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LVIL— ON THE INVERSION OF CENTROBARIO BODIES. 

 By THOMAS PRESTON, B.A. 



[Communicated by Professor Gr. F. Fitz Gerald.] 



[Eead, June 1, 1887.] 



If M be any figure, and a fixed origin, and if to each point P 

 of M a corresponding point P' be taken on the radius vector OP, 

 such that 



OP . OP' = a constant = a\ (1) 



a figure W is obtained, which is termed the inverse of M. 



Now if M be any continuous mass or any distribution of 

 matter, and if at P' an element of mass dm be placed, such that 



dm' = - dm, 



(2) 



where dm is the element of mass at P, and r = OP, we obtain a 

 distribution of matter M', the density p of which at any point P' 

 is connected with the density p of i!f at the corresponding point P, 

 by the equation 



P-\y)P 



P> 



(3) 



where / = OP'. This equation follows at once from the relation 

 do : dv' = r 6 : a 5 ; the elements of volume at P and P' being do 

 and dv. 



