circle in a; join B a, C a; then E a B and E a C will be equal to the angles made by the longer axis with X X and Y Y respectively; and the shorter axis will of course be perpendicular to the longer.

247. Feathering Paddle-Wheels exemplify a class of aggregate combinations in which linkwork is the means of producing the aggregate motion. Each of the paddles is supported by a pair of journals, so as to be capable of turning about a moving axis parallel to the axis of the paddle-wheel, while its position relatively to that moving axis is regulated by means of a lever and rod connecting it with another fixed axis. Thus, in fig. 199, A is the axis of the paddle-wheel; K the other fixed axis, or eccentric-axis; B, E, N, C, P, M, D the axis of a paddle at various points of its

revolution round the axis A of the wheel; BF, EH, N Q, CR, PS, M L, D G, the stem-lever of the paddle in various positions; K F, K H, KQ, K R, K S, KL, K G, various positions of the guide-rod which connects the stem-lever with the eccentric-axis. When the end of the paddle-shaft overhangs, and has no outside

The oblique-angled trammel is believed to be the invention of Mr. Edmund Hunt.

bearing, the eccentric-axis may be occupied by a pin fixed to the paddle-box framing; but if the paddle-shaft has an outside as well as an inside bearing, the inner ends of the guide-rods are attached to an eccentric collar, large enough to contain the paddle-shaft and its bearing within it, and represented by the small dotted circle that is described about K. One of the rods, called the driving-rod, is rigidly fixed to the collar, in order to make it rotate about the axis K; the remainder of the rods are jointed to the collar with pins.

The object of the combination is to make the paddles, so long as they are immersed, move as nearly as possible edgewise relatively to the water in the paddle-race. The paddle-race is assumed to be a uniform current moving horizontally, relatively to the axis A, with a velocity equal to that with which the axes B, &c., of the paddle-journals revolve round A. Let E be the position of a paddle-journal axis at any given instant; conceive the velocity of the point E in its revolution round A to be resolved into two components, a normal component perpendicular, and a tangential component parallel, to the face of the paddle. Conceive the velocity of the particles of water in the paddle-race to be resolved in the same way. Then, in order that the paddle may move as nearly as possible edgewise relatively to the water, the normal components of the velocities of the journal E and of the particles of water should be identical.

Let B be the lowest point of the circle described by the paddlejournal axes; that is, let A B be vertical. Draw the chord E B. Then it is evident that the component velocities of the points B and E along E B are identical. But the velocity of B is identical in amount and direction with that of the water in the paddle-race. Therefore the face of a paddle at E should be normal to the chord E B, or as nearly so as possible. Another way of stating the same principle is to say that a tangent, E C, to the face of the paddle should pass through the highest point, C, of the circle described by the paddle-journal axes, C A B being the vertical diameter of that circle.

It is impossible to fulfil this condition exactly by means of the combination shown in the figure; but it is fulfilled with an approximation sufficient for practical purposes, so long as the paddles are in the water, by means of the following construction:-Let D and E be the two points where the circle described by the paddle-journals cuts the surface of the water. From the uppermost point, C, of that circle draw the straight lines C E, C D, to represent tangents to the face of a paddle at the instant when its journals are entering and leaving the water. Draw also the vertical diameter C A B, to represent a tangent to the face of a paddle at the instant when it is most deeply immersed. Then draw the stem-lever projecting from the paddle in its three positions, D G, B F, E H. In the figure, that lever is drawn at right angles to the face of the paddle; but the

angle at which it is placed is to a certain extent arbitrary, though it seldom deviates much from a right angle. The length of the 3 stem-lever is a matter of convenience: it is usually about of the 5

depth of the face of a paddle. Then, by plane geometry, find the centre, K, of the circle traversing the three points, G, F, and H; K will mark the proper position for the eccentric-axis; and a circle described about K, with the radius K F, will traverse all the positions of the joints of the stem-levers.

From the time of entering to the time of leaving the water, paddles fitted with this feathering gear move almost exactly as required by the theory; but their motion when above the surface of the water is very different, as the figure indicates.

To find whether, and to what extent, it may be necessary to notch the edges of the paddles, in order to prevent them from touching the guide-rods, produce A K till it cuts the circle G F H in L; from the point L lay off the length, L M, of the stem-lever to the circle D B E, and draw a transverse section of a paddle with the axis of its journals at M, its stem-lever in the position M L, and its guide-rod in the position L K. This will show the position of the parts when the guide-rod approaches most closely to the paddle.

