184 Proceedings of the Royal Society of Edinburgh. [Sess. 
This, therefore, is the condition for “ normality ” ; and if it he satisfied, 
Dr Ehrenfest’s result applies, and our test of non-equipartition, i.e. 
M p a' 2 pr -M q a 2 qr (4) 
of the same sign, whatever he the value of r , when p and q are given, 
cannot be satisfied. If, on the other hand, equation (3) does not hold, the 
test is valid. 
6. Still presuming distinctness of periods, so that the first of equations 
(2) applies, we shall now use that equation to discriminate cases in which 
(3) will hold. 
Taking first the case of two freedoms, (2) gives 
1 4- k 12 A 21 A 22 = 0 , 
or, in the notation of § 3 of the first paper, 
1 - k l2 rr = 0 — b l - b 2 rr . 
But (3) becomes M^rr' — M 2 , so, since & 12 = M 1 /M 2 the conditions (2) and (3) 
are identical, and normality is necessary provided that = M 2 6 2 in 
accordance with the third law of motion. This verifies the conclusion in 
the former paper. 
7. In the case of three freedoms with distinct periods (2) gives 
1 + ^ 12 ^ 21^22 ■*“ ^ 13 ^ 31^32 = ^ > 
1 + ^ 12 ^ 22^23 + ^ 13 ^- 32^-33 = ^ ’ 
1 + ^12^-23^21 ^13^-33^-31 = ^ J 
while (3) gives 
M-l® 11^ 12 AE 2 C£ 2i^ 22 4" -M-3® 31® 32 == ^ 5 etc. 
Now, referring to the formulae of § 6 of the former paper, we have 
Aa' n = 
Aa' 22 = 
^22 
^32 
, Aa\ n = 
^23 
^33 
II 
\ 
<1 
^32 
1 
CO 
(M 
^-33 
j> 12 
^21 
^-31 
7 
CO 
< 
1 
^33 
1 
, Aa 0 -, = 
1 
^22 
, Aa’oo = 
1 
CO 
CM 
^•31 
1 
’ w 31 
1 
^23 
’ 32 
1 
^21 
Hence, using the first two of the three equations just deduced from (2), we 
obtain the results 
Similarly, 
v _ “22 
12 “A^ 
T. a 21 
12 “A 01 a\ 
^33 ~ 
■^•32^ 12 
31 
13 ^-3i a 11 
Therefore, substituting in the first of the three equations, we establish the 
first of equations (3), and the others can be established similarly. Thus 
normality obtains necessarily, and Dr Ehrenfest’s conclusion shows that, 
contrary to the expectation expressed in § 8 of the former paper, while 
