Vibrations elucidated by Simple Experiments. 255 

 Then (33) and (34) in (31) give 



|(2 + 2/3> 2 + (2 + /3)m 2 P = /3 2 m 4 , 



whence mi 



x=±mi or ± ^ti+py ' ' ' ( 35 ) 



Thus, using (35) in (33) and (34) and introducing the 

 usual constants, we may write the general solution in 

 the form 



(Tilt \ 



•(i+ff) + V ' ' (36) 



(TYlt \ 



,/(].+£) +< ft)' ' ( 37 ) 



where E, e, F, <f) are the arbitrary constants whose values 

 depend upon the initial conditions. 



Initial Conditions. — To obtain the special solution for any 

 concrete case we must state the initial displacements and 

 velocities and determine the four constants accordingly. As 

 a preliminary, we write the velocities from (36) and (37) by 

 differentiation with respect to the time. Thus 



1 = mE cos (mt + e > + V(TW) cos tTTTTTT) + V> (38) 



dz _ -r^ / . . x . mF 



di' 



«■«(«*+.)+ -^^-(-^ + *). (39) 



(i.) Single Velocity. — Take first the case of a single velocity 

 imparted to one bob by a blow when both are at rest in their 

 zero positions. Then we may write 



y=0, 0=0, j=u, g=0, for *=0. . (40) 



These conditions, put in (36)-(39), give equations which 

 are satisfied bv 



Hence for this case we have the special solution 



u . </{l + j3)u . mt 



y — s-sinmH ^ -sin . . . (42) 



w . . , V(l + &)u . mt /tn ^ 



z= — -~— sin mt+ — ^-5 —sin 7/1 ,> . . (43) 



2m 2m v(l + /3) v ' 



T2 



