27] CONSERVATION OF ENERGY. 69 



to follow any curve, reaches B always with the same velocity, 

 although the time occupied in the descent may be very different 

 from one curve to another. 



The equation 27) might have been applied to immediately give 

 us the integral equation 44) 21. (In that equation , the Z-axis is 

 drawn positively downward) 



The principle of virtual work, 25, may evidently be expressed 

 by saying that for equilibrium the potential energy of the system is 

 a maximum or minimum, and a little consideration shows that for 

 stable equilibrium it is a minimum. 1 ) 



For instance in the above example the potential energy 



W= mgz + const, 



z being measured positively upward. If the particle is in equili- 

 brium on a surface concave upwards, z and with it W is a 

 minimum, the equilibrium being stable. If the concavity is down- 

 wards, the equilibrium is unstable and W is a maximum. 2 ) The 

 question of stability of equilibrium will be discussed in 45. 



It is possible to have a force -function denned by equations 23), 

 which contains the time as well as the coordinates. The system is 

 not then conservative, and it is not customary to speak of its 

 potential energy. We have now 



dU dU ,(dUdx r dU dy r dU dz 



so that our equation of activity 20) is in this case 



dT _ dU dU 

 ~3f**~dt^W 



In certain cases we may be able to assign the term -^-~ to a potential 



dW 

 energy, as -- - rf -~.. 



If the forces depend on the velocities or on anything beside 

 the coordinates, the system is not conservative. Such a case is that 

 of motion with friction, where the friction, being a force that always 

 tends to retard the motion, not only changes sign with the velocities 

 but also depends upon the magnitudes of the velocities in such 

 resisting media as the air and liquids. 



The dynamical theory of heat accounts for the energy that 

 apparently disappears in non- conservative systems. 



1) Dirichlet, Uber die Stabilitdt des Gleichgewichts. Crelle's Journal, 

 Bd. 32, p. 85 (1846). 



2) See Kirchhoff, Mechcmik, p. 34. 



