Let A B be the curve. Join o A, O B, and with o as centre and any radius describe a circle A' B', cutting O A, O B in A' and B'. The ratio of the arc A'B' to the radius o A' is the same for all values of the radius o A', and is the measure of the angle A O B in 'radians'; if the radius o A' be unity, then the arc A'B' measures the angle. The A circular measure of an angle is the number of units of length in the arc of a circle of unit radius subtended by the angle. FIG. iii. B B' A corresponding method is employed to measure the 'solid angle' subtended at a point by a surface. FIG. iv. Let o be the point, A B C the surface (fig. iv). With o as centre and any radius describe a sphere, and consider a line, passing through o which B moves so as to trace out the boundary of the area A B C. It will thus describe a cone cutting the sphere in a closed curve A'B'C', and we can shew that the ratio of this area to the square of the radius o a' is the same for all values of the radius. C B' This ratio is adopted as defining the measure of the solid angle at o. If we take a sphere of unit radius, the ratio becomes the measure of the area A' B' c', and we thus find that the solid angle subtended by an area at a point is measured by the number of units of area intercepted from a sphere of unit radius by a cone with the given point as vertex and the given area as base. If the area as seen from the given point appears circular in form, the cone is a right circular cone and the boundary A'B'C' on the sphere is a circle. Let O L P (fig. v) be the axis of this cone, and let o A', the radius to any point on the circle, be inclined at an angle a to O P. Describe a cylinder with its axis parallel to o P touching the sphere. The circle A B C lies in a plane perpendicular to o P. Let this circle cut G the cylinder in the circle D E, and let a plane touching the unit sphere in L E cut the cylinder in FG. Then by an application of the method of projection. it may be shewn that the area of the belt of the cylinder between DE and F G is equal to the corresponding area LA'B'C' on the sphere, and this last measures the required solid angle at o. Let M be the centre of the circle A' B' c'. The solid angle = area of belt FDEG 2 D M. LM; LM LOOM=LO - O A' cos a = I- cos a ; for LOOA = 1, the sphere being of unit radius. Also DM I. This expression, of course, only holds when the solid angle in question is that of a right circular cone. It is clear from the above that a 'solid angle' is not an angle at all, but is only so named from analogy, being related to a sphere of unit radius in a manner similar to the relation between the circular measure of an angle and the circle of unit radius. MEASUREMENTS OF TIME. The time-measurements most frequently required in practice are determinations of the period of vibration of a needle. To obtain an accurate result some practice in the use of the 'eye and ear method' is required. The experi ment which follows (§ 11) will serve to illustrate the method and also to call attention to the fact that for accurate work any clock or watch requires careful 'rating,' i.e. comparison of its rate of going with some timekeeper, by which the times can be referred to the ultimate standard-the mean solar day. The final reference requires astronomical observations. Different methods of time measurement will be found in §§ 21 and 28. The 'method of coincidences' is briefly discussed in § 20. 11. Rating a Watch by means of a Seconds-Clock. The problem consists in determining, within a fraction of a second, the time indicated by the watch at the two instants denoted by two beats of the clock with a known interval between them. It will be noticed that the secondsfinger of the clock remains stationary during the greater part of each second, and then rather suddenly moves on to the next point of its dial. Our object is to determine to a fraction of a second the time at which it just completes one of its journeys. To do this we must employ both the eye and ear, as it is impossible to read both the clock and watch at the same instant of time. As the watch beats more rapidly than the clock, the plan to be adopted is to watch the latter, and listening to the beating of the former, count along with it until it can be read. Thus, listening to the ticking of the watch and looking only at the clock, note the exact instant at which the clock seconds-finger makes a particular beat, say at the completion of one minute, and count along with the watch-ticks from that instant, beginning o, 1, 2, 3, 4, and so on, until you have time to look down and identify the position of the second-hand of the watch, say at the instant when you are counting 21. Then we know that this time is 21 ticks of the watch after the event (the clock-beat) whose H time we wished to register; hence, if the watch ticks 4 times a second, that event occurred at 4 seconds before we took the time on the watch. We can thus compare to within sec. the time as indicated by the clock and the watch, and if this process be repeated after the lapse of half an hour, the time indicated by the watch can be again compared, and the amount gained or lost during the half-hour determined. It will require a little practice to be able to count along with the watch. During the interval we may find the number of ticks per second of the watch. To do this we must count the number of ticks during a minute as indicated on the clock. There being 4 or 5 ticks per second, this will be a difficult operation if we simply count along the whole way; it is therefore better to count along in groups of either two or four, which can generally be recognised, and mark down a stroke on a sheet of paper for every group completed; then at the end of the minute count up the number of strokes; we can thus by multiplying, by 2 or 4 as the case may be, obtain the number of watch-ticks in the minute, and hence arrive at the number per second. Experiment.-Determine the number of beats per second made by the watch, and the rate at which it is losing or gaining. Enter results thus: No. of watch-ticks per minute, 100 groups of 3 each. Clock-reading. Estimated watch-reading, 11 hr. 34 m. and 10 ticks Clock-reading. Difference Estimated watch-reading, 12 hr. 4 m. and 6 ticks Difference Losing rate of watch, 1·6 sec. per hour. 99 CHAPTER V. MEASUREMENT OF MASS AND DETERMINATION OF SPECIFIC GRAVITIES. 12. The Balance. General Considerations. THE balance, as is well known, consists of a metal beam, supported so as to be free to turn in a vertical plane about an axis perpendicular to its length and vertically above its centre of gravity. At the extremities of this beam, pans are suspended in such a manner that they turn freely about axes, passing through the extremities of the beam, and parallel to its axis of rotation. The axes of rotation are formed by agate knife-edges bearing on agate plates. The beam is provided with three agate edges; the middle one, edge downwards, supporting the beam when it is placed upon the plates which are fixed to the pillar of the balance, and those at the extremities, edge upwards; on these are supported the agate plates to which the pans are attached. The effect of hanging the pans from these edges is that wherever in the scale pan the weights be placed, the vertical force which keeps them in equilibrium must pass through the knife-edge above, and so the effect upon the balance is independent of the position of the weights and the same as if the whole weight of the scale pan and included masses were collected at some point in the knife-edge from which the pan is suspended. In order to define the position of the beam of the balance, a long metal pointer is fixed to it, its length being perpendicular to the line joining the extreme knife-edges. A small scale is fixed to the pillar of the balance, and the motion of the beam is observed by noting the motion of the pointer along this scale. When the balance is in good adjustment, the scale should be in such a position that the pointer is |