The asymptotic values of the variances and covariances for the distribution of the transition numbers AA, AB, BA and BB for a two-state Markoff chain have been given by Bartlett (l). Whittle (3) obtained the probability distribution and their factorial moments for the k-state chain under the restriction that the first and the last observations belong to the rth and the sth state. But the expressions for the factorial moments according to Whittle himself are of little immediate use. He obtains useful results for the low order factorial moments by approximating in a grosser fashion, and the evaluation of the cumulants from these factorial moments is rather cumbersome. The object of this note is to give the first four cumulants and product cumulants which are exact to the order nr (p11 – p21)n−r where r is the order of the cumulant, for the distribution of the transition numbers of a two-state Markoff chain. They have been calculated by using the method developed by Iyer and Kapur(2). This method can equally be used for evaluating the asymptotic values of the cumulants for the k-state also. This is done by differentiating directly the determinantal characteristic equation.

This paper is concerned with the small transverse displacement of a thin circular plate elastically restrained along the circular boundary and subject to uniform normal pressure distributed over the area of an ellipse concentric with the plate. The method of complex potentials is used and the general assumptions relating to the small deflexion theory of thin plates are adopted. Explicit formulae for the bending and twisting moments and shearing forces are given at any point on the boundary and at the centre of the plate.

An exact solution in finite terms is derived within the limitations of the classical thin-plate theory, for the problem of a thin circular plate acted upon normally by hydrostatic pressure distributed over the area of a concentric ellipse, and subject to boundary conditions covering the usual rigidly clamped and simply supported boundaries.

In this paper complex variable methods are used to derive exact solutions in closed forms for the small deflexions of certain thin elastic plates due to transverse concentrated forces or couples applied at arbitrary or specified points. The isotropic plates considered are bounded by curvilinear edges of certain types along which the plates are rigidly clamped. Plates bounded by quartic curves having the forms of the inverses of an ellipse with respect to its centre or its focus are included as special cases.