STABILITY OF A GBAYITATING- PLANET. 
173 
each of the cases referred to in the last section this vibration is found to l)e one of 
order n. = 1, i.a., one in which the displacement at every point is proportional to the 
first harmonic. This is the result which we should naturally expect -just as we 
expect a mass of liquid to become unstable through long surface waves sooner than 
through short ones. We shall, tlierefore, suppose it to lie true of the planets in 
general. It is conceivable that planets could be artificially consrracted for which this 
assumption would not he true, but, at present, since we have not a complete 
knowledge of the structure of the planets and are therefore compelled to make some 
assumptions, it seems as if the assumption just made is far and away the best to talvO 
as a working hypothesis. 
Application to the Nebulap>. Hypothesis, 
Theoretical Conclusions. 
§ 25. In the former paper, already referred to, the suggestion has been put forward 
that the instability of a nebula, sun or planet, which, upon the nebular hypothesis, is 
supposed ultimately to result in the ejection of a satellite, may be largely lirought 
about by a gravitational tendency to instability of the kind we have been investi¬ 
gating. Let us, for the moment, take an extreme hypothesis, and imagine that tliis 
agency is the preponderating agency, and that the rotational tendency to Instahillty 
may be disregarded in comparison. 
Upon this hypothesis let us consider the case of an approximately spherical planet 
or sun which is known to have thrown off a satellite. Before the ejection of this 
satellite commenced, the primary mass would have an approximately spherical foim, 
for which Pq~cc^I\^ would be below the critical value <f). Wlien this critical value is 
reached, a divergence from the spherical form occurs, and a series of new processes 
begins. We are not now concerned with the details of tliese processes, but they 
must be supposed ultimately to result in the ejection of a satellite. It must be 
noticed that we are not supposing the primary to be devoid of rotation—for this 
would be inconsistent with the ejection of a satellite—but are supposing the rotation 
to be so small that the rotational tendency to instability is small in comparison with 
the gravitational. 
If we suppose one or more satellites to have lieen ejected, and the primary to liave 
regained an approximately spherical form, the new value of PQ~a'’l\ must be less 
than fjj. Now every satellite of which we have any knowledge, in our own system 
at any rate, is small in comparison with the primary. A legitimate inference seems 
to he that the ejection of a satellite is only a small part of the life-history of tlie 
primary. We shall not, however, need to make any assumption so definite as tliis, 
hut shall suppose only that the values of Pq, a, Xq for the primary after ejection are 
