I. Various writers (Ferrar, 1927) have started out with different definitions of generalised derivatives. Essentially, the problem of the generalised derivative is a problem in interpolation. The values of the derivatives are known for all integer values of n; for all positive integers, being the ordinary derivatives; for zero, being the function itself; for negative integers, being repeated integrals. Any function of n which has the above values at the integers (i.e. any cotabular function) is a solution of the problem. Out of the infinite number of cotabular functions, there exists one discovered by E. T. Whittaker (Whittaker, 1915; Ferrar, 1925; Whittaker, 1935), called the cardinal function, possessing rather remarkable properties. In particular, if the cardinal series defining the cardinal function is convergent, then it is equivalent to the Newton-Gauss formula of interpolation.

The attempt to find reciprocal wave functions made in Part II led to some difficulties. We believe that these were due to an incorrect definition of the relativistic scalar product and Fourier coefficient. We shall show in this paper that a modification of these definitions leads to simpler equations, which allow us to determine the reciprocal wave functions and to find a transcendental equation for μ = aε/k, as was envisaged in Part I.

This paper is the continuation of the papers by Born (1939, Part I) and by Born and myself (1940, Parts II and III). Here the theory is extended to the case of spinor wave functions satisfying the Dirac equation.

The theory has often been put forward that cancer is a somatic mutation. The changes which differentiate the cancerous cell from the normal are reproduced in many or all of the cells derived from the original malignant cell, i.e. they show the properties by which we define a mutation. Thus, to a geneticist, the possibility which most readily suggests itself is (1) that of a mutation in the genetic material of the somatic cell. Other possible mechanisms that might simulate the results of (1) are (2a) the introduction into the cell of a malignant virus; (2b) the “activation” of some such virus pre-existing in the cell in an inactive state, a situation which would imply mutation in the virus since the change becomes reproduced; (3) the production of some autocatalytic substance other than gene or virus, inducing malignant changes in the cell which contains a certain amount of it.

Cytological studies carried out by McClung (1905, 1914) on various species of Orthoptera have shown that the male is the heterogametic sex. The male has only one X-chromosome, whereas the female has two. During spermatogenesis two kinds of gametes are produced, one with the X-chromosome and the other without it. It was also found that the segregation of the single sex chromosome takes place at first meiotic anaphase. The present paper describes the sex chromosome of the male Hexacentrus mundus Walker, from India. During the meiotic division this chromosome exhibits peculiarities which it is believed have not hitherto been seen in any species of the Orthopteræ.

The radial distribution of the atoms (or molecules) of liquids, as of solids, can be found from X-ray scattering photographs (Zernike and Prins, 1927; Warren and Gingrich, 1934), and in this way many such distributions have been determined experimentally (Harvey, 1938, 1939; Gingrich, 1940; references in Coulson and Rushbrooke, 1939). The results are generally given as distribution curves showing p(r) or r2p(r) as a function of r, where 4πr2p(r)dr measures the probability that two arbitrarily selected atoms (or molecules) of the liquid are distant r to r + dr apart. Such distribution curves show a sequence of peaks—for ρ(r) these are usually of diminishing amplitude—and for atomic liquids successive peaks may conveniently be ascribed to successive co-ordination shells (of atoms) about any (arbitrarily selected) central atom. The position and precise shape of the peaks depends, of course, upon the temperature as well as the substance of the liquid.

The object of this paper is to record certain temperatures which were measured in the deep Boreland Bore, near Dysart, in Fife, and to discuss the bearing of these and some other observations on the amount of heat which escapes from the crust, by conduction, in Britain.

The energy levels of a rotating vibrator have been calculated by the late J. L. Dunham (1932). The theory of the rotating vibrator is important, because it provides a method for direct quantitative study of molecular structure from the band spectra of diatomic molecules. The purpose of this paper is to extend Dunham's results and to express them in a form more convenient for numerical calculation.

The importance of the pituitary in relation to many body processes has long been recognised, and may be due either to its direct influence on the body or to its control over other endocrine organs, in particular the thyroids and gonads. It is evident that various hormones are elaborated in the gland, and it appears likely that they originate in particular regions or even in particular cell types. It is of interest therefore to ascertain at what period in the development of the animal these hormones first appear. This problem can be approached from two directions, experimental and histological.