From A grammar of the Latin language By Karl Gottlob Zumpt
Among the invocations of the gods, the following are particularly frequent: mehercule, mehercle, hercule, hercle, or mehercules, hercules, medius fidius, mecastor, ecastor, pol, edepol, per deum, per deum immortalem, per deos, per Jovem, pro (or proh) Juppiter, pro sancte (supreme) Juppiter, pro dii immortales, pro deum fidem, pro deum atque hominum fidem, pro deum or pro
deum immortalium (scil. fidem), and several others of this kind.
Note. Me before the names of gods must be explained by an ellipsis: the complete expression was : ita me (e. g. Hercules) juvet; or with the vocative: ita me Hercule juves. The interjection medius fidius arose, in all probability, from me dius (Διός) Julius, which is archaic for JUius, and is thus equivalent to mehercules, for Hercules is the son of that god. Mehercule is the form which Cicero (Oral. 47.) approves, and which, along with hercule, occurs most frequently in his writings. See my note on in Verr. iii. 62. The oath by Pollux (pol) is a very light one, and hence it is given especially to women in the comic writers. In edepol and edecastor the e is either the same as me, or it is a mere sound of interjection; de is deus.