The trace T of the metrical energy-momentum tensor Tkl of fields associated with particles of zero rest mass may be zero, either identically or as a consequence of the field equations. This property of Tkl is correlated here with the behaviour of the Lagrangian of the field under arbitrary conformal transformations. Certain classes of special fields are considered explicitly. It is shown in particular that T vanishes for all non-zero spin fields which correspond respectively to the two-component neutrino field or the photon field.

Suppose a preliminary set of m independent observations are drawn from a population in which a random variable x has a continuous but unknown cumulative distribution function F(x). Let y be the largest observation in this preliminary sample. Now suppose further observations are drawn one at a time from this population until an observation exceeding y is obtained. Let n be the number of further drawings required to achieve this objective. The problem is to determine the distribution function of the random variable n. More generally, suppose y is the r-th from the largest observation in the preliminary sample and let n denote the number of further trials required in order to obtain k observations which exceed y. What is the distribution function of n?

Let F(x) be a distribution function and denote by its characteristic function and by its moment of order k H. Milicer-Grużewska [2]has derived the following theorem: Suppose that F(x) has moments of all orders and that they satisfy the relations. where while C is a positive constant. Then f(t) is an entire function of order not exceeding two. The proof given by H. Milicer-Gruzewska is rather complicated, moreover the restriction (1.2) seems to be artificial and motivated only by the particular method of proof. We note that condition (1.1) means that the moments of F (x) are majorated by the moments of a normal distribution and we use this remark to generalize the problem.

We consider the finite dam model due to Moran, in which the storage {Zt} is known to be a Markov chain. The method of generating functions is used to derive stationary distributions of Zt in the two particular cases where the input is of geometric and negative binomial types.

The problem of the survival of a single mutant in a haploid genetic population when there exists selection is considered for a type of population model in which the generations ar overlapping. The results are compared with the previous work of Fisherand others for other models. The need is stressed for a solution of the same problem in a diploid population with general phenotypic selection coefficients.