THEORY OF VIBRATIONS 57 



conjugate pairs of the form m in, and we may assert that 

 the part of the solution corresponding to this pair of roots 

 will be given with sufficient generality if we make use of one 

 only of these, writing, for instance, 



x=Ce (m + in}t , ..................... (19) 



and taking the real part. 



We may apply these considerations, for example, to the 

 equation 



of resisted motion about an equilibrium position ( 11). If we 



put x = Ce , we have 



A. 2 + &X + /4 = ...................... (21) 



Hence \ = -\kiri, ..................... (22) 



where TO' = VO"-i& 2 ), ..................... (23) 



provided k 2 < 4//,. On the above principle a sufficient solution 



or, in real form, 



x=e~* kt (Acosn't-Bsmn't\ ......... (24) 



which is equivalent to 11 (8). 



The same method can be followed with regard to the 

 equation of forced oscillations, say 



?SBS /<8jtf ............. (25) 



Instead of this we take the equation 



g + ** + / u.^ ................ (26) 



the implied adjunct equation being of the type (25) with 

 fsinpt instead of /cos pt on the right hand. A particular 

 solution is 



z=Ce ipt , ........................ (27) 



provided (/z - p 2 + ikp) G =/. .................. (28) 



fj pt 

 Hence * = - ^ sr- ................... (29) 



* 



