152 V. OSCILLATIONS AND CYCLIC MOTIONS. 



73) 



It is to be noticed that the ratios of the As in any column 

 have been determined by the linear equations 64), so that there is 

 a factor which is still arbitrary for each column, that is to say, 

 2n in all. We may now replace the exponentials by trigonometric 

 terms. The appearance of the terms with conjugate imaginaries 



Are 1 * + A[e rt = 2e^ (a r cos vt - p r sin vt) 



leads to the disappearance of imaginaries from the result. Changing 

 the notation we will accordingly write 



= % e^ * cos V t - + B e^ cos 



74) 



If these be substituted in the differential equations it will be 

 found that the J5's satisfy the same linear equations as the As. 

 Each column then contains an arbitrary constant as before, in the .B's 

 and a second arbitrary constant in the 8 belonging to the column. 

 We may therefore state the general result: - - The motion of any 

 system, possessing n degrees of freedom, slightly displaced from a 

 position of stable equilibrium may be described as follows: Each 

 coordinate performs the resultant of n damped harmonic oscillations 

 of different periods. The phase and damping factor of any simple 

 oscillation of a particular period are the same for all the coordinates. 

 The absolute value of the amplitude for any particular coordinate is 

 arbitrary, but the ratios of the amplitudes for a particular period for 

 the different coordinates are determined solely by the nature of the 

 system, that is, by its inertia, stiffness and resistance coefficients. 

 The 2n arbitrary constants determining the n amplitudes and phases 

 are found from the values of the n coordinates q and velocities q' 

 for a particular instant of time. 



