In the present Treatise I have become myself an accomplice against my own design as at first conceived, by devoting a separate Volume to the republication, with considerable additions and improvements, of one part only of the original work. The disappearance of the Mechanical Philosophy has removed from the student-at any rate for the present, as no other work has yet appeared at Cambridge to supply the want-one subject of great importance and high interest, which that work first introduced into the UniversityI 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*." * See Article on Figure of the Earth, in the Encyclopædia Metropolitana, p. 192. 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 by Lieut.-Colonel James, R.E., Superintendent of the Ordnance Survey. J. H. P. CALCUTTA, February 15, 1860. LAPLACE'S COEFFICIENTS AND FUNCTIONS. ᎪᎡᎢ. 18. Formulæ for the attraction of a homogeneous mass on any particle. PAGE 14 Same equation, under first form, true of R, the reciprocal 27. Any function, which does not become infinite between the |