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CONTENTS.
ATTRACTIONS AND LAPLACE'S FUNCTIONS.
CHAPTER I.
THE ATTRACTION OF SPHERICAL AND SPHEROIDAL BODIES.
ART.
2.
Attraction of a spherical shell on an external particle
4. Ditto on an internal particle
7. Attraction of a spherical shell according to any law, on par-
ticles, external and (8) internal
9. Laws for which a shell attracts as if collected at its centre
12. Attraction of a homogeneous spheroid and of a spheroidal
shell on an internal particle .
15. Ivory's Theorem, for an external particle
PAGE
1
3
7
10
attracted particle is not or is part of the attracting mass,
14
15
22. First equation true also of R, the reciprocal of the distance
of the attracted particle from any point of the body
23, 24. Transformation of equations in R and V to polar co-ordi-
nates
25. Method of expanding R
26. Laplace's Coefficients, Equation, and Functions defined
27.
17
ib.
19
21
The definite integral of the product of two of Laplace's Func- tions of a different order always equals zero
28. Any function, which does not become infinite between the
limits used, can be expanded in a series of Laplace's Func-
tions. Remarks on this Proposition
34. A function can be arranged in only one series of Laplace's
Functions
37. Expansion of Laplace's coefficient of the ¿th order 39. Examples of arrangement in Laplace's Functions
22
30
31
33
CHAPTER III.
ATTRACTION OF BODIES NEARLY SPHERICAL.
41. Calculation of potential V for a homogeneous sphere
42. Attraction of a homogeneous body nearly spherical on par-
ticles, external and (43) internal .
44. By choosing the origin at the centre of gravity, and taking
the radius of the sphere of equal mass as a standard, the
general radius of the body is simplified
45. Attraction of a body consisting of nearly spherical shells on
particles, external and (46) internal
ATTRACTION OF TABLE-LANDS, MOUNTAINS, OCEANS, &c.
47. Object of this Chapter
48. Attraction of a slender prism on any particle .
50. Attraction of a slender pyramid on a particle at its vertex
45
ix
ᎪᎡᎢ.
51. Attraction of an extensive plain of given depth or thickness
on a point above it
53.
Correction for elevation above the earth's mean surface.
55. Attraction of a rectangular mass on a particle in the plane of
one of its sides. Examples
57. Method of calculating the attraction of extensive tracts of
mountain-country
58. Law of Dissection of the mountain-mass into compartments,
such that the attraction of each is proportional to the aver-
age height of the mass standing on it
60. Calculation of the dimensions of these compartments
62. Results arising from the Himmalayas, and the Ocean. At-
traction of a meniscus, and hemi-spherical shell .
46
47
48
50
51
54
57
64. Effect on the plumb-line of a slight but wide-spread defect or
excess of density in the interior of the Earth. Example 61
FIGURE OF THE EARTH.
FIGURE OF THE EARTH, CONSIDERED AS A FLUID MASS.
67. The figure of the earth more or less spherical
§ 1. The Earth considered to be a Fluid Homogeneous Mass.
68. A homogeneous mass of fluid, the law of attraction being the
inverse square, can revolve with a uniform velocity round
an axis, if it be in the form of an oblate spheroid of ellip-
ticity, the angular velocity and gravity being the same
as in the Earth. This is a stable form
72. If the central parts alone attract the same is true, but the
ᎪᎡᎢ .
§ 2. The Earth considered to be a Fluid Heterogeneous Mass.
74. The mass of the earth consists of strata nearly spherical
75. The Equation of Equilibrium of a mass consisting of nearly
spherical strata
77. The form of the strata is spheroidal
78. Conditions resulting from the fluid theory. Meaning of "sphe-
roid of equilibrium," and "surface of equilibrium
79. Centres of the earth's volume and mass coincident. The
earth's axis a principal axis.
80. Equation of ellipticities of strata
83. If the earth's surface be a spheroid of equilibrium, the mass,
whether solid or fluid, must lie arranged according to the
fluid law of density
71
81
§ 3. Tests of the Fluid Theory of the Earth.
88. Four tests enumerated
89. First Test: Law of gravity arising from the theory
93. Ellipticity deduced from them
94. Effect of three hypothetical rearrangements of the earth's
mass on the pendulum confirmatory of the fluid theory
100. Second Test: Perturbation of the Moon's motion in latitude,
and ellipticity thence deduced
102. Third Test: The Ellipticity of the surface
103. Law of density assumed
84
85
86
88
89
97
99
100
105. Expression for ellipticity of the surface: reduced to numbers 102
107. Ellipticity at certain depths
109. Fourth Test: Precession of the Equinoxes: and the ellipti-