4 34 Lord Rayleigh on the Fundamental 



I will remark finally that, when the paths of the points are not 

 closed, the transition from closed paths to those not closed can 

 be effected in precisely the same manner, whether we write the 

 Hamilton-Boltzmann equation in its primitive form or in the 

 logarithmic form as modified by Clausius. In truth the form of 

 the equation does not occasion, nor does it prevent, in this case 

 the difficulties mentioned by Professor Clausius at the end of his 

 memoir. 



For an easier view of the question, let us compare the dates of 

 the development of that dynamic equation to which the second 

 proposition of thermodynamics is reducible. Hamilton set up 

 the equation in the year 1834, but presupposed in the demon- 

 stration that the system of points is homogeneous, and the form 

 of the force-function invariable. In 1868 Boltzmann applies this 

 equation to a heterogeneous system, and, for the first time, brings 

 it into the well-known form of the second proposition of the 

 mechanical theory of heat. In 1871 Clausius proves that the 

 same equation is also valid when the form of the force-function 

 changes, but the mean value of the resulting variation =0. 

 Consequently, within these thirty-eight years, only the validity, 

 the applicability, of the equation has been extended, without the 

 equation itself undergoing any essential alteration, even inform. 

 I repeat the concluding words of my previous memoir : — "What 

 in thermodynamics we call the second proposition, is in dynamics 

 no other than Hamilton's principle, the identical principle which 

 has already found manifold applications in several branches of 

 mathematical physics." 



LVI. On the Fundamental Modes of a Vibrating System. 

 By Lord Rayleigh, M.A., F.R.S* 



THE motion of a conservative system about a configuration 

 of stable equilibrium may be analyzed into a series of nor- 

 mal component vibrations, each of which is entirely independent 

 of the others. When one of the components exists alone, the mo- 

 tion is simple harmonic, the period of the vibration and the phase 

 at any given moment of time being the same for all parts of the 

 system. In such a case the system is said to vibrate in a fun- 

 damental mode. In order to represent the most general kind of 

 motion, the whole series of normal components must be ima- 

 gined to exist together, the amplitude and phase of each being 

 arbitrary. 



The number of the normal components, which also expresses 

 the degree of freedom enjoyed by the system, may be either finite 

 or infinite, though, strictly speaking, every natural system be- 

 * Communicated bv the Author. 



