107, 108] System of Loaded Spheres 95 



Before impact the components parallel to x, y, z of the velocity of the 

 centre of the first sphere are 



u + r 



w 



Hence if a, ft, 7 denote the components of the velocity of the centre of 

 the second sphere, relatively to that of the first, we have 



= u u + r (^'Gr 2 / Z/Br/ /! w 2 + ^i), etc .......... (226). 



The components of relative velocity after impact can be deduced by 

 writing u, u .,. in place of u, u' ... . The quantities u ... are given by 

 equations (224), (225); the quantities u' ... are given by similar equations, 

 except that / must be replaced by /, since the impulse on the second 

 sphere in the direction X, //,, v is /. Substituting this value, and compar- 

 ing with equation (226), we obtain 



2/X r 2 / 



a = a 



I 



j- 2 i i 



+ li (li\ + w/yit + n/ 1/) 

 and there are similar equations for ft, 7. 



Multiplying these three equations by \, /*, v and adding, we obtain 



27 



\a + p0 + vy = Xa + pp + vj - -- (1 + Ar*) ......... (227), 



where A = -^ {(^X + m^ + n^) 2 + (1 2 \ + m^ + n 2 v) z 



+ (//X + m/yu + 7Z/J/) 2 + (1 2 \ + m 2 'fj, + n 2 ' v y>} ...... (228). 



At the moment of greatest compression, the components of relative velocity 

 will be |(a + a), %(& + ft), i (7 + 7), and therefore the relative velocity along 

 the line of centres will be 



4 (Xa + lift + vj + Xa + p/3 + vy). 

 This vanishes, and therefore, by equation (227), 



Xa + /*/3 + vj = (Xa + /j,ft + vj) 



= (l+Ar*)-. 

 ' m, 



Hence, since we are neglecting r*, 





(229). 



Substituting this value for I in equations (224) and (225), we obtain the 

 velocities of the first sphere after impact. 



