Steve,
    McKerlie has the following:--
    William Boyd,Laird of Myrtoun[2 miles west of Newton Stewart]
    minister of the parish of Penninghame had sassine[booked as
    owner] of the lands of HALSEGREEN Sept.1778.
    No other information given.

    The following with regards to the name FALL may be of interest.
    When I bought my first house in the early 60s the linear
    measure for the land in the title deeds was in old Scots
    measure as follows:--
    length-- so many Falls so many Ells & parts of an Ell
    breadth--do..    do..   do..    do....
    I knew what an Ell was but had no idea what a Fall was
    "Standard" Ell==39inches
    Scots weavers Ell==42inches
    English weavers Ell==45inches
    I measured the plot,reduced the measurement to inches &
    divided by these three measures----
    from the deed sizes--one Fall==5 Scots weavers Ells.
    In earlier times it seems a Fall was also an area of land.
    Could the surname have come from this?
    Joe.