mass below, that is, h+t, be so small that it may be neglected in comparison with the distance of the station from the furthest limit of the plain. 53. Ex. Suppose the height of the station above the middle of the mass below, that is, h+t, is a mile and c 10 miles. Then the second term within the brackets is less than 0.05, and the attraction is very much the same as if the plain were unlimited in extent. 54. COR. The result of this Proposition when the plain is unlimited in extent might have been foreseen from the result in the previous Proposition regarding the attraction of the frustum of a pyramid. Conceive an infinite number of slender pyramids to be drawn from the station intersecting the attracting plain; they will cut out of it an equal number of frustra, and the cosines of the angles they make with the perpendicular to the plain will be the thickness divided by the lengths of the frustra. But the attractions of the frustra are proportional to their lengths, and independent of the distance from the attracted point: (see Art. 49). Hence the resultant attraction of the whole will depend solely upon the thickness or depth of matter constituting the plain. PROP. To find the attraction of a rectangular mass, of small elevation compared with its length and breadth, upon a point lying in the plane of one of its larger sides. х 55. Let the attracted point be the origin of co-ordinates ; the axes of x and y parallel to the long edges of the tabular mass, the axis of z being measured upwards. Let x'y'z' be the co-ordinates to any point of the mass: xy co-ordinates to the nearest angle, XY to the furthest angle, H the height of the mass; p the density, supposed the same throughout. Then pdx'dy'dz' is the mass of the element; and the height being small, we may suppose the element projected on the plane of xy. Hence the whole attraction parallel to x ·x Y x'dx'dy' · [["x'dx'dy'dz' = PS "S"S" 12 21 P = H v {x'2+y'2}} and so of the rest. Hence, since 0.434 is the modulus of common logarithms, attraction = pH 0.434 {log tan (45o + 40,) + log tan (45o + 40,) - log tan (45° + † 0.) — log tan (45° +40.)}, which gives a remarkably simple rule for finding the attraction parallel to x: that parallel to y can be found in like manner. It is easy to show, that if the density be half the mean density of the earth, that is, about the same as granite, g be gravity, the radius of the earth 20923713 feet, and I be = expressed in feet, the coefficient above = This equals gH tan 1" gH 76127500* ("). Hence, since the tangent of 569 deflexion of the plumb-line caused by the attraction equals, by the parallelogram of forces, the ratio of the attraction to gravity, and the angle is very small, Deflexion of plumb-line caused by the Tabular Mass parallel to the axis of x = 1" 569 H log tan (45° +10) + log tan (45° +10) — log tan (45° + †0.) — log tan (45° + ‡0)} . It is evident that the Tabular Mass may be partly below and partly above the plane of xy, so long as the height or depth is not so great that its square may not be neglected in comparison with the square of the distance from the attracted point. In this case is the sum of the height and depth, above and below the plane of xy. 56. Ex. 1. The co-ordinates to the nearest and furthest angles of a tabular block of rock measured from the attracted point are 3 and 16, 40 and 30 miles, and the height of the mass from bottom to top is 628 feet. Show that the deflexion of the plumb-line parallel to the shorter side of the parallelogram=3" 172. Ex. 2. A table-land 1610 feet high, commencing at a distance of 20 miles from Takal K'hera, near the Great Arc of Meridian in India, runs 80 miles north, and 60 miles to the east and 60 to the west. Find the deviation of the plumbline at that station. It is about 5"; so considerable as materially to affect the Survey operations, and to have rendered it necessary to abandon that place as a principal station. In cases where the attracting mass is near, it is necessary to cut it up into prisms and calculate the effect of each separately and add the results. Examples of this are seen in the celebrated case of Schehallien, and more recently in the calculation of the deflexion at Arthur's Seat, Edinburgh, by Lieut. Colonel James, Superintendent of the Ordnance Survey. See Philosophical Transactions for 1856, p. 591. 57. The irregular character of the surface of the Earth over large tracts of country, consisting of mountain and valley and ocean, may in some instances have a sensible effect, by presenting an excess or deficiency of attracting matter, upon the position of the plumb-line, in such a way as to derange delicate Survey operations. Hindostan affords a remarkable example of this, as the most extensive and the highest mountain-ground in the world lies to the north of that continent, and an unbroken expanse of ocean stretches south down to the south pole. Both these causes, by opposite effects, make the plumb-line hang somewhat northerly of the true. vertical. In the following Propositions a method is laid down for calculating the attraction of an irregular superficial stratum of the Earth's surface, and making it depend altogether upon the contour of the surface. The method pursued is this: A law of geometric dissection of the surface is discovered which divides it into a number of four-sided spaces, such that if the height of the attracting mass were the same in them all, they would all attract the given station exactly to the same amount, whether far or near. In this case it would be necessary only to calculate for one space, then count the number of spaces in the country under consideration, and the final result is easily attained. The country being supposed irregular, the heights in the spaces will not be all alike. The principle, therefore, should be stated thus, that the attractions of the masses on the several compartments are in proportion to their mean heights. These mean heights are known by knowing the contour of the country. PROP. To discover a Law of Dissection of the surface of the earth into compartments, so that the attractions of the masses of matter standing on them, upon a given station, shall be exactly proportional to the mean heights of the masses, be they far or near. 58. Suppose a number of great circles to be drawn from the station in question to the antipodes, making any angle ß, each with the next, thus dividing the earth's surface (which we may in this calculation suppose to be a sphere, without incurring any sensible error) into a number of Lunes. Then, with the station as centre, describe on the surface a number of circles, at distances the law of which it is our object now to determine, dividing the whole into a number of four-sided compartments. We will begin by calculating the attraction of a mass of matter, standing on one of these compartments, at a uniform height throughout, upon the station in a horizontal direction. P. A. 4 1" = 569 ATTRACTIONS. Hlog tan (45° + angular distances from the station of this compartment; h the height of ar distance along the surface of an of the mass; a the radius of the - log It is evident thatch the plane of @ makes with the plane and partly above th depth is not so great comparison with th point. In this ca above and below distance = ne, and in which latter plane the retly acts. The area of the base of the prism (h) is supposed very small, the xtremities from the station may be taken 2a sin 10. Its attraction along the chord pa3h sin odo d↓ = 4 sin* 0 cos 0; on along the tangent to the mid-line of the lune |