56 BEPOKT — 1882. 



ellipsoid always remains similar to itself, the potential takes the very 

 simple form of 



The effective mass and the effective moment of inertia have been given 

 by Green and Clebsch respectively in not very comjDlicated forms, but it 

 does not seem worth while to reproduce them here. The function il also 

 serves to determine the stream-lines for a spheroid moving parallel to its 

 axis. They are given by Kirchhoff in his ' Vorlesungen ' in the form 



< C — +|-p > = const, (p.a;) being the cylindrical co-ordinates of any 



point, and the other equation being given by any plane through the axis. 



When the space in question is that inside an ellipsoid, the functions 



become extremely simple. For translation the velocity is, of course, a 



linear function of the rectangular co-ordinates, whilst for rotation about 



J2 g2 



the axis (a) the velocity-potential is given by ^ = w— -yz. "When the 



0" ~\~ c 



axes change, the fluid being incompressible, the volume must remain 



unaltered, or dja + Ijh -\- cjc = o. For this motion Bjerknes has shown 



that 



\a b c J 



He has also shown that if we suppose the density to change with the time 

 alone, yet so as to preserve the same mass of fluid within the ellipsoid, we 



may dispense with the condition S cl/a = o. If now E = 1 



2 



'-\ 



a,2 a.^ 



and D^ = n ( 1 + — 5 )) all the above results still hold for 11 variables. 



I have not been able to discover who first determined the potential 

 for the internal motion when the boundary rotates. It was given by 

 Bjerknes,' Beltrami,^ and Clerk Maxwell,^ all in 1873, but I believe tho 

 results must have been known before. Maxwell set it in a fellowship 

 examination at Trinity College, Cambridge, with a rider, that after a 

 certain number of revolutions, all the liquid particles would occupy the 

 same positions relatively to the boundary. 



Several other writers have discussed the motion of the ellipsoid, but 

 their work has either been based on that of Green or Clebsch, or their 

 results have been developed anew. A short notice, in order of the several 

 papers, will therefore suffice here. Ferrers ■• (1875) deduces the velocity- 

 potential for translation and rotation, and finds the vis viva of the fluid 

 motion by showing that the velocity-potential for a point just outside the 

 surface, bears a constant ratio to that just inside. This is a valuable re- 

 mark, and shortens the calculation very much, for since the normal motion 

 is the same in both cases, it follows that the energy of the motion outside 

 has the same ratio to that within. It is seen at once how this enables 

 us to deduce immediately the energy for translation. Sharpe^ (1876) 



' See above. = S\d Principii, <5'c., § 26. 3Iem. di Boloffiia, iii. 



' The question is given in the last edition of Besant's Hydromechauics amongst 

 the examples. 



•* ' On the motion of an infinite mass of water about a moving ellipsoid,' 

 Quart. Joiir. xiii. p. 3.30. 



'■• ' On fluid motion,' Mens. Math. v. p. 12j"). 



