216 
PROCEEDINGS OF SECTION A. 
If then T be the kinetic energy of the system, we have from 
Lagrange’s Equations the following : — 
&c., &c 
It can, however, be shown that T submits to an identical equation from 
the vanishing of the following emanant : — 
dT 
dx, 
x 
n 
Differentiating this with regard to x„ we have — 
dT • d 2 T • a* T 
+ X, — — -f- X 2 — + •* + # n 
dx, dx , dx, dx 3 dx. 
d* T 
dx n dx t 
0, 
but 
d id T\ d 2 T •• d 2 T •• d 2 T 
— I — - | = — — . x , -f — — — x 2 + . . . -j i — — 
dt \ dx, / dx 2 dx, dx 2 dx, dx n 
• d*T • ^ 2 T 
+ X, — T + X 3 — + .... 
dx , dx, dx 2 dx 2 
whence Lagrange’s. Equations become in this case — 
.. cP T .. d 2 T 
Xl dx 2 dx, dx 2 
.. d 2 T .. d 2 T 
00 1 dx, dx n X ' 2 dx, dx 2 
+ 
d 2 T 1 d T \ 
= 2Pz If + t I 
dx,dx n \ Pi dx, J 
d 2 T ( 1 <*T\ 
dx, dx n \ p n dx n J 
as, however, T can at any instant be expressed in this form — 
T = M ( n x + . . + n x 3 ) 
when the quantities — 
M, n\ . . n 
i n 
depend upon the masses and the disposition of the material system we 
have — 
df T 
dx 2 
2 Mu, 2 , 
&c., 
df T 
dx 2 
2 M n 2 . 
and 
d 2 T 
dx . dx. 
0, &c., 
d 2 T 
dx, dx, 
0, 
