ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 419 



Application of the preceding General Theory to the Determination of the 



Attractions of Ellipsoids. 



13. Suppose it is required to determine the attractions exerted by 

 an ellipsoid whose semi-axes are a', b', c' whether the attracted point 

 p is situated within the ellipsoid or not, the law of the attraction being 

 inversely as the w"*" power of the distance. Then it is well known 

 that the required attractions may always be deduced from the function 



j^ _ r p' dx' dy' dx 



{{x ~x'f + {y-yj + {x-%jy^ ' 



p being the density of the element dx' dy' d%' of the ellipsoid, and 

 X, y, % being the rectangular co-ordinates of p. 



We may avoid the breach of the law of continuity which takes 

 place in the value of V, when the point p passes from the interior of 

 the ellipsoid into the exterior space, by adding the positive quantity 

 M* to that inclosed in the braces, and may afterwards suppose u eva- 

 nescent in the final result. Let us therefore now consider the function. 



r=/ 



p' dx' dy' d%' 



{{X - x'y + (y- y'y + (z- zy + M^p ' ' 



this triple integral like the preceding including all the values of x', tf, »', 

 admitted by the condition 



,/2 ^-^ 



— + — + — Z 1 



If now we suppose the density /o' is of the form 



f^'^i^-T^^-h-z^ ' /(^',y.«') (34). . 



which will simplify / {x', y, »') 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. 



Vol. V. Part III. 3 1 



