Preston — On Centrobaric Bodies. 641 



centre of inertia, we have for the potential of M at A 



„ M 

 V= — . 

 AC 



Therefore, by equation (5) the potential V of M' at A' is given by 



V-— — = — - - a ' M 



a 'AC~A'Ca~A'C'.OC, 



where C is the inverse of C. But M/OC is the potential of if at 

 ; therefore by (4) we have 



V' = 



A' a 



that is, the potential of M' at any point A! is the same as if all its 

 mass were concentrated at C, the inverse of C. Hence we have 

 the following 



Theorem. 



If any mass M be centrobaric, the inverse mass M' is also centro- 

 baric, and the centre of mass of the latter is the inverse of the centre of 

 mass of the former. 



Thomson's Theorems. 



Since a uniform sphere is a centrobaric body, attracting any 

 external matter as if its mass were all concentrated at its centre, 

 we have at once the following theorems of Sir W. Thomson : — 



(1). A sphere, the density of which varies inversely as the 

 distance from a fixed point 0, attracts any other portion of matter 

 as if its mass were all collected at a certain point, viz. the inverse 

 of with respect to the sphere. 



[For the inverse of a uniform sphere is a sphere, the density of 

 which varies inversely as the fifth power of the distance from the 

 origin, and the inverse of its centre is the inverse of the origin with 

 respect to the inverse sphere.] 



(2). An infinitely thin shell, the density of which varies in- 

 versely as the cube of the distance from a fixed point 0, attracts 

 any other portion of matter as if the mass of the shell were all 

 concentrated at a point, viz. the inverse of with respect to the 

 shell. 



