76 



Proceedings of the Royal Irish Academy. 



moments of inertia ; x,y,z the co-ordinates of ; hm^ni, Lm^n^, liTn^ih the 

 direction cosines of the axes of P, Q, B; all referred to the principal axes 

 through the centre of gravity of A. Let the corresponding quantities for 

 A' be denoted by the same letters with dots. Then as A has been acted on 

 by a blow - P, - Q, - R at a^, y, z, we have : — 



M{u -U) = - {hP + hQ + kR). 



M{v - V) = - {iihP + m^Q + TriiB). 



J/(w - T-T) = - («,P + n.Q + 7hR). 



A {u>x - Q.x) = 



B(o 



Qrj) = - 



y, miP + iiuQ + m-iB 

 z, riiP + ihQ + ihB 



s, UiP + UtQ + n^B 

 X, hP + hQ + kU 



C'{Wy - Q,y) = 



Xi-P - XiQ — Aii?, 



fx,P - fi,Q - ^,h'. (1) 



viP — viQ - V3B. 



x\ UP + hQ + hB 

 y, wi,P + m^Q + iii^B 



The corresponding equations for A' are obtained by substituting 

 - P, - Q, - B for P, Q, B, and dotting all the other letters. 



These equations connect the velocities of translation and rotation of 

 A and A' with the total impulsive components P, Q, U at any stage of 

 the impact. 



At any such instant, let S inclined at 6 to the axis of P be the velocity 

 with which the point of ^ slides on A' ; let K be the velocity of com- 

 pression ; and let the initial values of these quantities be denoted by 

 (Sg, 0(1, K„ ; then we have 



iS cos 9 = li{u + zijiy - ywz) + m, {v + xu}. - zidi) + ?ii (w + ywx - xwy) — (similar 

 quantity with dots) 



= hu + iiiiV + moji + tL);r/\, + (iiyjUi + (,)zVi — (similar quantity with dots) 

 = So cos 0, - ^ - ^ (X.P + X,e + \R) - § (/x.P + ,hQ + fxdi) 



- jj ("i-P + v%Q + viR) 



-JJ (vi'P + Vt'Q f Z///0 



aS^o cos 9„ - aP - hQ - gR. 



