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
and the n th moment about the origin is given by
This population has a greater proportion of individuals of short llfe 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 Ks arising from Bessel's functions of an imaginary argument. A striking property is that the survivors at time 2" have an expectation of life which is given by T + (l/k)+ TI/2Ko(4kT)I/2/klI~KI(4kT) 1/~, which is greater by (2"/k)lt2Ko(4kT)II~/KI(4kT) 112 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.