278-279] mssiPATiVE SYSTEMS IN GENERAL. f 505 



The typical equation (10) then assumes the simple form 



a r q r + b r q r + c r q r = Q r (14), 



which has been discussed in Art. 275. Each coordinate q r now 

 varies independently of the rest. 



When F is not reduced by the same transformation as T and F, the 

 equations of small motion are 



where b ft =b sr . 



The motion is now more complicated ; for example, in the case of free 

 oscillations about stable equilibrium, each particle executes (in any fun- 

 damental type) an elliptic-harmonic vibration, with the axes of the orbit 

 contracting according to the law e~ at . 



The question becomes somewhat simpler when the frictional coefficients 

 b rs are small, since the modes of motion will then be almost the same as in the 

 case of no friction. Thus it appears from (i) that a mode of free motion is 

 possible in which the main variation is in one coordinate, say q r . The rth 

 equation then reduces to 



a r q r -f- b rr q r + c r q r = Q (ii), 



where we have omitted terms in which the relatively small quantities gj, </ 2 , ... 

 (other than q r } are multiplied by the small coefficients 6 rl , b r2 ,... We have 

 seen in Art. 275 that if b rr be small the solution of (ii) is of the type 



(iii), 



where T~ l =$b rr /a r , <r=(c r /a r )* (iv). 



The relatively small variations of the remaining coordinates are then given 

 by the remaining equations of the system (i). For example, with the same 

 approximations, 



a 8 q 8 + b rs q r + c s q s = Q (v), 



whence q t = _^ Ae~ tlr sin (at + e). 



** - 2 



Except in the case of approximate equality of period between two funda- 

 mental modes, the elliptic orbits of the particles will on the present supposi- 

 tions be very flat. 



If we were to assume that 



gV = acos(otf + ) ( vii )> 



where a- has the same value as in the case of no friction, whilst a varies slowly 



