Figare of the Earth. 6 S sin. cos SA3 If, as in Section 2. Art. 44. we suppose the spheroid to consist of a series of spheroidal shells of different d (bs e) Now to find a (MECHANICS, § XIX.) we must divide this by the moment of inertia, orf (x2 + y2) dm : which for a homogeneous spheroid C a2 b (a2 + b2) p = 2 C. b5. p nearly; and for a heterogeneous spheroid d (b3) =2C Sp. db. Call the general integral v (b), and the value which it has at the surface v (b), and we db have for the moment of inertia 2 C v (b). Physical Theory of Precession and Nutation, &c. Calculation of the momentum of inertia. Now this motion is perpendicular to the plane passing through the Sun and the Earth's axis. In fig. 49, let Direction of First, the resolved part of the velocity of B in the direction BP is D. sin B. cos B. sin B = . cos. sin. let the Sun be supposed to go (apparently) through the ecliptic in a year or T seconds, with a uniform motion, Change in (the effect of the inequality of motion is not sensible,) and let t be the time since the Sun was at . Then the inclina is plain that the last term is periodical: and, therefore, though the inclination is changed in the course of a year, it is the same in all successive years. The variable term is called Solar Nutation. Secondly, the resolved part of the velocity of B perpendicular to B is D. sin B. cos BO cos B. D = sin I. cos I (1 duced by - cost). Integrating with respect to & we have for the motion of B perpendicular the Sun. cos I. sin 2. The last term is periodic: it is considered a part of solar nutation. The first increases uniformly; it shows that is continually travelling towards, or D backwards: the regression in one year is cos I. T. This is called Solar Precession. The last investigations apply as well to the Moon as to the Sun. The only difference is that the Moon revolves in so short a time round the Earth that the periodic terms analogous to solar nutation are hardly sensible. So that putting M for the mass of the Moon, MA for her distance from the Earth's centre, (b) 3 M ω MA3. =E, I' the angular distance between the pole of the earth and the pole of the Moon's orbit, T' her v (b) E periodic time; the motion of the Earth's pole during one revolution of the Moon is sin I' cos I'. T' in a 2 direction perpendicular to the great circle joining the pole of the Earth, and the pole of the Moon's orbit. explana Now the pole of the Moon's orbit describes (nearly) a circle in 18 years round the pole of the ecliptic. And this change of place, altering the direction and magnitude of the motion of the Earth's pole, produces the cession and inequality called Lunar Nutation. In fig. 50, let p be the pole of the Moon's orbit. Then the motion of the Earth's pole in 1' will be sin is × (cos B p) × (sin Bp. sin PB p)=x (cos B P. cos Pp + sin B P. sin Pp. cos P) × (sin P p. 2 sin P). Let PpI", and let the periodic time of p round p in a retrograde direction be T": and let t be the tic pro duced by the Moon, These terms are one part of lunar nutation. They are evidently periodical; and, therefore, the inclination of the Earth's axis to the axis of the ecliptic is not permanently altered. 4 π T" E E 2 sin B p. cos B p. cos PB p cos Pp cos BP. cos B p = sin B p 2 E = sin BP. sin B p 2 sin BP 2 (cos I. cos I" + sin I sin' I. cos I"-sin I. cosI. sin I".cosP sin I cos I . sin2 I". cos P}. Putting for P, and integrating with respect to t, we find sin I cos I sin' I'. sin. The two last terms are evidently periodical : they are the remaining part of lunar nutation. The retrograde motion of 8 п The two last terms, which are periodical, constitute the equation of the To reduce this to numerical calculation, we must remark that (MECHANICS, Art. 65. where u corresponds E. T 1 cos I. (cos2 I"- sin' I"). 2 2 T 12 sin' I") } since w is the angular velocity of the Earth, Tw is evidently the angle through which the Earth revolves in a 2 : this gives for the annual precession in seconds, {1+T (cos3 I" — — sin2 I") } This, it is plain, would give us considerable assistance in determining the form and constitution of the observed value of annual precession (which is known very accurately) and might thence infer the value (b) of But as n is not very certainly known, it will be better to use also another observable quantity depending on n. cos I. sin I". cos I' v (b)' n T.w * Instead of n we ought in strictness to have put n + 1. See PHYSICAL ASTRONOMY, p. 653. The effect of this error is insensible. T Now suppose I = 23° 28′; I" = 5° 8′ 38′′; T Lunar nuta T 366.26 = 18.6; = ; also call the whole annual pre- tion in a T 27.32 cession a, and the term just mentioned b. Then on performing the calculations, we shall have these equations, form for cal culation. 864490 4869 + n). (b) v (b) 231460 y (b) b= n = 47.54 X 177.56 b y (b) Function depending on the Earth's form de The last value is the only one for which we have occasion at present. Now a is very nearly 50.3; the value of b given by Bradley is 9.0; by Mayer 9.65; by Maskelyne 9.55; duced from by Laplace 9.40; by Lindenau 8.99; by Brinkley 9.25. The last value is perhaps the most probable; (it has been adopted as such by Mr. Baily in the construction of the Astronomical Society's Tables, and we think properly.) Now it must be observed that we cannot from this deduce the Earth's ellipticity without assuming some law for the density of the strata, and solving approxi b. (b), (Section 2. Art. 63.) and, there- of ø (b) and v (b) will depend on the value of the constants in the law (whatever it may be ;) and, therefore, values above, be a complicated equation to