The Collaborative International Dictionary of English v.0.48:

Oversman \O"vers*man\, n.; pl. Oversmen.
   1. An overseer; a superintendent.
      [1913 Webster]

   2. (Scots Law) An umpire; a third arbiter, appointed when two
      arbiters, previously selected, disagree.
      [1913 Webster]

Bouvier's Law Dictionary, Revised 6th Ed (1856):

OVERSMAN, Scotch law. A person commonly named in a submission, to whom power
is given to determine in case the arbiters cannot agree in the sentence;
sometimes the nomination of the oversman is left to the arbiters. In either
case the oversman has no power to decide, unless the arbiters differ in
opinion. Ersk. Pr. L. Scot. 4, 3, 16. The office of an oversman very much
resembles that of an umpire.