m Homogeneous Solid Bodies. ' 513 



which is readily shown^ by substituting for 0%^^ + a^c^ + ti^c^ its 

 equal {a%''^a^c^ + b^c') {X + 'L + ^j,tobe equal to ?-^.-» 



Hence the expression for ^, given above, becomes 



an 



da-^^'^' abc P' 

 which agrees with (c). 



Attraction of a Homogeneous Ellipsoid on a Point within or 

 without it. 



If in (c) we put k^ = — ?, the value of — j- at any point will 



be the attraction on that point of a shell bounded by two similar 

 concentric ellipsoids, whose semiaxes are 



«„ ay{\-e''), «,^(1-/^), 

 and 



«, + </«!, (ai + c?«,)^/(l-e^), («j + 6?a,)v'(l-e'2), 

 where 



and ffl«-c2 = fl,2 



the density of the shell being unity. Now this attraction is in 

 a normal drawn through the point attracted to the surface of 

 the ellipsoid whose semiaxes are a, b, c. If we call u, /3, 7 the 

 angles which this normal makes with the coordinates x, y, z of 

 the point attracted, we have 



X 



2_^ 2g2^ 



.'2=«,V2J' . . . . (1) 



v/(l 



_px 



+ F + ?) 



and similarly. 



o P^ P^ 



cos/3 = ^, cos7=<^. 



Hence, calling </A, </B, </C the components of the attraction 

 parallel to the axes of coordinates, we have, from (c), 



ic^bc ^ ' 



^^ = ^'^^^^^'''^'' 



b c 

 dC = 47rr —j-^p^da^ 



Phil. Mag. S. 4. No. 48. Suppl. Vol. 7. 2 M 



(2 



