  The problem of kink stability of isothermal spherical self-similar flow in
newtonian gravity is revisited. Using distribution theory we first develop a
general formula of perturbations, linear or non-linear, which consists of three
sets of differential equations, one in each side of the sonic line and the
other along it. By solving the equations along the sonic line we find
explicitly the spectrum, $k$, of the perturbations, whereby we obtain the
stability criterion for the self-similar solutions. When the solutions are
smoothly across the sonic line, our results reduce to those of Ori and Piran.
To show such obtained perturbations can be matched to the ones in the regions
outside the sonic line, we study the linear perturbations in the external
region of the sonic line (the ones in the internal region are identically
zero), by taking the solutions obtained along the line as the boundary
conditions. After properly imposing other boundary conditions at spatial
infinity, we are able to show that linear perturbations, satisfying all the
boundary conditions, exist and do not impose any additional conditions on $k$.
As a result, the complete treatment of perturbations in the whole spacetime
does not alter the spectrum obtained by considering the perturbations only
along the sonic line.
