44 DYNAMICAL THEOKY OF SOUND 



The general case of m degrees of freedom hardly differs 

 except in the length of the formulae. We have then m equations 

 of the type 



a*i?i + a# 2 + . - + a sm q m + c gl ^ + c^ + . . . 4- C 8m q m = Q s , (15) 

 where s is any one of the integers 1, 2, 3,...m. 



16. Free Periods of a Multiple System. Stationary 

 Property. 



In the case of free vibrations we have Qi = 0, Q 2 = Q, and 

 the solution of 15 (12) then follows exactly the same course 

 as in the particular examples already given. We assume 



q 1 = A l cos(nt + e), q 2 = A 2 cos(nt + e), (1) 



and obtain (c u - n z a u ) A, -f (c 12 - n 2 a l2 ) A 2 = 0,1 

 (c 21 - ?i 2 (A 21 ) A l + (c sst - n^a^) A^ = 0.) 

 Eliminating the ratio A l :A 2) we obtain 



| c u -w 8 a n , C 12 -/i 2 a 12 

 C 21 -rc 2 a 21 , CJB 71*022 ~ 



where (it is to be noticed) the determinant is of the " sym- 

 metrical " type. This equation gives the two admissible values 

 of n*. Adopting either of these we obtain a solution in which 

 the ratio of A l to A 2 is determined by either of the equations 

 (2). The mode of vibration thus ascertained involves therefore 

 two arbitrary constants, viz. the absolute value of (say) A 1} and 

 the initial phase e. The second root of (3) leads to another 

 solution of like character. 



The extension of the method to the general case is obvious, 

 but it may be well to state the results formally. In any 

 conservative system of m degrees of freedom there are in 

 general m distinct " normal modes " of free vibration about 

 a configuration of stable equilibrium, the frequencies of which 

 are given by a symmetrical determinantal equation of the mih 

 order in n 2 , analogous to (3), and so depend solely on the con- 

 stitution of the system. In each of these modes ihe t system 

 oscillates exactly as if it had only one degree of freedom, the 

 coordinates q^,q^, ... q m being in constant ratios to one another, 

 and the description of 7 therefore applies. The directions of 

 motion of the various particles and the relative amplitudes are 



