, 2 D, 3 F, &c.; and when it coincides in direction with A C, the centre, A, being then at 6, the wheel will have performed one-quarter of a revolution, the outer edge of the board having described the cycloidal curve TE...C, and the inner edge the curve te... 11. The vessel continuing its rectilinear motion and the wheel its revolution, the edges of the board will describe the looped curves at C P X shown in the figure ; and when A has arrived at C, half a revolution of the wheel being performed, the board T will have the vertical position C X. The curves described by the points T and t during the second half of the revolution of the wheel will be symmetrical with those described during the first half; and the whole revolution will be completed when A has arrived at the point B. If P K represent the surface of the water, the oblique lines within the space PXK will show the positions of the several paddle-boards while in the water. The position of the point U may be found on dividing the velocity of the vessel, in feet per hour, by the number of revolutions of the wheel per hour (or by the number of double strokes made by the piston of the engine per hour); the quotient is the circumference, in feet, of the circle whose radius is AU; from this value of the circumference the radius A U may be obtained. A circle whose circumference is thus determined is called the circle of rotation. In vessels having the ordinary speed, the radius of the circle of rotation is equal to about two-thirds of the radius of the wheel, to the outer edges of the paddle-boards. The centre of pressure in any revolving plane is that in which, if the whole pressure were concentrated, the effect would be equal to that which takes place when the pressure is uniformly distributed over the plane. In a paddle-board the position of this centre varies with the depth of its immersion; and if, as an approximation to its position, its distance from the centre of the wheel of 24. If a spirals of the Indicdective power be considered as equal to re{b, representing this value by m, the expression (V – V' cos. O) V (r cos. 0 -- a) r' d 0 being integrated between the limits of o, the result would give the whole pressure on the wheel, and be the equivalent to the power of the engine. It is at present the practice to measure the effective power of a marine engine by means of the Indicator and Dynamometer. 24. If a spiral line were traced on the convex surface of a cylinder, so as to coincide with the hypotenuse of a right-angled triangle wound about it, the base of the triangle being equal to the circumference of the cylinder, and disposed in a plane perpendicular to the axis, then, if through every point in the spiral line straight lines are drawn perpendicular to the axis of the cylinder, those lines will be in the superficies of what is called the blade or feather of a screw. If all these perpendiculars are of equal length, their outer extremities will form the periphery of the helix : the distance between two points on this periphery, measured parallel to the axis of the cylinder or screw, is called the pitch of the screw. 25. If a screw thus formed is attached to a floating body, as a ship, with its axis in a horizontal position, and the screw is made, by means of machinery connected with a steam-engine, to revolve on that axis in the water, the pressure exerted by one surface of the blade on the water will be accompanied by a reaction of the water against that surface; and the force of this reaction, resolved in a direction parallel to the axis of the screw, will cause the ship to move in that direction. The reaction of the water against any point on the blade will depend on the velocity of the screw's rotation, on the depth of the point below the surface of the water, and on various other circumstances. 26. If the water, pressed by the posterior surfaces of the float-boards of a paddle-wheel, or by the posterior surface of the blade of a screw, could remain stationary so as to form a perfect fulcrum, the whole force of its |