188 
Proceedings of the Royal Society of Edinburgh. 
Similarly, selecting the m — 1 equations which involved X 22 
get 
and so on. Finally, 
k - 1 
^12 ~ T~ 
A 2] 
Generally, 
k - 1 
h l p - V- - 
Therefore the equation 
^12 
_ 1 
X22 
^22 
a 12 
a 21 
1 
f 
a 22 
1 
a 2m 
a n 
X22 
a 12 
X2TO 
& l m 
a ' pi 
1 
1 
^ pm 
a \\ 
X 
a 12 
Xp»i 
a 1 m 
[Sess. 
X m2J we 
1 + /q 2 ^-2r^-2s "t ^13X3^X35 + . . . . + k lm k. mr \ ms — 0 
becomes 
l _|_ ^ 2s _j_ _j}_ a 3 r a 3s 
^12 ^ 1 r & Is ^13 M lr & Is 
1 (l mr Cl ms _ 
• ~r y — — r 5 
"'lm ^ lr a Is 
or 
ls + M 2 a' 2r a , 2s + M S a 3r a 3s + . ... + M m a mr a ms = 0 , 
which shows that, in a system having m degrees of freedom with m distinct 
periods, the property of normality always exists, and therefore Dr 
Ehrenfest’s conclusion applies, and we can neither prove nor disprove the 
doctrine of equipartition of energy. If, on the other hand, even only two 
periods coincide, n r — n s , the above condition is not binding, and we may 
deduce infinities of cases where equipartition is not observed. At least this 
is so if we can postulate free response of the As and as to random impacts 
so as to satisfy ( 5 ). That condition being granted, the above numerical 
examples are conclusive. 
No doubt we can introduce, in an infinite number of ways, a linear 
transformation of co-ordinates, which would impose “ normality.” It would 
be impossible then to prove, by the present method, anything definite 
regarding the partition of energy amongst these new variables. But these 
variables are not those corresponding to the actual freedoms of the 
individual masses. Even if it could be otherwise proved that the energy 
is equipartitioned amongst them, it is not clear how any conclusion could 
be drawn from that fact regarding the partitioning amongst the actual 
freedoms. Unequal salaries, due to A, B, and C, can be equipartitioned, in an 
infinity of ways, amongst A ', B', and C' ; but law prevents the process. 
11 . In this connection it is important to examine analogous questions of 
partition in non- vibratory systems. 
Lord Kelvin has pointed out, and it seems to be fully admitted, that 
