tion will be negative. If then the point change its direction in the interval cd, the ordinates will decrease.

And, as in the former case, if the ordinates are taken in sufficient number, a continuous curve is obtained, as pPGKL, which will tend upwards when the point moves in one direction, and downwards when in the other direction.

Now since the space described in any interval of time is represented by the difference of the two ordinates corresponding to the beginning and end of that interval, so the velocity is proportional to that difference divided by the difference of the abscissæ. Thus in the interval bc (= fm), gm is the velocity, which is proportional to the tangent of gfm, or ultimately to the tangent of the angle which the curve makes with the axis Ax.

.

gm space described, and the fm

15. This method is better adapted for representing the motion of the parts of mechanism than the other, because the tendency of the sinuous line corresponds with the direction of the body, changing from upwards to downwards, and vice versa, as the direction changes; while its more or less rapid inclination indicates the change of velocity. Thus the line is a complete picture of the motion, as the line formed by the notes in music is a picture of the undulations of the melody; whereas by the first method where the ordinates represent the velocities, the directions are indicated by the situation of the curve above

or below the axis, which is a distinction of a different kind from the thing it represents, and requires an effort of thought for its comprehension. Sometimes the axis Ar of the time is drawn vertically, and the ordinates consequently are horizontal.

16. The two methods are compared in the following figure, which represents the motion of the lower extremity of a pendulum, the continuous line upon the first hypothesis, and the dotted line upon the second.

The axis of the abscissæ Ak is vertical, AM is the interval of time corresponding to one oscillation from left to right, and MN to the returning oscillation from right to left.

K

n

Fig. 2.
A

M

N

H

m

In the continuous line the horizontal ordinates represent the velocities, which beginning from zero at the left extremity of the vibration at A, reach their maximum values in the middle of each

oscillation at H and K, and vanish at the extremities of the oscillations at M and N. The right side of the axis is appropriated to the direction of motion from left to right, and the left side to the opposite direction.

In the dotted line the ordinates represent the distances from the middle or lowest point, which are greatest at the beginnings and ends of the oscillations at a, m, n. But the curve in this case moves from right to left, and vice versa, as the pendulum moves.* 17. In the varied motion of mechanical organs it generally happens that the changes of velocity recur perpetually in the same order, in which case the movement is said to be Periodic. The period is the interval of time which includes in itself one complete succession of changes, and the motion is made up of a continual series of similar periods. But the changes of velocity in the different periods may be similar in the law of their succession only, and may differ either in the actual values, or in the interval of time required for each period. In most cases, however, the periods are precisely alike in the law and value of the successive velocities, as well as in the interval of time assigned to each. Such motion is termed a Uniform Periodic Motion; of which examples are the motion of pendulums, or of the saws in a saw-mill, supposing the prime mover to revolve uniformly.

The complete set of changes in velocity included in one period may be termed the Cycle of Velocities. This phrase is, indeed, generally applicable to anything that is subject to recurring variations, whereas Period is applicable to time alone. The successive phenomena of motion in each period are sometimes termed its Phases, so that the periodic motion is thus a recurring series of phases. The choice of the phase in this series, which shall be reckoned as the beginning and end of the period, is arbitrary. Thus we may reckon the beginning of the periods of a pendulum, either from one of the extremities of its oscillation, or from the middle and lowest point.

* If a pencil be attached to the lower part of the pendulum so as to touch a vertical surface of paper behind it, and this surface travel by means of clockwork with a uniform motion upwards, the pencil will trace this very curve. This supposes that the circular are described by the pencil in each oscillation belongs to so small an angle that it may be taken as a horizontal right line.

Upon this principle apparatus is constructed for the registration of the motion of machinery, in which such motion curves are traced either by pencils or by the photographic image of some moving point of the machine upon paper applied to the surface of a revolving cylinder. The machines to which such apparatus is applied are those employed for measuring atmospheric phenomena, as barometers, hygrometers, windgauges, &c., or for the appreciation of magnetic variations, the recording of the variations of pressure in the cylinders of steam engines, and the like.

PART THE FIRST.

CHAPTER I.

ON TRAINS OF MECHANISM IN GENERAL.

18. MECHANISM may be defined to be a combination of parts so connecting two or more pieces, that the motion of one compels the motion of the others, according to a law of connection depending on the nature of the combination. The motion of elementary combinations are single or aggregate.

