On Francis Galton's Difference Problem

Francis Galton proposed to split the money available for the first two prizes in a competition according to some ratio X, depending on the marks of the three best competitors, but invariant under change of location or scale of the marks.Assuming normality, Galton found that EX is about .75 and empirically he observed that X is nearly uniformly distributed between 1/2 and 1.Our main purpose is to show that Galton was indeed right for a wide class of underlying distributions.As the number of competitors tends to ., the ratio X tends (in distribution) to a uniform random variable.