270 On Solids of greatest Attraction, or ReputsioTi. 

 value the equation becomes ax— a;^= — (y* + 2*), which 

 belonofs to an ellipsoid of revolution, or a sphere if d=:a„ 

 Again, if ^{x,y,z) - -f^l^jr^^^ we find the solid 



to be a sphere. 



Prop. 2. 



It is required to solve the reverse problem, or to find the 

 force when the figure of the solid of greatest aitraclion is 

 given. _______ 



By equation (l) x— -^^f^'''^—- Suppose then, that^ 

 from the nature of the given solid, a:= F(.r,y, z), we have 

 by making these values equal F[x, y,z) = ^ "^ ^ '^ ' 



^l-Tj y, »j 



whence, <J;(a;, y, z) - — ^^r^^^- 



Cor. It is plain that we may give F{x,y,z) a variety 

 of forms for the same solid ; and, consequently, that there 

 may be various laws of force, ^(a;, y, x), which give the 

 same solid for the solid of greatest attraction. 



If the solid be of revolution, there will be one of these 

 laws of force which is a function of the distance* ; and to 

 the finding this law I shall confine myself in the following 

 examples. 



Ex. 1. What function of the distance must the law of 

 force be, when the solid of greatest altraction is an ellip- 

 soid of revolution, whose axis of revolution coincides with 

 that of X, and terminates al the attracted point m ? 



Let a be this axis, the other, the equation of the super- 

 ficies is -Y-(ax—x*) = y^ + 2'; or, putting D^=x^+y* + 2:% 



!^Jax-x^)+x^-=D'-, whence x=aY/^^^^,+ ^ - 

 T{x, y, z) by substituting which we find the 



ab^- 



force; or, $(jr, y, z) 

 I) 



"V 47^11:];^. +^ziF-5(;^^=7o V ^"""t^—^'-'- 

 if the spheroid be oblong, this law of force is is always 

 possible. 



* Not, liowever, always posiible, as wc shall see. 



Ex, 



