858 M. Poinsot on the Percussion of Bodies. 



give 



Here also the first factor Ott' is the velocity of the point D', and 

 the other factor is the fraction 



of the body's whole mass ?n ; but this and the preceding fraction 

 together amount to unity, and by the foregoing relations their ratio 

 «^/3^ : {ot^x'^ + jQV^) is the same as that of x' to x or y' to y, and con- 

 sequently also of x^a/^ + y'^^ to \^x^ + y^; in short, the ratio in 

 question is the inverse of that of the distances u and u' of the 

 two points D and D' from the centre of gravity G. 



AVe may say, therefore, that during the movement of the body 

 the two reciprocal points D and D' divide, as it were, the whole 

 mass m into two parts //. and fu!, inversely proportional to their 

 distances from the centre of gravity G ; and, further, that the 

 percussions Q and Q', which these two points are capable of pro- 

 ducing, are the same as if these portions of the whole mass were 

 respectively concentrated therein. 



If we call A the semi-diameter of the central ellipse upon 

 whose direction the two points D and D' fall, we shall have 



and the two portions /j, and fJ of the mass m, which are 



u,=m i and uJ=m, — ; — ;, 



may be expressed by 



and m- 



so that the percussions Q and Q' will have the simpler expres- 



sions 



A2 



V + A'-^' 



These expressions are quite similar to those which we should 

 obtain in the case of a rigid immaterial rod DD', loaded at its 

 extremities by thfi two massive points /j, and fjJ, and, like the 

 body itself, animated with the same rotation 6 around the same 

 spontaneous axis OS. 



At the moment of the shock, therefore, we may imagine the 

 body to be replaced by this rigid rod DD', — or indeed by any 



