HISTORY OF PHYSICAL ASTRONOMY.

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in longitude will cause them to differ generally for any assignable time.The longest axis of the lunar ellipsoid will therefore deviate always a littlefrom the direction of the earth’s action, and hence will arise a series ofinequalities in the moon’s rotation corresponding to her various inequalitiesin longitude, and causing a real libration in her motion. The magnitudeof these inequalities manifestly depends on the figure of the moon, or in otherwords on the ratios of the three axes of the lunar ellipsoid. Hence, if theratios of these axes be known, we shall be enabled to compute the maxi-mum values of the inequalities, and, vice versa, if the latter be determinedby observation, we can readily derive from them the ratios of the axes, andconsequently the exact elongation of the moon’s figure. Bouvard andNicollet undertook for this purpose a series of careful observations of themoon’s libration in longitude at the Royal Observatory of Paris. TheConnaissance des Temps for 1822 contains a beautiful paper by Nicollet,in which he submitted these observations (amounting in number to 174)to a searching discussion. The only sensible inequality was that cor-responding to the annual equation in longitude: it appeared by observa-tion to have a maximum value equal to 4' 45". The results at which hearrived relative to the ratios of the axes do not accord with the generallyadmitted opinion respecting the primitive condition of the moon. Hefound, in fact, that the difference between the least and greatest axes wasgreater than what it would be on the supposition that the moon wasoriginally a fluid mass.

It has been well remarked by the illustrious Humboldt , that no inquiryin physical science can compare with that relating to the Figure of theEarth in the disproportion which exists between the ultimate object ofattainment and the number of ingenious and refined processes both inmathematical and astronomical science, which the long and arduous pursuitof it has given birth to. The theory of this subject remains nearly in thecondition in which Clairaut left it, for the researches of Laplace, notwith-standing the remarkable character of the analysis employed in their expo-sition, contributed only in a small degree to its advancement. We havementioned in one of the foregoing chapters that the attraction of ellipticspheroids is intimately associated with the question of the figures of thecelestial bodies. The most important improvement which this part of thetheory of gravitation has received in recent times is due to the late Mr.Ivory*. The problem for determining the attraction of a spheroid upon aparticle situated in its interior had yielded at an early period to theresources of the ancient geometry, and was afterwards found to admit of aneasy solution by analysis. The analogous problem for an exterior particle,on the other hand, offered difficulties which long seemed to be insuperable,and, although Laplace finally succeeded in devising its solution, his methodwas so incomplete as to leave ample scope for further research. The sub-ject continued to engage the attention of the ablest analysts until Mr.Ivory finally discovered the well-known beautiful theorem, by means ofwhich the attraction of a spheroid upon a point without it is immediatelyderived from its attraction on a point within itf. This theorem is remark-able for being the most important contribution to mechanical science which

* Born at Dundee in 1765 ; died at Hampstead, near London , in 1841. During theearly period of his career he was in all probability the only person in Britain who possessedan intimate acquaintance with the methods of analysis employed in the higher investiga-tions of Physical Astronomy.

t Phil. Trans., 1809.