[1] The headword is the aorist active infinitive (presumably quoted from somewhere) of ba/llw. For the many meanings of this verb cf. beta 70, beta 71, beta 72, beta 73, beta 74, beta 75, beta 76, beta 78, beta 79, beta 80; and see generally LSJ s.v. (web address 1). The particular distinction here, from verbs of close-range hitting, is repeated at nu 614, omicron 968, and tau 1240.
[2] Homer, Iliad 5.52, which reads ba/llein ["to hit"] for the lexicon's unmetrical ba/len: web address 2 below. The uses in Homer are well analyzed in Lexikon des frühgriechischen Epos II (fasc. 10, 1982) 23-37. The verb tugxa/nw is commonly used in Homer as a synonym of ba/llw "hit", e.g. Iliad 5.582 *)Anti/loxos de\ *Mu/dona ba/l'… xermadi/w| a)gkw=na tuxw\n me/son "Antilochus struck Mydon, hitting the middle of his elbow with a rock", cf. 23.857. See Cunliffe, Lexicon of the Homeric Dialect, and Dyer 660-61 (Bibliography below, citing A.W. Gomme on Thucydides' use of the word, A Historical Commentary on Thucydides III 488-89). It probably originally meant "hit the mark, succeed," an antonym for a(marta/nw "miss the mark, fail" (as at Iliad 5.287).