ATTRACTIONS, LAPLACE'S FUNCTIONS, AND THE FIGURE OF THE EARTH. BY JOHN H. PRATT, M.A. ARCHDEACON OF CALCUTTA, LATE FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE, AND AUTHOR OF “THE MATHEMATICAL PRINCIPLES OF MECHANICAL PHILOSOPHY." BIP) SECOND EDITION. Cambridge: London. 1861. PREFACE. This Treatise is in part a republication of those portions of my work on Mechanical Philosophy which treat of Attractions, Laplace's Functions, and the Figure of the Earth. The first edition, issued last year, consisted of a small number of copies. In the present issue the last Chapter has been rearranged and in part rewritten: other improvements have been made. The disappearance of the Mechanical Philosophy has removed from the student-at any rate for the present, as no other work has yet appeared in English to supply the want-one subject of great importance and high interest, which that work first introduced into the University ; I mean Laplace's Coefficients and Functions and the calculation of the Figure of the Earth by means of his remarkable analysis. The late Professor O'Brien subsequently published a Tract on the same subject; but it was incomplete. A Fourth Edition of Mr Airy's Tracts has been recently published, and in these is a treatise on the Figure of the Earth. But he adheres by choice (as stated in his Preface) to the “ geometrical and quasi-geometrical methods." There is still room, therefore, for the present Treatise; as no student of the Higher Branches of Physical Astronomy should be ignorant of Laplace's Analysis and its results “a calculus,” to use Mr Airy's language, “the most singular in its nature, and the most powerful in its application that has ever appeared*.” There are problems in the Figure of the Earth which the geometric and quasi-geometric methods cannot touch, and of which the student must remain ignorant, if he is ignorant of the method of potentials. It has been my endeavour to put the well-known difficulty in Laplace's analysis, arising from the use of a discontinuous function, in the clearest light, that the student may understand both what it is and how it is overcome. I have made use of Professor Stokes's valuable Paper in the Cambridge Philosophical Transactions of 1849 on the “ Variation of Gravity at the surface of the Earth.” I have also introduced some Propositions on the Geodetic Method of determining the Figure of the Earth, suggested by an acquaintance with the circumstances of the Great Trigonometrical Survey of India, and by the volume of the Ordnance Survey of Great Britain and Ireland recently published. J. H. P. CALCUTTA, 1861. * See Article on Figure of the Earth, in the Encyclopædia Metropolitana, p. 192. 2. Attraction of a spherical shell on an external particle 5. Attraction of a spherical shell according to any law, on par- 9. Laws for which a shell attracts as if collected at its centre 12. Attraction of a homogeneous spheroid and of a spheroidal 15. Ivory's Theorem, for an external particle 45 7 + + 18. Formulæ for the attraction of a homogeneous mass = 0, or -- 47p', according as the dga dh' V being the Potential 22. First equation true also of R, the reciprocal of the distance of the attracted particle from any point of the body. 23. Transformation of equation in R to polar co-ordinates 25. Laplace's Coefficients, Equation, and Functions defined 26. The definite integral of the product of two of Laplace's Func- 21 |