252 



Ellipsoidal Harmonic Analysis. 



for changing S into It may be well to note that there is no sine- 

 function when s is zero. 



The use of these factors does much to facilitate the laborious reduc- 

 tions involved in the whole investigation. 



It is well known that the Q-functions are expressible in terms of the 

 P-functions by means of a definite integral. Hence (S/ and Qj* must 

 have a second form, which can only differ from the other by a con- 

 stant factor. The factor in question is determined in the paper. 



It is easy to form a function continuous at the surface v which 

 shall be a solid harmonic both for external and for internal space. 

 Poisson's equation then gives the surface density of which this con- 

 tinuous function is the potential, and it is found to be a surface 

 harmonic of /x, <f> multiplied by the perpendicular on to the tangent 

 plane. 



This result may obviously be employed in determining the potential 

 of an harmonic deformation of a solid ellipsoid. 



The potential of the solid ellipsoid itself may be found by the con- 

 sideration that it is externally equal to that of a focaloid shell of the 

 same mass. It appears that in order to express the equivalent surface 

 density in surface harmonics it is only necessary to express the 

 reciprocal of the square of the perpendicular on to the tangent plane in 

 that form. This result is attained by expressing x 2 , y 2 , z 2 in surface 

 harmonics. When this is done an application of the preceding theorem 

 enables us to write down the external potential of the solid ellipsoid 

 at once. 



Since x 2 , y 2 , z 2 have been found in surface harmonics, we can also 

 write down a rotation potential about any one of the three axes in the 

 same form. 



The internal potential of a solid ellipsoid does not lend itself well to 

 elliptic co-ordinates, but expressions for it are given. 



If it be desired to express any arbitrary function of in surface 

 harmonics, it is necessary to know the integrals, over the surface of 

 the ellipsoid, of the squares of the several surface harmonics, each 

 multiplied by the perpendicular on to the tangent plane. The rest of 

 the paper is devoted to the evaluation of these integrals. No attempt 

 is made to carry the developments beyond /3 2 , although the methods 

 employed would render it possible to do so. 



The necessary analysis is difficult, but the results for all orders and 

 degrees are finally obtained. 



