DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
481 
Making this substitution for SV in equation (10), we have 
8Q = 2^8p + 8T I + ^-Sft.(11) 
Now let us suppose that only one coordinate, p x , changes, and that just as much 
heat is supplied or absorbed as is sufficient to prevent the temperature from changing; 
then, since the temperature is constant, ST L and §9 both vanish, and we have 
SQ = 
rfT, 
dl>i 
tyi- 
Now, if P x be the force which is required to keep p x constant when the system is in a 
steady state, 
p _ dN __ ( / : r l\ _ dT 2 
1 dp l dp x dp l 
= _ ( R. 
dPi f( r p) 1 d Pi 
Now, since dVjdp x and dTJdp x do not explicitly involve 6, and since T 2 is propor¬ 
tional to 6, we have 
e 
f O) rp 
f(P) 1 
= 
dpi 
/dP \ 
where, in finding (--N j, 0 is the only quantity which is supposed to vary. 
Thus equation (11) becomes 
m 
8 constant 
( 12 )-] 
This result can be obtained from the Second Law of Thermodynamics; it was so 
obtained by von Helmholtz, and applied by him to the very important case of the 
heat produced in the voltaic cell in his paper “ Die Thermodynamik chemischer 
Vorgange” ( c Wissenschaftliche Abliandlungen,’ vol. 2, p. 962), 
§ 4. It will be seen from the preceding work that the Second Law of Thermo¬ 
dynamics cannot be deduced from the principle of Least Action, unless we know 
a good deal about the distribution of energy among the molecules, and unless we 
make in addition a good many assumptions. For this reason, in discussing the 
applications of Dynamics to Physics, I prefer to apply the principle of Least Action 
directly to the various problems, and not to start from the Second Law as an 
intermediate stage. In the rest of the paper I shall endeavour to show how this can 
be done. 
3 Q 
MDCCCLXXXVII. — A. 
