DESCRIPTION OF STEPS.

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In general, if we have n steps oa,, oa,, oa,...oa,, it is always possible to find a point g such that

n. og = oa2+oa, +...+oa, Σoa,

2

as this sum may be conveniently written. The position of the point g will depend upon the points a,, а 2... α 2 but not in the least upon the point o. To prove this, suppose we take a point p, and draw the steps pa,, pa2...pan The resultant of these must be some step, which can be found by arranging them tandem as in our first process. Let pg be the nth part of this resultant, so that n.pg=Σpa. Now we know that

1

n

og = oppg, oa1 = op + pa1, ... oa1 = op + pa„. Therefore n. og = n.op+n.pg=n.op + Σpa=Σoa. Thus g being chosen so that n. pg=Σpa for a particular position of p, we see that n.og Zoa for any point o whatever. This point g is called the mean point, or mid-centre, of the points a,, a.........ɑ„:

Similarly, it may be shewn that there is a point g such that, if , ... are any numbers,

n

(112+ 1 + ... + 1) og = 11. oa1 + 1. oa2+

whatever point o is.

2

nn

And lastly, if we have n steps a,b,, a,b,...ab, anyhow situated, their resultant is n times the step from the mean point of a, a,...a to the mean point of b1, b...by. proof of this is left as an exercise for the reader.

RESOLUTION AND DESCRIPTION OF STEPS.

The

We have already seen that a step in a known direction may be completely specified by describing its length. This may be done in two ways. First, approximately, by stating the number of inches or centimeters and parts of an inch or centimeter; if the parts are expressed in decimal fractions, the approximation may be carried to any required degree of accuracy by taking a sufficient number of places of decimals. But as the length to be described is generally incommensurable in regard to an inch or a centimeter, this method is very rarely anything

more than an approximation. Second, graphically, by drawing the length to scale. A certain line being marked out upon the diagram to represent a centimeter, another line is drawn bearing the same ratio to this one that the length to be described bears to a centimeter. Thus at the side of a map there is a scale of miles, by which the distance between two places may be estimated. The actual distance bears the same ratio to a mile that the distance on the map bears to the representative length on the scale. This is the theoretically correct way of representing all continuous quantities, except angles, which should also be drawn; though it is sometimes convenient to describe an angle in terms of degrees, minutes and seconds; or in circular measure, which is the ratio of its arc to the radius.

When it is known that a step lies in a certain plane, it may always be resolved into two components which are in fixed directions at right angles to one another. Let oX, oY be two fixed lines at right angles to one another. Let op be the step which it is required to resolve. Draw

=

pm perpendicular to oX, then op om+mp; or the step op has been resolved into two, one of which is in the direction oX, and the other in the direction oY.

Let x be the number of units of length (e.g. centimeters) in om, and y the number in mp. Let also i represent a step of one centimeter along oX, and j a step of one centimeter along o Y. Then om is x times i, or xi; and mp is y times j, or yj. Hence the step op=xi+yj; and we may say that every step in the given plane may be described in the form xi + yj, where x and y are two numerical ratios, and i, j are fixed unit steps at right angles to one another.

When the lengths x and y are given either approximately or graphically, the step (known to lie in a given plane) is completely described in the same way.

It is to be understood that when m falls to the left of

RESOLUTION OF STEPS.

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o Y, x is a negative quantity; and when p falls below oX, y is a negative quantity.

When it is not known in what plane a step lies, we can still resolve it into three components along fixed directions at right angles to one another. Let oX, OY, OZ be three lines at right angles to one another, op the step to be resolved. Draw pn perpendicular to the plane Xo Y, and nm perpendicular to oX. Then op om+mn + np, or the step op has been resolved into three, which are respectively in the directions oX, oY, oZ.

Let, as before, x, y be the number of centimeters in om, mn, and let z be the number in np. Let also i, j, k be three steps of one centimeter each in the directions oX, o Y, oZ. Then om = xi, mn = yj, np=zk, and op = xi+yj + zk. Thus we see that any step whatever can be described in the form xi+yj+zk,

=

Y

where x, y, z are three numerical ratios, and i, j, k are fixed unit steps at right angles to one another.

When the lengths x, y, z are given approximately or graphically, the step is completely described in the same way. It is to be understood that z is reckoned negative when p lies on the further side of the plane Xo Y.

We shall find other quantities, besides steps, which can be resolved into components in three fixed directions, and completely described by assigning three lengths. All such quantities are called vectors, or carriers, from their analogy to a step of translation or carrying. They can always be described in the form xi+yj+zk, where i, j, k are fixed unit vectors at right angles to one another. Except these unit vectors, it is usual to represent a vector either by the beginning and end of the line representing it, as op, or by a single small Greek letter, as a, p.

When the position of a point p is described by means of the step from a fixed point o to it, the point o is called the origin, and the components x, y, z are called the co-ordinates of p. The lines oX, o Y, oZ are called axes of

co-ordinates, and the planes which contain them in pairs the co-ordinate planes. The step or vector op is called the position-vector of the point p.

REPRESENTATION OF MOTION.

We go on to describe more completely the translation of a rigid body. Hitherto we have considered only the step from the beginning to the end of the motion; we shall now take account of the path and of the time in which it is described. As before it will be sufficient to consider the motion of a single point of the body.

a

To describe completely the motion of a point p from a to b it would be necessary to assign the path and also the position of the point in the path at every instant of time. The path may be assigned by drawing it, or by stating its geometrical properties. The position of the point in the path may be assigned by giving the length ap measured along the path at every instant; and this may be done in two ways.

First, by the approximate or numerical method. We may construct a table, in the first column of which are marked seconds or fractions of a second, and in the second are written against them the number of centimeters in the length ap at that time. Tables on this principle are printed in the Nautical Almanac, giving the position at

any time of the Sun and the planets; principally of the Moon. The method is imperfect, because it only gives the position at certain selected moments, and then only approximately.

REPRESENTATION OF MOTION.

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Secondly, by the graphical method. In this, the seconds are marked off on a horizontal line oX, and above every point of this there is set up a straight line representing the distance traversed at that instant. Thus, at the instant t, about 3 seconds from the beginning of the motion, the distance traversed was tq, on the scale of centimeters marked on oY. Drawing qp horizontal to meet o Y, we find the distance about 7 centimeters.

The tops of all these lines form a curve oqr, which is called the curve of positions of the moving point. The figure is equivalent to a table with an infinite number of entries, each of which is exact. The line oX is the first column, and the lengths tq, etc., answer to the second column.

In certain ideal cases of motion, it is possible to get rid of one objection to the numerical method, and to make it partially describe the position of the point at every instant of time. This is when we can state a rule for calculating the number of centimeters passed over from the number of seconds elapsed; or, which is the same thing, when we can find an algebraical formula which expresses the distance traversed in terms of the time. Such motions do not occur accurately in nature; but there are natural motions which closely approximate to them, and which for practical purposes are adequately described in this way. We go on to consider some of these ideal motions.

UNIFORM MOTION.

When equal distances are gone over in equal times, the motion is said to be uniform.

In uniform motion, the distances gone over in unequal times are proportional to the times (Archimedes). For let t and Tbe unequal times in which the distances s and S are gone over. Take any two whole numbers m and n. Then if we take n intervals of time equal to t, there will be gone over in them n distances equal to s; that is, a distance ns is gone over in the time nt. Similarly, mS will be gone over in the time mT. Now if nt is greater than mT, ns is greater than mS; for in a greater time