The subject of this paper was studied by Orr (Camb. Phil. Trans., vol. xvii, 1898, pp. 171–199; 1899, pp. 283–290), and later by Barnes (Proc. Lond. Math. Soc., ser. 2, vol. v, 1906, pp. 59–116), in whose paper a number of references to earlier work on the subject are given. Formulæ equivalent to (9), (18), and (20) below were given by, Barnes, who derived them by his well‐known method of integrating products and quotients of Gamma Functions. In this paper the formulæ are deduced by induction from simpler formulæ which are established in section 2. In order to simplify the notation, four functions, denoted by P, E, Q, and H, are introduced, the first two in section 3, the last two in section 5. The P and Q functions are merely generalised hypergeometric functions multiplied by convenient factors. The E function, which is equivalent to Barnes's contour integral, is defined as a multiple integral, and from it the asymptotic expansion, with a useful form for the remainder, is easily derived. The H function is a multiple of the E function.

The preceding papers of this series have demonstrated (1) that there is a progressive increase of average Dp‐age in the subdivisions of Compositæ which can be correlated with geological age as determined from other data (Small and Johnston, 1937); (2) that these Dp‐ages, when placed on a time‐scale of years, follow a peculiar exponential curve, called the BAT curve (Small, 1937, Q.E. II); (3) that the measure of age in years which is given by the BAT curve can be used successfully for the Grasses (Small, 1937, Q.E. III).

The average ages in doubling periods (Dp‐ages) of the tribes and subtribes of Compositæ have been collated with geological time (Small, 1937, Q.E. II), but the Dp‐ages of individual tribes and sub‐tribes show considerable variation within each of the geological periods. When plotted as Dp‐ages against a time‐scale the tribes and the sub‐tribes show rather scattered positions, and the sub‐tribes within each tribe follow rather zigzag lines instead of being points on a straight line for each tribe.

Previous communications have established the formula for the Bat curve as - Dp.k + n.d = tk · 2n, where k is a constant Dp-age, d is a constant difference of Dp-age, tk is the time value in m.y. at the Dp-age k. For Compositæ (Small, 1937, Q.E. II) and apparently also for the Grasises (Small, 1937, Q.E. III) k = 0·6, d = 0·9, tk = 1·09375 m.y.

During recent decades the vital statistics of the more developed countries of the world have exhibited two outstanding features: the first is a decline in the death‐rate, and the second a similar decline in the birth‐rate, the latter setting in some time after the former. It is generally realised that, for an adequate study of the changes involved, it is necessary, not merely to consider the crude death‐ and birth‐rates—that is, the number of deaths and births respectively per 1000 inhabitants—but also to take into account the age distribution of the population. In the case of death‐rates, for instance, it is important to know the specific death‐rates for each age-group—that is to say, for example, the annual number of deaths of persons aged twenty, per 1000 individuals of that particular age. In the same way the crude birth‐rate can only be properly interpreted when analysed in reference to the age of the mothers.

Of the explanations advanced in respect of the observed inequality in the viability and longevity of the sexes in man, the sex‐linked lethal theory advocated by Lenz (1923), Gunther (1923), Geiser (1924–5), Huxley (1924), Schirmer (1929), amongst others, has attracted most support. One reason for this is undoubtedly that it is a genetic theory promulgated at a time when biological thought was being profoundly influenced by the many spectacular discoveries which had been made in this particular science.

The cellular processes of ovulation, maturation and fertilisation of the ovum in the albino mouse have been the subject of repeated and exact studies, and are now sufficiently well known. The biological phenomena of propagation in the mouse, i.e. the sexual cycle, œstrus, and copulation, have likewise been described in detail, but our knowledge concerning the period of gestation and the moment of parturition is still very inadequate.

