Tlie Torsion Prohlem for Bodies of Revolution. 
149 
dz _ dr 
— . dz -\- ^ . dr = 0. 
i. e. -^^ ~ constant. 
Again, in an isotropic solid the strain-energy function is an invariant 
for ail transformations from one set of orthogonal axes to another. Hence 
we conclude that if W is the strain -energy function, 
2W = (A -f 2/x) {err + eee + e.,y -f /x (^^.^ -r . . - 4^,^ . ee, -..,), 
where e^r , e^e , are the components of strain, and A and fx the 
elastic constants,* 
Further, rr — , - - -, - 
dW 
We therefore immediately get 
rr = A {err + + e.zz) -f 2f^err 
re = f^ere- 
For the strain-components Cre and eez we therefore have 
ere = — . re, eez = • 
If tie, '^^z components of displacement, then for lie we have 
the equations, 
hie Ka 1 
ere = ^ = . re. 
dr r [I 
«2 « 
(2) 
Eliminating lie from (2) we get, 
8z\rl drW/^' 
Making use of 1 we get, 
8z\V^' fzl^ fr\V^''8rl ' " " 
* Love, " Theory of Elasticity," § 59. 
