ART.

27. 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 ith order
39. Examples of arrangement in Laplace's Functions

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 internal

36

38

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 internal

40

41

CHAPTER IV.

ATTRACTION OF BODIES OF VARIOUS FORMS.

47. Object of this Chapter

48. Attraction of a slender prism on any particle

49. Attraction of a slender pyramid on a particle at its vertex

51. Attraction of an extensive plain of given depth or thickness

on a point above it.

43

55. Attraction of a rectangular mass on a particle in the plane of

one of its sides. Examples

44

46

ib.

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

61. Results arising from the Himmalayas, (62) the Ocean, and

(63) the interior of the Earth

64. Effect on the plumb-line of a slight but wide-spread defect or

excess of density in the interior of the Earth. Example

ART.

FIGURE OF THE EARTH.

CHAPTER I.

FIGURE OF THE EARTH, CONSIDERED AS A FLUID MASS, AND

THEREFORE CONSISTING OF STRATA NEARLY SPHERICAL.

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

ellipticity is

58

73, Equation of equilibrium, if the mass consist of nearly spheri-

cal strata of varying density

75. The form of the strata is spheroidal

76. An approximate law of density found

77. Ellipticity of the strata

80. Ellipticity of the surface in the case of the Earth

81. Conditions enumerated, which must be satisfied if the Earth

derived its form from being a fluid mass

PAGE

82. Test of the law of density used, by its application to calculate

Precession

84. Argument drawn by Mr Hopkins from Precession that the

crust is of considerable thickness .

85. Remarks on Professors Hennessy and Haughton's calcula-

tions

86. Argument drawn from the present aspect of the surface of the

Earth in mountains, plains, and oceans

CHAPTER II.

FIGURE OF THE EARTH, ON THE SOLE HYPOTHESIS OF THE

SURFACE BEING A SURFACE OF EQUILIBRIUM AND

NEARLY SPHERICAL.

89. The law of gravity at the surface

90. Clairaut's Theorem, generalized by Professor Stokes. Re-

marks on the result of Pendulum experiments, and the

ellipticity thence deduced

92. Ellipticity determined from the Moon's motion in latitude

94. The centre of the Earth's volume coincides with the centre of

its mass; and the axis of revolution is a principal axis

90

92

94

CHAPTER III.

THE FIGURE OF THE EARTH, DETERMINED BY GEODETIC

OPERATIONS.

ᎪᎡᎢ .

95. Geodesy affords a test of the earth's having been fluid or not

96. Length of an arc of meridian

98. Formulæ for finding the semi-axes and ellipticity by Two

Arcs arcs which are most likely to give best results.

99. Examples, from the English and Indian Arcs

100. The probable causes of deviation from the mean ellipticity

in the ellipticity deduced from particular arcs

101. A comparison of arcs on various parts of the Earth's surface

gives an average form agreeing with the hypothesis that

the Earth acquired its general form from being in a fluid

state

102. Propositions suggested by Captain A. Clarke's Article on

the Figure of the Earth in the Volume of the British

Ordnance Survey

104. Deviation of the mean arc from the elliptic form

105. Formula for correcting the observed latitudes to make

them suit the general curve. Examples.

107. Application of the Principle of Least Squares to find the

mean figure of the earth

108. Results from British Ordnance Survey

109. Ellipse which best suits the observations

PAGE

97

98

99

100

101

102

103

104

105

ib.

108

ib.

110. Local departures from the mean figure in India

109

112. Ellipse which most nearly accords with the mean ellipse and

with the calculated deflections of the vertical

116. Indian Arc compared with the mean ellipse

117. The difference between geodetical and astronomical ampli-

tudes arises solely from local attraction

118. Effect of local attraction on the mapping of a country

119. Geodesy gives no evidence regarding change of level

120. Effect of local attraction on the SEA-LEVEL

122. Examples, along the coast of Hindostan, arising from the

attraction of the Himmalayas and the deficiency of attrac-

tion of the Ocean lying on the south. The sea-level at

Karachi is nearly 600 feet higher than at Cape Comorin

from these causes

114. Difference in length, and distance at mid-latitude, of two

arcs having same extremities

113

ATTRACTIONS AND LAPLACE'S

FUNCTIONS.

1. THE Law of Universal Gravitation teaches us, that every particle of matter in the universe attracts every other particle of matter with a force varying directly as the mass of the attracting particle and inversely as the square of the distance between the attracted and the attracting particles. Taking this law as our basis of calculation, we shall investigate the amount of attraction exerted by spherical, spheroidal, and irregular nearly-spherical masses upon a particle, and apply our results in the second part of this Treatise to discover the Figure of the Earth. We shall also show how the attraction of irregular masses lying at the surface of the Earth may be estimated, in order afterwards to ascertain whether the irregularities of mountain-land and the ocean can have any effect on the calculation of this figure.

CHAPTER I.

ON THE ATTRACTION OF SPHERICAL AND SPHEROIDAL

BODIES.

PROP. To find the resultant attraction of an assemblage of particles constituting a homogeneous spherical shell of very small thickness upon a particle outside the shell: the law of attraction of the particles being that of the inverse square.

2. Let O be the centre of the shell, Pany particle of it, OP=r, dr the thickness, C the attracted particle, POC=0; mPMn a plane perpendicular to OC, the angle which the plane POC makes with the plane of the paper, PC=y.

P. A.

m

n

1

The attraction of the whole shell evidently acts in CO.

Let OP revolve about 0 through a small angle de in the plane MOP; then rde is the space described by P. Again, let OPM revolve about OC through a small angle do, then r sin @ do is the space described by P. And the thickness of the shell is dr. Hence the volume of the elementary portion of the shell thus formed at P equals rdo.r sin Odp. dr ultimately, since its sides are ultimately at right angles to each other.

Then, if the unit of attraction be so chosen, that it equals the attraction of the unit of mass at the unit of distance, the attraction of the elementary mass at P on C in the direction CP

=

pr2 sin e drdedo
y2

,p the density of the shell;

.. attraction of P on C in CO pr2 sin 0 drd0dp c — r cos

y2

y

To obtain the attraction of all the particles of the shell we integrate this with respect too and y, the limits of O and 2π, those of y being c-r and c+r;

being

dy do