DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
513 
Let £ y), £, e, o), be the number of molecules of hydrogen, oxygen, carbonic oxide, 
carbonic acid, and water respectively; then the value of L will be 
0 {m, ( R, log + ,„ 2 v r 3 i 0 g AS + m s ( R 3 log 
PjTQ 
. T> , Po"Q , -R 1 Po""Q 
4- m. e R 4 , log c -h m K w ft- log- 
4 4 & ™ - 0 0 & 
m 4 e 
where m l5 m 3 , m 3 , m 4 , m 5 are respectively the masses of the hydrogen, oxygen, 
carbonic oxide, carbonic acid, and water molecules respectively ; R 1} R 3 , . . . R 5 the 
value of p/pd for these gases; w the mean potential energy of the mixture of gases, 
and Q the volume of the vessel in which they are contained. 
Since the quantity of hydrogen in the mixture remains constant, 
$ + oj = a constant; 
and, since the quantity of oxygen remains constant, 
V + + 6 + = a constant ; f 
since the quantity of carbon remains constant, 
£ + e — a constant. 
( 68 ) 
Since there are three equations between these five quantities, we may take two of 
them as independent variables. Let us take £ and e as our independent variables ; 
then we have from the above equations 
d v _ , 
•s 
O 
II 
^ 1 ^ 
dw 
d% ~ 2 ’ 
d£ = 
dr] _ j 
rU 
dw 
de ~ 2 ’ 
de 
By the Hamiltonian principle 
we have 
(69) 
dL 
d£ 
dL 
de 
0 , 
0 , 
so that 
0 \m, R, log^ 
[ x 1 6 m x £ 
Po Q 
m 1 R : + ^ ( m % Ro log -m 2 R. 
— (m 5 R 5 log —— m 5 | = ( c 'y.\ 
\ ) J \ constant. 
(70) 
Since wq R : = m 3 R 3 = m 3 R s = m 4i R 4 = m 5 R 5 = k, we may write this equation as 
MDCCCLXXXVII.-A. 3 U 
