(The), in races, is not a race for a prize subscribed for by ladies, but a race run for by women.
“On the Monday succeeding St. Wilfred's Sunday, there were for many years at Roper's Common [a race] called the Lady's Plate, of 15 value, for horses, etc., ridden by women.” —Sporting Magazine, vol. xx., New Series, p. 287.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Ladies' Plate from Infoplease: