1902.] Figure of Eqtdlibrmm of a Rotating Mass of Liquid. 1ST 



the pear above the kinetic energy which it would have had if it rotated 

 with the angular velocity of the critical Jacobian. If a> denotes the 

 latter angular velocity, and (o) 2 + 6V 2 )* the actual angular velocity of the 

 pear; if Aj, A r denote the moments of inertia of J, and of E considered 

 as filled with positive density, we have 



E = JJJ - JE + CE - WC + WD + J (Aj - A,) S<o 2 * 



The co-ordinates of points are determined by reference to the ellip- 

 soid J which envelopes the whole pear. The size of J is indeterminate, 

 and therefore the formulae must involve an arbitrary constant expres- 

 sive of the size of J. But the final result for E cannot in any way 

 depend on the size of the ellipsoid which is chosen as the basis for 

 measurement, and therefore the arbitrary constant must ultimately 

 disappear. Hence it is justifiable to treat it as zero from the begin- 

 ning, and we may use the formula for the internal gravity throughout 

 the investigation.! 



Although the constant expressive of the size of J is put equal to 

 zero — which means that the pear is really partly protuberant beyond 

 the ellipsoid — yet there is a considerable amount of mental conveni- 

 ence in continuing to discuss the subject as though the ellipsoid com- 

 pletely enveloped the pear. 



When an ellipsoid is deformed by an harmonic inequality, the 

 volume of the deformed body is only equal to that of the ellipsoid, to 

 the first order of small quantities. In the case of the pear, all the 

 inequalities, excepting the third zonal one, are of the second order, and 

 as far as concerns them the volumes of J and of the pear are the same. 

 But it is otherwise as regards the third zonal harmonic term, and the 

 first task is to find the volume of such an inequality as far as e 2 . 

 When this is done, we can express the volume of J in terms of that 

 of the pear, which is of course a constant. 



By aid of ellipsoidal harmonic analysis we may now express the first 

 four terms of E in terms of the mass of the pear and of certain definite 

 integrals which depend on the shape of the critical Jacobian ellipsoid. 



The energy JDD presents much more difficulty, and it is especially 

 in this that M. Poincare's insight and skill have been shown. The 

 system D consists of a layer of negative volume density coated on its 

 outer surface with a layer of surface density of equal and opposite 

 mass. His procedure virtually amounts to regarding this system as 

 consisting of an infinite number of magnetic layers, whose energy may 

 be evaluated and summed. The reduction of this part of the energy 

 to calculable forms is not very simple. 



* A term depending on the shift of the centre of inertia proves to be negligible, 

 f Compare with M. Poincare's treatment of the same point, ' Phil. Trans./ A, 

 vol. 198, p. 352. 