Some engineers prefer to treat the paddle-race as undergoing a gradual acceleration from the point where the paddle enters the water to the point of deepest immersion. The following is the consequent modification in the process of designing the gear:-Let the final velocity of the paddle-race be, as before, equal to that of the point B in the wheel, and let the initial velocity be equal to that of the point b, at the end of a shorter vertical radius, A b. Let D be the axis of a paddle-journal in the act of entering the water, and E the same axis in the act of leaving the water. Join b D and B E; draw the face of the paddle at D normal to D b, the face of the paddle at B vertical, as before, and the face of the paddle at E normal to E B. Then draw the stem-lever in its three positions, making a convenient constant angle with the paddleface; and find the centre of a circle traversing the three positions of the end of the stem-lever; that centre will, as before, mark the proper position for the eccentric-axis.

248. spherical Epitrochoidal Paths—Z-Crank.—A point rigidly attached to a cone which rolls on another cone describes a spherical epitrochoid, situated in a spherical surface whose centre is at the common apex of the two cones. This sort of aggregate motion is illustrated by Mr. Edmund Hunt's Z-crank.

In fig. 200, A A is a rotating shaft, carrying at B, B, two crankarms, which project in opposite directions, and are connected with each other by means of a cylindrical crank-pin, B B. The shaft,

crank-arms, and crank-pin, are all rigidly fastened together; and they rotate as one piece about the axis A A.

The crank-pin is contained within a hollow cylindrical sleeve or

ED

Fig. 200.

tube, fitting it accurately, but not tightly, and free to rotate, relatively to the pin, about the axis B B, but not to shift longitudinally. From that tube, and in a plane normal to B B, and traversing C, the intersection of B B and A A, there project any required number of arms, such as CD, CD. Those arms move as one piece with the tube; and each of them, at its end D, is connected, by means of a ball-and-socket joint, with a link, and through the link with a piston-rod, which has a reciprocating sliding movement parallel to A A.

The lower part of the figure shows the position of the mechanism after a quarter of a revolution has been made from the position represented in the upper part of the figure.

The motion of the sleeve, with its arms, is compounded of a rotation about the fixed axis A A, and of a rotation with equal speed in the contrary direction about the revolving axis B B.

Therefore, in the plane of those axes at any instant draw C E, bisecting the obtuse angle B C A, and C E will be the instantaneous axis of the sleeve and arms.

T

Draw C F, making the angle ACF ACE; and C G, making the angle BCG = BCE. Then ECF will be the trace of a fixed cone, having A C A for its axis; and ECG will be the trace of a rolling cone, having B C B for its axis; and the motion of the sleeve with its arms will be the same as if it were rigidly attached to the rolling cone.

Each of the points marked D describes a spherical epitrochoid, shaped like a long and slender figure of 8, and situated in the surface of a sphere of the radius C D, whose trace in the figure is marked by a dotted circle. The trace of the fixed cone on that sphere is projected in the figure by the straight line E F; that of the rolling cone, in the upper part of the figure, by the straight line E G, and in the lower part by a dotted ellipse. The centre of the base of the rolling cone is marked H and h in the two parts of the figure respectively.

SECTION V.-Parallel Motions.

249. Parallel Motioas in General.—A parallel motion is a combination of turning pieces in mechanism, usually links and levers, designed to guide the motion of a reciprocating piece either exactly or approximately in a straight line, so as to avoid the friction which arises from the use of straight guides for that purpose. Its most common application is to the heads of piston-rods.

Some parallel motions are exact; that is, they guide the pistonrod head or other reciprocating piece in an exact straight line; but these parallel motions cannot always be conveniently made use of. Other parallel motions are only approximate; that is, the path of the piece which they guide is near enough to a straight line for the practical object in view; and these are the most frequent. They are usually designed upon the principle, that the two extreme positions and the middle position of the guided point shall be exactly in one straight line; care being taken, at the same time, that the deviations of the intermediate parts of the path of that point from that straight line shall be as small as possible.

There are purposes for which no merely approximate parallel motion is sufficiently accurate; such as the guiding of the tool in a planing machine, whose motion ought to be absolutely straight.

250. Exact Parallel Motions.-When a wheel rolls round inside a ring of exactly twice its radius, any point in the pitch-circle of the wheel traces a straight line, being a diameter of the pitchcircle of the ring (Article 245, page 266). This combination, then, has sometimes, though seldom, been used as an exact parallel motion for a piston-rod; the head of the piston-rod being jointed to a pin at the pitch-circle of the rolling wheel, and the crank to another pin at the centre of that wheel.