solve. If, for instance, we assume the law mentioned in Section Art. 68. and take Brinkley's value of nutation, we shall have this equation to solve Solving this by approximation, with the value m = .0034672, we find q b = 143° 53'; and substituting this in the expression out a knowledge of the law of density. Calculation with Brink ley's nuta tion. of the Motion Doubts such a determination, as in all probability the observa- Inequaliti tions which are compared have been made by different persons and in different manners. The small lunar Moon's inequalities, besides, are involved among a mass of terms much greater than themselves; but an error in their determination has less influence on the value of e than an the certai equal error in the determination of nutation. We mention of these c these things merely tc show that these deductions could clusions. not be put in competition with those derived from geodetic measures, or pendulum observations, if the discordancies among the latter were not so great as to make every confirmation of their results desirable. The coincidence of the results is however satisfactory, as it gives us a strong confidence that the result deduced from the measures is not far from the truth, and that our theory is in the main correct. same arc. in com parts o Section 11.-Observations which show that the Attraction of Masses comparatively small is sensible; and Determinations of the Earth's mean Density. Among these we may place the discrepancies which Anoms have been observed in comparing different parts of the ing dif Of these it must be confessed, that in some instances no distinct explanation can be given. At same a Arbury Hill, for instance, one of the stations nearly meridi bisecting the English arc of meridian, the latitude was observed, and was found to differ about 5" from any that could be admitted, on any supposition of the Earth's form; at Dodagoontah, on the great Indian arc, an unexplained disturbance to nearly the same amount was observed; at Takal Khera the same thing was ob served; but Captain Everest appears to have accounted perfectly for this by the attraction of a range of mountains at the distance of 15 miles, (the range running Eastward from the termination of the Western Ghauts.) The magnitude of these disturbances, it must be observed, is much greater than any error that could possibly happen in the use of the astronomical instruments or in the geodetic measures, without the most unreasonable neglect. And in inferring the latitude of one place from the observed latitude of another not on the same meridian, through the medium of a geodetic measure, results have frequently been obtained which differ much from the observed latitudes. Thus (Conn. des Temps, In co 1827, Additions) the latitude of Turin deduced from ing that of Milan differs from the observed latitude by 8".9; tudes that of Venice by 9".5; that of Rimini by 27.4. place We regret that in the statement which we have copied, tance there is no mention of the dimensions of the Earth which have been used in deducing the difference of latitudes from the geodetic measure. small arcs same grou Another class of observations leading to the same con- In di clusion is the difference between the values of degrees given by different arcs passing over nearly the same ground. We have seen that in the Swedish arcs it is necessary to suppose one disturbance to the amount of 12", or two disturbances whose sum = 12". In the arc of parallel between Beachy Head and Dunnose, as compared with that between Dover and Falmouth, there is evidence (though less perfect) of disturbance nearly as great. arc. In the Piedmontese arc, if we wish to reconcile it in Int any degree with the arcs in greater and smaller lati- mot tudes, we must suppose that the effect of the disturbances is more than 40". But no one who considers the situation of the alluvial basin of Piedmont, with the re of highest part of the chain of the Alps to the North, and Earth the Apennines (of no inconsiderable height) to the South, will doubt that this is perfectly explained by their attractions. for These observations, though they render the application of our complete theory doubtful, yet serve to give the strongest confirmation of its fundamental principle. We must not reject them because they disagree with our theory; we must endeavour to ascertain whether they are consistent with the principle of universal gravitation, and if we find them to be consistent, we must examine what alteration must be made in the other suppositions of our theory to make it represent the facts of observation. Now all that is necessary is, to imagine that, after see the Earth had assumed a form of equilibrium and become solidified, parts of the external crust were elevated by some internal force. And if successive coatings were deposited from a fluid covering the whole, we must suppose irregularities of deposition, in many cases connected with the former, to have taken place. These inequalities, on the principle of gravitation, might account for disturbances in the Earth's form, or rather in the form of the sea, great in themselves, but small in comparison with the magnitude of the Earth. The circumstance, however, of islands being found scattered over every part of the Ocean seems to justify us in the Lelief that the Earth was originally fluid, and that its form has not been much altered by posterior convulsions. The disturbances alluded to have shown the expediency, and suggested the possibility, of determining by observation the attraction of mountains; and the desire of throwing some new light