460 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



p only. There is further the relation of POISSON, 



V 2 V = -4^, ........... (4) 



so that on operating on (3) with V 2 we obtain 



2w\ .......... (5) 



the differential equation which must be satisfied by p in any configuration of 

 equilibrium under a rotation w. 



5. In general a solution of equation (5) will involve negative and zero values of p- 

 In the physical problem p will be limited as to values, and this limitation will 

 determine the physical boundary of the rotating mass. 



Let V m denote the gravitational potential at any point in space of the finite mass 

 determined in this way. We have found a configuration of equilibrium under a 

 potential V, the potential of the mass is V m , so that for equilibrium we require an 

 additional field of potential V V m . We can say that the configuration found will 

 be a true configuration of equilibrium under an external field of force of potential V 

 such that 



(6) 



And, inasmuch as V 2 V m = 4^/0 = V 2 V, it is clear that V 2 V = 0, so that the 

 external field has poles only at the origin or at infinity. The condition that any 

 solution shall lead to a configuration of equilibrium for a mass rotating free from 

 external influence is, of course, V = 0. 



6. The simplest solution of equation (5) is obviously that in which p is a function 

 of r only, but it is clear from (3) that this cannot give a free solution except when 

 w = 0. 



7. The next simplest form of solution is that in which p is a function of z and w 

 only, and this can give a free solution. It includes, of course, as a particular case 

 the system of Maclaurin spheroids. For this class of solutions every section at right 

 angles to the axis of z is circular, and in any such section the lines of equal density 

 are circles. The density at any point is of the form p =/(&, z). 



Let O denote colatitude measured from Oz, and let \}r be azimuth measured from 

 the plane of xz. The most general configuration which can be obtained by displace- 

 ment of that just considered will have a law of density of the form 



GO 



P=fo ( w , *)+ 2/5 (w, z) COS S\fr. 



It is easily seen that the separate cosine terms lead to independent displacements, 

 and we shall for the moment only consider the displacement of the first order, for 

 which the law of density is 



P =/o(w, z)+/(w, z)cos^, ........ (7) 



where/! (ra, z) is a small quantity of the first order. 



