In this paper the real analyticity of all conically self-similar free-vortex solutions to the Navier-Stokes equations is proven. Furthermore, it is mathematically established that such solutions are uniquely determined by the values of three derivatives on the symmetry axis, and hence a numerical method, invented and successfully used by Shtern & Hussain (1993,1996), is justified mathematically. In addition, it is proven that these results imply that for any conically self-similar free-vortex solution to the Navier--Stokes equations there exists a second order non-swirling correction term. For this term it is also shown that the second order contribution to the total axial flow force vanishes in the cases of the entire space and a half-space, but that it need not vanish for general conical domains. In doing so an old claim by Burggraf & Foster (1977) is established mathematically, however not for Long's problem but for Shtern & Hussain's (1996) extension of this problem to the full Navier-Stokes equations and the entire space.