Illustrations of Certain Optical Phenomena. 237 



By division and reduction, we have the quadratic in k, 



pk* + k(a a *-a* + £-r\-!j=0, . . . (20) 



which has a positive root k x and a negative root — k 2 . 



Let the corresponding values of 2tt/T for the fundamental 

 modes be called f^ and fl 2 . For both of them we have, 

 by equations (19), 



n 2 = <+/.(!-*) =®2 3 +f(i-]i)- • • • C 21 ) 



Putting —k 2 for k, we have 



n 2 2 = V+Kl+* 2 ) = « 2 2 +^(l+^), • • (22) 



showing that fl 2 is greater than either o> x or &) 2 ; in other 

 words, that the fundamental mode in which the displacements 

 are opposite has a higher frequency than the vibration of 

 either pendulum alone. 

 Putting k ] for k, we have 



<V = V+Mi-*i) = « 2 2 +^(i-£). . . . (23) 



When k v is different from unity, one of the two quantities 

 \ — k x and \ — \jk x is positive and the other negative ; hence 

 Hj is intermediate between co l and <w 2 . Equation (20) shows 

 that, if £j is unity, gjj is equal to co 2 ; that is, the pendulums 

 are of equal length. Whenever they are unequal, the funda- 

 mental mode in which their displacements are similar is 

 intermediate in frequency between the vibrations of the two 

 pendulums singly. 



The actual vibration of the system will be either that 

 corresponding to k x and O,, or that corresponding to — k 2 

 and fl 2 , or a combination of the two in an arbitrary ratio ; 

 according to the initial circumstances. 



If the pendulums are nearly equal, both in length and 

 mass, the coefficient (o^—co^ + fi/s—fA of k in the quadratic 

 is small and the positive and negative roots are nearly equal, 

 also their product — 1/s is —1 nearly ; hence the roots are 

 approximately + 1. The values of X2 2 from (21) are therefore 

 approximately 



D 1 2 = cof, n 2 2 = <V + 2At, 

 the latter being always the greater, as proved above. 



In practice //, is usually exceedingly small ; hence 12 l 

 and I2 2 are nearly equal. The general equations 



x= A cos (IV- «i) -f B cos (n 2 t-a 2 ), 1 



f = & 2 A cos (12^— c^)— & 2 B cos (I2 2 £— a 2 ), J 

 Phil Mag. S. 5. Vol. 46. No. 279. Aug. 1898. S 



