A DR. LARDNER'S LECTURES ON MECHANICS. Let the line AB (fig. 1) represent the direction and force of the body A, and let CE be the plane against which it strikes at B. Now, by forming the parallelogram CADB, of which the line AB is the diagonal, it is evident from the principle of the resolution of forces that the force AB may be also expressed by the two forces CB and DB; but the force DB is destroyed by the nonelasticity of the bodies, while that of CB is not at all lessened; consequently it will move in the direction BE. But if the body and the plane be both perfectly elastic, the motion will not be produced by the resolution alone, but also by the re-composition of forces. It is this which explains the angle of reflection. A perfectly elastic body striking another perfectly elastic body or plane, is rebounded with the angle of reflection equal to that of incidence. It is a wellknown fact (or property) that two bodies of this kind when struck together, will return with exactly the same force. 107 tinued to BF, consequently the motion will then be described by the lines BF and BD, and, by completing the parallelogram BDCF, the force may be represented by the diagonal BC, and the angle FBC will be found to be equal to EBA, or, in other words, the angles of reflection and incidence are equal. This is if the bodies are both perfectly elastic, but the angle of reflection diminishes with their elasticity, until they are both non-elastic, when the angle is lost altogether as before shewn, The next proposition is, if three forces act on the same point, it is required to discover under what circumstances they will keep that point in equilibrium. Thus, suppose that point to be at A (fig. 3), and the three forces to be expressed by the three lines AB, AC, and AD. Draw the lines EF perpendicular to AB, also EG to AC, and FG to AD. Now, if the side EF of the triangle EFG, bears the same proportion to AB as EG does to AC and FG to AD, then these three forces will keep the point A, on which they act, in equilibrium. To prove this, complete the parallelogram FHEG, by drawing FH parallel to GE and HE to FG; therefore the diagonal FE, will be equal in effect to the two forces FG and FH, or (as FH is parallel to GE) to FG and GE. In the same manner, it may be proved that any other side is equal in effect to the two opposite sides; therefore, the three forces AB, AC, and AD, neutralise each other's effects, and will, therefore, keep any point on which they act (A) in equilibrium. Thus, a point will be at rest, if the three forces which act on it are in proportion to the sides of a triangle, which are perpendicular to the direction of the respective forces; 108 DR. LARDNER'S LECTURES ON MECHANICS. or, as tins rule may be otherwise expressed-three forces will keep a point in equilibrium, if they are in proportion to the sides of a triangle, which are equally inclined to the direction of those forces. This figure is called the triangle of forces. It may be observed, that the limits of a resultent, produced by the composition of two forces, are the sum (when they act in the same,) and their difference, when they act in opposite directions, or as it is otherwise expressed, the resultent of two forces, can never be less than the difference, or greater than the sum of those forces. The next thing is when three forces are given in the same plane, in different directions, to find a single force, whose effect shall be equal to those three. A Fig. 4. B Let AB, BC, and CD, (fig. 4), be the three given forces. Complete the parallelogram ABCD, by joining DA, and AD will be the required force. To prove this, draw the line AC, and as AC would be the dia gonal, were a parallelogram to be described on the two sides AB and BC, it may be substituted for those two forces, consequently there are only two forces remaining, to be attended to, viz. AC and CD; when, it is evident, that if a parallelogram be formed on these two lines, the diagonal AD would be a single force, producing the same effect as the two AC and CD; but AC was proved to be equal to AB and BC, therefore, AD is the single force whose effect is equal to that of the three given forces, AB, BC, and CD. The next proposition, viz. When will four forces acting on a point, keep it in equilibrium? will only require a repetition of the last case, as it is well known that a force, whose effect is equal to that of three others, will, if opposed to them, neutralise their effects, and keep a body in equilibrium. The same rules also apply to motion Suppose a body receives three impulses, the one moving along AB (fig. 4) in one second, and the others along lines parallel and equal to BC and CD, in the same time. The effect produced will be that the body will move along AD in one second. By the same method of proceeding as in the last proposition, a single force may be found equal in effect to any number of given forces, and also under what circumstances any number of given forces will keep a body in equilibrium. Hitherto, all the forces have been supposed to be acting on the same point, but as that is a thing, which seldom or never occurs, the next proposition must be, what effect will be produced by several forces acting