The relation between the maximum term and the maximum modulus of an entire function is exhibited by means of general theorems and specific examples. Functions of zero order and of infinite order are mainly considered.

A modified form of the centroid method used in factor analysis is described. Various large sample results are obtained, including a test of significance of the residuals. The method is compared with the corresponding form of maximum likelihood estimation and its efficiency is investigated. A numerical illustration is given of some of the foregoing theory.

The object of this paper is not to produce a chart of co-tidal and co-range lines which is more accurate than an existing one, but to investigate methods of computing such charts on the supposition that there are no observations of tidal streams such as were used to produce the existing chart. Only coastal observations of tidal elevations are supposed to be known, for such conditions would exist in many parts of the world. The methods used are similar to the so-called “relaxation methods”, using finite differences in all variables and attempting to satisfy all the conditions of motion within the sea, proceeding by successive approximations. There are many difficulties, peculiar to the tidal problem, in the application of these methods, due to the very irregular coast-lines and depths, gaps in the coasts, shallow water near the coasts, frictional forces, and the very serious complication due to the fact that the tides are oscillating and thus require two phases to be investigated simultaneously owing to their reactions one upon the other. One very important point in testing the methods is that no use whatever should be made of existing charts in obtaining first approximations of heights to commence the processes, not even where there are wide entrances to the sea. The resulting chart is shown to be very closely the same as the existing chart, thus proving the validity of the method.

The methods of S. N. Lin (1943) and B. Friedman (1949) for approximating to the factors of a polynomial by iterated division are studied from the point of view of convergence. The general theory, hitherto lacking, is supplied. The matrices which transform the errors in coefficients from one iterate to the next are explicitly found, and the criterion of convergence derived. Numerical examples are given. The tentative conclusion is that the methods are less simple in theory and less adaptable than the method of penultimate remainder, which admits of accelerative devices.

By examining certain connections between the derivatives and the powers of a Lie algebra, bounds are obtained for the indices of nilpotent Lie algebras over an arbitrary field. The results are used to obtain bounds for the indices of solvable Lie algebras over a field of characteristic zero.

The Bragg two-hologram method for eliminating the unwanted image in Diffraction Microscopy is here thoroughly examined. It is shown to be of general validity for objects containing both amplitude and phase-contrast terms.

Other two-hologram methods are briefly discussed. In principle any two sufficiently distinct holograms may be used, but the Bragg 2:1 focal ratio is the simplest. The pair of holograms at ±f is also of interest, since it allows moderate degrees of phase-contrast and amplitude-contrast terms in the original object to be separated. A “three-wavelength” technique for obtaining the Bragg records is briefly outlined.