1906-7.] Partition of Heat Energy in Molecules of Gases. 201 
and only in the case of rest ( A 2 = A 2 . . . . = A 3/! = 0) would both expres- 
sions be simultaneously zero. Let us use farther 
WM (20) 
to denote the number average of the time average, taken over all molecules 
of the second kind. 
For molecules of such structure, if they could exist, equipartition for 
the momentoids Jm h £ h would never occur, even with arbitrary collisions 
throughout. Equipartition would demand 
= (21) 
which, according to 19, cannot occur. 
To attain, free from contradiction, to a specification of molecules of such 
structure, would, moreover, be to contradict a much more fundamental con- 
sequence of Boltzmann’s theory, which says : Given a multitude of multiply - 
atomic molecules, of arbitrary structure, which obeys the law of distribution 
e -h. Energy^ equipartition holds for every system of momentoids (Boltzmann, 
Gasth., b. ii. § 42). For the molecules here discussed there is no distri- 
bution which can permit of equipartition for the momentoids Jm h £ h . 
§ 7. Having regard specially to this contradiction of one of the weightiest 
fundamental laws of Boltzmann’s theory, I would show in what follows 
that the 3 n inequalities (18) can never be fulfilled simultaneously. If, by 
means of the substitutions (12), the kinetic energy 
3 n 
T = i2 H? '.£ 2 A (22) 
1 
be transformed to the form 
3 « 2 
t =j 2 x < ( 23 ) 
i 
the quantities a hi must satisfy the following orthogonal conditions, 
3 n 
^4 vi h a 2 hi— 1 i— I, . . . . ,3 n . . . (24) 
i 
and 
3 n 
m h a M a 
But, from (24) and (25) it follows that * 
3 n 
»»»2 a2 « =1 7t=1 Sn ■ ■ ■ ■ (26) 
and 
3 n 
i 
* To see this readily, one introduces new quantities j hi = V m h ahi. 
*=0 1 3m. . . (25) 
( 27 ) 
