ON OUR KNOWLEDGE OF THERMODYNAMICS. 109 



The right-hand side is equal to d log (s-), where 5 is the angular 

 momentum, thus agreeing with Helmholtz's result (§ 21, equation 27). 



Boltzmann shows how such a machine may be made to undergo a 

 series of transformations analogous to Carnot's cycle. In an isothermal 

 transformation the angular velocity and the distance of the bead from 

 the shaft are varied in such a manner that the kinetic energy of rotation 

 remains constant ; in an adiabatic transformation no work is performed 

 on the shaft, and thei-efoi'e the angular momentum, mr'-u), as also the 

 corresponding entropy, remains constant. 



The author gives other models of monocyclesin which several movable 

 rods and beads are attached to the same shaft. A Watt's governor is 

 another simple example of a monocycle. Other examples of 'kinetic 

 engines ' were given by Professor Osborne Reynolds in a lecture delivered 

 on November 15, 1883.' 



36. An attempt is also made by Boltzmann to extend the dynamical 

 analogy to irreversible processes, by showing that for a cycle of changes 

 which do not take place infinitely slowly we must have fclQ/T <0. Un- 

 fortunately, however, this generalisation does not hold good if the system 

 is frictiouless, and, as already remarked, the introduction of friction is 

 not allowable in forming a purely dynamic analogue of the pi'operties 

 of heat. Boltzmann assumes that ichen the head is sliding outwards along 

 the spoke, the tension in the string is always slightly less than the centrifugal 

 force, and that when the head is sliding inwards the tension is alivavs sliglithj 

 greater than the centrifugal force ; for otherwise (he says) the bead and 

 suspended weights would never start moving. Thus if p denote the ten- 

 sion in the string, we may put 



p ^ mrtji^ — e, 



where e always has the same sign as dr. 



But the statements in italics are not true if the spoke is frictiouless, 

 for the equation of motion of the bead is 



so that 



d^r 



e=m — ,. 



dt- 



If the bead be allowed to slide outwards, starting at distance r^ and 

 stopping at distance r^, then d'^r/dt^ must be at first positive and afterwards 

 negative, for otherwise the outward velocity dr/dt would continually 

 increase. Hence e cannot always bave the same sign as dr, and Boltz- 

 mann's argument fails. 



37. Boltzmann's mechanical representation of a system in whicb dQ 

 has no integrating divisor consists of two parallel revolving vertical 

 shafts, which we will call A, B, each similar to that described in § 35 and 

 figured in § 38, each provided with a horizontal revolving spoke, along 

 which a bead is capable of being made to slide. The motions of the 

 two shafts are connected together through the following mechanism .- — 

 The motion of A is transmitted by means of bevelled cog-wheels to a 

 horizontal shaft C, carrying at its other end a i-ough disc Gr, which of 

 course revolves in a vertical plane. Attached to the vertical shaft B is a 



' JVature, vol. sxix. p. 113, 



