hence 



M. Poinsot on the Percussion of Bodies. 257 



K'-^=K^ + i^=il-i^ + i^=il=P.^, 

 K'=I.^ and 2K'=/1, 



TT'=2K'* 



14. If in the formula? of art. 13 we assume fj, infinite with 

 respect to M, in order to pass to the mathematical hypothesis of 

 a fixed point in the rigid rod OC, which is loaded at C with a 

 massive point M, we shall find for the moment of inertia 

 (M + /ti)K'^ around the fixed point the value 



(M+ya)K'2=M/2; 



the same as the simple body M would give around the same 

 point. But the arm of inertia becomes 



K'2 

 though — T- represents a finite line equal to /. Thus, although 



K' is infinitely small, it is infinitely greater than i ; in other 

 words, the arm of inertia K' bears the same relation to the di- 

 stance i of /i from the centre of gravity G of the system, as the 

 sine of an infinitesimal arc does to its versed sine. 



When C is the centre of gravity of a mass M of any form, 

 fi remaining a inassive point, the moment of inertia of the 

 system of /i and M around its centre of gravity G is 



(M + ^)K2= 



"C^"'^)' 



D being the arm of inertia of the simple body M around its 

 centre of gravity C. Making /* infinite, in order to pass to the 

 hypothesis of a fixed point at jx, we deduce the moment of inertia 



which is the same as if the point ytt were annihilated. 



In conclusion, it results from what has been said of the motion 

 of a body M around a fixed point 0, that, as in the case of a free 

 body, the ordinary centre of percussion is not the centre of the 

 greatest percussion which the body is capable of producing 

 against a fixed point T, suddenly opposed to its actual motion. 



* This result is general, sinee from the two values of a'o the distance TT' 

 is always c(iual to 2 '^i7=2K'. 



Phil, May. S. 4. VjI. 18. No. 120. Oct. 1859. S 



