1890.] 



On Ellipsoidal Harmonics. 



can be first expressed in terms of spherical harmonics in x', y', z', by 

 Laplace's expansion, and then in ellipsoidal harmonics in x, y, z. 



A sei'ies of ellipsoidal harmonics can thus be found having an 

 arbitrary value at the surface of the ellipsoid. 



5. External Harmonics. The leading proposition in this part of 

 the subject is as follows : 



If TrabcVn denote the potential at an outside point xyz due to a 

 solid ellipsoid, whose semi-axes are a, 6, c, such that the density at 

 any internal point fgli is of the form 



f g* 

 l<?~~b* 



then the harmonic of degree n and order a, suitable to the space 

 outside of the ellipsoid, is given by 



where 



r 



- J e (*i- 



X) 



This result may primarily be regarded as a means of reducing the 

 integral on the left-hand side, when the values of 6 are known, into 

 simpler forms, which can be actually evaluated when the surface is 

 one of revolution. It is a result of some importance in the subject, 

 as containing within itself the numerous expressions into which the 

 external harmonics of spheroids can be thrown. 



6. Spheroids. The foregoing formulae admit of easy reduction when 

 two of the axes of the ellipsoid are equal, say a b. It may then 

 be shown that the spherical harmonics H of 2 are the ordinary 

 spherical harmonic conjugate system. It is therefore convenient to 

 adopt the definitions and specifications in Thomson and Tait's 

 ' Natural Philosophy,' and thus to harmonise the spheroidal system 

 with the spherical. Accordingly, if H be now used to express a 

 spherical harmonic according to the definitions in that work, the new 

 signification of Gr will be in accordance with the relation in 3. 



Taking the results contained in 3, 5, and effecting reductions 

 suitable to the prolate spheroid, we obtain the following : 



cos aO dO, 



r" +1 -^P00 }^~ 



dp? 



where 



B 2 



