640 Scientific Proceedings, Royal Dublin Society. 



Hence, if the density of M be uniform, the density at any point 

 of 31' will vary inversely as the fifth power of its distance from 0. 

 So also, if M be a uniform superficial distribution, M' will be a 

 superficial distribution of density varying inversely as the cube of 

 the distance from 0. 



It may be remarked that if, in equation (3), p varies inversely 

 as the fifth power of r, then p' is uniform ; that is, if M' be inverted 

 from the inverse mass will be of uniform density. However, if 

 any other origin 0' be taken, and if M ' be inverted with respect to 

 0' and a radius a' , then the density p" of this second derived dis- 

 tribution will be given by the equation 



_fa gay 



where 0" is the inverse of with respect to O '. Hence the density 

 of M", the inverse of M with respect to 0', varies inversely as the 

 fifth power of the point from 0", the inverse of 0'. 



Or a body, the density of which varies inversely as the fifth power 

 of the distance from a fixed point inverts into a body of density, vary- 

 ing inversely as the fifth poiver of the distance from the inverse point. 



Again, for the total mass of M' we have 



M = 



dm' = I - dm = a 

 r 



[dm 



— = aV , (4) 



r 



where V is the potential of the mass M at the origin 0. 



It is also easily seen that V, the potential of JK at any point A, 

 is connected with V ' , the potential of M! at the inverse point A!, 

 by the equation. (Thomson and Tait, Natural Philosophy, Part n. 

 Art. 516). 



OA 



V'=—V. (5) 



a 



Case of a Oentrobaric Body. 



If the mass 3f be centrobaric, that is, such that it attracts any 

 other portion of matter as if all its mass were collected at C } its 



