where the arctangents are all taken between — and, and the integer k is

2

chosen so as to make <T ST. Since the polygon is not overlapping, we have

For a fixed value of m, we consider the set of all the values of a1,

., μm for which (6) maps |z|<1 on a simple polygon, and it is first shown that this set is closed. On this set, and for -≤0≤ the expression (7) has therefore a maximum and a minimum, which are found by observing that

=

has a maximum = arcsin r for = arccos r and a minimum arcsin r for= arccos r. In the discussion of (7), it is necessary to distinguish the case where all μ are negative, corresponding to a convex polygon, from the general case where some μ are positive; for this reason, the convex regions appear separately in the statement of the theorem. By considerations of continuity, and the use of elementary properties of harmonic functions, it is finally shown that the upper and lower bounds in (1) and (4) are reached in the cases (2) and (5) only, and that no closer bounds than (3) can be found for 2-≤r<1 in the non-convex case.

Regarding (2) and (5), we observe that by rotating the z- and w-planes through the angle -α about their origins, we may make a = 0; in this case, the circle <1 is mapped by (2) on the w-plane slit along the straight line segment

and by (5) on the half plane in which the real part of wei is greater than — §.

1 Koebe, Göttingen, Nachr. Ges. Wiss., 1909, (73).

2 Gronwall, Paris, C. R. Acad. Sci., 162, 1916, (249).

SEVENTEEN SKELETONS OF MOROPUS; PROBABLE HABITS OF THIS ANIMAL

BY HENRY FAIRFIELD OSBORN

AMERICAN MUSEUM of Natural HISTORY, NEW YORK CITY

Read before the Academy, April 29, 1919

Moropus is an aberrant perissodactyl, closely related to the family of the Titanotheres and more remotely to that of the Horses. It occurs in the Lower Miocene age in France and North America, and its ancestors have been

traced back to the Upper Eocene in both countries; it is thus of Holarctic distribution, and while very rare, it must have been perfectly adapted to its environment, because it survived the majority of perissodactyls and occurs in the Pliocene of Europe and England and will not improbably be found in the North American Pliocene.

The habits and habitat of the animal have always presented a very difficult problem. The skeleton presents the most noteworthy exception to Cuvier's law of correlation. All the foot bones which were discovered since Cuvier's

FIG. 1

Mounted skeleton of Moropus cookei in The American Museum of Natural History. One of the seventeen. Drawing one twenty-sixth natural size.

time consisted of large deeply-cleft terminal phalanges and were grouped with the edentates, especially the plantigrade sloths. All the teeth which were discovered, on the other hand, were grouped with the perissodactyl ungulates. It was not until H. Filhol discovered the nearly complete skeleton of Macrotherium that he was enabled to prove that the chalicotheres were of composite adaptive structure, with the teeth of perissodactyls and the claws of edentates. Macrotherium is very similar to the American Moropus.

Great light was thrown upon the structure of Moropus through the explorations of the Carnegie Museum by Holland and Peterson, described in 1914, from materials collected in the famous Agate Spring Quarry, Sioux County,

Nebraska, discovered by James H. Cook in 1897. After the lapse of the Carnegie researches and explorations, the American Museum entered this quarry and through five years of continuous exploration (1911–1916) an irregular area within a square of about 36 feet yielded nearly complete skulls of ten individuals and skeletal parts of seventeen individuals all together. From this wonderful material it has been possible to supplement the descriptions of Holland and Peterson and to present for the first time the proportions and pose, by which we may estimate the habits of this animal. We reach the conclusion that the Moropus type was not plains living, but forest living; that it was the seclusion of the forests which protected this type and which accounts for its great rarity in fossil deposits, for it is characteristic of forestliving forms that they are not readily entombed. We form an entirely different conception of the habits of the animal when we observe the extremely long fore limbs, which are not of the digging or fossorial type, and which thus belie the apparently fossorial or digging structure of the terminal phalanges. It appears more probable that these terminal claws were used partly for purposes of offense and defense, but largely for the pulling down of the branches of the trees. The animal was probably forest living like the African okapi, with which in its general body and head proportions it has many analogies. Like the okapi it survived through retreat to the recesses of the forests.

THE STRUCTURE OF ATOMS AND THE OCTET THEORY OF

VALENCE

BY IRVING LANGMUIR

RESEARCH LABORATORY, GENERAL ELECTRIC COMPANY, SCHENECTADY, NEW YORK

Read before the Academy, April 29, 1919

In a paper soon to be published in the Journal of the American Chemical Society, I will give a new theory of the structure of atoms and molecules based upon chemical data. This theory, which assumes an atom of the Rutherford type, and is essentially an extension of Lewis' theory of the 'cubical atom," may be most concisely stated in terms of the following postulates.

1. The electrons in atoms are either stationary or rotate, revolve or oscillate about definite positions in the atom. The electrons in the most stable atoms, namely, those of the inert gases, have positions symmetrical with respect to a plane called the equatorial plane, passing through the nucleus at the center of the atom. No electrons lie in the equatorial plane. There is an axis of symmetry (polar axis) perpendicular to the equatorial plane through which four secondary planes of symmetry pass, forming angles of 45° with each other. These atoms thus have the symmetry of a tetragonal crystal.

