Suppose that λ is the average density of virus particles per unit volume. If x is a dilution of this and unit volume is applied to an egg (or plate in other problems) the probability that the egg remains sterile is

provided that if a particle is present, it will infect the egg. To make a dilution assay we choose dilutions x1, …, xm (m levels) and apply these to n1, …, nm eggs. If these result in r1, …, rm sterile eggs we can estimate λ by maximum likelihood. The theory has been given by Barkworth & Irwin (1938), and full references to work on this problem will be found in Finney (1952). If we plot the quantities r1/n1, …, rm/nm against x1, …, xm we get a set of point whose fit to the curve (1) can be tested by a χ2 test. In a number of situations, however, it is found that (1) does not give a good fit. The estimation of λ is then completely invalid. In the present paper we consider why this happens, what types of curve may be fitted to the data and what they imply, and we also give a simple rapid test for such data fitting an exponential curve.