MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
37 
remains true, if the new meaning is given to the symbols in each case. We conclude 
that the question of stability is not affected by the potential — P a V'. 
The remaining potential term is the spherically symmetrical term V'. The total 
potential may now be taken to be V + V', and this potential, besides being spherically 
symmetrical, satisfies the condition which was postulated in the determination of the 
criterion of stability ; namely, that its radial differential coefficient shall vanish at 
infinity to the order of 1/r. The value of the derived function V' 7 (equation (109)) is 
V" = Li y- (r 2 -y- ) — -In 3 , by equation (] 14). 
Hence the stability function is given by (cf. equation (113)) 
, H 2 
V zz: 1 4- - 
' ^ 3\ 2 T 2 ’ 
W T e have therefore found that when an infinite nebula is rotating, with such 
angular velocities that the linear velocities at infinity have the limiting value H, the 
value of Mj, is greater than unity no matter how small H may be. This result has 
only been obtained on the supposition that ofi may be neglected. We have obtained 
no information as to what happens when ofi is taken into account, i.e., when the 
square of the “ ellipticity ” of the nebula is taken into account. 
Influence of Viscosity. 
§ 32. No account has so far been taken of the viscosity of the gas. The terms 
arising from viscosity which may be supposed to occur in the true equations of 
motion, will contain the coefficient of viscosity (y), and will in each case depend on 
velocities and not on displacements. Hence viscosity enters the equations of motion 
through the factor yip. The vibrations for which p = 0 are accordingly unaffected 
by viscosity, and since it is upon the existence of such vibrations that the whole 
question of stability turns, it is clear that the results already obtained must remain 
true even in the presence of viscosity. 
It can be shown that equations (24) to (26) specify a principal vibration, whether 
the gas is viscous or not. The result is stated without proof, as the proof is rather 
lengthy, and has no bearing upon the main question under discussion. 
A Nebula in Process of Cooling. 
§ 33. In the mathematical investigation we have been concerned with vibrations 
about a position of absolute equilibrium. In nature, no such position of absolute 
equilibrium will occur ; the condition of the nebula will be incessantly changing. 
