45] 



SMALL OSCILLATIONS. 



159 



62) 



dT 



- 



H ----- h a r *qn 



H h C<?K, 



Inserting these values in our differential equation we have, performing 

 the differentiation by t, 



r oN ^ 2 & , ^2* , **" 



O^; <-*rl 7,3 TT r2 j.g- T ' 



llr " 6?^ 2 



dt 



+ --+c rn q n = 



a linear differential equation of the second order with constant 

 coefficients. We have one such equation for each coordinate q r . As 

 in the case of one variable we may find a particular solution by 

 assuming for each coordinate 



the I being the same constant for all the g's. Inserting these values 

 in 63) we obtain, after dividing by the common factor e u , 



A 1 (a n 



h 



64) 



c lw ) = 0, 



0, 



\- A n 



C nn ) = 0. 



These are linear equations in the A's and suffice to determine their 

 ratios when A is known. In order that they may be satisfied by 

 other than zero values of the A's, the determinant of the coefficients 

 must vanish, namely, 



65) 



4- 



-f 



This is Lagrange's determinantal equation for A. It is_ol.degr_ee._22i 

 and jfouL3B&&&du^^ We shall denote 



its roots by ^, A 2 , . . . A 2n , which we shall suppose are all different. 

 We shall first prove that none of the roots are real and positive. 



