366 POPULAR LECTURES AND ADDRESSES. 



give rise to corresponding forces of restitution 

 according to the following condition. The system, 

 starting from rest in its displaced configuration, 

 will, under the influence of these forces, move so 

 as to diminish the displacements of all its parts in 

 the same proportion. Thus all the displacements 

 will come to zero simultaneously ; and therefore 

 the system will move precisely through its con- 

 figuration of equilibrium. There being no fric- 

 tional or other resistance, it will oscillate each 

 displacement varying from maximum positive to 

 maximum negative according to the simple har- 

 monic law ; the system passing, twice in each 

 period, through its configuration of equilibrium, 

 and being twice for an instant at rest in the con- 

 figuration of extreme displacement on cither side. 

 This is called a fundamental mode of vibration. 

 There are as many such fundamental modes as the 

 system has of degrees of freedom to move (inde- 

 pendent variables). Every possible motion of the 

 system may be resolved in simple harmonic vibra- 

 tions according to these fundamental modes ; or 

 the superposition of simple harmonic vibrations, 



