  This paper contains a suite of results concerning the problem of adding $m$
distinct new points to a configuration of $n$ distinct points on the Riemann
sphere, such that the new points depend continuously on the old. Altogether,
the results of the paper provide a complete answer to the following question:
given $n \ne 5$, for which $m$ can one continuously add $m$ points to a
configuration of $n$ points? For $n \ge 6$, we find that $m$ must be divisible
by $n(n-1)(n-2)$, and we provide a construction based on the idea of cabling of
braids. For $n = 3,4$, we give some exceptional constructions based on the
theory of elliptic curves.
