KECENT PROGRESS IN HYDRODYNAMICS. 57 



uses the methods of Green developed in his memoir on the determination 

 of the attractions of elHpsoids of variable densities, to obtain the velocity- 

 potentials for translation and rotation, but he does not refer to Green's 

 memoir on the same subject. In 1879 GreenhilP discussed the motion of 

 an ellipsoid in general, and in particular of a spheroid. Amongst the 

 results obtained may be mentioned the condition that a prolate spheroid 

 projected through a fluid may keep its point in front. He found that it 

 must have an angular velocity about the axis 



> 2^ (C33C44(1 — C33/C1,)} /Cg5 



where Cn, C33 are the effective masses along and perpendicular to the axis, 

 and Cgg, c^^ the effective momenta of inertia about the axis, and a line 

 perpendicular to it.^ In this same paper he has determined the initial 

 motion of an ellipsoidal solid within a confocal ellipsoidal shell, when the 

 shell has any motion of translation or rotation impressed on it, also the 

 small oscillations of such a body about the jjosition of confocality. In 

 the same year also Craig-* published a paper dealing with the same 

 questions with reference to a single ellipsoid, and containing transforma- 

 tions to the notation of elliptic functions. 



By making one of the axes of an ellipsoid indefinitely small we arrive 

 at a solution of the equation of continuity with conditions over a jDlano 

 elliptic disc, but which does not satisfy the hydrodynamic conditions that 

 the pressure must be everywhere finite. The solution of the discon- 

 tinuous motion which ensues when a disc is moved perpendicular to itself 

 through a perfect fluid has yet to be found. 



Another motion in connection with surfaces of the second degree is 

 that where the stream-lines are the lines of curvature on a family of one 

 kind of confocal quadrics — or are the intersections of two families and 

 orthogonal to the third. By supposing the hyperboloid of one sheet to 

 degrade into the space outside the focal ellipse we get the solution of the 

 equation of continuity for fluid flowing through an elliptic hole in a 

 plane. 



Fluid ElUpsoid and Sphere under their oion Attractions. — The problem 

 of the ellipsoidal forms of equilibrium of a rotating fluid, under the attrac- 

 tion of its own particles, is naturally the next object for consideration. 

 Since Maclaurin's discovery of the spheroidal form of equilibrium, and 

 Laplace's discussion of it, little seems to have been done until Jacobi 

 announced to the French Academy, in 1834, that particular ellipsoids, 

 with three unequal axes, could also be forms of equilibrium for fluid rota- 

 ting about the least axis. The fact being discovered, several proofs were 

 given by different writers, Liouville,"* Ivory, '^ Pontecoulant,^ and others. 

 The first to discuss the case with any fulness was C. O. Meyer,'' of 

 Koenigsberg, who set himself to do for Jacobi's case what Laplace had 

 done for Maclaurin's. If w be the angular velocity, and if the ratio of 

 w-/27r to the force between two unit volumes of the fluid at unit distance, be 



' ' Motion of liquid between two confocal ellipsoids,' Quart. Jour. xvi. p. 234. 



= For a simpler proof of this see a paper by the same author : ' Steady motion of 

 a top and of a solid of revolution moving in an infinite fluid,' Quart. Jour. xvii. p. 

 86 (1880). 



* ' On the motion of an ellipsoid in fluid,' Amer. Jour. Math. ii. p. 260. 



* Journal de Vecolo 'polytechniqiw, T. xiv. p. 289. 

 5 PUl. Trans, pt. i. for 1838, p. 57. 



^ Siixtrme dn 3Iondc, T. ii. , 



' ' l)e Aequilibrii formis ellipsoidicis, Crellc, xxiv. p. H. 



