OO EEPORX— 1882, 



case of a liquid, and he showed that the effect of an incompressible fluid 

 is to increase the inertia of the sphere by one-half the mass of the fluid 

 displaced by the sphere. His investigation was published in 1832, and, 

 the year after, Green read a paper on the same subject before the Royal 

 Society of Edinburgh, in which he considered the ball of the pendulum to 

 be ellipsoidal. His investigation was carried out without a knowledge of 

 Poisson's work, but he gives the same result as to the effect of the sur- 

 rounding fluid. In 1835 Plana • also took up the subject in a long paper 

 of 166 pages, in which he attempted to take account of the friction, and 

 of small differences from the spherical form of the pendulum. He con- 

 sidered the friction on the spherical surface to be proportional to the 

 relative motion of the fluid over the surface, and that this relative motion 

 was the same as if there were no friction. For frictionless fluid his 

 results agree with Poisson's, as do Stokes^ deduced in 1843 as a special 

 case of a more general one. 



In 1852 the same problem was solved by Dirichlet ' independently. 

 He found, in addition, the equation to the stream lines. The chief im- 

 portance of this paper lies in the impulse it gave in Germany to the study 

 of hydrodynamics, forming as it does the first of a series of important 

 papers by himself, Clebsch, Riemann, Helmholtz, and Kirchhoff . Clebsch * 

 (1856) also gives the stream lines as a particular case of those for the 

 ellipsoid, and discusses with more detail the motion of a spherical ball- 

 pendulum. Amongst new results may be mentioned the equations to the 

 paths of particles, the co-ordinates being expressed very elegantly in 

 terms of elliptic functions of one arbitrary parameter. The motion of a 

 sphere in fluid when its centre of gravity is eccentric is the subject of a 

 paper by G. J. Michaelis.^ 



It is well known how Thomson discovered that the electrification 

 induced on a sphere by a quantity of electricity at a point outside it, 

 produces the same effect on an external point as another portion of elec- 

 tricity at the optical image of the first, and how from this he developed 

 his theory of electric images. This theory suggested to Stokes to search 

 for an analogous theorem in fluid motion, and he found ^ that a doublet 

 (n) outside a sphere, with its axis directed to the centre of the sphere, 

 has an image also at the inverse point, whose magnitude is — fia^jr^, 

 \7here a is the radius of the sphere and r the distance of the external 

 doublet from the centre. The importance of this lies in the fact that the 

 motion of a sphere produces the same motion in the fluid as a doublet at 

 its centre, and thus it gives the means of solving the case of two spheres 

 moving along the line joining their centres. The general case, of which 

 the preceding is a particular instance, for the image of a source of fluid 



1 ' Memoire sur le mouvement d'uu pendule dans un milieu resistant,' il/t'»t. d. r . 

 Ace. di Sc. Turin, xxxviii. p. 209. 



'^ On some cases, &c., see below. 



» Monatsber. d. herl. Ahad. 1852- 



It is curious how, even down to the present moment, Dirichlet is regarded on 

 the Continent as the first investigator in this region, and how the work of Green and 

 ycokes is ignored. 



•• ' Ueber die Bewegung eines Ellipsoids in eiaer tropfbaren Fliissigkeit,' Crelle, 

 lii. p. 103 (1856). 



s ' Over eenige gevallen van beweging in eene onsamendrukbare vloeislof,' 

 Nicem. Arch. iii. p. 163. 



6 Brit. Ass. Ec]). 1817, ii. p. 6. ' On the resistance of a fluid to two oBoillating 

 spheres.' Be^rint, vol. i. p. 230. 



