111-112] TRANSLATION OF AN ELLIPSOID. 163 



The corresponding solution when the ellipsoid moves parallel 

 to y or z can be written down from symmetry, and by superposition 

 we derive the case where the ellipsoid has any motion of translation 

 whatever*. 



At a great distance from the origin, the formula (8) becomes 

 equivalent to 



which is the velocity-potential of a double source at the origin, of 

 strength J C, or 



f 6cu/(2 - o). 

 Compare Art. 91. 



The kinetic energy of the fluid is given by 



where I is the cosine of the angle which the normal to the surface 

 makes with the axis of x. The latter integral is equal to the 

 volume of the ellipsoid, whence 



C 



(11). 



The inertia-coefficient is therefore equal to the fraction /(2 OQ) 

 of the mass displaced by the solid. For the case of the sphere 

 (a=b =c) we find or = f ; this makes the fraction equal to J, in agree- 

 ment with Art. 91. If we put b = c, we get the case of an ellipsoid 

 of revolution, including (for a = 0) that of a circular disk. The 

 identification with the results obtained by the methods of Arts. 

 102, 103, 105, 106 for these cases may be left to the reader. 



112. We next inquire whether the equation V 2 </> = can 

 be satisfied by 



* This problem was first solved by Green, "Researches on the Vibration of 

 Pendulums in Fluid Media," Trans. R. S. Edin., 1833, Math. Papers, p. 315. The 

 investigation is much shortened if we assume at once from the Theory of Attrac- 

 tions that (8) is a solution of V 2 0. being in fact (save as to a constant factor) 

 the ^-component of the attraction of a homogeneous ellipsoid on an external 

 point. 



112 



