272 On Solids of greatest Attrctction, or Repulsion. 



radius r,and the attracted point and origin of the coordiriafes 

 to be at its centre. Then »^= f (x, yY= r^—x'^ — y^, and 



the equation of the curve becomes a:=C (x*+ y^) : which 

 belongs to a circle having the attracted point at the ex- 

 tremity of its diameter. So that the portion, cut out by 

 the cylinder, in Viviani's celebrated problem, is a solid of 

 greatest attraction of this kind. 



jEx. 3. The surface still being a sphere, if the attracted 

 point, instead of being at the centre, is at the extremity of 

 a diameter, which is also the axis of x, we shall have 

 z~=: i' {x,yy = 2rx—x^—y^; whence the equation of the 



X ' 



curve IS r= + C = 0, or a: = C (x^+ y')*; which 



(a-* + y-) tjlrx^ 



is the same curve which generates, by its revolution, the 

 solid of greatest attraction, when the force is inversely as 

 the cube of the distance. Vide Ed. Trans, vol. vi. p. 203. 

 In the first proposition, the mtLSS of the solid was sup- 

 posed to be given, in which case it appeared that the figure 

 of that solid is independent of the density: but if it is the 

 volume, instead of the mass, that is given, the ease will be 

 different. 



Prop. 4. 



It is required to solve Prop. 1. but with this difference^ 

 that now the volume ( fff^ V ^jf ^"'^ "^o^ *^^ mass, is 



supposed given. 



We find immediately, for the equation of the solid of 

 greatest attraction, 



; : . — -[- K^ ^ O. 



_ (j;- + y" + ^-)' 



If the density is constant, this, of course, enters into the 

 first proposition ; bm, in other cases, there may be an in- 

 finite variety of functions of the distance which represent- 

 ing the law of force, will give the same solid of greatest 

 attraction J provided a suitable density be supposed. 



Ex. 1. What laws of force and density will give the 

 solid of greatest attraction, of this kind, a sphere? It is 

 plain we have only to satisfy the equation (^{x, y, z) x 



A [x, y, z) = r. Thus, if the force is to be a 



function of the distance, we may make c^ {x, y, z) = 

 -^ i r, A{x,y,z) =f(a;H2/' + »')i or the 



reverse. 



