86 KEPORT— 1894, 



Also H cannot become minus infinity ; therefore H diminishes to a 

 minimum, and in the ultimate state of the system when H attains this 

 minimum we have (30) 



The quantity H has been called BoltzmanrHs minimiim /unction,^ and 

 the above theorem may therefore be called Bollzmann's minimum theorem. 



Rate of Subsidence of Disturbances. 



44. Equations (25), (26) have been applied to calculate in certain cases 

 the rate of subsidence of a disturbance in which Yf is initially unequal 

 to F/. In the paper already referred to Burbury has employed them to 

 investigate the rate of subsidence of disturbance in the case of a medium 

 of two sets of elastic spheres, the masses and numbers of spheres per unit 

 volume of the two sets being M, m and N, n respectively, and the disturb- 

 ance consisting in an initial small difierence between the h constants in 

 the two sets. He arrives at the result 



D oc e-^'-', d oci e-'", 



c=i^_(N + «) ^^Mm ,If! . . . (36) 



Here the h constants for the two sets are supposed to be h (1 +D) and 

 h {\+d) : their arithmetic mean is /t, and s is the sum of the radii of two 

 spheres. 



The same results had been previously found by quite independent 

 methods by Tait - and Natanson,-* both working under different assump- 

 tions. 



"Watson ^ has applied the same method to a number of lop-sided discs 

 in one plane, supposing that in the disturbed state the average kinetic 

 energy of rotation differs slightly from the two components of translational 

 energy, so that the law of disti'ibution is 



tfi 2 du dv dnj 



where /li is different from unity. The result is given by 



where 



«=-,,- \^n 



and 



K=N log fc^^ 

 3=/' 



=N < ^—^ '- -f higher powers of /:i — 1 > 

 C being a known numerical quantity. 



1 Burbury, Nature, Dec. 14, 1893. 



- Trans. B.S.E., 1886, pp. 82, &c. Sec vay first Report, §49, for his numerical 

 results. 



' L. Natanson, Wied. Ann., xxxiii. 1888, p. 683. 

 * Kinetic Theory of Gases, p. 49. 



