24 
MR. J. H. JEANS ON THE INSTABILITY OF THE PEAR-SHAPED 
The corresponding rotation is given by 
2 
■£- = n + e 2 n" = 014200+ 0'007423e 2 . 
2irp 
We notice at once that w increases as we pass along the pear-shaped series, whereas 
in Sir G. Darwin’s solution « was found to decrease. Thus the present solution 
diverges in essentials from that of Darwin. 
On the other hand the present solution is similar to that for rotating cylinders* in 
which the rotation was also found to increase as we passed along the pear-shaped 
series, and as we shall now see, the increase is at a very similar rate. 
Our present figure, as far as the first order of small quantities, is 
2 2 2 
^■ + -^i + f Sr A +e{-0-079x 3 + 0-127xy 2 + 0-106xz 2 +0’U2x) = 1 , 
o oo 0 oo4 0 424 
while the cylindrical figure, on replacing the parameter lCf O used in the second half of 
the two-dimensional investigation by e, was found to be 
2 2 
- + -&— + e(-0-063x 3 +0190.^ 2 + 0-211x) = 1. 
5 u‘555 
A comparison of these two figures shows that the two e’s may be taken to be very 
approximately the same. As regards angular velocity, the value in the present three- 
dimensional problem has been found to be 
2 
AL - = n + n"e a = 0H4200 (l + 0'05227e 2 ), 
2-7T f> 
while that in the two-dimensional problems wasf 
= - + l -^e 2 = 0-3750 (l + 0'0513e 2 ). 
2t r P 8 448 v ’ 
Thus, in so far as it is possible to compare the three-dimensional problem with its 
two-dimensional analogue, we may say that the two rotations are in very close 
agreement. 
Calculation of the Moment of Inertia. 
19. We know that the pear-shaped figure will be stable or unstable according as 
the angular momentum increases or decreases as we pass from the critical Jacobian 
ellipsoid along the series of pear-shaped figures. 
* “On the Equilibrium of Rotating Liquid Cylinders,” ‘Phil. Trans.,’ A, vol. 200 (1902), p. 67. 
f I have recently repeated the calculations of the two-dimensional problem as far as the third order of 
small quantities, including the evaluation of w 2 , and find that the numbers given originally for the figure 
and angular velocity are absolutely correct. See, however, § 21 below. 