on several points? This problem would be an inde terminate one, as the data on which the results would proceed would be insufficient. As for instance, if a force were to act at each end of an inflexible rod in one direction, and another force were to act in the centre of the same rod, but in a direction exactly opposite, the rod, by reason of its inflexibility, would continue straight; but, the same effect would not be produced, were the rod made of whalebone, or were the forees applied to a piece of ropeconsequently, it would be impossible to determine the exact effect produced by different forces on different points of any body, or system, unless those points are connected by some known law. But, when forces are acting on the same point, it is immaterial whether it is a particle of a solid, a fluid, or of gas-the same principles are applicable to all descriptions of bodies. It may be proper to explain the proper meaning of the word "System" used above. "System" is a more general phrase than "Body"-a body is a mass of one constitution or texture, as a rod of iron, &c. But, when it is sepa OCCULTATION OF ALDEBARAN. rated in different places, and joined at one place by a hinge, at another, by a rope, &c., then it will be called a system, although, when the parts of a system are connected by a law, in virtue of which, the same effect will be produced on all its parts, then it partakes of the properties of a body. A solid body is the most perfect system, and is supposed not to change its figure. (To be continued.) OBSERVATIONS OF THE OCCULTATION OF ALDEBARAN, ON AUGUST 22ND, WITH REMARKS SHOWING THE IMPORTANCE OF THESE PHENOMENA AS THE MEANS OF DISCOVERING THE TRUE FIGURE OF THE EARTH. Owing to the unfavourable state of the atmosphere on the morning of the 22nd of August, the occultation of Aldebaran by the moon was only in part observed here; for the moon was obscured by clouds till 42m. after 6, when she became visible, and continued so to the time of emersion, which happened at 7h. 1m. 32s. M. S. T. according to the meridian of the place of observation. From the intervention of a web-like stratus, the dark part of the moon was rendered invisible, and the star much fainter than it otherwise would have been in a bright cerulean sky. Under these circumstances, the phenomena which have been observed to take place on the near approach of the moon's limb to this star could not be observed; and for the same reason, the exact time of the re-appearance of the star might not have been quite so satisfactorily observed as could be wished. From our more northerly situation, and the relative position of the moon with respect to the star, the absolute time of emersion here, must have been a few seconds later than at Greenwich. The error of the clock was accurately found by altitudes of the sun's lower limb, taken before and after the occultation, with an excellent reflecting circle made by Troughton. These altitudes were taken alternately by 109 the direct and reverse state of the instrument, and afterwards corrected for the sun's semi-diameter refraction and parallax; thus was obtained the true altitude of the sun's centre at the instant of each observation. From which, with the known latitude of the place, and sun's declination, the time was accurately determined to the fraction of a second. Dollond's achromatic, with a power of 50 only, was used for observing the emersion. It is well known, that occultations of the fixed stars by the moon, if correctly observed, afford the most accurate means, of any yet known, for determining the true longitude of places on the earth's surface; and, accordingly, much attention has of late been paid to this subject by the Astronomical Society, though not solely with this view, but likewise for the purpose of calling the attention of astronomers to other important circumstances connected with the actual appearance of these phenomena. There is, however, one very important purpose to which many of these occultations could be applied with advantage to science, but which, in a great measure, seems to have been overlooked; I mean that of determining the true figure of the earth. It is now nearly ten years since that indefatigable astronomer, Francis Baily, Esq., gave to the world a translation of Cagnoli's memoir, entitled, "A New and Certain Method of ascertaining the True Figure of the Earth, by means of Occultations of the Fixed Stars;" yet little or no notice seems to have been taken of the means afforded by these celestial phenomena for solving this interesting problem; owing probably to Mr. Baily's views of the subject, limiting, in a great measure, the occultations suitable for this important purpose, to those only, where the apparent path of the star is perpendicular to the vertical circle passing through the moon's centre at the time, of what may be termed the middle of the occultation. That such occultations are not in general the most proper for determining with precision the variation in the curvature of the earth's surface, may be readily shown by an example. For this purpose let us take the occultation of Aldebaran In the foregoing figure, S is the place of the star, C that of the moon's centre at true