250 ]\I. Poinsot on the Percussion of Bodies. 



remains immoveable under the action of sucli forces, and consti- 

 tutes in reality what we call a fixed point. 



4. But if the fmxe of inertia or 7nass of the system M + /i is 

 infinite, the moment of inertia around an axis passing through 

 the centre g has a finite value which, as will be seen, is exactly 

 the same as the moment of inertia, with respect to the same axis, 

 of the simple body M. Hence, although the centre of gravity </, 

 in consequence of the infinite mass M + fi with which it is 

 chai-ged, remains immoveable under the direct action of all the 

 forces of the system removed, parallel to themselves, to the 

 point ff, the body will not remain immoveable under the action 

 of the couples which such a removal originates, but will receive 

 a finite rotation 6 around the centre (/, in consequence of the 

 finite value of its moment of inertia with respect to an axis 

 through this point (/. 



Hence to solve dynamical questions with respect to a body 

 compelled to turn around a fixed point, it will suffice to apply 

 the solutions already found for a free body, provided that in so 

 doing we regard the fixed point as the centi'c of gravity of the 

 body, suppose the mass of this body to be infinite, and give to 

 its moment of inertia the true finite value, which we shall now 

 proceed to determine. 



5. For greater clearness let us suppose, in the first place, that 

 this material point which wc attach at T to the proposed body M, 

 has only a certain finite mass /i, and let us seek the moment of 

 inertia of the system with respect to its centre of gravity g ; we 

 shall then see what the expression (/u. + ]M)K'^ of this moment 

 becomes when fj, is supposed to be infinite. 



6. Let G be the centre of gravity of the simple body M and 

 d the length of the line GI. The centre of gravity (/ cuts this line 

 IG into two segments, i and d—i, inversely proportional to the 

 masses M and /t, so that 



Now the moment of inertia of the material point /u, with respect 

 to the centre ff is evidently /xr ; and that of the body M with 

 respect to the same point is composed, Jirsf, of its moment of 

 inertia around its own centre of gravity G, which may be repre- 

 sented by MD-; and secondly, of the product M(f/— z)^ of the 

 mass of the body into the square of the distance [d—i] of its 

 own centre G from the point g. Adding these values, wc 

 have for the moment of inertia (ya + M)K- of the system the 

 value 



d 



