54 ' REPORT — 1882. 



however, that the actions are the opposite of those of electricity and 

 magnetism — for instance, like oscillations attract — nor is it easy to see 

 how the rotatory effects of magnetism can be illustrated in this manner. 

 In his ' Vorlesnngen iiber mathematische Physik' (1876), Kirchhoff 

 has given a short treatment of the question of two moving spheres, and 

 this has been carried somewliat further by Lamb in his treatise on the 

 motion of fluids. The present writer' has also applied the theory of 

 images, referred to above, to the solution of the same problems, and has 

 attempted^ to sketch out an explanation of gravitation on Thomson's 

 vortex-atom theory of matter. When two spheres intersect at an angle 

 which is a submultiple of two right angles, the number of successive 

 images is finite, and the velocity-potential has a finite form. Stokes,^ in 

 the reprint of his papers, has worked out in detail the case when two 

 spheres cut at right angles. 1£ r,d; r', 6' • r,, 8^, be the polar co-ordinates 

 of any point referred to the centres of the spheres, and the middle point 

 of the common chord respectively, then the velocity-potential when they 

 move along the line of centres is — ^Y {a^ cos jr'^ — a%^ cos d^/ch^^ 

 + b^ cos 0' /)•'-}, c being the distance between the centres. In this case 

 the mass of the equivalent solid is 



1 Trpc-^ {4c3(«3 + P) -2(a^ + Z-6) - 3a%''(a + by} . 



The Ellipsoid. — The solution of the problem of the most general motion 

 of an ellipsoid in fluid is due to the successive labours of Green (1833), 

 Clebsch (1856), and Bjerknes (1873). To the first we owe the velocity- 

 potential for a motion of translation, to the second that for a motion of 

 rotation and the stream-lines both for translation and rotation, whilst 

 the third has given us the solution when the axes of the ellipsoid change 

 in any manner with the time. 



Green's^ paper was read in 1833. In this he finds the velocity 

 potential for translation only, and the effective momentum of the fluid. 

 In finding the effective momentum Green neglected the term in the ex- 

 pression for the pressure at a point which depends on the square of the 

 velocity, and he supposed, therefore, that his result was only true for 

 small vibrations. It was not till ten years later, when Stokes proved that 

 this term produces no effect on the i-esultant pressure on a single body in 

 an infinite fluid, that it could be seen that Green's value of this momentum 

 was rigorously true. His solution for the sphere has already been men- 

 tioned ; he also gave the analogous expressions for the spheroids. In 1835 

 Plana, in his before-mentioned paper, showed how to determine the velocity, 

 potential for a surface of revolution only slightly differing from a sphere 

 and moving parallel to its axis. Nothing more seems to have been done 

 for twenty- three years, until Clebsch's^ investigations were published in 

 1856, although the paper seems to have been finished in 1854. The first 

 part deals with the general theory of fluid motion, and has already been 

 referred to in the portion of this report presented to the Association last 

 year. Here we confine ourselves to his results bearing directly on the 



' ' On the motion of two spheres in a fluid,' Trans. Roy. Soc, pt. ii. p. 455 (1880). 



'^ ' On the problem of two pulsating spheres in a fluid,' Proc, Camb. Phil. Soc. iii. 

 p. 276, and iv. p. 29. 



3 ' On the resistance of a fluid to two oscillating spheres,' reprint, vol. i. p. 2.30 

 (1880). 



■• ' Researches on the vibration of pendulums in fluid media,' Trans. Eoy. Soc, Edin., 

 vol. xiii. ; also reprint, p. 315. 



* ♦ Ueber die Bewegung eipes Ellipsoids in einer Fliissigkeit,' Crelle, Hi. p. H9. 



