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is their extreme pugnacity. In preparing the skull of Archaotherium wanlessi, I was surprised to find a large antorbital vacuity on the right side, shaped much like an obliquely inclined figure eight (Fig. 21), completely perforating the surface of the lachrymal and adjacent anterior tongue of the frontal. A slight indentation occurs also on the frontal above the orbit, and on the opposite side of the skull are two shallow depressions toward the upper edge of the lachrymal, one on the maxillo-lachrymal suture and the other farther back (Figs. 5, 6). All of these seem to be correctly interpretable as battle scars and to indicate that this individual was gripped in front of the eyes by the powerful jaws of an opponent.

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FIG. 22. Archæotherium scotti sp. nov. Holotype, No. 10885. Left jugal process showing injury to anterior margin, one half the natural size.

In the specimen of Archæotherium scotti, Princeton Geological Museum 10885, not only is there a fractured rib which never healed, leaving a large callus on each of the broken ends, with an irregular suture separating them, but the left cheek flange has had the front margin bitten off (Fig. 22), showing clearly by the scalloped outline where the canines and adjacent incisors penetrated (compare with Fig. 1).

Function of the Dependent Malar Processes.

Incidentally this has a bearing on the function of the jugal process. Mr. Troxell thinks that it "in all probability gave origin to the masseter muscle which generally arises from the jugal and is inserted broadly on the wide angle of the ramus," and that "from the tip of the process the fibers of the muscle might have given the forward, the backward and even a sideward movement of the mandible...."12 It seems to me equally probable that the process in question extended over the masseter without giving origin to it, and projected sufficiently beyond the outer surface of the cheek to afford a handy grip to an antagonist. In A. scotti, No. 10885, both outward curvature and relative length of the cheek process are somewhat increased by the distortion of the specimen.

RÉSUMÉ.

Restricting the survey to the Big Badlands of South Dakota and combining data from the Yale and Princeton collections, we have the following range of forms in time:

I. Titanotherium beds.

Archæotherium scotti sp. nov.
Archæotherium marshi Troxell.18

II. Oreodon beds.

Archæotherium wanlessi sp. nov.

Archæotherium mortoni Leidy.
Archæotherium ingens Leidy.
Archæotherium crassum? (Marsh).

III. Protoceras beds.

Megacharus zygomaticus Troxell.
Megacharus latidens Troxell.14

12 Loc. cit., p. 255.

13 North bank of Cheyenne River between French Creek and Battle Creek, South Dakota, associated with Brontotherium and Hyracodon. Troxell, loc. cit., p. 386, footnote.

14 Probably Upper Oligocene, near Cheyenne River, South Dakota. Troxell, loc. cit., p. 437.

Pelonax bathrodon (Marsh) Peterson.15
Scaptohyus altidens gen. et sp. nov.
Unnamed form a.

Unnamed form b.

It will be noticed at once that Archæotherium is not listed from the Protoceras beds. Whether it is really absent or merely lacking from the collections so far examined is uncertain, nor is it yet possible to say to what extent the species listed from the Titanotherium and Oreodon beds respectively are confined to these levels. A. wanlessi, A. mortoni and A. crassum? are found in the zone of rusty nodules in the upper part of the "turtle-oreodon layer" of the lower Oreodon beds, and were certainly contemporary.

The origin of the group as a whole is uncertain. The Eocene achænodonts, as Professor Osborn points out,16 are too specialized in the teeth to be regarded as directly ancestral. The European genus Entelodon and the American Archæotherium both appear in the lower Oligocene, and, as Professor Osborn suggests, may have sprung alike from an unknown northern or Holarctic form. A significant fact bearing on this general subject is the sudden appearance in the Protoceras beds of several types of entelodonts, both large and small, in which the shape of the fourth upper premolar agrees more closely with the character of that tooth in the European genus than in Archæotherium. Perhaps this is to be explained as a new faunal invasion. Megacharus zygomaticus, the small form which I have not named, and a still larger individual represented in the Princeton collection by some teeth and other fragments, all show this character. On the other hand, Megacharus latidens Troxell and Scaptohyus altidens gen. et sp. nov. have the anterior border of the tooth in question indented, as in Archæotherium. Further discussion of the affinities of entelodonts in general and the forms from the Protoceras beds in particular may be postponed until the collection of the American Museum of Natural History has been studied. This contains excellent material of one or more undescribed large 15 Protoceras sandstones? Big Badlands of South Dakota, Peterson, loc. cit., pp. 57, 58.

16" The Age of Mammals," pp. 217, 218.

forms from the Protoceras beds, perhaps the same as some of the fragmentary specimens in the Princeton collection already referred

With the kind permission of Professor Osborn and Dr. Matthew, I hope to pursue these studies farther, on the collections in their charge.

The facts presented regarding habits amplify Professor Scott's published observations and have an important bearing on the supposed function of the dependent process of the jugal which projected far enough beyond the cheek to be grasped and badly injured by the teeth of an adversary.

ON MEAN RELATIVE AND ABSOLUTE PARALLAXES.

BY KEIVIN BURNS.

(Read April 22, 1921.)

In computing the mean parallax of a group of stars by comparing the radial velocities with the proper motions, it has been the custom to proceed in one of two ways. Knowing the apices of the sun's way, a great circle is passed through these points and the star. The total proper motion is then divided into two parts, one at right angles to the plane of this great circle, and the other in the direction of the circle. The former is called the tau component and the latter the upsilon component. The tau component is evidently free from any motion due to the motion of the sun, while the upsilon component contains all of the effect of the solar motion. Knowing the sun's velocity, the mean parallax of a group of stars distributed at random over the whole sky can be derived from a study of the mean algebraic upsilon component taken for each part of the sky. The formulæ used in this and the following method are found in "Stellar Motions," by W. W. Campbell, page 214 and following. It is seen that for the average of a group of stars Vr4.74(7/π), where V, denotes the radial velocity freed from the motion of the sun. For each star Vm For each star Vm=4.74(μ/π), V m being the total velocity across the line of sight. Let V, be the total radial velocity, then for the mean of a group V=1.57Vr. For, denoting the cross velocity freed from the motion of the sun by Vm, Campbell shows that in the mean, Vm=1.57Vr, and Sm=1.57Sr, S being the velocity of the sun with respect to any star, the subscripts denoting cross and radial motion as above. The individual values of S, and V, unite by addition and subtraction to form the values of Vr, and the quantities Sm and Vm unite in the same manner to form Vm. Hence we have m = F(Sm, Vm)=1.57F X (Sr, Vr)=1.57Vr. The relationship deduced by Campbell, Vm= 1.57Vr, holds equally well if we choose a coördinate system fixed

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