write down the average heights of the masses standing on all the several compartments of any one lune; add them together, multiply the sum by 1".1392 sin 6, and the equation (2) shows that we have the deflection caused by the mass on the whole lune in the vertical plane of its middle line. Multiply by the cosine and then the sine of the azimuth of that middle line, and we have the deflections in the meridian and the primevertical. The same being done for all the lunes, and the results added, we have the effects in meridian and primevertical produced by the whole country under consideration. PROP. To calculate the dimensions of the successive compartments from the law of dissection. 60. For this purpose we should solve the equation of last Proposition, viz. sin 14 cos2 (a+10) 1 21 .... (1). But this cannot be done. We must therefore approximate, which will equally well suit our purpose. In order to afford a test of the values we arrive at the equation may be written under the following form: log sin 14° 18.6777807 + log sin (a+16). = Equation (1) can be solved by expansion so long as a and o are not too large. a and being expressed in degrees. Let a,,...,,,... be the successive values of a and for the several compartments of a lune, beginning with the antipodes*. These are connected by the following relations: 21 sin" 1, cos1, or cot3 1,+cot1- 21 = 0, which gives = = cot 12.6379, or 41 = 83° 2′, .'. α1 = 96° 58′. For the second,1 (α, +6,) = 1α, = 48° 29'. Putting this in equation (3) we must by trial find the value of 4, which satisfies it; and so of .... This process brings out the series of values of and a,, 4, and a,, &c. as far as the 22nd, gathered together in the following Table: * In the last edition the compartments were counted from the station, not from the antipodes. After the 22nd each of the remaining values of & is obtained by dividing the next preceding values of a by 11, as the small term in equation (4) then becomes insignificant, and the succeeding value of a is easily deduced by means of the formulæ (5). The Table may be carried on to any extent, the only restriction on its use being that the height of the mass on any compartment must not be so great relatively to its distance from the station that the square of the ratio cannot be neglected. 61. COR. The relative effect of the same or an equal and similar mass, situated on different parts of the earth's surface, is easily obtained as follows. As the effects of the compartments into which any lune is divided are all the same, the height of the mass standing on them being the same, the effect of a given mass standing on any area will vary inversely as the area of the particular compartment in which it is situated. Now if a and a + be the distances of the nearer and further sides of any compartment, and be the width of the lune, the area of the compartB {cos a cos (a+)}. Hence the relative attraction of the same mass in different situations will vary inversely as ment {cos a - cos (a+b)}. HIMMALAYAS. OCEAN. 57 For example; the centre of the Island of Australia is about 36° and 63° from Singapoor and Calcutta ; it stands therefore, with reference to those places, on the 9th and 4th compartments, reckoning from their antipodes, and the ratio of the horizontal attractions of the Island on those places 62. The formulæ above deduced may be applied to find the effect on the plumb-line of any mountain-region, or hollow (as in the case of the ocean), so long as the angle subtended at the station by any vertical line in it is such as to allow its square to be neglected. Ex. 1. In the Philosophical Transactions for 1855 (p. 85) and 1859 (p. 770) the author has applied these principles to find the effect of the Himmalayas and the mountain-region beyond them on the plumb-line in India, and has found that the meridian deflection caused in the northern station of the Great Arc of Meridian (lat. 29° 30′ 48′′, and long. 77° 42') is nearly 28", as far as the data regarding the contour of the mass have been ascertained; and that the astronomical amplitudes between that and the next principal station (lat. 24o 7' 11"), and between that and the third (lat. 18° 3′ 15′′), are diminished by the quantities 15".9 and 5".3. He has also shown that the meridian deflection at points between the first and third stations varies very nearly inversely as the distance from a point in the meridian in latitude 33° 30'. General Chodzko states that at Tiflis, Douchet, Wladikawkas, Alexandrowskaja, and Mosdok, which are severally 70, 35, 35, 55, 70 miles from the central line of the Caucasus, the deflections are (taking that at Tiflis to be zero) 25".1, — 28′′.6, -12".0,5".6 North. (See Monthly Notices of Astron. Soc. April, 1862.) Ex. 2. The effect of the deficiency of matter in the Ocean south of Hindostan down to the south pole is also calculated (Phil. Trans. 1859, p. 790) by the author, upon an assumed but not improbable law of the depth, and found to produce a meridian deflection northwards at the three stations of the Indian Arc of 6", 9", 10".5 respectively; and 19".7 at Cape Comorin. The deflections at Karachi, and a point half way between Cape Comorin and Karachi, arising from this cause, are shown to be 10" and 13".8. It is not difficult to show from the last three, that the horizontal attraction northwards, at points along the west coast of India, arising from deficiency of matter in the ocean, may be approximately represented by the formula (0.000095556839 -0.000002836162X+0.000000004072x2) g, in which is the difference of latitude of the station and Karachi, expressed in degrees and parts of a degree. (Phil. Trans. 1859, p. 793.) Ex. 3. The formulæ may be applied also to obtain the attraction of thin sections of the earth's surface of a regular form which the Integral Calculus does not enable us to calculate. The following is an example which the reader may work out the result is here given because it will be used in the last chapter of this treatise. The horizontal attraction of a slender hemi-spheroidal meniscus of matter at the earth's surface on points 90°, 120o, 135, 150°, 180° from the pole of the meniscus is = h = greatest thickness of the meniscus, a radius of the earth. If 0 be the distance of any point in the further hemisphere from the pole of the meniscus the above quantities lead to the following formula. Horizontal attraction = (0.1446 sin + 0.0958 sin 20 +0.0244 sin 30) h 9 ... .. (1), which may be taken as representing generally the attraction at any point of the hemisphere of the meniscus. By means of Art. 14 it may be shown, that the attraction of the difference of two spheroids of different small ellipticity |