For Steiner $n$-chains of circles of radii $r_1,\dots,r_n$ tangent to an inner circle of radius $r_-$ and an outer circle of radius $r_+$, is there a Soddy-type relation between the $n+2$ quantities $r_1,\dots,r_n$,$r_-$, and $r_+$?

For Steiner $n$-chains of circles of radii $r_1,\dots,r_n$ tangent to an inner circle of radius $r_-$ and an outer circle of radius $r_+$, is there a Soddy-type relation between the $n+2$ quantities $r_1,\dots,r_n$,$r_-$, and $r_+$?

   
   _{(Image from Wikipedia added by J.O'Rourke)}