236 Prof. J. D. Everett on Dynamical 



X being either 2ma or any submultiple of it exceeding 2a. 

 This gives m — 1 modes. 



As an example, if the distance between the fixed ends 

 is 4a, the values of A- are 



Sa, 4a, 8a/3. 



In the first mode the amplitudes of the three free particles 

 are as the sines of 



tt/4, tt/2, 3tt/4. 



In the second mode, as the sines of 



tt/2, tt, 3tt/2. 



In the third, mode, as the sines of 



3tt/4, 3tt/2, 9tt/4. 



§ 9. Another mechanical illustration that has been often 

 mentioned as analogous to certain optical phenomena is the 

 mutual influence of pendulums. 



First, suppose two simple pendulums of masses 1 and s, 

 with natural frequencies G>,/27r and <o 2 /27r, their bobs being 

 at the same level, and elastically connected so that there is a 

 mutual push or pull according as their distance is less or 

 greater than when both pendulums are vertical. The con- 

 nexions are supposed to be of negligible mass, and the 

 vibrations to be so small that the vertical component of 

 the push or pull is negligible. The movements are supposed 

 to be confined to one vertical plane. 



Let x denote displacement of the mass 1 from the vertical 

 through its point of support, reckoned positive when towards s, 

 and f the displacement of the mass s, reckoned positive when 

 away from the mass 1. Then x— j is their approach, and 

 £ — x their recess. Let fi(x — f) be the push, and jJ<{i;—x) 

 the pull. 



If the masses were unconnected, we should have 



£=-0)^, f= r © 2 2f (17) 



When they are connected, we have 



X = - CD^X + fX^ — x), ^ 



l — -rt+fc-e, } (18) 



When the system is vibrating in a fundamental mode, 

 x and f are in a constant ratio. Assume %=kx ; then 

 equations (18) become 



x = -{w l 2 + /Lt(l — k)}x, -, 



*,=-.; ^(i-A)!,, }• • • ^ 



