forming k revolutions, where k may be a whole number or a fraction. Then, since 2 is the circumference whose radius is unity; 27k will be the space described in k revolutions by the point whose radius is unity, but A is the space described by the same point in the unit of time;

.. A: 2πk :: 1 : T; .'. T= (1); k =

2πk
A

TA

2π

(2);

hence the number of turns in a given time varies as the angular velocity.

Let R be the radius of a wheel and Vits perimetral velocity;

whence the number of turns in a given time varies directly as the perimetral velocity, and inversely as the radius or diameter of the wheel."

Let the time in which a wheel performs one complete revolution be termed its Period (= P); .'. P= {putting k-1 in

2π

A

(1)}; and the period varies inversely as the angular velocity;

T

Also from (2) k=1; whence the period varies inversely as the number of turns in a given time. When the rotations of two wheels are to be compared, the number of turns they respectively make in a given time may be termed their synchronal rotations.

12. When the velocity is not uniform, these expressions can no longer be applied, because the velocity is different at different times. In this case, then, the velocity at every instant is measured by the space that would be described in the succeeding unit of time, were the velocity with which that unit is commenced continued uniformly throughout it.

If the velocity of a body increase, it is said to be accelerated, and if the velocity diminish, to be retarded.

* In practice linear velocity is commonly referred to seconds and feet, but angular velocity to minutes and revolutions or turns; thus a millwright will define the velocity of a given wheel by either saying that it performs twenty revolutions in a minute, or that its circumference moves at the rate of three feet per second. In the expression (3) if k and T be expressed in minutes, and V is to be expressed in seconds, we must put 60V for V;

60 TV 10 TV

.. k=

very nearly.

Railroad velocities are so high that they are stated in miles per minute.

13. Varied motion admits of convenient graphical representation, by which its characteristic points and general laws are rendered much more easy of comprehension than they are by the use of formulæ alone.

Thus to represent the motion of a point of which the velocities at certain given intervals of time are known, take an indefinite straight line AX, and from A set off abscissæ Ab, Ac, Ad...... proportional to the given intervals of time as measured from the

beginning of the motion. Upon A, b, c, d...... erect ordinates Ae, bf, cg, dh, respectively proportional to the velocity of the point at the beginning of the motion and after each interval of time. By joining the extremities of these ordinates, a polygon efgh...... is obtained, which, if the intervals of time be taken with their differences sufficiently small, will become a curve as hPGKL, of which the abscissa AN at any given point P, will represent the time elapsed from the beginning, and the ordinate NP the corresponding velocity of the point.

If the motion of the point cease, its velocity becomes zero, and the curve meets the axis, as at G and L. If the point change its direction in its path, this is indicated by the change of sign in the velocity; for either direction being assumed positive, the other will be negative; and so in this curvilinear representation, the ordinates representing the velocity for one direction being set off upwards from the line, as from e to G, those of the opposite direction will be set off downwards as from G to L.

14. By another method a curve is constructed of which the abscissæ shall represent the time as before, but the ordinates the space described by the point. Thus, if the last figure be supposed to be constructed on this second hypothesis, Ae will represent the distance of the point at the beginning of the motion from that point of its path whence the space is to be measured; bf its distance from the same point at the end of the time Ab; its distance after the time Ac; and so on. But the motion in one direction being accounted positive, that in the opposite direc

cg

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 pPG KL, 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 be (= fm), gm is the space described, and fm 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.

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 versâ, 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 Ax 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

H

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 versâ, 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.

HANISM

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.