CHAPTER II.

GENERAL THEORY OF COMPOSITION OF MOTIONS OF

TRANSLATION.

16. DEFINITION.-If A, B and c are any three bodies, the motion of A relative to c is called the resultant of the motion of a relative to в and the motion of в relative to c.

If c is regarded as at rest, the motion of A will be called the resultant of the motion of A relative to B and the motion of B.

In the present treatise we shall only have to discuss the motions of points, and we shall regard two points as having the same motion if their motions are equal and parallel; in other words, all the points of a rigid

FIG. 5.

body which has a motion of translation will be regarded as having the same motion.

17. CONSTRUCTION FOR COMPOSITION OF MOTIONS.If A and B are any two points whose motion is to be compounded, we may take any fixed point o (Fig. 5) and construct the parallelogram of which O A, O B are two sides. If OR be the diagonal of this parallelogram, the motion of

RESULTANT OF TWO MOTIONS.

13

R is the required resultant. For since AR is constantly equal and parallel to o B, the motion of R with respect to A is the same as the motion of в about the fixed point o, and the motion of R is by definition the resultant of this motion and the motion of A.

If we choose different positions for o, the paths obtained for R will differ in position, but will be equal and similar, and may be regarded as the paths of different points of a rigid body which has a motion of translation.

It is not necessary to suppose the paths of a and в to lie in the plane of the paper. The construction is applicable to the movements of any two points in space.

HÁLF

18. THE MOTION OF THE MIDDLE POINT OF THE LINE JOINING ANY TWO POINTS A B IS HALF THE RESULTANT OF THEIR MOTIONS.

This is obvious from the figure in the preceding section; for since the diagonals of a parallelogram bisect each other, the middle point of A B is the middle point of O R, and its motion about the fixed point o is similar to that of R, but on half the scale.

A

19. Let x1 x, x, be the distances of three points A B C from a fixed plane. Then, since x-x, is constantly equal to the sum of x,-x, and x,-x, the motion of a relative to c resolved in a direction normal to the plane is the sum of the motions of a relative to в and of в relative to c, similarly resolved. Hence, whenever one motion is the resultant of two others, in the sense of the definition at the head of this chapter, its component in any direction

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 x1 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....P, 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 sum of the motions

n

of the n points similarly resolved. Hence the motion of the centre of mean position, if magnified times, is the resultant of the motions of the n points. This proposition 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

MEAN OF TWO MOTIONS.

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

gravity of a weight a at A and a

G

B

FIG. 6.

20 that a such motions as that of A and 6 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 a 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

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

carried over the original is at a definite point B in one of the arms. The pencil or pen is at G, which is not a definite point but depends on the scale of reduction required, the bar D G 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.