STABILITY OF A GKAYITATIXG PLANET. 
KLI 
Hence the force at x -r y r], z i in the direction of x increasing, found by 
differentiating the foregoing expression with respect to is, neglecting squares of the 
displacements. 
cV , , 0-V , BE 
ad + ^ aha,// + ^ a^ar a.- 
( 0 ). 
Let us suppose that, in addition to its own gravitation, tlie sj)here is acted upon 
by an external field of force of potential Vy, and let us, in the usual notation, denote 
the six stresses by P, Q, R, S, T, U. Then the equations of motion of the element 
at :c + ^, y + p, 2 + ^ in the displaced configuration are three of the form 
ap , ar , aT , /aw , .ffw , a^v , ^ffw 0 e\ , , 
^a^^a.+ a^/+a; + /^(ad + ^aA+^aray + ^a.a.“^j- • 
in which W = A" -|- A^y, and all the terms such as ^ 
aw anv 
, ^ , , . . . are evaluated at 
IX a.';" 
X, y, 2, but p, P, Q ... are calculated in the displaced configuration at + f, y + y, 
s + C. 
§ 8. Now the only case in vdiich we have any accurate knowledge as to the values 
of P, Q, R, S, T, U is when the whole strain is small, i.e., when AV is small. In the 
case of the earth, A^ is not, in this sense, small.^ The only way in which we '^an 
proceed with any certainty is, therefore, by taking A^y = — A", or W = 0. That is 
to say, we must artificially annul gravitation in the equilibrium configuration, so that 
this equilibrium configuration may be completely unstressed, and each element of 
matter be in its normal state. In this case it seems justifiable to suppose both the 
density and rigidity to be constant throughout tlie sphere, and, indeed, it is only 
with the help of this simplification that the equations become at all manageable. 
The vibrations of this system will be of two kinds. First there are “spherical” 
vibrations in which the displacement is purely radial at every point, so that the solid 
remains spherically symmetrical after displacement, and, secondly, there is the larger 
class of vibrations in which the di.splacement is not of tins simple type, so that the 
displaced configuration is not one of spherical symmetry. 
We hope, by discussing the vibrations of this system, to obtain some Insiglit into 
the corresponding vibrations of a natural non-homogeneous solid, say the earth. Now 
it is extremely doubtful whether the spherical vibrations of our artificial system have 
much in common with those of the natural system, but it will be seen later that this 
is of no importance. AVe shall not be in any way concerned with tliese vibrations. 
What we shall require is a knowledge of the imsymmetrlcal vibrations, and this, it is 
hoped, can be obtained with fair accuracy from a consideration of the corresponding 
vibrations in the artificial case. There must be some uncertainty even in the case of 
unsymmetrical vibrations, and, unfortunately, this seems to be inevitable; our 
* Lovr, ‘ Elasticity,’ L, p. 220. 
YOL. CCI.—A. 
Y 
