62 EBPOST — 1882. 



foregoing investigations, but gives approximations when the eftcentricitiea 

 are very small or very large. The motion of a sphere under its own 

 attraction vrhen slightly deformed, according to a spherical harmonic of 

 order n, is the subject of a paper by Thomson.' The time of oscillation 

 is shown to be 2ir\/ {(2n + l)a/2«(ii — 1)</}, where g is the acceleration 

 at the surface produced by the gravitation. This is independent of the 

 size, and depends only on the density of the sphere. The forced oscilla- 

 tions of liquid spheres have been shortly treated by G. H. Darwin ^ in a 

 similar manner. 



All the foregoing go upon the supposition that the density throughout 

 the mass is constant, but this is not the case with the planetary masses, 

 at least with the earth. This led Betti^ to take up the question of the 

 equilibrium of heterogeneous ellipsoids, the surfaces of equal density 

 being similar to the external surface. The investigation is not pressed to 

 qualitative results, and is chiefly of mathematical interest. 



Other Surfaces. — When the meridian curve of a surface of revolution 

 can be expressed in the form V = S cjr'^ = 1 where the r denotes the dis- 

 tances of a point from each of a set of fixed points on the axis, the 

 velocity-potential for a motion of translation pai-allel to the axis is easily 

 wi'itteu down. The solution is duo to Hoppe,"* and takes the simple 

 forms (^ — X [x + \ '^ c{x — a)jr^} and i// = p^ (1 _ y)^ in which X is a 

 constant, a the distance from the origin of the point from which r is 

 measured, and p the distance of the variable point from the axis. 

 He has drawn figures of the lines of flow for the particular case 

 81 /r^ — 16 /r'^ = 1. Another sui-face of revolution is that formed when 

 a circle rotates about a line in its own plane. An investigation of this, 

 based on notes taken at a course of lectures delivered by Riemann, has 

 been published by Godecker,-^ the velocity-potential being obtained as 

 an infinite series. The velocity-potential when the ring moves per- 

 pendicularly to its plane was given independently by myself" in 1881. 

 For an infinitely small wire of any foi'm with cyclic motion through the 

 opening the sokition flows at once from Helmholtz's theory of the vortex 

 filament. This has been treated of by Kirchhoff ^ and Boltzmann.^ 



c. Viscous Fluid. 



Motion in Tubes and Canals. — Naturally the first problem to which 

 the equations of viscous motion were applied was that of the flow of 



1 ' Oscillations of a liquid sphere," PMl. Trans. 15.3 (1863) p. 608. 



' ' On problems connected with the tides of a viscous spheroid,' PMl. Trani. 

 Part. II. 1879, p. 585. 



' ' Sopra i moti che conservando la fignra ellissoidale a una massa fluida etero- 

 genea,' Annali di Matem. (2) X. p. 173 (1881). 



■* ' Vom Widerstande der Fltissigkeiten gegen die Bewegung fester Kiirper,' 

 -P()^,9. ^7)«. xciii. 1854. 'Determination of the motion of conoidal bodies through 

 an incompressible fluid,' Quart. Jour. i. p. 301. 



^ ' Die Bewegung eines kreisformigen Einges in einer unendlichen incompresBiblen 

 Fliissigkeit,' Pr. Gdttiiiffeii, 1870. 



< On Toroidal Functions,' Trans. Bay. Soc. Part. III. 1881, p. 609. 



' ' Ueber die Kriltte, welche zwei unendlich diinne, starre Kinge in einer Flussig- 

 keit scheinbar auf einander ausiiben konnen,' Crelle, Ixxi. and Keprint, p. 404. 



' ' Ueber die Druckkriifte, welche auf Pdnge wirksam sind, die in bewegte Fliis- 

 sigkeit tauchen,' Crelle, Ixxiii. p. 111. 



See Part I. of this report, p. 74. 



