The Collaborative International Dictionary of English v.0.48:

Eigne \Eigne\, a. [OF. aisn['e], ainsn['e], F. a[^i]n['e], fr.
   L. ante natus born before. Cf. Esnecy.]
   1. (Law) Eldest; firstborn. --Blackstone.
      [1913 Webster]

   2. Entailed; belonging to the eldest son. [Obs.]
      [1913 Webster]

   Bastard eigne, a bastard eldest son whose parents
      afterwards intermarry.
      [1913 Webster]

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

EIGNE, persons. This is a corruption of the French word aine, eldest or
first born.
     2. It is frequently used in our old law books, bastard eigne. signifies
an elder bastard when spoken of two children, one of whom was; born before
the marriage of his parents, and the other after; the latter is called
mulier puisne. Litt. sect. 399.