148 
Tnw tactions of the Royal Society of South Africa. 
The stress -components acting upon an elementary cylindrical element 
will therefore be as shown in the figure. 
The equations may immediately be deduced as follows : 
Since the isotropic body is in equilibrium, and since no body forces are 
acting, the equation of equilibrium * that contains ri) and hz reduces to 
rB 
8r 
— 0, 
A functon ;// therefore exists such that 
rO 
I. e. 
8z 
r9 - 
and .0z = 
1 8xp 
' 8z 
8xp 
8r 
r- or 
(1) 
At any point the axis of the resultant of the rO, 9z stresses is normal 
to lines given by 
dz _ dr 
9z vB ' 
which lines are therefore lines of zero stress as far as Bz and rB are con- 
cerned, and which will be completely free from stress if the other equations 
rr BB = zz = rz = 0 
also hold. 
The theory, however, given below is independent of these last equations. 
Hence we . have for the stress lines (lines of zero stress) dependent on 
rB and Bz, 
* Love, " Theory of Elasticity," § 59. 
