and the Partition of Thermal Energy. 675 



each other. On each of these, beside the two at the poles, 

 are placed four others, one at each extremity of a pair of 

 diameters making an angle of 60° with the axis. There are 

 thus six molecules on each of these meridian circles, one at 

 each vertex of a regular hexagon inscribed in the circle. 

 This arrangement gives the 14 molecules arranged about the 

 central one. It is symmetrical through the centre and has 

 an axis. 



4. The Potential Energy of displacement. 



Let us consider the motion of a molecule in the direction 

 AB which we take as the direction of the axis of the model 

 (fig. 1). Let two molecules be placed at A and B and let 



Fig. 1. 



C, D, E, F be the four other molecules in the same meridian 

 plane, so that OA=OB = OC = AE... = 1. For simplicity, 

 we shall assume that the central one is the only one vibrating, 

 the others being stationary in their mean positions. Let L 

 be the extreme position of 0. Take OL = a, and r = the 

 radius of the molecule. Then 



l=.a+2r. 



Let f(l) be the potential energy of two molecules at a 

 distance I. Then for a displacement x taken small, along 

 the axis of the model, the potential energy due to A and B is 



f(l + x )+f(l- x) =2/(1) +**f"(l). 



The potential energy due to a pair like C, D 



=f( r i) +f( r 2) where r 1 = CL, and r 2 = DL. 

 Now rf = P + a?.+xl. 



neglecting higher powers of x, 



2X2 



