172 
MR. J. H. JEANS ON THE YIBRATIONS AND 
but it will be seen that, with slight alterations, the proof of the exchange of stabilities 
for the spherical nebula, which was given in § 28 of the earlier paper, can be made to 
apply to the present case. Admitting this, it appears that the spherical system 
which is at present under discussion will be stable so long as yp^a~j\ is less than (/>, 
and becomes unstable so soon as exceeds </>. 
§ 23. The next question is as to the exact value of </>, and as to the vibration through 
which instalality enters at the point of l)ifurcation. To the first part of the question 
we have not l^een able to olitain a very definite answer. This matters the less, since 
the numerical data which would have to be used in making any applications of our 
i-esults are not themselves very definite. On the whole, the uncertainty in the value 
of (f) is not much greater tlian the uncertainty in tlie value of the numerical data (or, 
wliat comes to tlie same tiling for our present purpose, the uncertainty in our 
knowledge of the law of compressil)illty and distriliution of density in tiie planets of 
our system). 
Tlie general argument of § 3 showed that (f) must, except in extreme cases, l)e 
comparable witii unity. We then examined an artificial case : a planet in which the 
density and elasticity were constant tlirougliout—this system being made mechanicallv 
possilde by introducing an external field of force, of amount just sufficient to annul 
gravitation in the equililnium configuration. For this system was, of course, 
taken equal to p, tlie uniform density, and was taken to be equal to X + 2p, in tlie 
notation of LovE, or ni + n in the notation of Thomson and Tait. The value of (f, 
depends, of course, on the ratio p,/X or n/m. For /i/X = 0 we found (^ = IT); for 
p X = 1 ive found = r27 ; for intermediate value of p/X we saw that the value of (j) 
was intermediate between these two values. 
'fhe iilauets to ivhicli we wish to apply our results do not possess uniform density : 
it IS almost certain that in every case the mean density is much greater than the 
surface density. The general argument of § 3 shows that there is still a point of 
bifurcation corresponding to a value of ^ which is comparable with unity, but affords 
no evidence as to the change which a concentration of density ivill effect in the value 
of We therefore examined a case in which there is an infinite concentration of 
density—the case of a spherical nebula extending to infinity—and found that in tliis 
extreme case tlie value of becomes infinite. It therefore seems probable that a 
concentration of density is attended by an increase in the value of </>. As a working 
hypothesis ive shall assume for the planets of the solar system the uniform value 
<t> — 2. It must be left to the reader to form a judgment as to the amount of error 
involved ill this assumption, Imt it will, perhaps, be admitted that results dependiiig 
U])on it will at least be right as regards order of magnitude. It will lie seen laten 
that considerable variation in the value of cf> is possible liefore the astronomical 
evidence which we are going to bring forward is seriously invalidated. 
§ 24. As reprds the nature of the vibration through which instability of the 
epheiical conflguiation enteis, we are able to come to a more definite conclusion. In 
