528 Mr. W. Sutherland 



on a 



to consider E as possibly a function of both temperature and 

 pressure. But we can first see how the supposition that E is 

 constant will work. The equation (1) then states that, given 

 E the size of the molecules, their total energy, and their law 

 of force, then e, the mean distance apart of the molecules free 

 from external force, is determinable. The equation specifies 

 the law of expansion of the simplest solid free of external 

 stress. For example, imagine that we know only that 

 solids expand very slightly with rising temperature, then 



~22r<£(V) is probably nearly constant, and therefore a must 



be nearly proportional to the kinetic energy, or the expansion 

 from absolute zero is nearly proportional to the absolute 

 temperature, which is the definite experimental result. But 

 before discussing this equation any further, we may as well 

 make it more general by taking account of external stress 

 and removing the condition of isotropy. 



Imagine the properties of our molecules to be different in 

 different directions, and in such a manner that all the circum- 

 stances can be specified with reference to three rectangular 

 axes. Let the paths and velocities of the molecules be such 

 that they produce the same collisional pressure in the direc- 

 tion of the x axis as if they were an isotropic system with 

 mean kinetic energy mif/% per molecule, and similarly kinetic 

 energy mv 2 /2 and mw 2 /2 along the y and z axes, so that 

 {mu 2 -\- mv 2 -{- mw?) /2 = 3D, where D is the mean kinetic 

 energy per molecule of the actual system. Let the mean 

 positions of the molecules correspond to a parallelepiped dis- 

 tribution with sides f, tj, £ to the parallelepiped, so that the 

 mean distance apart of the molecules along the x axis is f, 

 and so on. Let the external stress be specified by pressures 

 P, Q, B parallel to the three axes, and let the mean compo- 

 nents of the swing of each molecule in its domain be a, /3, <y, 

 then for equilibrium parallel to the x axis we have merely to 

 add P to the molecular pressure in equation (1), and make 

 the necessary slight alterations to get 



mu 



1 



P-«ET^(r) = 0, .... (2) 



with two similar equations. 



If SHZ are the values of ?, rj, f when the molecules are in 

 contact, we have three equations of the form f — S"='». Hence, 

 given H, H, Z, u 2 y v 2 , iv 2 , P, Q, R, and the law of force, the 

 problem of finding f , r), f is determinate ; that is, the law of 

 expansion in the three directions with increase of kinetic 



