291-293] Ellipsoidal Harmonics 247 



We cannot solve this equation by methods of the kind used in developing 

 the theory of spherical harmonics, but it is easy to obtain solutions of limited 

 generality in which 



8 * 



7" ^ o > a/r ^ /-) ***vi ^ri- 



Z 9a 2 M 9/3 2 JV^ 



are rational integral functions of X, p and v respectively. These will be 

 found to correspond to the solution, in spherical polar coordinates, in a 

 series of rational integral harmonics. 



293. Assume general power series of the form 



-r -^r-T = A + B\ + C\ 2 + . . 



Li COL" 



L&M = A> B' C' 2 



N dy 2 



then on substitution in equation (211), it will be found that we must have 



A" = A' =A, 



... . 



So that we must have 



^ = (A+B\)L (212), 



c/cr 



and similar equations, with the same constants A and B, must be satisfied 

 by M and N. 



Equation (212), on substituting for a in terms of X, becomes 



a differential equation of the second order in X, while M and ^V" satisfy 

 equations which are identical except that //, and v are the variables. 



The solution of equation (213) is known as a Lame"s function, or ellip- 

 soidal harmonic. The function is commonly written as E%(\), where p, n 

 are new arbitrary constants, connected with the constants A and B by the 

 relations 



n(n+l) = B, and(6 2 + 



Thus E%(\) is a solution of 



and a solution of equation (211) is 



p n 



