Rev. J. Challis on Ihe Ellipticities of the Planets. 



ferentiatmg, R- 4> (R) + ^^^^.^^ ( -j^ 



Hence finally, , „ / + 5'^ R g = 0, q"^ being 



The integral of this gives -J- = ^ — , j being the value 



of § at the centre. Legendre has calculated (Man. Acad. Scien. 

 An. 1789), that according to this law of density, if c = the 



earth's radius, and qc = -- , the ellipticity would be — ; and 



if qc = TT, the ellipticity would be — — . But it is plain from the 



.379 



nature of solid substances that qc cannot be so great as tt, for 

 then g would be = 0. In proportion as the value of 7 R is near 

 to TT, the density is small, and decreases rapidly as R increases: 

 and because solid substances do not admit of this rapid change 

 of density, and at their surfaces possess a certain limited den- 

 sity, therefore it is probable that we shall not be far wrong by 



assuming qc = —. In fact, this value gives an ellipticity 



of — - to the earth, which is very near the experimental de- 

 termination. 



If now we calculate the attractive force of Jupiter at his 

 equator, by means of the period of his fourth satellite, and 

 the law of the inverse square of the distance, we shall find 

 that the ratio of the centrifugal force at his equator to this 



force is . But because the law of the inverse square 



does not accurately hold by reason of the planet's spheroidal 

 shape, this value requires a correction. When the correction 

 has been made according to the formula given in the Mec. 



Celeste, liv. iii. art. 35, the ratio becomes -— . The ratio 



1 2*348 

 of the centrifugal force to gravity at the equator of Saturn, 

 calculated from the period (79-33 days) and mean distance 

 (64*36 equatorial radii) of the extreme satellite, and from the 



time (10'' ITi) of Saturn's rotation, will be found to be • 



7*76 



Correcting as before, the ratio becomes |^ nearly. Calcu- 



N.S. Vol. 10. No. 57. Sept. 1831. 2D lating 



