Ellipsoidal Harmonic Analysis. 



249 



giving to the functions such forms as shall render numerical results 

 accessible. 



Throughout the work I have enjoyed the immense advantage of 

 frequent discussions with Mr. E. AV. Hobson, and I have to thank him 

 not only for many valuable suggestions but also for assistance in 

 obtaining various specific results. 



My object in attacking this problem was the hope of being thereby 

 enabled to obtain exact numerical results as to M. Poincare"s pear- 

 shaped figure of equilibrium of rotating liquid. But it soon became 

 clear that partial investigation with one particular object in view was 

 impracticable, and I was led on to cover the whole field, leaving the 

 consideration of the particular problem to some future occasion. 



The usual symmetrical forms of the three functions whose product is 

 a solid ellipsoidal harmonic are such as to render purely analytical 

 investigations both elegant and convenient. But it seemed that 

 facility for computation might be gained by the surrender of sym- 

 metry, and this idea is followed out in the paper. 



The success attained in the use of spheroidal analysis suggested 

 that it should be taken as the point of departure for the treatment 

 of ellipsoids with three unequal axes. In spheroidal harmonics we 

 start with a fundamental prolate ellipsoid of revolution, with imaginary 

 semi- axes k J - 1, k J - 1, 0. The position of a point is then defined 

 by three co-ordinates ; the first of these, v, is such that its reciprocal is 

 the eccentricity of a meridional section of an ellipsoid confocal with 

 the fundamental ellipsoid and passing through the point. Since that 

 eccentricity diminishes as we recede from the origin, v plays the 

 part of a reciprocal to the radius vector. The second co-ordinate, /x, 

 is the cosine of the auxiliary angle in the meridional ellipse measured 

 from the axis of symmetry. It therefore plays the part of sine of 

 latitude. The third co-ordinate is simply the longitude <£. The 

 three co-ordinates may then be described as the radial, latitudinal, 

 and longitudinal co-ordinates. The parameter k defines the absolute 

 scale on which the figure is drawn. 



It is equally possible to start with a fundamental oblate ellipsoid 

 with real semi-axes k, k, 0. We should then take the first co-ordinate, 

 £, as such that £ 2 = - v 2 . All that follows would then be equally 

 applicable ; but in order not to complicate the statement by continual 

 reference to alternate forms, the first form is taken as a standard. 



In the paper a closely parallel notation is adopted for the ellipsoid 

 of three unequal axes. The squares of the semi-axes of the funda- 

 mental ellipsoid are taken to be - k 2 \^ , - k 2 , 0, and the' three 



co-ordinates are still v, <£. As before, we might equally well start 

 with a fundamental ellipsoid whose squares of semi-axes are 



£2L±i, h\ 0, and replace v 2 by £ 2 where £ 2 = - v 2 . All possible 



