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:

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 bc, &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 A U; 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

be considered as equal to r-b, representing this value by r', the expression

being integrated between the limits of 0, 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

reaction would be effective in propelling the ship; but this is not the case-the water pressed by the paddle or blade recedes aftward, and therefore the reaction of the water is that only which is due to the difference between the velocity of rotation in the paddle or screw and that of the water's recession.

It should be observed that the action of the water on the anterior surface of a paddle, or on the anterior surface of a screw-blade, is also a cause of retardation in a ship's motion.

If the pitch of a screw were 10 feet, one revolution of the screw would, if it were not for these impediments, cause the ship to be moved forward 10 feet; whereas, in ordinary circumstances, it is moved only about 8 feet."

27. In the early days of screw-propulsion the propelling surface consisted of a single and continuous blade or feather, making at least one entire revolution about the spindle or axis of the screw; but this formation was soon found to be defective in practice, on account, first, of the great cross-strain it gave to one side of the axis, and the disturbance it occasioned in the different parts of the system; secondly, the severe vibratory motion it caused in the stern, which is the weakest part of the vessel. It was at first supposed that all parts of the

a TABLE I.-Showing the elements of several vessels.

It therefore appears that the average loss by slip is about one-third of the whole effective speed: the ratio of the loss to the whole speed varies very sensibly for the several vessels considered individually; but, in general, we believe that one-third will be very near the truth.

surface of a screw are equally effective in producing propulsion, and this opinion led to the formation of a screw having an entire, or more than one, convolution about the axis; but the supposition is erroneous, and so much the more as the screw revolves on its axis with greater velocity.

28. The most effective part of a screw-blade is that which is near the periphery of the spiral, and the action of any part on the water becomes less as the part is nearer the axis or spindle. Science has, however, as yet done little in investigating the propelling properties of a screw in water; and the intricacy of the subject is such that the formulæ expressing those properties are too complex to admit of being practically applicable except under very limited conditions.

29. If the screw were supposed to consist of a feather or blade forming two or three convolutions in the length of the axis, the reaction of the water on it would be precisely the same as on a screw with one convolution only in its length, since corresponding parts in each of the two or three portions are in corresponding positions in the water, and all are in action at the same time. The water between the posterior part of one turn of the blade and the anterior part of the next, towards the stern of the ship, is nearly quiescent, the water at the posterior surface of the last turn of the blade alone receding; and there appears to be no foundation for the opinion that the water between every two turns of the blade is made to revolve with the screw, or even to be in any state of commotion.

30. Experiment soon showed that, when the length of the screw was diminished so that the feather had successively three-quarters, one-half, and even less than one-third of a turn in the length, there did not appear, with equal engine-power, to be any diminution in the speed of the ship. This apparently anomalous circumstance caused at first much surprise, and the cause of it is not, even now, free from uncertainty; but the explanation, for which the author is indebted to Mr. Lloyd, Director-General of the Steam Department