Properties of the negative exponential distribution of exponential populations are calculated as an illustration of the behavior of a population in which the average extension of life for the survivors increases as a function of the time. The equation specifying the distribution is dN dt =N o ∫ o ∞ ke − kl− t l dl l and the n th moment about the origin is given by μ′=(n!) 2 k n This population has a greater proportion of individuals of short life than a corresponding simple exponential; at intermediate times there are fewer observations, and for a time of ten or twenty average lives the number of survivors is orders of magnitude larger than for the simple exponential distribution. The properties of the population surviving at time T are computed using the K n arising from Bessel's functions of an imaginary argument. A striking property is that the survivors at time T have an expectation of life which is given by T + ( 1 k ) + T 1 2 K 0(4KT) 1 2 k 1 2 K 1(4kT) 1 2 which is greater by ( T k ) 1 2 K 0(4kT) 1 2 K 1(4kT) 1 2 ) than the result obtained for the exponential. A distribution obtained in experimental work, which appears to be adequately represented by the postulated distribution, is compared with it.