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):

BASTARD EIGNE', Eng. law. Elder bastard. By the old English law, when, a man
had a bastard son, and he afterwards married the mother, and by her had a
legitimate son, the first was called a bastard eigne, or, as it is now
spelled, aine, and the second son was called puisne, or since born, or
sometimes he was called mulier puisne. See Mulier; Eigne, 2 Bl. Com. 248.