198 
Proceedings of the Royal Society of Edinburgh. [Sess. 
systems, the molecules of the second kind possess 3 n normal-vibrations. 
The normal co-ordinates may (in arbitrary order) be denoted by 
X l5 X 
2» 
i X 3n 
For the kinetic and potential energies we have the following representations : 
3 n 3 n g 
2T=2»<A=2x < (3) 
1 1 
3 n 3 n 3 n 
(4) 
i i i 
There are, farther, the following substitutions : — 
= hl^l "t" h.2^2 "t ^13^3 "t" • • • • h ) 3n $ 3 n 
X * =, ^ + - . . . (5) 
X3 n = hn > 1 ^1 “1” h n ) 3 n £3 n. 
The substitution coefficients l ih depend on the mechanical structure of the 
molecule. We make now the following special assumption regarding this 
structure. Postulate that 
hi = h2 = hs = ® • • • • • • ( 6 ) 
All other coefficients l ih shall have any values, different from zero, so far as 
they are compatible with positive values of m and real values of /3 7l *. The 
physical meaning of this assumption is this : — 
In the expression for the energy connected with the first normal vibration 
E 1 = i(^ 2 i + B 1 X 2 1 ), (7) 
the speed components ^ 2 , (* 3 , of the atom A do not appear (nor also its co- 
ordinates). 
5. One sees easily now that this assumption regarding the structure, 
together with the restricting assumption regarding the collisions, brings 
in non-equipartition. 
To this end we show next that the quantity E x retains for each molecule 
its original value (E x is an integral of the type defined in equ. 2). 
(a) During the free motion the quantities E p . . . , E sn remain constant. 
( b ) Ej also does not change by the blows. For the quantities 
£4 » • • • • » £3 n > f !»■••• £3 n 
possess immediately after the blow the same values as they had immediately 
before the blow. Only the quantities 
£1 > £2 > £3 
suffer certain sudden changes. But, since the quantities f 2 , £ 3 , do not 
