462 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



harmonic, this configuration must be one in which /J (w, z) vanishes at the boundary, 

 so that the boundary must be a figure of revolution about the axis of rotation. 



It now follows from (3) that V must be a function of w and z only on the 

 boundary, and hence also (since V is harmonic) at all external points. It follows 



that -7 , and hence also -- , are functions of is, z only at the boundary. Whence 



rin en 



again, by equations (4) and (5), it follows that - and -^ are functions only of TS 



and z at the boundary. And, by successive differentiation of equations (4) and (5), 

 it is seen that all the differential coefficients of V and p are functions only of rs and z 

 at the boundary. 



It can be seen from this* that the configuration must be one of revolution through- 

 out. In other words, there can be no configuration of equilibrium which differs from 

 the configuration of revolution by first harmonic terms only. 



LAPLACE'S Law. 



8. I have not found that any progress worth recording can be made with the 

 general relation^? =f(p), so that progress can only be hoped for by examining special 

 cases. 



The case that suggests itself as most important is that of the gas law p = Kp, 

 satisfied in a perfectly gaseous nebula at uniform temperature. The difficulty is that 

 such a nebula extends to infinity in all directions, and so cannot rotate as a rigid 

 body. Or rather, when it is caused to rotate, it throws off its equatorial portions 

 and the remainder rotates in the shape of an elongated spindle of infinite length. In 

 this connection I have worked out the purely two-dimensional problem of a rotating 

 gaseous cylinder of infinite length. The results are too long to be worth printing ; 

 it will, perhaps, suffice to record that the analysis bears out in full the conclusions 

 arrived at in this paper. 



The law which is most amenable to mathematical treatment is LAPLACE'S law 



or, as it is more convenient to write it, 



(12) 



in which c, p, K, and a are constants, <r being the value of the density at the free 



* 



I have not succeeded in obtaining a rigorous proof of this. It might be objected that 

 nothing in the above argument precludes first harmonic terms proportional to such a function as 



~f(w, :) , where /(cr, g) = is the equation of the boundary. The pure mathematician may not, 

 although the astronomer will, be influenced by the consideration that such functions never occur in natural 

 problems. If such a function did occur, it would involve an extremely fantastic relation between p and p. 



