498 VISCOSITY. [CHAP, xi 



As the frictional coefficient b is increased, the two quantities 

 i, 2 become more and more unequal; viz. one of them ( 2 , say) 

 tends to the value b/a, and the other to the value c/b. The 

 effect of the second term in (8) then rapidly disappears, and the 

 residual motion is the same as if the inertia-coefficient (a) were 

 zero. 



276. We consider next the effect of a periodic extraneous 

 force. Assuming that 



Q <<"> (10), 



the equation (1) gives 



Q - ...(ii). 



c - <r 2 a +<r 

 If we put 



1 - o- 2 a/c = R cos lf \ 

 ab/c^Rsme, } 



where e x lies between and 180, we have 



Taking real parts, we may say that the force 



Q=Ccoa(<rt + ) ........................ (14) 



will maintain the oscillation 



C 

 q = - cos (at + ^ ................... (15). 



Since 



(16), 



it is easily found that if 6 2 < 4ac the amplitude is greatest when 



...(17), 



(18). 



a \ ac 

 its value being then 



In the case of relatively small friction, where 6 2 /4ac may be 

 treated as of the second order, the amplitude is greatest when the 

 period of the imposed force coincides with that of the free 

 oscillation (cf. Art. 165). The formula (18) then shews that the 

 amplitude when a maximum bears to its 'equilibrium-value' 

 (C/c) the ratio (ac)^/6, which is by hypothesis large. 



