On Solids of greatest Attraction, or Repulsion. 271 



Ex. 2. What function of the distance must the law of 

 force be, when the solid is half of an ellipsoid of revolution, 

 with its centre at the point m, and its axis of revolution 

 ci)inciding with that of a;? 



The equation of the solid now (if a and h denote the 



m 



halves of what they did before) is -^(a^— x-)=y^ + a;'; or, 

 — (a*— x'-) + a?" = D^; whence, x= ■ —¥{x,y,z)j 



and the required force, or ^(x, y, s) a — • 



If the spheroid be oblate, D is always less than h, and 

 the law of force here found will be impossible: or, in 

 other word>;, no force, which is a function of the distance, 

 will have the oliule spheroid for a solid of greatest attrac- 

 tion, with respect to a point at its centre. 



But a portion of an ohlovg spheroid thus situated, may 

 be a solid of gieatest attraction, with this law of force, 

 provided there be a distance greater than h between the 

 solid and attracted point. 



Prop. 3. 



The force being inversely as the square of the distance, 

 what must be the base of a homogeneous cylinder erected 

 perpendicularly on the plane of x and y, and intercepted, 

 above this plane, by a given curve surface, in order that 

 the intercepted cylindric portion may exercise the greatest 

 poisible attraction, on a point m at the origin of the co- 

 ordinates, in the direction of x; its mass being given ? 



The attraction is // '-^ rj the mass is 



rfz y X ; therefore, if a; = f (x, y) express the nature of 



the given surface, the following expression must be a maxi- 

 mum; viz. 



rr -^-y-'l^ H- C /7f (x, y) yx; so that 



J J (^' + y^'^' + 2/'- + f {x, y"-)^ '^•^ 



the equation of the required curve, bounding the base, is 



-r^, + C = 0. 



Ex. 1. Let f (x, ?/) be fa), a function of x only, and 

 we get the same result as in Prop. 34 of a late paper in the 

 Phil, Trans, which is only a particular case of this. 



JL'x. 2. Suppose the intercepting surface to be a sphere, to 



radius 



