  The conjugate locus of a point on a surface is the envelope of geodesics
emanating radially from that point. In this paper we show that the conjugate
loci of generic points on convex surfaces satisfy a simple relationship between
the rotation index and the number of cusps. As a consequence we prove the
`vierspitzensatz': the conjugate locus of a generic point on a convex surface
must have at least four cusps. Along the way we prove certain results about
evolutes in the plane and we extend the discussion to the existence of `smooth
loops' and geodesic curvature.
