388 Prof. L. Natanson on the 



§ 3. Lagrangian Equations. — From (I.), remembering the 

 definitions laid down, we obtain by a well-known calculation 



These equations, a thermokinetic extension of Lagrange's 

 well-known dynamical equations, have been given implicitly 

 by Helmholtz and explicitly by M. Duhem ; the form they 

 take in an important particular case had been previously ex- 

 plained by Lord Hayleigh. 



§ 4. Conservation of Energy. — Considering a real trans- 

 formation dq v ds { , multiply each of these equations by sflt 

 respectively, and add ; we find 



dT + dU-2P^-dQ = (1) 



The principle of conservation of energy in its general form is 

 thus seen to follow from the thermokinetic principle. That 

 inversely the thermokinetic principle cannot be deduced from 

 conservation of energy is an obvious proposition which scarcely 

 requires special mention. 



§ 5. Free Energy. — We shall suppose in the following 

 (except when the contrary is expressly stated) that one inde- 

 pendent variable is the temperature ; and accordingly we 

 shall use g t to indicate all the other variables. That work is 

 not required for merely changing the temperature of a system 

 is an experimental fact ; hence, when the variables •&, q., and 

 s. receive increments &$, 8q { , Bs v the work done on the system 

 will be still %T i 8q i (in our present modified notation) and no 

 term including && will appear. Variables with such properties 

 attributed to them have been employed by Lord Kelvin as 

 long ago as 1855 ; they have been often adopted in general 

 thermodynamical investigations. M. Duhem calls them 

 " normal " variables. 



Let us suppose that 3, g. represent a system of u normal " 

 variables. Write 



2 H % " 2B - %i=2 ii &?i - • • • (1) 



The function V, if it exists, will be called the free energy of 

 the system, because, as we shall find hereafter, V defined by 

 equation (1) will agree in the case of Reversible Thermo- 



