40 Prof. Barton and Miss Browning on Coupled 



When small quantities are further neglected, these will 

 simplify to 



* = E*- rf sin(p* + €)+Fe- rf sin(gtf + <£), . (21) 

 and 



y = Ge~ rt sin (pt + e) + He~ st sin (qt + (/>). . (22) 



(21) and (22) are each equations of two superposed 

 vibrations, of which the frequency ratio is 



ff_ p + a+x/{la-c) 2 + 4p& 2 } -] 1 / 2 ( 



?~L + a-*/{(a-c) 2 + 4^ 2 }J ' ' ' V } 



where a, b, and c are given by (14). 



Initial Conditions. — Let the heavy bob of mass Q on 

 the pendulum of length /. be pulled aside a distance / by 

 a horizontal force, the light bob on the other pendulum 

 hanging freely at rest. The displacement of the light bob 

 can then be written at once from equation (31) page 67 of 

 the January paper, since the length of this pendulum makes 

 no difference to the quantity in question. Hence we have 



For * = (), 



'=* »=TT^=TTefZ%20 I (24) 



Then, introducing these conditions in (21) and (22) and 

 into the differentiations of these with respect to the time 

 and, as before, omitting small quantities, we find 



/=E sin e + F sin <f>, 



M_ =0lhl ..— 



1+0 



+ Hsin0. C 



= Ep cos € + Fq cos fa 

 = Gp cos e + H<7 cos (p. 



Equations (26) are satisfied by 



2' ^=2 



e=-r. ^=f (27) 



From (17), (20), and (22) we have 



G= ~f + C E= C ~ a ~ ^{(«-") 2 + V> 2 } E; "1 



f-- (28) 



h- -y + c f - c ~ a+ ^i^-c y+ipV} w I 



b 26 r - J 



