1905 - 6 .] Dr W. Peddie on Vibrating Systems . 
141 
phase for which the conditions of encounter with the fixed obstacle 
are satisfied, but they are not subject to the equations of momentum. 
It is difficult in a case of such extreme complexity to arrive at a 
thoroughly satisfactory conclusion, but we may with considerable 
confidence assert that, except for particular forms of the surface of 
the fixed obstacle, the system will sooner or later, after a sufficient 
number of encounters, pass through every phase consistent with 
the equation of energy.” 
These remarks of Maxwell seem to apply as conclusively to 
material systems constituted in the manner postulated in this 
paper. At least this is so if we presume that, in a collision, 
individual masses of the system may suffer independent impacts. 
If that view be correct, Maxwell’s assumption is satisfied while his 
conclusion is departed from. It seems that, before we can arrive 
at his conclusion, we must at least assume, when periodic motions 
are concerned, that no phase-cycle is to be continuously completed 
by any individual member of the system. Thus, in the liquid 
and isotropic solid states, equi-partition of energy amongst all 
freedoms may be possible, while in sparse gases it is impossible. 
But the statement, that “ any law of force which is consistent 
with the equation of energy ” can be postulated, is in general 
impossible because of the restricting assumption. In general the 
expression of the law involves relations amongst the momenta and 
the co-ordinates, which relations prevent the simplification that 
arises in the Boltzmann -Maxwell treatment, or in Jeans’s treat- 
ment, and leads to the law of equi-partition. The usual methods 
will then lead to results such as those obtained above. 
As a simple example, which includes Maxwell’s example as a 
special case, we find, in §5, the relations (& 2 £ x + a 2 £ 2 + — 
( a i£i + ^ 1 ^ 2 + e ^ c * These relations, like the energy con- 
dition, subsist despite encounters and collisions. Similar conditions 
will hold with laws other than the generalised Hooke’s law. In 
this connection an interesting paper by Bryan, published in the 
volume of the Arch. Neerlandaises already referred to, § 1, would 
have been more fully quoted had I observed it before the preceding 
sections were written. In 1900, Bryan, by a different process, had 
indicated the preponderance of conditions for unequal partition 
{Issued separately May 24 , 1906 .) 
