486 
PROFESSOR J. J. THOMSON ON SOME APPLICATIONS OF 
so that 
dL 
dx 
Rtf - -- 
p clx 
Now. during the changes that we contemplate, tf remains constant; hence we see 
that the change in L is the same as the change in 
Rtf log ^°> 
where p 0 is some constant density, so that we may put for the positional part of the 
kinetic energy, and that part of Y which depends on the density, 
L = L 0 + Rtf log — > 
P 
where L 0 is the value of L when the density is p 0 . As the energy vanishes at the 
zero of absolute temperature, L 0 will contain tf as a factor, so that we may put for the 
mean kinetic energy, and that part of V which depends on the density, 
L= tf (A + Rlog^,. (14) 
where A may be a function of tf, but not of p. 
This is the value of the aforesaid part of L for unit mass of the gas; if the mass 
of the gas were m, the value of this part of L would be 
mtf.(A+ Rlog^ > 
and we may treat the gas as if it were a dynamical system whose positional 
Lagrangian function contained the term 
mtf |a -f Rlog — 
all the variations being made at constant temperature. 
We have next to consider the liquid. The expansion of solids and liquids by heat 
shows that there must be some terms in the expression for the energies of a solid or 
liquid which indicate the existence of a stress depending on the temperature. In 
order to find such terms, let us suppose that v is the volume of the solid or liquid at 
the absolute temperature tf. The dilatation per degree of temperature is 
1 dv 
v dO' 
