where 41.. m denote numbers chosen at random in (a, x1)... (Xm1, b). Riemannt takes the limit of the sum (2) as his definition of the definite integral of any function f(x) in the interval (a, b). A bounded function f(x) will therefore be said to be integrable in the Cauchy sense if the limit on the right in (1) is unique for all modes of subdivision of the interval (a, b) in which the limit of the largest sub-interval is zero; and in the Riemann sense if the like is true of the limit on the right in (2). It is the obj ect of this note to prove that these two definitions are equivalent. Since the sum (1) is included among the sums (2), if f(x) is integrable in the Riemann sense it is obviously integrable in the Cauchy sense. It is therefore only necessary to prove that if f(x) is not integrable in the Riemann sense it is not integrable in the Cauchy sense. The necessary? and sufficient condition that f(x) be integrable in the Riemann sense is that every closed setlj contained in the set of points at which the oscillation** of f(x) is greater than any positive number k that