Manchester Memoirs, Vol. Ivi. (1912), No. 10. 



= ^+^--—^, 



dy dz^ dydz 



witli two analogous equations ; 

 ^ \ d / da db dc\ a"e 



2 ^/.v\ dx dy dzj dydz 

 with two analogous equations .... (6). 

 On comparison of the right-hand side of these equa- 

 tions with the right-hand side of the equations (5), it is 

 seen that they become identical on the substitution of 



Gi for €, e„ for/, 63 for g ; 

 2^1 for a, 2^„ for /% 2'^^ for c, 



This general relation will be made use of at a later 

 period in the paper. 



Within the limits to which Hooke's Law applies, the 

 elements of the strain can be replaced by a linear function 

 of the elements of the stress with constant coefficients, 

 and if the coefficients are allowed to be variable the 

 substitution may be made, even though Hooke's Law no 

 longer obtains. I shall, however, here limit myself to the 

 case of a homogeneous isotropic elastic solid, and write 



qe={l+a)P-fr{P+Q + Ji), 

 with two analogous relations ; 



(/a = 2(1 + (t)S, 

 with two analogous relations, 



where q and rr are constants. 



Then for the relations affecting the stresses in such a 

 solid, we shall have 



^ l^r- dz- dydz) \dy- dz'J I ^ J 



with two analogous relations, 



. , .(d / dS dT dU\ d-P] _^ a- \p_^r,A.h>\ 

 {ax\ dx dy dz ) dydz) dyaz{ ^ J 



with two analogous relations . . . . (7). 



