40 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



1 1 



d8z 



(10). 



--%*-ft 



dz 



dx 



dz 



m 



.(11). 



By substituting in Equations (3) the values of the forces given in Equa- 

 tions (8) and (9), they become 



d /< 



z/J 2 

 d8z\ m/c? 



d&x 



These are the general equations of elasticity, and are identical with those 

 of M. Cauchy, in his Exercices d'Analyse, Vol. in., p. 180, published in 1828, 



m 



where k stands for m, and K for p , and those of Mr Stokes, given in the 



2 



Cambi-idge Philosophical Transactions, Vol. vin., part 3, and numbered (30) ; 



in 



in his equations A = 3/x, B = . 



8 



If the temperature is variable from one part to another of the elastic 

 solid, the compressions , , , - , -r- , at any point will be diminished by a 



quantity proportional to the temperature at that point. This principle is applied 

 in Cases X. and XI. Equations (10) then become 



