215 
196-198] Ellipsoidal Cori figurations 
of the surface to the next may be regarded as a sort of wave-length of the 
vibration. The stability or instability of a vibration depends on which is the 
greater—the gain in o 2 I or the gain in gravitational energy when the 
vibration takes place. But as between a vibration of great wave-length and 
one of short wave-length there is this important distinction : for vibrations 
of equal amplitude the gravitational disturbance caused by the vibration 
of great wave-length is much greater than that caused by the vibration of 
short wave-length, since the elements of the latter very largely neutralise 
one another. Thus the change in gravitational energy is enormously the 
greater for vibrations of great wave-length, while it is easily seen that the 
changes in ft» 2 / are approximately the same. It follows that when a 
rotating mass first becomes unstable, instability will set in through a vibra 
tion of the greatest possible wave-length. 
Any vibration or disturbance on the surface of an ellipsoid may be 
analysed into series of ellipsoidal harmonics. The harmonics of order n have 
wave-lengths, in the sense loosely defined above, of the order of 27 ra/n, where 
a is the semi-major-axis of the ellipsoid. Thus the vibrations of greatest 
wave-length are those in which the surface-displacements are proportional to 
harmonics of the lowest orders. Thus we should expect a- rotating incom 
pressible mass first to become unstable through harmonic displacements of 
the lowest order which is physically possible. 
Displacements of order n = 1 are not physically possible, for they displace 
the centre of gravity of the mass off the axis of rotation and so are prohibited 
from the outset. 
The lowest order of harmonics available, then, is the second. The effect 
of superposing a second order displacement onto a spheroid is to transform it 
into an ellipsoid of three unequal axes. Hence we should anticipate that if 
the series of Maclaurin spheroids ever becomes unstable, it will be by deforma 
tion into an ellipsoid. This is precisely what we have found, the ellipsoidal 
figures in question being the Jacobian ellipsoids. 
When we apply the same train of thought to the Jacobian ellipsoids, we 
find that there must be a possibility of these becoming unstable in turn 
through the deformation of greatest wave-length which is possible for them. 
This is represented by a third harmonic, since a second harmonic deformation 
merely changes an ellipsoid into another ellipsoid and so only represents a 
step along the Jacobian series of ellipsoids. Our analysis so far has been 
definitely restricted to the consideration of configurations which cannot leave 
the ellipsoidal shape, and so has not been capable of disclosing an instability 
which enters through a third harmonic displacement. 
Stability of the Jacobian Ellipsoids. 
198. The problem of determining the stability of the Jacobian ellipsoids 
was undertaken by Poincaré *, who proved rigorously that instability first 
* Acta Math. vn. (1885), p. 259, and Phil. Trans. 198 A (1902), p. 333.