204? J. B. on the Attraction of an Homogeneotis Ellipsoid. 



the same density, whose semiaxes are «, h^ c and a, /3, y re- 

 spectively, such that a^—c^ = a^-— y* = F and c^^b^ = y^ 

 — /3- = /i^; and let two particles be assumed, p and -cr, whose 

 coordinates are /a, ?w /S, w y, and la^mb, nc\ l^-\-m^-\- n^ being 

 = 1, it can be easily shown that the particles jf? and -crare on 

 the two ellipsoids. Calling the external ellipsoid E and the 

 internal one I, then the attraction of E on -sr is to the at- 

 traction of I on jD, in a direction perpendicular to one of the 

 principal planes, as the product of the axes of E in that plane 

 is to the product of the axes of I in the same plane ; which is 

 the celebrated theorem given by Mr. Ivory. 



First, let the particles p and txr be situated at the surfaces 

 in the axes of y and c, then Z = 0, m = 0, and « = 1, and the 

 attraction of I on p is by the formula as 



4^/g/;cf, 3 ( 2c'-a^-¥ y^'\ 

 3y^ f'^loV y" q 



{i+ To (-/-)}> 



p is as 

 ""37 |^"^10^ y^ if 



" " 3y^ Y'^ loV y^ J I' 



Hence the attraction of I on ^ is to the attraction of E on 

 the same particle p as abc: oL^y. Now, if through the in- 

 ternal point -cT an ellipsoid similar to the external E concen- 

 tric, and similarly situated, be drawn, the attraction of E on 

 p : the attraction of E on -cr : : y : c ; therefore the attraction of 

 1 onpi the attraction of E on tzr : : « Z> : a |3. 



Now, if the particles be not at the extremities of c and y, let 

 their coordinates to the plane o^ xy he.nc and ny, and let the 

 particles be^' and ts' ; then the attraction of E on the particles 

 p and p^ is as their perpendicular distance from a principal 

 plane as y and n y : i \ : n. In the same way the attraction of 

 I on -cr and -or' : : c : w c : : 1 : w. Hence attraction of I on p' : 

 attraction of E on -or' ::« 6 : a |3. 



6. Should the ellipsoid become an oblate spheroid a = c, 

 and let a^—b'^ 



_ ^'Kfabc 

 ~ 3y^ 

 and the attraction of E on p is as 





{'+i^4^(— v)f 



7. When cos'^ ju. = -— , the attraction is the same as that of 



