278-279] D1SSIPATIVE SYSTEMS IN GENERAL. , 505 



The typical equation (10) then assumes the simple form 



Qi ...................... (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 = 



f 



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~^. 



The question becomes somewhat simpler when the frictional coefficients 

 b rg 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,.V + &n-j- + &amp;lt;V2,. = .............................. (ii), 



where we have omitted terms in which the relatively small quantities y 1} y 2 , ... 

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

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



(iii), 

 where T -i=$b rr /a r , &amp;lt;r = (c&amp;gt;,)* ........................... (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, 



whence q s = ~ &amp;lt;r ~~ Ae~ t/T sin (at + e) . 



-* 



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 



3V=acos((r* + e) .............................. (vii), 



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



