## O'Brien's Mathematical Tracts

MATHEMATICAL TRACTS
PART I.

ON
LAPLACE'S COEFFICIENTS,
THE FIGURE OF THE EARTH,
THE MOTION OF A RIGID BODY ABOUT ITS CENTER OF GRAVITY,

AND

PRECESSION AND NUTATION.

BY

MATTHEW O'BRIEN, B.A.,
MATHEMATICAL LECTURER OF CAIUS COLLEGE.

CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS,
FOR J. & J. J. DEIGHTON, TRINITY STREET;
AND
JOHN W. PARKER, LONDON.

M.DCCC.XL.

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PREFACE

The subjects treated of in the following Tracts are, Laplace's Coefficients; the Investigation of the Figure of the Earth on the Hypothesis of its Original Fluidity; the Equations of Motion of a Rigid Body about its Centre of Gravity; and the Application of these Equations to the case of the Earth. The first of these subjects should be familiar to every Mathematical Student, both for its own sake, and also on account of the many branches of Physical Science to which it is applicable. The second subject is extremely interesting as a physical theory, bearing upon the original state of the Earth and of the planetary bodies; it is also well worthy of attention on account of the important and extensive observations which have been made in order to verify it. The Author has put both these subjects together, commencing with the Figure of the Earth, and introducing Laplace's Coefficients when occasion required the; this being perhaps the best and simplest way of exhibiting the nature and use of these coefficients.

The Author has treated some parts of these subjects differently from the manner in which they are usually treated, and he hopes that by so doing he has avoided some intricate reasoning and troublesome calculation, and made the whole more accessible to students of moderate mathematical attainments than it has hitherto been.

In calculating the attractions of the Earth on any particle, he has arrived at the correct results, without considering diverging series as inadmissible; and this he conceives to be important, because there is evidently no good reason why a diverging series should not be as good a symbolical representative of a quantity as a converging series; or why there should be any occasion to enquire whether a series is diverging or converging, as long as we do not want to calculate its arithmetical value or determine its sign. Instances, it is true, have been brought forward by Poisson in which the use of diverging series appears to lead to error; but if the reasoning employed in Chapter III of these Tracts be not incorrect, this error is due to quite a different cause; as will be immediately perceived on referring to Articles 33, 34, 35, and 37.

The Author has deduced the equations of motion of a rigid body about its centre of gravity by a method which he hopes will be found, less objectionable than that in which the composition and resolution of angular velocities are employed, and less complex than that given by Laplace and Poisson ; he has also endeavoured to simplify the application of these equations to the case of the Earth.

In the First Part of these Tracts he has confined himself to the most prominent and important parts of each subject. In the Second Part, which will shortly be published, he intends, among other things, to give some account of the controversies which Laplace's Coefficients have given rise to; to investigate more fully the nature and properties of these functions; to give instances of their use in various problems; for this purpose to explain the mathematical theory of Electricity; to consider more particularly the Equations of motion of a rigid body about its centre of gravity, and the conclusions that may be drawn from them; to give the theory of Jupiter's Satellites, and of Librations of the Moon; and to say something on the subject of Tides.

The Author has not given the investigation of the effect of the Earth's Oblateness on the motions of the Moon, but he has endeavoured to prove that this effect does not afford any additional evidence of the Earth's original Fluidity beyond that which may be obtained from the Figure of the Earth, and Law of Gravity.

JOC/EFR February 2016