The distribution of xn, the number of occurrences of a given one of k possible states of a non-homogeneous Markov chain {Pj} in n successive trials, is considered. It is shown that if Pn → P, a positive-regular stochastic matrix, as n → ∞ then the distribution about its mean of xn/n½ tends to normality, and that the variance tends to that of the corresponding distribution associated with the homogeneous chain {P}.

If the temperature in an elastic rod is not uniform and if it varies with time, dynamic thermal stresses are set up in the rod. This paper is concerned with the calculation of the distribution of temperature and stress in an elastic rod when its ends are subjected to mechanical or thermal disturbances. Simple waves in an infinite rod are first discussed and then boundary value problems for semi-infinite rods and rods of finite length. The paper concludes with an account of an approximate method of solving the equations of thermoelasticity.

The presence of a non-uniform distribution of temperature in an elastic solid gives rise to an additional term in the generalized Hooke's Law connecting the stress and strain tensors and to a term involving the time rate of change of the dilatation in the equation governing the conduction of heat in the solid. The present paper is concerned with the effects produced by these additional terms in two simple situations. In the first, the elastic solid is regarded as being of infinite extent and the distribution of temperature in the solid is produced by heat sources whose strength may vary with time. In the second, the solid is supposed to be semi-infinite and to be deformed by prescribed variations in the temperature of the bounding plane and by heat sources within itself.

The commutative and entropic congruence relations determine a homomorphism on the free logarithmetic , the arithmetic of the indices of powers of the generating element of a free cyclic groupoid. A necessary and sufficient condition that two indices should be concordant (i.e. congruent in the free commutative entropic logarithmetic) is that the bifurcating trees corresponding to these indices should have the same number of free ends at each altitude. It follows that the free commutative entropic logarithmetic can be represented faithfully by index ψ-polynomials (or θ-polynomials) in one indeterminate.

In the concluding section enumeration formulæ are obtained for the number of non-concordant indices of a given altitude and for the number of indices concordant to a given index.

The roots of the equation zez = a are of importance in several theories. Various authors have studied certain of their properties over more than a century. Here we solve the equation, in the sense that we define the sequence {Zn} of roots and, except for a small, finite number of values of n, find a rapidly convergent series for Zn. The terms in this series are alternately real and purely imaginary and so the series is very convenient for calculation. For the few remaining roots, we give practicable methods of numerical calculation and supply an auxiliary table.

The main results of this article have been announced without proof or details in Wright 1959.