506 VISCOSITY. [CHAP, xi 



with the time, and that the variations of the other coordinates are relatively 

 small, we should find 



T+V=$a r q r * + & r q r * = ^a r a* .................. (viii), 



nearly. Again, the dissipation is 



the mean value of which is ^o- 2 b rr a 2 .................................... (ix), 



approximately. Hence equating the rate of decay of the energy to the mean 

 value of the dissipation, we get 



da , b r 



di = -* a r 



rr 

 a 



whence a = a e~^ T .................................... (xi), 



if r-^i&^/a, .................................. (xii), 



as in Art. 275. This method of ascertaining the rate of decay of the oscilla 

 tions is sometimes useful when the complete determination of the character 

 of the motion, as affected by friction, would be difficult. 



When the frictional coefficients are relatively great, the inertia of the 

 system becomes ineffective ; and the most appropriate system of coordinates 

 is that which reduces F and V simultaneously to sums of squares, say 



The equations of free-motion are then of the type 



b r q r + c r q r = Q ................................. (xiv), 



whence q r Ce~ t r .................................... ( xv )j 



if r = b r /c t ..................................... (xvi). 



280. When gyrostatic as well as frictional terms are present 

 in the fundamental equations, the theory is naturally more com 

 plicated. It will be sufficient here to consider the case of two 

 degrees of freedom, by way of supplement to the investigation of 

 Art. 198. 



The equations of motion are now of the types 



%&+*nft+(*is+0)f s +i?r-$i,) (i) 



ift+(*it-)&+*si+tyft&amp;lt;?! J 



To determine the modes of free motion we put Q l = ) Q 2 = 0&amp;gt; and assume that 

 q l and q. z vary as e* 1 . This leads to the biquadratic in X : 



t 6 22 ) X 3 + ( 2 c t + a^+ft* + b n 6 2 . 2 - 6 12 2 ) X 2 



c 2 = ...... (ii), 



