1908-9.] On Energy Accelerations and Partition of Energy. 353 
Similar relations are, of course, true for d? 2 (a3 2 , Vv z z)’ an d fh e mean values 
of 1 1 and clearly vanish. 
dx 1 dy 1 ox 2 dy 2 
Also 
but 
-0Y 
dr _ 
~/0V\ 2 " 
~/0Y \ 2 ~ 
~A> 
_w _ 
A dy) - 
A dzJ _ 
[0 2 Y1 
r 0 2 y _ 
r0 2 vi 
_ dx 2 _ 
Ay 1 ] 
Jdl 1 J 
0 2 V~ 
_dxdy_ 
= 0, 
0 2 V 
_d0d(p 
4=0 in general, although 
= 0, 
0 2 V 
dxdO 
= 0 , etc., 
0Y' 
-dx 
and the other mean force com- 
ponents vanish. 
(5) Let us form what Dr Bryan calls the accelerations of the energy 
components in the Arch. Neerlandaises to which I have referred. From 
Lagrange s equations we obtain 
j t \ b( n h + m 2 )x 2 
Differentiating again with respect to the time, we have 
£ { «•»! + { £ f ' - 4© ’ 
where ) represents the total change due to the motion of the particles 
and the variation of the field ; and thus 
d /0 Y\ 
dt\dx J 
00.0 .0 .0 A 0 ■ AJ ^ \0Y 
— + x— +y + z — + r — + rtf — + r sin 0<t > — 7 — - — — , 
dt dx dy dz dr rdO r sm Odcfi/dx 
where — represents the part due to the variation of the field alone. As, 
dt 
however, we are considering a statistically stationary state, positive and 
negative values of g^(g-") are equally probable independently of the value 
of x, and consistently with our assumed conditions the mean value of this 
term is to be put equal to zero. Therefore, taking mean values, we have in 
accordance with our assumptions 
d 2 i / \ . o 
jpU™l+ m 2) X 
1 
/0Y\ 2 
- r £c 2 i 
-02 V - 
m l 4 m 0 
i\dx ) J 
j 
_ dx 1 _ 
■ ( 2 ), 
and the corresponding equations in y and 0 will give the same result. 
vol. xxix. 23 
