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XLIV. On the Vibrations of Approximately Simple Systems. 

 By Lord Rayleigh, F.R.S* 



THE meaning of the words "approximately simple" in the 

 above title will be most easily explained by an example. 

 Suppose that the " system M is a perfectly flexible string stretched 

 between two fixed points. If the longitudinal density is uniform, 

 the problem of determining the vibrations can be thoroughly 

 solved. We know that in each of the fundamental modes the 

 string vibrates as a curve of sines, and that the periods form a 

 harmonical progression. These fundamental modes of vibration 

 may coexist; and we know how to determine their amplitudes 

 and phases so as to suit arbitrary initial circumstances. But 

 when the density of the string varies from point to point, the 

 solution of the problem cannot in general be effected ; for it de- 

 pends on a differential equation which has hitherto proved intrac- 

 table. The question arises whether, when the string is nearly 

 uniform, an approximate solution cannot be obtained which shall 

 be correct so far as the first power of the deviation from uni- 

 formity. In this case the system may be called approximately 

 simple, in the sense that a small alteration would make it simple, 

 and bring it within the domain of exact analysis. The object of 

 the present paper is to show that such cases admit of a perfectly 

 general treatment by means of Lagrange's method of generalized 

 coordinates. 



Since the vibrations are supposed to occur in the neighbour- 

 hood of a configuration of stable equilibrium, and to be infinitely 

 small, the potential and kinetic energies are expressed by homo- 

 geneous quadratic functions of the coordinates and velocities. 

 By a suitable choice of coordinates (Thomson and Tait's ' Na- 

 tural Philosophy, § 337) the terms involving the products of 

 the coordinates and velocities may be got rid of, and the ener- 

 gies expressed as a sum of squares, with each term positive. 

 Thus if 0j, (j> 2) &c. be the normal coordinates, 



T=i[l]#+i[2]$ + 



V=i{l}£ 





Now suppose that the system is slightly varied. The energies 

 become 



T + ST=4(Ll]+8[l]>tf+...+S[12]<^+... ) 



V + SV=i({l}+S{l})#+-.-+S{13}^ 2 + ..., 



where, if new coordinates appear, their coefficients are small 



* Communicated by the Author. 