conjunction d in right ascension, G the apparent place of the same at that time, and M when nearest the star; then will GM represent a portion of the apparent orbit of the moon for the visible latitude, parallel to which draw EC, and EG, FM, perpendicular to the same. For the reduced latitude, let d be the apparent place of the moon's centre at true conjunction ά, and m the same when nearest the star; hence in the latter case b'e' will be the chord of duration, and in the former be. Now, without going over the process of computation, it will be sufficient to give the resulting values of the lines and angles of the figure upon both hypotheses, and first, for the visible latitude 50° 49' 30", and equatorial horizontal parallax of the moon=3600-27". Here we have CS the difference of declination of the moon and star at true conjunction 6 in right ascension=1011", the angle FCS 82° 21' 59.7", FC=134.296", and FS-1002-04". Again, the pa rallax of the moon in altitude CG= 2588-34", the angle ECG-50° 6' 6.1"; hence EC-1660-23", and EG FM 1985.76"; but the moon's apparent semi-diameter=991-05", therefore MS-983-7, and the chord of duration be=240.952". Now, the visible horary motion of the moon= 1754.4", hence the time of describing be 8m. 14 4s. OCCULTATION OF ALDEBARAN. altitude) 2578 27", the angle ECd 49° 53' 30·76"; hence EC-1661", and Ed-FM-1971 94"; then tak ing the moon's visible semi-diameter as above 991 05", we have mS= 969 89, from which the chord of duration b'e' is found-4074", and the time of describing the same=13m. 564s. Hence it appears that if the polar compression amount to of . the earth's radius, the duration of this occultation will be 13m. 56 4s. but if there be no variation in the curvature of the earth's surface, the duration will be only 8m. 14 4s. making a difference of 5m. 42s! and which I believe is very near the truth, but as the visible orbit is seldom or ever a right line even for a very short period, for this reason had the parallactic effects been computed at the time of nearest approach, for the two latitudes, the result might have been a few seconds different from the above. Again, if we suppose that in this occultation, the chord of the lunar disc which passes over the star to be perpendicular to the vertical circle cutting the moon's centre, the difference in the time of duration would have been only 3m. 51s; making in favour of an oblique direction of the visible lunar motion, 1m. 51s! That the example here chosen is not by any means more favourable for showing the effects of compression upon the duration, than many others of these phenomena that yearly happen, may be readily shown. Indeed, most of the lunar occultations, where the star is of sufficient magnitude for observing its immersion and emersion correctly, are equally fit for the purpose here proposed to some parts or other of the globe. And as Cagnoli very justly observes:" It is in the power of every principal academy materially to assist in such a discovery, by two methods. First, in regard to times past, to collect together, from all quarters, the observations of occultations stated to have been made in a given interval; for instance, in the last ten years; and to employ some calculator to select and compute all those which are proper for showing the variation at different places. Secondly, with respect to the future, to insert in the ephemerides accurate 111 notices of those places or districts where it would be most important that any occultation should be observ ed (particularly of the principal stars) in order that it might serve to apprise and excite the attention of such astronomers as might be favourably situated themselves, or contiguous to more advantageous situations." It is evident, that if this subject were taken up by the different observatories with that degree of spirit which its importance so justly demands, we should, ere long, obtain sufficient data for determining, independent of any hypothesis, the true figure of the earth, about which there is at present so much uncertainty. From the unknown density of the internal strata of the earth, and the variableness of the atmospheric refraction near the horizon, it is pretty evident that this end is not likely to be accomplished either by the pendulum or triangulation, if we may judge from the discordant conclusions already obtained from these methods, although conducted with great care, ingenuity, labour, and expence. In the present example I have supposed m to fall on the line MF, as it will do very nearly, the times of nearest approach, in the two cases not differing more than 3", and EC, without any sensible error, may be considered constant. Hence EG:CG CG:EG, or, as the decrement of CG (a G) is to the decrement of EG (dG). But EG EGCG × CG, =CG x CG, or EG= CG EG XCG, and as CG is greater than EG, the decrement of EG will always be greater than the contemporary decrement of CG. Much more might be said on the fluxionary relations of the lines and angles arising from the effects of the lunar parallax, by which investigation the most favourable occultations might be pointed out for the purpose here intended. I remain, Sir, Your's, &c. THOMAS SQUIRE. Epping, Sept. 24, 1829, |