From Plutarch's Life of Aristides (http://ancienthistory.about.com/library/bl/bl_text_plutarch_aristides.htm):

For taking hold of the spears with their bare hands, they broke many of them, and betook themselves not without effect to the sword; and making use of their falchions and scimitars, and wresting the Lacedaemonians' shields from them, and grappling with them, it was a long time that they made resistance.

The Arms and Armour Glossary of Terms defines it thusly:

Falchion: A short, heavy, broad-bladed sword with a single edge, bearing a similarity to a heavy scimitar.

http://www.chronique.com/Library/Glossaries/glossary-AA/arms_f.htm#falchion


Bingley

Dear Bingley: I searched for "falx" and found this in the Smith dictionary:
FALX, dim. FALCULA (a#rph, dre/panon, poet. drepa/nh, drepa/nion), a sickle; a scythe; a pruning-knife, or pruning-hook; a bill; a falchion; a halbert.

As CULTER denoted a knife with one straight edge, "falx" signified any similar instrument, the single edge of which was curved (Dre/panon eu)kampe/j, Hom. Od. xviii.367; curvae falces, Virg. Georg. i.508; curvamine falcis aenae, Ovid, Met. vii.227; adunca falce, xiv.628). By additional epithets the various uses of the falx were indicated, and its corresponding varieties in form and size. Thus the sickle, because it was used by reapers, was called falx messoria; the scythe, which was employed in mowing hay, was called falx foenaria; the pruning-knife and the bill, on account of their use in dressing vines, as well as in hedging and in cutting off the shoots and branches of trees, were distinguished by the appellation of falx putatoria, vinitoria, arboraria, or silvatica (Cato, De Re Rust. 10, 11; Pallad. i.43; Colum. iv.25), or by the diminutive falcula (Colum. xii.18). "

Early English foot soldiers carried a "billhook" which was like a heavy axe with forward curving tip, "J" shaped.