on (one's) oath: under the obligation of an oath; in the position of having sworn a binding oath. Similarly upon (one's) oath, under oath.

on (also upon) one's faith: on one's word of honour. Frequently used parenthetically as an oath or to emphasize the truth of a statement: truly…

With solemn or binding assurances.

In construction with a verb, in sense A. 7a or A. 7b: on the security of, or as bound by, one's promise or affirmation.

By way of oath or solemn obligation. Obsolete.

with or by means of a vow or prayer; solemnly.

In a promissory manner; in the manner of a promise.

So as to involve a pledge; cf. engaging, adj. 2.

In a binding manner, so as to impose obligation; so as to force or constrain. Obsolete.

As promised in a vow; in fulfilment of a vow. Also: solemnly, devoutly.