1906-7.] Dr W. Peddie on Vibrating Systems. 
181 
XIX. — On Vibrating Systems which are not subject to the 
Boltzmann- Maxwell Law. (Second Paper.) By Dr W. Peddie. 
(Read January 7, 1907. MS. received March 7, 1907.) 
1. In the first part of this paper ( Proc . Roy. Soc. Edin., vol. xxvi., 1905-6, 
p. 130), a method was suggested by means of which it might be found 
possible to discriminate amongst certain oscillatory dynamical systems 
according as they did, or did not, follow the law of equipartition of energy 
in their respective vibratory freedoms. The fundamental assumption, 
implicitly made in the proposed process, was that the mean value of the 
kinetic energy of the system could be expressed as a sum of mean squares 
of the time-rates of circular functions of the time ; which functions were 
not “ normal ” co-ordinates, for their products in the expression for the 
kinetic energy were supposed only to vanish on the average over sufficient 
time. The results obtained are therefore subject to such limitations as 
that postulate may be found to impose. It is of importance to settle the 
matter decisively ; for the method, if valid, gives a test which is independ- 
ent of any question as to difference of the time-average for a single system 
and the number-average for many similar systems, and it is also applicable 
to dissipative systems, provided that the dissipation function is quadratic. 
The investigation is continued in the present paper, and the result 
confirms the previous general conclusion as to the existence of systems 
which do not exhibit equipartition of energy amongst their freedoms. 
2. A distinction was drawn in the previous paper between cases in 
which the third law of motion held and those in which it did not hold. It 
was pointed out in § 4 that equipartition did not hold in a biperiodic 
system exempt from the third law, and this conclusion is quite general. It 
applies equally to a system possessing any number of freedoms, for the 
masses can always be chosen so that their ratio is either greater than the 
greatest, or less than the least, of the ratios b' / a', etc., as in § 6. 
It is important to remark that this absence of equality between action 
and reaction merely necessitates the existence of a suitably constituted 
medium through which that action is propagated with finite speed. 
3. Dr Paul Ehrenfest has kindly communicated to me an elegant proof 
showing that, in the case of normal co-ordinates, the expression (see § 6 of 
the former paper) 
2 {m P a 2 pr - m q a* qr )n r 2 A r 2 
