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X XVI. On the Theory of the Compressibility of the Matter com- 

 posing the Nucleus of the Earth, as conjirined by 'what is knoison 

 of the Ellipticitics of the Planets. By the Rev. J. Challis, 

 Fello-ii) of the Cambridge Philosophical Society*. 



T~\R. YOUNG suggested that the increase of density towards 

 -*-^ the centre of the earth, might be owing to the com- 

 pressibility of the material of which it is composed. Laplace, 

 adopting the suggestion, obtained, in an addition to a Memoir 

 on the Figure of the Earth (Mem. Acad. Scien. An. 1818) the 

 law of the increase of density in proceeding from the surface 

 to the centre, on the suppositions that the relation between the 



pressure {p) and density (g) is expressed by ^^ = /r ^ -^ 1 j , 



(§ being the density at the surface), and that the chemical 

 composition of the nucleus of the earth is the same throughout. 

 He found that on these suppositions the requisite degree of 

 compressibility, and the proportion of the density at the sur- 

 face to the mean density, were not by any means at variance 

 with what we know on these points by experience. The cause 

 assigned in this theory for the increase of density towards the 

 centre, and the relation between p and g, are of so simple a na- 

 ture, that I have been induced to inquire how far the theory 

 is confirmed by what is known of the ellipticities of the planets. 

 I here repeat the investigation of Laplace, modified for the 

 purposes I have in view. Suppose the mass of the planet to 

 be spherical. If r be any distance from its centre, and q = 

 (J) (/•), the attraction on a particle of the mass at a distance R 



r , . f^:'Kr^ (t> h-) dr ., . i i • , i 



from the centre is ^ Wi~ — ' ^"^ integral bemg taken 



from r = to r = R. To express this force in the manner 

 in which terrestrial gravity is usually expressed, let M = the 

 earth's mass, a = its radius, and g = 32;^^ feet, the measure 

 of the accelerative force of gravity at the earth's surface: 



then -^r^ . „j ^ — , is the force expressed as re- 

 quired. Suppose thatyr'fZr <{) (r) = \J/ (r). Then attraction 

 (A) = i^|^x -KRyW . Now-ip=AsrfR; 



and -dp=- -^ §d§. 



Therefore * '^ - ''' ^ ^i^)-MO) __2/c-§dg 

 ineretore ^ . j^^ - gs^R • 



* Communicated by the Author. 



Hence 



