132 Proceedings of Boy al Society of Edinburgh. 
4. It is worthy of note that the impossibility of obtaining a 
conclusion in the case of two freedoms depends on the observance 
of the third law of motion. If the third law be not observed in 
sub-atomic dynamics, there could be bi-periodic atoms exempt 
from the Boltzmann-Maxwell Law. There is no a priori reason 
why the third law should hold. Steady vibrations could take 
place until a collision, sufficiently violent to alter the equations of 
motion, occurred. At that stage, non-observance of the law could 
give rise to effects analogous to some of those made evident in the 
phenomena of radioactivity. 
5. The next simplest case is that of three freedoms as typified 
by the equations 
£l = ^1^1 "k ^1^2 "h ^1^3 > 
^2 = ^ 2^1 "k a 2^2 4 " ^ 2^3 > 
^3 = C 3^1 d* d & $2 + a 3^3 • 
The third law of motion gives the conditions b 1 m 1 — b' 2 m 2i 
c 1 m 1 = c 3 m 3 , d 2 m 2 = d 3 m 3 ; whence 
b\d 2 c 3 = cf> 2 d 3 ( 1 ) 
This is Tait’s condition ( Scientific Papers , vol. ii., art. cxx.), 
for the reality of the roots of the cubic 
u - x b , c n 
d c 
= 0 
( 2 ) 
u 3 3 3 
regarded as determining the non-rotated lines in homogeneous 
strain possible in matter. Indeed, the form of the above equations 
of motion exhibits the analogy between homogeneous three- 
dimensional strain and the vibrations of a three-period system. 
The problem of determining non-rotated lines in the one is the 
problem of determining fundamental periods in the other. Tait’s 
method of reducing the general differentially irrotational strain to 
a pure strain in the one is the method of reducing the general 
three-period system to an equal-mass system in the other. This 
will he farther discussed in § 9. 
Tait’s limitation of the strain problem to strain possible in 
matter corresponds to the subjection of the vibrating system to 
the third law of motion. We can still have the condition 
b 1 d 2 c 3 = cf 2 d 3 if we take, for example, 5 1 m 1 = & 2 m 2 , c 3 m 3 = kc 1 m v 