on the constitution of the Earth has prompted several experiments for ascertaining the mean density of the Earth. In the last century, too, it was doubtful whether the Earth's ellipticity was less than (the value assigned by Newton,) and the determination of the Earth's mean density might (assuming the principle of gravitation) assist in settling that point. For, as has been seen, if the principal attracentive mass of the Earth were at its centre, the ellipticity 's BOT of the 1 230' 1 230' 1 580 would be not much greater than ; and therefore if the central part of the Earth were more dense than the parts near the surface, the ellipticity might be expected to be less than the value which it would have if the Earth were homogeneous. The first experiment of this sort was that of Bouguer, mentioned in our first Section; of which we take no further notice, as the person who made it did not think its results worthy of any confidence. The second was that of Dr. Maskelyne on the attrac * In the Phil. Trans. 1812, is a paper by Don. J. Rodriguez, (we believe one of the gentlemen who assisted Delambre in his survey,) in which observed latitudes are compared with assumed dimensions of the Earth, and the difference is at once set down as an error of observation. The reader, who has taken the trouble to examine our Section on Meridian Measures, and its results, will judge how uncertain the dimensions of the Earth still are, and how certain it is that, using any dimensions whatever, we must still suppose some disturbing cause to affect the latitudes with errors far greater than the errors of observation. The paper in question has been severely criticised by Delambre in the Conn. des Temps, 1816, Additions. For our own part, we can scarcely imagine how any one who had been concerned with geodetic surveys, or who even knew their object, could compose such a memoir as that to which we allude. VOL. V. Earth's mean Density. mical obser tion of Schehallien. The history of this is given in our Determinafirst Section; the astronomical observations are in the tion of the volume of the Phil. Trans. for 1775. The sector was made by Sisson, with the plumb-line passing over a dot at the centre of the instrument; it was divided by taking an arc (7° 9′ 59′′.917) whose chord = 4th of Attraction the radius, and continually bisecting this arc: it was of Schehalused with a micrometer-screw as we have described in lien found Section 3. At each of the stations it was reversed but by astrono once. The error of collimation, as determined by a vations. mean of the observations on the different stars, appears to have changed only a fraction of a second between making the observations on the North and on the South sides of the mountain. On the South side 76 observations were made, face East, and 93, face West; and on the North side 68 observations, face West, and 100, face East: the whole number of stars observed was 43. The situation of the observatory on each side was about half way up the hill. The difference of astronomical latitude was found to be 54".6. The distance in feet between the parallels passing through the two observatories was found to be 4364.4 feet. This was determined by a survey of the mountain founded on two bases, one of 3012 feet and another of 5897 feet, measured with deal rods that were compared with the Royal Society's brass standard. From the extremities of these bases two cairns on the ridge of the mountain were observed, and the distance between them found. These cairns were not visible at the observatories, but signals were fixed at distant points where a cairn and an observatory appeared in the same vertical; then the angle between these signals, as seen from the observatory, was the same as that between the cairns. The same signals were observed at the cairns, instead of the observatories. The distance 4364.4 feet, at the rate of 101.64 feet to one second, (according to Bouguer's table,) gave for the difference of geodetic latitude 42".94. The difference between this and the observed difference, or 11".6, is to be attributed to the attraction of the mountain. For determining the figure and dimensions of the mountain, stations were chosen all round it; then poles were fixed in the hill-side in vertical planes, as determined by observations with a theodolite in one station, and the azimuth and altitude of each was observed at tain and calculation another station. A few, however, were placed in horizontal planes; and some in different manners. The calculation of the attraction was made by Dr. Charles Survey of made, concentric circles were described with one obHutton. (Phil. Trans. 1778.) An accurate map being the mounservatory for the centre, and with radii in arithmetical of its attracprogression, their common difference being 666 feet. tion. Each of the rings between these was divided into 48 parts, according to the following law: a line being drawn in the direction of the meridian, radii were drawn making angles with it whose sines were successively 1 2 3 &c. Then it is easily seen that the attrac12' 12' 12' tion (in the direction of the meridian) of each of the prisms thus formed is a constant multiplied by the sine by an easy practical method, for which, with many of the angular altitude of its top. This was determined other details, we must refer to the original Memoir. Indeed the various contrivances* of calculation in this Most of these, it appears, were suggested by Mr. Cavendish (whose experiments we shall shortly describe.) And it appears that nearly all the preliminary calculations of the attraction of Skiddaw, * 21 |