Vibrations by Forces of Double Frequency. 151 



(c—ik) in place of c, and (® — F) in place of ® - Accordingly 



costt(c-^) = 1-V / (0), .... (27) 



V'(0) differing from V(0) only by the substitution of ® — & 2 

 for ® . 



If 1 — V'(0) lies between +1, (c—ik) is real, so that 



c = ik±ct+2n (28) 



In this case both solutions are affected with the factor e~ u , 

 indicating that whatever the initial circumstances may be, the 

 motion dies away. 



It may be otherwise when 1 — V'(0) lies beyond the limits 

 + 1. In the case of most importance, when © is nearly equal 

 to unity, 1 — V'(0) is algebraically less than —1. If 



cos27r/3=-l + V / (0), (29) 



we may write 



e=l + i(*+0)+2fi (30) 



Here again both motions die down unless /3 is numerically 

 greater than k, in which case one motion dies down, while the 

 other increases without limit. The critical relation may be 

 written 



cos{i7rk) = -l + \/ / (0) (31) 



From (30) we see that, whatever may be the value of k, the 

 vibrations (considered apart from the rise or subsidence indi- 

 cated by the exponential factors) have the same frequency as 

 if k, as well as S v © 2 , &c. vanished. 



Before leaving the general theory it may be worth while to 

 point out that Mr. Hill's method may be applied when the 

 coefficients of d 2 wjdt 2 and dw/dt, as well as of w, are subject 

 to given periodic variations. We may write 



where 



®=t® n e* nt , V=2V n e 2int , ®=2®«0*'"'. (33) 



Assuming, as before, 



w=Snb n J e ' + * nt , (34) 



we obtain, on substitution, as the coefficient of e ict+2imt , 



-t n b n (c + 2ny® m -n + it n b n (c + 2n)V m _ n + $b n ® m ^ 



which is to be equated to zero. The equation for c may still 

 be written 



S(c)=0, (35) 



