420 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Now by 73, 86), 



06 2x /I x* y* z* \ 



dx a* + 6/ j(a 2 +<>) 2 ~*~ (6 2 +<0 2 (c 2 -f<0 2 j 



Forming , and -^ and adding, 



QO 



C\ 1 1 1 I 



a 



The integration may be at once effected. 

 Since 



7/ s (du , dv , div] 



d(UVW)=UVW\- ->; 



, 1 u v w I 

 we nave 



45) d\ , l = =} 



(V(a* + u)(b*-\-u)(c* + u)l _ 



_ _ du _ f ya*+u } 



~ y(a* 4- u] (& 2 + u} (c 2 -f u) I 2 |/(a 2 -f u) s I 



M 1 i 1 I 1 1 du 



2 |a 2 +tt 6 2 + ^ c 2 + wjy^2^ w )( & 2_^ w ) 

 The integral becomes then 



which cancels the second term, and z/F=0. 

 For an internal point 



/* 



I - 



J ( 



At infinity tf = oo, and V and its derivatives accordingly vanish. 

 Therefore the value of V found satisfies all the conditions. 



159. Ivory's Theorem. If x, y, z is a point on the ellipsoid (1) 



4T ) $ + + # = 1, 



the point 



(%i Oj Cj 



lies on the ellipsoid (2) 



