390.] CONSTRAINED SYSTEM. 219 



elementary works for a finite time, say from t=o to t=t, repre- 

 sents a certain finite amount of work W \ 2 (Xdx+ Ydy + Zdz), 

 .so that equation (6) gives always 



^\mv*-^\mv*= W. 



This means that if the conditions are independent of the time, 

 the increase of kinetic energy during any interval of time is equal 

 to the work done during this time by all the external and internal 

 forces. 



But when a force-function exists, this work is W= U 7 , 

 where U is a function of the co-ordinates only. The work done 

 by the forces depends therefore only on the initial and final 

 values of these co-ordinates ; i.e. on the initial and final con- 

 figuration of the system, but not on the character of the motion 

 by which the system is brought from the initial to the final 

 position. 



390. It has been shown in Art. 222 that, for an invariable 

 system of n points, i.e. for a free rigid body, the number of condi- 

 tions is k = $n 6 ; hence the number of independent equations 

 of motions of a free rigid body is 372 ($n 6) =6. 



A rigid body with a fixed axis (Art. 291) has but one degree 

 of freedom and 5 constraints ; i.e. its position is given by a 

 single variable, say the angle of rotation, 0, about the fixed 

 axis. The motion of such a body is therefore given by a single 

 equation. 



A rigid body that can turn about and also slide along a 

 fixed axis has 4 constraints and 2 degrees of freedom ; it has 

 therefore 2 equations of motion, and 2 variables are sufficient to 

 determine any particular position of the body, say the angle 6 

 and the distance x measured along the axis of rotation. 



A rigid body with one fixed point (Art. 311) is an example of 

 an invariable system with 3 constraints and 3 degrees of free- 

 dom. Three variables are necessary and sufficient to determine 

 a particular position, and the number of independent equations 

 of motion is 3. 



