SURFACES OF DISCONTINUITY 545 



velocities or the coordinates and momenta of its discrete parts 

 (the molecule, in the case of a physico-chemical system). The 

 most usual method of attack on the problem of how its con- 

 figuration will change in time is by the use of a group of differ- 

 ential equations which involve an important function of the 

 coordinates and momenta which is called the Hamiltonian 

 function. There is another method, however, actually devel- 

 oped by Lagrange before Hamilton's memoirs were written, 

 which involves another group of differential equations asso- 

 ciated with a function of the coordinates and velocities called 

 the Lagrangian function. J. J. Thomson has made a brilliant 

 application of this analysis to the discussion of the broad 

 development of physico-chemical systems. Before the present- 

 day methods of statistical mechanics had developed, he showed 

 how to convert the actual Lagrangian function of a system 

 into a "mean Lagrangian," expressed in terms of the physical 

 properties of the system which are open to measurement, and 

 by the aid of it to use the Lagrange equations so as to deduce 

 macroscopic results. His work on this subject is summ.arized 

 in his Applications of Dynamics to Physics and Chemistry (1888), 

 a book that has never received the attention which it justly 

 merited. By this method he deduced the following result for 

 adsorption from a solution at its surface: 



P 

 p 



= '''p{ii} ('«' 



In deducing it he assumes that we have a thin film whose area is 

 s and surface tension cr connected with the bulk of the liquid by a 

 capillary tube. The quantity ^ is the mass of the solute in the 

 thin film itself, while p and p' are the densities of the solute in 

 the film and in the liquid, respectively. R is the gas constant 

 for unit mass of the solute, i.e., the gram-molecular gas constant 

 divided by the molecular weight of the solute. Now on study- 

 ing Thomson's work we realize that his mean Lagrangian 

 function is formulated for dilute solutions in which ideal laws 

 are satisfied. This limitation enables us to transform (16) into 

 the approximate form of Gibbs' relation. Provided p'/p is 



