  We present a general formula for the Atiyah-Sutcliffe determinant function,
which holds for any integer $n \geq 2$, as a global factor times a sum of
terms, with each term similar to a higher degree cross-ratio. The formula is to
our knowledge new.
  We also conjecture that the Atiyah-Sutcliffe determinant is a rational linear
combination of products of factors of only two simple types, each of them
manifestly $SO(3)$-invariant. This allows us to obtain a conjectural purely
angular formula for the determinant for $n=4$, as an illustration of how our
conjecture can be applied.
