  We determine all helical surfaces in three-dimensional Euclidean space which
possess a constant ratio $a:=\kappa_1/\kappa_2$ of principal curvatures (CRPC
surfaces), thus providing the first explicit CRPC surfaces beyond the known
rotational ones. A key ingredient in the successful determination of these
surfaces is the proper choice of generating profiles. We employ the contours
for parallel projection orthogonal to the helical axis. This has the advantage
that the CRPC property can be nicely expressed with the help of the involution
of conjugate surface tangents. The arising ordinary differential equation has
an explicit parametric solution, which forms the basis for a further study and
classification of the possible shapes and the singularities arising for $a>0$.
