Illustrations of Certain Optical Phenomena, 239 



^ Assume f=&# for a fundamental mode. Then the equa- 

 tions become 



k[v=- 



9 



■- h {k-\)x 



*=-f{s^(l+s)-ks} 



(28) 



By division and reduction, we obtain the quadratic for k 



ask 2 -k{b-a+(b + a)s}-a=;0; . . . (29) 



which, as in the previous example, has a positive root k x 

 and a negative root — k 2 



As before, denoting the acceleration-factor — a/xov (2tt/T) 2 

 by O 2 , the two equations give 



«-f(i-J)-J{ 1+ ' + ?( l -«}- • • (30) 



Putting — k 2 for k all the terms are positive ; hence the first 

 expression for X2 2 2 is greater than g/b, and the second is 

 greater than g/a ; that is to say : — The mode in which the 

 displacements are opposite is quicker than either pendulum alone. 

 For the other mode, we have 



V-f(l-#-£{l + . + ?(l-*>}. • • (3D 



The first value shows that f2 x 2 is less than g/b. The second 

 value shows that it will be less than g/a if s + (l — k } ) as/b is 

 negative, that is if ki is greater than 1 4- b/a ; and this con- 

 dition is always fulfilled, for the substitution of this value of 

 k in the quadratic gives an opposite sign to the substitution 

 of a very large positive quantity. Hence the mode in which 

 the displacements are similar is slower than either pendulum 

 alone. 



The physical meaning of the result k l >l + b/a is that, in 

 the mode in which the displacements are similar, the lower 

 string is more inclined to the vertical than the upper. 



When a is infinite, —k 2 is — 1/s, and 2 2 is (l + s)g/b. 



When b is infinite, k is zero, and X2 2 is (l + s)g/a. 



In the former case, the lower string rotates about the 

 common centre of gravity as a fixed point, so that the virtual 

 length of the pendulum is b/(l + s). In the latter, the lower 

 string simply alters the downward force on the upper mass 

 from g to (l + s)g. 



§ 11. When the ratio s of the lower to the upper mass is 

 very small, and a and b are not approximately equal, the 

 quadratic (29) may be written 



S 2 



