must be the sum of their components in the same direction. Conversely, this property may be taken as the definition of the resultant of two (or any number of) motions.

20. If x, y11 be the distances of a point P, from three fixed planes at right angles to each other, with similar notation for the distances of other points P2 P3....P1 from the same planes, the point whose distances are

is called the centre of mean position of the n points, and is identical with the centre of gravity of equal masses at the n points. Its motion resolved normally to any one of the three planes is obviously of the

n

sum of the motions

of the n points similarly resolved. Hence the motion of

the centre of mean position, if magnified

the resultant of the motions of the n points.

times, is This pro

position reduces to that of § 18 when there are only two points.

The motion of the centre of mean position may with propriety be called the arithmetical mean of the motions of the n given points.

21. If the line joining two points A B be unequally divided in a constant ratio, the motion of the point of

section may be called a mean with unequal weights.' Let the point of section be called G, and let AG be to G B as b to a, so that G is the centre of

gravity of a weight a at A and a

weight at B. Then if a and b are integers, it follows from

A

G

FIG. 6.

B

§ 20 that a such motions as that of a and b such motions as that of B would have for their resultant the motion of G magnified a+b times. Whether a and b be integers or not, a+b times the motion of G will be the resultant of a times the motion of A and 6 times the motion of B.

22. These principles can be illustrated by the pantagraph, an instrument used by engravers for reducing

drawings. It may be regarded as consisting of a jointed parallelogram A B, divided by two bars parallel to its sides into four smaller jointed parallelograms, two of which, AG, G B, are similar to the whole parallelogram and are therefore about the same diagonal. The intersection G of the two cross-bars thus divides the diagonal A B in a constant ratio.

In the ordinary use of the instrument A is fixed, and a pen at G draws a reduced copy of the curve traced by a style at B, the scale of the copy being to that of the original as AG to A B, or as A D to A E, a ratio which the operator has the power to adjust at pleasure.

If A and B are simultaneously moved, G will have a motion which is a mean of their motions, and if the instrument is set for reducing one-half (in other words, if I be the middle point of A E) the motion of G will be the resultant of the motions of A and B reduced one-half. This application of the pantagraph is, we believe, new.1

We have described the pantagraph in the manner which is simplest from a theoretical point of view. Its actual construction for the purposes of the engraver is as shown in the annexed figure. The tracing point to be

The

carried over the original is at a definite point B in one of the arms. pencil or pen is at G, which is not a definite point but depends on the scale of reduction required, the bar DG being graduated for this purpose; and D A is graduated to correspond with it in such a way that when G and A are at similarly marked divisions, B, G and A will be in one straight line. A is pivoted to a heavy weight to prevent it from moving, and there are castors at the ends and corners of the frame, to roll over the paper.

If G is fixed, the motion of в will always be opposite in direction to that of a, and if a and G be simultaneously moved in given ways, it is obvious from the last sentence of § 21 that 6 times the mòtion of B will be the resultant of a+b times the motion of G, and a times the reversed motion of A.

23. The arithmetical mean of the motions of three points A, B, C can be found by employing one pantagraph to give, by the intersection G of its cross-bars, the arithmetical mean of the motions of A and B ; and employing a second pantagraph, with one corner jointed to G and the opposite corner to c, to give a mean in which G has double the weight of c.

To obtain the arithmetical mean of the motions of four points, we may employ one pantagraph to give the mean of the motions of two of them, a second to give the mean of the motions of the other two, and a third to give the mean of these two means.

111

It is thus always possible to obtain by a combination of n--I pantagraphs the arithmetical mean of the motions of n points. The resultant of the motions of the n points will be this mean magnified n times.

с

CHAPTER III.

COMPOSITION OF VIBRATIONS OF THE SAME PERIOD.

24. Two uniform circular motions of the same period and in the same direction compound into a single uniform circular motion.

For if A and B (Fig. 9) revolve with the same period and therefore with the same angular velocity round o, the angle between the revolving radii o A, O B will be constant, and the parallelogram O ACB will revolve as a rigid figure round o. The uniform circular motion of c is the resultant of the two given motions.

FIG. 9,

25. Two S.H. motions of the same period in the same straight line compound into a single s.H. motion of the same period.

For the two components are the projections (upon the given line) of two uniform circular motions of the same period; and the resultant will be the projection of the resultant of these two circular motions. It will there

1 Here and elsewhere we use the initial letters S.H. as an abbreviation for 'Simple Harmonic.'