100 102.] SPECIAL CASES OF MOTION. 113 



These two values of -& agree provided we put % t = Ayz, and make 



We have then, finally, 



vy -f wz 



Example 2. A cavity in the form of a rectangular parallel 

 epiped. If the axes of co-ordinates be taken parallel to the edges 

 of the cavity, it is plain that the conditions (3) are satisfied by 

 making ^ a function of y, z only, &c., -so that the problem be 

 comes one of two dimensions. For the complete solution, effected 

 by means of Fourier s series, we refer the student to Stokes*, or 

 to Thomson and Taitf. 



Example 3. An ellipsoid moves ia an infinite mass of liquid 

 which is at rest at infinity. 



This problem, the only one of its class which has been com 

 pletely worked out, was solved by Green J in 1833, for the par 

 ticular case where the motion of the ellipsoid is one of pure 

 translation. The complete solution was published by Clebsch, 

 in 1856; it is here reproduced much in the form given to it by 

 Kirchhoff j . 



The principal axes of the ellipsoid being taken as axes of co 

 ordinates, let the component attractions which would be exerted 

 at the point (x, y, z] by an ellipsoid of unit density, coincident in 

 shape, size, and position with the given one, be denoted by X, Y, 

 Z. It is known that X is the potential of the ellipsoid when 

 magnetized uniformly with unit intensity parallel to x negative, 

 and therefore that it is the potential of a distribution of matter, 



7 -rr 



of surface-density L over the surfaced. Hence -/- is discon- 



dn 



* Camb. Phil. Trans., Vol. vni., pp. 131, 409. 

 t Natural Philosophy, Art. 707 (B). 



J Researches on the Yibratien of Pendulums in Fluid Media, Trans. R.S. 

 Edin. 1833. Reprinted in Mr Ferrers edition of Green s works, pp. 315 et seq. 

 Crelle, it. 52, 53 (18567). 



II Vorlesungen iiber Math. Physik. Median ik. , c. 18. 

 11 See, as to these points, Maxwell, Electricity and Maguetiam, Art. 437. 



r, 8 



