332 Mr. L. C. Jackson on 



and by the use of (15) these become 



m 2 -b+ ^(b-m*)* + ±a*p } 

 r ~ 2 v /{(&-m 8 ) 3 + 4fl 2 />} ' * " K } 



_ b-m* + x /{( b-m*) + ±a*p\ , 



S ~ 2yf{[b-m*)* + 4a'p\ * * ' " l ; 



Using (6) and (8), and introducing the usual constants, 

 the general solution may be written in the form 



z-e- rt (AeV it -hBe-P ii )+e- st (Ce^ t +'De-^ t ), . (18) 

 and 



J ap K ' up v 



+ ?2!2^-r*(-A^H-Be-^0 + -— (-C«^ + D^« w ). (19) 

 ap ap ' v y 



If small quantities are further neglected, these will 

 simplify to 



z = Ee~ rt sin (pt + e) + F*r s * sin (^ + </>),. . (20) 



and y= G«" rt sin Qrt + e)+H.e~ st sin (gtf + (/>). . (21) 



(21) and (22) are equations representing two superposed 

 simple harmonic vibrations, of which the frequency ratio is 



p r b + m*+ Si(b-m'V + ±a*p\ -\l 



q b + m a - x/{{b-m 2 ) 2 + 4,a' 2 p}J ' ' ' ^ > 



Initial Conditions. 



Let us consider now the form of the general solutions 

 (20) and (21) for various conditions of starting the vibra- 

 tions of the pendulums. 



(i.) Upper bob struck. 

 We may here write 



y-0, *=0, J=0, J=t. for i = 0. . (23). 

 Introducing these conditions in (20) and (21) and into the 



