54 KINETICS OF A PARTICLE. [100. 



100. Owing to the fact that, by combining with the impressed 

 forces the reversed effective force, we obtain at any given in- 

 stant a system in equilibrium, it becomes possible to apply to 

 kinetical problems the statical conditions of equilibrium. 



Since in the case of a single particle the forces are all con- 

 current, the conditions of equilibrium are obtained by equating 

 to zero the sums of the components of the forces along each 

 axis. This gives 



and these are the ordinary dynamical equations of motion (see 

 (26), Art. 98). 



101. The conditions of equilibrium of a system of forces can 

 also be expressed by means of the principle of virtual work 

 (Part II., Art. 239). Thus, let &r, Sj, 8z be the components of 

 any virtual displacement Bs of the particle ; then the principle 

 of virtual work applied to our system of forces gives the single 

 condition 



' (27) 



which is of course equivalent to the three equations (26) on 

 account of the abitrariness of the displacement Bs. 



The equation (27), which may also be written in the form 



(28) 



expresses d'Alembert's principle for a single particle : for any 

 virtual displacement the sum of the virtual works of the im- 

 pressed forces is equal to that of the effective force. 



102. The advantage of using the equations of motion in the 

 form given to them by d'Alembert arises mainly from the appli- 

 cation of the principle of virtual work which thus becomes 



