KECBNT PROGRESS IN HYDRODYNAMICS. 61 



sistent motions. Suppose the particles to describe similar ellipses in 

 parallel planes perpendicular to a principal section of tlie ellipsoid in the 

 same way as if attracted to the centres of their paths. Then the actual 

 motion may be represented by supposing the whole system to have an 

 extra motion of uniform rotation about an axis in that principal plane. 

 In other words we may suppose the motion set up in the same way as 

 imagined by Greenhill, referred to below. In the latter part of his paper 

 he shows that when the axis of rotation is not a principal axis the motion 

 is unstable, that Jacobi's and Dedekind's ellipsoidal forms are stable, and 

 that in the other cases the motions are not stable if the two rotations 

 which represent the motion are in the same direction. 



In the same year as Riemann's researches appeared, Brioschi ' pub- 

 lished a paper in which he investigated equations for the moving axes 

 of the ellipsoid and the molecular rotations. Dedekind's reciprocal law 

 follows also easily from his forms of the differential equations. The in- 

 vestigations of Dirichlet and Riemann have been co-ordinated and pub- 

 lished in Italian by Padova^ with some extensions of his own, when the 

 ellipsoid changes its form periodically. 



In 1879 and 1880, in the ' Proceedings ' of the Cambridge Philosophical 

 Society, GreenhilP took up the same problem so far as it relates to 

 motions with persistent form, but from quite a different point of view 

 from that of Dirichlet. Instead of the Lagrangian equations he uses 

 the Eulerian referred to moving axes. The fluid is supposed enclosed in 

 a rigid envelope without mass, the whole system to have a rotation as a 

 rigid body communicated to it about one line through the centre, and 

 then a rotation of the shell alone about another. This is quite as general 

 as Dirichlet's, and has the advantage of expressing the motion in terms 

 of quantities whose dynamical meanings are evident. As the velocity- 

 potentials for the last rotations are known, the equations are obtained 

 with the greatest ease, and it only remains to find the condition that the 

 pressure of the fluid on the shell may be everywhere the same, in which 

 case the shell may be removed and the fluid mass will move as before. 

 In the first paper the case of rotation about the principal axis is alone 

 considered, and the relations between the axes and two rotations deduced. 

 This is case (3) of Riemann. In the second paper he takes up the 

 general question, and gives the condition for a free surface, but does not 

 discuss the equation giving the condition. The calculations can be even 

 here much simplified by considerations adduced at the end of the paper. 



On the same subject reference may be made to papers by Lipschitz^ 

 (1874) and Hagen^ (1879). The former uses Riemann's form of 

 Dirichlet's expressions for the positions of the particles, finds the action, 

 and applies Hamilton's principle. The latter does not refer to any of the 



> ' Devcloppements relalifs au § 3 des Eecherches de Dirichlet sur un probl&me 

 d'hydrodynamiqiie,' Borrh. lis. p. 63. 1861 (dated Nov. 1860). 



- ' Sul moto di un ellissoide fiuido ed omogenco,' Ann d. Sc. Korni. Pisa. 18G8-9. 



3 ' On the rotation of a liquid ellipsoid about its mean axis,' Proe. Camh. Phil. 

 Soc. iii. p. 233. 



' On the general motion of a liquid ellipsoid under the gravitation of its own 

 parts,' Proo. Camb. Phil. Soc. iv. p. 4. 



■• ' Reduction der Bewegung eines fliissigen homogenen ellipsoids auf das Varia- 

 tioiispi'oblcm eines einfachen Integrals, und Bestimmung der Bewegung fiir den 

 Qjen^^^'ill eines unendlichen elliptischen Cylinders,' Porch. Ixxviii. p. 245. 



i 'Zur Theorie der drei elUpsoidischen Gleichgewiclitsfiguren frei rotirender 

 homoSener Fliissigkeiten,' ScMdm. Zeits Math. sxiv. p. 104. 



