Vibrations by Forces of Double Frequency. 153 



Reverting to (24), we have as the approximate particular 

 solution, when there is no dissipation, 



e (c--2)it e cit e (c + 2)it 



w = (C _2) 2 -0 O + W, + Xe + 2f-® ' ' ' (41) 



If c be real, the solution may be completed by the addition 

 of a second, found from (41) by changing the sign of c. Each 

 of these solutions is affected with an arbitrary constant mul- 

 tiplier. The realized general solution may be written 



_ R cos (c — 2)t + S sin (c-2)t 

 (c-2f-® 3 

 _i_ R cos ct + S sin ct Rcos'(c + 2)£ + Ssin(c + 2)£ { .^ 

 + ~ ~@T ' + (c + 2) a -0 o ' ' +**> 



from which the last term may usually be omitted, in conse- 

 quence of the relative magnitude of its denominator. In this 

 solution c is determined by (26). 

 When c 2 is imaginary, we take 



4 S 2 = © 1 2 -((H) -1) 2 ; (43) 



so that 



c 2 = l + 2is, c=l+is, c— 2=— 1+is. 



The particular solution may be written 



w=e- st {® i e- it + (l-® -2is)e it \ ; . . . (44) 



or, in virtue of (43), 



w = e- 8t \(l—® + ® 1 )cost + 2ssmt\; . . (45) 

 or, again, 



^= e -^{ N /(e 1 +i-e ).cos^+ v /(e 1 -i+@ ).siiu}. . (46) 



The general solution is 



w ==]&-**{ (I— © + e 1 )cosi + 2ssin$}l 



+ Se s ; {(1— ® + ©i) cos t — 2s sin*} J' 



R, S being arbitrary multipliers. 



One or two particular cases may be noticed. If ® = 1, 

 25 = ®!, and 



ic=zWe- st {cost+ sin a ) 



V (48) 



+ SV {cos*- sin*} J 



Again, suppose that 



1 2 = (0„-l) 2 , (49) 



so that s vanishes, giving the transition between the real and 



