no pinching. The slot and the piece A в in which it is cut will evidently be compelled to take a simple harmonic motion.

This plan involves a large amount of friction; and it would be difficult to preserve a good fit, as the parts would wear loose. Instead of allowing the crank to rub against the slot, it is better, as suggested by the Rev. F. Bashforth, in 1845, to make the crank work in a circular hole in the centre of a sliding-piece which travels in the slot. Fig. 30, which is copied from Mr. Bashforth's

FIG. 30.

drawing, shows the slot and sliding-piece with the hole in its centre.

85. A motion approximately simple harmonic can be obtained by the arrangement shown in Fig. 31. A B is a

B

FIG. 31.

C

crank revolving uniformly round the centre A.

CD is a

rod guided to move in a straight line passing through a,

and B c is a connecting-rod jointed at B and C to the other two pieces. CD will have a reciprocating motion, which will be more nearly simple harmonic as the ratio of the length of the connecting-rod to that of the crank is greater. The arrangement here described is met with in many of the commonest forms of steam-engine, CD being the piston-rod and A B the crank, which it drives with a nearly uniform velocity of rotation; the uniformity being maintained by means of a fly-wheel, or in the locomotive by the inertia of the engine and train. A similar arrangement is also generally employed for working the slide-valves, the crank AB being usually replaced by an eccentric mounted on the axle of the fly-wheel. If в be the centre of this eccentric and a the fixed point round which в revolves, the motion of the connecting-rod BC and of the valve-rod C D will be precisely the same as if A B were a crank.

B

86. In steam-engines which admit of being reversed, the apparatus for reversing consists usually of a combination of two eccentrics, each having its own connectingrod for giving an approximately simple harmonic motion to the slide-valves. The principle of its action is illustrated by Fig. 32, where A and B are the two eccentrics, their centres revolving in one and the same circle (the

1 If the projection of BC upon the straight line A C D can be regarded as of constant length, the motion of c is the same as the motion of the projection of B, and is therefore simple harmonic. In order that the motion of c may be sensibly simple harmonic, the difference between the greatest and least projections of BC (the former being BC itself) must be negligible in comparison with the amplitude A B.

dotted circle in the figure) round a fixed centre. The line joining the centres of the two eccentrics is always a diameter of this circle, so that when one is in the extreme position to the right the other is in the extreme position to the left. CD is the valve-rod, which is constrained to travel in what is very nearly a straight line, passing through the centre of the dotted circle. The ends, E F, of the two connecting-rods are joined by a piece in which

a slot is cut for the purpose of receiving a button G, which moves with the valve-rod. In the figure the button is represented as midway along the slot. In this position, the opposite motions of the points EF combine to leave the button and valve-rod nearly at rest. This accordingly

is the position for stopping the engine.

By means of the bar FH, which contains three holes

for fixing it by a pin L, the slot can be made to travel along the button. When the hole a is brought down to 1, the upper end of the slot will be brought down to the button, and the movement of the button will be governed by the eccentric A. This is the position for driving the engine in one direction-say forwards. Then, to reverse the engine, the hole 6 is brought to L, and the lower end of the slot is thus brought to the button, which will now be driven by the eccentric в instead of by A, and the

phases of motion for the valve-rod (and consequently fo the piston) will be reversed.

87. New method of obtaining S.H. motion.

A rigorous simple harmonic motion can be obtained without the friction of guides by employing a pantagraph (as described in § 22) to give the arithmetical mean of two equal and opposite uniform circular motions (see § 26), as

illustrated by Fig. 33. ADBE is a jointed rhombus, with two cross-bars jointed to the middle points of its sides. A pair of opposite corners, A B, are to be carried with equal and constant velocities in opposite directions round two equal circles in the same plane, so that the line joining AB is always parallel to the line of centres. Then the other diagonal DE will always bisect the line of centres at right angles, and the point c, in which the cross-bars intersect, being identical with the intersection of the two diagonals, will be the projection of A or B upon the fixed line D E. The point c will therefore have simple harmonic motion, of amplitude equal to the radius of either circle. The motion of c could be magnified or diminished in any required ratio by means of a second pantagraph with one corner fixed, and either the intersection of its cross-bars or the opposite corner attached to c. The dotted line m n in the figure represents the path of c, and the dotted parallelogram represents another position of the rhombus.

We have for simplicity supposed the parallelogram A D BE to be a rhombus and the diagonal A B to be parallel to the line of centres; but it is evident from §§ 22 and 26 that these restrictions are not necessary. The general

ised construction will be as follows:

Let ADBE be any parallelogram with two cross-bars jointed to the middle points of its sides; and let a pair of opposite corners AB be carried, with equal and constant velocities in opposite directions, round two equal