OF ELLIPSOIDS OF VARIABLE DENSITIES. 211 



Application of the preceding General Theory to the Determination 

 of the Attractions of Ellipsoids. 



13. Suppose it is required to determine the attractions ex- 

 erted by an ellipsoid whose semi-axes are a', &', c whether the 

 attracted point p is situated within the ellipsoid or not, the law 

 of the attraction being inversely as the n' th power of the distance. 

 Then it is well known that the required attractions may always 

 be deduced from the function 



~j< 



p'dx'dydz 



p being the density of the element dxdy'dz of the ellipsoid, 

 and x, y, z being the rectangular co-ordinates of p. 



We may avoid the breach of the law of continuity which 

 takes place in the value of F, when the pointy passes from the 

 interior of the ellipsoid into the exterior space, by adding the 

 positive quantity w 2 to that inclosed in the braces, and may after- 

 wards suppose u evanescent in the final result. Let us therefore 

 now consider the function 



pdxdydz 



_ f 



this triple integral like the preceding including all the values 

 of x' 9 y', z, admitted by the condition 



*1 y" z 



tF*V**'is* 



If now we suppose the density p is of the form 

 V 2 7/ 2 g' 2 \ w '- 2 



-i- / KV '^ (34)> 



which will simplify f(x, y , z'} when p is constant and n = 2, 

 and then compare this value with the one immediately deducible 

 from the general expression (28) by supposing for a moment 

 n = n, viz. 



142 



