112 SYSTEMS OF FORCES. [CHAP. VII. 



forms in terms of the velocities and accelerations with which six 

 particular quantities increase. 



In order that a system may be rigid it is necessary that the 

 component accelerations contributed to any particle by the actions 

 of the other particles should satisfy such conditions that the 

 resultant acceleration of any particle of the system is expressible 

 in the proper form in terms of six quantities such as those above 

 described. It follows that the internal forces between the 

 particles of the system must be so adjusted as to produce such 

 accelerations. The internal forces in question are not subject to 

 any other condition, and we may arbitrarily assume any system of 

 internal forces provided the assumed system obeys this condition. 



115. Forces applied to a rigid system. Suppose a rigid 

 system is in motion in any manner, and suppose, accordingly, that 

 the internal forces are adjusted to satisfy the condition just described. 



Let two forces equal in magnitude and opposite in sense be 

 applied to two particles, the line of action of the forces being the 

 line joining the particles. To fix ideas, let m, m be the masses 

 of the particles, and let the force applied to m be of magnitude R 

 in the sense from m to m . 



The internal forces will be altered in some way consistent with 

 the rigidity of the system, and it is clear that, if the internal force 

 assumed to act between the two particles m and m before the 

 application of the forces R is modified by compounding with it a 

 repulsion of magnitude R between these particles, the resultant 

 forces on all the particles will be the same as before the application 

 of the forces R to m and m , and the system will still be rigid, and 

 will be moving exactly as before. 



It follows that the motion of the rigid system is unaffected by 

 the application to any two particles of the system of a pair of 

 equal and opposite forces in the line joining the particles. 



Now let F be any external force applied to a particle m of the 

 system, and let m be a particle in the line of action of F. Suppose 

 applied at m a force of the same magnitude, direction and sense 

 as jP, and suppose applied at m a force of the same magnitude and 

 in the same direction as F but opposite in sense. The motion 

 of the system is unaffected. 



Thus the effect of a force applied to any particle of a rigid 



