250 Methods for the Solution of Special Problems [CH. vm 



SPHEROIDAL HARMONICS. 



296. When any two semi-axes of the standard ellipsoid become equal 

 the method of confocal coordinates breaks down. For the equation 



reduces to a quadratic, and has therefore only two roots, say X, jj,. The 

 surfaces X = cons, and /u, = cons, are now confocal ellipsoids and hyperboloids 

 of revolution, but obviously a third family of surfaces is required before the 

 position of a point can be fixed. Such a family of surfaces, orthogonal to 

 the two present families, is supplied by the system of diametral planes 

 through the axis of revolution of the standard ellipsoid. 



The two cases in which the standard ellipsoid is a prolate spheroid and 

 an oblate spheroid require separate examination. 



Prolate Spheroids. 

 297. Let the standard surface be the prolate spheroid 



* +> " 



a 2 + 6 2 

 in which a>b. If we write 



y = OT COS (j), 2 = 1& Sin (j), 



then the curvilinear coordinates may be taken to be X, /*, <, where X, //, are 

 the roots of 



In this equation, put a 2 6 2 = c 2 and a 2 -J- 6 c 2 0' 2 , then the equation 

 becomes 



~ 



If 2 , rf are the roots of this equation in 6"*, we readily find that # 2 

 so that we may take 



a? = cfiy ........................... . .............. (219), 



p)(^-l) ........................ (220) 



in which 77 is taken to be the greater of the two roots. 



The surfaces f = cons., 77 = cons., are identical with the surfaces 6 = cons., 

 and are accordingly confocal ellipsoids and hyperboloids. The coordinates 

 f , 77, <f> may now be taken to be orthogonal curvilinear coordinates. 



