108 THEORY OF NEWTONIAN FORCES. [PT. I. CH. II. 



This equation may be called the fundamental equation of 

 dynamics, and is the analytical statement of what is known as 

 d'Alembert's Principle. Lagrange made it the basis of the entire 

 subject of dynamics*. Interpreted by means of the principle of 

 virtual work, equation (19) states: 



If, the motion of a system of particles being given, we find the 

 acceleration of every particle, and apply to each particle a force 

 whose components are 



then the system of forces X', Y, Z', together with the impressed 

 forces X, Y, Z, will form a system in equilibrium. 



The forces X', Y, Z' are called the forces of inertia, or the 

 reversed effective forces. D'Alembert's principle is thus only 

 another form of stating Newton's third law of motion. 



We have now a measure of the inertia of a body, namely the 

 force of inertia above defined f. We may now define matter as 

 whatever can exert forces of inertia. 



58. Energy. Conservative Systems. If in the equation of 

 d'Alembert's principle, (19), we put for &c, Sy, Sz the displacements 

 which take place in the actual motion of the system in the time dt, 



s- d&r 7 . cs dy r -, % dz r , 



r== ~dt y r== ~3t r== ~dt ' 

 we obtain 



/ x v f (d 2 x r dx r d?y r dy r d*z r dz r \ 



(20) r jm r ^jrjf + -fa ~dt + W ~dt) 



x dx r Y dy r dzr[ , Q 

 ~ Xr ~dt Yr dl Zr dt} d 



d*x r dx r d dx r 



Smce 



the sum of the first three terms is the derivative of the sum 



* Lagrange, Mecanique Analytique. (Euvres, t. 11, p. 267. 



f The inertia of a body is sometimes considered as the factor of the negative 

 acceleration in the expression for the force of inertia, thus making inertia 

 synonymous with mass. 



