Manchester Memoirs, Vol. Ix. (1916), No. 14. 3 



finally, to determine it, v, w from the three equations 

 given above, and to leave the definition of the, as yet, 

 arbitrary vector for experiment and observation. 



I have limited these remarks to the statical case, in 

 which there is supposed to be an instantaneous change 

 from one distribution of load to another, or in which, if 

 there is a varying distribution of load, we have a statical 

 question at each moment, and not a dynamical question 

 from moment to moment. 



There is, I imagine, no truly dynamical theory of 

 Elasticity. 



There remains the very interesting question of small 

 oscillations about a position of statical equilibrium. Here 

 Ave have, as Stokes pointed out, the evidence of isochronism 

 of forced vibrations, and I give the Stress relations as 

 applicable either to the statical case or to that of small 

 oscillations. 



_ Stress Relations. 



(1) For Cartesian coordinate systems. 



d*P mlm - n) , 



r .ct~ yn-11 v v ' 





yn — ;/ ox 

 °*Q o ^ m(m — n) nj n 



2//171 0- , 



yn - ;/ oy v ^ h 



d 2 R „ „ m(m- n) a/ 



or $m — n x ^ ' 



2mn 0" 



P^r - ft\y 2 S = — -^(P+Q + R), 



ot 2> m ~ n cycz x ^ 7 ' 



2mn 



