202 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Equations (26) give (see eq. 16) the relation 
3 n 
2 pi «=° ( 28 > 
i 
for each pair of indices h, k. Therefore the Sn inequalities (18) can never be 
satisfied simultaneously, and the system postulated in § 6 cannot exist. 
We remark further, that the conclusion of this paragraph, and also 
equation (28), are independent of the frequencies p { being different from 
each other, or some of them being equal. 
§ 8. Equations (28) and (16) show farther: If equipartition exists for 
the momentoids X p . . . . X 3?l , and all the frequencies p t are different 
from each other, equipartition also holds for the momentoids Jm h £ h . 
For if 
• -{x**}. 
• • (29) 
%.e. if 
p\A\ = p 2 2 A\= . . . 
— n 2 A 2 
* ~~ t 3 n a 3 n J 
■ ■ (30) 
equations (16) and (28) give 
{mif 2 1 } = {m 2 f 2 2 }= . 
• • • ={^3n4 2 3n} - 
• • (31) 
The case in which not all the p i are different from each other requires some 
complementary presumptions. Take, e.g., p 1 =p 2 - Then the relation (14) 
no longer suits for i = 1, j = 2. We have instead 
{ X 1 X 2 } = A : A 2 cos </> 12 (32) 
where <p is the phase-difference between the two fundamental oscillations 
X p X 2 , which remains constant during the free motion. In order to arrive 
at equation (16) again, we make the farther assumption regarding the 
distribution, that 
A 2 cos <f> 12 = 0 ..... (33) 
It is always fulfilled if the distribution referred to allows contrary phase 
differences to occur, with equal frequency, in the various molecules. But 
assumption (33), together with (29), then leads to equation (31) in the case 
of partly equal frequencies also. Consequently it is possible to arrange 
equipartition for every structure of the vibrating molecules, between every 
set of momentoids. 
§ 9. The above discussion is intentionally confined to the two definite 
questions which were formulated in the introduction ; for my intention in 
this connection was merely to refer to the restricting remarks which 
Boltzmann himself made on the H-theorem. On this account the remaining 
questions and objections which have recently been formulated in reference 
to the equipartition law can be left aside in the meantime. 
{Issued separately August 28 , 1907 .) 
