the screw. The fractions of a turn are read off on a scale, F, on the edge of this cap. The pitch of the screw ordinarily employed is 0.5 mm., and the edge of the cap is divided into 50 parts. Hence as turning the screw through a whole turn or 50 divisions advances the point A by 0.5 mm., one division on the scale F corresponds to a motion of the point of 1/50 of 0.5 mm. or 0.01 mm. 19. The Comparator.-For comparing together two very nearly equal lengths, as, for instance, a standard metre with a copy, an instrument called a comparator is used. The principle on which this instrument works will be seen from Fig. 6. Two stone pillars, A and B, which are firmly embedded in the ground, carry two microscopes, C and D. The cross wires of these microscopes, instead of being fixed, can be moved E A FIG. 6. B through a small distance in the direction of the line joining the pillars by means of micrometer screws, E, which have divided heads. The two bars to be compared are supported on a carriage running on two rails fixed between the pillars, so that first one bar and then the other can be brought underneath the microscopes. In order to keep the bars at a constant temperature, they are immersed in a water bath. When using this instrument one bar is brought beneath the microscopes, and the cross wires are adjusted till they exactly coincide with the image of the division lines on the bar. The other bar is then substituted, and the number of turns and fractions of a turn of the micrometer screws necessary to bring the cross wires into coincidence with the image of the division lines is noted. Preliminary experiments are made to determine the magnification of the microscopes and the pitch of the micrometer screws, so that from the number of revolutions the difference in length of the two bars can be calculated. In the instrument in use for comparing the standard metres at the Bureau International des Poids et Mesures at Paris, one division on the micrometer heads corresponds to a difference in length of the bars of 0.001 millimetre, i.e. to one-millionth of a metre. 20. The Cathetometer.-In order to measure a vertical height, an operation of frequent occurrence in Physical measurements, an instrument called a cathetometer is usually employed. One form of cathetometer is shown in Fig. 7. A vertical metal pillar, AB, is fixed in a B T heavy tripod-stand in such a When using the instrument to measure the vertical distance between two points, the pillar is first set vertical by means of the levelling screws, this adjustment being complete when on rotating the pillar the position of the bubble of the spirit-level L does not alter. The carriages are then moved till the lower end of the object to be measured is seen through the telescope. The carriage D is then clamped, and by turning the screw E the intersection of the cross wires of the telescope is made to coincide with the image of the lower point. The position of the carriage C having been read by means of the vernier, the carriage is moved till the image of the upper point coincides with the intersection of the cross wires. The FIG. 7. difference between the two readings gives the vertical distance between the two points. In order to obtain a correct result it is very important that the axis of the telescope in the two positions should be exactly parallel. This will be evident from Fig. 8, where the axis of the telescope when at T, is shown inclined to its position when at T1. The distance between x and y would then be read off as T1T2 instead of TC as it ought to be. The error due to this cause is minimised by always setting the axis of the telescope horizontal, as shown by the delicate spirit - level 1, by means of the screw F before making the final adjustment of the slide in both positions of the telescope. T FIG. 8. 21. Units of Surface.—For all scientific purposes the unit of surface is a square of which each side is of unit length. In the c.g.s. system, therefore, the unit of surface is a square, each side of which is one centimetre, and is called a square centimetre. A square centimetre is sometimes written sq. cm. and sometimes cm2. For measuring such surfaces as would in the British system be measured in acres, the unit employed in the metric system is the hectare, which is 10,000 square metres; one hectare is equal to 2.471 acres. The dimensions of surface are [Z2]. 22. Measurement of Surface. The area of certain figures can be readily calculated by geometry. Some of the commonly occurring cases are given in the following table : The surface of an irregular plane figure may be determined experimentally by tracing the outline of the figure on a sheet of cardboard or tinfoil, then cutting it out and weighing. The weight of a square centimetre of the same card or foil is then measured, and from this the area of the figure calculated from its weight. Another method in common use is to trace the figure on paper (called curve or squared paper) which is subdivided by two series of parallel lines at right angles to one another into a number of equal small squares. The number of these squares in cluded within the figure is then counted, and by multiplying this number by the area of each of the squares, the area of the figure is determined. For an account of the rules for approximately calculating the area of certain figures, and for a description of instruments for mechanically obtaining the area of plane figures, reference must be made to text-books on mensuration and the integral calculus, since they cannot be profitably described without assuming a knowledge of the calculus. 23. Units of Volume.-The unit of volume for all scientific purposes is the volume of a cube each edge of which is of unit length. Thus in the c.g.s. system the unit is the volume of a cube each edge of which is one centimetre in length. This unit is called the cubic centimetre, and is generally written c.c. or cm3. For commercial purposes the unit of volume in the metric system is the litre, which is the volume of a kilogram of pure water at the temperature of its maximum density (4° C.). The litre is thus for all practical purposes equal to 1000 cubic centimetres or one cubic decimetre. One litre is equal to 1.76077 imperial pints, or 0.220097 gallon. The following table is convenient for converting pints to litres, or vice versa : The following table gives the volumes of some of the simpler geometrical figures : The discussion of the methods of measuring mass is for the present 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 |