I03-] D'ALEMBERTS PRINCIPLE. 55 



possible ; this will be seen more clearly later on, in the treat- 

 ment of constrained motion. For the present it may suffice 

 to notice that, if the actual displacement ds of the particle in 

 its path be selected as the virtual displacement &s, equation (28) 

 becomes 



This is the equation of kinetic energy (Art. 71); for the left- 

 hand member is the exact differential d(\mv*) of the kinetic 

 energy, while the right-hand member represents the element 

 of work of the impressed forces. 



In the particular, but very common, case of conservative im- 

 pressed forces, the right-hand member is likewise an exact 

 differential dU\ hence, in this case a first integration can at 

 -once be performed, and we find, as in Art. 75, 



U- U Q . (30) 



103. There is an essential distinction between the principle of 

 -d'Alembert on the one hand, and the principles of kinetic energy and 

 of areas on the other. D'Alembert's principle merely gives a con- 

 venient form and interpretation to the dynamical equations of motion, 

 through the application of the principle of virtual work ; but it does 

 not show how to integrate these equations. 



The principle of kinetic energy and the principle of areas are really 

 -methods for integrating the equations of motion under certain con- 

 ditions. If we enquire why these particular methods of combining 

 the differential equations so frequently furnish the solution of physical 

 problems, we are led to the conclusion that the quantities whose exact 

 differentials are introduced by the combination correspond to some- 

 thing really existing in nature. It is thus made probable on purely 

 theoretical grounds that kinetic and potential energy are not mere 

 abstractions, but have an objective reality, and that the conservation 

 of energy is a law of nature. 



