158 
MR. J. H. JEANS ON THE VIBRATIONS AND 
Pi 'elimin arij A'ppi 'oximo.tion. 
§ 3. A rough and very simj^le calculation will give an approximate answer to this 
latter question. 
Let Cl be the radius of a sphere, which will ultimately be taken to be the earth, 
M its mass, and the mean density given by M = ^vp^a^. 
Let us use the elastic constants X, and let Xo be the mean value of X. Since 
the sphere is supposed to be spherically symmetrical, X, p, and p will be functions of 
the single co-ordinate r, the distance from the centre. Imagine X/X^, p/Xq, and p/p^ 
each expressed as functions of r/a, and let c^, Co, , . . be the coefficients which occur 
in these functions, these coefficients being mere numbers and independent of the 
system of units in which X, p, and a are measured. 
Imagine a linear series of equilibrium configurations obtained by varving any one 
of the quantities X^, p^, or ci, while keejDing the remaining two quantities and the 
coefficients c^, c^, . . . constants. The points of bifurcation on this series will occur 
when the varying parameter becomes equal to some definite function of the remaining 
quantities and of y, the gravitational constant. 
Hence, however the linear series are arrived at, the points of bifurcation will be 
given by an equation of the form 
fiyr- K Po^ 0^. Cn.c'o. . . .) = 0.(1). 
Now the coefficients c^, Cg, . . . are mere numbers, and the only way in which y, 
Pof and a can be combined so as to give a mere number is through the term 
ypo'«7^o- Hence equation (1) can be expressed in the form 
e have seen that the spherical configuration must be unstable for some values of 
y, po> and X (c.p., it is always unstable for yp^~a^l\ = oo ), hence equation (2) must 
have at least one real root between yp^^ci^jX^y = 0 and yp^^a^|\Q = oo , Let the lowest 
root he 
ypo^^lK = ^ .( 3 ), 
where (fi is a function of Cj, Cg, . . , ; then a .spherical configuration is stable so long as 
yp^hi~l\^ < and becomes unstable as soon as yp/aV^o > 
I he coefficients c^, Cg, . . . will, on the average, he comparable with unity, because 
X, p are 1 ‘eferred to their mean values ; they are as likely (speaking somewhat loosel}") 
to be above as to be below unity. Hence cf) itself will be comparable with unity, and 
* The notation is that of Love’s ‘ Theory of Elasticity.’ The m, n of Thomson and Tait are given 
A -f /n = M 
/i = H. 
