186 Proceedings of the Royal Society of Edinburgh. [Sess. 
The signs of the various quantities (4) , § 5 , have now to he determined. 
The chosen values of the As give 
Aa' n = 4 , A«' 12 = 1 , 
So we find 
Aa' 13 = -5/2; Aa' 21 = -7/2, Aa' 22 =l 
Aa' 31 = 1 , A a 32 = - 1 , Aa' 33 = 0 . 
Aa' 23 = 5/2 ; 
A 2 (M 1 a' 2 11 - M 2 a' 2 21 ) = 3f ; A 2 (M 1 a' 2 12 - M 2 a' 2 22 ) = 0 ; A 2 M 1 a' 2 13 
A 2 (M 9 a' 2 - Mo«' 2 ) = 1 If ; A 2 (M 2 a' 2 „ - M s a' 2 ) = i ; A 2 M 2 a' 2 28 
A 2 (M 3 a' 2 31 - M 1 a' 2 11 ) = - 15*; A 2 (M 3 a' 2 32 - M,a 2 12 ) = - i; A 2 M 3 a' 2 33 
^23) — ^ ; 
- M g a 2 33 ) = 6^ ; 
- M 3 a' 2 13 ) = -6J. 
The sign of every one of the three quantities in any one of these three rows 
is similar. Therefore there is no equipartition of energy between any pair 
of masses. The average energy of M x exceeds that of M 2 , which exceeds 
that of M 3 . 
If the postulate (5) were not granted, it is not easy to see how the quan- 
tities A and a , which are the arbitrary constants of integration, could be 
regarded as truly arbitrary within their allowable ranges. The arbitrari- 
ness is supplied by the randomness of collisions. 
9. We shall now take an example in four freedoms with the conditions 
n 2 = n^ = n^n v With the values 
Aj] — 1 J ^31 — f > ^41 — f ) 
X 22 = — 1 > A 32 = 1 , A 42 = 2 , 
^"23 = 1 J ^33 = ^ > ^43 = 1 , 
A 2 4 = — 3 , /V 34 = 0 , A 44 = 3 , 
the conditions (2) for inequality of periods give 
1 Zq 2 & 43 + 27 c 14 — 0, 
1 — 37^ 2 + 3& 44 = 0 , 
^^12 — ^13 ~ ^14 = 6 • 
Hence we have Jfe u =2/3,* u ,= l ,k u = 1/3 ; and also, by the expression iom r 
in § 4, if we take a 12 = — a ls = a u = 1 , we have — = a n -f 2 , — n 2 2 — — n s 2 
= — n 2 = a 11 — 1. Taking a u = — 3 , these give n x = l ,n 2 ~n z = = 2 ; 
and the conditions (2) for equality of periods are identically satisfied. 
Taking a 22 = — 10/3 ,a 23 = —2/3 , a 24 = 2/3 , a 33 = — 3 , <x 34 = — 1 , <x 44 = — 11/3 , 
and remembering a pg k lg = a gp k lp , we find that equations (1) connecting the As, 
and the <xs are identically satisfied. The quartic in n 2 is therefore 
- 3 + n 2 
1 
- 1 
1 
2 
3 
- ^ + n 2 
2 
3 
2 
3 
- 1 
- 1 
-3 + n 2 
- 1 
1 
3 
i 
_ 1 
3 
- V+w 2 
