cluded within the figure is then counted, and by multiplying this nu by the area of each of the squares, the area of the figure is determin

For an account of the rules for approximately calculating the of certain figures, and for a description of instruments for mechan obtaining the area of plane figures, reference must be made to texton mensuration and the integral calculus, since they cannot be prof described without assuming a knowledge of the calculus.

23. Units of Volume.-The unit of volume for all scientific pur is the volume of a cube each edge of which is of unit length. Th the c.g.s. system the unit is the volume of a cube each edge of wh one centimetre in length. This unit is called the cubic centimetre is generally written c.c. or cm3.

For commercial purposes the unit of volume in the metric syste the litre, which is the volume of a kilogram of pure water at the tem ture of its maximum density (4° C.). The litre is thus for all pra purposes equal to 1000 cubic centimetres or one cubic decimetre. litre is equal to 1.76077 imperial pints, or 0.220097 gallon.

The following table is convenient for converting pints to litre vice versa:

The following table gives the volumes of some of the simpler metrical figures :

The discussion of the methods of measuring mass is for the pre deferred (see $ 95).

PART II-KINEMATICS

CHAPTER IV

POSITION

24. Province and Subdivisions of Mechanics.-The title mechanics is generally given to that part of physics which deals with the effects of force on matter, without in any way considering how the force originates. For the present we may regard force as typified by muscular exertion. When we exert our muscular powers to overcome some obstacle we derive, by means of our sense organs, a certain sensation which we describe as due to the fact that we are exerting a force. When any inanimate agency produces effects on bodies which are similar to those which we produce by muscular exertion, it is in the same way said to exert force.

As far as mechanics is concerned, the effects of force on matter are of two kinds—(1) change of motion, and (2) change of size or shape.

Before studying the effects of force on the motion of bodies, which constitutes the branch of mechanics called Dynamics, it is advantageous to study motion in the abstract, i.e. without reference to the cause of the motion. This branch of mechanics is called Kinematics.

25. Material Particle.-A portion of matter so small that, for the purposes of the discussion in hand, the distances between its different parts may be neglected, compared to the other lengths we are considering, is called a material particle.

The limiting size of a material particle varies very much in different investigations. Thus in some astronomical problems the earth and the other planets are treated as material particles, while if we attempt to account for the different kinds of light emitted by glowing gases, by a consideration of the vibrations of the molecules or even of the atoms, it is no longer permissible to regard an atom as a material particle.

26. Position. The definition of a material particle amounts to a statement that the position of such a material particle can be represented by a geometrical point, which has position but not magnitude. This at once leads to the question of position.

In order to define the position of a point, we require to know its distance from some fixed point of reference, called the origin, and also the

P

direction in which we must go in order to pass from the origin to the given point. In order to be able to specify this direction, it is necessary that we have some fixed direction. Suppose we first take the case of the definition of the position of a point on a plane surface. Let P (Fig. 9) be such a point, and let o be the origin, and ox (called in geometry the initial line) be the fixed direc

FIG. 9.

X

tion. Then it is evident that if we know the angle 0, which the straight line joining P to the origin makes with Ox, and also the distance (†) we have to travel along this line from 0 to reach P, then the position of P is completely defined. The quantities and 6, which serve to define the position of P, are called the co-ordinates of P.

Another method of defining the position of a point in a plane is to

have two fixed intersecting straight lines, called the axes, inclined at any angle to one another, and refer the position of the point to these lines. Thus, suppose we have two fixed straight lines, X xox' and YoY' (Fig. 10), intersecting at o (the origin), and through any given point P we draw two lines, PN PM, parallel respectively to the axes, then if we are given the distances NP and MP, the position of

P is defined. For if we measure off from o along ox a distance OM equal to NP, and through N draw a line parallel to YY', the point P must lie somewhere on this line. In the same way P must lie somewhere on the line NP, and hence must lie at the only point which is common to the two, that is at their point of intersection. It is usual to indicate the distance OM or NP by the symbol x, and ON or MP by the In almost symboly, so that the co-ordinates of the point P are x and y. all practical applications of this method of defining the position of a point (called the Cartesian method) the two axes are taken at right angles to one another. In order to define the position of a point in space we require three co-ordinates. In the Cartesian method three axes are taken which are at right angles to each other, and the co-ordinates of a

point are then the distances from the origin of the feet of the perpendiculars drawn from the point to the three axes.

27. Vectors and Scalars.-Suppose we have the positions of two points (O and P) given. Then the position of P relative to o is given by the length and the direction of the straight line OP drawn from 0 to P. That is, starting from O you will arrive at P if you go in the direction of the line OP for a distance equal to the length of this line.

In geometry the expression OP is used simply to designate a line. When, however, it is used to designate the operation by which the line is drawn, ie. the motion of a tracing point in a certain definite direction for a certain definite length, it forms an example of a quantity called a vector. To emphasise this fact we shall indicate a line such as OP, when it is used as a vector, by an arrow placed over the letters which define the ends of the line, thus OP. The arrow will here remind us of the distinctive property of a vector, namely, that in addition to a definite magnitude, it has also a definite direction, for we are constantly in the habit of indicating a direction by means of an arrow-head. The expressions OP and PO represent two different vectors, for although the distance is the same in the two cases, yet in one the tracing point is supposed to move from 0 to P, and in the other from P to 0. Where we use a single symbol to represent a vector quantity, and we want to emphasise that it is a vector, we shall use a thick fount of type, while for scalar quantities the ordinary type will be employed. Thus v will represent a vector of which the magnitude is v units in some definite direction.

A quantity which has only magnitude and not direction is called a scalar. Thus mass and density are scalars, but velocity and force, as we shall see, are vectors, for they have not only magnitude, but have associated with this magnitude a certain direction.

28. Motion.—If the position of a material particle is changed, then if we only consider its state before and after the process of change, and take no account of the time during which this change takes place, we are said to study the displacement of the particle. When a particle is displaced, however, from one point to another, it must travel over a continuous path from one position to the other; and further, it must take a certain time in travelling over this path, so that it has occupied in succession every point along this path. When we consider the actual process of change of position as occurring during a certain time, we are said to study the motion of the particle, while that branch of mechanics which is concerned with the motion of bodies treated in the abstract, i.e. without considering what causes the motion or change of motion, is called Kinematics.

29. Different Kinds of Motion.-The motion of a material particle, taken with reference to some fixed point as origin, can consist either in change in the distance of the particle from the origin, the

direction of the straight line joining the particle to the origin rema fixed, i.e. motion can take place along this straight line either away or towards the origin, or in a change in the direction of the line jo the particle to the origin, the length of this line remaining fixed motion along the circumference of a circle having the origin as co or in a combination of these two. In the case of a material particle, it has no parts, the above are the only kinds of motion possible, and form of motion is called motion of translation. If, however, inste dealing with a material particle, we are dealing with a body of app able size, so that its different parts can have different motions, we h further kind of motion possible. Thus in addition to a motion of tra tion, in which the body moves so that the line joining any two poin the body is always parallel to some fixed line, the body may spi rotate. In the case of a pure translation, the motion of all the part of which we may consider the body to be built up, is exactly the s while when the body rotates the motions of the different parts of the are different. The most general kind of motion of which an exte body is capable is a combination of a rotation with a translation.

As an example of a motion of translation, if we neglect the curva of the earth's surface, we may take the case of a boat sailing in a stra line. The fly-wheel of a stationary engine is an example of a moti pure rotation. The motion of the screw propeller of a ship, the whe a locomotive, and a ball rolling along the ground are obvious exan of the combination of a motion of translation with one of rotation.