FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
809 
density — in conjunction with equal and opposite surface density + C. This 
double system, say D, is therefore C — Ii. The lost energy \RR may be considered 
as consisting of three parts; first the energy of — C with itself, say hCC; 
secondly that of D with itself, say \DJ) \ thirdly that of — C with D. This third 
item is obviously equal to — (7(7 + and therefore \RR is equal to — \CG 
+ CR + ^DD. 
It follows that the gravitational lost energy of the pear may he wi’itten symboli¬ 
cally in the form 
1.7.7 - JR -GCR- 1(7(7 + WD. 
In this discussion no attention has as yet been paid to the rotation, 1)ut fortunately 
^t haj^pens that the introduction of this consideration actually simplifies the problem, 
for if we suppose \ JJ and JR to mean the lost energies of J with itself and with R 
on the supposition that the mass is rotating with the angular velocity of the critical 
Jacobian, the formulae become much more tractalde than would have l)een the case 
otherwise. 
The inclusion of part of the angular velocity in this portion of the function /f, 
only leaves outstanding the excess of the kinetic energy of the pear above the 
kinetic energy, which it would have if it rotated with the angular velocity of the 
critical Jacobian. If co denotes the latter angular velocity, and {ur §aj^)“ the actual 
angular velocity of the pear; if Aj be the moment of inertia of 7, and A,- that of R 
considered as filled with positive density, we have 
E = IJJ -JR A- CR - \CC + ADD + i (Aj - A,) SojA 
In this statement I have omitted a term which arises from the displacement of 
the centre of inertia from the centre of the ellipsoid; it is duly considered in the 
paper, but is shown to vanish to the requisite order of approximation (§§ 2 , 14 ). 
The co-ordinates of points are determined by reference to the ellipsoid 7, which 
envelopes the whole pear, and the formula for the internal gravitation of .7, inclusive 
of the rotation w, is of a simple character. The size of 7 is indeterminate, and 
therefore the formulae must involve an arbitrary constant expressive of the size of 7. 
But the final result E cannot in any way depend on the size of the ellipsoid which 
is chosen as a basis for measurement, and therefore this arbitrary constant must 
ultimately disappear. Hence it is justifiable to treat it as zero from the beginning. 
It appears then that we are justified in using the formula for internal gravity 
throughout the investigation. If the artifice of the enveloping ellipsoid had not 
been adopted, it would have been necessary to take note of the fact that the pear 
is in part protuberant above and in j^art depressed below the ellipsoid of reference. 
M. Potncahe did follow this last plan, and then proceeded to prove the justifiability 
of using the formula for internal gravity throughout. The argument adduced above 
seems, however, sufficient to prove the point. 
