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XL. Note on the Annual Precession calculated on the Hypo- 

 thesis of the Earth's Solidity. By Henky Hennessy, F.R.S., 

 Professor of Applied Mathematics in the Royal College of 

 Science for Ireland *. 



IN discussing the influence of the internal structure of the 

 Earth upon precession, it has been frequently assumed 



that with the ellipticity ^-r the annual precession of a homo- 

 geneous solid shell or completely solid spheroid would be 57". 

 This was the result of Mr. Hopkins's calculations ; and the 

 difference, amounting to between six and seven seconds, between 

 it and the observed value, formed the basis of all his con- 

 clusions relative to the Earth's internal condition. Hitherto 

 I have not seen any reason for doubting the above numerical 

 result; but on looking more closely into the question, it appears 

 probable that we must reduce the precession for the hypo- 

 thetical solid spheroid to about 55". If the Earth were a 

 spheroid perfectly rigid, the amount of precession can be 

 calculated from formulae given in Airy's ' Tracts/ Pratt's 

 ' Mechanical Philosophy/ Pontecoulant's The'orie Analytique 

 du Systeme clu Monde, or ResaPs Traitd de Mdcanique Celeste. 

 In the two latter works, Poisson's memoir on the rotation of 

 the Earth about its centre of gravity is very closely followed, 

 and the formulae are those which I have generally employed. 

 From these writings we find 



where I is the inclination of the equator to the ecliptic, y the 

 ratio of the Moon's action on the Earth compared to that of 



the Sun, m the Earth's mean motion around the Sun, — the 

 ; 7 n 



ratio of this mean motion to the Earth's rotation, and A, B, C 



the three principal movements of inertia of the Earth. When 



the Earth is supposed to be a spheroid of revolution A=B, 



and the above becomes 



(1) P = 27T-C~- (1 + 7)C0SL 

 Pratt gives the formula 



* Communicated by the Author. 



