134 
Proceedings of Royal Society of Edinburgh. [sess. 
*^[^2 (^2^3 C '$ a Y ) 4 ^3^2) ^3] 
+ A 2 [& 2 (a ^3 - JjCg) + (oq - a 2 )(b 2 d 3 - c 3 a 2 ) - d 3 (a s b 2 - c 3 d 2 ) + 
c 3 {afa Y - a 2 ) + d 2 d 3 - c x c 3 )] 
+ A[(flq — a^){afL 3 - b Y c^) — bfb 2 d ?i — c 3 a 2 ) + cfc\ d z — af-^) — 
d ^ a s( a i ~ a 2 ) + d 2 d z ~ c 2 c s)] 
-W«A-W + #A- fl s y] =0 .... (5) 
The roots of this equation being regarded as known, the first of 
equations (3) gives the corresponding values of g. 
The condition (1), for reality of the roots of the cubic (2), gives 
the single definite relation which subsists amongst the constants, 
and further limiting relations are imposed through the three 
equations of type (4). Tor, if the three quantities in brackets on 
the right-hand side of one of equations (4) be of like sign, there 
can never be equi-partitioning of energy between the masses with 
which that equation deals. 
The Boltzmann-Maxwell Law would, if it were applicable to 
vibrational freedoms obeying the generalised Hooke’s Law 
expressed by the equations of motion given in § 5, constitute a 
proof of the impossibility of satisfying all the above relations 
simultaneously. 
8. To obtain a definite test, we may choose X x = — 1, A 2 = 0, 
*3 = 1; whi ch give g 1 = [{b 2 - \) - (a 1 - a 2 )]/ (c 3 + d z ), g 2 = - bjd 3 , 
g 3 = [(6 2 - + (a 1 - a 2 )]/ (d 3 - c 3 ), and also 
a i~ ~ Uo 1 ® 2 = Mi 4 ^3 j a 3 = — g 2 , 
b 1 = ^2 — /^3 5 ^ 2 = /^3 — Al ’ ^ 3 ~ Al — » 
C l= 1 , C 2~ ~ > C 3 =1 - 
Hence the inequalities are 
m 1 
> 
(H ~A 3 \ 2 
> 
/ A*”3 AiV 
> 
/Ai 
~IH\ 
m 2 
= h 
< 
\ H ) > 
< 
VA 3 + Ai/ » 
< 
l 
H ) 
m 3 
m 2 
! I 
aJ a. 
CO Ito 
> 
< 
(h - hY 
AV 
^3 ~ Aiy 
> 
< 
(/*i 
~ A 2 ) 2 j 
Vh 
:<a 
> 
9 
> 
/Ai + /aA 2 
> 
2 
m 1 
C 3 
< 
/V » 
< 
l 2 j, 
< 
/V- 
If we take g 1 + g 3 = 2g 2 , we get necessarily g 3 — g 1 = — 2(g 2 — g 3 ) 
= 2(g 2 - gf, and the inequalities reduce to 
hj. ( H ~ /bA 2 
b ] \ J 3 
+6h -/•*)*. 
a 3 
