Sidebilder
PDF
ePub

Mdr a - r cos 0
ī

= a + gol – 2ar cos 0,

yo

[ocr errors]

Mdr a sin? 0 cos 0 (r - a cos )

ī {a® + pol – 2ar cos 0}' ... attraction of whole prism

[merged small][merged small][ocr errors][merged small]

M
7

M al Na + P – 2al cos e at 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.

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.

49. Let l 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; ark is its area; p the density of the matter: then ar pdr is the mass of an element of the pyramid, and this divided by mi is its attraction;

::. attraction of pyramid on vertex = apdr = apl.

If d is the length of any frustum of the pyramid, and l=l+d, then

[ocr errors]

0

attraction of pyramid, length l', = a.pl";

i. 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.

50. 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 ablk is the area of the base; and the attraction of the whole pyramid on the vertex =pßk. Hence, all pyramids having the same angular width and the same linear depth at the base attract their vertex alike, whatever their lengths be.

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 plain equally. Hence attraction of the particle 27 pr (h+z) dr dz 2p const.

dr {72 + (h+z)?}"

92 +(h+z)

[ocr errors]

dr

[ocr errors]

+2*p [e-vo+*{(1 + 1}]
2wpt {1 ....

h + 1 /

+ +i

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 + 4t, 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 + ft, 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. 29). 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'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 a

A c'doc'dy'dz Ep

H =p

Н
{at" + y'}"

.
{x^2 + y'?}"

[merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small]

To simplify the formula put
Y
y

Y
tan , = tan 0 = tan 6s, in

X

= tan 0,; X

[ocr errors]
[ocr errors]
[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

and so of the rest. Hence, since 0.434 is the modulus of common logarithms,

pH attraction

0:434

{log tan (45° + 30,) + log tan (45° +10)
log tan (15° +10.) – log tan (159 +30)},

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, 9 be gravity, the radius of the earth = 20923713 feet, and ' be expressed in feet, the coefficient above

gH

76127500

1" This equals gH tan

,

569 deflexion of the plumb-line caused by the attraction equals, by the parallelogram of forces, the ratio of the attraction tó gravity, and the angle is very small,

Deflexion of plum b-line caused by the Tabular Mass parallel to the axis of x

1"

Hlog tan (45° + 10) + log tan (45° +10) 569

{log

° +

-1 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 H 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

« ForrigeFortsett »