A recent paper (2012 emph{J. Phys. A} textbf{45} 374018) is extended by investigating the behavior of the regularized quantum scalar stress tensor near the axes of cones and their covering manifold, the Dowker space. A cone is parametrized by its angle $theta_1$, where $theta_1=2pi$ for flat space. We find that the tensor components have singularities of the type $r^gamma$, but the generic leading $gamma$ equals ${4pi over theta_1} - 2$, which is negative if and only if $theta_1>2pi$, and is a positive integer if $theta_1={2piover N}$. Thus the functions are analytic in those cases that can be solved by the method of images starting from flat space, and they are not divergent in the cases that interpolate between those. As a wedge of angle $alpha$ can be solved by images starting from a cone of angle $2alpha$, a divergent stress can arise in a wedge with $pi <alpha le 2pi$ but not in a smaller one.
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