568 Lord Rayleigh on Iso-periodic Systems. 



where X 1? X 2 , &c. are constants for each point. These 

 equations indicate elliptic motion in the plane 



*(Y ) Z,-Z 1 Y,)+y(Z 1 X 1 -X 1 Z s ) +*(X 1 Y 2 -Y 1 X s )=0. (5) 



Thus every point of the system describes an elliptic orbit in 

 the same periodic time. 



An interesting case is afforded by a line of similar bodies 

 of which each is similarly connected to its neighbours'*. 

 The general formula for n 2 is 



2 _ C — 2Ci cos ka — 2C 2 cos 2Jca — . . . 

 ~ A — 2A 1 cos£a — 2A 2 cos 2ka— . . .' * ^ ' 

 in which the constants C , Gj . . . refer to the potential, and 

 A 1? A 2 . . . to the kinetic energy. Here C^ A 1 represent the 

 influence of immediate neighbours distant a from one another, 

 C 2 , A 2 the influence of neighbours distant 2a, and so on. 

 Further, k denotes 27r/X, X being the wave-length. If 

 Ci, C 2 . . ., Aj, A 2 . . . vanish, each body is uninfluenced by 

 its neighbours, and the case is one considered by Reynolds of 

 a number of similar and disconnected pendulums hanging side 

 by side at equal distances. It is obvious that a vibration of 

 any type is normal and is executed in the same time. If 

 we consider a progressive wave, its velocity is proportional to 

 X. A disturbance communicated to any region has no ten- 

 dency to propagate itself; the " group velocity" is zero. 



Although the line of disconnected pendulums is interesting 

 and throws light upon the general theory of wave and group 

 propagation, one can hardly avoid the feeling that it is only 

 by compliment that it is regarded as a single system. It is 

 therefore not without importance to notice that there are 

 other cases for which n assumes a constant, and the group- 

 velocity a zero, value. To this end it is only necessary that 



C :C 1 :C 2 :...=A :A 1 :A 2 : (7) 



If this condition be satisfied, the connexion of neighbouring 

 bodies does not entail the propagation of disturbance. Any 

 number of the bodies may remain at rest, and all vibrations 

 have the same period. 



We might consider particular systems for which C 2 , C 3 . . . 

 A 2 , A 3 . - . vanish, while Ci/C = Aj/A ; but it is perhaps more 

 interesting to draw an illustration from the case of continuous 

 linear bodies. Consider a wire stretched with tension T l5 

 each element dx of which is urged to its position of equilibrium 

 (y = 0) by a force equal to fjuydx. The potential energy! 



is given by y = ^^ + iTl j( JJ ^ (8) 



* Phil. Mag. vol. xliv. p. 356, 1897. 



t See ' Theory of Sound,' §§ 122, 162, 188. 



