30 
MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
vary K as much as we please. But on examining the problem in detail it is found that 
K is always infinite except for one special value of £ For this special value of £ we 
can eliminate Iv altogether by a new choice of parameter. But for other values of £ our 
solution is only valid if e 3 K is small, and, replacing e 3 K by a new parameter e 1 , the 
solution reduces to 
<p= 0.P+0.Q+eT 
and so returns to the original first-order solution (108). 
Thus, except for one special value of f, it is impossible to carry the linear series 
outside the second order rectangle ; if we attempt to do so, the solution keeps lapsing 
back into the first-order solution. 
In the previous paper I gave reasons for believing that Sir G. Darwin, in his well- 
known investigation of this problem, had introduced a spurious equation of equilibrium. 
This extra equation could only be satisfied by assigning to f a special value, namely 
f = -(F015988e 3 . 
Sir G. Darwin accordingly gave this value to £, so that the value of w 2 decreased on 
passing along his series of pear-shaped figures, and, assuming this value for £, he showed 
his series to be stable. * 
But the investigation of the previous paper showed that there was no need to assign 
this special value to £, and the present investigation has further shown that with this 
value of £ it is impossible to extend the series beyond second-order terms at all. 
There is only one value of £ which leads to a real linear series of configurations, and 
this is shown in the present paper to be 
f = +0-007423e 2 . 
Thus as we pass along the true linear series u> 2 continually increases. The angular 
momentum is however found to decrease, so that the pear-shaped figure is shown to be 
unstable. 
24. The amount of computation involved in the problem lias proved to be very 
great, and as the whole question of stability or instability depends on the sign of a 
single term at the end of all this computation, the question of numerical accuracy 
becomes one of great importance. 
The difference between my second-order figure and that of Darwin arises solely 
from the difference in the value of £. The moment of inertia of such figures is a 
linear function of C, and a very simple calculation gives the rate at which it ought to 
vary with £. Allowing for this difference in £, I find that my computations give for 
the moment of inertia of Darwin’s figure (in terms of Darwin’s parameter e D ), 
M& 0 2 (l + OT5 777e D 2 ), 
while Darwin calculated as the value of the same quantity 
M& 0 2 (1 + 0'157786e D 2 ). 
