§ 1. In his treatise on Fourier Series and Integrals Carslaw quotes without proof Sommerfeld's theorem that

when the limit on the right-hand side exists. In applied mathematics, he remarks, it is this limit, rather than the corresponding Fourier repeated integral which occurs.

In the present paper I propose to extend this result in various ways. After proving Sommerfeld's result on the general hypothesis, not considered by him, that the integral is a Lebesgue integral, I show that the limit in question is whenever the origin is a point at which f(u) is the differential coefficient of its integral, and I obtain the corresponding results for

In all their generality these statements are only true when the interval (0, p) is a finite one. I then show how, under a variety of hypotheses with respect to the nature of f(x) at infinity, they can be extended so as to be still true when p = + ∞ . These hypotheses correspond precisely to those which have been proved f to be sufficient for the corresponding statements as to the Fourier sine and cosine repeated integrals in their usual forms.

So far back as 1867, Flower and Murie (1), in giving an account of the dissection of a Bush woman, stated that “observations upon the comparative anatomy of the different races of Man have hitherto been confined too exclusively to the external characters and to the skeleton. With very few exceptions the arrangement of the muscles, vessels, viscera, and even of the brain and nervous system, constitute at present an unexplored field; and numerous well-marked races of our species are passing away from the face of the earth without the slightest record being left on any one of these points. And yet in discussing questions, daily becoming of greater interest, relating to the unity or plurality of Mankind, and the amount of divergence of races, data such as these afford, whether their testimony be negative or positive, whether they tend to show absence or presence of variation from a given standard, cannot be neglected by the conscientious inquirer.”

Some of the relationships, such as those between toxins and antitoxins, discovered during the last twenty years have lately been investigated by Arrhenius. It has been found by this observer that many of these processes can be adequately represented by the equations which govern homogeneous mass reactions; for instance, the rates of the disappearance of diphtheria antitoxin and of typhoid agglutinin from the circulation are described at least empirically by such a formula (Diagram I.)

In this paper are collected and discussed the observations of the specific gravity of saline solutions of moderate and high dilution made during the course of the last ten years. The instrument employed was the hydrometer, of the type used by me during the cruise of the Challenger in the years 1872 to 1876. The results then obtained were such as to convince me that the instrument, in capable hands, and with strict attention to the obligatory conditions of experiment, gives more exact results than any other instrument in use for such work. In the course of the last forty years the method has been improved in the sense that more and more attention has been paid to the conditions, especially those of temperature, under which the experiment is made, because it was found that the method is so delicate that it responds to every increase of care in experimenting by greater exactness in the results which it furnishes.

1. The rate of multiplication of fast-growing micro-organisms is proportional to the number of organisms, and to the concentration of foodstuff.

2. The initial rate of multiplication affords a factor of comparison both of efficiency for media and of reproductive properties of organisms.

3. Vaccines may be prepared in large quantities on the basis that a maximum number of organisms is attained to, this maximum being dependent on the concentration of nutriment, and independent of the amount of culture inoculated.

This paper describes a new form of rotating photometric “sector,” capable of very accurate adjustment to any desired light-transmission value while running, and possessed of the great advantage of having its rotating part in one piece only.

A mathematical discussion of this new type of apparatus follows, in which are deduced formulæ for the intensities of the light transmitted under different conditions, and for the greatest width of the beam of light that can be employed; and a graphical tabulation of the values of these formulae in different cases is provided.

The mounting and details of an actual instrument are described; together with an additional mechanism for the purpose of automatically recording the photometric measurements obtained.