1906-7.] Partition of Heat Energy in Molecules of Gases. 199 
enter into E v that quantity possesses, immediately before and after the blow 
(therefore in general always), the same value. 
We consider at present the simplest case : E x originally zero for all 
molecules. It remains then lastingly zero, and with it also the value of 
X x 2 . So the mean value of X x 2 taken over all molecules is lastingly zero. 
( 8 ) 
however one may change the temperature of the first gas.* 
On the other hand, the law of equipartition would demand that 
F 1 = F 2 =. . . . =F 3w = a.T (9) 
for, according to (3), the kinetic energy is made up of sums of squares of 
F which are the so-called “ momentoids ” (Boltz., b. ii. § 33). 
For certain questions it would be interesting to settle : Does this 
manifest non-equipartition for the momentoid X x necessitate that non- 
equipartition holds also for Jm h £ ? To be able to deduce this consequence, 
one must exclude all the special structures, which, besides the integral 
E x = const., admit other integrals of the form (2). Yet that would lead as 
far. It is sufficient for the preceding object to settle that at any rate 
tke momentoid X x no longer complies with ike law of equipartition. 
Throughout, it is essential for the attainment of the result that we make 
the assumption, previously taken into consideration by Boltzmann, regard- 
ing the limitation of the collisions. 
If one, on tke contrary, made regarding tke blows the assumption I., 
tke H-tkeorem would give equipartition for both series of momentoids 
( and X*). 
§ 6. We pass now to the consideration of the test of that class of non- 
equipartition cases which, in the introduction, we denoted as the second 
class of non-equipartition cases. Apart from making an assumption re- 
garding the limitation of the blows to the atom A, equipartition can only 
be excluded by an assumption regarding the structure of the oscillating 
molecule. 
We make now the assumption that the frequencies of the 3 n normal 
vibrations possess distinct values p v .... , p 3n . f We have 
Xi = A t sin pi(t + ti) (10) 
= cos Pi(t 4- 1) (11) 
* One can also proceed from any other original value of E r Only one must then 
specially prove first that X\ remains constant with E r There exists here an analogy to 
the case of rigid molecules with rotational symmetry, 
t In § 8 this restriction will be removed. 
