As this is symmetrical with respect to a and b, it shows that the particle is attracted equally towards the two extremities of the prism; and that therefore the resultant attraction acts in a line bisecting the angle which the prism subtends at the attracted point. 49. COR. A uniform bar of very great length attracts a point not far from its centre with a force varying inversely as the distance from the bar. For let xy be the co-ordinates to the point from the centre measured along the bar and at right angles to it: 27 the length of the bar, M its mass. The bar is divided into two parts by y, and they attract the point towards the bar with forces The sum of these, when 7 is very large in comparison with x and y, is M÷yl, which varies inversely as y. The following is an approximate illustration of this. The Himmalayas resemble a very long prism running W.N.W. and E.S.E. through a point in latitude 33°30′ in the longitude of Cape Comorin. They attract three places in the meridian of Cape Comorin-viz. Kaliana (lat. 29°30′48′′), Kalianpur (24°7'11"), and Damargida (18°3'15"), so as to produce deflections in the plumb-line in the meridian equal to about 28", 12", 7" (see Art. 61). The deflections towards the prism or axis of the Himmalayas may be taken to bear the same UNIFORM BAR. PYRAMID. 45 proportion to each other as those in the meridian. Now the distances of the three stations from the point where the axis crosses the meridian are 3°39', 9°23', 15°27′ or 219', 563', 927'; in the same proportion are the distances of the stations from the prism or axis. It will be found that the reciprocals of these numbers are not far from being in the same proportion as the deflections. If Kalianpur were removed 20′ north the comparison would be exact. PROP. To find the attraction of a slender pyramid of any form upon a particle at its vertex; and also of a frustum of the pyramid. 50. Let be the length of the pyramid, a the area of a transverse section at distance unity from the vertex; r the distance of any section; ar2 is its area; p the density of the matter: then ar2pdr is the mass of an element of the pyramid, and this divided by " is its attraction; = ['apdr = apl. .. attraction of pyramid on vertex = If d is the length of any frustum of the pyramid, and l=l'+d, then attraction of pyramid, length l', c.pl'; = .. attraction of frustum = apd. It is observable that this is quite independent of the distance of the frustum from the vertex; and therefore all portions of the pyramid of equal length, any where selected, attract the vertex equally. COR. Suppose the angular width of the pyramid to be ß and to remain constant, while the angular depth varies; and let k be the linear depth of the transverse section of the base; then Blk is the area of the base; and the attraction of the whole pyramid on the vertex pẞk. Hence, all slender pyramids having the same angular width and the same linear depth at the base attract their vertex alike, whatever their lengths be: or, which is the same thing, the angular width being the same the attraction varies as the linear depth of the = base, and is independent of the length. Thus, suppose it is required to find the effect of the deficiency of matter in the sea on a place on the sea coast, the shore of which shelves gradually. By dividing the sea into slender horizontal pyramids, the attraction of the shelving portion of it can be calculated by knowing only the depth at the extremities of the pyramids without knowing their lengths. PROP. To find the attraction of an extensive circular plain of given depth or thickness upon a station above its middle point. 51. Let t be the thickness or depth; h the height of the particle from the nearer surface, c the radius, r the radius of any intermediate elementary annulus of the attracting mass, z its depth. The several elements of this annulus of matter will attract the particle towards the plane equally. Hence attraction of the particle = = {r2 + (h+ z)2} ŝ r + h2 √j2 + (h+t) · 2πρ {√c2 + h2 − h − √ c2 + (h + t)2 + h +t} 52. If the plain be of infinite extent, the attraction equals 2pt; and this remarkable result is true, that it is independent of the distance from the plain. The same will be the case if the height of the station above the middle of the attracting 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. Thus, for example, 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 ex tent. 53. If p is the density of rock, taken to be half the mean 8 density of the earth, g=pa. Hence the attraction of an extensive plain = 2πpt 3 t 4 a g. Suppose, owing to geological changes of level, a continent is lifted up above the mean surface of the earth through a space t. Then gravity at a station on the continent will be diminished from this cause by the amount 2t 9{1-(1+)-9 nearly. But the attraction of the underlying mass of thickness t must be taken into account. Hence the real diminution of gravity by the upheaval will be 5 t g. 4 a The ratio of this to the correction for increase of distance = 0.625*. If the station be at the height h above the level of the continent, then the diminution This correction will depend upon the kind of rock of which the continent is made, whether of a dense or light description. Thus also for a station at sea, like St Helena, the correction would be different for a similar height above a continent, as sea water is only half the density of rock. The importance of these considerations will be seen in Art. 91, when we come to consider the vibration of a pendulum as a measure of gravity. * Dr Young takes the ratio of the density of the surface to the mean density to be 5: 11. In this case the correction would be 58÷88=0'66. See Phil. Trans. 1819, p. 93. 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. 50). 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. 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' de' 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 |