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For the comparison of maps of different regions, two geometrical concepts are of fundamental importance: the distortion = dw/dz, or the ratio of the lengths of corresponding line elements in the w- and z-planes, and the twist imaginary part of log (dw/dz), or the angle between corresponding line elements, this angle being always taken between (excl.) and (incl.). Koebel has shown that on the circle |z|= r, 0<r<1, both ❘dw/dz] and [z] lie between positive bounds which depend on r alone, and the writer2 has determined the exact values of these bounds.

It is the purpose of the present note to state the corresponding result in regard to the twist:

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maps the circle z|<1 on the interior of a simple region D in the w-plane, the twist 7 satisfies the following inequalities for |z|= r and 0<r<2-*.

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= 1.

In this case, r attains the upper or lower


+arc cos r or B


arc cos r.

where a and B are real, and cos B
bound in (1) for z = re-ai according as ß
For 2 <r<1, we have only the inequality included in the definition of T

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and no single class of functions analogous to (2) reaching the upper and lower bounds can be assigned.

When the region D is CONVEX, we have for |z|= r in the whole interval 0<r<1

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the upper and lower bounds being then attained as above.

The proof is similar to that of the distortion theorem outlined before. The region D is approximated by rectilinear polygons, for which we have the formula of Schwarz

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

Σμy = -2, -3≤ ≤ 1 (v= 1,2, . . .,m)

For a fixed value of m, we consider the set of all the values of α1,


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., 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

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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 -a about their origins, we may make a = 0; in this case, the circle z<1 is mapped by (2) on the w-plane slit along the straight line segment

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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).





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.





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,"1 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

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