526 Mr. W. Sutherland on a 



in common, then, between the kinetic theory and the old static 

 ones, with the great difference, however, that the kinetic state 

 of the molecules is taken account of and regarded as the 

 fundamental cause of the effective repulsions, and these repul- 

 sions act only between immediate neighbours, and do not 

 extend to all distances as the attractions do. 



What are the conditions of equilibrium in our system of 

 molecules ? First, let us take the case of a solid homoge- 

 neous and isotropic body free from external force. Suppose 

 it divided into two parts by a plane, and on unit area in the 

 plane suppose a right cylinder erected in one part of the 

 body ; this cylinder as a whole is attracted by the half of the 

 body on the other side of the plane, and this attraction can 

 be equilibrated only by the repulsion due to the collisions 

 occurring across the base of the cylinder. Accordingly we 

 must evaluate these two forces. Let the distribution of 

 molecules be a cubical one with edge of cube e ; then, in the 

 manner of Poisson (Journal cle I'Ecole Poly tech. Cahier xx.), 

 the attraction on the cylinder is easily proved to be 



^-gS^(r), where the summation extends to all molecules 



within reach of appreciable action of one fixed molecule. 



To evaluate the collisional repulsion on the base of the 

 cylinder, we will first take a short cut justified by the theory 

 of gases, and then verify the result by another method. The 

 short cut consists in assuming that the collisional transfer of 

 momentum goes on as if all the molecules had the average 

 velocity square, and as if one third of the number always 

 travelled parallel to each of three rectangular axes. Take 

 one axis to be perpendicular to the unit-base of the cylinder, 

 and let n be the number of molecules per unit area, so that 

 n = 1/e 2 , then n/3 molecules collide against the base of the 

 cylinder with velocity v, and the number of times per second 

 that each strikes the base is v/2ct, where a is the mean swing 

 of a molecule inside its domain. As each molecule has its 

 velocity reversed by a collision, the force of the collisions on 



the base of the cylinder is ^ 2mv =- or mv 2 /3e*a, and for equi- 

 librium we have the equation 



j£-w**«-o, (1) 



or, as we have been dealing with average values, we may as 

 well sum for all the molecules of the body and put our result 

 in the form 



^ m« 2 111 ^^ . / N n M x 



