THEOKY OF VIBKATIONS 41 



smaller the greater the inertia of the support. This is however 

 more easily seen directly. 



15. General Equations of a Multiple System. 



The general theory of the small oscillations of a multiple 

 system can only be given here in outline. In the case of one degree 

 of freedom ( 7) it was possible to base the theory on the equation 

 of energy alone, but when we have more than one dependent 

 variable this is no longer sufficient, and some further appeal 

 must be made to Dynamics. For brevity of statement we will 

 suppose that there are two degrees of freedom, but there is 

 nothing in the argument which cannot at once be extended to 

 the general case. 



We imagine, then, a system such that every configuration 

 which we need consider can be specified by means of two 

 independent geometric variables or "coordinates" q l9 q 2 . If in 

 any configuration (q l} q^) the coordinate q 1 (alone) receive an 

 infinitesimal variation 8q lt any particle ra of the system will 

 undergo a displacement 88 1 = a 1 8q 1 in a certain direction. 

 Similarly if q z alone be varied m will be displaced through a 

 space Bs 2 = *q* in a certain direction, different in general from 

 the former. The resultant displacement Bs when both variations 

 are made is given by 



8s* = Bs, 2 + 2&! &? 2 cos 6 + &? 2 2 



= di 2 Bqi* + 2a 1 2 cos 6 8q 1 Bq^ + c^ 2 fy 2 2 , (1) 



where 6 denotes the angle between the directions of 8s lt Bs 2 . 

 If we divide by Bt*, we obtain the square of the velocity v 

 of the particle m, in any motion of the system through the con- 

 figuration (q lt q 2 ), in terms of the generalized "components of 

 velocity" q lt q 2 , thus 



v z = ct^ 2 + 2^0, cos Oq-fa + a 2 2 ? 2 2 (2) 



The total kinetic energy of the system is therefore given by 



2T = 2 (mi; 2 ) = a u tf + 2a 12 ^ 2 4- o^ 2 , (3) 



where 



a n = 2, (rav), a 12 

 the summation 2 extending over all the particles m of the 



