ME, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
19 
If we deduce the values of B and C from the solution (76), and substitute in the six 
boundary equations the values so obtained, we shall be left with six linear and 
homogeneous equations between the six E’s. Eliminating the six E’s, we have a 
single relation between n, the constants of the nebula and p. Now it will be seen 
that it will always be possible to pass to the limit p = 0 in this equation, since this 
amounts only to finding the ratio of the values of f- a or f 6 at the two boundaries. 
The equation obtained in this manner will give us a knowledge of the configurations 
at which a change from stability to instability can take place. 
§ 19. It therefore appears that it is not sufficient to consider vibrations of frequency 
p — 0 as represented by positions of “limiting equilibrium.” The method of 
PoincarB # for determining points of transition from stability to instability is not 
sufficiently powerful for the present problem; indeed it appears that it is liable to 
break down whenever there are boundary-equations to be satisfied.! 
It is of interest to notice that this complication is not (as might at first sight be 
suspected) a consequence of our having taken thermal conductivity into account. 
For we can put C = 0 and remove the equation of conduction of heat without causing 
any change in our argument, except that the right-hand member in equation (74) must 
be replaced by a determinant consisting only of the minor of the bottom right-hand 
member in the present determinant. The value of A is now 
A — p~ D 4 + D 2 , 
and the number of boundary-equations is of course reduced from six to four. Thus 
an exactly similar situation presents itself, although we are now dealing with 
a strictly conservative system. 
The consequences of this result are more wide-reaching than would appear from 
the present problem, inasmuch as all problems of finding adjacent configurations of 
equilibrium are affected. For instance, it appears that an equilibrium theory of tides 
is meaningless except in very special cases ( e.g ., when the elements of the fluid in 
which the tide is raised are physically indistinguishable). 
If we attempt to calculate by the ordinary methods the tide raised in a mass of compressible fluid by 
a small tide-generating potential, we reach a number of equations which are (except in special cases) 
contradictory. To take a simple case, suppose we have a planet of radius E 0 covered by an ocean of 
radius Rj, the whole being surrounded by an atmosphere which maintains a constant pressure n at the 
surface of the ocean. Let the law of compressibility be w = op, where c varies from layer to layer of the 
ocean. Let the tide generating potential be a 0 r n S, t . Then the equations of this paper will hold if we 
write p = 0, C = 0, ignore the equation of conduction of heat, replace AT everywhere by c, and include in 
* “ Sur l’Equilibre d’une Masse fluide . . . ‘ Acta. Math.,’ 7, p. 259. 
t There is not, of course, a flaw in Poincare’s analysis, but he works on the supposition that the 
potential-function is a holomorphic function of the principal co-ordinates, and this supposition excludes a 
case like the present one. 
D 2 
