M. Poinsot on the Percussion of Bodies. 243 



to M by the shock of the material point m the value 



2. From this expression we learn that the quantity of motion 

 imparted to M diminishes when x increases, so as to become 

 zero when the shock takes place at an infinite distance from the 

 centre of gravity. When a; = 0, that is to say, when the shock is 

 given at the centre of gravity G, the imparted quantity of motion 



becomes, of course, r- : it is the greatest value miv — v) can 



n+1 ' ° ^ ' 



have. 



Let us now conceive m and v to vary in such a manner as 

 always to give the same ])roduct mv, and let us inquire how m 

 and V ought to vary with the distance x in order that the quan- 

 tity of motion imparted to M may always remain the same. 

 Since the numerator mvY^ in the expression for this quantity of 

 motion is, by hypothesis, constant, it is clear that the denomi- 

 nator nuist be so also ; and consequently, neglecting in it the 

 constant quantity K^, we must have 



n(KHa'2) = const. =B2; 



whence it follows that 



MB2 



, P(K2 + ^2) 



and v= -Vli2— ', 



Iv2 + A'2 MB2 



where P simply represents the constant product mv. 



Thus m and v must vary as these two reciprocal functions of 

 X in order that the material point m, endued with the velocity v, 

 may impart to M the same quantity of motion, whatever may be 

 the distance x of the point of impact C. 



Instead of the constants B^ and P, two others may be intro- 

 duced more closely related to the data of the question. Thus if 

 OTq and Vq be the mass and velocity of the material point when C 

 corresponds to a;' = 0, we shall have 



whence we deduce 



_ ]\IB2 _ PK2 



'o— Y^i ^iid V(j— j^^^2J 



B2='-^i^%ndP = ,v'o 



MO that the two variables m and v have the values 



Thus at the distance x from the centre of gravity the mass m 

 112 