Routine observations on the growth of chickens from a flock of Brown Leghorns maintained at this Institute indicated that factors, apparently closely associated with season of the year, caused considerable variations in the rate of growth. It is possible to conceive of two distinct means by which seasonal factors may produce their effect (1) directly on the growing chick, or (2) indirectly in that their influence may be exerted on the hen and transmitted to the egg from which the chick subsequently develops; either or both may contribute to the final variations obtained.

Shortly after Juhn, Faulkner and Gustavson (1931) had shown that natural differentials in the growth‐rate of individual feathers on one and the same fowl determined, in part, the threshold of their response to œstrin, the faster the growth‐rate the greater being the amount of œstrin required to produce female feathering, Lillie (1931) extended their conclusions to a re‐examination of the cases of lateral gynandromorphism in birds recorded in the literature up to that time. He pointed out that this condition seemed always to be associated with lateral size asymmetry, and argued therefore that, since the male side was the larger and presumably the faster growing, to account for lateral gynandromorphism it was necessary only to postulate that the ovary of the bird should be malfunctioning, producing a sub‐normal amount of œstrin, sufficient to evoke henny‐feathering on the smaller, more slowly growing side, but insufficient to do so on the larger, faster growing side.

Ophelia cluthensis is a Polychæte worm which has been known for some years from specimens taken from the sands around Millport, and particularly from Kames Bay. It was described for the first time by McGuire in 1935, and its general appearance may be seen in the Plate, figs. A, B.

In an earlier paper (Aitken, 1932) the author put forward a method of polynomial interpolation which was designed to make full use of the speed and power of modern calculating machines in forming and summing products of two factors. The present paper is devoted to two further applications of the method. It is necessary to give a brief review and explanation of the process used, in order that these applications may be properly understood.

Green's function for a plane outside an ellipse and for a plane between two confocal ellipses are evaluated.

The motion at a point vortex outside an ellipse is found.

The Lochs of Stenness and Harray on the mainland of Orkney are the only known stations in Scotland for Neritina fluviatilis L. The Loch of Stenness has a salinity varying between 9 and 26·8 parts per thousand; 24 species of invertebrates live in the loch, as well as several kinds of fish. The distribution of H. jenkinsi Smith and Peringia ulvœ Pennant is peculiar and only partly explicable by the salinity. The Loch of Harray is almost fresh and only slightly affected by spring tides, the salinity varying between 0·6 and 4·3 parts per thousand. Twenty-seven species of invertebrates are to be found in it, most of which are fresh-water species. The calcium content of the lochs varies from 56 to 139 mg. per litre.

Karyological studies on spermatogenesis in species of aquatic bugs of the family Corixidæ have brought to light several examples of an unusual type of chromosome behaviour during the course of meiosis. It is considered to be an association of non‐homologues in a manner unexampled in other animal species.

In theoretical analyses of the hereditary composition of human populations it is often assumed that crossing occurs at random and that individuals with different hereditary make‐up are equally fertile. This state of things has been called amphimixis. Sometimes such assumptions tally so closely with the actual state of things that it is possible to employ calculations carried out from this point of departure. In other cases processes present in populations cause larger or smaller deviations from amphimixis.

Only a small number of salmon smolts (Salmo salar) that leave the river for the sea return for spawning (Calderwood, 1930, p. 180; 1931, pp. 36, 56; Menzies, 1931, p. 60; and others). The large loss of adolescent fish seriously affects the stock of salmon in the rivers, so that the question of possible causes of the death of salmon in the sea is one of some practical interest.

This communication is largely founded on a paper entitled “Stresses in a Plate due to the presence of Cracks and Sharp Corners,” delivered by Professor C. E. Inglis, to the Institution of Naval Architects (1913). The importance of this contribution to the theory of fracture is recognized, and the results seem, among other things, to have a bearing on some geological problems. They may help, for instance, to explain the inception and development of sheet intrusions, under which class one may include ordinary vertical dykes (but not ring‐dykes), horizontal sills, and other hypabyssal bodies whose thickness bears only a very small proportion to their length.