The Chair which I have been elected to occupy, in succession to Professor Darwin, is associated with the name of a great scholar of our fathers' generation, Peter Guthrie Tait. This name has been familiar to me from the time when I first began to study mathematical physics. At that time Felix Klein was the leading figure in a group of outstanding mathematicians at Göttingen, amongst them Hilbert and Minkowski. I remember how Klein, ever eager to link physics with mathematics, missed no opportunity of pointing out to us students the importance of studying carefully the celebrated Treatise on Natural Philosophy of Thomson and Tait, which became a sort of Bible of mathematical science for us.

In two former papers (Proc. Roy. Soc. Edin., vol. liv, 1934, pp. 135–144; vol. lv, 1935, pp. 85–90) a number of integrals and series, involving Associated Legendre Functions of the First Kind regarded as functions of their degrees, were evaluated. In this paper similar integrals and series for the Associated Legendre Functions of the Second Kind, and also for Bessel Functions, are discussed. The latter are deduced from Bessel's Integral in its generalised form and from the corresponding integral for the Modified Bessel Function of the First Kind; the former from an analogous formula for the Associated Legendre Functions of the Second Kind.

Any reasonable generalisation on possibly fundamental rules of specific and generic differentiation would appear to be worth consideration. Defining our main terms briefly, we may here state that the genera and species of quantitative evolution are the genera and species recognised as such by responsible systematists who have considered entire groups in detail: the relatively few “critical” genera do not yield comparable data.

The isochronism of a pendulum is only true to a first approximation. If we make the arc large, the period increases, reaching infinity just before the pendulum makes complete circuits. Assuming the full value of the Circular Error, the actual calculation is given in No. 2 of this series (Proc. Roy. Soc. Edin., vol. xxxviii, 1918, p. 169) for the first few degrees, where it is seen that even with variations that may occur in practice, the change of daily rate may easily amount to several seconds. The defect was recognized before regulation of pressure and temperature were considered feasible, as early, for instance, as in 1673, when Huygens published his Horologium Oscillatorium. In fact Huygens invented the cycloid to prevent it, and gave an ingenious device to make the pendulum describe true cycloids—a device that has been given up for various reasons, among others, that it involves disturbance of the pendulum which it is very desirable to avoid.

The albino rats used in this study were obtained originally from the Wistar Institute of Philadelphia and belong to a strain in which microphthalmia was first observed by Dr Helen Dean King in the F4 generation extracted from a cross between an albino female of this stock and a wild grey Norway male. The defect occurred only in the “S” strain which formed a small proportion of the main stock. Its inheritance was observed by King (1931) through eleven generations, and 538 rats having either both or a single eye microphthalmic were obtained in a strain comprising 1884 individuals. The greater frequency of microphthalmia in the American stock is a marked feature, since only 154 rats with eye defects have been observed in the Edinburgh branch of Wistar rats in fourteen generations comprising over 5000 rats, and this total includes animals with eyes of normal size but possessing defects such as aniridia, cataract, etc., and also rare cases of anophthalmia and buthalmos. It is probable that 80 per cent, of the total affected were microphthalmics, and apparently the defect was generally more pronounced than in the parent stock, that is if the rat illustrated in King's monograph is to be taken as typical. According to King, the eye-defect could easily be detected at birth, “since a small eye did not protrude from the socket.” In the total given here no still-borns are included, as it was found that this method of diagnosis, as also the opacity of the lens in such rats immediately or soon after birth, were unreliable criteria of abnormality. We are in agreement that prenatal mortality did not play an important role in determining the ratio of normal to abnormal individuals, because litters containing affected were of the average litter size and generally contained some normals (Table I). However, where a large number of still-born rats occurred in litters containing defectives, it is possible that the abnormality involved proved lethal, since still-births are not of common occurrence in this stock (Table II).

While observations were being carried out on the fecundity of certain experimental Drosophila pseudo-obscura females, it became apparent that there was a regular alternation of periods of rapid laying with periods of no laying. Since an irregularity in rate of oviposition in this species had been noticed by other investigators, and since, under similar circumstances, Drosophila melanogaster lays approximately the same number of eggs each day, it was decided to investigate more closely the possibility that differences in laying habit existed among the various species.

The purpose of the present paper is to give as simple an account as possible of the general-relativity theory of two-component spinors, and to investigate its geometrical and analytical consequences. The work was suggested by courses of lectures given at Edinburgh in 1932 and 1935 by Professor E. T. Whittaker, who, on the basis of the special-relativity spinor theory of van der Waerden (1929), obtained the completely tensorized form of Dirac's equations given by him in a recent paper (1937).

