424 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Therefore 



64) 



65) y = 



(a'-& 2 )2 



..! 



In all these formulae, 6 is the larger root of the quadratic 



66) _-L + __!, 



for an outside point, and 6 = for an inside point. In the latter 

 case, we have functions only of the ratio -v-- 1 ) 



161. Development of Potential of Ellipsoid of Revolution. 



We may develop the expression 58) for the potential of an oblate 

 ellipsoid of revolution in a series of spherical harmonics. Considering 

 first a point on the axis of revolution, let us put 



x = r > a, y = 0, 

 so that we have by 66), 61), 



67) -^- = 1, J = 



2 -|-(7 



and using the tan" 1 instead of sin" 1 in 58) and 60), we obtain 



, 



2 tan~ ] 

 2 



~ Q x Jr 

 68) Vk = 



Remarking that M the mass of the ellipsoid is -nab 2 , and 

 developing the antitangent, we have, if r > a > ]/6 2 a 2 , 6 2 < 



3M 



= 3 M I ^, (- 1)" (fe 2 - ay ^, (- 1)* (& 2 - a 2 )" 

 2 }^/2w + l r 2w + A ^-/2n + 3 



1) Thomson and Tait, Natural Philosophy, Part H, 527. 