2. The electrons in any given atom are distributed through a series of concentric (nearly) spherical shells, all of equal thickness. Thus the mean

radii of the shells form an arithmetric series 1, 2, 3, 4, and the effective areas are in the ratios 1: 22: 32: 42.

3. Each shell is divided into cellular spaces or cells occupying equal areas in their respective shells and distributed over the surface of the shells according to the symmetry required by postulate 1. The first shell thus contains 2 cells, the second 8, the third 18, and the fourth 32.

4. Each of the cells in the first shell can contain only one electron, but each other cell can contain either one or two. All the inner shells must have their full quotas of electrons before the outside shell can contain any. No cell in the outside layer can contain two electrons until all the other cells in this layer contain at least one.

5. When the number of electrons in the outside layer is small, these electrons arrange themselves over the underlying ones, being acted on by magnetic attractive forces. But as the charge on the kernel or the number of electrons in the outside layer increases, the electrostatic repulsion of the underlying electrons becomes predominant and the outer electrons then tend to rearrange themselves so as to be as far as possible from the underlying ones. 6. The most stable arrangement of electrons is that of the pair in the helium atom. A stable pair may also be held by: (a) a single nucleus; (b) two hydrogen nuclei; (c) a hydrogen nucleus and the kernel of another atom; (d) two atomic kernels (very rare).

7. The next most stable arrangement of electrons is the octet; that is, a group of eight electrons like that in the second shell of the neon atom. Any atom with atomic number less than twenty, and which has more than three electrons in its outside layer tends to take up enough electrons to complete its

octet.

8. Two octets may hold one, two, or sometimes three pairs of electrons in common. One octet may share one, two, three or four pairs of its electrons with one, two, three or four other octets. One or more pairs of electrons in an octet may be shared by the corresponding number of hydrogen nuclei. No electron can be shared by more than two octets.

The inert gases are those having atoms in which all the cells in the outside shell have equal numbers of electrons. Thus according to the first four postulates the atomic numbers of the inert gases should be 2, 10, 18, 36, 54, and 86 in agreement with fact.

that of helium, have as their The line connecting the two Neon has in its second shell

All atoms with an atomic number greater than first shell a pair of electrons close to the nucleus. electrons establishes the polar axis for the atom. eight electrons, four in each hemisphere (i.e., above and below the equatorial plane), arranged symmetrically about the polar axis. The eight electrons are thus nearly at the corners of a cube. In argon there are eight more electrons in the second shell.

The eight electrons in the third shell of the atom of iron are arranged over the underlying electrons in the second shell. The two extra electrons in the

atom of nickel are placed in the polar axis. Beyond nickel the electrons in the atom cannot be held by magnetic forces, and thus tend to rearrange themselves so as to be placed as far as possible from the underlying electrons. This leads to an explanation of the chemical and magnetic properties of copper, zinc, etc.

Krypton has in its third shell nine electrons in each hemisphere, symmetrically placed with respect to the polar axis and to the four electrons in the second shell. The ninth electron in each hemisphere goes into the polar axis. Xenon is like krypton, except that it has twice as many electrons in its third shell. Beyond Xenon eighteen electrons in the fourth shell can be held by magnetic forces over the eighteen cells of the third shell, so that lutecium, the eighteenth element beyond Xenon marks the last of the rare earth elements. The electrons in the outside shell of the atoms beyond this element are arranged as far as possible so as to leave eighteen empty spaces over the underlying electrons. In this way it is possible to explain the chemical and magnetic properties of tantallum and tungsten as contrasted to those of the rare earths.

Niton has sixteen electrons in each hemisphere of its fourth shell. These are placed symmetrically with respect to the polar axis and the eight underlying electrons.

This theory of atomic structure explains in a satisfactory way most of the periodic properties of all the elements including those of the eighth group and the rare earths. It lends itself especially well to the explanation of the socalled physical properties, such as boiling-points, freezing-points, electric conductivity, etc. For the details of its application to specific elements the paper in the Journal of the American Chemical Society should be consulted.

Postulates 6, 7 and 8 lead directly to a new theory of valence which we may call the Octet Theory. This theory may be stated in terms of the equation

where e is the total number of available electrons in the shells of all the atoms in a molecule; n is the number of octets forming the outside shells of the atoms and p is the number of pairs of electrons held in common by the octets (Postulate 8). If we let E be the number of electrons in the 'shell' of an atom then e = (E). The value of E for a given atom, at least in case of the first twenty elements, corresponds to the ordinal number of its group in the periodic table. Thus we have the following values of E:-one for hydrogen, lithium, sodium, etc., two for magnesium, three for boron, aluminum, etc.; four for carbon and silicon, five for nitrogen and phosphorus; six for oxygen and sulphur; seven for the halogens, and zero for the inert gases.

The above equation expresses the fact that every pair of electrons held in common between two octets results in a decrease in two in the total number of electrons needed to form the shells of the atoms in the molecules. It also implicitly expresses the fact that all the electrons in the shells of the atoms forming a molecule form part of one or two of the octets in the molecule.