In view of the considerable ground covered by the Commission at its Paris meetings and the fairly complete record of the activities of institutes and observatories, etc. published in the Minutes, it has not been deemed profitable by the president to call for further reports in advance of the Stockholm meeting. At the Paris meeting it was agreed that such reports be printed independently before each meeting of the Union and that reprints of or references to the published reports be sent to the president. It is hoped that all such reports if ready will be made available before the Stockholm meeting so that they may be summarized by the representatives in attendance or by the president and recorded in the Minutes. With reference to the pronouncement at the Paris meeting “that it is eminently desirable that more attention be given to the development of accurate general perturbations and mean elements on the basis of accurate osculating elements”, the president has visited the Planeten-Institut at Frankfurt and the Rechen-Institut at Berlin and has been in correspondence with the Leningrad Institute. From these sources particularly valuable material has been received.

The distribution of the length of a typical chord of a stationary random set is an interesting feature of the set's whole distribution. We give a nonparametric estimator of the chord length distribution and prove its strong consistency. We report on a simulation experiment in which our estimator compared favourably to a reduced sample estimator. Both estimators are illustrated by applying them to an image sample from a yoghurt ferment. We briefly discuss the closely related problem of estimation of the linear contact distribution. We show by a simulation experiment that a transformation of our estimator of the chord length distribution is more efficient than a Kaplan-Meier type estimator of the linear contact distribution.

For applications in spatial statistics, an important property of a random set X in ℝ k is its first contact distribution. This is the distribution of the distance from a fixed point 0 to the nearest point of X, where distance is measured using scalar dilations of a fixed test set B. We show that, if B is convex and contains a neighbourhood of 0, the first contact distribution function F B is absolutely continuous. We give two explicit representations of F B , and additional regularity conditions under which F B is continuously differentiable. A Kaplan-Meier estimator of F B is introduced and its basic properties examined.

An optimum or a high heat treatment of milk before yogurt preparation was applied according to an experimental design with the deliberate aim of making stirred yogurts with varying properties. Images of sections of yogurt samples were obtained using a transmission electron microscope. These were digitized and segmented via a threshold determination to separate the protein structure from the background. Four spatial statistical estimates were determined in order to evaluate their applicability for characterization of yogurt microstructure. The volume content was estimated and compared with the protein content obtained through chemical analysis. Heat treatment had no significant effect on volume content, as expected. The surface area of the protein aggregates was a useful measure to describe the difference between casein networks obtained by an optimum and a high heat treatment. The star volume gave a measure of the average local size of pores in the yogurt gel. The optimum heat treatment resulted in yogurt with small values for the star volume, indicating a less coarse structure of the casein network compared with yogurt prepared from high heat milk. The covariance function was able to differentiate between yogurts made using two types of heat treatment.

In the statistical analysis of random closed sets the distribution of the distance from a fixed point to the nearest point of the random set (the empty space function) is an important tool, and therefore development of efficient estimators is crucial.