Lord Rayleigh on Iso-periodic Systems. 569 



If the u rotatory inertia " be included, the corresponding ex- 

 pression for the kinetic energy is 



in which p is the volume density, w the area of cross section, 

 and k the radius of gyration of the cross section about an 

 axis perpendicular to the plane of bending. In waves along 

 an actual wire vibrating transversely the second term would 

 be relatively unimportant, but there is no contradiction in the 

 supposition that the rotatory term is predominant. The differ- 

 ential equation derived from (8) and (9) is 



where 



a 2 = T 1 /pco, c 2 = fJL/pco (11) 



If we suppose that there is no tension and no rotatory 

 inertia, a=0, k = Q, and the solution of (10) may be written 



y = cos ct ,y x + sin ct . y 2 , .... (12) 

 y ly y 2 being arbitrary functions of x. If y 1 = cosmx, 

 y 2 = sinmx. (12) becomes 



y— cos (ct — mx), (13) 



and the velocity of propagation (c/m) is proportional to \, 

 equal to 27r/w. This is the case of the disconnected 

 pendulums. 



On the other hand w T e may equally w r ell suppose that c is 

 zero and that the rotatory inertia is paramount, so that (10) 

 reduces to 



j #9 |a i<fo-0. 



dx 2 dt 2 dx 2 



The periodic part of the solution is again of the form (12), 

 and has the same peculiar properties as before. 



In the general case we have the solution for stationary 

 vibrations 



y = sin mx cos ?xt 1 (14) 



where m = i7r/l, i being an integer, and 



c 2 + a 2 m 2 



" ~l+,c 2 m 2 ( 15 ) 



This gives the frequencies for the various modes of vibration 

 of a wire of length / fastened at the ends. 



If « 2 = a 2 /c 2 , n becomes independent of m as before. 

 If k 2 < d 2 /c 2 , n 2 increases as i and m increase and approaches 

 a finite upper limit a 2 /* 2 . The series of frequencies is thus 

 analogous to those met with in the spectra of certain bodies*. 

 * Compare Schuster, "Nature/ vol. lv. p. 200 (1890). 



