of the Obliquity of Planets to their Orbits. 189 



mutual gravitation of their parts. This application involves a 

 large assumption, viz. that the precession of a nebulous mass 

 is nearly the same as though it were rigid. In defence thereof 

 I can only quote Sir W. Thomson, who says, " Now, although 

 the full problem of precession and nutation, and what is now 

 necessarily included in ifr — tides, in a continuous revolving 

 liquid spheroid, whether homogeneous or heterogeneous, has 

 not yet been coherently worked out, I think I see far enough 

 towards a complete solution to say that precession and nuta- 

 tions will be practically the same in it as in a solid globe, and 

 that the tides will be practically the same as those of the 

 equilibrium theory " *. 



I therefore once for all make this assumption. 



The coefficient p depends solely on the orbit of the planet 

 and of its satellites, and during the contraction of the mass 

 will have been constant, or very nearly so. To determine 

 the other quantities involved, we have the three following 

 principles : — ■ 



(1) The conservation of angular momentum. 



(2) The constancy of mass of the planet. 



(3) That the form of the planet is one of equilibrium. 



(1) is expressed by the equation Cn=H, a constant ; and, 

 if p, a be the mean radius and density of the planet at any 

 time, (2) by ^7rpa 3 = M, the mass. Then, if the law of internal 



density during contraction be that of Laplace, viz. — ±- , if 



k be the ratio of the surface-density to the mean density, e the 

 ellipticity of the surface, and m the ratio of the centrifugal 

 force at the distance a to mean pure gravity, the third prin- 

 ciple gives f 



2e~M(qa-l) 



Also 



C_A= l(e-j) Ma>, 



3n 2 



m= -. 



4z7Tjjip 



* Address to Section A. of the British Association at Glasgow, ' Na- 

 ture/ September 14, 1876, p. 429. 



t Compare Thomson and Tait's < Natural Philosophy,' § 824 (14), § 827 

 (20). 



