A study is made of faithful representations of the free cyclic (non-associative) groupoid by means of bifurcating root-trees and by means of index polynomials in two indeterminates emphasizing the intimate connection between these representations. The properties of trees and index polynomials are investigated and the concept of lattice of trees is introduced.

Given a power series with real non-negative coefficients and having radius of convergence p, a summability method P is defined as follows:

The main concern of this note is to establish conditions sufficient for one such method to include another.

The National Institute of Oceanography owes its formation to a growing need for more detailed and systematic knowledge of the physics as well as the biology of the oceans, and the feeling that the United Kingdom, in co-operation with the Commonwealth countries, should play a part in marine science more in keeping with its tradition of interest in the oceans and their navigation. The Institute is controlled by the National Oceanographic Council, which was granted a Royal Charter in 1949 and operates with the aid of Government grants.

The emphasis of the programme is on long-term research under the main headings:

Interchange of energy between the atmosphere and oceans

Response of the sea surface to wind and pressure changes

General circulation of the oceans

Free and forced oscillations in the oceans

Distribution of marine organisms

Organic production in the oceans

Bionomics of whales.

There are many other activities and as much as possible is done to work out the practical implications of the research, and to help other marine scientists. The headquarters of the Institute are at Wormley, near Godalming, Surrey, and its research ship, the R.R.S. “Discovery II”, is based at Plymouth.

In this note we study the asymptotic behaviour of a product of matrices where Pj is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P(n) tend to identity and that (ii) P(n) tends to a limit matrix with identical rows.

The temperature gradient in the lower atmosphere can be directly determined by measuring the optical refractive index of the air. This method is suitable for use on the Greenland ice sheet where errors introduced by water vapour are small, and where the strong solar radiation reflected by the snow surface makes it difficult to measure temperature differences over height differences of about I metre.

The refraction was measured by observing the apparent vertical angle of each of a set of targets at distances up to 4 km. from a theodolite. The refraction was found to vary linearly with the distance of the target. The true vertical angle to the targets was determined when a second theodolite was available and reciprocal sights could be taken with it from the site of target to the fixed theodolite. The true vertical angle varied with time due to slow descent of the theodolite as the firn slumped; a correction for this was made. The standard error of the temperature gradient measurements was about 1.5 × 10−2 C.° per metre. It is considered that the method could be developed and improved so that over a range of only 100 metres temperature gradients could be measured to an accuracy of about 0·1° C. per metre.