200 
Proceedings of the Royal Society of Edinburgh. [Sess. 
There subsist farther the substitutions * 
^ “1” ^ 12^2 • 
. . . + Q>y , 2>n 1^3 n 
+ 
e 
II 
<N 
► 
n — ®'3n 5 1 + • • • • 
. ( 12 ) 
with analogous substitutions for £ h . We introduce now the following time 
average symbol 
| - m k £ 2 k | = f (m h i 2 h - m k £ 2 k )dt . . . (13) 
Here 0 denotes a time which is large relatively to the longest of the periods 
of the normal vibrations. The integration is extended over a free motion 
of the molecule of the second kind. We shall correspondingly make for 
the following application, the assumption that a molecule of the second kind 
makes numerous oscillations between two collisions.*]* From the assumption 
that all frequencies, p iy are different from each other, it follows that 
{X,XJ = 0 for i=¥j . 
On the other hand we have 
. (14) 
. (15) 
One has also, for the difference of the time averages of the kinetic energies 
2 m h£h 2 and %m k £ k 2 the expression 
3 n 
- {mj 2 *} = P 2 i A2 i( m h a2 hi - m k a 2 ki ) . . . (16) 
i 
The quantities 
Pta = m h a 2 M - m k a 2 ki (17) 
are determined by the structure of the molecule, the quantities A 2 { by the 
occasional excitation. 
One can now put the question J — Is it possible so to choose the structure 
of the molecule, that is, the quantities m 1 . . . . m 3n and f3 k , that, at least 
for one pair of indices h , k, all the 3^ quantities P shall satisfy 
PU>0 i=l, . . . . , 3 f> . . . . (18) 
Let us assume that this might actually be possible. Then, from equation 
(16), for each arbitrary excitation (for each value of the quantities 
A ± ... . A 3n ) we would have 
(19) 
* Here, naturally, the earlier assumptions regarding l n , £ 12 , ? 13 are no longer made, 
t Cf. Peddie, Z.c., § 13. 
t Cf. Peddie, U, §§ 2, 3, 13, 14. 
