214 MOTION OF A VARIABLE SYSTEM, [383. 



II. System Subject to Conditions. 



383. The constraints and conditions to which a variable 

 system is subject may be of very different kinds. In general, 

 however, they can be imagined as replaced by certain forces, 

 called constraining- forces or reactions, by the introduction of 

 which the system becomes free. On the other hand, it may be 

 noticed that internal forces, such as tensions of connecting rods 

 or strings, can sometimes be regarded as constraining conditions. 



If all conditions and constraints be expressed by means of 

 forces, and these forces be included among the forces X, Y, Z, 

 the equations of motion of the particle m have again the form 

 (i), Art. 376, and the principle of virtual work gives the equa- 

 tion of d'Alembert (2), Art. 377. But it should be noticed that, 

 in general, the constraints will do no work if the displacements 

 &r, By, z are properly selected ; in other words, if the displace- 

 ments be taken so as to be compatible with the conditions to 

 which the system is subject, the constraining forces will not 

 enter into the equation (2). This is d'Alembert's principle. 



384. Before further developing this idea it may be well to 

 indicate here the considerations by which d'Alembert himself 

 (and, in more exact language, Poisson) explained his celebrated 

 principle. 



Any particle m of the system is acted upon at any time t 

 by two kinds of forces, the given external and internal forces, 

 whose resultant we denote by F, and the internal reactions 



and constraining forces whose 

 resultant we call F' (Fig. 55). 

 The resultant of .Fand F' must 

 be geometrically equal to the 

 effective force mj, where j is 

 the acceleration of the particle 

 at the time t. 



Now, if we introduce at m 

 the equal and opposite forces mj, mj, the action of ^and F\ 



