ON OUR KNOWLEDGE OF THERMODYNAMICS!. 117 



author employs the method of redaction to sums of squares and subse- 

 quent use of Lagrange's equations — in short, most of the steps that are 

 erroneous in Maxwell's work ; the proof is therefore invalid except in 

 certain special cases. One result is, however, interesting; for the case of 

 a system whose configuration is determined by a single coordinate, and 

 ■whose period of oscillation is t, Boltzmann finds 



SQ=2T8 log, (TO .... (72) 



thus giving for the entropy the expression found by Clausius, and 

 described in the first section of this Report (§ 12, equation (14) ). 



47. Statistical Construction of Monocyclic Systems. — A very interest- 

 ing and suggestive paper has been published by Boltzmann,' who has 

 shown how systems possessing monocyclic properties can be built up by 

 combining a large number of systems which are similar to one another, 

 but not individually monocyclic. This is the paper to which reference 

 has been made in § .37. 



A single particle moving in an elliptic orbit about a centre of foi'ce in 

 the focus is not monocyclic in itself, but a monocyclic system may be 

 built up by taking a very large number of such particles, thus forming a 

 stream or a kind of Saturn's ring, whose density at any point of the 

 orbit is independent of the time. Here, if the attraction at distance r be 

 fl/r'^, Boltzmann finds 



■where 2Ts a 



'^=T' '=T-i- 



Moreover, if jx is the total flux across any section up to the time t, and 

 ni the mass of the ring, ■we have 



27r dfL 



and, therefore, fqdt may be taken as a generalised coordinate of the 

 system. 



Another example is afforded by a stream of particles of total mass m 

 performing rectilinear oscillations under a conservative system of forces. 

 In this case Boltzmann finds 



dQ=2Td]o^,iT (73) 



■which agrees with Clausius' result (equations 14, 72). Here we may 

 take for the generalised velocity and momentum of the system respec- 

 tively, 



q = mji, s = 2Tlq=2iT/ui . . . (74) 



A particular case is that of a stream of particles reflected backwards 

 and forwards between two fixed perfectly elastic parallel walls at a dis- 

 tance a apart. If ^m is the mass of the stream going in either direction, 

 V the velocity, and H the kinetic potential, we have 



dQ=mvdv + viv-— =qds .... (75) 



' Crelle, Journal, xcviii. p. 68. 



