To rest with confidence, as when fully satisfied of the veracity, integrity, or ability of persons, or of the certainty of facts or of evidence; to have confidence; to trust; to depend; -- with on, formerly also with in.
Rely was a brand of superabsorbent tampons made by Procter & Gamble starting in 1975. It was recalled from the market in September 1980 because of links to Toxic Shock Syndrome.