185 
1906-7.] Dr W. Pecldie on Vibrating Systems. 
equipartition may not hold in the case of three freedoms with distinct 
periods, we cannot prove by the present method that it does not hold. 
8. We shall next take the case of three freedoms with two periods 
identical and the third distinct. In this case the three left-hand alternatives 
of (2) may still hold. If they do hold, no conclusion can be drawn ; but if 
one does not hold, we cannot establish the condition for normality, and so 
Dr Ehrenfest’s result does not apply. 
We have now to make the triple sum, expressed in § 5, vanish by 
extending the average so as to include many collisions, and making the 
assumption 
{ A r A s COS (a r - a s ) } = 0 (5) 
with regard to the term in which n r = n s . 
Because of the random nature of the collisions relatively to the actual 
co-ordinates, this seems to be the proper assumption to make. Yet it must 
be recognised as an assumption to be verified by tests. If we can construct 
a three-freedom system, having two periods coincident, and satisfying the 
conditions which secure unequal partition of kinetic energy on the average 
amongst the freedoms, the assumption will be justified. 
We shall take n 1 = n^^n 2 ] X 21 = l, A 22 = l, X 23 = 2 ; A 31 = 2, A 32 = — 
A 33 = 3. With these values, the conditions 
1 ^ 12^’21^'22 "h ^'13^’31^'32 == 9 
1 + ^ 12 V 2 V 3 “t ^13^32^33 = 0 
give k 12 = 1 , k 13 = 2 , and the remaining quantity 1 + k 12 \ 2S \ 21 + & 13 A 33 A 31 
is not zero, so 'that the condition for normality is not satisfied. Going 
back now to equation (1), we find 
A 25 ( a n + A 2q a 21 + A 3g a 31 ) = a l2 + X 2g a 22 4- A 32 a 32 
A 3g («n + A 2q a 21 + A 32 a 31 ) = a 13 + X 2g a 23 + A 39 a 33 
in which q takes the values 1,2,3. Taking the values a n = — 3 , a 12 — — 2 , 
a 13 = 1 , a 22 = — 3 , a 2Z = 1 , a 33 = — 2 , and remembering the relations a 21 k n 
= a 12 k 12 , etc., it is easily found that the above equations are identically 
satisfied. Therefore the cubic in n 2 (see § 3 of the first paper) is 
- 3 + n 2 
_ 9 . 
-2 
-3 + rc 2 
2 
1 
1 
— 2 + n 2 
= 0 
the roots of which are n x 2 = n s 2 = 1 , n 2 — 6 . This may be verified from the 
expression — n 2 = a n + A 29 a 21 + A 32 <x 31 . If we put M 1 = 1 , we get M 2 = 1 , M 3 = J , 
and the dynamical system is completely specified.* 
* The numerical example given in the first paper was fallacious. 
