106-107] ELLIPSOID WITH UNEQUAL AXES. 155 



107. In questions relating to ellipsoids with three unequal 

 axes we may use the method of Lamp's Functions*, or, as they 

 are now often called, ' Ellipsoidal Harmonics.' Without attempting 

 a complete account of this, we will investigate some solutions of 

 the equation 



V 2 = (1), 



in ellipsoidal coordinates, which are analogous to spherical 

 harmonics of the first and second orders, with a view to their 

 hydrodynamical applications. It is convenient to begin with the 

 motion of a liquid contained in an ellipsoidal envelope, which can 

 be treated at once by Cartesian methods. 



Thus when the envelope is in motion parallel to the axis of x 

 with velocity u, the enclosed fluid moves as a solid, and the velocity- 

 potential is simply <f> = u#. 



Next let us suppose that the envelope is rotating about a 

 principal axis (say that of x) with angular velocity p. The 

 equation of the surface being 



/p2 y2 ^2 



Q? ff* C 2 ~ ^ '' 



the surface condition is 



x d(f> y dcj) z d<f> 

 a 2 dx 6 2 dy c 2 dz 



We therefore assume cf> = Ayz, which is evidently a solution of (1), 

 and obtain 





Hence, if the centre be moving with a velocity whose com- 



* See, for example, Ferrers, Spherical Harmonics, c. vi.; W. D. Niven, "On 

 Ellipsoidal Harmonics," Phil. Trans. , 1891, A. 



