2io-2i2] Ejection in Filaments 213 



mass. Thus this angular velocity is connected with p, the mean density of 

 the matter of the main mass, by equation (534), 



a> 2 =0'35 x 27TJ5, 



so that Poincare's result takes the form that p A , the mean -density of the 

 matter in the ring, must be not less than 0*35 times p, the mean density of 

 the main mass otherwise the ring cannot form at all. 



To appreciate the full meaning of this result, we must remember that it 

 presupposes the formation of a sharp edge on the main mass, and that this 

 sharp edge cannot form at all unless the matter of the main mass has a degree 

 of compressibility comparable with that of a perfect gas. The ring, if ever 

 formed, will be a structure in equilibrium under gravitation and its own 

 pressure. Clearly the matter of a ring of small mass will expand under its 

 own pressure until its density becomes very small, and the condition that the 

 mean density shall be as great as 0'35p cannot be satisfied at all the ring 

 will be disintegrated by its own rotation. Thus we see that a ring of this 

 type, if it forms at all, must have a mass comparable with that of the main 

 central body for a ring of mass much less than that of the central body 

 would have a mean density much less than that of the central body. This 

 result seems to dispose of Laplace's theory of the formation of the solar system, 

 for this supposed the planets to have formed out of a ring whose mass must 

 have been small compared with that of the central mass. 



EJECTION IN FILAMENTS 



212. The formation of Laplace's ring required perfect symmetry of the 

 mass about its axis of rotation. To ensure this the mass was supposed to be 

 rotating freely in space, unaffected by the presence of any other masses. The 

 distances of adjacent masses in space will in general be so great that their 

 gravitational influence will be extremely small. For most problems there 

 would be no question that this gravitational influence might legitimately be 

 neglected, but the problem we now have under consideration is peculiar in 

 that even the slightest external gravitational field is sufficient to alter entirely 

 the nature of the solution. 



In the neighbourhood of the mass of gas under consideration the gravi- 

 tational potential of all external masses will be a spherical harmonic, and 

 therefore will be capable of expansion in the form 



where S , S l} S 2t ... are harmonics of degrees 0, 1, 2, ... respectively. As in 

 previous discussions of the value of V T (cf. for instance, 47), the constant 

 term $ may be omitted as giving rise to no forces, the term Si may be 

 neutralised by supposing the areas of reference to have the same acceleration 



