ASTRONOMY
739
His radius at geocentric latitude <j> is P=6367368 m. +10868 m. cos 2 <£' + 14 m. cos 4 <£'. we For sin (t)' Bien fiave £ = 6370980. Helmert’s general discussion of the length of the seconds pendulum gives for its value in terms oi the geographical latitude 0, L = 0-990918 m. (1 + 0-005310 sin2 <£)._ 2 This multiplied by tt gives for the intensity of gravity 0=9-77997 m. (1 + 0-005310 sin- <£). _ This is the earth’s attraction diminished by the vertical component of the centrifugal force, of which the value is 0 -03392 p cos 0' cos 0. This expression for the force of gravity gives ^for the actual attraction of the earth at the parallel of 35 26 43 9-79743 m. + -02253 m. = 9 -81996 m., which may be taken as the attraction of the whole mass of the earth, if concentrated at its centre or reduced to a sphere, upon a body at distance 6370980 metres. Taking the metre as the unit of length, we have for the mass of the 2earth in the astronomical units already defined 9-81996 x 6370980 , which gives for the total mass of the earth in gravitational measure : Logarithm of the earth's mass = 14"600522. If, instead of the metre, we take the centimetre as the unit, we have Logarithm of the earth's mass=20 -600522. In other words, this is the logarithm of the gravitation of the earth’s mass at 1 centimetre distance expressed in C.G.S. units. The corresponding attraction of 1 gramme of matter being the number already stated, of which the logarithm is 8 823326, it follows that the logarithm of the earth’s mass in grammes is 20-600522-8-823326 = 27-777196. Clarke’s dimensions give for the logarithm of its volume 27 034711. It follows that we have— Logarithm of earth's density — t)'i whence density of earth = 5 "5270. • , j j This conclusion as to density supposes the whole mass included in the geoid. It will be diminished by allowing for the elevation of the continents, and increased if the ocean be excluded from the matter taken into account. Some of the results for the density found by the other methods are as follows :— Method of Weighing. D = 5-4934 Poynting ,1 5 "69 Jolly ...... „ 5-505 Iticharz and Menzel Method of Pendulum. D = 5-579 Wilsing One of the most important astronomical applications of the preceding results is the determination of the mean p ra ax distance of the moon from the earth. Knowing the f “ masses m and m' of the earth and moon, and the mean o ® motion n of the latter in one second, its mean distance moon, a follows at once from the well-known equation of the elliptic motion, may therefore be accepted as the last word on the subject. Cto — m +.->m'1 ‘ r_o r_ From this may be derived the mean density of the earth by a process which we shall include in a general deter- p. being the ratio of the masses. In one second of mean time the mination of the astronomical constants which pertain to moon moves through an arc whose logarithm is 4 425159 —10. We shall presently find /x = l+-81-65, and have just given the the mass, figure, and dimensions of the earth. value of log. m from the seconds pendulum. We then find from The latest complete investigation of the dimensions and figure the above equation log. a in metres = 8-585164 in question is that of Clarke, to be lound in the whence a = 384737 kilometres. Mass of Brit., ninth edition, article Earth, Figure of. With his numbers we give, for comparison, those of Bessel, The motion of the moon is so affected by the action of the sun jreoid. which are still to a certain extent in use : that this number does not rigorously represent the actual mean Clarke. Bessel. C-B. distance of the moon. Moreover, what is used in astronomical Polar axis . 6356515 m. 6356079 m. +436 practice is the horizontal parallax of the moon. Equatorial axis 6378249 m. 6377397 m. +852 Gravitational theory shows that the constant of this quantity, Ellipticity . 1+-293-46 1=299-15 which we call ttq, is connected with the above value of a by the reAccording to Helmert, the most recent measures of arcs in Europe lation and Asia indicate a diminution of Clarke’s ellipticity to Bessel s sin 7r0u = 1-000907 —a > value, but tend to confirm his larger value of the equatorial semiaxis. The datum which we need for the solution of our problem p being the diameter of the earth. Using Clarke’s equatorial . is the attraction which the earth would exert on a point at its diameter we find :— Equatorial horizontal parallax of the moon = 57 2"76 . ibis surface if it were a perfect sphere composed of spherical layers ot equal density. In deriving this quantity a theorem in the attraction result is in good agreement with that of direct observations.. It is interesting to remark that, if we regard the dimensions of of spheroids is introduced by which the force in question is lound. to be approximately equal to that of the actual earth at a point the earth as unknown, observations of the seconds pendulum, combined with measures of the moon’s parallax, would enable us the sine of whose geocentric latitude is For this point we to determine them. The form of the equations we have used to determine the earth’s mass and the moon’s distance^ show that, if have— we express the earth’s radius in terms of the moon s parallax, it = 35° 15' 52" Geocentric latitude .... will come out in the form = 35° 26' 43" Geographical latitude p = & sin3 tt0, is 6370997 metres. The geometric mean of Clarke’s three axes
is impossible within the space of the present article to describe in detail the methods by which these determinations have been made. There is probably no other physical experiment involving so many difficulties or requiring attention to such a multitude of minute details. Speaking in a general way, three methods have been applied in recent times. In the first, use is made of the torsion balance, consisting of a light rod suspended by its centre and carrying a ball at each end, which is attracted by leaden masses. This is known as the Cavendish experiment. The apparatus was described in the article on Astronomy in our ninth edition. The apparatus for the application of the second method is a pendulum suspended very slightly above its centre of gravity. This method is new, having so far only been employed by Wilsing of Potsdam. The third method, which is also new, consists in determining the changes in the weight of bodies produced by the attraction of the leaden masses. In the application of this method a change of weight of the small fraction of a millionth part is not only to be made evident, but actually measured; yet it has been successfully carried out by Poynting at the Cavendish Physical Laboratory, Cambridge. Two determinations by this method have also been made in Germany, one by Jolly and the other by Iticharz and Krigar-Menzel. Notwithstanding the extraordinary delicacy of Poynting’s work, the torsion balance seems better adapted to the purpose, owing to the horizontal direction of the minute force measured. The results reached by Mr C. Y. Boys, F.R.S., at the Clarendon Physical Laboratory, Oxford, and Dr Carl Braun, S.J., at Mariaschein, Bohemia, are wonderfully accordant as well as self-consistent. Defining the gravitational constant as the attraction in C.G.S. units of one gramme of matter at one centimetre distance, they are :— Boys: G.C. =6-65760-4-1088 Braun =6-65786-4-10 As to accuracy, Boys conceives that his factor 6"6576 cannot be more than O’OOl, or at the outside 0"002, in error, while Braun estimates his probable error at ±0-00168. The agreement of the two results is much closer than we should expect from these probable errors, which we may regard as practically equal. The mean result— 6-65773-4-108