I have previously (Graham, 1934) noted the fact that in the bivalve mollusc Solecurtus scopula the structure of the outer gills is peculiar in that they possess no groove along the free edge as do the inner and the great majority of lamellibranch gills. In Solecurtus (fig. 1) the gills are rather long and narrow, the outer shorter than the inner on each side, leaving a small triangular area of the latter exposed at the anterior end as well as a narrow strip along the ventral margin. The outer gill is provided with a supra-axial extension reaching to the dorsal part of the body on each side. The lamellæ are plicate, and the filaments lying on the summits of the folds on the outer gill project at the free edge further than do those at the bottom of the folds, so that the entire gill has a wavy or scalloped edge. It was of interest to investigate the effect of this departure from the normal structure of the gill on the ciliary mechanisms found there.

The generalisation to the stationary case of the solution given by Weyl (1918) of Einstein's gravitational equations in a statical axially symmetric universe, has been attempted by various writers. It has not, however, so far been possible to reduce the equations to linear form and thus to find the most general solution. A set of special solutions has been obtained by Lewis (1932), valid in a region free of matter, which contain Weyl's solution as a particular case. They depend upon an arbitrary solution of the equation

This is a direct continuation of an earlier communication by the author, on “The Revised Prepared System of the Quadratic Complex,” which appeared in vol. lvi (1936) of the Society's Proceedings, pp. 38–49. It will be convenient to utilize the same notation and references to earlier works I, II and III as before, with IV to denote the 1936 paper. A complete system of concomitants for a quadratic complex is here worked out; omitting polar forms, it consists of 210 forms. The sign ≡ in a relation A ≡ B is here used to mean “may be replaced by”: thus A is equal to a non-zero numerical multiple of B plus reducible terms.

(1) It is generally assumed that the vacuum photo-cell has no appreciable time lag. The electrons start out from the surface as soon as the light falls on it. The most important investigation on the subject is that of Marx and Lichtenecker (1913). By means of a mirror rotated 170 times per second an image of a narrow slit was swept over the cathode of a photo-cell at a distance of 8·3 metres from the mirror; the cathode was a potassium one sensitised with hydrogen. The electrometer reading was noted, and then the mirror was rotated at a much slower speed and the electrometer read again. No appreciable difference was recorded. In the words of the authors an alteration of the time of illumination from 4·38 × 10-3 to 1·46 × 10-7 sec., the intensity of radiation being 0·56 erg/sec. cm.2, had no effect on the result to an accuracy of one or two per cent. Theoretical deductions have been drawn from this statement which have been much quoted. It seemed to me that the statement itself was worth verifying, and this paper is the record of an attempt to verify it and push the limit of accuracy further.

The solution of simultaneous linear algebraic equations, the evaluation of the adjugate or the reciprocal of a given square matrix, and the evaluation of the bilinear or quadratic form reciprocal to a given form, are all special cases of a certain general operation, namely the evaluation of a matrix product H′A-1K, where A is square and non-singular, that is, the determinant | A | is not zero. (Matrix multiplication is like determinant multiplication, but exclusively row-into-column. The matrix H′ is obtained from H by transposition, that is, by changing rows into columns.) The matrices H′ and K may be rectangular. If A is singular, the reciprocal A-1 does not exist; and in such a case the product H′(adj A) K may be required. Arithmetically, the only difference in the computation of H′A-1K and H(adj A) K is that in the latter case a final division of all elements by | A | is not performed.

The fact that the atomic weight of an element is not invariable, but that the same element may exist in two or more forms which are chemically identical even though their atomic weights differ, was first noted by Soddy (1910, p. 285) in connection with radioactive elements. A little later Thomson (1913) showed, by the method of positive ray analysis, that neon, with an average atomic weight of 20·2, was a mixture of two kinds of atoms with masses 20 and 22 respectively. The brilliant extension of this method of attack by Aston has established that the majority of the common elements are similarly inhomogeneous. Chlorine (35·457) is a mixture of Cl35 and Cl37; lead (207·22) exists in as many as eight forms, with atomic masses ranging from 203 to 210. These chemically-identical species of the same element with diverse masses are known as isotopes.

