420 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Now in the case of our dumb-bell figure it appears, from calculations referred to in 
the Appendix, that e= '0925, i = '4873, and E— ’5798. Hence in the dumb-bell 
figure the kinetic energy is less, but the intrinsic energy is so much greater that the 
total energy is about a half per cent, greater. These numbers are, of course, computed 
from the approximate formulae, and must not be taken as rigorously correct for the 
dumb-bell figure of equilibrium. 
With reference to a figure of transition from the Jacobian ellipsoid, Sir William 
Thomson has remarked :—* 
“We have a most interesting gap between the unstable Jacobian ellipsoid, when 
too slender for stability, and the case of smallest moment of momentum consistent 
with stability in two equal detached portions. The consideration of how to fill up 
this gap with intermediate figures is a most attractive question, towards answering 
which we at present offer no contribution.” t 
Figs. 2 and 3 are intended to form such a contribution, but it is certain that the 
matter is far from being probed to the bottom. 
M. Poincare has made an admirable investigation of the forms of equilibrium of a 
single rotating mass of fluid, and has especially considered the stability of Jacobi’s 
ellipsoid, j He has shown, by a difficult analytical process, that when the ellipsoid is 
moderately elongated (he has not arrived at a numerical result) instability sets in by a 
furrowing of the ellipsoid along a line which lies in a plane perpendicular to the 
longest axis. It is, however, extremely remarkable that the furrow is not symmetrical 
with respect to the two ends, and thus there appears to be a tendency to form a 
dumb-bell with unequal bulbs. 
If M. Poincare’s result shall appear to be not only true, but to contain the whole 
truth concerning the mode in which instability of the ellipsoid supervenes, then there 
must be some other transitional form between the unsymmetrically furrowed Jacobian 
* Thomson and Tait’s ‘Natural Philosophy,’ 1883, § 778" ( i ). 
f In 778" (g) he remarks that “ a deviation from the ellipsoidal figure in the way of thinning it in 
the middle and thickening it towards the end would, with the same moment of momentum, give less 
energy.” I conceive that the energy referred to throughout this paragraph is kinetic only, and we have 
seen that the kinetic energy is less for the dumb-bell than for the ellipsoid. 
[If we write U for a quantity proportional to the excess of kinetic above intrinsic energy, so that 
U = e + (1 — i), then figures of equilibrium are to be determined by making U stationary for variations 
of the parameters involved in it. This course is actually pursued in the Appendix below, the function 
(viii.) being, in fact, this U; and the variations of it, being made stationary, afford a controlling solution 
of the problem of this paper. The similar method may easily be applied to the case of Jacobi’s ellip¬ 
soids. From this point of view the interesting function to tabulate is e + (1 — i ), and we observe that 
in the case of the Jacobian ellipsoid referred to on the last page it is '6052, and for the dumb-bell it is 
'6156. Is not the energy referred to by Sir W. Thomson this function U ? (Addition to foot-note, 
dated October 10, 1887.)] 
J ‘Acta Mathematics,’ 7, 3 and 4, 1885. 
