272 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



The velocity of P, considered as a point of w, has components 

 V &(r) y\ V+ Q(g-x) before impact, and 

 u - a) (77 - y\ v + o> ( - x) after impact. 

 The velocity of P, considered as a point of m', has components 

 V - fl' (77 - y'\ V + Q'(- x'} before impact, and 

 u' a>'(r)- /), v' + o>' ( - x'} after impact. 



The equation provided by the generalised Newton's Kule is accordingly 

 u-(ri-y)-u' + '(i(-tf)=-e{U-a(ri-y)-U'+ti(ri-tf)}. 



The equations of motion of the two bodies by resolving parallel to the 

 axis of x are 



m(u-U}=-R, m'(u'-U'}=R. 



The equations of motion by resolving parallel to the axis of y are 

 m(v-F) = 0, m'(v'-F') = 0. 



The equations of moments about axes through the centres of inertia per- 

 pendicular to the plane of motion are 



where k and k' are the radii of gyration of the bodies about the axes in 

 question. 



On substituting for u, u', o>, o>' in the equation containing e, we find 



and this equation shows that the impulsive pressure with any value of e is 

 (1 4-e) times what it would be if e were zero. 



The result of this Article can be expressed in the statement that the 

 generalised Newton's rule and the rule derived from Poisson's hypothesis 

 are equivalent for any two smooth bodies moving in one plane. 



*241. Impulsive action between rough bodies. The impulsive action 

 between two rough bodies which come into contact, when there is sliding at 

 the point of contact, is assumed to be expressible by means of an impulsive 

 pressure of the kind we have met with in the case of smooth bodies, and 

 an impulsive friction tending to resist sliding, the friction and the pressure 

 having a constant ratio, the coefficient of friction. We shall suppose the 

 geometrical condition as regards the relative velocity to be the same as in 

 the case of smooth bodies, viz. the generalised Newton's rule. 



In the case of bodies rough enough to prevent sliding, the elastic action 

 cannot be so simple as in the case considered by Poisson, and accidental 

 circumstances probably play an important part. 



We shall now show that when there is sliding at the points that come 

 into contact the rule deduced from Poisson's hypothesis is equivalent to the 

 generalised Newton's rule, for the impulsive action between rough bodies. 



