Kinetic Theory of Solids. 545 



only of the second order, so that %r<j>(r)/6e? may be regarded 

 as unaltered, and after the shear we have 



B + BT> 1 , _ 1 



{e + Sn)e{e + BS -E) + P - 5?Sr*M=°- 



Replace the last term by its value 2D/3e 2 (e — E), and expand 

 the first term, then 



_D /SD,_ Bv__ Sg \ 



e\e-E)\D e e-E/ 



_P— 2 



JL Q 



2- 



D /ggi , gg % 



S> 



_3 e 2 (e-E)V D ' e e-E, 

 P is the shearing stress, and 2S£/e is the shear, so that the 

 rigidity n is Pe/2Si*, 



This equation gives a statement of what rigidity is according 

 to the kinetic theory ; it depends on the rate of change of 

 the effective kinetic energy in a direction with the change of 

 the distance apart of the molecules in that direction. This 

 ratio of SD 1 to 8| is fundamental in the theory of solids ; the 

 condition for zero rigidity or fluidity is 



eSD 1 /DSf=^/(e-E)-l. 



It would be possible to calculate the ratio of SD X to S£ on 

 purely theoretical ground with the aid of suppositions as to 

 the distribution of kinetic energy in different directions and 

 its relation to temperature, but this is not worth doing at 

 present; it will only be shown shortly that even in solids 

 8D t /S| differs only from the large number e/(e— E) by a 

 small number, that is, a number not much greater than 1. 

 But first it may be as well to determine Young's modulus 

 from our equations in the same way as we have just found 

 the rigidity. It is only necessary to put — P for P 1} Q = 0, 

 Z = 0, £ = e + Sf, Sf= 8t)=— crSf, and replace %r(j>(r)/6e 3 by 

 Bm 2 /£Vf 2 j and to expand, to get the following equations in 

 which the variables are accented to distinguish the conditions 

 of variation from the former ones : 



V(e-E)U "•" e e-ilj' 



oD 2 '_SD 3 < / <re \of 



Phil. Mag. 8. 5. Vol. 32. No. 199. Dec. 1891. 2 



