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quired by a corpuscle attracted towards this circle in a line passing through its centre and perpendicular to its plane, the

I attractive force of each particle varying as

D2

6. Find the attraction of a rectangle on a corpuscle situated in one of its sides produced, in a direction perpendicular to the other side; the force tending to each particle of the rectangle varying inversely as the square of the distance. .

7. If two given spheres touch each other internally, and 1824 the interior be taken away, find a point within the remainder such that a particle being placed there shall remain at rest.

1 dist.2

(Attraction of each particle a

8. If the particles of two spheres attract with forces varying as the distance, the force with which the spheres attract each other is as the distance between their centres.

9. Let Q be a point in a semicircle whose diameter is AB, join AQ and in AQ produced, take AP a mean proportional between AB and AQ; find the equation of the curve which is the locus of the point P; its area; the content of the solid generated by its revolution, and the radius of curvature at its vertex.

10. Prove that the curve in the last problem possesses these
properties.
(1.) That a particle placed anywhere in its perimeter as at

P, will attract a particle at A, in the direction AB, with
a constant force; supposing the force of attraction to

1
vary as

dist. 2

(2.) That if it be made the revolving, orbit in the ninth

F
section of Newton, and G the orbit traced out

2'
in fixed space will be the lemniscate of Bernoulli.
11. In what latitude will a ring surrounding the earth, and
parallel to the equator, attract a particle placed in the earth's
centre with the greatest force possible?

-12. If a circle whose diameter is equal to the whole tide in any given latitude be placed vertically, and so as to have the lower extremity of its diameter coincident with the level of low water, prove that the tide will rise or fall over equal arcs in equal times.

13. Supposing a comet of the same magnitude and density as the moon, on its nearest approach to the earth, to be distant thirty radii from the earth's centre; required the magnitude of the tide raised by the comet. 14. If particles of a spherical shell attract with forces

1 varying as

D2

and a cylindrical rod of uniform density whose length equals n times the radius of the sphere pass through the shell; find the pressure on the shell when the rod is at rest, the part of it within the shell being equal to the radius of the sphere.

15. If the inscribed sphere be taken away from the earth, find the time in which a particle situated in the plane of the earth’s equator within the space occupied by the inscribed sphere will reach the inner surface of the remaining meniscus ; the earth being supposed an oblate spheroid of small ellipticity.

16. A second's pendulum is carried to the height of one radius above the earth, and another is sunk to the depth of half a radius. Compare the times of their oscillations.

17. A corpuscle placed within a circle is attracted to every particle in the circumference with a force that varies inversely as the square of the distance; prove, when the distance of the corpuscle from the centre is small, that the attraction on the corpuscle varies nearly as its distance from the centre, and draws it from the centre.

18. Given the ratio of the periodic time of the moon to the time of the earth’s revolution about its axis, and the ratio of the mean distance of the moon to the mean semi-diameter of the earth, to find the ratio of the polar and equatorial diameters of the earth nearly.

19. Given the heights of the spring and neap tides, to compare the densities of the sun and moon; their apparent diameters being considered as equal.

1825

20. Given the declination of the moon, to find the duration of the superior or inferior tide occasioned by her action alone.

21. If cos (X + 8): cos (a 8):: 1:3, where is the lati- 1826 tude of the place and & the declination of the moon; prove that the time of the ebbing or flowing of the superior tide : the time of the ebbing or flowing of the inferior tide :: 2:1.

22. Find that section of a sphere which attracts a corpuscle placed at a given point in the axis produced with the greatest

1 possible force, the force of each particle o

(dist) 23. Shew that Saturn's ring cannot be a homogeneous and 1827 regular solid of revolution.

24. Of all conical surfaces of equal altitudes, determine that which exerts the greatest attraction on a particle at its

vertex.

25. Explain clearly, from elementary principles; why the moon's attraction causes a tide on the opposite side of the earth.

26. Prove that there are generally either two homogeneous fluid spheroids of equilibrium or none, for the same time of rotation ; and supposing the eccentricity of the one spheroid very small, find the ratio of the axes in the other.

27. A particle is placed anywhere within a triangle, the 1828 sides of which are composed of particles attracting with forces

1 varying as ; find the direction in which it will begin to

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move.

28. A given quantity of matter is to be formed into a cone; find its form, that its attraction on a particle at its vertex may

1 be a maximum, the attraction of each particle varying as

D2 29. If the whole force at the pole of an oblate spheroid be to that at the equator as the equatorial radius to the polar, and to any point within the spheroid canals of any form be drawn, the pressure on that point will be the same whatever be the form or direction of the canal.

30. Find the attraction of a spheroid of finite eccentricity on a particle in its equator.

31. Construct for the place of high water in a given position of the sun and moon, and find an expression for the actual height of the compound tide.

32. Give an analysis of the reasoning by which Newton explains the theory of the tides, and deduce a numerical comparison between the force of the sun on the tides, and the force of gravity. 33. Find the attraction of a spherical shell in which the

1 attraction of each particle of

(dist.)2

according to Newton's method, and analytically.

34. Find the attraction on a particle placed within a heterogeneous spheroidal shell of small eccentricity, the density being the same throughout concentric spheroidal surfaces of different eccentricities, and the internal and external surfaces being of given eccentricity, and the density uniform throughout them.

1829

D2;

35. Find the length of the tide-day, the sun and moon being in the equator, and shew how the densities of the sun and moon may be compared, by observing the lengths of the greatest and least tide-days. 36. A spherical surface being constituted of particles the

1 forces of which vary as shew that the attraction of the whole surface on a particle without it, varies inversely as the square of the distance of the particle from the centre. (Newton, Book I. Prop. 71.)

37. The elevation of the summit of the spheroid produced by the attraction of the sun and moon on a fluid sphere is double of the depression of the equator below the sphere.

38. Investigate the motion of the pole of the earth produced by the moon in one sidereal revolution.

39. Determine the attraction of an oblate spheroid on a particle situated in its equator.

k (4a

vary as

r

40. In a homogeneous spheroid attracting a point on the surface, the effect of the force parallel to the equator is as the distance from the axis.

41. Find the attraction of a homogeneous spheroid of small eccentricity on a particle situated in its pole.

42. If the earth be an oblate spheroid of small ellipticity 1830 with semi-axes a and b, the ratio of the mean density to that at the surface is 3

- b)
1
ka2

3 tan kb,

very nearly, assuming the density to be uniform throughout each spheroidal stratum at the same distance from the earth's surface, and to sin kr

at different distances, where k is a constant quantity and r the polar semi-axis of the surface of equal density.

43. Find the attraction of an oblate spheroid on a particle in its equator.

44. Having given the declination of the moon, compare the magnitudes and durations of the superior and inferior tides in any latitude, the effect of the sun on the tide being neglected.

45. When the force at the pole of a revolving fluid spheroid is to the force at the equator as the equatorial radius is to the polar radius, any two canals drawn from any points in the surface and meeting within it, will balance each other.

46. Supposing the earth to be spherical, and the matter in 1831 its interior to be compressed according to the law p=k(p2-82), p being the pressure and p the density at any distance r from the centre, and the density at the surface; shew that pa 9 being a certain constant.

47. If a plumb-line be drawn from the vertical by a small quantity of matter, at a small elevation m above the earth's surface; shew that the deviation will be a maximum, when sin 0

where is the angular distance at the earth's centre of the plumb-line and attracting point, and r the radius of the earth supposed spherical.

sin q?

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m

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TV2'

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