Aggregate motions are produced by combining in a peculiar manner two or more single combinations, as will hereafter appear in Part II. All that follows in this Part relates to the single combinations alone.

19. The motion of every point of a given piece in a machine being defined, as in the Introduction, by path, direction, and velocity, it will be found that its path is assigned to it by the connection of the piece with the frame-work of the machine; but its direction and velocity are determined by its connection with some other moving piece in the train. Thus the points of a wheel describe circles, because its axis is supported by holes in the frame; but they describe them swiftly or slowly, backwards or forwards, by virtue of its connection with the next wheel in the train, which lies between it and the receiver of power.

This connection affects the ratio of the velocities, and the relative direction of motion of the two pieces in question, but its action is independent of the actual velocities or directions of either piece, as in the familiar example already quoted of the two hands of a clock, where the connection by wheel-work is so contrived, that while one hand revolves uniformly in an hour, the

*

* We shall find a few contrivances in which this is not strictly true with respect to the direction, but they are not of a nature to vitiate the generality of the principle.

other shall revolve uniformly in twelve. But this connection has this more general property, that it will also compel the latter to revolve with an angular velocity of one-twelfth of the former, whatever be the actual velocity communicated to either; as, for example, when we set the clock by moving the minute-hand rapidly to a new place on the dial, and similarly with respect to direction, the two hands will always revolve the same way, whether that be backwards or forwards.

Since Mechanism is a connection between two or more bodies, governing their proportional velocities and relative directions, and not affected by their actual velocities or directions; it follows that a systematic arrangement of the principles of mechanism must be based upon the proportions and relations between the velocities and directions of the pieces, and not upon their actual and separate motions.

20. Proportional velocities may be divided into those in which the ratio is constant, and those in which it varies.

V

Let Vand be the velocities of two bodies, then is the

v

velocity ratio; and if the velocities are uniform, let S, s be the V S spaces described in the same time T; ..

v

= a constant

ratio; consequently between uniform velocities the velocity ratio is constant, which indeed is sufficiently obvious.

If, however, the velocities be not uniform, and yet the velocity ratio constant, let the bodies in any successive intervals of time T, T,, T,, ... move with velocities V, V, V,,.. and v, v, vu respectively, of any different magnitudes, but so that the two velocities at the same instant always preserve the same ratio;

And as this is true whatever be the magnitude of the intervals of time, it is also true when they are taken so small that the changes of velocity become continuous, and therefore when the velocity ratio is constant it is obtained by comparing the entire

spaces described in the same interval of time, whatever changes the actual velocities of the bodies may have undergone during that time.

And in the same manner it may be shown that in revolving bodies the angular velocity ratio, if constant, is equal to the ratio of the synchronal revolutions, notwithstanding the velocities of rotation may vary, and also to the inverse ratio of the periods if the angular velocities be uniform.

When the velocity ratio varies, the relations of motion between two pieces may often be more simply defined by means of the law of their corresponding positions than by the ratio of their velocities.

21. With respect to actual direction we have seen that it has only two values, but the relation of direction between two bodies moving in given paths may be conveniently divided into two classes. In the first, while one continues to move in the same direction, the other shall also persevere in its own direction; but if one change the other shall change. To this class belongs the clock-hands; and in this instance both hands move the same way round the circle. But this is not necessary; it may be that when one piece revolves to the right the other may revolve to the left, and vice versâ, as in a pair of flatting rollers; or again in the old simple mangle, so long as the handle is turned in one direction, the bed of the mangle will travel forwards, but when the motion of the handle is reversed, the bed of the mangle also returns. In all these cases the directional relation is constant. In another class the connection is of this nature, that while one body perseveres in the same direction, the other shall change its direction; as, for example, in a saw-mill. The saw-frame moves up and down, changing its direction periodically, but the piece from which it derives this motion revolves continually in the same direction. In cases of this kind the directional relation changes.

22. We have thus two kinds of directional relation, and two of the velocity ratio, by means of which all the simple combinations of mechanism, for the modification of motion, may be conveniently grouped under the following heads or divisions.

DIVISION 1. Directional relation and Velocity ratio con

stant.

DIVISION 2. Directional relation constant-Velocity ratio

varying.

DIVISION 3. Directional relation changing periodically—
Velocity ratio either constant or varying.*

* The third division might have been separated into two by arranging the constant and varying velocities under different heads, but it will be found that the contrivances