In the male chromosome complement of man, Painter (1923) found that one chromosome pair is heteromorphic and forms an asymmetrical or unequal bivalent during meiotic metaphase, and consequently two kinds of gametes are produced. One member of this heteromorphic pair is about three times longer than the other and is designated the X, the other is described as the Y chromosome. Though the sex chromosomes of man were carefully studied and identified with certainty throughout the nuclear cycle of mitosis and meiosis by several investigators (cf. Shiwago and Andres, 1932), no further attempt was made to correlate the peculiar pairing properties of the heteromorphic chromosomes with their qualitative and structural differentiation.

The ages of the various tribes and sub-tribes of Compositæ have been calculated in doubling periods, according to Yule's formulæ; these are known briefly as Dp-ages (Small and Johnston, 1937; Udny Yule, 1924). Yule advises caution in interpreting Dp-ages in terms of geological time, but if his conceptions are to be applied to realities there should be some method of comparing Dp-ages with actual ages. Further, in the case of Compositæ, an almost complete parallel seriation has been found for Dp-ages and order of origin of both tribes and sub-tribes (Small and Johnston, 1937). Since the order of origin has been provisionally correlated with geological time (Small, 1919) it is possible to compare the calculated Dp-ages with suggested points of origin in geological time. This has been done by Small and Johnston (1937, Tables, VI, VII).

Applying Udny Yule's formulæ (1924) to the Compositæ, Small (1937) found that the average ages in doubling periods (Dp-ages) of the tribes of Compositæ, when plotted against a time-scale, gave points on an exponential curve called the BAT curve. If this curve is characteristic of average families of Angiosperms it should be possible to place the Dp-ages of tribes within other families on this curve as plotted against geological time, and thus obtain an order of geological origin which is quite independent of actual fossil records and which can be checked against any facts known concerning the evolutionary history of the family.

It is often a matter of considerable importance to decide whether the fluctuations exhibited by a series of observations are random in character, or whether some factor operating according to a definite law must be assumed to exist. One general method of testing a statistical hypothesis is to work out results which would be expected theoretically, and then to compare these with the observations. A statistic such as the mean or standard deviation to be observed on a particular hypothesis may be calculated and compared with the actual mean or standard deviation as evaluated from the observed figures. Agreement does not necessarily imply that the hypothesis is a true one, or even that it is consistent with the observations, because it might be found that when another statistic was calculated, the agreement was no longer satisfactory. Obviously the greater the number of tests applied and found satisfactory, the greater the confidence which can be placed in the conclusions.

The order Amphipoda, of the Crustacea Malacostraca, which consists of the 3 sub-orders Hyperiidea, Gammaridea, and Caprellidea, is highly important in the economics of the sea as a source of fish food. Hyperiids figure prominently in the diet of species which feed pelagically. Hardy (1924), for example, found them amongst the principal items in the food of East Anglian herring. Gammarids are preyed upon by bottom feeders in general. Todd (1905, 1907) and Blegvad (1916) record them in varying percentages from the stomachs of adult cod, haddock, whiting, plaice, dabs, flounders, sole, and skate. Their chief importance, however, would seem to be in the diet of fishes in post-larval and adolescent stages. Todd (1907, 1914) found Gammarids the main support of cod under 15 cm. and of plaice and dabs under 10 cm. Blegvad (1930) states that in Danish waters, plaice in their first year feed almost exclusively upon them. Clark (1922) found that in the early stages of 4 species of skate, Amphipods occurred in from 50 to 84 per cent of the stomachs examined, and Steven (1930) found the food of young stages of R. clavata and R. maculata to consist chiefly of Gammaridea. Gammarid species, as recorded by Hunt (1924), are also eaten by a variety of bottom-dwelling invertebrates.

In the course of a study of fecundity in Drosophila funebris (Donald and Lamy, 1937), it was noted that a particular female was laying very few eggs, and that the shape of these was peculiar. Among her descendants other females of similarly low fecundity and laying eggs of the same abnormal shape were encountered, and it was assumed therefore that a mutation affecting fecundity and egg shape had been recognised. In this paper evidence concerning the type of heredity of these characteristics is presented, and a statistical description of the morphological aspect of the new mutation, the size and shape of the egg, as compared with those of the normal D. funebris egg, is given.

In many branches of applied mathematics problems arise which require for their solution a knowledge of the latent roots of a matrix A, sometimes only the root of greatest modulus but often the second and other roots as well, and the corresponding latent vectors. A few examples, among many that might be cited, are problems in the dynamical theory of oscillations, problems of conditioned maxima and minima, problems of correlation between statistical variables, the determination of the principal axes of quadrics, and the solution of differential or other operational equations. It is important, therefore, to have a choice of methods for obtaining latent roots and latent vectors without undue labour, and the object of the present paper is to augment the existing store of such methods.