INFLUENCE OF SUSPENSION ON WATCH. 367 



according to the fundamental modes, will give any 

 possible motion of the system. The arbitrary 

 circumstances of displacement and projection by 

 which any possible motion of the system may be 

 instituted are producible by giving proper values 

 to the energies and proper times to the epochs of 

 maximum displacement of the component funda- 

 mental modes. The squares of the periodic times 

 of the fundamental modes are the roots of the 

 algebraic equation referred to above. In particu- 

 lar cases, some of these periods may be equal to 

 one another ; or all may be commensurable. In 

 general, however, the periodic times of the funda- 

 mental modes are all different and incommen- 

 surable ; and then none of the compound motions 

 that is to say, no motion except one or other of 

 the fundamental modes is periodic. The mathe- 

 matics of the problem, including proofs of these 

 results, will be found in the first volume (now on 

 the point of appearing) of Thomson and Tait's 

 Elements of Natural PJiilosopJiy. 



The theory is not limited to systems presenting 

 a finite number of independent variables, such as 



