THEORY OF VIBRATIONS 19 



case in which the initial conditions are definite. Thus, in the 

 case of (8), the formula may be written 

 / sin \ ( p n) t . 



and as p approaches equality with n this tends to the limiting 

 form 



(10) 



This may be described (roughly) as a simple vibration 

 whose amplitude increases proportionally to t. For a reason 

 just indicated this is only valid as a representation of the earlier 

 stages of the motion. 



The case of a disturbing force of more general character 

 may be briefly noticed. The differential equation is then of 

 the form 



+ *=/() ................... (11) 



The method of solution, by variation of parameters, or 

 otherwise, is explained in books on differential equations. The 

 result, which may easily be verified, is 



x = - sin nt I f(t) cos nt dt cos nt I f(t) sin nt dt. (12) 



It is unnecessary to add explicitly terms of the type 

 A cos nt + B sin nt, which express the free vibrations, since 

 these are already present in virtue of the arbitrary constants 

 implied in the indefinite integrals. 



If the force f(t) is only sensible for a certain finite range of 

 t, and if the particle be originally at rest in the position of 

 equilibrium, we may write 



x = - sin nt I f(t) cos nt dt -- cos nt I f(t) sin nt dt, (13) 

 n J -ao n J _QO 



since this makes x 0, dx/dt = for t = - oo . The vibra- 

 tion which remains after the force has ceased to be sensible is 

 accordingly 



x = A cos nt + B sin nt, ............... (14) 



where 



=-- 1 f(t)smntdt, B = -T f(i)cosntdt. (15) 

 nj -oo nj _aj 4 



22 



