1905 - 6 .] Dr W. Peddie on Vibrating Systems. 
131 
2. Rayleigh (> Scientific Papers, vol. iv., no. 253), in defending 
Maxwell’s argument, desires “some escape from the destructive 
simplicity of the general conclusion.” Kelvin {Balt. Led., App. B) 
speaks of the difficulty as one of the two “nineteenth century 
clouds over the dynamical theory of heat and light,” and concludes 
that the doctrine should be denied. He gives a number of tests 
tending to show considerable deviations from equi-partition of 
energy between translational freedoms and also between trans- 
lational and rotational freedoms. He also points out an, at least, 
ideal case in which this equi-partition is impossible. 
The object of the present communication is to determine cases 
in which equi-partition of vibrational energy cannot take place. 
3. It is convenient to suppose that the motions are limited to 
one-dimensional space. The simplest case is that of two freedoms 
expressed, say, by the conditions 
$2 = ^ 2^1 — ^* 2^2 * 
These give 
(r + r')^ = rk sin {nt + a) + rk ' sin {n't + a) , 
{r + /)£ 2 = A sin {nt + a) - A' sin ( n’t + a') , 
where n 2 — aq — rb 2 , n 2 = a 1 + rb 2 , the quantities r and - r being re- 
spectively the positive and negative roots of b 2 \ 2 + (a 2 - a- l )\ — b l = 0. 
Hence we obtain 
2 (r + /) 2 {m 1 i 1 2 - m 2 £ 2 2 } == n 2 (m 1 r' 2 - m 2 )A 2 + n 2 {m 1 r 2 - w 2 )A 1 ' 2 
where the bracket indicates an average taken over a time which 
is long relative to the longer of the two periods, and m x , m 2 , are 
the masses. If it is possible to determine conditions under which 
tbe quantity on the right-hand side of the equation is either always 
positive or always negative, the Boltzmann-Maxwell distribution of 
energy is impossible. 
Now the third law of motion necessitates the condition 
/; 1 m 1 = 6 2 m 2 , so that rV 2 = m 2 2 /m 1 2 . Hence it is impossible to 
have both brackets on the right-hand side always of one sign. 
Thus, so far as this investigation goes, we can make no assertion 
as to the observance or non-observance of the Boltzmann-Maxwell 
Law, except in the case of numerical equality of the two roots, in 
which case it certainly is observed. 
